Homogenization of Coupled Phenomena in Heterogenous Media

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Author: Jean-Louis Auriault | Claude Boutin | Christian Geindreau

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Homogenization of Coupled Phenomena in Heterogenous Media

This page intentionally left blank

Homogenization of Coupled Phenomena in Heterogenous Media

Jean-Louis Auriault Claude Boutin Christian Geindreau

First published in France in 2009 by Hermes Science/Lavoisier entitled: Homogénéisation de phénomènes couplés en milieux hétérogènes volumes 1 et 2 © LAVOISIER 2009 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2009 The rights of Jean-Louis Auriault, Claude Boutin and Christian Geindreau to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Auriault, J.-L. (Jean-Louis) [Homogénéisation de phénomènes couplés en milieux hétérogènes. English] Homogenization of coupled phenomena in heterogenous media / Jean-Louis Auriault, Claude Boutin, Christian Geindreau. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-161-2 1. Inhomogeneous materials--Mathematical models. 2. Coupled problems (Complex systems) 3. Homogenization (Differential equations) I. Boutin, Claude. II. Geindreau, Christian. III. Title. TA418.9.I53A9513 2009 620.1'1015118--dc22 2009016650 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-161-2 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

Contents

Main notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

PART ONE . U PSCALING M ETHODS . . . . . . . . . . . . . . . . . . . . . . .

27

Chapter 1. An Introduction to Upscaling Methods

29

. . . . . . . . . . . . . .

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Heat transfer in a periodic bilaminate composite . . . . . . . . 1.2.1. Transfer parallel to the layers . . . . . . . . . . . . . . . . 1.2.2. Transfer perpendicular to the layers . . . . . . . . . . . . 1.2.3. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4. Characteristic macroscopic length . . . . . . . . . . . . . 1.3. Bounds on the effective coefﬁcients . . . . . . . . . . . . . . . 1.3.1. Theorem of virtual powers . . . . . . . . . . . . . . . . . . 1.3.2. Minima in the complementary power and potential power 1.3.3. Hill principle . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4. Voigt and Reuss bounds . . . . . . . . . . . . . . . . . . . 1.3.4.1. Upper bound: Voigt . . . . . . . . . . . . . . . . . . 1.3.4.2. Lower bound: Reuss . . . . . . . . . . . . . . . . . 1.3.5. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6. Hashin and Shtrikman’s bounds . . . . . . . . . . . . . . . 1.3.7. Higher-order bounds . . . . . . . . . . . . . . . . . . . . . 1.4. Self-consistent method . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Boundary-value problem . . . . . . . . . . . . . . . . . . . 1.4.2. Self-consistent hypothesis . . . . . . . . . . . . . . . . . . 1.4.3. Self-consistent method with simple inclusions . . . . . .

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29 30 31 33 35 35 36 36 38 39 40 40 42 44 45 46 46 47 48 49

6

Homogenization of Coupled Phenomena

1.4.3.1. Determination of βα for a homogenous spherical inclusion 1.4.3.2. Self-consistent estimate . . . . . . . . . . . . . . . . . . . . 1.4.3.3. Implicit morphological constraints . . . . . . . . . . . . . 1.4.4. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 51 52 53

Chapter 2. Heterogenous Medium: Is an Equivalent Macroscopic Description Possible? . . . . . . . . . . . . . . . . . . . . . . . .

55

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Comments on techniques for micro-macro upscaling . . . . . . . . . 2.2.1. Homogenization techniques for separated length scales . . . . 2.2.2. The ideal homogenization method . . . . . . . . . . . . . . . . 2.3. Statistical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Method of multiple scale expansions . . . . . . . . . . . . . . . . . . 2.4.1. Formulation of multiple scale problems . . . . . . . . . . . . . 2.4.1.1. Homogenizability conditions . . . . . . . . . . . . . . . 2.4.1.2. Double spatial variable . . . . . . . . . . . . . . . . . . . 2.4.1.3. Stationarity, asymptotic expansions . . . . . . . . . . . . 2.4.2. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Parallels between macroscopic models for materials with periodic and random structures . . . . . . . . . . . . . . . . . . 2.4.3.1. Periodic materials . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2. Random materials with a REV . . . . . . . . . . . . . . . 2.4.4. Hill macro-homogenity and separation of scales . . . . . . . . 2.5. Comments on multiple scale methods and statistical methods . . . . 2.5.1. On the periodicity, the stationarity and the concept of the REV 2.5.2. On the absence of, or need for macroscopic prerequisites . . . 2.5.3. On the homogenizability and consistency of the macroscopic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4. On the treatment of problems with several small parameters . .

. . . . . . . . . . .

55 56 57 59 60 61 61 61 62 64 65

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68 68 68 69 69 69 70

. .

71 72

Chapter 3. Homogenization by Multiple Scale Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Separation of scales: intuitive approach and experimental visualization 3.2.1. Intuitive approach to the separation of scales . . . . . . . . . . . 3.2.2. Experimental visualization of ﬁelds with two length scales . . . 3.2.2.1. Investigation of a ﬂexible net . . . . . . . . . . . . . . . . 3.2.2.2. Photoelastic investigation of a perforated plate . . . . . . . 3.3. One-dimensional example . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Elasto-statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 75 78 78 81 84 85

Contents

3.3.1.1. Equivalent macroscopic description . . . . . . . . . . . . 3.3.1.2. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Elasto-dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1. Macroscopic dynamics: Pl = O(ε2 ) . . . . . . . . . . . 3.3.2.2. Steady state: Pl = O(ε3 ) . . . . . . . . . . . . . . . . . 3.3.2.3. Non-homogenizable description: Pl = O(ε) . . . . . . 3.3.3. Comments on the different possible choices for spatial variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Expressing problems within the formalism of multiple scales . . . . 3.4.1. How do we select the correct mathematical formulation based on the problem at hand? . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Need to evaluate the actual scale ratio εr . . . . . . . . . . . . 3.4.3. Evaluation of the actual scale ratio εr . . . . . . . . . . . . . . 3.4.3.1. Homogenous treatment of simple compression . . . . . 3.4.3.2. Point force in an elastic object . . . . . . . . . . . . . . . 3.4.3.3. Propagation of a harmonic plane wave in elastic composites . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.4. Diffusion wave in heterogenous media . . . . . . . . . . 3.4.3.5. Conclusions to be drawn from the examples . . . . . . . PART TWO . H EAT AND M ASS T RANSFER

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7

86 89 91 92 95 95

. 97 . 100 . . . . .

100 101 102 103 104

. 104 . 105 . 106

. . . . . . . . . . . . . . . . . . . 107

Chapter 4. Heat Transfer in Composite Materials . . . . . . . . . . . . . . . 109 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Heat transfer with perfect contact between constituents . . . . . . 4.2.1. Formulation of the problem . . . . . . . . . . . . . . . . . . . 4.2.2. Thermal conductivities of the same order of magnitude . . . 4.2.2.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . 4.2.2.2. Macroscopic model . . . . . . . . . . . . . . . . . . . . 4.2.2.3. Example: bilaminate composite . . . . . . . . . . . . . 4.2.3. Weakly conducting phase in a connected matrix: memory effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . 4.2.3.2. Macroscopic model . . . . . . . . . . . . . . . . . . . . 4.2.3.3. Example: bilaminate composite . . . . . . . . . . . . . 4.2.4. Composites with highly conductive inclusions embedded in a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . 4.2.4.2. Macroscopic model . . . . . . . . . . . . . . . . . . . . 4.3. Heat transfer with contact resistance between constituents . . . . 4.3.1. Model I – Very weak contact resistance . . . . . . . . . . . .

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109 109 110 113 113 117 119

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121 122 124 125

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126 127 129 130 132

8

Homogenization of Coupled Phenomena

4.3.2. 4.3.3. 4.3.4. 4.3.5. 4.3.6. 4.3.7.

Model II – Moderate contact resistance . . . . . . . . . . Model III – High contact resistance . . . . . . . . . . . . . Model IV – Model with two coupled temperature ﬁelds . Model V – Model with two decoupled temperature ﬁelds Example: bilaminate composite . . . . . . . . . . . . . . . Choice of model . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5. Diffusion/Advection in Porous Media

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133 135 138 140 141 142

. . . . . . . . . . . . . . . 143

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Diffusion-convection on the pore scale and estimates . . . . . . . . 5.3. Diffusion dominates at the macroscopic scale . . . . . . . . . . . . . 5.3.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1.1. Boundary value problem for c∗(0) . . . . . . . . . . . . . 5.3.1.2. Boundary value problem for c∗(1) . . . . . . . . . . . . . 5.3.1.3. Boundary value problem for c∗(2) . . . . . . . . . . . . . 5.3.2. Macroscopic diffusion model . . . . . . . . . . . . . . . . . . . 5.4. Comparable diffusion and advection on the macroscopic scale . . . 5.4.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.1. Boundary value problems for c∗(0) and c∗(1) . . . . . . . 5.4.1.2. Boundary value problem for c∗(2) . . . . . . . . . . . . . 5.4.2. Macroscopic diffusion-advection model . . . . . . . . . . . . . 5.5. Advection dominant at the macroscopic scale . . . . . . . . . . . . . 5.5.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1.1. Boundary value problem for c∗(0) . . . . . . . . . . . . . 5.5.1.2. Boundary value problem for c∗(1) . . . . . . . . . . . . . 5.5.1.3. Boundary value problem for c∗(2) . . . . . . . . . . . . . 5.5.2. Dispersion model . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Very strong advection . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Example: Porous medium consisting of a periodic lattice of narrow parallel slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1. Analysis of the ﬂow . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2. Determination of the dispersion coefﬁcient . . . . . . . . . . . 5.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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143 143 146 146 146 147 148 148 149 149 149 149 150 151 151 151 151 153 154 154

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155 156 157 159

Chapter 6. Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2. Effective thermal conductivity for some periodic media . . . . . . . . 162 6.2.1. Media with spherical inclusions, connected or non-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Contents

6.2.1.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . 6.2.1.2. Solution to the boundary value problem over the period 6.2.1.3. Effective thermal conductivity . . . . . . . . . . . . . . . 6.2.2. Fibrous media consisting of parallel ﬁbers . . . . . . . . . . . . 6.2.2.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . 6.2.2.2. Solution to the boundary value problem over the period 6.2.2.3. Effective thermal conductivity . . . . . . . . . . . . . . . 6.3. Study of various self-consistent schemes . . . . . . . . . . . . . . . 6.3.1. Self-consistent scheme for bi-composite inclusions . . . . . . . 6.3.1.1. Granular or cellular media . . . . . . . . . . . . . . . . . 6.3.1.2. Fibrous media . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.3. General remarks on bi-composite models . . . . . . . . . 6.3.2. Self-consistent scheme with multi-composite substructures . . 6.3.2.1. n-composite substructure . . . . . . . . . . . . . . . . . . 6.3.2.2. Treatment of a contact resistance . . . . . . . . . . . . . 6.3.3. Combined self-consistent schemes . . . . . . . . . . . . . . . . 6.3.3.1. Mixed self-consistent schemes . . . . . . . . . . . . . . . 6.3.3.2. Multiple self-consistent schemes . . . . . . . . . . . . . 6.4. Comparison with experimental results for the thermal conductivity of cellular concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1. Dry cellular concrete . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2. Damp cellular concrete . . . . . . . . . . . . . . . . . . . . . . .

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9

162 163 163 168 168 169 170 175 175 175 178 179 181 181 183 184 185 185

. 188 . 189 . 190

PART THREE . N EWTONIAN F LUID F LOW T HROUGH R IGID P OROUS M EDIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Chapter 7. Incompressible Newtonian Fluid Flow Through a Rigid Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.1. 7.2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-state ﬂow of an incompressible Newtonian ﬂuid in a porous medium: Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1. Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. Comments on macroscopic behavior . . . . . . . . . . . . . . . 7.2.2.1. Physical meaning of the macroscopic quantities . . . . . 7.2.2.2. Structure of the macroscopic law . . . . . . . . . . . . . 7.2.2.3. Study of the underlying problem . . . . . . . . . . . . . 7.2.2.4. Properties of K∗ . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.5. Energetic consistency . . . . . . . . . . . . . . . . . . . . 7.2.3. Non-homogenizable situations . . . . . . . . . . . . . . . . . . 7.2.3.1. Case where QL = O(ε−1 ). . . . . . . . . . . . . . . . .

. 197 . . . . . . . . . .

199 201 203 203 204 205 205 206 206 207

10

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7.2.3.2. Case where QL = O(ε−3 ) . . . . . . . . . . . . . . . 7.3. Dynamics of an incompressible ﬂuid in a rigid porous medium . 7.3.1. Local description and estimates . . . . . . . . . . . . . . . . 7.3.2. Macroscopic behavior: generalized Darcy’s law . . . . . . 7.3.3. Discussion of the macroscopic description . . . . . . . . . . 7.3.3.1. Physical meaning of macroscopic quantities . . . . . 7.3.3.2. Energetic consistency . . . . . . . . . . . . . . . . . . 7.3.3.3. The tensors H∗ and Λ∗ are symmetric . . . . . . . . 7.3.3.4. Low-frequency behavior . . . . . . . . . . . . . . . . 7.3.3.5. Additional mass effect . . . . . . . . . . . . . . . . . 7.3.3.6. Transient excitation: Dynamics with memory effects 7.3.3.7. Quasi-periodicity . . . . . . . . . . . . . . . . . . . . 7.3.4. Circular cylindrical pores . . . . . . . . . . . . . . . . . . . 7.4. Appearance of inertial non-linearities . . . . . . . . . . . . . . . 7.4.1. Macroscopic model . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Macroscopically isotropic and homogenous medium . . . . 7.4.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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208 209 209 211 213 213 213 215 215 215 216 216 216 220 221 224 226 226

Chapter 8. Compressible Newtonian Fluid Flow Though a Rigid Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Slow isothermal ﬂow of a highly compressible ﬂuid . . . 8.2.1. Estimates . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. Steady-state ﬂow . . . . . . . . . . . . . . . . . . . . 8.2.3. Transient conservation of mass . . . . . . . . . . . . 8.3. Wall slip: Klinkenberg’s law . . . . . . . . . . . . . . . . 8.3.1. Pore scale description and estimates . . . . . . . . . 8.3.2. Klinkenberg’s law . . . . . . . . . . . . . . . . . . . 8.3.3. Small Knudsen numbers . . . . . . . . . . . . . . . . 8.3.4. Properties of the Klinkenberg tensor Hk . . . . . . . 8.3.4.1. Hk is positive . . . . . . . . . . . . . . . . . . 8.3.4.2. Symmetries . . . . . . . . . . . . . . . . . . . 8.4. Acoustics in a rigid porous medium saturated with a gas 8.4.1. Harmonic perturbation of a gas in a porous medium 8.4.2. Analysis of local physics . . . . . . . . . . . . . . . . 8.4.3. Non-dimensionalization and renormalization . . . . 8.4.4. Homogenization . . . . . . . . . . . . . . . . . . . . 8.4.4.1. Pressure and temperature . . . . . . . . . . . . 8.4.4.2. Velocity ﬁeld . . . . . . . . . . . . . . . . . . .

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229 229 230 231 235 238 238 240 241 243 243 244 245 246 247 249 251 251 252

Contents

11

8.4.4.3. Macroscopic conservation of mass . . . . . . . . . . . . . 252 8.4.5. Biot-Allard model . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Chapter 9. Numerical Estimation of the Permeability of Some Periodic Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Permeability tensor: recap of results from periodic homogenization 9.3. Steady state permeability of ﬁbrous media . . . . . . . . . . . . . . 9.3.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Transverse permeability . . . . . . . . . . . . . . . . . . . . . . 9.3.2.1. Mesh, velocity ﬁelds and microscopic pressure ﬁelds . . 9.3.2.2. Transverse permeability KT . . . . . . . . . . . . . . . . 9.3.3. Longitudinal permeability . . . . . . . . . . . . . . . . . . . . . 9.3.3.1. Mesh, velocity ﬁelds . . . . . . . . . . . . . . . . . . . . 9.3.3.2. Longitudinal permeability KL . . . . . . . . . . . . . . . 9.4. Steady state and dynamic permeability of granular media . . . . . . 9.4.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Steady state permeability . . . . . . . . . . . . . . . . . . . . . 9.4.4. Dynamic permeability . . . . . . . . . . . . . . . . . . . . . . . 9.4.4.1. Effect of frequency . . . . . . . . . . . . . . . . . . . . . 9.4.4.2. Low-frequency approximation . . . . . . . . . . . . . . . 9.4.4.3. High-frequency approximation . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

257 259 259 259 260 261 262 264 264 264 267 267 267 269 269 269 270 272

Chapter 10. Self-consistent Estimates and Bounds for Permeability . . . . 275 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1. Notation and glossary . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Intrinsic (or steady state) permeability of granular and ﬁbrous media 10.2.1. Summary of results obtained through periodic homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1.1. Global and local descriptions – energetic consistency . . . 10.2.1.2. Connections between the micro- and macroscopic descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2. Self-consistent estimate of the permeability of granular media . 10.2.2.1. Formulation of the self-consistent problem . . . . . . . . . 10.2.2.2. General expression for the ﬁelds in the inclusion . . . . . 10.2.2.3. Boundary conditions . . . . . . . . . . . . . . . . . . . . . 10.2.3. Solution and self-consistent estimates . . . . . . . . . . . . . . . 10.2.3.1. Pressure approach: p ﬁeld . . . . . . . . . . . . . . . . . 10.2.3.2. Velocity approach: v ﬁeld . . . . . . . . . . . . . . . . .

275 277 278 279 280 281 281 281 283 285 288 288 289

12

Homogenization of Coupled Phenomena

10.2.3.3. Comparison of estimates . . . . . . . . . . . . . . . . . . . 10.2.4. From spherical substructure to granular materials . . . . . . . . . 10.2.4.1. Cubic lattices of spheres . . . . . . . . . . . . . . . . . . . 10.2.4.2. Bounds on the permeability of ordered or disordered granular media . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4.3. Empirical laws . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5. Intrinsic permeability of ﬁbrous media . . . . . . . . . . . . . . . 10.2.5.1. Periodic arrangements of identical cylinders . . . . . . . . 10.2.5.2. Permeability bounds for ideal ordered and disordered ﬁbrous media . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Dynamic permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1. Summary of homogenization results . . . . . . . . . . . . . . . . 10.3.1.1. Global and local description – Energetic consistency . . . 10.3.1.2. Frequency characteristics of dynamic permeability . . . . 10.3.2. Self-consistent estimates of dynamic permeability . . . . . . . . 10.3.3. Formulation of the problem in the inclusion . . . . . . . . . . . . 10.3.3.1. Expressions for the ﬁelds . . . . . . . . . . . . . . . . . . . 10.3.3.2. Boundary conditions . . . . . . . . . . . . . . . . . . . . . 10.3.4. Solution and self-consistent estimates . . . . . . . . . . . . . . . 10.3.4.1. P estimate: p ﬁeld . . . . . . . . . . . . . . . . . . . . . . 10.3.4.2. V estimate: v ﬁeld . . . . . . . . . . . . . . . . . . . . . 10.3.4.3. Commentary and comparisons with numerical results for periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4.4. Bounds on the dynamic permeability of granular media . 10.3.4.5. Bounds on the real and imaginary parts of K(ω) . . . . . . 10.3.4.6. Bounds on the real and imaginary parts of H(ω) . . . . . . 10.3.4.7. Low-frequency bounds . . . . . . . . . . . . . . . . . . . . 10.3.4.8. High-frequency bounds for tortuosity . . . . . . . . . . . . 10.4. Klinkenberg correction to intrinsic permeability . . . . . . . . . . . . 10.4.1. Local and global descriptions obtained through homogenization 10.4.2. Self-consistent estimates of Klinkenberg permeability . . . . . . 10.5. Thermal permeability – compressibility of a gas in a porous medium 10.5.1. Dynamic compressibility obtained by homogenization . . . . . 10.5.2. Self-consistent estimate of the thermal permeability of granular media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3. Properties of thermal permeability . . . . . . . . . . . . . . . . . 10.5.4. Signiﬁcance of connectivity of phases . . . . . . . . . . . . . . . 10.5.5. Critical thermal and viscous frequencies . . . . . . . . . . . . . . 10.6. Analogy between the trapping constant and permeability . . . . . . . 10.6.1. Trapping constant . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 291 291 292 296 297 298 298 299 300 300 302 304 304 305 306 307 308 309 310 314 315 316 317 318 318 318 319 322 322 323 324 326 327 328 328

Contents

10.6.1.1. Comparison between the trapping constant and intrinsic permeability . . . . . . . . . . . . . . . . . . . 10.6.1.2. Self-consistent estimate of the trapping constant for granular media . . . . . . . . . . . . . . . . . . . . . . . 10.6.2. Diffusion-trapping in the transient regime . . . . . . . . . . . 10.6.3. Steady-state diffusion-trapping regime in media with a ﬁnite absorptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

. . 330 . . 331 . . 332 . . 333 . . 334

PART FOUR . S ATURATED D EFORMABLE P OROUS M EDIA . . . . . . . . . 337 Chapter 11. Quasi-statics of Saturated Deformable Porous Media . . . . . 339 11.1. Empty porous matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1. Local description . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2. Equivalent macroscopic behavior . . . . . . . . . . . . . . . . . 11.1.2.1. Boundary-value problem for u∗(0) . . . . . . . . . . . . 11.1.2.2. Boundary-value problem for u∗(1) . . . . . . . . . . . . 11.1.2.3. Boundary-value problem for u∗(2) . . . . . . . . . . . . 11.1.3. Investigation of the equivalent macroscopic behavior . . . . . . 11.1.3.1. Physical meaning of quantities involved in macroscopic description . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3.2. Properties of the effective elastic tensor . . . . . . . . . . 11.1.3.3. Energetic consistency . . . . . . . . . . . . . . . . . . . . 11.1.4. Calculation of the effective coefﬁcients . . . . . . . . . . . . . 11.2. Deformable saturated porous medium . . . . . . . . . . . . . . . . . 11.2.1. Local description and estimates . . . . . . . . . . . . . . . . . . 11.2.2. Diphasic macroscopic behavior: Biot model . . . . . . . . . . . 11.2.2.1. Boundary-value problem for u∗(0) . . . . . . . . . . . . 11.2.2.2. Boundary-value problem for p∗(0) and v∗(0) . . . . . . . 11.2.2.3. Boundary-value problem for u∗(1) . . . . . . . . . . . . 11.2.2.4. First compatibility equation . . . . . . . . . . . . . . . . 11.2.2.5. Second compatibility equation . . . . . . . . . . . . . . . 11.2.2.6. Macroscopic description . . . . . . . . . . . . . . . . . . 11.2.3. Properties of the macroscopic diphasic description . . . . . . . 11.2.3.1. Properties of macroscopic quantities and effective coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3.2. The coupling between (11.31) and (11.32) is symmetric, α = γ. . . . . . . . . . . . . . . . . . . . . . 11.2.3.3. The tensor α∗ is symmetric . . . . . . . . . . . . . . . . 11.2.3.4. The coefﬁcient β ∗ is positive, β ∗ > 0 . . . . . . . . . . . 11.2.3.5. Speciﬁc cases . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

340 340 342 342 343 344 345

. . . . . . . . . . . . . .

345 346 348 348 349 350 352 352 352 353 354 355 355 355

. 355 . . . .

356 356 357 357

14

Homogenization of Coupled Phenomena

11.2.3.6. Homogenious matrix material . . . . . . . . . . . . . 11.2.3.7. Homogenous and isotropic matrix material and macroscopically isotropic matrix . . . . . . . . . . . 11.2.3.8. Diphasic consolidation equations: Biot model . . . . 11.2.3.9. Effective stress . . . . . . . . . . . . . . . . . . . . . . 11.2.3.10. Compressible interstitial ﬂuid . . . . . . . . . . . . . 11.2.4. Monophasic elastic macroscopic behavior: Gassman model 11.2.5. Monophasic viscoelastic macroscopic behavior . . . . . . . 11.2.6. Relationships between the three macroscopic models . . .

. . . 357 . . . . . . .

. . . . . . .

. . . . . . .

358 359 361 361 362 363 365

Chapter 12. Dynamics of Saturated Deformable Porous Media . . . . . . . 367 12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. Local description and estimates . . . . . . . . . . . . . . . . . . . . . . 12.3. Diphasic macroscopic behavior: Biot model . . . . . . . . . . . . . . 12.4. Study of diphasic macroscopic behavior . . . . . . . . . . . . . . . . . 12.4.1. Equations for the diphasic dynamics of a saturated deformable porous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2. Rheology and dynamics . . . . . . . . . . . . . . . . . . . . . . . 12.4.3. Additional mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4. Transient motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.5. Small pulsation ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.6. Dispersive waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5. Macroscopic monophasic elastic behavior: Gassman model . . . . . . 12.6. Monophasic viscoelastic macroscopic behavior . . . . . . . . . . . . . 12.7. Choice of macroscopic behavior associated with a given material and disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1. Effects of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1.1. Transition from diphasic behavior to elastic behavior . . . 12.7.1.2. Transition from viscoelastic behavior to elastic behavior . 12.7.2. Effect of rigidity of the porous skeleton . . . . . . . . . . . . . . 12.7.3. Effect of frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.3.1. Low-dispersion P1 and S waves . . . . . . . . . . . . . . . 12.7.3.2. Dispersive P2 wave . . . . . . . . . . . . . . . . . . . . . . 12.7.4. Effect of pore size . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.5. Application example: bituminous concretes . . . . . . . . . . . .

367 368 370 374 374 375 376 376 376 376 377 378 380 382 382 383 384 384 384 385 385 385

Chapter 13. Estimates and Bounds for Effective Poroelastic Coefﬁcients . 389 13.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 13.2. Recap of the results of periodic homogenization . . . . . . . . . . . . 389 13.3. Periodic granular medium . . . . . . . . . . . . . . . . . . . . . . . . . 391

Contents

13.3.1. Microstructure and material . . . . . . . . . . . . . . . . . . . 13.3.2. Effective elastic tensor c . . . . . . . . . . . . . . . . . . . . . 13.3.2.1. Methodology . . . . . . . . . . . . . . . . . . . . . . . 13.3.2.2. Compressibility and shear moduli . . . . . . . . . . . . 13.3.2.3. Degree of anisotropy . . . . . . . . . . . . . . . . . . . 13.3.2.4. Young’s modulus and Poisson’s ratio . . . . . . . . . . 13.3.3. Biot tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. Inﬂuence of microstructure: bounds and self-consistent estimates 13.4.1. Voigt and Reuss bounds . . . . . . . . . . . . . . . . . . . . . 13.4.2. Hashin and Shtrikman bounds . . . . . . . . . . . . . . . . . . 13.4.3. Self-consistent estimates . . . . . . . . . . . . . . . . . . . . . 13.4.4. Comparison: numerical results, bounds and self-consistent estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Comparison with experimental data . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

15

391 392 392 394 396 396 398 398 399 399 400

. . 401 . . 403

Chapter 14. Wave Propagation in Isotropic Saturated Poroelastic Media . 407 14.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2. Comments on the parameters . . . . . . . . . . . . . . 14.2.2.1. Elastic coefﬁcients . . . . . . . . . . . . . . . . 14.2.2.2. Dynamic permeability . . . . . . . . . . . . . . 14.2.3. Degrees of freedom and dimensionless parameters . . 14.3. Three modes of propagation in a saturated porous medium 14.3.1. Wave equations . . . . . . . . . . . . . . . . . . . . . . 14.3.2. Elementary wave ﬁelds: plane waves . . . . . . . . . . 14.3.2.1. Homogeneous plane waves . . . . . . . . . . . . 14.3.2.2. Inhomogenous plane waves . . . . . . . . . . . 14.3.3. Physical characteristics of the modes . . . . . . . . . . 14.3.3.1. Low frequencies: f fc . . . . . . . . . . . . 14.3.3.2. High frequencies: f fc . . . . . . . . . . . . 14.3.3.3. Full spectrum . . . . . . . . . . . . . . . . . . . 14.4. Transmission at an elastic-poroelastic interface . . . . . . . 14.4.1. Expression for the conditions at the interface . . . . . 14.4.2. Transmission of compression waves . . . . . . . . . . 14.5. Rayleigh waves . . . . . . . . . . . . . . . . . . . . . . . . . 14.6. Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1. Source terms . . . . . . . . . . . . . . . . . . . . . . . 14.6.2. Determination of the fundamental solutions . . . . . . 14.6.3. Fundamental solutions in plane geometry . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

407 408 408 410 410 410 411 412 413 416 416 417 419 419 421 423 423 426 428 430 432 432 433 437

16

Homogenization of Coupled Phenomena

14.6.4. Symmetry of the Green’s matrix, and reciprocity theorem 14.6.5. Properties of radiated ﬁelds . . . . . . . . . . . . . . . . . 14.6.5.1. Far-ﬁeld – near-ﬁeld – quasi-static regime . . . . . 14.6.5.2. Decomposition into elementary waves . . . . . . . 14.6.6. Energy and moment sources: explosion and injection . . 14.7. Integral representation . . . . . . . . . . . . . . . . . . . . . . . 14.8. Dislocations in porous media . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

438 439 441 442 442 445 448

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Main notations

Operators ⊗ : · × , x div, divx grad, gradx curl, curlx a aΩα

tensor product doubly contracted tensor product scalar product vector product Laplacian, Laplacian with respect to the variable x divergence, divergence with respect to the variable x gradient, gradient with respect to the variable x curl, curl with respect to the variable x mean of a over the period Ω mean of a over the domain Ωα

Dimensions, bases, spatial variables Ω Ωα ∂Ω Γ l L ε = l/L ei n X = Xi ei x∗ = X/Lc = x∗i ei y∗ = X/lc = yi∗ ei

period domain occupied by constituent α boundary of the period interface between two constituents microscopic length [m] macroscopic length [m] separation of scales parameter unit vector – orthonormal basis unit vector normal to the interface Γ physical spatial variable [m] macroscopic dimensionless variable [-] microscopic dimensionless variable [-]

18

Homogenization of Coupled Phenomena

Dimensionless numbers B Kn Pe Re Rt S

Biot number [-] Knudsen number [-] Péclet number [-] Reynolds number [-] transient Reynolds number [-] Strouhal number [-]

Properties and physical quantities a = aijkl ei ⊗ ej ⊗ ek ⊗ el c cα Cα c = cijkl ei ⊗ ej ⊗ ek ⊗ el D(v) = Dij (v) ei ⊗ ej dif ei ⊗ ej Ddif = Dij dis dis ei ⊗ ej D = Dij mol Dmol = Dij ei ⊗ ej e(u) = eij ei ⊗ ej E h H = Hij ei ⊗ ej H(ω) = Hij (ω) ei ⊗ ej H R , HI H(ω) = K−1 = HI + iHR H(ω) = Hij (ω) ei ⊗ ej k ei ⊗ ej Hk = Hij I K K = Kij ei ⊗ ej k ei ⊗ ej Kk = Kij K(ω) = KR + iKI K(ω) = Kij (ω) ei ⊗ ej K R , KI M p t Tα us = usi ei

elastic tensor [MPa] concentration [-] volume fraction of constituent α [-] heat capacity of constituent α [J/(Kg.K)] effective elastic tensor [MPa] strain rate tensor [s−1 ] effective diffusion tensor [m2 /s] effective dispersion tensor [m2 /s] molecular diffusion tensor [m2 /s] strain tensor [-] Young’s modulus [MPa] inverse of the thermal contact resistance inverse of the intrinsic permeability tensor [m2 ] inverse of the dynamic permeability tensor K [m−2 ] real and imaginary parts of the H tensor inverse of the dynamic permeability of an isotropic medium inverse of the dynamic hydraulic conductivity tensor Λ = ηK(ω) Klinkenberg tensor [m2 ] identity tensor intrinsic permeability of an isotropic medium [m2 ] intrinsic steady state permeability tensor [m2 ] Klinkenberg permeability tensor [m2 ] dynamic permeability of an isotropic medium [m2 ] dynamic permeability tensor [m2 ] real and imaginary parts of the K tensor form factor ﬂuid pressure [Pa] time [s] temperature of constituent α [˚K] solid displacement [m]

Main notations

v = vi ei α δ η λα λeﬀ = λeﬀ ij ei ⊗ ej λ Λv Λ(ω) = Λij (ω) ei ⊗ ej μ ν ρ σ = σij ei ⊗ ej τ0 , τ∞ φ ω

ﬂuid velocity [m/s] Biot tensor [-] boundary layer thickness (thermal, diffusion...) dynamic ﬂuid viscosity [Pa.s] thermal conductivity of constituent α [W/(m.˚K)] effective thermal conductivity tensor [W/(m.˚K)] Lamé coefﬁcient [MPa] viscous length hydraulic conductivity tensor shear modulus [MPa] Poisson’s ratio [-] ﬂuid density [kg/m3 ] Cauchy stress tensor low- and high-frequency tortuosity porosity [-] pulsation (angular frequency) [rad/s]

Subscripts αc α∗ Ql QL

characteristic value of α dimensionless quantity (α = αc α∗ ) dimensionless number Q estimated from the microscopic viewpoint dimensionless number Q estimated from the macroscopic viewpoint

19

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Introduction

The result of a fusion of mathematical and physical concepts, homogenization has established itself as a method of overcoming the usual framework based on a description of elementary phenomena in a homogenous medium, to achieve the objective of a global description of coupled phenomena in heterogenous media. This book aims to present the key methodological points and their relevance to engineering science in a pedagogical format. What is the nature of the problem? Even brief observation of natural or industrial materials reveals that they often consist of a combination of different constituents in various structures, and they are therefore heterogenous. For example, take the behavior of civil engineering materials. The descriptions of the properties that they exhibit – and consequently the design rules for structures built using these materials – are, for the most part, issues related to the mechanics of continuous media applied to homogenous media. This theory has been widely proven, and a huge number of constructions designed using these principles can attest to its success, as can the accuracy of modeling performed using this approach. This simple observation leads us to believe that heterogenous materials can, at least subject to certain constraints, be treated similarly. But why and to what extent is this concept useful? Furthermore, although the heterogenous nature of the material may not be obvious for this apparently continuous medium, it is on the other hand clear that its behavior depends on the characteristics of the heterogenities. How then do we proceed if we are to account for the properties of the constituents when deﬁning the behavior of an equivalent continuous homogenous medium? These two points are of great practical importance. On one hand, understanding of the limits of a method is an important safety consideration, and on the other hand determination of the equivalent continuous medium allows us to better understand the

22

Homogenization of Coupled Phenomena

parameters that govern its behavior (for a natural material) or to adapt the constituent parts to achieve the desired performance (for an artiﬁcial material). The homogenization methods have been developed to answer these questions. They make it possible – under well-speciﬁed conditions – to obtain a description of the behavior of heterogenous materials starting from the behavior of the heterogenities. A condition essential for the existence of an equivalent continuum is that the physical mechanism under study should vary on a length scale which is very large compared to the scale of the heterogenities. This requirement for a difference in length scales gives rise to the expressions “upscaling method” and “method of multiple scales”. The term “homogenization” also arises from this, because considering the heterogenities to be of inﬁnitessimal size compared to the effects under study naturally leads us to consider the medium as a homogenous or, more precisely, homogenized continuum. Linking the large-scale observable behavior to microscopic mechanisms is an ageold preoccupation of physicists. One famous example is that of elasticity, where Navier (1821) and Poisson (1829) obtained a single macroscopic isotropic elastic coefﬁcient from a particular molecular model: the two Lamé coefﬁcients are equal. Cauchy (1828) obtained a two-coefﬁcient isotropic elastic model starting from a more sophisticated molecular model. From among these well-known names we also draw attention to the preliminary work of Rayleigh (1892) on the conductivity of media containing impurities present in a parallelopiped lattice, and that of Einstein (1906) on the viscosity of suspensions and sedimentation rates. These attempts remained intermittent until the 1950s when the needs of industrial development demanded a detailed understanding of the behavior of natural materials (the oil industry), manufactured materials (in particular steels and alloys), and the design of new materials (mainly for aeronautics). In order to understand the signiﬁcance of small scale mechanisms on global behavior, scientiﬁc approaches at the time involved phenomenological micromechanical models built on thermodynamic principles. The pioneering works of Biot (1941) and of Hill (1965) took this approach. It was in the 1970s that a new school of thought was born, started by Keller who used a rather different angle to tackle the question of the change of scales. This involved the method of asymptotic expansions at multiple scales. Initially built on an approach which was more mathematical than mechanical, this method was a true conceptual leap forwards in terms of its rigor and formalism. The works of Bensoussan et al. (1978) and Sanchez-Palencia (1980) are still important references on this subject, with similar ideas developed in Russia by Bakhvalov and Panasenko (1989).

Introduction

23

Initially conﬁned to very specialist circles, these methods blossomed considerably in the 1990s. During this period their ﬁelds of application were broadly diversiﬁed across all traditional engineering ﬁelds, and also in life sciences, particularly biomechanics. This undeniable success is thanks to the effectiveness of asymptotic methods for treating complex physics on a microscopic scale, and their ability to include coupling between different phenomena. However, while their use has become almost routine for some research groups, it remains poorly documented at present in a form suitable for engineers and researchers working in related ﬁelds. These considerations convinced us there was a demand for a book which would set out a coherent picture of these approaches, and render them accessible to a wider audience than just specialists in this research area. Rather than be exhaustive (which would not be an easy task), we have chosen to pick a few problems where the main points can be presented in a simple manner. The aim is also – using a uniﬁed treatment – to illustrate the common thought processes connecting the issues addressed in this book. In keeping with this approach, the bibliography does not attempt to be exhaustive, but shows the reader the seminal works in the ﬁeld, and the references corresponding to the main steps of the problems we consider. This volume, which has grown out of the Mechanics of Heterogenous Media course taught at the University of Grenoble by J.L. Auriault, is intended to be a basic course in upscaling methods, aimed at advanced students, engineers and graduate students. With this pedagogical aim, we have used a progressive approach to each subject, starting out with traditional problems and then following them with recent developments. We also thought it useful to illustrate the potential applications of the results of homogenization. With this in mind, for each of the themes we treat, the theoretical results are followed by an example of the development through homogenization which provides a concrete example of the advances in a particular ﬁeld of application. This book is divided into four parts. Part 1 is an introduction to the philosophy of homogenization methods. We discuss methods aimed at periodic and random materials while emphasizing their physical signiﬁcance and their potential applications to real materials, which are often neither perfectly periodic or perfectly random. The basic examples given in Chapter 1 give an understanding of the fundamental tools underlying both methods. Chapter 2 goes into more detail on the techniques and discusses connections between them and the details which distinguish them. There is a detailed discussion of conditions for their application, delineating the range of validity of these approaches. This overview of methods underlines the power of the asymptotic

24

Homogenization of Coupled Phenomena

method at multiple scales for the treatment of complex physics with many coupled effects in materials with simple or heirarchical morphologies. Combining rigorous formalism and intuitive reasoning, Chapter 3 presents the methodology of the multiple scale approach which will be used throughout the rest of the book. The emphasis is on the systematic use of dimensional analysis combined with the separation of length scales. We also detail the means of expressing a practical problem involving real materials in the context of homogenization. This methodological basis is applied in the following sections where we speciﬁcally treat the physical mechanisms involved in coupled phenomena. Part 2 presents a ﬁrst ﬁeld of application of homogenization. We study the physics of transport by diffusion, convection and advection, phenomena which allow us to apply the basic tools of upscaling methods to engineering problems. Chapter 4 focuses on thermal transfer in heterogenous media. Going beyond the classical model of thermal transfer in a composite, we ﬁnd a diverse range of macroscopic models depending on the level of contrast in the conductive properties of the constituents and their interfaces. In particular, memory effects arise from the presence of local non-equilibrium of a very weakly conducting phase, and twotemperature models can be developed for quasi-insulating interfaces. The transport of solutes in porous media is examined in Chapter 5. We highlight the different descriptions associated with the local physics of pure diffusion and then with diffusionadvection. This second situation, which is reached at high transport rates, results in a macroscopic dispersion. The range of validity of each of these models is explicitly speciﬁed. Chapter 6 makes use of, and extends, these results, focusing on speciﬁc materials. The numerical procedure of periodic homogenization is illustrated by determining the coefﬁcients for ﬁbrous and granular materials. By way of comparison, we recall the classical self-consistent analytical estimates. Finally, comparison with experimental results enables us to judge the appropriateness of these models for describing the properties of materials. Part 3 is dedicated to the modeling of Newtonian ﬂuid ﬂows in rigid porous media. Chapter 7 discusses incompressible ﬂuids using multiple-scale asymptotic expansions. It starts with the canonical problem of Darcy’s law (in the regime of steady-state laminar ﬂow). It continues taking into account inertial effects, both in the dynamic linear regime which leads to memory effects through visco-inertial coupling, and in the steady-state advective regime, where the correction due to weak nonlinearities is established. The ﬂow in porous media of compressible ﬂuids such as gases is the subject of Chapter 8. Using the asymptotic method, we treat in succession high pressure steady-state ﬂows, wall slip effects in rareﬁed gases and, in the dynamic regime, the acoustic description under weak pressure perturbations with thermal coupling. The transfer of theoretical results for homogenization to their numerical formulation is illustrated in Chapter 9. The solution to local problems derived by

Introduction

25

periodic homogenization is given for calculation of the Darcy permeability of granular and ﬁbrous materials. Finally, Chapter 10 returns to the same problems, which are discussed in the context of a self-consistent approach. We use this to establish analytical estimates and bounds for steady state and dynamic permeabilities, thermal effects, wall slip corrections and – by analogy – for the trapping constant. Part 4 focuses on the behavior of deformable saturated porous media. Chapter 11 considers the behavior in the quasi-static regime, ﬁrst examining that of the empty porous medium (a speciﬁc case of an elastic composite) and then that of the saturated medium, introducing the ﬂuid-solid coupling. Depending on the level of contrast between the shear properties of the ﬂuid and the solid, the asymptotic method of multiple scales leads to three distinct behaviors whose properties are discussed. The study of poroelastic behavior is extended to the dynamic regime in Chapter 12. The characteristics of the three possible behaviors – including the Biot biphasic model – are analyzed in detail, particularly properties of the effective coefﬁcients. The range of validity of each of the descriptions is speciﬁed. Chapter 13 puts the homogenization results to numerical use in order to carry out a parametric analysis of the elastic and coupling coefﬁcients in the biphasic model. At the same time these results, obtained for cohesive granular media, are compared to traditional selfconsistent estimates and to bounds. In Chapter 14, the homogenized biphasic behavior is used with the aim of describing the propagation of waves in saturated porous media. After specifying the properties of the three propagation modes, the transmission of waves across a poroelastic interface is examined. We also establish the expression for Green’s functions in the context of poroelasticity, the integral formulation, and the ﬁelds radiated by abrupt dislocations. To complete our summary of this text, it is worth mentioning certain important subjects which are not treated here (or only discussed brieﬂy). One of these subjects is complex microstructures. In fact, we will only consider media whose local geometry is sufﬁciently simple that it can be characterized by a typical length scale of the heterogenity, and whose local problems can be formulated in terms of continuous media. This choice means that we omit: – Media whose architecture involves very different characteristic sizes (such as double porosity media). These can give rise to various interacting physical effects on each length scale. These many possible couplings vastly increase the diversity of the possible macroscopic behaviors, with some behaviors only being possible in such media as; – Microstructure whose behavior can be reduced to that of various interacting points within the material (for example the nodes in trellis structures). For these it is preferable to use a locally discrete description, and to move to a continuum description through homogenization. This alternative approach will not be discussed here.

26

Homogenization of Coupled Phenomena

A second aspect only outlined is that of the corrections to macroscopic descriptions which have been established to ﬁrst order. Indeed, for the most part, the results presented here are restricted to the ﬁrst signiﬁcant term, and lead to descriptions involving a continuous medium which is materially simple, and descriptions valid in the bulk of the heterogenous medium. There are two corrections which can usefully be applied to these descriptions: – those which appear on the boundary of the medium. They lead to a boundary layer with a thickness of the order of the size of the representative elementary volume. This makes it possible to reconcile local boundary conditions and boundary conditions used at the macroscopic scale; – those which make it possible to treat situations with weak separation of scales, obtained by including higher-order terms within the homogenized descriptions. These correctors to simple continuum models introduce non-local effects whose spatial extent is of the order of the size of the representative elementary volume. Finally, we will not discuss the taking into account of non-linearities. All the cases that we present involve linear effects, or sometimes weakly non-linear ones where the non-linearity can be treated as a perturbation of the linear solution. Whether they have material or geometric origins, non-linearities introduce considerable theoretical difﬁculties compared to linear situations. While the establishment of criteria through local limit analysis – the rheology of elastic composites with a non-linear power law, or the ﬂow of non-Newtonian power-law ﬂuids in porous media – has been successfully achieved, in general non-linearities present a real challenge to upscaling methods. These three omitted themes – complex microstructures, corrections and nonlinearities – are very rich and interesting, and they deserve further discussion on their own. We hope that this volume will offer a sufﬁciently clear and solid basis to guide the reader who may wish to explore these ﬁelds. This work is the fruit of a long collaboration between its authors. It has of course been supported by the work and suggestions of numerous friends, colleagues and research students, whom we are delighted to thank for the assistance that they have given us, and in particular: P. Adler, I. Andrianov, L. Arnaud, P. Y. Bard, J. F. Bloch, G. Bonnet, L. Borne, L. Dormieux, H. Ene, M. Leﬁk, T. Levy, J. Lewandowska, C.C. Mei, X. Olny, L. Orgéas, P. Royer, E. Sanchez-Palencia, T. Strzelecki. We extend particular gratitude to P. Adler, whose sound advice and criticism has added a great deal to this work.

PART ONE

Upscaling Methods

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Chapter 1

An Introduction to Upscaling Methods

1.1. Introduction In engineering applications, media with small-scale heterogenities are often used in volumes which contain a very large number of these heterogenities. Despite progress in numerical modeling, the small-scale description of such volumes remains difﬁcult (Figure 1.1). The aim of homogenization techniques – also know as upscaling methods – is to model the heterogenous medium as a simpler equivalent continuous medium, whose description is valid at a scale which is very large compared to that of the heterogenities. The aim of this chapter is to introduce the problem and the basics of three common upscaling methods, speciﬁcally the technique of homogenization of periodic structures, the determination of bounds to the effective coefﬁcients, and self-consistent estimates. By way of example, we will consider steady-state heat transfer in a heterogenous medium and for this simple physical phenomenon we will discuss: – the explicit calculation of macroscopic conduction coefﬁcients for a periodic medium with a known microstructure. Here we will consider a bilaminate composite, for which analytic solutions can be found (section 1.2); – the determination of limits which bound the macroscopic conduction coefﬁcients for heterogenous media, regardless of the microstructure (section 1.3); – the self-consistent estimate of a macroscopic conduction coefﬁcient, valid for media with random microstructures which obey certain morphological constraints (section 1.4).

30

Homogenization of Coupled Phenomena

Figure 1.1. Two examples of heterogenous materials: concrete and polyurethane foam. These photographs illustrate the large number of heterogenities contained within the volumes that are typically used

1.2. Heat transfer in a periodic bilaminate composite Consider the steady-state heat transfer in a periodic bilaminate composite (Figure 1.2). The composite is periodic, with period l in the direction e2 . The volume fraction of medium a is denoted ca , and that of medium b is cb = 1 − ca . The constituents are isotropic and homogenous, with thermal conductivities λa and λb . Γ designates the interfaces. We consider a sample consisting of a large number of periods. The equations describing the temperature ﬁeld T (x) are: divQ = 0 Q = −λα gradT,

(1.1) λα > 0,

α = a, b

(1.2)

An Introduction to Upscaling Methods

31

e2 Q·n=0

L2 T0

T1

l

e1

(a)

Q·n=0

L1

e2

1 (1 cb )l

a

(b)

cb l

b

e1

Figure 1.2. (a) Macroscopic sample undergoing a transfer parallel to the layers and (b) period of the bilaminate composite

[Q · n]Γ = 0

(1.3)

[T ]Γ = 0

(1.4)

where Q(x) is the heat ﬂux. In the above equations, [φ]Γ represents the jump in φ across the interface Γ. Can we replace this description with a simpler equivalent continuum? 1.2.1. Transfer parallel to the layers Consider a sample (Figure 1.2) subjected at x1 = 0 and x1 = L1 to temperatures T0 and T1 . The boundaries x2 = 0 and x2 = L2 are perfectly insulating: Q · n = 0 Clearly T only depends on x1 , and the same is therefore true of Q. Since: d dT (λα )=0 dx1 dx1

32

Homogenization of Coupled Phenomena

the temperature gradient is constant in each phase. And with (1.4), it is constant everywhere. However Q1 = −λα ddxT depends on the material. If we take the average, 1 with in this case: Q1 =

1 L2

L2

Q1 dx2 0

we ﬁnd that: Q1 = −λα

dT dx1

where . represents the average volume. We will make the following hypothesis (H) which we will use throughout the rest of this chapter: (H): the macroscopic behavior is described by the same formulation as the local behavior, with a Fourier law and a balance equation. In the particular case we are considering here, we have: d Q1 = 0 dx1 Q1 = −λeﬀ 11

(1.5) dT dx1

(1.6)

Hence the macroscopic conductivity λeﬀ 11 is such that: 1 L2

λeﬀ 11 = λα =

L2

λα dx2 0

The length L2 is a number (not necessarily integer) of periods: L2 = N l + l0 ,

l0 < l,

N integer

Thus we have: λeﬀ 11

1 = N l + l0

N l + l0 eﬀ λ11 Nl

=

0

Nl

1 λα dx2 + N l + l0

1 Nl

Nl

λα dx2 + 0

l0

λα dx2 0

1 Nl

l0

λα dx2 0

An Introduction to Upscaling Methods

= =

l0 1 λα dx2 + λα dx2 Nl 0 0 l0 1 int λ11 + λα dx2 Nl 0 1 l

33

l

where λint 11 , a constant independent of L2 , is intrinsic to the material. It is the arithmetic mean of the conduction coefﬁcients of the constituents of the period. λint 11 = ca λa + (1 − ca )λb Deﬁning: ˜ 11 = 1 λ l

l0

0

λα dx2 < λint 11

it follows that: λeﬀ 11 =

Nl Nl ˜ 11 λ λint 11 + N l + l0 (N l + l0 )N

λeﬀ 11 is not intrinsic, and depends on L2 . If N is large, we have: int λeﬀ 11 ≈ λ11 −

1 int l0 ˜ (λ + λ11 ) N 11 l

When N tends to inﬁnty, N ≈

L2 and: l

N →∞

int λeﬀ 11 −→ λ11 int eﬀ int is intrinsic. Since λeﬀ 11 = λ11 [1+O(l/L2 )], the approximation of λ11 by λ11 becomes better when the separation of scales is good. In other words the ratio l/L2 tends to 0. λint 11 is also known as the effective coefﬁcient.

1.2.2. Transfer perpendicular to the layers Consider a sample (Figure 1.3) subject at x2 = 0 and x2 = L2 to temperatures T0 and T1 . The boundaries at x1 = 0 and x1 = L1 are perfectly insulated: Q · n = 0. T and Q now only depend on x2 . We have: d dT (λα )=0 dx2 dx2

34

Homogenization of Coupled Phenomena

e2 T1

L2

l

Q·n=0

Q·n=0

e1 L1

T0

Figure 1.3. Macroscopic sample subject to transfer perpendicular to the layers

Q is constant in each material, and with (1.3) it is constant throughout. We therefore have: Q dT =− dx2 λα and by integration: T1 − T0 Q =− L2 L2

L2 0

dx2 λα

For the boundary-value problem we are considering here, the left hand side represents the macroscopic temperature gradient. With the hypothesis (H) made in the previous section, it follows that: −1 (λeﬀ 22 )

1 = L2

L2

0

dx2 1 = λα N l + l0

N l+l0 0

dx2 λα

Thus λeﬀ 22 is not intrinsic and, as in the preceding section, we can write: 1 −1 int −1 1 + O( (λeﬀ ) , ) = (λ ) 22 22 N

−1 (λint = 22 )

1 l

0

l

dx2 λα

λint 22 is the geometric mean of the conduction coefﬁcients within the period: ca 1 − ca 1 = + int λ λb λ22 a λint 22 is intrinsic.

or alternatively

λint 22 =

λa λb ca λb + (1 − ca )λa

An Introduction to Upscaling Methods

35

1.2.3. Comments The above results lead to various observations: – The effective coefﬁcients are only an approximation. The error is O(l/L2 ). When l/L2 tends to zero, λeﬀ tends to the intrinsic value λint . – When the stratiﬁed material has a random structure, we get the same kind of result. The period is then replaced by the representative elementary volume. This is a volume containing a sufﬁciently large number of layers that its properties can be considered as equivalent to that of the macroscopic sample. l represents the thickness of the representative elementary layer. – It would be useful if we could abandon the prerequisite (H) on the macroscopic scale. – Here the geometry of the heterogenities is very simple. Can we extend certain results to more complex geometries? – Our demonstration here makes use of very simple conditions in the macroscopic limit (homogenous). What happens in more general cases? – We have studied the transfer in the steady state regime. What are the effective coefﬁcients in the transient regime? Do they still exist?

1.2.4. Characteristic macroscopic length The two examples above demonstrate the role played by the sample length L1 or L2 . This macroscopic length characterizes the size of the sample. We note that L = Lgeom = L1 , L2 , for geometric L. We can also deﬁne a phenomenological macroscopic length Lphen which characterizes the heat ﬂux [BOU 90]: Lphen =

T |gradT |

(1.7)

In the two cases considered here: Lphen = Lgeom

(1.8)

The geometric and phenomenological macroscopic lengths are the same in our case. This is a very special case, though. In many problems: Lphen = Lgeom

(1.9)

and the macroscopic length scale that must be used will be: L = min(Lphen , Lgeom )

(1.10)

36

Homogenization of Coupled Phenomena

In particular, this will be the case for transient phenomena where Lphen can be compared to a “wavelength”. We note also that Lphen may vary from one point to another in the sample, thus introducing a separation of scales which is not constant across the sample. We will return in detail to this issue in Chapter 3. 1.3. Bounds on the effective coefﬁcients We now consider arbitrary local geometries, either periodic or random. In contrast with the layered structure where the microstructure was explicitly deﬁned, we assume here that the only information available on the material is the concentration of the constituents and their properties. Naturally, depending on the morphology of the medium, the effective coefﬁcients will take different values: without additional assumptions we cannot therefore offer estimates of the effective coefﬁcients. We can however show with the help of (i) the principle of virtual powers, (ii) minimization of potential and complementary energies and (iii) the Hill principle, that the range of possible values of the effective conditions is bounded. These energetic methods using variational approaches make up a powerful theoretical tool with which to frame the macroscopic properties of the material [WIL 81]. Only requiring the bare minimum of information on the medium (concentrations and coefﬁcients of the constituents), these bounds create a very widely-applicable result. In order to present this approach, we will again study steady-state heat transfer, and we will assume the constituents to be isotropic. 1.3.1. Theorem of virtual powers In the steady-state regime, heat transfer in a heterogenous medium, with a symmetric, positive deﬁnite thermal conductivity tensor λ(x) at all points, which is described by: divQ = 0

(1.11)

Q = −λ gradT

(1.12)

with continuity conditions on the temperature and ﬂux on the surface Γ of any discontinuities which may be present: [Q · n]Γ = 0

(1.13)

[T ]Γ = 0

(1.14)

We will consider the boundary-value problem shown in Figure 1.4. A volume V of material is bounded by the surface S = ΓQ ∪ ΓT . On ΓQ we impose a normal ﬂux Qn (x) and on ΓT we impose a temperature T0 (x).

An Introduction to Upscaling Methods

37

Q · n = Qn

n

V

Q T (x)

T0 T Figure 1.4. Macroscopic boundary conditions

We deﬁne two virtual ﬁelds: – a ﬁeld of admissible ﬂux Q∗ , satisfying: divQ∗ = 0

within V

Q∗ · n = Qn

over ΓQ

– a ﬁeld of admissible temperature T , satisfying: T = T0

over ΓT

In general Q∗ = −λ gradT , except if we choose the actual ﬁelds Q and T , which are the correct solutions to the boundary-value problem for Q∗ and T . We will now calculate the virtual power developed by the ﬁelds T and Q∗ . We have: T divQ∗ dV = 0 V

so that: ∗ div(T Q )dV − Q∗ · gradT dV = 0 V

V

And using the divergence theorem: T Q∗ · ndS − Q∗ · gradT dV = 0 S

V

Finally, decomposing S to reﬂect the type of boundary condition imposed: Q∗ · gradT dV = T Qn dS + T0 Q∗ · n dS (1.15) V

ΓQ

ΓT

38

Homogenization of Coupled Phenomena

The virtual power dissipated equals the exterior virtual inputs. This equality, which forms the theory of virtual power, is satisﬁed in particular for the solution Q∗ = Q and T = T . 1.3.2. Minima in the complementary power and potential power Consider the following quadratic form: 1 ∗ (Q + λgradT ) · λ−1 (Q∗ + λgradT ) dV Φ(Q∗ , T ) = 2 V Clearly: Φ(Q∗ , T ) 0 and Φ = 0 for Q∗ = Q, and T = T , since Q = −λ gradT . Making use of the properties of λ, expansion of Φ gives: 1 ∗ −1 ∗ 1 Q ·λ Q dV + λ gradT · gradT dV + Q∗ · gradT dV Φ(Q∗ , T ) = V 2 V 2 V The last integral is given by the theory of virtual powers. Φ can then be written in the form: Φ(Q∗ , T ) = Φ1 (Q∗ ) + Φ2 (T ) ∗

Φ1 (Q ) = V

Φ2 (T ) =

V

1 ∗ −1 ∗ Q · λ Q dV + 2

T0 Q∗ · n dS ΓT

1 λ gradT · gradT dV + 2

T Qn dS

ΓQ

The complementary dissipation Φ1 (Q∗ ) and the potential dissipation Φ2 (T ) vary independently with Q∗ and T , and their sum is minimized for Q∗ = Q and T = T . As a result, Φ1 is minimized for Q∗ = Q and Φ2 is minimized T = T . From this we deduce two inequalities: Φ1 (Q) Φ1 (Q∗ )

(1.16)

which states that the minimum in the complementary dissipation is attained for the ﬂux ﬁeld Q which is the solution to the boundary-value problem, and: Φ2 (T ) Φ2 (T )

(1.17)

An Introduction to Upscaling Methods

39

which states that the minimum in the potential dissipation is attained for the temperature ﬁeld solution T . In conclusion, since Φ(Q, T ) = Φ1 (Q) + Φ2 (T ) = 0, for any admissible temperature ﬁeld T and admissible ﬂux ﬁeld Q∗ , we arrive at the following framework: −Φ2 (T ) −Φ2 (T ) = Φ1 (Q) Φ1 (Q∗ ) 1.3.3. Hill principle The Hill principle makes it possible to establish a link between the microscopic and macroscopic descriptions. It is therefore fundamental to the upscaling methods for which the micro-macro connection does not fall out of the homogenization procedure. This principle is based on two hypotheses: – hypothesis (H) (see section 1.2) which, we recall, stipulates (i) that the global variables are the volume means of the local ﬂux and gradients, and (ii) that the conservation and constitutive equations have the same structure at microscopic and macroscopic scales; – the assumption of energetic consistency – known as the Hill principle – which imposes equality on the energy contained within the medium, whether it is expressed in local variables or using macroscopic variables deﬁned according to (H). In the case of heat transfer, the Hill principle applies to dissipation expressed in terms of the microscopic variables – in other words the thermal conductivity tensor λ, gradT and Q. It is also expressed in terms of the dissipation relating to the macroscopic variables which are, according to (H), the effective thermal conductivity tensor λeﬀ , gradT and Q: gradT · Q dV = gradT · λgradT dV = V gradT · λeﬀ gradT (1.18) V

V

Q · λ−1 Q dV = V Q · λeﬀ−1 Q

= V

where . represents the volume mean: 1 . dV . = V V In the following section, we will show that when the conditions at the boundary of an arbitrary volume V are homogenous, either in terms of temperature gradient or ﬂux, then: V gradT · Q dV = gradT dV · Q dV V

V

V

40

Homogenization of Coupled Phenomena

Under these conditions (and these conditions alone), and in the context of (H), the Hill principle becomes the Hill theorem. In addition, we will see many times in subsequent chapters that even if (H) is not always respected – the Darcy law is a blatant example – energetic consistency can be demonstrated through the method of asymptotic expansions, without the requirement of homogenous conditions at the boundary. More precisely, we show that in its present form the Hill principle is just an approximation, which becomes better when the separation of scales is large. 1.3.4. Voigt and Reuss bounds 1.3.4.1. Upper bound: Voigt Consider the case where the medium is subject over its entire boundary S to an imposed temperature T0 , corresponding to a uniform gradient G (Figure 1.5) so that: ΓQ = ∅,

and over

ΓT = S,

T0 = G · x

This load induces a temperature ﬁeld T in the medium. We can show that this satisﬁes the Hill hypothesis. This indicates that in any arbitrary heterogenous material, under this type of load, the mean gradient in the medium is identical to that followed by the temperature imposed at the boundary: gradT = G

V

T0 = G · x

T (x)

T Figure 1.5. Temperature imposed over the entire boundary S, corresponding to a uniform gradient G

This results from: V gradT = T ndS = T0 ndS = (G · x)ndS S

S

S

xi ndS = Gi

= Gi S

ei dV = V G V

An Introduction to Upscaling Methods

41

where we have adopted, as we will throughout the rest of this work, the Einstein convention: summation over all repeated indices in a term. Also, applying the theorem of virtual powers, the dissipated power takes the form: gradT · Q dV = T0 Q · n dS = (G · x)Q · n dS = Gi xi Qj nj dS V

S

S

S

and, using the divergence theorem as well as the property of zero divergence of ﬂux: (∂xi Qj ) Gi xi Qj nj dS = Gi dV = Gi δij Qj dV = Gj Qj dV ∂xj S V V V It follows from this that: gradT · Q dV = V gradT · Q V

Assuming (H), and further supposing that λeﬀ is homogenous (necessarily) and eﬀ isotropic (out of convenience), λeﬀ ij = λ δij , the macroscopic description is given by: Q = −λeﬀ gradT = −λeﬀ gradT This ﬁnally leads us to the quoted result for this ﬁeld: λ gradT · gradT dV = V λeﬀ gradT · gradT

(1.19)

V

Consider now the temperature ﬁeld T with homogenous gradient gradT = gradT . This ﬁeld is admissible in temperature since over V : T = gradT · x = G · x

and hence

T = T0

over S

Furthermore, the ﬁeld T satisﬁes the minimum of the potential dissipation Φ2 (T ) Φ2 (T ), in other words, given that ΓQ = ∅: λ gradT · gradT dV λ gradT · gradT dV V

V

λ gradT · gradT dV = gradT · gradT

= V

λ dV V

Using the energy property (1.19), it follows that: V λeﬀ gradT · gradT gradT · gradT

λ dV V

42

Homogenization of Coupled Phenomena

which gives: λeﬀ λ

(1.20)

The arithmetic mean λ is the upper bound of Voigt [VOI 87]. In the case of a composite with two homogenous constituents with volume fractions ca and cb : λeﬀ ca λa + cb λb We recover the value established for transfer parallel to the layers in a stratiﬁed sample. In fact, the bound of Voigt [VOI 87] is reached when gradT = gradT = gradT , in other words if the solution ﬁeld presents a homogenous gradient. 1.3.4.2. Lower bound: Reuss Now consider the case where the medium is subject across its entire boundary S to a uniform imposed ﬂux Q (Figure 1.6), so that: ΓT = ∅,

and over ΓQ = S

Qn = Q · n

Qn = Q · n

n

V T (x)

Q Q

Figure 1.6. Uniform ﬂux imposed over the entire boundary S

This load induces a ﬂux ﬁeld Q in the medium. We will show that it also satisﬁes the Hill hypothesis of energetic consistency. First of all we will prove the following remarkable property, which states that in any arbitrary heterogenous medium, under this type of load, the mean ﬂux within the medium is identical to that imposed at the boundary: Q = Q For each ﬂux component: V Qi =

δij Qj dV =

V

Qj V

∂xi dV = ∂xj

V

∂(xi Qj ) − xi divQ dV ∂xj

An Introduction to Upscaling Methods

43

The last term of the integral is zero because Q has zero divergence. The remaining term can be transformed successively using the divergence theorem and the boundary conditions: xi Qj nj dS = xi Qj nj dS = Qj xi nj dS V Qi = S

S

S

div(xi ej )dV = Qj

= Qj V

δij dV = V Qi V

Having achieved this result, we now apply the theorem of virtual work:

gradT · QdV = V

T Q · n dS = S

T Q · n dS = Q · S

T n dS S

gradT dV = V Q · gradT

=Q· V

Invoking (H), and assuming once again that λeﬀ is homogenous (out of necessity) eﬀ and isotropic (for convenience) then λeﬀ ij = λ δij , and the macroscopic description is given by: gradT = −λeﬀ−1 Q from which we deduce the following energetic property: V

Q·Q Q · Q dV = V λ λeﬀ

(1.21)

Now consider the ﬁeld Q∗ deﬁned by Q∗ = Q. This ﬁeld is ﬂux admissible since on the one hand Q∗ is uniform, and so has zero divergence, and on the other hand, since Q = Q, we have over S: Q∗ · n = Q · n = Q · n = Qn

(1.22)

The ﬁeld Q∗ therefore satisﬁes the minimum of the complementary dissipation: Φ1 (Q) Φ1 (Q∗ ) which, since ΓT = ∅, gives: V

Q·Q dV λ

V

Q∗ · Q∗ dV = Q · Q λ

V

1 dV λ

44

Homogenization of Coupled Phenomena

and, using the energetic property (1.21) of Q: Q · Q V Q · Q λeﬀ

V

1 dV λ

Finally: 1 1 λeﬀ V

V

1 dV, λ

or alternatively

λeﬀ λ−1 −1

(1.23)

The geometric mean λ−1 −1 is the lower Reuss bound [REU 29]. In the case of two homogenous constituents, this lower bound can be written as: λeﬀ

λa λb cb λa + ca λb

This expression is identical to that obtained for transport in a stratiﬁed medium perpendicular to the layers. Indeed the Reuss bound [REU 29] is reached when Q = Q∗ = Q, in other words if the solution ﬁeld presents a homogenous ﬂux.

1.3.5. Comments – The same comments still apply as for the bilaminate composite. In particular, the above results only have meaning in the presence of a good separation of length scales. In this sense λeﬀ is only an approximation, and is not therefore intrinsic to the material. – In the case of homogenous constituents with known properties, the Voigt and Reuss bounds only require a knowledge of their volume fractions. – The preceding results were, for convenience, established for the isotropic case. They can be extended without difﬁculty to the anisotropic case, where they become: ∂T ∂T ∂T (λeﬀ 0, ∀ ij − λij ) ∂xi ∂xj ∂xi which corresponds to the upper bound of Voigt, and: − (λ−1 )ij )Qj 0, Qi (λeﬀ−1 ij

∀Qi

which corresponds to the lower bound of Reuss. – Since the Voigt and Reuss bounds are in fact attained by certain stratiﬁed materials, they cannot be improved upon without the addition of further information on the medium. – By way of example, consider a composite with two isotropic constituents with: λa = 10 λb , ca = cb = 0.5. We obtain: 0.182 λa λeﬀ 0.55 λa .

An Introduction to Upscaling Methods

45

– The bounds are clearly looser when the contrast in the properties of the constituents – which can be quantiﬁed by the product λλ−1 – is larger. When ca is varied from 0 to 1, we obtain the graph in Figure 1.7, known as Hill’s diagram. 1

e& / a

0.8

0.6

Reuss

HS+ HS-

0.4

Voigt 0.2

0

0

0.2

0.4

0.6

0.8

1

ca Figure 1.7. Hill’s diagram, and bounds of [HAS 63] for a composite with λa /λb = 10

1.3.6. Hashin and Shtrikman’s bounds By advancing the additional hypothesis that the material is isotropic, and using other variational principles, we can obtain improved bounds. In the case of two constituents, Hashin and Shtrikman [HAS 63] derive the following bounds when λa > λb : λb +

ca +

1 λa −λb

cb 3λb

λeﬀ λa +

cb +

1 λb −λa

ca 3λa

(1.24)

In the above example, λa = 10λb , ca = cb = 0.5, and we obtain: 0.28λa λeﬀ 0.47λa Of course, the calculation made by Hashin and Shtrikman leads implicitly to restrictive assumptions on the spatial distribution of the two phases. However it is not possible to explicitly formulate these morphological constraints using this method. Thus, even when the morphology of the microstructure is known, it is not generally possible to make a judgement on the macroscopic isotropy and hence on the validity of the bounds. However, for periodic media, the isotropy of the conduction tensor is assured if the period presents three orthogonal planes of symmetry: in this case Hashin and Shtrikman’s bounds are known to be valid.

46

Homogenization of Coupled Phenomena

In addition, Hashin showed that these bounds give effective conduction for materials formed of aggregates of bi-composite homothetic spheres ﬁlling space (see Chapter 6). The upper bound is reached when each sphere consists of a core of the less conductive constituent surrounded by a shell of the more conductive material, and the lower bound is reached in the reverse situation. Since the bounding values can be reached, these bounds are optimal for the level of information available (the properties of the constituents, their proportions, macroscopic isotropy). 1.3.7. Higher-order bounds It is possible to deﬁne higher-order bounds. Their deﬁnition requires other morphological parameters in addition to the volume fraction of each phase. On this subject, the reader is referred for example to [BOR 01; TOR 02; and MIL 02]. 1.4. Self-consistent method The aim of the self-consistent method [BRU 35; BRI 49; HIL 65; BUD 65; HAS 68] is to estimate the effective coefﬁcients using minimal information, in other words the properties and concentrations of the constituents. Naturally, this estimate requires us to introduce an additional hypothesis known as self-consistency. This assumes that each inclusion, or more generally each generic substructure [ZAO 87] “sees” the rest of the medium not in terms of the actual heterogenous material, but as the equivalent homogenous material whose effective properties we are trying to determine. The idea of generic inclusion is to approach the mechanisms at a local scale. The inclusion leads to assumptions on the morphology, but these do not appear in an explicit manner. The simplicity of this approach, and the ﬂexibility in the choice of generic substructures – on which the macroscopic model will depend – make this method widely used in practical terms. We will present the self-consistent approach by drawing on the same physics as in the earlier sections, that is to say steady-state thermal transfer in a composite: divQ = 0 Q = −λ gradT,

(1.25) λ>0

(1.26)

with: [Q · n]Γ = 0

(1.27)

[T ]Γ = 0

(1.28)

over any surfaces Γ of discontinuity.

An Introduction to Upscaling Methods

47

1.4.1. Boundary-value problem We will consider a composite with N homogenous and isotropic constituents α = 1...N (Figure 1.8) occupying a volume V . We will denote as Vα the volume occupied by phase α and its volume fraction as cα = Vα /V . In order to apply the

kn k2 k1

kn k1

k1

k2

k1

kn k1

k2

kn

k1

kn kn

V

k2

T0 = G · x

k2 k1 T

Figure 1.8. Boundary-value problem

self-consistent method we must assume that the medium is subject at its boundary S to homogenous conditions either in terms of temperature gradient or ﬂux (identical to those introduced for the Voigt and Reuss bounds). While the type of boundary condition does not matter, homogenity is essential because then – as previously seen – the solution must respect Hill’s condition of energetic consistency. To give a concrete example, we will assume that the medium is exposed at its boundary S to an imposed temperature T0 corresponding to a uniform gradient G of magnitude G in the direction ej , so that: T0 = G · x. We have shown that the solution T also satisﬁes: gradT = G (see the Voigt bound). The macroscopic ﬂux Q is given by: 1 Q = V

V

with: QVα =

1 Vα

N N 1 QdV = QdV = ci QVα V Vα α=1 α=1 QdV Vα

Since our boundary-value problem is linear, the solution T is a linear function of the amplitude G = |gradT | of the temperature gradient imposed at the boundary. The same applies to the ﬂux. As a result, the means over each constituent can be expressed in the form: QVα = −aα |gradT |

48

Homogenization of Coupled Phenomena

where the vector aα only depends on the microstructure and physical properties. We will adopt the aforementioned hypothesis (H) and will assume the material to be macroscopically homogenous (out of necessity) and isotropic (out of convenience). Thus Q, gradT and ej are collinear and: Q = −λeﬀ gradT so that: −

N

cα aα |gradT | = −λeﬀ |gradT | ej

α=1

Setting: aα · ej = βα and projecting the preceding equality onto ej , we ﬁnd: λeﬀ =

N

cα βα

α=1

λeﬀ is thus determined once the βα are known. However, a rigorous calculation of these coefﬁcients would need to take into account all the complexities of the interactions between the constituents, and would therefore be very involved. The selfconsistent method relies on a physically acceptable hypothesis which allows us to carry out a simple estimate of its value. 1.4.2. Self-consistent hypothesis As already mentioned, the very shrewd point of the self-consistent method [BRU 35; BRI 49; HIL 65; BUD 65; HAS 68], is to assume that each inclusion (or more generally each generic substructure) “sees” the rest of the medium not as the actual heterogenous material, but as its equivalent homogenous material whose conductivity λeﬀ we are seeking to determine. To put it another way, βα is determined by considering the inclusion α to be embedded in a material of conductivity λeﬀ subject to the same homogenous boundary conditions (Figure 1.9). Thus the βα appear as functions of λeﬀ and λα : βα = βα (λeﬀ , λα ) and λeﬀ is then determined by solving the following equation: λeﬀ =

N α=1

cα βα (λeﬀ , λα )

(1.29)

An Introduction to Upscaling Methods

T0 = G · x

T0 = G · x

V K

49

V

1

…

K

T

n

T

Figure 1.9. Self-consistent scheme with simple inclusions: boundary-value problems for inclusions 1 and n, treated as spheres

For simple geometries of inclusions, and considering the volume V to be inﬁnite, the calculation can then be performed analytically.

1.4.3. Self-consistent method with simple inclusions The original self-consistent method consists of considering homogenous spherical inclusions. Other structures will be considered in Chapter 6. 1.4.3.1. Determination of βα for a homogenous spherical inclusion In order to determine βα , the spherical inclusion represented by constituent α is placed in a volume V of the equivalent medium, which is assumed inﬁnite. The ensemble is subject to a homogenous gradient G at inﬁnity (Figure 1.10). We will work in spherical coordinates (r, θ, ϕ), with the origin at the center of the inclusion deﬁned by r < R, and the θ = 0 axis deﬁned by G. Bearing in mind the symmetries of the problem, the temperature ﬁelds in the inclusion α (Tα for r < R) and in the equivalent homogenous medium (T for r > R) take the form: Tα = fα (r)cos(θ),

T = f (r)cos(θ)

and their gradients are given by: gradTα = fα cos(θ)er − gradT = f cos(θ)er −

fα sin(θ)eθ r

f sin(θ)eθ r

Consider the external medium (r > R). The Fourier equation can be written: divQ = div(λgradT ) = λeﬀ ΔT = 0

50

Homogenization of Coupled Phenomena

G ez

ez

er

M

r R

Equivalent medium V

ey

O

Constituent V

ex

Figure 1.10. Spherical inclusion representing the constituent α placed in a volume V (assumed inﬁnite) of the equivalent medium

where: λeﬀ 2 f ) − 2f =0 (r r2 which has solutions of the form: f (r) = A

R2 r +B 2 r R

for r > R

Following the same line of reasoning in the inclusion, we have:

fα (r) = a

R2 r +b r2 R

for r < R

Since the temperature must take ﬁnite values throughout, a = 0 and as a consequence the ﬂux is uniform within the inclusion. In order to determine this ﬂux, we write the continuity condition on the temperature and the ﬂux normal to the interface (r = R), as well as the condition of homogenous gradient at inﬁnity: T (R) = Tα (R) =⇒ λeﬀ gradT (R) · er = λα gradTα (R) · er

=⇒

gradT (∞) = G

=⇒

A+B =b λα λeﬀ (−2A + B) = b R R B=G

An Introduction to Upscaling Methods

51

Solving this system of equations leads us to: A=G

λeﬀ − λα , λα + 2λeﬀ

b=G

3λeﬀ kα + 2λeﬀ

This means that the ﬂux within the inclusion is: Q = λα

3λeﬀ G λα + 2λeﬀ

And as a result: βα (λeﬀ , λα ) = 3

λeﬀ λα λα + 2λeﬀ

1.4.3.2. Self-consistent estimate Using these values of βα (λeﬀ , λα ), equation (1.29) which deﬁnes λeﬀ becomes: N 1 λα cα = 3 α=1 λα + 2λeﬀ

(1.30)

This equation (which can be written in the form of a polynomial of degree N ) only had a positive root of λeﬀ > 0, the second term decreases Nsince, in the domain eﬀ monotonically, from α=1 cα = 1 for λ = 0, to 0 for λeﬀ = ∞. Returning to the example of the bi-composite already considered in the preceding sections, λa = 10 λb , ca = cb = 0.5. λeﬀ is given by: 1 1 = 3 2

10λb λb + 10λb + 2λeﬀ λb + 2λeﬀ

and so: (λeﬀ )2 −

11 λb λeﬀ − 5 λb 2 = 0 4

Rejecting the negative solution to λeﬀ we ﬁnd: λeﬀ = 4 λb = 0, 4 λa The different results, the bounds of Voigt (V), of Reuss (R), of Hashin and Shtrikman (HS) and the self-consistent estimate (SCE) (which of course falls within the bounds) are summarized in Figure 1.11.

52

Homogenization of Coupled Phenomena 1

a / b = 10

e& / a

0.8

0.6

R

0.4

SCE HS-

HS+

V

0.2

0

0

0.2

0.4

0.6

0.8

1

ca

1

a / b = 100

e& / a

0.8

0.6

0.4

R HS+

0.2

SCE HSV

0

0

0.2

0.4

0.6

0.8

1

ca Figure 1.11. Comparison between self-consistent estimates and the bounds for a bi-composite for two conductivity contrasts

1.4.3.3. Implicit morphological constraints Although the self-consistent method can be applied to general microstructures (here simple inclusions), the morphological constraints which these impose are not explicitly formulated. Because of this, it is not possible to deﬁne the class of microstructures to which the self-consistent result can be applied. This is an inherent limitation of the method: a result is obtained, but its conditions of applicability are not speciﬁed. To illustrate these implicit constraints, consider the case of media A and B with two constituents. One has ﬁnite conductivity λ, the other of concentration c has

An Introduction to Upscaling Methods

53

zero conductivity (medium A) or inﬁnite conductivity (medium B). The self-consistent model gives: λeﬀ A =λ

2 − 3c 2

for c < 2/3,

λeﬀ A = 0 for

λeﬀ B =λ

1 1 − 3c

for c < 1/3,

λeﬀ B =∞

c > 2/3

for c > 1/3

Thus, the self-consistent method with a simple inclusion imposes – without this being speciﬁed a priori – a connectivity threshold at a concentration of 1/3: – any constituent with a concentration greater than 1/3 is connected (medium B is inﬁnitely conductive when c > 1/3); – any constituent with a concentration of less than 1/3 is non-connected, but is dispersed within the other phase (medium A is perfectly insulating when the concentration of the conductive phase 1 − c falls below 1/3). Many materials do not respect this constraint. In order to avoid this difﬁculty, other generic layouts tailored to speciﬁc microstructures can be imagined. We mention in particular the bi-composite spheres method of Kerner [KER56b, a] or Hashin [HAS 68] (see Chapter 6) which allows us to ensure the connectivity of one of the phases, whatever the concentration, with the other necessarily being disperse. Particularly well suited to media formed of inclusions within a matrix, this is also one of the few models with which we can associate an exact or approximate explicit morphology.

1.4.4. Comments – The result obtained with the self-consistent model is only meaningful if the material and the temperature ﬁeld exhibit a good separation of scales. If we calculate the βα over ﬁnite volumes V , they depend on the position of the inclusion in the volume. They are not therefore intrinsic to the material, and neither is λeﬀ . The inﬁnite medium and the macroscopic homogenous thermal ﬂux ﬁeld correspond to an ideal scale separation l/L = 0. In this case the effective conductivity is intrisic. The question is: under what conditions does the result obtained remain valid for inhomogenous macroscopic ﬂux ﬁelds or non-steady-state ﬂows? – The bounds presented above, those of Voigt, Reuss and Hashin and Shtrikman, as well as the self-consistent effective conductivity, can be superseded in the general context of a systematic theory of random heterogenous materials [KRÖ 72]. A more and more precise understanding of the material, given by correlation functions of increasing order n, offer tighter and tighter bounds on the effective conductivity. For n = 1, in other words when we only know the volume fractions of the constituents,

54

Homogenization of Coupled Phenomena

we obtain the bounds of Voigt and Reuss. For n = 2, for uncorrelated conductivities of neighboring inclusions and an isotropic material, we obtain the bounds of Hashin and Shtrikman. Finally, in the case of perfect disorder (zero correlations at all orders, up to inﬁnite n) we obtain the self-consistent result. – All the models discussed up to now rely on conservation of potential energy, and the absence of any other type of energy. They therefore apply to elliptical problems. As already mentioned, it remains to be seen if the results are applicable to hyperbolic or parabolic problems.

Chapter 2

Heterogenous Medium: Is an Equivalent Macroscopic Description Possible?

2.1. Introduction Heterogenous materials represent an important and expanding ﬁeld. In the ﬁeld of mechanics, we can think of composite materials consisting of several solids, of mixtures of ﬂuids, of suspensions of solids in a ﬂuid, or of porous media. In fact all materials are heterogenous on one scale or another, at least if this only occurs when they are considered on an atomic scale. Such materials are often used in volumes consisting of a very large number of heterogenities (Figure 2.1) so that the description of the physical processes involved becomes difﬁcult, or indeed impossible, if we take into account every heterogenity. The idea – a very old and traditional one – is then to determine, where possible, a macroscopically equivalent medium, known also as the homogenized medium. This is a continuous medium which behaves “on average” like the heterogenous material. The description thus obtained must be intrinsic to the material and the perturbation under consideration, and independent of the macroscopic boundary conditions. In what follows, the scale of the heterogenities will be referred to as the microscopic, or local, scale – as opposed to the macroscopic scale, the scale on which the equivalent continuum is deﬁned. The continuum macroscopic description of the heterogenous materials introduces behavior, conservation equations, physical quantities and effective coefﬁcients, in place of the behavior, the conservation equations, physical quantities and coefﬁcients valid on the scale of the heterogenities. The study of the relationship between the local and macroscopic descriptions is of very great interest.

56

Homogenization of Coupled Phenomena

1200 μm

Figure 2.1. Microstructures of heterogenous materials: aluminum with aluminum oxide reinforcement (left) and SiC-Titanium composite (right)

The macroscopic description can be obtained either by phenomenological or experimental investigation directly at the macroscopic scale, or by a homogenization technique, in other words by working from a microscopic description to a macroscopic description. It is this second line of attack which we will describe here. It is not possible to give a complete account of all the techniques of homogenization, of which there are many. We will therefore mainly restrict ourselves to presenting two of them: homogenization using multiple scale expansion for small-scale periodic structures (HPS) technique, with the foundation laid by Sanchez-Palencia [SAN 74] and introduced by others [KEL 77; BEN 78; SAN 80], and statistical modeling (SM) developed by Kröner [KRÖ 86], for materials with random structures. These are without doubt the most effective techniques and we will by analogy be able to draw a number of general concepts from them. Section 2.2 makes some general comments on homogenization techniques, as well as the relationships between some of them. SM is then brieﬂy discussed in section 2.3. Similarly, the technique of multiple scale expansions is the subject of section 2.4 followed in section 2.5 by a comparison of the usefulness of these two techniques. 2.2. Comments on techniques for micro-macro upscaling To start with, we observe (and this will be understood throughout the introduction below), that the homogenization of a medium with a high density of heterogenities is only possible if we consider regions containing a large number of these heterogenities. If l is a characteristic dimension of the heterogenities and L is a characteristic dimension of the volume of material or of the phenomenon of interest, the condition of separation of scales can be expressed as: ε = l/L 1

Is an Equivalent Macroscopic Description Possible?

57

Two main classes of material can then be deﬁned, depending on whether these two scales are effectively separated or not. For the ﬁrst class there is no other intermediate scale. Homogenization is then possible. The materials are then characterized by a translational invariance (or quasiinvariance) at the microscopic scale, in a sense that we will explain later on. There may alternatively be more than two coexisting scales, but only if they are well separated from each other. We then move from the smallest scale to the larger scale by homogenization, but with the possibility of strong interaction between the different length scales depending on the different degrees of separation [AUR 92; AUR 93a]. When there is a continuum of non-separated length scales (the second class), results can be obtained when the structure is length-scale invariant. For example, in the case of a regular lattice of conducting or non-conducting rods organized in a random fashion, the structure of the conducting part of the lattice is length-scale invariant when the probability of a rod being conductive is close to the critical probability which forms the conductive–non-conductive threshold of the lattice. Percolation theory [BRO 57; CLE 83] then makes it possible for such structures to obtain the change in conductivity of the lattice close to this threshold. As we will see, this is also a type of homogenization since an “average” property, here the effective conductivity of the lattice is obtained. Nevertheless, we will in general reserve the term homogenization to micro-macro transition techniques involving structures with separated length scales. In what follows we will only consider this last class of materials.

2.2.1. Homogenization techniques for separated length scales We will start by giving a few examples of heterogenous materials in the context of civil engineering. The characteristic microscopic length scale l is only “microscopic” in comparison to the characteristic macroscopic length scale L. Thus l can take values which are not remotely microscopic compared to the human scale. If l is the size of the pores in a clay, maybe 50 angstroms, a glass ﬁber-epoxy resin composite would have l ≈ 1 mm, concrete l ≈ 1 cm, reinforced earth or a bank drained with geotextiles l ≈ 50 cm, and ﬁnally a pile foundation l ≈ 5 m. While it is not possible to give an exact cutoff, it is generally considered that homogenization becomes effective for L > 10 l or ε < 0.1. As we will see in the examples below, two more classes of material will become evident: materials with periodic structures, and such random structures as possess a representative elementary volume. The former can be studied using the technique of homogenization for periodic structures (HPS), which uses multiple scale expansions and which was developed by Bensoussan et al. [BEN 78] and Sanchez-Palencia [SAN 80]. Nowadays it is applied to many ﬁelds of physics. We will revisit the method of multiple scale expansions in section 2.4 where we will present a

58

Homogenization of Coupled Phenomena

methodology [AUR 91] which in particular allows us to investigate the conditions of homogenizability of a material subject to a given excitation. It will be used in speciﬁc examples in the following chapters. The second class of materials, those with random structures, are the subject of many techniques, and we cannot discuss all of them here. We will refer to Kröner’s method of statistical modeling (SM) [KRÖ 86], the methods developed by Gelhar [GEL 87] or Matheron [MAT 67], selfconsistent methods (see for example [HAS 68] and [ZAO 87]) and, in general terms, the averaging techniques [NIG 81; BED 83; GIL 87; QUI 93; WHI 99; ELH 02]. In section 2.3 we will brieﬂy present the technique introduced by Kröner [KRÖ 86]. The study of materials with random structures presupposes the assumption of stationarity or quasi-stationarity. Even if it is not always stated, this assumption underlies all the techniques and is required for homogenization to be possible. It corresponds to the property of periodicity in periodic structures, and captures the property of translational invariance which is required for separation of length scales, as opposed to the invariance by self-similarity which is used for certain materials with non-separable length scales. Another general characteristic of all the methods is that they use mean values to deﬁne macroscopic quantities. These may be introduced in a phenomenological manner however – and this constitutes an assumption on the macroscopic scale which leads to the deﬁnition of the macroscopic physical quantities. Alternatively they may be introduced in the micro-macro transition, which guarantees that we will obtain the “correct” macroscopic physical quantity. We note on this subject that the volume averaging theorem, which allows us to take the mean of a system of partial differential equations, and which is often used to determine the macroscopic description, is valid whether or not the problem is homogenizable. It is interesting to note the points in common between the most effective homogenization techniques. On one hand the self-consistent techniques appear to be equivalent to the SM method in the case of perfect disorder [KRÖ 72; KRÖ 86]. For an elastic composite, the (macroscopic) effective elasticity coefﬁcients have an identical formal structure, whether they are obtained by the SM technique for a random composite, or using the HPS method for a periodic structure [KRÖ 80]. For ﬁltration problems, it can easily be seen that a similar result is valid for a porous medium, by comparing the formal structure of the effective permeability coefﬁcient obtained by Matheron [MAT 67], and that obtained by the HPS method. More recently, Bourgeat and Piatnitski [BOU 04] established a connection between homogenization of periodic and random materials in the context of heat transfer. In general terms, though, the larger the separation of scales, the better the results. That gives a clear advantage to the HPS method. Indeed the periodicity introduces a perfect separation of scales, with l deﬁned as the characteristic dimension of the period. Such a precise deﬁnition is not possible for materials with a random structure. The results will therefore be more accurate for periodic materials.

Is an Equivalent Macroscopic Description Possible?

59

2.2.2. The ideal homogenization method Starting from a complete description at the microscopic scale, the ideal homogenization method is intended to determine the complete description at the macroscopic scale without additional assumptions. Thus at the microscopic scale we assume that we know: – the conservation equations, – the rheologies and the values of the parameters, – the physical quantities which describe the phenomena of interest, – the geometry. In addition, the process of the micro-macro upscaling must give an equivalent macroscopic description, intrinsic to the material and to the phenomenon of interest, speciﬁcally: – the conservation equations, – the rheologies and the effective parameters, – the correct physical quantities which describe the phenomena on the macroscopic scale. In short, the ideal process should be independent of any assumption on the physics of the model on the macroscopic scale in order to ensure the quality of the result. When we consider that the structures of the macroscopic model can be very different to the structures of the corresponding microscopic model, we can understand the importance of this point. One example is ﬁltration in a rigid porous medium: on the scale of the pores, the description for a Newtonian ﬂuid matches the Stokes description with the corresponding rheology. On the macroscopic scale the model is that of Darcy’s law, which is of course also an expression of the fundamental principles of mechanics, but with a very different structure, where the rheology is now intimately combined with the dynamics [AUR 80]. The ideal procedure must also permit localization, that is the determination of the local ﬁelds of physical quantities starting from the values of macroscopic physical quantities. It is then clear that the only method which meets these requirements is the HPS method, since its periodicity ensures a complete description of the material. On the other hand, it is never possible to completely describe a random material (even in the limiting case of perfect disorder). Thus the methods associated with random structures cannot be as effective. Since the local description is incomplete, it is necessary to introduce assumptions on the macroscopic scale. The HPS technique also makes it possible to treat regions close to macroscopic boundaries where, perpendicular to these boundaries, the separation of length scales, as well as the stationarity, are violated. The introduction of matching boundary layers then makes it possible to complete the solution to the problem.

60

Homogenization of Coupled Phenomena

2.3. Statistical modeling This technique, developed by Kröner [KRÖ 86], is mainly used to obtain information about the effective coefﬁcients of an elastic composite subject to a static load. Its assumptions are: – the medium, in the steady state, has inﬁnite dimensions. The scales are separated: ε = l/L ≈ 0. This assumption underlies the technique, but systematic use of this assumption is not made during the micro-macro transition; – the material has a stable, random structure; – the assumption of ergodicity – the mean of the ensemble is equal to the volume mean – is made; – the Hill principle is applied: σ : e = σ : e, where the . operator represents the volume mean, σ the stress and e the strain. This implies that: – the macroscopic stress is σ, – the macroscopic strain is e, – the material has a macroscopically elastic behavior, – the mean of the local elastic energy density is equal to the macroscopic elastic energy density. As we indicated above, the local description of a material with a random structure is never perfectly known. The point of Kröner’s statistical modeling is to perfectly adapt to this situation: the technique provides upper and lower bounds for the effective conditions, with these bounds becoming tighter as the information available is increased. The technique can be presented starting from the integral formulation of the boundary-value problem for an inﬁnite medium, taking into account all the heterogenities and using the modiﬁed Green’s function for the strain. Since this Green’s function is clearly unknown, bearing in mind the complexity of the problem, Kröner [KRÖ 86] introduces the modiﬁed Green’s function for the strain, associated with a ﬁctitious homogenous elastic medium. This function is exactly known. Kröner then shows that the true Green’s function can be expressed based on this ﬁctitious Green’s function and on the ﬁeld c of the elastic tensor. Then, using the following deﬁnition for the effective elastic tensor Ceﬀ which follows from the aforementioned assumption: σ = c : e = Ceﬀ : e Ceﬀ is represented as a series expansion with respect to correlation functions of increasing order in c. Then, considering energy, we can show that the truncated series, which only requires the knowledge of correlation functions up to a given order n, corresponds

Is an Equivalent Macroscopic Description Possible?

61

to an upper bound on Ceﬀ , which decreases with n. The same approach with s = c−1 leads in the same way to an increasing series of lower bounds. The least precise bounds are those of Voigt, c, and Reuss, c−1 −1 . They only require a knowledge of the means of c and s. The introduction of second-order correlation functions leads to tighter bounds. If the material is macroscopically isotropic, and the moduli of elasticity of two neighboring grains are uncorrelated, these are the bounds of Hashin and Shtrikman [HAS 63], and so on. The knowledge of all the correlation functions leads in theory to the actual value of the effective coefﬁcient. 2.4. Method of multiple scale expansions The method of multiple scale expansion, whose foundations were laid by SanchezPalencia [SAN 74], was introduced [KEL 77; BEN 78; SAN 80] in order to study problems involving the homogenization of structures with small-scale periodicity. The methodology presented here [AUR 91] makes use of these developments and allows us to answer the question of homogenizability. It also demonstrates the common points between periodic and random materials. When homogenization is possible, the structure of the equivalent macroscopic behaviors is identical in both cases. Only the effective coefﬁcients require different treatments to obtain them. 2.4.1. Formulation of multiple scale problems Before embarking on this subject, we must deﬁne the concept of the macroscopically equivalent medium. We would like to obtain an equivalent macroscopic boundary-value problem, in other words relations between the macroscopic quantities (in practice these are average quantities, whose meaning will be speciﬁed later on) and the effective parameters. These relations are either constitutive laws or conservation laws. The description is intrinsic to a class of media subject to a given type of disturbance the macroscopic description must be valid for all boundary-value problems belonging to this class. The macroscopic description is continuous, as opposed to the microscopic description which can be discontinuous on a small scale. Here we will only consider media and disturbances which are piecewise continuous on the microscopic scale. It is clear that the macroscopic descriptions obtained are also valid for macroscopic problems which are piecewise continuous, on the condition that each macroscopic region of continuity is itself homogenizable. 2.4.1.1. Homogenizability conditions We will now investigate the homogenizability conditions. It is clear, based on the considerations developed above, that the macroscopic domain must contain a (very) large number of heterogenities and that the size of these and their organization must be such that the representative elementary volume (REV) for the medium is

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small compared to the macroscopic volume. In order to be representative of the heterogenities’ geometry and the constituents’ properties, the REV must contain a sufﬁciently large number of heterogenities. Its size does not however tend to exceed 10 heterogenities in each spatial direction [CHE 88; ANG 94; ROL 07]. Thus we can normally compare the characteristic size of the heterogenities to that of the REV: the separation of scales parameter is not signiﬁcantly modiﬁed. The separation of scales is a property which depends as much on the geometry of the medium as on the phenomenon. The REV reduces to the unit cell when the medium is periodic. Let lc be a characteristic length of the REV or period, and Lc a macroscopic length. Lc represents either a characteristic length of the volume of material under consideration, or a macroscopic characteristic length of the phenomenon. Separation of scales requires that: lc =ε1 Lc

(2.1)

We again emphasize that the physical quantities must satisfy this condition of separation of scales. For example, consider a periodic elastic composite satisfying (2.1) from the geometric point of view, but subject to a dynamic excitation of wavelength O(lc ). Now diffraction comes into play. The Lc of the perturbation is O(lc ) and condition (2.1) is not satisﬁed. We cannot obtain a macroscopic description which satisﬁes the conditions of homogenization presented above. This example will be revisited in more detail in Chapter 3. Furthermore, periodic and random media behave in very different manners. In the case of periodic media, diffraction introduces forbidden frequency ranges, and if the structure is random then the excitation is conﬁned close to the sources. 2.4.1.2. Double spatial variable Condition (2.1) is taken as a base assumption for all homogenization processes, even if most of them do not make systematic use of it. The two characteristic length scales lc et Lc introduce two dimensionless spatial variables y∗ = X/lc and x∗ = X/Lc where X is the physical spatial variable (an asterisk superscript indicates a dimensionless quantity and the subscript c indicates a characteristic value). Due to the separation of scales, each quantity Φ appears as a function of these two dimensionless variables rather than anything else. (In the literature, it is common to use physical spatial variables, in other words X and Y = X/ε.) The variable x∗ is the macroscopic (or slow) spatial variable and y∗ is the microscopic (or fast) spatial variable. The fast variable y∗ describes the short-range interactions O(lc ), whereas the slow variable x∗ describes the long-range interactions O(Lc ). Two equivalent notations are therefore possible: – the ﬁrst corresponds to the macroscopic viewpoint, since the analysis is conducted with the spatial variable x∗ : Φ = Φ(x∗ , y∗ ),

y∗ = x∗ /ε,

Φ = Φ(x∗ , ε)

(2.2)

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63

– the second corresponds to the microscopic viewpoint, with the analysis conducted with the spatial variable y∗ . Φ = Φ(x∗ , y∗ ),

x∗ = εy∗ ,

Φ = Φ(y∗ , ε)

(2.3)

Let Φ be the mean of Φ. For a random medium, the mean is taken over a representative elementary volume with respect to the variable y∗ , whereas for a periodic medium the volume considered is the unit cell. Separation of scales implies that for Φ (see Figure 2.2): Φ = Φ + (Φ − Φ) where Φ varies with x∗ and (Φ − Φ) varies with y∗ , and: ∂Φ =O ∂y ∗

∂Φ ∂x∗

(2.4)

The symbol O(.) must be interpreted relative to ε: A = O(B)

ε1/2 |A/B| ε−1/2

if

lc

Lc (x , y )

(x , y )

h i h i x = X/Lc

1

y = X/lc

Figure 2.2. Macroscopic and local variation of Φ

1

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Homogenization of Coupled Phenomena

The local gradient of Φ is of the same order of magnitude as the macroscopic gradient of Φ. In general terms, lc is known for a given material. Lc is determined a posteriori, after solving the macroscopic boundary-value problem, using [BOU 89b]: ⎞ ⎛ ⎜ Φ ⎟ Lc = O ⎝ ⎠ ∂Φ ∂X

(2.5)

The separation of scales condition is then written as: ⎞ ∂Φ lc ⎟ ⎜ ε= = O ⎝ ∂X ⎠ 1 Lc Φ ⎛

lc

(2.6)

2.4.1.3. Stationarity, asymptotic expansions Figure 2.2 shows that the variation of Φ as a function of x∗ over a distance O(lc ) is small, and is in fact zero in the limit ε → 0. This means that Φ satisﬁes the property of y∗ -stationarity at the local scale. The y∗ -stationarity of Φ is deﬁned as follows. Let Φ and ΦS be the volume and surface means of Φ, over a representative elementary volume and surface respectively. The choice between a volume or surface mean depends on the physical meaning of Φ. If Φ is for example a density, the volume mean is the appropriate choice. If Φ is a stress, then it is the surface mean which should be considered, etc. Φ is y∗ -stationarity if its local mean is invariant under a local translation, of order lc and whatever the actual structure of a given instance of the random medium (ergodic hypothesis). Consider for example a surface mean. Let Σ∗1 and Σ∗2 be two arbitrary straight, parallel sections of a REV (see Figure 2.3). The stationarity condition can be written: ΦdS ∗ = ΦdS ∗ (2.7) Σ∗ 1

Σ∗ 2

A similar property is satisﬁed when the microstructure is periodic, with the REV replaced by the unit cell. In this case the property is stronger: Φ is y∗ -periodic. It is now clear that the small parameter ε is the keystone of the homogenization process. This leads us to look at ﬁelds for unknown physical quantities in the form of asymptotic expansions at multiple scales [BEN 78; SAN 80] in powers of ε. For a dimensionless quantity Φ∗ , the expansion can be written: Φ∗ (x∗ , y∗ ) = Φ∗(0) (x∗ , y∗ ) + εΦ∗(1) (x∗ , y∗ ) + ε2 Φ∗(2) (x∗ , y∗ ) + · · ·

(2.8)

The equivalent macroscopic behavior is estimated to be ﬁrst order by the behavior of Φ∗(0) . The stationarity or the periodicity then implies the y∗ -stationarity or the

Is an Equivalent Macroscopic Description Possible?

P 1

y = X/lc

65

P 2

1

Figure 2.3. Representative elementary volume for a randomly-structured material

y∗ -periodicity of Φ∗(i) . In return, this property ensures a good separation of scales, and hence homogenizability: if the unknowns can be found in form (2.8), where Φ∗(i) are y∗ -stationarity or y∗ -periodic, then homogenization is possible. In the converse case, the medium and excitation are not homogenizable. The method is consistent with itself (self-consistent in this sense). 2.4.2. Methodology More precisely, the method to be followed is as described here. We assume the local description to be given, and we are looking for the equivalent macroscopic description. – First of all we choose the macroscopic or microscopic viewpoint. These lead to equivalent results, and the choice depends purely on convenience for the problem being studied. The expansions are then made in form (2.8) with y∗ = x∗ /ε or x∗ = εy∗ respectively. The variable x∗ (y∗ ) is the directional spatial variable and Lc (lc ) is the characteristic length to be used to non-dimensionalize the various quantities which appear in the description. The processes of homogenization corresponding to the two viewpoints are illustrated schematically in Figure 2.4. – We then proceed to the non-dimensionalization of the local description. The dimensionless numbers are evaluated as a function of powers of ε (other types of evaluation are possible, depending on the problem being considered). A dimensionless quantity Φ∗ is said to be O(εp ) if: εp+1/2 |Φ∗ | εp−1/2 The normalization is an important stage during which the physics of the problem is taken into account. It is required before using expansions in powers of ε. After this

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lc

X Lc

1

y

x

1

1

y

1

x

X Lc Figure 2.4. The two possible viewpoints: microscopic (left), normalized with lc and Φ∗ = Φ∗ (ε−1 y∗ , y∗ ) with x∗ = ε−1 y∗ and macroscopic (right), normalized with Lc and Φ∗ = Φ∗ (x∗ , εx∗ ) with y∗ = εx∗

operation, the equations appear in the form: εqp Ψ∗p = 0

(2.9)

p

where the Ψ∗p operators are dimensionless. – Finally we substitute the asymptotic expansions in the form (2.8), where Φ∗(i) are y∗ -periodic and Φ∗(0) is O(1), into the normalized local description. The identiﬁcation of terms of the same power in ε leads us to solve the various problems thus obtained over the unit cell. This last stage is that described in Bensoussan et al. [BEN 78] for periodic media when the macroscopic viewpoint is adopted. In the case of linear problems with convex energy, it is generally possible to show the existence and uniqueness of the solutions to these successive problems and to obtain a numerical solution to them.

Is an Equivalent Macroscopic Description Possible?

67

In the case of random media things are not the same. Nevertheless, considering the boundary-value problem over a macroscopic volume (Figure 2.1) as well posed (although difﬁcult to solve in practice due to the large number of heterogenities), each representation of the medium possesses one solution, and only one solution. The local stationarity then implies that the surface (or volume mean) of the quantity under consideration is independent of the representation used, which leads to the uniqueness of the macroscopic description. The cornerstone of homogenization is a necessary and sufﬁcient condition, often known as the compatibility condition, for the existence of solutions to each problem which needs to be solved in succession over the period. The equations to solve take the form (they follow from conservation laws!): divy∗ Φ∗(i) = −divx∗ Φ∗(i−1) + W ∗

(2.10)

where the indices x∗ and y ∗ indicate that the derivatives are taken with respect to x∗ and y∗ respectively. This equation represents the local conservation of Φ∗(i) , where −divx∗ Φ∗(i−1) + W ∗ appears as a source term. Since Φ∗(i) are locally periodic or stationary, the source must have a mean of zero: divx∗ Φ∗(i−1) − W ∗ = 0,

or

divx∗ Φ∗(i−1) − W ∗ = 0

(2.11)

This property corresponds to the Fredholm alternative: the operator in y∗ on the left-hand side of of the equation to be solved, given appropriate boundary conditions (periodicity and possibly others depending on the problem at hand), has an eigenvalue of zero, associated with an eigenfunction independent of y∗ . The necessary and sufﬁcient condition for the existence of a solution lies in the orthogonality of the righthand side of the equation to this eigenfunction, which leads to the result. Due to the non-dimensionalization, the ﬁrst non-zero terms of the asymptotic expansions must be such that the orders of magnitude of the dimensionless numbers estimated during normalization are respected. In other words, the ﬁrst non-zero terms of Ψ∗p must satisfy (2.9). This will be made clearer in the one-dimensional example considered in the following chapter, in the context of elasticity (Chapter 3). In this case, the compatibility conditions lead either to the equivalent macroscopic description or to non-homogenizability when the ﬁrst non-zero terms of Ψ∗p do not satisfy (2.9); what happens is that such values will end up modifying the order of magnitude of the dimensionless numbers characterizing the medium and the phenomenon under study. We will return to this argument in a systematic manner in all the physical problems discussed in the following sections. In conclusion, we note that convergence problems for solutions when ε −→ 0 will not be covered in this work. On this subject, the reader may wish to refer to: [SPA 68; GIO 75; BEN 78; MUR 78; TAR 78; MUR 95; ALL 92; ALL 96].

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Homogenization of Coupled Phenomena

2.4.3. Parallels between macroscopic models for materials with periodic and random structures Let us recall the characteristics of these two types of material when they exhibit a separation of scales, as well as the phenomena to which they are subjected. We will also take a viewpoint that is distant from any macroscopic boundary. 2.4.3.1. Periodic materials The geometry is periodic. The physical quantities, and the derived quantities, are also periodic with respect to the fast variable y∗ . The dependence of the macroscopic physical quantities with respect to the boundary conditions of a given macroscopic problem are expressed in terms of the slow variable x∗ alone. 2.4.3.2. Random materials with a REV The geometry is random. The physical quantities, and derived quantities, are stationary with respect to the fast variable y∗ . The dependence of the macroscopic physical quantities with respect to the boundary conditions of a given macroscopic problem, for a given realization of the structure, is expressed in terms of the slow variable x∗ alone. For both types of material, we will consider macroscopic samples subjected to phenomena, for which the modeling at the local scale leads to a well-posed boundaryvalue problem (even if in practice the numerical solution is difﬁcult to obtain due to the large number of heterogenities contained within the sample). We can therefore be sure that a unique solution does exist in both cases, for the periodic and the random material. If we assume that the asymptotic expansions are unique, the same then applies for successive terms in the expansions. These successive terms are solutions to the boundary-value problem across the period or the REV, with the same partial derivative equations and boundary conditions where discontinuities occur. The only differences are boundary conditions on the surface of the period or the REV, which specify either periodicity or stationarity. Also, the variable of integration is y∗ , with the slow variable x∗ only playing the role of a parameter here. The macroscopic boundary conditions are not therefore involved in these problems, and the structure of solutions at each order of expansions is the same for both types of material. Finally, the determination of compatibility conditions such as the symmetric properties of the effective coefﬁcients only involves integrals over the boundary of period or REV. Periodicity and stationarity are equivalent from the macroscopic viewpoint. Thus, if we restrict ourselves to looking for the structure of the macroscopic description, random and periodic media are equivalent when homogenization is possible. Another means of comparing media with period and random structures is to consider a rectangular parallelepiped REV of the random medium. We will

Is an Equivalent Macroscopic Description Possible?

69

apply the three symmetries to the three orthogonal faces of this REV so that we can ensure the connectivity of the phases. We thus obtain a period equivalent to the REV in terms of ε. The two media will have the same macroscopic behavior, except the anisotropy behavior which may be affected by the planar symmetries we have introduced.

2.4.4. Hill macro-homogenity and separation of scales Hill [HIL 63; HIL 67] deﬁnes the representative volume (RV) as a volume of material with the following two properties: – The RV is on average structurally representative of the whole material; – Macro-homogenity. The RV contains a sufﬁciently large number of heterogenities that the effective coefﬁcients are independent of the boundary conditions for strain or load applied to the RV. The contribution of the matching boundary layer along the edges of the RV is thus negligible. If d is the characteristic size of a heterogenity and L the characteristic size of the RV, d/L 1. Property a corresponds to the deﬁnition of the REV introduced above. When applied to the asymptotic expansion method, property b implies a large number of REV in the RV: in this way the effective coefﬁcients are deﬁnitely independent of the boundary conditions on the RV. The RV is a volume which ensures the separation of length scales and L may thus be compared to a macroscopic length Lc .

2.5. Comments on multiple scale methods and statistical methods 2.5.1. On the periodicity, the stationarity and the concept of the REV Both methods deal with materials with separated length scales: they have a small parameter ε = lc /Lc 1. Of course, the lack of systematic use of this property in the statistical methods explains certain weaknesses that it suffers from, which we will mention later. The same formal structure of the effective coefﬁcients of an elastic composite shows that the two methods are similar, and it is reasonable to assume this property extends to all elliptical problems. The translational invariance (periodicity or stationarity) represents a third point in common. Finally, although the result is obtained by very different means, both methods make it possible to study the physical characteristics of the volume means. The solution follows from the assumption of ergodicity in the statistical methods, and from the periodicity or stationarity, and the local solenoidal nature of the quantities, in the case of the multiple scale method (see [AUR 86; AUR 91] for some simple examples). The method of multiple scales however allows the consideration of examples where the volume mean is different to the surface mean [LEV 81; AUR 87a; AUR 89].

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Homogenization of Coupled Phenomena

Now we return to selection of the representative elementary volume (REV). While there is no issue for periodic media, because REV is the period, the REV is necessarily approximate for random materials [ROU 88; DRU 96; GUS 97; KAN 03; KAN 06; FOR 06; ROL 07]. For a porous structure the optimal REV for porosity differs from the optimal REV for permeability. A good investigation of this topic involving unsaturated porous media can be found in Rouger [ROU 88]. The works of Cherel et al. [CHE 88] and Kanit et al. [KAN 03; KAN 06] demonstrate that the optimal REV depends on the physical quantity of interest (thermal conductivity, modulus of elasticity, etc.), on the contrast between the different phases and the precision required for the property being investigated. They also show that it is possible to obtain a precise estimate of the effective properties of a material with a random microstructure, starting from its “apparent” properties identiﬁed over a number of volumes the size which is much smaller than the REV. This leads to the question of whether we can transfer some of the advantages of the multiple scale method applied to periodic media to the study of random materials, to the calculation of effective coefﬁcients. When homogenization is possible, we have seen that the structure of the macroscopic description is independent of whether the material is periodic or random. So, applying the method of multiple scales to a ﬁctitious periodic material which is “analogous” to the random material, we can obtain effective coefﬁcients which we then need to relate back to those of the random material. Various works [CHE 88; ANG 94; BOU 04; KAN 03; KAN 06] show, for example, that the assumption of periodicity is not an obstacle in estimating the effective properties of a material with a random microstructure. The same issue approached from a different angle has led to estimates for the permeability of periodic and/or random porous media [BOU 00; BOU 08] (see Chapter 10).

2.5.2. On the absence of, or need for macroscopic prerequisites One crucial difference between the methods is the absence (multiple scale methods) or need (statistical methods) for assumptions in order to establish the macroscopic description. These assumptions do not help when it comes to determining “exotic” descriptions, in other words descriptions which do not follow the normal macroscopic phenomenological rules. Assumptions impose a priori restrictions on the macroscopic description and limit the generality of the statitical theory and most other homogenization methods. The method of multiple scales, which does not require any starting assumptions, stands out as the only entirely satisfactory method in this respect: – In the ﬁeld of elastic composites the work of Duvaut [DUV 76] can be consulted. He shows without any starting assumptions that steady state microscopic and macroscopic behaviors have the same structure to their description (see Chapters 3 and 11).

Is an Equivalent Macroscopic Description Possible?

71

– The study of plastic composites and materials with non-linear behavior is another more complicated example [deB 87; deB 86; PAS 86; SUQ 87; deB 91; SUQ 97; PON 98; BOR 01]. – The study of ﬁltration in a rigid porous structure represents one situation where the structures of the microscopic and macroscopic descriptions are very different (the Navier-Stokes equations and Darcy’s law) even though they represent the expression of fundamental principles of mechanics on both scales [ENE 75; AUR 80]. The case of dynamic perturbations [LEV 79; AUR 80] shows the unusual property that the macroscopic ﬁltration, or generalized Darcy’s law, has “exotic” dynamics: the ﬁltering ﬂuid has a memory of past accelerations (see Chapter 7). – The study of suspensions in a Newtonian liquid [FLE 83; LEV 83b; SAN 85] as well as ﬂuid mixtures [LEV 81; AUR 89; BOU 93] is a clear example, as is the study of any such mixture, of the microscopic causes of mono- or pluriphasic macroscopic behavior. – For porous media (see Part 4), references to the homogenization of various physical problems can be found in Hornung [HOR 97]. More recently, Moyne and Murad [MOY 02; 03; 06] showed that the consideration of electrochemical effects in a saturated porous medium (ﬂuid with dissolved solute) leads to the concept of osmotic stress on the macroscopic scale. 2.5.3. On the homogenizability and consistency of the macroscopic description The question of homogenizability has been considered in Auriault [AUR 91]. The reader is also referred to Auriault and Boutin [AUR 92; 93a; 94] for porous media with double porosity [AUR 93a], and [AUR 95] for Taylor dispersion in porous media, as well as [BOU 90] for the acoustics of bituminous concretes and [BOU 93] for that of Newtonian liquids with high concentrations of gas bubbles. Physical consistency of the results is a very important consideration. The macroscopic description is obtained through conservation equations and rheological laws, with macroscopic quantities whose physical meaning needs to be speciﬁed. It is also necessary to ensure energetic consistencies. Thus for an elastic composite under static load, the mean of the local elastic energy density must equal the macroscopic elastic energy density deﬁned using the correct macroscopic physical quantities. The method of multiple scales is well suited to investigating this. As far as the statistical method is concerned, the consideration does not apply, since consistency is included as a prerequisite (at least as far as elastic composites are concerned). Localization is another advantage of the method of multiple scales, as applied to periodic media, whether it is used to obtain local ﬁelds in the bulk of the material or to escape the simplistic restriction of an inﬁnite material. Close to macroscopic boundaries, the separation of scales (and the stationarity) is broken perpendicular

72

Homogenization of Coupled Phenomena

to these boundaries. The introduction of matching boundary layers means that the method of multiple scales can be applied to boundary-value problems on ﬁnite regions and can be used to investigate the problems associated with them [ENE 75; AUR 87; TUR 87; LEV 75; LEV 77; SAN 87]. The method of multiple scales makes it possible to deduce whether the point measurement of a physical quantity Φ is reasonable or not within the heterogenous material. The dimensionless macroscopic behavior is given to ﬁrst order by that of Φ∗(0) (x∗ , y∗ ) and the whole problem can be reduced to that of determining whether the macroscopic quantity Φ∗(0) equals the quantity Φ∗(0) (x∗ , y∗ ) at all points across the period. Or to put it another way, whether Φ∗(0) (x∗ , y∗ ) is independent of the variable y∗ . This is, for example, the case for a temperature T in heat transfer problems in a composite of materials with conductivities of the same order of magnitude with respect to ε: T ∗(0) as a function of the slow variable x∗ alone, where point measurement is permitted [AUR 83]. In the problem of ﬁltration of a Newtonian liquid in a porous medium, the pressure is such that p∗(0) = p∗(0) (x∗ ) while the velocity (clearly) depends on the fast variable y∗ , v∗(0) = v∗(0) (x∗ , y∗ ). Point measurement of the pressure is permitted, whereas measurement of the velocity must be global (a surface mean!). Such conclusions are also possible for other homogenization methods, and the reasoning relies to a lesser or greater extent on the presence of separated length scales. On this topic the reader is referred to: [HAS 79; WHI 86; NOZ 85]. 2.5.4. On the treatment of problems with several small parameters The systematic use of the small parameter ε offers a decisive advantage to the method of multiple scales. It is not possible to describe the dynamics of an elastic composite using the statistical method [KRÖ 86]. On the other hand, the method of multiple scales rapidly demonstrates that the macroscopic rheology obtained in the steady state regime is useful as a macroscopic dynamic description, at least in most normal situations [AUR 85a]. Other examples involve the possibility of treating problems with several length-scale separations using the method of multiple scales, and problems where, as well as the small homogenization parameter ε, there are other small parameters involving the internal geometry of the period or rheological coefﬁcients of the composite materials. Thus many situations can be studied depending on the respective levels of various small parameters. Double porosity rigid structures can also be added to this category [LEV 88; ROY 92; ROY 94]. Three well-separated length scales are used (Figure 2.5). The ﬁrst scale is that of the pores, where behavior can be described using Stokes equations. The second scale is that of ﬁssures. On this scale the Stokes equations are valid in the ﬁssures, and the medium equivalent to the micro-porous material on the preceding scale is described by Darcy’s law. Finally, on the macroscopic scale

Is an Equivalent Macroscopic Description Possible?

73

Figure 2.5. Doubly porous material: macroscopic scale of the same, mesoscopic scale of the fractures and microscopic scale of the pores

homogenization leads to a Darcy law. When the matrix is deformable, [AUR 92; 93a; 94a] interaction between the length scales becomes strong, and the macroscopic description depends on the relative values of the separation of scales as well as the contrast in rheological properties of the matrix and the ﬂuid. This makes it possible to conﬁrm or discount [ROY 94] various macroscopic descriptions introduced directly on this length scale through phenomenological studies, such as those of Barrenblatt et al. [BAR 60] or Warren and Root [WAR 63]. Along similar lines, double conductivity media were modeled in [AUR 83]. There again, the study made it possible to conﬁrm or discount the existing models. Finally the case of porous media with strong heterogenities on the macroscopic scale, which introduces a third characteristic length, are homogenized on a larger scale [MEI 89]. Filtration in a standard porous medium (where the geometry is only described in terms of one small parameter) is governed by Darcy’s law. When the matrix consists of particles (or ﬁbers) which are small compared to the period, by adding a small geometric parameter, the description becomes that of Brinkman’s law [LEV 83a]. Along similar lines, a study can be found of the Dirichlet problem for a region containing small holes [CIO 85]. Banks drained using geotextile sheets [AUR 82] introduce, in addition to ε, small parameters characterizing the geometry of the period (ratio of the thicknesses of geotextile and soil) and also the hydraulic properties (ratio

74

Homogenization of Coupled Phenomena

of permeabilities of soil and geotextile). The study of reinforced earth [PAS 86], steady state diffusion in a body with small highly-conducting bodies [CAI 83], the behavior of elastic bodies reinforced with thin highly-rigid ﬁbers [CAI 81], or the steady state analysis of thin elastic layers with a small-scale periodic structure [CAI 82], are all also problems involving three small parameters. Finally we mention homogenization of reticulated structures, which makes it possible to replace these structures with an equivalent continuum. This sort of problem, treated as a continuous medium on the local scale, introduces a second small parameter characterizing the geometry of the period [CIO 86; 88] in addition to ε. It can also be treated using a description based on beam elements on the local scale, which leads to the methods of homogenization for discrete structures [CAI 89; BAK 89; BOU 03]. Recent developments in structural mechanics are summarized in Andrianov et al. [AND 04].

Chapter 3

Homogenization by Multiple Scale Asymptotic Expansions

3.1. Introduction Following discussion of the multiple scale method and its formalism, in this chapter we will explain in detail how it can be implemented. We will begin by using basic experiments to show how the concepts presented in the previous chapters apply in reality and how they match up to physical intuition. We will then show how the homogenization process is carried out for a one-dimensional example with an analytical solution. Finally, the last section focuses on the translation of physical problems into the framework of the multiple scale method.

3.2. Separation of scales: intuitive approach and experimental visualization The concept of multiple scales, and its use in homogenization methods, may appear an abstract one that could be taken as a mathematical trick. It is no such thing, because in fact this idea represents an actual physical reality. Here we will try to help the reader grasp this by using an intuitive approach illustrated with simple experimental examples.

3.2.1. Intuitive approach to the separation of scales We have already seen that homogenization involves the search for a given phenomenon in a given heterogenous material, for an equivalent – or “homogenized” – global description, which does not make any explicit reference to local ﬂuctuations.

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Homogenization of Coupled Phenomena

This statement incorporates, as a subtext, the concept of separation of scales, since a global description has no meaning if the phenomenon of interest only varies on a local scale. As indicated in the preceding chapters, it is this crucial concept of separation of length scales which makes it possible to look for a homogenized description. The concept can be described in terms of two requirements: – the ﬁrst involves the medium, which must be such that we can deﬁne a characteristic length l, which is only possible if the material has a representative elementary volume (without a REV, there is no characteristic length!); – the second involves the phenomenon: a quantity associated with it must exhibit a characteristic length L, which is large compared to l.

The graphics in Figure 3.1 (top) give a visual depiction of the separation of scales required for homogenizability: as long as the phenomenon of interest (the runner) has a scale of motion (his stride) which is large compared to the REV of the material (a sand or pebble beach), a global description (of the speed and trajectory of the runner) which ignores local ﬂuctuations (the exact positions of the grains of sand or pebbles) is possible. It can be clearly seen in this example that for the phenomena involved a homogenized description is more efﬁcient – and also more realistic – than a description which incorporates every last detail of reality without removing all but the essential parts of the picture. Outside this framework, in other words without a separation of scales, the search for a macroscopic description is doomed to failure (Figure 3.1 below): on a route consisting of meter-sized rocks, neither the trajectory nor the speed of the runner can be known independently of the distribution of the blocks. This would also be the case for an insect on the pebble beach (despite the fact that it is homogenizable for the runner). This illustrates the fact that homogenizability is a property not intrinsic to the material or the phenomenon, but which depends on the material/phenomenon pair. The role of the periodic or random nature of the microstructure (when the constituents follow the same connectivity conditions) was discussed in Chapter 2. The images in Figure 3.2 (top) illustrate the main conclusions: when the separation of scales is obvious, whatever the organization on the local scale (pebbles arranged periodically or laid out randomly), the mechanisms (the determination of the runner’s trajectory) are the same, and as a result the macroscopic behavior of the material (what the runner experiences) will be qualitatively the same. Here we justify the use of the method of periodic media for treating real aperiodic materials when there is a separation of scales. Conversely, the closer the macroscopic scale gets to the microscopic scale, the more sensitive it becomes to local ﬂuctuations, and consequently the organization of the microstructure. At the limit of the homogenizable domain, Figure 3.2 (bottom – the runner striding across separate blocks), the

Homogenization by Multiple Scale Asymptotic Expansions

77

Figure 3.1. The separation of scales is the sine qua non condition for a global description. Here it is only the case in the top picture: the property of homogenizability only has a meaning for the combination of the material and the phenomenon together (illustrations by Jacques Sardat)

phenomena in periodic media (where the runner can jump from block to block) and random media (where the runner falls between blocks that are too far apart) diverge: without the separation of scales, homogenization loses its meaning, and the type of organization within the microstructure becomes critical.

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Homogenization of Coupled Phenomena

Figure 3.2. The role of the microstructure layout is more signiﬁcant when there is not a good separation of scales (illustrations by Jacques Sardat)

3.2.2. Experimental visualization of ﬁelds with two length scales Here we will investigate, with the help of basic experiments on two periodic twodimensional media: the manifestation of local and global scale variations, and the (quasi-)periodicity or local periodicity of the ﬁelds. 3.2.2.1. Investigation of a ﬂexible net The photos in Figure 3.3 show a net with a diamond mesh (period Ω) ﬁxed at its edges to a square framework consisting of four rigid, articulated rods. If we apply a distortion to the frame, we impose a homogenous distortion to the net: – photo (a) in Figure 3.3 shows the starting position, where the net is undistorted; – photos (b) and (c) in Figure 3.3 show the geometries obtained when a moderate, and then considerable, distortion is applied to the supporting framework. It is clear that the mesh is distorted but that the structure remains periodic. So, for homogenous deformations, the property of periodicity of the initial medium is preserved by the perturbations, even for large deformations.

Homogenization by Multiple Scale Asymptotic Expansions

79

(a)

(b)

(c) Figure 3.3. Visualization of a periodic net (a) of the periodicity under (b) moderate and (c) large deformations

What happens for loadings which lead to inhomogenous distortions? – In photo (a) in Figure 3.4 the net is dragged in the plane by a rigid rake which applies a tension across several units of the mesh. The deformation produced in the net is not homogenous. Nevertheless, a local (quasi-)periodicity (i.e. Ω-periodicity relative to the microscopic variable) is visible. What we mean is that all the meshes adjacent to mesh A have an almost identical geometry. The same is true of mesh B. However, meshes A and B, which are fairly far apart from each other, have a very different geometry. We also point out that the geometries of the deformed meshes are the same as we have already seen under homogenous distortion. This situation shows

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the two-length-scale variations, where each cell is deformed, but the amplitude of this deformation varies gradually across distances corresponding to many mesh cells. – If instead of being spread out the tension is applied at one point (photograph (b) in Figure 3.4), a new effect appears which is characterized by a violation of the local periodicity on either side of the line of the pull. In these regions where there is a high gradient of deformation, there is no longer a separation of scales because the phenomenon is concentrated on the local scale (which leads to the loss in periodicity): perpendicular to the direction of the pull the problem is not homogenizable.

Figure 3.4. Inhomogenous load: the quasi-periodicity relies on a separation of length scales: (a) a load which respects the separation of scales: the local quasi-periodicity is modulated by large-scale variations; (b) localized loading: the periodicity is lost along the line of the pull

This net makes it possible to directly observe the deformed geometry of the lattice. However, in many homogenization problems the period is considered to be ﬁxed, as is effectively the case of ﬂow in rigid porous media, cases of heat transfer or diffusive solute transport, etc. or where it is an approximation which can be justiﬁed by the low level of deformation such as when considering elastic composites, poroelastic behavior, etc. In this case the homogenizability conditions apply to the ﬁelds which develop within this periodic geometry. We will consider such a case in the next example.

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3.2.2.2. Photoelastic investigation of a perforated plate Consider a plexiglass plate drilled with oblong windows distributed in a periodic staggered pattern, which we will subject to small deformations in plane. Through the photoelastic effect we can visualize the deviatoric stresses which develop in the plate under different loads: – when the plate is loaded uniformly in its plane (photograph (a) in Figure 3.5) it is very obvious that the ﬁeld is periodic, matching up with the periodicity of the plate; – if the loading area on the upper edge is reduced (photograph (b) in Figure 3.5) while maintaining the entire contact surface on the lower edge, the local quasiperiodicity and the global ﬂuctuations can both be seen; – “large-scale” intensity variations are even clearer in photograph (c) in Figure 3.5 where the load is pointlike. It is clear in this case that close to where the force is applied the phenomenon is not homogenizable, but that it becomes so outside a region around the point of loading (which extends for around one period). We will return to this aspect of the problem at the end of this chapter. We also remark that outside the areas of concentrated load, the local distribution of the deviatoric stresses looks the same, but its global evolution depends on the load. If on the other hand the same load is applied in a different orientation relative to the plate (photographs (a) and (b) in Figure 3.6) the local and global distributions of the deviatoric stresses are completely changed (but of course the periodicity is still retained). Hence the anisotropy of the distribution of the perforations has a direct impact on the local, and hence global, strains: this illustrates the fact that the macroscopic description is tightly linked to the microscopic structure. Also, across all the pictures, an edge effect can be seen at the boarder of the plate, which rapidly fades towards the middle of the periodic medium. This rapid decrease can be conﬁrmed in photograph (c) of Figure 3.6 where, under homogenous compression, the periodicity remains obvious when the plate only consists of oneand-a-half periods! The edge effects which result from the loss of periodicity at the boundary can be treated from a theoretical point of view by the introduction of a boundary layer [see for example SAN 87; AUR 87a]. In conclusion, these two examples show how the principles of homogenization have a basis in physical reality. They also show that these principles apply even some way from the ideal separation of scales which the theoretical developments require. Indeed, in situations of inhomogenous loading, the actual scale ratio εr is, at very best, in the order of the inverse of the number of periods contained in the smallest dimension of the experiment, so εr 0.1 for the net and εr 0.3 for the plate. This possibility of extending the ﬁeld of applicability is also one of the main reasons

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Homogenization of Coupled Phenomena

(a)

(b)

(c) Figure 3.5. Condition of separation of scales, and quasi-periodicity of the ﬁelds in a periodic medium. Perforated plate subject to: (a) homogenous compression exerted by pressure across the width of the plate, (b) inhomogenous compression exerted by a pressure from above across a narrower width, (c) inhomogenous compression under point loading of the upper surface

Homogenization by Multiple Scale Asymptotic Expansions

(a)

(b)

(c) Figure 3.6. Role of the microstructure in the distribution of local and global forces: (a) inhomogenous compression parallel and (b) perpendicular to the holes, (c) quasi-periodicity and edge effects under homogenous compression for a plate which only consists of one-and-a-half periods

83

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Homogenization of Coupled Phenomena

why the results of homogenization are so good at describing real situations: results established rigorously in the context of ideal assumptions retain their pertinence for real physical situations corresponding to weakened hypotheses. From a theoretical viewpoint, this observation is analogous to proof that the results converge when the scale ratio approaches zero. To clarify what is meant by this, we will return to these issues, and to the importance of the actual separation of scales in a real-life problem, after we have demonstrated application of the method to a simple example.

3.3. One-dimensional example Now, and in what follows, we will systematically apply the method of multiple scales, following the methodology laid out in Chapter 2, section 2.4.2. In order to present the various stages of the process, we have selected a one-dimensional example which has an analytical solution. Due to its simplicity, this example cannot include all the problems inherent to homogenization techniques. We will encounter them in subsequent chapters during the study of multi-dimensional problems. Here we will consider a one-dimensional elastic Galilean medium with an oedometric modulus E and density ρ, subject to a dynamic perturbation. The medium is periodic, with a small period lc , and we will consider a sample of length Lc lc . The displacement u is governed by the equation of dynamic equilibrium: divX (E gradX u) = ρ

∂2u ∂t2

(3.1)

where divX and gradX are the divergence and gradient operators with respect to the spatial variable X, which are the same here since the problem is one-dimensional. We recall that E(X) is a positive quantity, as is the density ρ. They are both periodic with period lc , and may exhibit discontinuities. Figure 3.7 shows an example of the variation of E.

E

lc Figure 3.7. Periodic variation of E

X

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85

Across the discontinuities Γ, the stress σ and the displacement u are continuous: [σ] = [E gradX u]Γ = 0

(3.2)

[u]Γ = 0

(3.3)

In the above equations, [φ]Γ indicates the jump in φ across the interface Γ. We will ﬁrst consider the steady state problem, where the second member of (3.1) is zero. We will then treat the dynamic case.

3.3.1. Elasto-statics Equation (3.1) now becomes: divX (E gradX u) = 0

(3.4)

Equations (3.2, 3.3, 3.4) do not introduce any dimensionless numbers. We will take the microscopic viewpoint. With: X = lc y ∗ ,

E = Ec E ∗ ,

u = u c u∗

where Ec and uc are characteristic values, we have: σ = E gradX u = σc σ ∗

with σc = Ec uc /lc

Equation (3.4) becomes: divlc y∗ (Ec E ∗ gradlc y∗ uc u∗ ) = 0 Carrying out the same change of variables in (3.2, 3.3), we ﬁnd after simpliﬁcation of the terms referring to the same characteristic values: divy∗ (E ∗ grady∗ u∗ ) = 0,

[E ∗ grady∗ u∗ ]Γ∗ = 0,

[u∗ ]Γ∗ = 0

The unkown u∗ must be found in the form of the following expansion: u∗ (x∗ , y ∗ ) = u∗(0) (x∗ , y ∗ ) + εu∗(1) (x∗ , y∗ ) + · · · ,

x∗ = εy ∗

(3.5)

where ε = lc /Lc and u∗(i) are periodic with respect to the local variable y ∗ = X/lc , of period 1. The differential operators are therefore operators with respect to the variable y ∗ , and in (3.5) x∗ = εy ∗ . The equivalent macroscopic description will be valid when the perturbation satisﬁes the condition of separation of scales (we assume that the condition on the separation of geometric scales is met).

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3.3.1.1. Equivalent macroscopic description The method involves the introduction of the expansion (3.5) into the dimensionless system and identifying the powers of ε. We note that due to the two spatial variables and the choice of the microscopic viewpoint, the spatial derivative takes the following form: ∂ ∂x∗ ∂ ∂ ∂ + ∗ ∗ = +ε ∗ ∗ ∗ ∂y ∂y ∂x ∂y ∂x The local description becomes: (

∂ ∂ ∂ ∂ ∗ ∗ ∗ + ε ) E (y )( + ε )u =0 ∂y ∗ ∂x∗ ∂y ∗ ∂x∗

with: [E ∗ (y ∗ )(

∂ ∂ + ε ∗ )u∗ ]Γ∗ = 0 ∗ ∂y ∂x

[u∗ ]Γ∗ = 0 across the discontinuities. Substituting the expansion (3.5) into these expressions, we obtain in succession the following results, separating out terms of the same power of ε: First order in ε0 : the system deﬁning u∗(0) is the following: ∂ ∂y ∗

∂u∗(0) E (y ) ∂y ∗ ∗

[E ∗ (y ∗ )

∗

=0

(3.6)

∂u∗(0) ]Γ∗ = 0 ∂y ∗

[u∗(0) ]Γ∗ = 0 where u∗(0) is 1-periodic in y ∗ . By successive integration of (3.6) it follows, making use of the conditions at the discontinuities, that: E ∗ (y ∗ )

∂u∗(0) = σ ∗(0) (x∗ ) ∂y ∗

u∗(0) (x∗ , y ∗ ) = σ ∗(0) (x∗ )

y∗ 0

E ∗−1 (y ∗ )dy ∗ + u∗(0) (x∗ , 0)

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87

where the constants of integration σ ∗(0) (the zero-order stress) and u∗(0) (x∗ , 0) are functions of x∗ alone. Also, the periodicity can be expressed as: u∗(0) (x∗ , 1) = u∗(0) (x∗ , 0) which leads us to: 1 E ∗−1 (y ∗ )dy ∗ = 0 σ∗(0) (x∗ ) 0

which means that σ ∗(0) = 0, since E ∗ > 0. Finally: u∗(0) (x∗ , y ∗ ) = u∗(0) (x∗ ) proving that at the dominant order, the displacement is a function of x∗ alone. In other words it does not ﬂuctuate over the course of a period. Second order in ε: the following order gives us: ∂ ∂y ∗

∂u∗(1) du∗(0) + ) E (y )( ∂y∗ dx∗ ∗

[E ∗ (y ∗ )(

∗

=0

∂u∗(1) du∗(0) + )]Γ∗ = 0 ∂y ∗ dx∗

[u∗(1) ]Γ∗ = 0 where u∗(1) is 1-periodic in y ∗ . The general solution to the differential equation can be obtained as before: E ∗ (y ∗ )(

∂u∗(1) du∗(0) + ) = σ ∗(1) (x∗ ) ∗ ∂y dx∗

u∗(1) (x∗ , y ∗ ) = σ ∗(1)

y∗

E ∗−1 (y ∗ )dy ∗ − y ∗

0

du∗(0) + u∗(1) (x∗ , 0) dx∗

where σ ∗(1) (ﬁrst-order stress) and u∗(1) (x∗ , 0) are functions of x∗ alone. Again the periodicity of the unknown u∗(1) (x∗ , 1) = u∗(1) (x∗ , 0) allows us to determine σ ∗(1) : σ ∗(1) (x∗ )

1

E ∗−1 (y ∗ )dy ∗ −

0

σ ∗(1) (x∗ ) = E ∗−1 −1

∂u∗(0) ∂x∗

du∗(0) =0 dx∗

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Homogenization of Coupled Phenomena

where . represents the mean operator over the period, here:

1

. =

. dy ∗

0

Third order in ε2 : Compatibility condition. At this order we have: ∂ ∂y ∗

∂u∗(2) ∂u∗(1) + ) E (y )( ∂y ∗ ∂x∗ ∗

∗

[E ∗ (y ∗ )(

∂ =− ∗ ∂x

∂u∗(1) du∗(0) + ) E (y )( ∂y ∗ dx∗ ∗

∗

(3.7)

∂u∗(2) ∂u∗(1) + )] = 0 ∗ ∂y ∂x∗

[u∗(2) ] = 0 where u∗(2) is 1-periodic in y ∗ . We do not need to calculate u∗(2) as we did for u∗(0) and u∗(1) , at least if we limit ourselves to studying the ﬁrst macroscopic order. In fact the differential equation represents the conservation of the periodic quantity: σ ∗(2) = E ∗ (y ∗ )(

∂u∗(2) ∂u∗(1) + ) ∗ ∂y ∂x∗

in the presence of the source term: ∂ − ∗ ∂x

∗

∂u∗(1) du (0) + ) E (y )( ∂y∗ dx∗ ∗

∗

=−

∂σ ∗(1) (x∗ ) ∂x∗

In accordance with the analysis presented in section 2.4.2, equation (3.7) is the exact analog of equation (2.10) with W ∗ = 0. By integrating this conservation equation over the period, we have:

∂σ ∗(1) (x∗ ) ∂σ ∗(2) = − ∗ ∂y ∂x∗

But, due to the periodicity, the left hand side is zero:

∂σ ∗(2) = ∂y ∗

1 0

∂σ ∗(2) ∗ dy = σ ∗(2) (y ∗ = 1) − σ ∗(2) (y ∗ = 0) ∂y ∗

Thus we have established the compatibility condition requiring the source to have a mean of zero (see equation (2.11)):

∂σ ∗(1) =0 ∂x∗

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89

so that, swapping the derivation with respect to x∗ and integration with respect to y ∗ , and introducing the expression for σ∗(1) : d dx∗

E

∗−1 −1 du

∗(0)

=0

dx∗

(3.8)

This compatibility equation represents, in dimensionless form and to ﬁrst order of approximation, the equivalent macroscopic description that we were looking for. With: x∗ =

X , Lc

E∗ =

E , Ec

u∗ =

u , uc

¯ u = u(0) + O(ε)

the stress can be written in dimensional variables: σ = σc σ ∗(1) = (Ec uc /lc )E ∗−1 −1

du∗(0) du = E −1 −1 dx∗ dX

In the same way, the model can be written in dimensional variables: d −1 −1 du ¯ E = O(ε) dX dX ¯ where O(ε) is a term of relative order ε. 3.3.1.2. Comments 3.3.1.2.1. Effective coefﬁcient The structure of the macroscopic description is identical to that of the local description. In particular, the property E ∗ > 0 is preserved because the macroscopic effective elastic coefﬁcient is such that: E eﬀ∗ = E ∗−1 −1

>0

This result is incidentally a classical one, and does not require any particular homogenization technique to prove it (see Chapter 1 where the equivalent thermal problem was treated). We ﬁnd in this one-dimensional steady state problem that the stress is constant: σ=E

du = constant dX

Taking the mean of σ/E over the period, we ﬁnd:

1 du σ = σ = E E dX

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Homogenization of Coupled Phenomena

which leads to the result when it is observed that the mean strain is the macroscopic strain. Finally we note that E(y∗ ) tends to E(y ∗ ) when ε tends to zero, weakly in L2 [SAN 80], but that in general terms: E eﬀ = E(y ∗ ) 3.3.1.2.2. Macroscopic physical quantities The dimensionless physical quantities – the displacement u∗ and the stress σ ∗ – are given to ﬁrst order by: u∗ = u∗(0) (x∗ ) σ ∗ = εσ ∗(1) = εE ∗ (y ∗ )(

du∗(0) ∂u∗(1) du∗(0) + ) = εE ∗−1 −1 ∗ ∗ ∂y dx dx∗

They are independent of the local variable y ∗ and represent macroscopic quantities, without any mean operator. The physical signiﬁcance of the macroscopic quantities does not therefore pose any problem here because it is identical to those introduced locally. 3.3.1.2.3. Accuracy of the macroscopic description Returning to the displacement u∗ , the dimensionless macroscopic description (3.8) can be written: d dx∗

E ∗−1 −1

du∗ dx∗

= O(ε)

In practice the small parameter ε is non-zero and the equivalent macroscopic description is only approximate. This is the case for any macroscopic description of a heterogenous material. 3.3.1.2.4. Quasi-periodicity: macroscopically heterogenous material The case of quasi-periodicity where the modulus E ∗ is not only a function of y∗ but also of x∗ does not pose any difﬁculty, as long as the variations are sufﬁciently slow that a separation of scales is retained. The effective coefﬁcient is still written as E ∗−1 −1 , but now it depends on the variable x∗ . What happens is that x∗ plays the role of a parameter in the process: we recall that the differential systems that must be solved involve the variable y ∗ . This observation can of course be applied to all homogenization problems, thus making it possible to systematically extend the results to slightly macroscopically heterogenous media.

Homogenization by Multiple Scale Asymptotic Expansions

91

Finally, when the material is not strictly periodic, in other words when the period Ω∗ depends on x∗ , (3.8) becomes: d dx∗

|Ω∗ |E ∗−1 −1

du∗ dx∗

= O(ε)

3.3.2. Elasto-dynamics We will now include the inertial term. The local description is then given by the system of equations (3.1, 3.2, 3.3). The change is that this system introduces a dimensionless number denoted P, the ratio of the inertial term to the elastic term: ∂2u | ∂t2 P= |divX (EgradX u)| |ρ

We will again adopt the microscopic viewpoint here, so that the characteristic length for non-dimensionalization is lc . With: X = lc y ∗ ,

E = Ec E ∗ ,

u = u c u∗ ,

ρ = ρc ρ ∗ ,

t = t c t∗

it follows in dimensionless form that: divy∗ (E ∗ grady∗ u∗ ) = Pl ρ∗

∂ 2 u∗ ∂t∗2

(3.9)

[σ ∗ ]Γ∗ = [E ∗ grady∗ u∗ ]Γ∗ = 0

(3.10)

[u∗ ]Γ∗ = 0

(3.11)

with: Pl =

ρc lc2 Ec t2c

Typically the time tc is linked to the period of the wave, or to its pulsation ωc by tc = 1/ωc . The physical signiﬁcance of the dimensionless number Pl , the value of P using lc as the characteristic length, should be clariﬁed. We can anticipate that the effective elastic modulus E eﬀ , if it exists, is of the order of magnitude of the characteristic modulus Ec . The wave velocity is then: c=O

Ec ρc

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Homogenization of Coupled Phenomena

and the wavelength λ for pulsations of order ωc is: 2π Ec Ec λ=O = O 2πtc ωc ρ c ρc Finally, Pl is the squared product of the wavenumber (2π/λ) and the length of the geometric period: 2 2πlc (3.12) Pl = O λ We will again look for a displacement u∗ of the form: u∗ (x∗ , y ∗ , t∗ ) = u∗(0) (x∗ , y ∗ , t∗ ) + εu∗(1) (x∗ , y ∗ , t∗ ) + · · ·

(3.13)

with x∗ = εy ∗ , where ε = lc /Lc and u∗(i) are periodic with respect to the local variable y ∗ , of period 1. Before beginning any homogenization, we must evaluate Pl as function of powers of ε. Different values of Pl can in fact be imagined, which reveals whether the situation can be homogenized or not. We will begin with the local description which leads to an equivalent macroscopic description of the dynamics. This situation corresponds to a Pl = O(ε2 ). Then we will consider values close to Pl = O(ε3 ) which lead to a macroscopic description which is steady state to ﬁrst order of approximation, the case investigated in section 3.2, and ﬁnally Pl = O(ε) which corresponds to a local description which cannot be homogenized. 3.3.2.1. Macroscopic dynamics: Pl = O(ε2 ) 3.3.2.1.1. Normalization We are looking for the local description corresponding to macroscopic dynamics. It must be homogenizable, and so the geometry and disturbance must exhibit a separation of scales. We will assume that this is the case for the geometry. As far as the perturbation goes, λc /2π is a good candidate to deﬁne a characteristic macroscopic length Lc , as we will demonstrate in the following section. The separation of scales then requires that: 2πlc =ε1 λ and with (3.12): 2 2πlc = O(ε2 ), Pl = O λ

so that Pl = ε2 Pl∗ with Pl∗ = O(1)

Homogenization by Multiple Scale Asymptotic Expansions

93

We can reasonably hope that this estimate of Pl represents a homogenizable local description which will lead to a macroscopic description of the dynamics. This is proven below. We observe that the condition of separation of scales, in imposing a wavelength which is large relative to lc , implies as a consequence a frequency ω which must be sufﬁciently low: ω < ωdif (diffraction becomes signiﬁcant for frequencies O(ωdif ) such that λ is of the order of lc ). Equation (3.9) becomes: divy∗ (E ∗ grady∗ u∗ ) = ε2 Pl∗ ρ∗

∂ 2 u∗ ∂t∗2

(3.14)

3.3.2.1.2. Homogenization Substituting the expansion (3.13) into the dimensionless equations, it is easy to see that the way the ﬁrst two problems we solved, for unknowns u∗(0) and u∗(1) , are identical to those obtained for the steady state case. We therefore have: u∗(0) = u∗(0) (x∗ , t∗ ) u∗(1) (x∗ , y ∗ , t∗ ) = σ ∗(1) (x∗ , t∗ )

y∗

0

E ∗−1 (y ∗ )dy ∗ − y ∗

du∗(0) + u∗(1) (x∗ , 0, t∗ ) dx∗

with: σ ∗(1) (x∗ , t∗ ) = E ∗−1 −1

∂u∗(0) ∂x∗

By way of contrast, the next order is modiﬁed, with the appearance of the inertial term −ω ∗2 ρ∗ Pl∗ u∗(0) in the source term: ∂ ∂y ∗

E ∗ (y ∗ )(

∂ =− ∗ ∂x [E ∗ (y ∗ )(

∂u∗(2) ∂u∗(1) + ) ∗ ∂y ∂x∗

2 ∗(0) ∂u∗(1) du∗(0) ∗ ∗∂ u + ) + ρ P E (y )( l ∂y ∗ dx∗ ∂t∗2

∗

∗

∂u∗(2) ∂u∗(1) + )]Γ∗ = 0 ∂y ∗ ∂x∗

[u∗(2) ]Γ∗ = 0 Once again we ﬁnd an equation analogous to equation (2.10), where W ∗ is the inertial term. Setting the mean of the source to zero, this leads us to the compatibility condition which gives the macroscopic description: ∗(0) d ∂ 2 u∗(0) ∗−1 −1 du E = ρ∗ Pl∗ (3.15) ∗ ∗ dx dx ∂t∗2

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Homogenization of Coupled Phenomena

in dimensionless form, with: x∗ =

d dX

X E ω ρ u(0) u , E∗ = , ω∗ = , ρ∗ = , u∗(0) = = + O(ε) Lc Ec ωc ρc uc uc

∂2u du Ec t2c ¯ ρPl∗ 2 + O(ε) E −1 −1 = 2 dX ρ c Lc ∂t

and since: Pl = O

2πlc λ

2

= ε2 Pl∗ = O(ε2 )

we have: PL = O

2πLc λ

2 = O(1)

It follows that: d dX

du ∂ 2u ¯ E −1 −1 = ρ 2 + O(ε) dX ∂t

The return to dimensional variables then occurs without ambiguity. In the next sections, the dimensionless numbers will be taken as equal to their order ε estimate (which is equivalent to taking Pl∗ = 1). 3.3.2.1.3. Comments – The estimate does indeed correspond to a homogenizable situation which leads to a macroscopic description of the dynamics. – The effective elastic modulus to be used in the dynamic regime is the same as that in the steady state regime! – The effective density is the mean volume of the local density. – The dynamic description incorporates the steady state situation as a special case. We just need to set ω ∗ = 0. – The macroscopic description is an approximation of order O(ε). – The considerations in section 3.2 about the physical meaning of the macroscopic quantities still apply here.

Homogenization by Multiple Scale Asymptotic Expansions

95

3.3.2.2. Steady state: Pl = O(ε3 ) The normalization of equation (3.9) is obvious: divy∗ (E ∗ grady∗ u∗ ) = ε3 ρ∗

∂ 2 u∗ ∂t∗2

with the relations at the discontinuities remaining unchanged. It is clear that now, up to third order, the problems to be solved are identical to those obtained in section 3.2 for the steady state case. There is now an equivalent macroscopic description given by (3.8): d dx∗

E

∗−1 −1 du

∗(0)

dx∗

=0

As with the other macroscopic descriptions obtained up to now, this is only an approximation. The investigation of the next order (the fourth problem), gives a second approximation of order ε. As can easily be anticipated, this approximation includes an inertial term. For Pl = O(εp ), p 2, the dynamics appear at the (p−2)th order of approximation. 3.3.2.3. Non-homogenizable description: Pl = O(ε) Again the normalization is clear: divy∗ (E ∗ grady∗ u∗ ) = ερ∗

∂ 2 u∗ ∂t∗2

with the relations at the discontinuities remaining unchanged. But now only the ﬁrst problem is the same as that obtained above, with: u∗(0) = u∗(0) (x∗ , t∗ ) The dynamics appear in the second problem, which can be written: ∂ ∂y ∗

E ∗ (y ∗ )(

[E ∗ (y)(

∂u∗(1) du∗(0) + ) ∂y∗ dx∗

= ρ∗

∂ 2 u∗(0) ∂t∗2

∂u∗(1) ∂u∗(0) + )]Γ∗ = 0 ∂y ∗ ∂x∗

[u∗(1) ]Γ∗ = 0 where u∗(1) is 1-periodic in y ∗ . The ﬁrst equation is the conservation of a periodic quantity, and includes the source term ρ∗ ∂ 2 u∗(0) /∂t∗2 . The compatibility condition

96

Homogenization of Coupled Phenomena

implies that this term must have a mean of zero (Fredholm alternative): ρ∗

∂ 2 u∗(0) =0 ∂t∗2

so, since ρ∗ > 0: ∂ 2 u∗(0) =0 ∂t∗2

4

3

2

1

c 2 Pl = ( 2$l )

Di&raction Not homogenizable

Static 2$

1

À Lc =

2$

lc À lc

2

1

1

lc

2

c 2 PL = ( 2$L )

Dynamic 2$

= Lc = 1 lc À lc

Figure 3.8. Macroscopic descriptions that may or may not be valid depending on the values of Pl or PL

This result is impossible since ∂ 2 u∗(0) /∂t∗2 is O(1) by construction. The estimate Pl = O(ε) is a non-homogenizable description. It corresponds to: Pl = O

2πl λ

2 = O(ε)

and hence to: lc λ = √ Lc ε The dynamic excitation does not fulﬁll the condition of separation of scales. To conclude, the different situations are shown in Figure 3.8 as a function of the values of Pl . We observe that the richest macroscopic description, which corresponds to dynamic behavior Pl = O(ε2 ), lies at the limit of the homogenizable situations.

Homogenization by Multiple Scale Asymptotic Expansions

97

3.3.3. Comments on the different possible choices for the spatial variables In order to analyze the previous example we transformed the dimensional spatial variable X into dimensionless spatial variables x∗ = X/Lc and y∗ = X/lc . In addition, the normalization was carried out by adopting the microscopic viewpoint. The problem was then examined in the space of dimensionless variables, with the return to dimensional variables being carried out at the end of the process. In the literature, the change into the variables x∗ and y ∗ is often omitted. The treatment is carried out directly in a system of variables x and y, where in general x refers to the normal unit of length, the meter. Alternatively the normalization is carried out by adopting either the micro- or macroscopic viewpoints. The use of these different approaches, although equivalent, is sometimes a source of confusion. It is for this reason that we will now revisit these different methods. We recall that variables x∗ and y ∗ are particularly well suited to the analysis of problems with a double length scale because, by construction, x∗ is the measure of the distance X when using the distance Lc as unit length, and y∗ is the measure of the same distance X using the distance lc = εLc as the unit length. Thus: – x∗ varies by 1 over the macroscopic length Lc (and hence by ε over lc ); – y ∗ varies by 1 over the microscopic distance lc (and hence by ε−1 over lc ). We also note that as a measure of distance in some systems of units, x and y are both dimensionless variables. We will choose x for metric value X, and will designate c and lc = εL c as the metric values of lengths Lc and lc . Denoting a respectively L meter by “1m ”, we have the following: c 1m = x 1m X = x∗ Lc = x∗ L and: X = y ∗ lc = y ∗ lc 1m = x 1m = yε 1m whence it follows that: x y c = ∗ =L ∗ x y which shows that the variables x and y are homothetic to x∗ and y ∗ . We note that here y is a measurement of X in ε m (for example in millimeters for ε = 10−3 , etc.). From these we deduce that the derivative operator can take the following equivalent forms: 1 ∂ 1 ∂ ∂ = = ∂X Lc ∂x∗ 1m ∂x

98

Homogenization of Coupled Phenomena

or: ∂ 1 ∂ 1 ∂ = = ∗ ∂X lc ∂y ε1m ∂y To illustrate this we will return to the preceding problem in the dynamic regime. We will return to the initial equation, written for convenience in the harmonic regime: divX (E gradX u) = ρω2 u which can also be written in terms of the variable y ∗ (microscopic viewpoint): 1 divy∗ (E grady∗ u) = ρω 2 u (εLc )2

(3.16)

or alternatively, in terms of x∗ (macroscopic viewpoint): 1 divx∗ (E gradx∗ u) = ρω 2 u L2c

(3.17)

Physical analysis showed us that the dynamic regime was characterized by: Pl =

lc2 ρc ωc2 = O(ε2 ) Ec

or alternatively

PL =

L2c ρc ωc2 = O(1) Ec

Substituting these expressions into (3.17) and (3.16), and changing to the double variable operators, i.e. for the microscopic viewpoint: ∂ ∂y ∗

becomes

∂ ∂ +ε ∗ ∗ ∂y ∂x

and for the macroscopic viewpoint: ∂ ∂x∗

becomes

∂ ∂ + ε−1 ∗ ∂x∗ ∂y

we obtain the dimensionless formulations established starting with the microscopic viewpoint (already given in the previous section) and the macroscopic viewpoint. We can show that they of course lead to the same equations: (divy∗ + εdivx∗ ) E ∗ (grady∗ + εgradx∗ )u∗ = ε2 ρ∗ ω ∗2 u∗ or: (divx∗ + ε−1 divy∗ ) E ∗ (gradx∗ + ε−1 grady∗ )u∗ = ρ∗ ω ∗2 u∗

Homogenization by Multiple Scale Asymptotic Expansions

99

Also, using the expressions for the derivative operators, we can transform the equations by writing them in terms of the variable y (microscopic viewpoint): 1 divy (E grady u) = ρω 2 u (ε1m )2 or x (macroscopic viewpoint): 1 1m 2

divx (E gradx u) = ρω 2 u

As before, these two equations are normalized in order to describe the dynamic regime. For the mathematical treatment, the unit (1m ) is neutral (since all of the variables and parameters are expressed in the metric system) and so we can abstract ourselves from it. Thus we obtain the formulations in x and y resulting from the microscopic and macroscopic viewpoints: (divy + εdivx ) E(grady + εgradx )u = ε2 ρω 2 u or: (divx + ε−1 divy ) E(gradx + ε−1 grady )u = ρω 2 u which, again, are the same. We observe that the use of variables x and y is inconvenient because we lose the unit variation over the micro- or macroscopic distances. The advantage is we can continue to use the normal system of units (metric), and maintain the dimensional physical parameters throughout the treatment. At the end of the process all that needs to be done is to restore the meter as the unit. In other words, replace the value x with the distance X in order to obtain the dimensional formulation. We also note that the equations in x∗ , y ∗ or x, y are formally identical and lead to an identical treatment. As a ﬁnal example, consider the quasi-static case corresponding to: Pl =

lc2 ρc ωc2 = O(ε3 ) Ec

or alternatively

PL =

L2c ρc ωc2 = O(ε) Ec

The normalizations are, in terms of the variable y ∗ (microscopic viewpoint): 1 divy∗ (E grady∗ u) = ερω 2 u (εLc )2 and, in terms of the variable x∗ (macroscopic viewpoint): 1 divx∗ (E gradx∗ u) = ερω 2 u L2c

100

Homogenization of Coupled Phenomena

Transforming the derivative operators, we ﬁnd in terms of the variable y (microscopic viewpoint), and after canceling out the meter units: divy (E grady u) = ε3 ρω 2 u and, in terms of the variable x (macroscopic viewpoint): divx (E gradx u) = ερω 2 u After introducing derivative operators for the double variables, we again reach the same conclusions about the equivalence of the different approaches.

3.4. Expressing problems within the formalism of multiple scales The above example shows the general approach to be taken in order to establish various behaviors depending on the assumptions made. However, when a macroscopic description is sought for a real phenomenon within a given material, one of the difﬁculties is that of expressing the assumptions within the formalism of homogenization, in accordance with the problem under investigation. In the previous example, the question would be the following: if a material (for example a soft rock) has the following characteristic values: lc = 1 mm, Ec = 8 × 109 Pa, ρc = 2 × 103 kg/m3 , and cycles of testing at a frequency of 3 kHz are performed on a lattice of size H = 10 cm, which of the models that we obtained is the appropriate one to use?

3.4.1. How do we select the correct mathematical formulation based on the problem at hand? The macroscopic description will only be valid if the physics at the microscopic scale is described correctly. The physical analysis of the problem is thus a crucial stage that must occur before the process of homogenization. We have seen that dimensional analysis is an extremely useful tool for carrying out this process correctly. The problem is expressed in dimensionless form and, in order to correctly account for the importance of each term, the dimensionless numbers are expressed in powers of ε. This normalization phase is a key point in the process because that is where the physics of the phenomenon is taken into account. We emphasize that normalizing the dimensionless numbers in terms of powers of ε ensures that the various physical effects are accounted for to the same order, independent of the value of ε 1. Thus when a description is normalized it retains the nature of the physics that applies to the situation, but does not contain any reference to the effective value of ε which, although small, is still not zero.

Homogenization by Multiple Scale Asymptotic Expansions

101

Nevertheless, it is rare that the normalization follows naturally from the problem under consideration. In particular, when several small parameters are involved (ratios of properties, characteristic times, etc.), several possibilities are available and one should be chosen which applies to the situation being examined. Examples include bituminous concretes, whose behavior varies strongly with temperature and frequency of the load [BOU 89b; BOU 90] (see Chapter 9), or cement pastes which change from a ﬂuid to solid state when they set [BOU 95]. We will show later that this difﬁculty can be overcome by analyzing the value taken by the scale ratio εr in the actual problem. This idea is clear in the previous example where the choice of model depends on the value of Pl as a function of ε. For the problem in question, with the numerical values given above, Pl can be estimated objectively: Pl =

ρc lc2 2 103 (10−3 )2 = 10−4 Ec t2c 8 109 (2π 3 103 )−2

However ε is not speciﬁed. We should also point out that if we assume ε to be inﬁnitely small, this is equivalent to considering Pl = O(1), which is a situation that cannot be homogenized! Also, considering an arbitrary value of ε to give a scale to Pl is equivalent to making an arbitrary choice in the constitutive model. To avoid this impasse we are therefore forced to come up with a realistic estimate of ε for the problem being considered. 3.4.2. Need to evaluate the actual scale ratio εr The difﬁculty here is the gulf between: – the mathematical view, where ε = lc /Lc → 0 is inﬁnitely small and the macroscopic description in this limit is inﬁnitely accurate, corresponding to heterogenities which are inﬁnitely small compared to the macroscopic scale, or alternatively to macroscopic dimensions which are inﬁnitely large compared to the heterogenities; – the physical reality where this ideal situation is not reached because the size of the REV is ﬁnite (lc = 0) and the macroscopic scale is not inﬁnite (Lc = ∞) so that the actual scale ratio takes a value which is small but non-zero (0 < εr 1). We can reconcile these two viewpoints by evaluating εr . Indeed if εr can be estimated, the dimensionless numbers of the real problem can be evaluated in terms of powers of εr . Thus we can deﬁne a normalization which is consistent with the physics of the problem. If, with this normalization, we carry out homogenization, we obtain a macroscopic description in which all the physical mechanisms act with the same strengths as in the actual problem. Because of this, the problem being considered is only an imperfect example (for ε = εr ) of the macroscopic description we have

102

Homogenization of Coupled Phenomena

developed, with the discrepancy being smaller when εr is small, in other words when the separation of scales is clear. In this case, the zero-order description matches the actual behavior up to order O(εr ). To summarize, there are two reasons we need to evaluate εr : the correct description of the local physics and the estimation of error in the macroscopic description.

3.4.3. Evaluation of the actual scale ratio εr For a given problem, lc is known for the medium, but the characteristic macroscopic size Lc which, as we have just seen, is crucial for selection of the correct model, is one of the unknowns. The literature is still rather unclear on this issue: this dimension is often associated with the size of the medium under study but, depending on the problem of interest, Lc might alternatively depend on the boundary conditions imposed or on a characteristic dimension of phenomenon such as a wavelength, or a thickness of a viscous layer, etc. In order to evaluate Lc (and εr ), we will follow the approach proposed by [BOU 89b; 89a] which consists of observing that the process of homogenization must necessarily lead to a quantity – in the case considered above, the displacement u(0) – with the following dimensional form: u(0) (X) + εu(1) (X, ε−1 X) + ...

with

O(u(0) ) = O(u(1) )

Turning the problem around, we can say that results of the homogenization will only be applicable to the real situation if this (necessary) condition is satisﬁed when ε takes the value εr . In other words if the variations in u(0) are effectively negligible (i.e. O(εr )) over the period. If we consider for example the growth of u(0) over a period in the direction X1 , we must therefore necessarily have: |u(0) (X1 + lc ) − u(0) (X1 )| O(εr |u(0) (X1 )|) On the macroscopic scale lc is very small and we can write: (0) ∂u (0) −1 |u | O(εr ) lc . ∂X1 This gives an underestimation of εr . However since εr is a measure of macroscopic accuracy, the optimum value is the smallest one that is permissible, which means we can write: (0)

ε r lc

| ∂u ∂X1 | |u(0) |

or

Lc

|u(0) | (0)

| ∂u ∂X1 |

(3.18)

Homogenization by Multiple Scale Asymptotic Expansions

103

In the general case where the displacement ﬁeld is three-dimensional, we have: ⎛ εr lc max ⎝

(0)

⎞

∂ui ∂Xj | ⎠ (0) |uj |

|

⎛

Lc

and

⎞

(0) |u | min ⎝ i(0) ⎠ ∂u | ∂Xi j |

Locally, an order of magnitude for εr is thus given by relative variation of the displacement ﬁeld over a period. This is equivalent to the estimate that would be obtained by dimensional analysis carried out directly on the macroscopic scale: the slower (or faster) the spatial variations in u(0) the larger (or smaller) Lc is, and the “smaller” (larger) εr is (in other words the accuracy is greater (or smaller)). We note that εr depends on the geometry of the ﬁeld, and because of this it is not generally constant in the material, but can vary depending on load, boundary conditions, etc. Our estimate of (3.18) can answer the questions of accuracy and validity of the zeroorder macroscopic description. We will now give an evaluation of εr in several familiar situations. 3.4.3.1. Homogenous treatment of simple compression The displacement in a sample of height H takes the form (Figure 3.9): u(0) = aX, (0) = a, so that: from which it follows that O(u(0) ) = aH and ∂u ∂X Lc =

|u(0) | (0) | ∂u ∂X |

=

aH =H a

and

εr =

lc H

It follows that an accuracy of order 10% for the constitutive law requires samples with dimensions which are around 10 times larger than the size of the heterogenities. u(0)

u(0)

Lc

Figure 3.9. Estimate of the physical scale ratio εr in the case of simple compression: εr = O(lc /Lc )

104

Homogenization of Coupled Phenomena

3.4.3.2. Point force in an elastic object This is a case where the value of εr is not constant in the medium. In fact the displacement ﬁeld varies as ∼ 1/r2 (Figure 3.10), which gives: Lc

|u(0) | (0) | ∂u | ∂X

=

r 2

and

εr = 2

lc r

P

r

Figure 3.10. Estimate of the physical scale ratio εr for a point source in a porous medium: εr = 2lc /r

where r is the distance to the point force. From this we can deduce that close to the point force is applied, the phenomenon is not homogenizable. Taking into account the effects of the microstructure, it becomes homogenizable beyond a radius R ≈ 10lc . The simple continuum description becomes acceptable at distances greater than 200lc (with an accuracy in the order of a few percent). 3.4.3.3. Propagation of a harmonic plane wave in elastic composites The displacement created by a plane wave in an inﬁnite medium (Figure 3.11) has the form: u(0) (X, t) = |u(0) | exp(2iπ(t/T − X/λ)) and consequently: ∂u(0) = −(2iπ/λ)|u(0) | exp(2iπ(t/T − X/λ)) ∂X

Homogenization by Multiple Scale Asymptotic Expansions

105

whence: Lc =

|u(0) | (0) | ∂u ∂X |

=

λ 2π

and

εr =

2πlc λ

Again we ﬁnd that the closer we get to the diffraction regime, the poorer the zeroorder description performs, so that we require higher order corrections [BOU 96b; AUR 05a]. For wavelengths shorter that 2πlc , homogenization is no longer applicable.

Lc u(0) r u(0) lc = r L c

Figure 3.11. Estimate of the physical scale ratio εr for wave propagation: εr = 2πlc /λ

3.4.3.4. Diffusion wave in heterogenous media For a harmonic plane wave of thermal diffusion, temperature takes the form: √ θ(0) (X, t) = |θ(0) | exp(2iπ(t/T − X i/δt )) where δ is the wavelength of thermal diffusion: λ δt = ρCω λ is thermal conductivity and ρC is heat capacity. As a result: √ √ ∂θ(0) = −(2 iπ/δt )|θ(0) | exp(2iπ(t/T − X i/δt )) ∂X so that: Lc =

|θ(0) | (0) | ∂θ ∂X |

=

δt 2π

and

εr =

2πlc δt

106

Homogenization of Coupled Phenomena

The same applies for diffusive waves as for elastic waves, and it is the wavelength which determines the macroscopic scale. 3.4.3.5. Conclusions to be drawn from the examples Let us return to the example of dynamic measurements of a rock sample. At a frequency of 3 kHz we have the following estimate of the wavelength: Ec 1 8.109 1 λ c = = 0.1m 2π ω ρc ω 2.103 2π3.103 which corresponds here to the height H of the sample. The value of εr is thus (with lc = 1 mm): εr =

2πlc lc = = 10−2 λ H

so that, following the value of Pl estimated above: Pl = 10−4 = ε2r It is thus natural that we should use the model corresponding to Pl = O(ε2 ), in other words the dynamic description. If, for the same material, tests were carried out at f = 300 Hz, we would have: P l =

ρc lc2 2 103 (10−3 )2 = 10−6 Ec t2c 8 109 (2π3 102 )−2

and

λ 1m > H 2π

In this case the macroscopic size is no longer deﬁned by the wavelength but by the dimensions of the sample, and this time we have: εr =

lc = 10−2 H

so that:

P l = 10−6 = ε3r

which leads us to use the model for Pl = O(ε3 ), in other words the quasi-static description. Conversely, tests carried out at 30 kHz give Pl 10−2 , λ/2π √ 1 cm < H, and εr = 10−1 , so that Pl = O( εr ), putting the tests in the dynamic regime, at the limit of what is homogenizable. As for the diffractive regime where homogenization is no longer applicable due to the absence of a separation of scales, this is reached at frequencies where (λ/2π) lc so that εr 1.

PART TWO

Heat and Mass Transfer

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Chapter 4

Heat Transfer in Composite Materials

4.1. Introduction The ﬁrst works which made it possible to determine the effective conductivity through upscaling techniques were those of Rayleigh [RAY 92], for materials with low concentrations of inclusions and with a weak thermal conductivity contrast. The study of thermal transfer in heterogenous media has been the subject of many investigations, an excellent summary of which can be found in the works of Kaviany [KAV 99; 01]. Here we focus on thermal transfer in a composite material with two constituents. In order to obtain an equivalent macroscopic description we will use the method of multiple scale expansions described in Part 1. In order to apply this technique, we will assume the medium to be periodic. (The main results were obtained by [AUR 83; LEW 04; 05; and AUR 94a].) Section 4.2 focuses on thermal transfer with perfect contact between the constituents. A contact resistance will be introduced in section 4.3. The theoretical developments are systematically illustrated in the case of a bilaminate composite.

4.2. Heat transfer with perfect contact between constituents In this section we assume perfect contact between the constituents. Section 4.2.2 covers composites with thermal properties of the same order of magnitude. Section 4.2.3 treats composite media consisting of a matrix with inclusions of low thermal conductivity. Finally, section 4.2.4 considers thermal transfer for composites consisting of highly conductive inclusions embedded in a matrix.

110

Homogenization of Coupled Phenomena

4.2.1. Formulation of the problem The composite material is periodic, with a period Ω. Material a occupies the volume Ωa , and material b occupies the volume Ωb , with their interface being Γ (Figure 4.1). We assume that |Ωa |/|Ωb | = O(1); and Ω, Ωa and Ωb have the same characteristic size lc . In the transient regime, the heat transfer equation in each of the constituents α = a, b can be written: divX (λα gradX Tα ) = ρα Cα

∂Tα ∂t

(4.1)

Lc

lc

n

b

Sb Sa

a (a)

(b)

Figure 4.1. Composite medium: (a) macroscopic sample, (b) representative elementary volume (REV) with period Ω

By way of simpliﬁcation, we assume the two conductivities to be isotropic and constant: λa = λa I and λb = λb I. We will also assume their densities ρα and their heat capacities Cα to be constant. In addition, it is convenient to consider heat transfer in the harmonic regime, with a temperature of the form: Tα (X)eiωt where the temperature Tα is a function of the spatial coordinate X only. When ω = 0, the heat transfer is permanent. Since the problems we will consider are all linear, the time dependence eiωt will be omitted throughout the following text. In general, heat transfer in the composite is governed by the Fourier law, which leads to the following system of equations: divX (λa gradX Ta ) = ρa Ca iωTa divX (λb gradX Tb ) = ρb Cb iωTb

in Ωa in Ωb

(4.2) (4.3)

Heat Transfer in Composite Materials

111

with continuity in temperature and normal ﬂux across Γ: T a − Tb = 0

over Γ

(4.4)

(λa gradX Ta − λb gradX Tb ) · n = 0

over Γ

(4.5)

where n is the outward unit normal to constituent a. We introduce into the system of equations (4.2-4.5), Ta = Tac Ta∗ ,

Tb = Tbc Tb∗ ,

ρa = ρac ρ∗a ,

λa = λac λ∗a ,

λb = λbc λ∗b ,

Ca = Cac Ca∗ ,

ρb = ρbc ρ∗b Cb = Cbc Cb∗

where the quantities with the subscript c and the superscript ∗ are respectively the characteristic and dimensionless magnitudes, and take the macroscopic view X = x∗ Lc . Then the dimensionless microscopic description can be written in the form: divx∗ (λ∗a gradx∗ Ta∗ ) = PL ρ∗a Ca∗ iω ∗ Ta∗

in Ω∗a

L divx∗ (λ∗b gradx∗ Tb∗ ) = PL C ρ∗b Cb∗ iω ∗ Tb∗

in Ω∗b

Ta∗ − Tb∗ = 0 over Γ∗ (λ∗a gradx∗ Ta∗ − Lλ∗b gradx∗ Tb∗ ) · n = 0

(4.6) (4.7) (4.8)

over Γ∗

(4.9)

The above system involves three dimensionless quantities: PL =

ρac Cac ωc L2c |ρa Ca iωTa | taL = = |divX (λa gradX Ta )| λac tc

L=

λbc λac

C=

ρbc Cbc ρac Cac

which characterize heat transfer in the composite. The quantity PL is the inverse of the Fourier number. It represents the ratio between the characteristic times taL and tc in material a. taL = ρac Cac L2c /λac is the characteristic time associated with conductive transfer phenomena in the sample with characteristic dimension Lc . tc is the characteristic time associated with the harmonic or transient regime. In order to simplify the treatment, we will limit the following analysis to constituents with heat capacities of the same order of magnitude, C = ρbc Cbc /ρac Cac = O(1).

112

Homogenization of Coupled Phenomena

We will study harmonic regimes where the effects of macroscopic conduction will equal the thermal inertia, so that PL = O(1). If PL = O(εp ), p > 0, the macroscopic behavior is permanent to ﬁrst order. If PL = O(εp ), p < 0, the regime is not homogenizable, because there is not enough of a separation in length scales between the wavelength of thermal diffusion and the size of constituents. A similar situation has already been analyzed in section 3.3.2.3 for the example of dynamic behavior of elastic composites. The interesting cases are the following: – Composite materials, without any restriction on connectivity of the constituents, and whose conductivities are of the same order of magnitude L = O(1) (Figure 4.2 (a) and (b), section 4.2.2). In this case, we recover the classical heat transfer equation [SAN 80; AUR 83]. It is simple to show that L = O(ε) is a particular case of the classic example with negligible λb . – Composite materials with high conductivity contrast, with the strongly conducting b phase embedded in a matrix formed by the a phase, in other words L = O(ε2 ) (Figure 4.2 (a) and (b), section 4.2.3). We obtain a non-standard description with memory effects. This means that transfer time in the macroscopic structure is of the same order of magnitude as the transfer time in weakly conducting inclusions [AUR 83]. – Composite materials with high conductivity contrast, with the strongly conducting b phase embedded in a matrix formed by the a phase, in other words L = O(ε−1 ) (Figure 4.2 (b), section 4.2.4). We once again recover a classical heat transfer equation, with the effective conductivity arising from local non-standard effects along the lines of those treated by Levy [LEV 90] for the ﬂow of a Newtonian ﬂuid in a porous medium; and Lewandowska and Auriault [LEW 04] and Lewandowska et al. [LEW 05] for non-linear diffusion problems.

lc

n

lc

b

Sb

n

Sa

b b n

Sa a a

a (a)

(b)

Figure 4.2. Unit cell of a composite material: (a) the two phases a and b are embedded, (b) inclusions buried in the matrix a. Phase a is connected, phase b is non-connected

Heat Transfer in Composite Materials

113

4.2.2. Thermal conductivities of the same order of magnitude We assume PL = O(1) and L = O(1). Media a and b are either connected or non-connected (Figure 4.2). The thermal transfer time in Ωa and Ωb are of the same order of magnitude. The dimensionless description (4.6-4.9) can be written as: divx∗ (λ∗a gradx∗ Ta∗ ) = ρ∗a Ca∗ iω ∗ Ta∗

in Ω∗a

(4.10)

divx∗ (λ∗b gradx∗ Tb∗ ) = ρ∗b Cb∗ iω ∗ Tb∗

in Ω∗b

(4.11)

Ta∗ − Tb∗ = 0 over Γ∗

(4.12)

(λ∗a gradx∗ Ta∗ · n − λ∗b gradx∗ Tb∗ ) · n = 0

over Γ∗

(4.13)

We look for solutions to the unknowns Ta∗ and Tb∗ of the form: Ta∗ (x∗ , y∗ ) = Ta∗(0) (x∗ , y∗ , ) + εTa∗(1) (x∗ , y∗ ) + · · · ∗(0)

Tb∗ (x∗ , y∗ ) = Tb

∗(1)

(x∗ , y∗ ) + εTb

(x∗ , y∗ ) + · · ·

∗(i)

where Tα are Ω∗ -periodic in y∗ and with y∗ = ε−1 x∗ . The technique involves the introduction of these expansions into the dimensionless equations (4.10-4.13) and the identiﬁcation of the powers of ε. We recall that, because of the two spatial variables and adoption of the macroscopic point of view, the spatial derivation takes the following form: gradx∗ −→ gradx∗ + ε−1 grady∗ Also, henceforth all un-subscripted variables deﬁned for Ω∗ will take values subscripted by a in Ωa and by b in Ωb . For example, the conductivity λ∗ will be given as λ∗a in Ω∗a and λ∗b in Ω∗b . 4.2.2.1. Homogenization ∗(0)

4.2.2.1.1. Boundary value problem for Ta

∗(0)

and Tb

Equations (4.10-4.11) of order ε−2 , (4.12) of order ε0 and (4.13) of order ε−1 give ∗(0) ∗(0) the boundary value problem required by the Ω∗ -periodic ﬁelds Ta and Tb : divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)

divy∗ (λ∗b grady∗ Tb

in Ω∗a

(4.14)

) = 0 in Ω∗b

(4.15)

114

Homogenization of Coupled Phenomena ∗(0)

Ta∗(0) − Tb

over Γ∗

=0

(4.16) ∗(0)

(λ∗a grad∗y∗ Ta∗(0) − λ∗b grad∗y∗ Tb

)·n=0

over Γ∗

(4.17)

This is a system of differential equations in terms of the variables y∗ , where x∗ is only present as a parameter. We note that T ∗(0) is only deﬁned up to an additive constant, which is independent of y∗ . In these terms, the issue in the unit cell is a steady-state conduction problem without source terms. An obvious solution is a constant temperature ﬁeld. Over a given period: ∗(0)

Ta∗(0) (x∗ , y∗ ) = Tb

(x∗ , y∗ ) = T ∗(0) (x∗ )

To formalize the solution to this problem, as well as subsequent ones, we introduce the space V of regular functions θ deﬁned over Ω∗ and Ω∗ -periodic, satisfying the condition of having zero mean, a condition which has no bearing on the problem at hand: 1 θ dΩ∗ = 0 (4.18) |Ω∗ | Ω∗ that has been introduced to solve the issue of the indeterminate nature of T ∗(0) . The bilinear, symmetrical positive deﬁnitive form: (θ1 , θ2 )V =

Ω∗

λ∗ grady∗ θ1 · grady∗ θ2 dΩ∗

(4.19)

deﬁnes a scalar product over V. Multiplying (4.14) and (4.15) by θ ∈ V, integrating by parts over each constituent, we ﬁnd: λ∗a grady∗ Ta∗(0) · grady∗ θ dΩ∗ − λ∗a grady∗ Ta∗(0) θ · na dS ∗ + Ω∗ a

∂Ω∗ a

∗(0)

Ω∗ b

λ∗b grady∗ Tb

· grady∗ θ dΩ∗ −

∗(0)

∂Ω∗ b

λ∗b grady∗ Tb

θ · nb dS ∗ = 0

where na and nb are the outward normals to Ω∗a and Ω∗b . Decomposing the surface integrals into integrals over the interface Γ∗ and over the cell boundaries Sa∗ = ∂Ω∗a ∩ ∂Ω∗ and Sb∗ = ∂Ω∗b ∩ ∂Ω∗ , we have: ∂Ω∗ a

λ∗a grady∗ Ta∗(0) θ

∗

· na dS +

∂Ω∗ b

∗(0)

λ∗b grady∗ Tb

θ · nb dS ∗ =

Heat Transfer in Composite Materials

Sa∗

Sb∗

λ∗a grady∗ Ta∗(0) θ · na dS ∗ +

∗(0) λ∗b grady∗ Tb θ

∗

· nb dS +

Γ∗

λ∗a grady∗ Ta∗(0) θ · na dS ∗ +

∗(0)

Γ∗

115

λ∗b grady∗ Tb

θ · nb dS ∗

Also, over Γ∗ (Figure 4.2), we have na = −nb = n, and using the condition of ﬂux continuity (4.17), the sum of the two integrals over Γ∗ cancel out. As for the integrals over Sa∗ and Sb∗ , they come to zero because of their periodicity. Thus, the equivalent variational formulation of the initial problem over space V can be written, ∀θ ∈ V: Ω∗ a

λ∗a grady∗ Ta∗(0)

∗

· grady∗ θ dΩ +

Ω∗ b

∗(0)

λ∗b grady∗ Tb

· grady∗ θ dΩ∗ = 0

Alternatively, in a more compact form: ∀θ ∈ V,

(T

∗(0)

, θ)V =

Ω∗

λ∗ grady∗ T ∗(0) · grady∗ θ dΩ∗ = 0

(4.20)

Making use of the Lax-Milgram theorem, this formulation ensures the existence and uniqueness of solution T ∗(0) which belongs to V. Here it is clear that this solution is exactly zero. The solution to the initial problem can be obtained by adding a constant ﬁeld which is independent of y∗ . After all this, we obtain T ∗(0) in the form: ∗(0)

Ta∗(0) (x∗ , y∗ ) = Tb

(x∗ , y∗ ) = T ∗(0) (x∗ )

As a result of this, to ﬁrst order the temperature, independent of y∗ , is therefore constant over the period. 4.2.2.1.2. Boundary value problem for T ∗(1) and the canonical problem P(λ∗a , λ∗b ) Equations (4.10-4.11) of order ε−1 , (4.8) of order ε and (4.9) of order ε0 lead to ∗(1) ∗(1) the following problem over the period Ω∗ for Ta and Tb , which are Ω∗ -periodic: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = 0 in Ω∗a ∗(1)

divy∗ (λ∗b (grady∗ Tb ∗(1)

Ta∗(1) − Tb

=0

+ gradx∗ T ∗(0) )) = 0

over Γ∗

in Ω∗b

(4.21) (4.22) (4.23)

116

Homogenization of Coupled Phenomena ∗(1)

(λ∗a (grady∗ Ta

+ gradx∗ T ∗(0) ) − ∗(1)

λ∗b (grady∗ Tb

+ gradx∗ T ∗(0) )) · n = 0

over Γ∗

(4.24)

This is a steady-state conduction problem, but different to the preceding problem in that there is a forcing term gradx∗ T ∗(0) . In order to solve this problem, we follow the same approach as before. Multiplying (4.21) and (4.22) by θ ∈ V, and integrating by parts and using the conditions on Γ∗ and the periodicity, the equivalent variational formulation to the initial problem over space V can be written: ∀θ ∈ V

(T ∗(1) , θ)V =

Ω∗

λ∗ grady∗ T ∗(1) · grady∗ θ dΩ∗ =

−

Ω∗

λ∗ gradx∗ T ∗(0) · grady∗ θ dΩ∗

(4.25)

In the same way as before, this formulation ensures the existence and uniqueness of the solution in V. The linearity of the problem makes it possible to decompose the solution into three elementary solutions t∗i (y∗ ) associated with unit macroscopic gradients in the three directions gradx∗ T ∗(0) = ei . In this way, the solution in V can be written t∗i (y∗ )∂T ∗(0) /∂x∗i . The solution to the initial problem can be obtained by adding a constant ﬁeld T¯ ∗(1) which is independent of y∗ . We arrive at T ∗(1) in the form: T ∗(1) (x∗ , y∗ ) = t∗i (y∗ )

∂T ∗(0) + T¯ ∗(1) (x∗ ) ∂x∗i

The speciﬁc solutions t∗i are obtained by solving three distinct boundary value problems over the period, which form the canonical problem P(λ∗a , λ∗b ): ∗ ∂ ∗ ∂tai (λ ( + Iij )) = 0 in Ω∗a a ∂yj∗ ∂yj∗

(4.26)

∗ ∂ ∗ ∂tbi + Iij )) = 0 ∗ (λb ( ∂yj ∂yj∗

(4.27)

t∗ai − tbi = 0 (λ∗a (

in Ω∗b

over Γ∗

∗ ∂t∗bi ∗ ∂tbi + I ) − λ ( + Iij )).nj = 0 ij b ∂yj∗ ∂yj∗

(4.28) over Γ∗

(4.29)

Heat Transfer in Composite Materials

1 |Ω∗ |

Ω∗ a

t∗ai dΩ∗

+

117

Ω∗ b

t∗bi dΩ∗

=0

(4.30)

For convenience, we combine the three solutions into a vector t∗ (y∗ ), writing: T ∗(1) (x∗ , y∗ ) = t∗ (y∗ ) · gradx∗ T ∗(0) + T¯ ∗(1) (x∗ )

(4.31)

and with the help of vector t∗ the three problems can be written in the following compact form: divy∗ (λ∗ (grady∗ t∗ + I)) = 0 in Ω∗ [t∗ ]Γ∗ = 0 over Γ∗ [λ∗ (grady∗ t∗ + I)]Γ∗ · n = 0

over Γ∗

t∗ = 0 where [φ∗ ]Γ∗ represents the step change in φ∗ at the interface Γ∗ and t∗ is deﬁned by equation (4.30). 4.2.2.2. Macroscopic model Equations (4.10-4.11) of order ε0 , (4.12) of order ε2 and (4.13) of order ε lead us ∗(2) to the following boundary valve problem over the period of Ω∗ -periodic ﬁelds Ta ∗(2) and Ta : divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = ρ∗a Ca∗ iω ∗ T ∗(0) ∗(2)

divy∗ (λ∗b (grady∗ Tb

∗(1)

divx∗ (λ∗b (grady∗ Tb ∗(2)

Ta∗(2) − Tb

=0

∗(1)

+ gradx∗ Tb

in Ω∗a

(4.32)

in Ω∗b

(4.33)

))+

+ gradx∗ T ∗(0) )) = ρ∗b Cb∗ iω ∗ T ∗(0)

over Γ∗

(λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) )− ∗(2)

λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

)) · n = 0

over Γ∗

118

Homogenization of Coupled Phenomena

The existence of a periodic solution imposes a compatibility condition which can be establised by integrating (4.32) and (4.33) over Ω∗a and Ω∗b . First we note that: Ω∗ a

∗(2)

divy∗ (λ∗a (grady∗ Ta

Γ∗

since: Sa∗

∗(1)

+ gradx∗ Ta

))dΩ∗ =

∗(2)

λ∗a (grady∗ Ta

∗(1)

+ gradx∗ Ta

) · ndS ∗

(4.34)

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · ndS ∗ = 0

due to the periodicity of the ﬂuxes. Similarly, since n is the outward normal to Ω∗a , we have: ∗(2) ∗(1) divy∗ (λ∗b (grady∗ Tb + gradx∗ Tb ))dΩ∗ = Ω∗ b

−

∗(2)

Γ∗

λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

) · ndS ∗

(4.35)

As a result, using the continuity of the ﬂux over Γ∗ , the sum of integrals (4.34) and (4.35) is zero. We also note that: grady∗ T ∗(1) + gradx∗ T ∗(0) = (grady∗ t∗ + I)gradx∗ T ∗(0) From this it follows, swapping the integration with respect to y∗ and derivation with respect to x∗ , that: Ω∗

divx∗ (λ∗ (grady∗ T ∗(1) + gradx∗ T ∗(0) ))dΩ∗ =

divx∗

(

Ω∗

λ∗ (grady∗ t∗ + I)dΩ∗ )gradx∗ T ∗(0)

All this leads us to the following macroscopic description: divx∗ (λeﬀ∗ gradx∗ T ∗(0) ) = (ρ∗ C ∗ )eﬀ iω ∗ T ∗(0)

(4.36)

where: λeﬀ∗ ij

1 = ∗ |Ω |

∂t∗ λ∗a (Iij + ai∗ ) dΩ∗ + ∂yj Ω∗ a

∂t∗ λ∗b (Iij + bi∗ ) dΩ∗ ∂yj Ω∗ b

(4.37)

Heat Transfer in Composite Materials

∗

∗ eﬀ

(ρ C )

1 = ρ C = ∗ |Ω | ∗

∗

Ω∗ a

ρ∗a Ca∗

∗

dΩ +

Ω∗ b

119

ρ∗b Cb∗

dΩ

∗

(4.38)

The variational form (4.25) associated with the symmetry and positivity of the scalar product makes it possible to show the symmetry and positivity of the effective thermal condictivity λeﬀ [AUR 83]. For an outline of this proof, which we will not discuss here, the reader is referred to Chapter 7, section 7.3.3 and Chapter 11, section 11.1.3.2, where the same arguments are used to show the symmetry and positivity of the permeability and elasticity tensors. Returning to the dimensional variables, the macroscopic model can be written: divX (λeﬀ gradX T (0) ) = ρC

∂T (0) ∂t

(4.39)

¯ But T = T (0) + O(ε) and as a result, for the physical variable T , the model can be written: divX (λeﬀ gradX T ) = ρC

∂T ¯ + O(ε) ∂t

(4.40)

¯ where O(ε) is a small term, of order ε relative to the other terms in the equality. The order 0 model is accurate up to ε. A result of the separation of length scales is that the presence of transient effects does not have any effect on the effective coefﬁcients. Also, the form of the result that we have established for bi-composites can be applied to all types of composite where properties of the constituents are of the same order. 4.2.2.3. Example: bilaminate composite Using this method we will return to the case of the bilaminate composite that we considered in Chapter 1 (cf. Figure 4.3). The speciﬁc solutions t∗i are obtained by solving the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) over the period Ω∗ . The calculation is performed using dimensionless variables. The stratiﬁcation perpendicular to e2 requires that the solutions t∗i are functions of y2∗ only. If we impose a macroscopic gradient, gradx∗ T ∗(0) = e1 , the problem can be written for t∗a1 and t∗b1 as: dt∗ d (λ∗a a1 )=0 ∗ dy2 dy2∗

in Ω∗a

d dt∗ (λ∗b b1∗ ) = 0 in Ω∗b ∗ dy2 dy2 t∗a1 − t∗b1 = 0

over Γ∗

120

Homogenization of Coupled Phenomena

e2 L2 l2

e1

(a)

L1 e2

1 (1 cb )

a

cb (b)

b

e1

Figure 4.3. Bilaminate composite material: (a) macroscopic structure, (b) dimensionless period Ω∗

λ∗a

dt∗a1 dt∗ − λ∗b b1∗ = 0, ∗ dy2 dy2

over Γ∗

The obvious solution to this is: t∗a1 (y2∗ ) = t∗b1 (y2∗ ) = 0 In the same way, with a macroscopic gradient along e3 : t∗a3 (y2∗ ) = t∗b3 (y2∗ ) = 0 If we impose a macroscopic gradient, gradx∗ T ∗(0) = e2 , the problem can be written for t∗a2 and t∗b2 as: dt∗ d (λ∗a a2 + 1) = 0 in Ω∗a ∗ dy2 dy2∗ ∗ d ∗ dtb2 (λ + 1) = 0 b dy2∗ dy2∗

t∗a2 − t∗b2 = 0

over Γ∗

in Ω∗b

Heat Transfer in Composite Materials

λ∗a (

dt∗a2 dt∗ + 1) − λ∗b ( b1∗ + 1) = 0, ∗ dy2 dy2

121

over Γ∗

The solution to this satisfying the condition t∗a2 + t∗b2 = 0 (see equation 4.30) is:

t∗a2 =

cb (λ∗b − λ∗a ) cb λ∗a + (1 − cb )λ∗b

t∗b2 =

(cb − 1)(λ∗a − λ∗b ) ∗ cb (y − ) cb λ∗a + (1 − cb )λ∗b 2 2

y2∗ −

(1 + cb ) 2

in Ω∗a

in Ω∗b

Substituting these expressions into equation (4.82), we obtain the effective conductivity tensor in dimensionless form, which is consistent with the results of Chapter 1: eﬀ eﬀ λeﬀ 12 = λ13 = λ23 = 0 eﬀ λeﬀ 11 = λ33 = cb λb + (1 − cb )λa

λeﬀ 22 =

λb λa cb λa + (1 − cb )λb

4.2.3. Weakly conducting phase in a connected matrix: memory effects We still assume that PL = O(1), but we now consider a strong conductivity contrast: L = (λbc /λac ) = O(ε2 ). In addition, medium a is always connected, but medium b may be connected or non-connected (Figure 4.3). With these assumptions, we have: ρbc Cbc lc2 ρac Cac L2c =O λac λbc This indicates that thermal transfer time in the macroscopic structure is of the same order of magnitude as the transfer time in Ωb . Bearing in mind these estimates, the description in (4.6-4.9) becomes: div∗x∗ (λ∗a grad∗x∗ Ta∗ ) = ρ∗a Ca∗ iω ∗ Ta∗

in Ω∗a

ε2 div∗x∗ (λ∗b grad∗x∗ Tb∗ ) = ρ∗b Cb∗ iω ∗ Tb∗

in Ω∗b

Ta∗ − Tb∗ = 0 over Γ∗ (λ∗a grad∗x∗ Ta∗ − ε2 λ∗b grad∗x∗ Tb∗ ) · n = 0

(4.41) (4.42) (4.43)

over Γ∗

(4.44)

122

Homogenization of Coupled Phenomena

4.2.3.1. Homogenization ∗(0)

4.2.3.1.1. Boundary value problem for Ta

∗(0)

and Tb

Substituting the asymptotic expansions into the equations (4.41-4.44), equations (4.41) of order ε−2 and (4.44) of order ε−1 make up the boundary value problem for ∗(0) Ta (Ω∗ -periodic): divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 λ∗a grady∗ Ta∗(0) · n = 0

in Ω∗a

(4.45)

over Γ∗

which has an obvious solution of the form: Ta∗(0) (x∗ , y∗ ) = Ta∗(0) (x∗ ) This is obtained in the same way as in the previous section, with functions in space V now deﬁned over Ω∗a (c.f. equations (4.19) and (4.18)). ∗(0)

The boundary value problem for Tb (4.42) and (4.43) of order ε0 : ∗(0)

divy∗ (λ∗b grady∗ Tb ∗(0)

Tb

= Ta∗(0)

(Ω∗ -periodic) is obtained from equations

∗(0)

) = ρ∗b Cb∗ iω ∗ Tb

in Ω∗b

(4.46)

over Γ∗

∗(0)

∗(0)

where Ta (x∗ ) is known. Setting Tb problem becomes:

∗(0)

= Ta

divy∗ (λ∗b grady∗ W ) = ρ∗b Cb∗ iω ∗ (Ta∗(0) + W ),

(x∗ ) + W , the boundary value in Ω∗b

W = 0 over Γ∗

(4.47) (4.48)

This problem is very different to the preceding case, because it involves both transient effects and a source term resulting from the temperature imposed at the boundary. To solve this problem, we introduce W, the space of complex θ-regular functions, deﬁned over Ω∗b , Ω∗ -periodic, zero over Γ∗ , and with the Hermitian product: (θ1 , θ2 )W =

Ω∗ b

(grady∗ θ1 · grady∗ θ˜2 + θ1 θ˜2 ) dΩ∗

Heat Transfer in Composite Materials

123

where θ˜ is the complex conjugate of θ. Multiplying (4.47) by θ˜ ∈ W, integrating by parts and making use of the condition on Γ∗ as well as the periodicity, we reach the equivalent formulation: ˜ dΩ∗ = ∀θ ∈ W, (λ∗b grady∗ W · grady∗ θ˜ + ρ∗b Cb∗ iω ∗ W θ) Ω∗

(W, θ)W = −

Ω∗

ρ∗b Cb∗ iω ∗ Ta∗(0) θ˜ dΩ∗

(4.49)

This formulation implies the existence and uniqueness of the solution. The last ∗(0) part depends linearly on the forcing term associated with Ta : ∗(0)

W (x∗ , y∗ ) = Tb

(x∗ , y∗ ) − Ta∗(0) (x∗ ) = −τ ∗ (y∗ , ω∗ )Ta∗(0) (x∗ ) ∗(0)

= 1. This is a complex function which where τ ∗ is the speciﬁc solution for Ta depends on ω ∗ . It is clear that τ ∗ (ω ∗ = 0) = 0 and τ ∗ (ω ∗ → ∞) = 1. To ﬁrst order, the temperature is constant in the matrix, but not in the inclusions. ∗(1)

4.2.3.1.2. Boundary value problem for Ta

Equations (4.41) of order ε−1 and (4.44) of order ε0 make up the boundary value ∗(1) problem for the Ω∗ -periodic Ta : divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = 0 in Ω∗a λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n = 0

(4.50)

over Γ∗

∗(0)

where gradx∗ Ta appears as a forcing term. This system can be studied using the space V associated with Ω∗a . Once again the linearity allows us to write the solution in the form: Ta∗(1) (x∗ , y∗ ) = m∗a (y∗ ) · gradx∗ Ta∗(0) + T¯ ∗(1) (x∗ ) where T¯ ∗(1) is an arbitrary function of x∗ , m∗a (y∗ ) is y∗ -periodic and has a mean of zero in Ω∗a . The speciﬁc solutions m∗ai are solutions to three distinct problems associated with the unit macroscopic temperature gradients gradx∗ T ∗(0) = ei : ∗ ∂ ∗ ∂mai ∗ ∗ (λa ( ∗ + Iij )) = 0 in Ωa ∂yj ∂yj

λ∗a (

∂m∗ai + Iij ).nj = 0 ∂yj∗

over Γ∗

(4.51)

(4.52)

124

Homogenization of Coupled Phenomena

1 Ω∗

Ω∗ a

m∗ai dΩ∗ = 0

(4.53)

which can be written in the following form: divy∗ (λ∗a (grady∗ m∗a + I)) = 0 in Ω∗a λ∗a (grady∗ m∗a + I) · n = 0

over Γ∗

m∗a = 0 We note that we have recovered the canonical problem P(λ∗a , λ∗b ) (4.26-4.30), where here we have λ∗b = 0, as a result of the assumption we have made about the contrast. 4.2.3.2. Macroscopic model ∗(2)

The boundary value problem for Ω∗ -periodic Ta of order ε0 and (4.9) of order ε:

is obtained from equations (4.6)

divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = ρ∗a Ca∗ iω ∗ Ta∗(0)

(4.54)

with continuity of normal ﬂux across Γ∗ : ∗(0)

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n = λ∗b grady∗ Tb

·n

(4.55)

∗(2)

The existence of a periodic solution Ta requires a compatibility condition which we establish by integrating (4.54) over Ω∗a . Using in succession the divergence theorem, conditions (4.55) and (4.46), we ﬁnd: divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) )) dΩ∗ Ω∗ a

=

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n dS ∗

Γ∗

=

∗(0)

λ∗b grady∗ Tb

Γ∗

=−

Ω∗ b

· n dS ∗ = −

∗(0) ρ∗b Cb∗ iω ∗ Tb

∗

dΩ = −

∗(0)

Ω∗ b

Ω∗ b

divy∗ (λ∗b grady∗ Tb

) dΩ∗

ρ∗b Cb∗ iω ∗ (1 − τ ∗ )Ta∗(0) dΩ∗

Heat Transfer in Composite Materials

125

The volume average of (4.54) can then be written: ∗(0) divx∗ (λeﬀ∗ ) = (ρC)eﬀ∗ iω ∗ Ta∗(0) a gradx∗ Ta

(4.56)

which gives the macroscopic description, with the following effective coefﬁcients: λeﬀ∗ aij

1 = ∗ |Ω |

Ω∗ a

λ∗a (Iij +

(ρC)eﬀ∗ = ρ∗ C ∗ −

1 |Ω∗ |

∂m∗aj ) dΩ∗ ∂yi∗ Ω∗ b

(4.57)

ρ∗b Cb∗ τ ∗ dΩ∗

(4.58)

Equation (4.56) represents the macroscopic behavior of the composite at constant ∗(0) frequency, to ﬁrst order. In this equation, Ta represents the temperature in medium eﬀ∗ ∗(0) the macroscopic heat ﬂux. We note that λeﬀ∗ is identical a and λa gradx∗ Ta a to that of a composite where material b is a perfect insulator and that the effective speciﬁc heat capacity is complex and depends on the frequency. This unusual property results from the local non-equilibrium in the inclusions. This follows from the similar transfer times in the inclusions and in the macroscopic volume, and leads to a memory effect. We note that in the steady-state regime the effective speciﬁc heat capacity is the mean of that of the two constituents, and that at high frequency, only the speciﬁc heat capacity of material a has an effect on the macroscopic scale. We write macroscopic model (4.56) in dimensional form and in the time domain:

(0)

(0) divX (λeﬀ a gradX Ta ) = ρC

∂Ta ∂t

− ρb Cb

t

−∞

M (t − t )

where the memory function M (t) is the inverse Fourier transform of The properties of M can be found in Auriault [AUR 83].

(0)

∂ 2 Ta ∂t2

dt (4.59)

1 iω|Ω|

τ dΩ. Ωb

4.2.3.3. Example: bilaminate composite Let us return to the composite depicted in Figure 4.3. Medium a (the more conductive) is connected in directions e1 and e3 . The results derived above can therefore be applied to thermal ﬂuxes in these directions. In this speciﬁc case, the solution to the boundary-value problem (4.51-4.53) over Ω∗ means that m∗a = 0. As a result, using equation (4.57), the effective dimensional thermal conductivity becomes: eﬀ eﬀ λeﬀ a12 = λa13 = λa23 = 0 eﬀ λeﬀ a11 = λa33 = (1 − cb )λa

126

Homogenization of Coupled Phenomena

λeﬀ a22 = 0 The function τ (y2∗ ) can be determined from the following system of equations: d ∗ dτ ) = ρ∗b Cb∗ iω ∗ (1 + τ ), ∗ (λb dy2 dy2∗ τ =0

at

y2∗ = 0 and

in 0 < y2∗ < cb

y2∗ = cb

After integration and return to dimensional variables: 1 |Ω|

τ dΩ = cb Ωb

tanh(i1/2 β) 1− , i1/2 β

β=

ωρb Cb λb

1/2

cb h 2

The function M can be found by inverse Fourier transform: M (t) = 8cb

2 2 ∞ e−(2p+1) π α/4

p=0

(2p +

1)2 π 2

,

α=

4λb t ρb Cb c2b h2

4.2.4. Composites with highly conductive inclusions embedded in a matrix We still assume PL = O(1), and also a high conductivity contrast. Compared to the preceding situation, the more conductive phase is embedded in the matrix, so that L = (λac /λbc ) = O(ε−1 ). Medium a is therefore connected and b is not (Figure 4.4). Here we look at a speciﬁc traditional case considered in section 4.2.2. However λb λa and the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) with λ∗b −→ ∞ is singular. The method of asymptotic expansions makes it possible to remove that singularity. A similar treatment can be found in [LEW 04] and [LEW 05] for solute diffusion. With these assumptions, the description (4.6-4.9) becomes: divx∗ (λ∗a gradx∗ Ta∗ ) = ρ∗a Ca∗ iω ∗ Ta∗ divx∗ (λ∗b gradx∗ Tb∗ ) = ερ∗b Cb∗ iω ∗ Tb∗

in Ω∗a in Ω∗b

Ta∗ − Tb∗ = 0 over Γ∗ (ελ∗a gradx∗ Ta∗ − λ∗b gradx∗ Tb∗ ) · n = 0

over Γ∗

Heat Transfer in Composite Materials

e2

127

S a2

l2

n

S a1

b b n a a

e1 l1

Figure 4.4. Period Ω∗ of a composite material consisting of inclusions Ω∗b embedded in a matrix Ω∗a

4.2.4.1. Homogenization ∗(0)

4.2.4.1.1. Boundary value problems for Ta

∗(0)

and Tb

∗(0)

The boundary value problems for Ω∗ -periodic Tb ∗(0)

divy∗ (λ∗b grady∗ Tb ∗(0)

λ∗b grady∗ Tb

are:

) = 0 in Ω∗b

·n=0

over Γ∗

The solution has the form: ∗(0)

Tb

∗(0)

(x∗ , y∗ ) = Tb

(x∗ )

It can be obtained as before using space V associated with Ω∗b (see equations (4.19) ∗(0) and (4.18)). The boundary value problem for Ω∗ -periodic Ta can be written: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)

Ta∗(0) = Tb

(x∗ )

in Ω∗a

over Γ∗

This is a conduction problem without source terms, where the temperature imposed at the boundary of the inclusions is uniform. The obvious solution is: ∗(0)

Ta∗(0) (x∗ , y∗ ) = Tb

(x∗ ) = T ∗(0) (x∗ )

To ﬁrst order, the temperature is constant across the whole period.

128

Homogenization of Coupled Phenomena ∗(1)

4.2.4.1.2. Boundary value problem for Tb ∗(1)

The Ω∗ -periodic temperature Tb ∗(1)

divy∗ (λ∗b (grady∗ Tb ∗(1)

λ∗b (grady∗ Tb

satisﬁes:

+ gradx∗ T ∗(0) )) = 0

+ gradx∗ T ∗(0) ) · n = 0

in Ω∗b

(4.60)

over Γ∗

where gradx∗ T ∗(0) appears as a forcing term. Multiplying the two terms from (4.60) ∗(1) by (y · gradx∗ T ∗(0) + Tb ), integrating over Ω∗b , then by parts, and using the ∗ condition over Γ , we ﬁnd: ∗(1) λ∗b (grady∗ Tb + gradx∗ T ∗(0) )2 dΩ∗ = 0 Ω∗ b

∗(1)

The positivity of λ∗b implies that grady∗ Tb ∗(1)

Tb

+ gradx∗ T ∗(0) = 0, and then:

(x∗ , y∗ ) = −y∗ · gradx∗ T ∗(0) + T¯∗(1) (x∗ )

(4.61)

where T¯ ∗(1) is an arbitrary function of x∗ . Thus we obtain an explicit solution for ∗(1) Tb . We note that the gradient is order 0 in temperature: ∗(1)

grady∗ Tb

+ gradx∗ T ∗(0) = 0

as a consequence of our assumption of a high conductivity in the inclusion. ∗(1)

4.2.4.1.3. Boundary value problem for Ta ∗(1)

The Ω∗ -periodic temperature Ta

satisﬁes:

divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = 0 in Ω∗a ∗(1)

Ta∗(1) = Tb

= −y∗ · gradx∗ T ∗(0) + T¯∗(1) (x∗ )

(4.62)

over Γ∗

where gradx∗ T ∗(0) appears as a forcing term. T¯∗(1) (x∗ ) can also be considered as ∗(1) a forcing term, and the obvious solution associated with it is Ta = T¯ ∗(1) (x∗ ). The above boundary-value problem is non-singular and the solution can be written in the form: Ta∗(1) (x∗ , y∗ ) = s∗a (y∗ ) · gradx∗ T ∗(0) + T¯ ∗(1) (x∗ ) The vector s∗a (y∗ ) is the solution to the following boundary-value problem: divy∗ (λ∗a (grady∗ s∗a + I)) = 0 s∗a + y∗ = 0 over Γ∗

in Ω∗a

(4.63) (4.64)

Heat Transfer in Composite Materials

129

4.2.4.2. Macroscopic model ∗(2)

Let us consider the equations satisﬁed by Ta

∗(3)

and Tb

:

divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = ρ∗a Ca∗ iω ∗ T ∗(0) ∗(3)

∗(2)

divy∗ (λ∗b (grady∗ Tb

+ gradx∗ Tb

∗(2)

divx∗ (λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

in Ω∗a

(4.65)

in Ω∗b

(4.66)

)+ )) = ρ∗b Cb∗ iω ∗ T ∗(0)

with the condition of normal ﬂux continuity over Γ∗ : ∗(3)

λ∗b (grady∗ Tb

∗(2)

+ gradx∗ Tb

)·n=

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n

over Γ∗

(4.67)

We integrate (4.65) and (4.66) over Ω∗a and Ω∗b respectively. After applying the divergence theorem, using (4.67) and dividing by |Ω∗ |, we ﬁnd: 1 ∗ ∗(1) ∗(0) ∗ λa (grady∗ Ta + gradx∗ T ) dΩ + divx∗ |Ω∗ | Ω∗ a 1 divx∗ |Ω∗ |

Ω∗ b

∗(2) λ∗b (grady∗ Tb

+

∗(1) gradx∗ Tb )

dΩ∗

ρ∗ C ∗ iω ∗ T ∗(0)

=

(4.68) ∗(2)

We cannot make direct use of the macroscopic behavior in (4.68) because Tb is not known. We therefore need to to transform (4.68). The heat ﬂuxes in each constituent can be written: q∗a = λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) ∗(2)

q∗b = λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

)

They satisfy: ∗ = qαk

∗ ) ∂(yk∗ qαi ∗ ∂yi

where

q∗α = divy∗ (y∗ ⊗ q∗α )

130

Homogenization of Coupled Phenomena

Thus the integrals in (4.68) can be written, after use of the divergence theorem, as: ∗ ∗ ∗ ∗ ∗ qa dΩ = (y ⊗ qa ) · ndS + (y∗ ⊗ q∗a ) · ndS ∗ Ω∗ a

Ω∗ b

Γ∗

q∗b dΩ∗ = −

∂Ω∗

Γ∗

(y∗ ⊗ q∗b ) · ndS ∗

where n is the unit external normal to Ω∗a , and ∂Ω∗ is the contour Ω∗ (Figure 4.4). We emphasize that these expressions make use of the fact that the inclusions are buried in the matrix. Substituting these expressions into (4.68), and making use of the condition of ﬂux continuity over Γ∗ : (q∗a − q∗b ) · n = 0 we ﬁnd that: divx∗ (λ+eﬀ∗ gradx∗ T ∗(0) ) = ρ∗ C ∗ iω ∗ T ∗(0)

(4.69)

where the effective conductivity λ+eﬀ∗ is deﬁned by: ∂s∗aj 1 ∗ = λ ( + δij )yk∗ ni dS ∗ λ+eﬀ∗ kj |Ω∗ | ∂Ω∗ a ∂yi∗ The macroscopic dimensional model can be written to ﬁrst order of approximation as: divX (λ+eﬀ gradX T ) = ρC

∂T ¯ + O(ε) ∂t

(4.70)

Thus we recover the classical model of conduction. To ﬁrst order, the effective conductivity is that which would be obtained by assuming the inclusion inﬁnitely conductive (so that it does not support any temperature gradient and the ﬂux across it is constant). Our treatment gives the solution to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) when λ∗b −→ ∞. 4.3. Heat transfer with contact resistance between constituents In this section we investigate the inﬂuence on the equivalent macroscopic model of having imperfect contact between the constituents, which leads to a contact resistance (Figure 4.5) [AUR 94b]. The system describing thermal transfers is the same as introduced above. Over Γ the continuity of temperatures is replaced by a Biot boundary condition (in this section we will revert to explicit notation of the time t): divX (λa gradX Ta ) = ρa Ca

∂Ta , ∂t

in Ωa

(4.71)

Heat Transfer in Composite Materials

131

lc

n

Sb

Sa

b

a

contact resistance

Figure 4.5. Period Ω of a composite material with contact resistance

∂Tb , ∂t

in Ωb

(4.72)

λa gradX Ta · n = −h(Ta − Tb )

over Γ

(4.73)

divX (λb gradX Tb ) = ρb Cb

(λa gradX Ta − λb gradX Tb ) · n = 0 over Γ

(4.74)

We recall that n is the normal unit of Γ exterior to Ωa . The scalar h > 0 measures the contact conductivity. The contact resistance is characterized by 1/h. We introduce into the equations (4.71-4.74): Ta = Tac Ta∗ ,

Tb = Tbc Tb∗ ,

ρa = ρac ρ∗a ,

ρb = ρbc ρ∗b

λa = λac λ∗a ,

λb = λbc λ∗b ,

Ca = Cac Ca∗ ,

Cb = Cbc Cb∗ ,

h = hc h∗

and take the macroscopic viewpoint X = x∗ Lc . Then the microscopic description, in dimensionless form, can be written: divx∗ (λ∗a gradx∗ Ta∗ ) = PL ρa Ca∗

∂Ta∗ , ∂t∗

Ldivx∗ (λ∗b gradx∗ Tb∗ ) = PL Cρ∗b Cb∗

in Ω∗a

∂Tb∗ , ∂t∗

λ∗a gradx∗ Ta∗ · n = −BL h∗ (Ta∗ − Tb∗ )

in Ω∗b

over Γ∗

(λ∗a gradx∗ Ta∗ − Lλ∗b gradx∗ Tb∗ ) · n = 0 over Γ∗

(4.75)

(4.76) (4.77) (4.78)

132

Homogenization of Coupled Phenomena

This description involves four dimensionless numbers: ∂Ta | ρac Cac L2c ∂t PL = = |divX (λa gradX Ta )| λac tc |ρa Ca

L=

λbc λac

C=

ρac Cac ρbc Cbc

BL =

|h(Ta − Tb )| h c Lc = |λa gradX Ta · n| λc

For simplicity, we assume the following orders for the usual parameters: PL = O(1), L = O(1) and C = O(1). We still need to estimate the value of the Biot number BL , which depends on the conductivity h. The situations of interest are: BL = O(εp ),

p = −2, −1, 0, 1, 2

They correspond to different macroscopic models which we denote respectively by I, II, III, IV and V. The criteria for choosing between these different models will be discussed in section 4.3.7. We will look for solutions to Ta∗ and Tb∗ of the form: Ta∗ (x∗ , y∗ , t∗ ) = Ta∗(0) (x∗ , y∗ , t∗ ) + εTa∗(1) (x∗ , y∗ , t∗ ) + · · · ∗(0)

Tb∗ (x∗ , y∗ , t∗ ) = Tb ∗(i)

where Tα

∗(1)

(x∗ , y∗ , t∗ ) + εTb

(x∗ , y∗ , t∗ ) + · · ·

(4.79) (4.80)

are Ω∗ -periodic in y∗ and y∗ = ε−1 x∗ . As a result, we have:

gradx∗ −→ gradx∗ + ε−1 grady∗ 4.3.1. Model I – very weak contact resistance In this ﬁrst case we have: PL = O(1), L = O(1), C = O(1) and BL = O(ε−2 ). It is easy to see that by substituting (4.79-4.80) into the system (4.75-4.78), the Biot condition (4.77) leads to temperature continuity in the ﬁrst two orders: ∗(0)

Ta∗(0) − Tb

=0

Heat Transfer in Composite Materials ∗(1)

Ta∗(1) − Tb

133

=0

As a result, the analysis of section 4.2.2 remains appropriate, and model I corresponds to the model deﬁned by (4.36): divx∗ (λIeﬀ∗ gradx∗ T ∗(0) ) = (ρ∗ C ∗ )eﬀ with: eﬀ∗

(ρC)

λIeﬀ∗ ij

1 = ρ C = ∗ |Ω |

1 = ∗ |Ω |

∗

∗

Ω∗ a

ρ∗a Ca∗

∂T ∗(0) ∂t∗

dΩ +

∂t∗ λ∗a (Iij + ai∗ ) dΩ∗ + ∂yj Ω∗ a

∗

Ω∗ b

ρ∗b Cb∗

dΩ

∗

∂t∗ λ∗b (Iij + bi∗ ) dΩ∗ ∂yj Ω∗ b

(4.81) (4.82)

where vectors t∗a and t∗b are solutions to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30). They depend on y∗ , λ∗a , λ∗b and are independent of h∗ . Finally, we can see that the Biot numbers O(εp ), p < −2 lead to the same model. In dimensional variables, this model can be written: divX (λIeﬀ gradX T ) = (ρC)eﬀ

∂T ¯ + O(ε) ∂t

4.3.2. Model II – moderate contact resistance Contact resistance (1/h) is increased by a factor of ε−1 . We have: PL = O(1), L = O(1), C = O(1) and BL = O(ε−1 ). The ﬁrst order in temperature, Ω∗ -periodic in y∗ , satisﬁes: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0, ∗(0)

divy∗ (λ∗b grady∗ Tb

) = 0,

in Ω∗a

(4.83)

in Ω∗b

(4.84) ∗(0)

λ∗a grady∗ Ta∗(0) · n = −h∗ (Ta∗(0) − Tb λ∗a grady∗ Ta∗ · n = λ∗b grady∗ Tb∗ · n

∗(0)

)

over Γ∗

over Γ∗ ∗(0)

(4.85) (4.86)

respectively, and integrate over We multiply (4.83) and (4.84) by Ta and Tb their respective domains of deﬁnition. Integration by parts and use of the divergence theorem for each of these integrals leads with (4.86) to: ∗(0) ∗ ∗(0) 2 ∗ λa (grady∗ Ta ) dΩ + λ∗b (grady∗ Tb )2 dΩ∗ = 0 Ω∗ a

Ω∗ b

134

Homogenization of Coupled Phenomena

The thermal conductivities λ∗a and λ∗b are positive, and as a result grady∗ T ∗(0) = 0. We deduce from this that: ∗(0)

Ta∗(0) (x∗ , y∗ , t∗ ) = Tb

(x∗ , y∗ , t∗ ) = T ∗(0) (x∗ , t∗ )

The second order in temperature, Ω∗ -periodic in y∗ , satisﬁes: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = 0 in Ω∗a ∗(1)

divy∗ (λ∗b (grady∗ Tb

+ gradx∗ T ∗(0) )) = 0

(4.87)

in Ω∗b

(4.88) ∗(1)

λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) · n = −h∗ (Ta∗(1) − Tb

) over Γ∗

(4.89)

λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) · n = ∗(1)

λ∗b (grady∗ Tb

+ gradx∗ T ∗(0) ) · n,

over Γ∗

(4.90)

where gradx∗ T ∗(0) appears as a forcing term. The boundary conditions are linear. ∗(1) ∗(1) The unknowns Ta and Tb are linear functions of gradx∗ T ∗(0) , up to an arbitrary ∗ function of x : ∗ ∗(0) + T¯ ∗(1) (x∗ , t∗ ) Ta∗(1) (x∗ , y∗ , t∗ ) = χII∗ a (y ) · gradx∗ T ∗(1)

Tb

∗ ∗(0) (x∗ , y∗ , t∗ ) = χII∗ + T¯ ∗(1) (x∗ , t∗ ) b (y ) · gradx∗ T

(4.91) (4.92)

II∗ ∗ The superscript II refers to the model type. Vectors χII∗ a and χb , Ω -periodic in ∗ ∗ ∗ ∗ y , depend on y , λa , λb and h . They are solutions to: ∗

divy∗ (λ∗a (grady∗ χII∗ a + I) = 0

in Ω∗a

(4.93)

∗ divy∗ (λ∗b (grady∗ χII∗ b + I) = 0 in Ωb

(4.94)

∗ II∗ II∗ ∗ λ∗a (grady∗ χII∗ a + I) · n = −h (χa − χb ) over Γ

(4.95)

∗ II∗ ∗ λ∗a (grady∗ χII∗ a + I) · n = λb (grady ∗ χb + I) · n over Γ

(4.96)

II∗ χII∗ a + χb = 0

(4.97)

When h → ∞, this boundary condition is equivalent to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30). Finally, the temperature T ∗(2) , Ω∗ -periodic

Heat Transfer in Composite Materials

135

in y∗ , satisﬁes: divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) = ρ∗a Ca∗ ∗(2)

divy∗ (λ∗b (grady∗ Tb

∗(1)

divx∗ (λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

∂T ∗(0) ∂t∗

in Ω∗a

(4.98)

∂T ∗(0) ∂t∗

in Ω∗b

(4.99)

))+

+ gradx∗ T ∗(0) ) = ρ∗b Cb∗

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n = ∗(2)

λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

)·n

over Γ∗

(4.100)

Integrating the ﬁrst two equations over their volume of deﬁnition, using the divergence theorem, equation (4.100) and the periodicity properties lead to the following macroscopic description: divx∗ (λIIeﬀ∗ gradx∗ T ∗(0) ) = (ρ∗ C ∗ )eﬀ

∂T ∗(0) ∂t∗

(4.101)

where: λIIeﬀ∗ = ij

1 |Ω∗ |

Ω∗ a

λ∗a (Iij +

(ρC)eﬀ∗ = ρ∗ C ∗ =

1 |Ω∗ |

∂χII∗ 1 aj ) dΩ∗ + ∗ ∗ ∂yi |Ω |

Ω∗

Ω∗ b

λ∗b (Iij +

∂χII∗ bj ) dΩ∗ ∂yi∗

ρ∗ C ∗ dΩ∗

The structure of model II is identical to that of model I. It is however important to note that λIIeﬀ∗ depends on h∗ and: λIeﬀ∗ = λIIeﬀ∗ (λ∗a , λ∗b , h∗ → ∞) In dimensional variables, the model becomes: divX (λIIeﬀ gradX T ) = (ρC)eﬀ

∂T ¯ + O(ε) ∂t

4.3.3. Model III – high contact resistance We increase the contact resistance (1/h) still further, by a factor of ε−1 . We have: PL = O(1), L = O(1), C = O(1) and BL = O(1). The ﬁrst order in temperature,

136

Homogenization of Coupled Phenomena

Ω∗ -periodic in y∗ , now satisﬁes: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)

divy∗ (λ∗b grady∗ Tb

∗(0)

·n=0

(4.102)

in Ω∗b

) = 0,

λ∗a grady∗ Ta∗(0) · n = 0 λ∗b grady∗ Tb

in Ω∗a

(4.103)

over Γ∗

(4.104)

over Γ∗

(4.105) ∗(0)

The boundary value problems for Ta before, it is easy to show that: Ta∗(0) (x∗ , y∗ , t∗ ) = Ta∗(0) (x∗ , t∗ ),

∗(0)

and Tb

∗(0)

and

Tb

are decoupled. Proceeding as ∗(0)

(x∗ , y∗ , t∗ ) = Tb

(x∗ , t∗ )

The second order in temperature, Ω∗ -periodic in y∗ , satisﬁes: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) = 0 ∗(1)

divy∗ (λ∗b (grady∗ Tb

∗(0)

+ gradx∗ Tb

)) = 0

in Ω∗a

(4.106)

in Ω∗b

(4.107) ∗(0)

λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n = −h∗ (Ta∗(0) − Tb

) over Γ∗ (4.108)

λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n ∗(1)

= λ∗b (grady∗ Tb

∗(0)

+ gradx∗ Tb

)·n ∗(1)

over Γ∗

(4.109)

∗(1)

Thus we have two decoupled linear problems for Ta and Tb . System (4.1064.109) imposes a continuity condition obtained by integrating (4.106) over Ω∗a . Using the divergence theorem and equation (4.108), we ﬁnd: ∗(0) ∗(0) ∗ ∗(0) ∗ ∗(0) h (Ta − Tb )dS = (Ta − Tb ) h∗ dS ∗ = 0 Γ∗

Γ∗

The condition h∗ > 0 then implies: ∗(0)

Ta∗(0) = Tb

= T ∗(0) (x∗ , t∗ ) ∗(1)

(4.110) ∗(1)

∗(1)

The system for Tb gives the same condition. The unknowns Ta and Tb are both linear functions of gradx∗ T ∗(0) , up to an arbitrary function of x∗ and t∗ . ∗ ∗(0) + T¯a∗(1) (x∗ , t∗ ) Ta∗(1) (x∗ , y∗ , t∗ ) = χIII∗ a (y ) · gradx∗ T

Heat Transfer in Composite Materials ∗(1)

Tb

∗(1)

∗ ∗(0) (x∗ , y∗ , t∗ ) = χIII∗ + T¯b b (y ) · gradx∗ T

137

(x∗ , t∗ )

and χIII∗ are independent of h∗ . They are solutions We note that the vectors χIII∗ a b ∗ ∗ to the canonical problems P(λa , λb = 0) and P(λ∗a = 0, λ∗b ) respectively (see equations 4.26-4.30), so that: divy∗ (λ∗a (grady∗ χIII∗ + I) = 0 in Ω∗a a divy∗ (λ∗b (grady∗ χIII∗ + I) = 0 b

in Ω∗b

λ∗a (grady∗ χIII∗ + I) · n = 0 a

over Γ∗

λ∗b (grady∗ χIII∗ + I) · n = 0 b

over Γ∗

III∗ χIII∗ a = χb = 0

Macroscopic behavior is obtained from the continuity equation which forms part of the boundary value problem (4.98-4.99) for T ∗(2) . It is clear that, just as was the case for model II, this model also has the same structure as model I: divx∗ (λIIIeﬀ∗ gradx∗ T ∗(0) ) = (ρC)eﬀ∗

∂T ∗(0) ∂t∗

(4.111)

where: λIIIeﬀ∗ = λIIIeﬀ∗ + λIIIeﬀ∗ a b λIIIeﬀ∗ aij

1 = ∗ |Ω |

λIIIeﬀ∗ = bij

1 |Ω∗ |

Ω∗ a

Ω∗ b

∂χIII∗ aj ) dΩ∗ ∂yi∗

λ∗a (Iij +

∂χIII∗ bj

λ∗b (Iij +

(ρC)eﬀ∗ = ρ∗ C ∗ =

1 |Ω∗ |

∂yi∗

Ω∗

) dΩ∗

ρ∗ C ∗ dΩ∗

The effective conductivities λIIIeﬀ∗ and λIIIeﬀ∗ of constituents a and b can be a b obtained by considering each constituent as perfectly insulated from its neighbor. Thus we have: λIIIeﬀ∗ = λIIeﬀ∗ (λ∗a , λ∗b , h = 0)

138

Homogenization of Coupled Phenomena

In the speciﬁc case where phase b is dispersed (non-connected) within matrix a, the condition of zero ﬂux at the interface Γ∗ requires that χIII∗ = −y∗ . As a result, b phase b does not participate in the conductive heat transfer: λIIIeﬀ∗ =0 b In dimensional variables, the model can be written: divX (λIIIeﬀ gradX T ) = (ρC)eﬀ

∂T ¯ + O(ε) ∂t

4.3.4. Model IV – model with two coupled temperature ﬁelds We increase the contact resistance (1/h) still further by a factor of ε−1 . We have: PL = O(1), L = O(1), C = O(1) and BL = O(ε). The ﬁrst order in temperature, Ω∗ -periodic in y∗ , satisﬁes, as in case II: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)

divy∗ (λ∗b grady∗ Tb

in Ω∗a

) = 0 in Ω∗b

λ∗a grady∗ Ta∗ · n = λ∗b grady∗ Tb∗ · n = 0

over Γ∗ ∗(0)

Once again, the boundary value problems for Ta this we deduce that:

∗(0)

Ta∗(0) (x∗ , y∗ , t∗ ) = Ta∗(0) (x∗ , t∗ ) and Tb

∗(0)

and Tb

are decoupled. From ∗(0)

(x∗ , y∗ , t∗ ) = Tb

(x∗ , t∗ )

The second order in temperature, Ω∗ -periodic in y∗ , satisﬁes: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = 0 in Ω∗a ∗(1)

divy∗ (λ∗b (grady∗ Tb

∗(0)

+ gradx∗ Tb

)) = 0

λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n = 0 ∗(1)

λ∗b (grady∗ Tb

∗(0)

+ gradx∗ Tb

)·n=0

in Ω∗b

over Γ∗ over Γ∗ ∗(1)

∗(1)

We obtain two decoupled linear problems for Ta and Tb . These two problems do not have any continuity requirements, as can be seen by integrating the partial ∗(1) differential equations over their respective domains of deﬁnition. The unknowns Ta

Heat Transfer in Composite Materials ∗(1)

∗(0)

and Tb are linear functions of gradx∗ Ta include an arbitrary function of x∗ and t∗ .

∗(0)

and gradx∗ Tb

139

respectively, and

∗ ∗(0) Ta∗(1) (x∗ , y∗ , t∗ ) = χIV∗ + T¯a∗(1) (x∗ , t∗ ) a (y ) · gradx∗ Ta ∗(1)

Tb

∗(0)

∗ (x∗ , y∗ , t∗ ) = χIV∗ b (y ) · gradx∗ Tb

∗(1) + T¯b (x∗ , t∗ )

= χIII∗ and χIV∗ = χIII∗ are independent of h∗ and are solutions Vectors χIV∗ a a b b to the canonical problems P(λ∗a , λ∗b = 0) and P(λ∗a = 0, λ∗b ) respectively. Finally, the temperature T ∗(2) , Ω∗ -periodic in y∗ , satisﬁes: divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = ρ∗a Ca∗ ∗(2)

divy∗ (λ∗b (grady∗ Tb

∗(1)

divx∗ (λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

∗(0)

+ gradx∗ Tb

∗(0)

∂Ta ∂t∗

)) = ρ∗b Cb∗

∗(0)

∂Tb ∂t∗

∗(0)

∗(2)

∗(1)

+ gradx∗ Tb

(4.112)

in Ω∗b

(4.113)

))+

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n = −h∗ (Ta∗(0) − Tb λ∗b (grady∗ Tb

in Ω∗a

∗(0)

) · n = h∗ (Ta∗(0) − Tb

)

) over Γ∗ (4.114) over Γ∗

(4.115)

Integrating (4.112) over Ω∗a , making use of (4.114), gives our ﬁrst macroscopic equation (4.116). We obtain our second macroscopic equation (4.117) by treating (4.113) and (4.115) in the same way: ∗(0)

(ρC)eﬀ∗ a

∂Ta ∂t∗

(ρC)eﬀ∗ b

∂Tb ∂t∗

∗(0)

= divx∗ (λIVeﬀ∗ gradx∗ Ta∗(0) ) − H ∗ (Ta∗(0) − Tb a

∗(0)

∗(0)

= divx∗ (λIVeﬀ∗ gradx∗ Tb b

where: λIVeﬀ∗ = λIIIeﬀ∗ , a a (ρC)eﬀ∗ a

1 = ∗ |Ω |

λIVeﬀ∗ = λIIIeﬀ∗ b b

Ω∗ a

ρ∗a Ca∗ dΩ∗

∗(0)

) + H ∗ (Ta∗(0) − Tb

)

(4.116)

)

(4.117)

140

Homogenization of Coupled Phenomena

(ρC)eﬀ∗ = b 1 H = ∗ |Ω | ∗

1 |Ω∗ | Γ∗

Ω∗ b

ρ∗b Cb∗ dΩ∗

h∗ dΓ∗

In the same way as for model III, in the speciﬁc case where phase b is dispersed (non-connected) within matrix a, we have: = λIIIeﬀ∗ =0 λIVeﬀ∗ b b The macroscopic behavior now involves two temperature ﬁelds. In dimensional form, the model becomes: (ρC)eﬀ a

∂Ta ¯ = divX (λIVeﬀ gradX Ta ) − H(Ta − Tb ) + O(ε) a ∂t

(ρC)eﬀ b

∂Tb ¯ = divX (λIVeﬀ gradX Tb ) + H(Ta − Tb ) + O(ε) b ∂t

4.3.5. Model V – model with two decoupled temperature ﬁelds The contact resistance (1/h) is very large. We have: PL = O(1), L = O(1), C = O(1) and BL = O(ε2 ). The analysis used for model IV can be applied step by step as before, except for the right-hand side of equations (4.114) and (4.115) which are now zero. This leads to decoupled macroscopic equations: ∗(0)

(ρC)eﬀ∗ a

∂Ta ∂t∗

(ρC)eﬀ∗ b

∂Tb ∂t∗

= divx∗ (λVeﬀ∗ gradx∗ Ta∗(0) ) a

∗(0)

∗(0)

= divx∗ (λVeﬀ∗ gradx∗ Tb b

)

(4.118)

(4.119)

with: λVeﬀ∗ = λIVeﬀ∗ = λIIIeﬀ∗ , a a a

λVeﬀ∗ = λIVeﬀ∗ = λIIIeﬀ∗ b b b

It can immediately be seen that if we assume BL = O(εp ), p > 2 then we end up with model V . In physical quantities the model can be written in the form: (ρC)eﬀ a

∂Ta ¯ = divX (λVeﬀ a gradX Ta ) + O(ε) ∂t

(ρC)eﬀ b

∂Tb ¯ = divX (λVeﬀ gradX Tb ) + O(ε) b ∂t

Heat Transfer in Composite Materials

141

4.3.6. Example: bilaminate composite Consider again the composite depicted in Figure 4.4. As noted above, the symmetries of the problem require that for any model: eﬀ λeﬀ 22 = λ33 ,

eﬀ eﬀ λeﬀ 12 = λ13 = λ23 = 0

For model I, we have the classical results: λIeﬀ 22 =

λb λa cb λa + (1 − cb )λb

Ieﬀ λIeﬀ 11 = λ33 = (1 − cb )λa + cb λb

Now consider model II. Conductivity in the plane of layers remains the same as that of model I: IIeﬀ λIIeﬀ 11 = λ33 = (1 − cb )λa + cb λb

In the direction perpendicular to the layers contact resistance takes effect, and as a result we ﬁnd: −1 2 λb λa 1 IIeﬀ λ22 = + , with λIeﬀ 22 = Ieﬀ lh c λ + (1 − cb )λb λ22 b a which approaches the conductivity λeﬀ 22 as h rises. Model III introduces a discontinuity in the medium in the direction perpendicular to the layers: the macroscopic equivalent material does not conduct in this direction, at least to the ﬁrst order of approximation that we are considering here. (The conductivity is hl/2, a quantity which is negligible to ﬁrst order.) Conductivity in the plane of the layers is that of model I, with the contact resistance playing no role in this direction: λIIIeﬀ = 0, 22

λIIIeﬀ = λIIIeﬀ = (1 − cb )λa + cb λb 11 33

Finally, models IV and V introduce the following effective parameters: IVeﬀ Veﬀ Veﬀ λIVeﬀ a11 = λb11 = λa11 = λb11 = 0 Veﬀ λIVeﬀ a22 = λa33 = (1 − cb )λa Veﬀ λIVeﬀ b22 = λb33 = cb λb

H=

2h l

Model IV approaches model V as h approaches zero. In contrast, there is no direct transition from model IV to model III.

142

Homogenization of Coupled Phenomena

4.3.7. Choice of model Here we look for a dimensionless number which can characterize the different models, independent of details of perturbation they are subjected to. It is enough to consider Bl in this case: Bl =

h c lc = εBL λc Composite B

O(3 )

O(2 )

Two temperature fields

Composite A

O()

O(1)

O(1 )

Bl =

hc lc c

One single temperature field

Figure 4.6. Composite materials with contact resistance: different models as a function of the number Bl

The different models are characterized by: – model I: Bl O(ε−1 ) ; – model II: Bl = O(1) ; – model III: Bl = O(ε) ; – model IV: Bl = O(ε2 ) ; – model V: Bl O(ε3 ). Thus we can split the composite materials into two categories: – Composites A such that Bl 1. The only possible estimates of Bl are Bl O(ε−1 ) and O(1). These materials can only be described by models I or II. – Composites B such that Bl 1. The only possible estimates of Bl are Bl = O(1), O(ε), O(ε2 ) and O(ε3 ). These materials can only be described by models II, III, IV and V. The different models and their domains of validity are summarized in Figure 4.6. The scale separation parameter ε depends on the applied disturbance and on the point under consideration in the material. A given material can thus be described by different models depending on the nature of the disturbance and the spatial position.

Chapter 5

Diffusion/Advection in Porous Media

5.1. Introduction The transport of a solute by diffusion-convection in a porous medium is involved in a number of ﬁelds such as environmental engineering, process engineering, and even metallurgy. The works of Taylor [TAY 54] and Aris [ARI 56] were undoubtedly the ﬁrst to demonstrate coupling between the ﬂow and solute transport equation in elementary geometries. These works were extended to porous media by Brenner [BRE 80] using the method of moments and with the help of periodic homogenization [MAU 91; MEI 92; AUR 93b; AUR 95; AUR 96]. This chapter concerns the study of diffusive and advective transport of a lowconcentration solute in a saturated porous medium. The macroscopic transport equations are derived from the Fick and Navier-Stokes equations at the microscopic scale. 5.2. Diffusion-convection on the pore scale and estimates Consider a rigid porous Galilean medium with connected pores. We assume it to be periodic with period Ω (Figure 5.1). The pores Ωf are saturated with a viscous, incompressible Newtonian ﬂuid containing a low concentration of solute c. The ﬂuid is in slow steady-state isothermal ﬂow, so that the solute is transported by diffusion and convection. We ignore adsorption of the solute onto the surface Γ of the pores, and absorption into the solid matrix Ωs . We denote as lc and Lc the characteristic lengths of the pores and macroscopic length. The solute transport is described by conservation of mass: ∂c + divX (−Dmol gradX c + vc) = 0 ∂t

in Ωf

(5.1)

144

Homogenization of Coupled Phenomena

and the condition on the surface of the pores that: Dmol gradX c · n = 0

over Γ

(5.2)

where c is the solute concentration (mass of solute per unit volume of ﬂuid), Dmol is the (positive) molecular diffusion tensor which is in general isotropic, t is the time, n the unit vector giving the normal to Γ exterior to Ωf , and v is the ﬂuid velocity. This is analyzed in section 7.2 in Chapter 7 and here we will make use of the results obtained in that section. Introducing into (5.1-5.2): c = cc c∗ ,

v = vc v ∗ ,

t = t c t∗ ,

Dmol = Dc Dmol∗

where the quantities with subscript c and exponent ∗ are the characteristic and dimensionless quantities respectively, and then adopting the microscopic viewpoint X = x∗ lc , we can write the microscopic description in dimensionless form as: Pl

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + Pel v∗ c∗ ) = 0 ∂t∗

Dmol∗ grady∗ c∗ · n = 0

within Ω∗f

over Γ∗

(5.3) (5.4)

Lc

lc

n

s

f (a)

(b)

Figure 5.1. Porous medium: (a) Macroscopic structure, (b) Representative Elementary Volume (REV) with period Ω

This system introduces two dimensionless numbers: ∂c | lc2 ∂t = Pl = |divX (−Dmol gradX c)| Dc tc |

(5.5)

Diffusion/Advection in Porous Media

145

and the Péclet number deﬁned by: Pel =

|divX (vc)| v c lc = mol |divX (−D gradX c)| Dc

(5.6)

where Dc and vc are the characteristic values of molecular diffusion and the velocity of the ﬂuid. The characteristic time tc is the time over which we intend to describe the solute transport: tc is the characteristic time of the observation. On the macroscopic scale these numbers become: PL =

L2c , Dc tc

PeL =

vc Lc Dc

It is useful to introduce the characteristic times associated with advection and diffusion [AUR 95], which on the scale of the pores are: = tadv l

lc , vc

tdif = l

lc2 Dc

and on the macroscopic scale are: = tadv L

Lc , vc

tdif L =

L2c Dc

The ratio of characteristic times for diffusion and advection is the Péclet number: Pel =

tdif l , tadv l

PeL =

tdif L tadv L

These considerations lead us to consider the following situations: adv – Diffusion dominates at the macroscopic scale: tdif L = εtL . To study this case dif we use the macroscopic diffusion time tc = tL as our characteristic time. This means that Pl = O(ε2 ) and Pel = O(ε2 ). – Diffusion and advection are of the same order of magnitude on the macroscopic adv dif adv scale: tdif L = tL . In this case we use tc = tL = tL as our characteristic time. This 2 means that Pl = O(ε ) and Pel = O(ε). = εtdif – Advection dominates on the macroscopic scale: tadv L L . In this case we adv use tc = tL as our characteristic time. This means that Pl = O(ε) and Pel = O(1). – We will also consider the case where advection is overwhelmingly dominant on adv = ε2 tdif the macroscopic scale: tadv L L . In this case the characteristic time is tc = tL −1 and Pl = O(1) or Pel = O(ε ).

146

Homogenization of Coupled Phenomena

5.3. Diffusion dominates at the macroscopic scale The characteristic time is the time take for diffusion at the macroscopic scale: adv tc = tdif L = εtL

then Pel = O(ε2 ) and Pl = O(ε2 )

Equations (5.3) and (5.4) take the following dimensionless form: ε2

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + ε2 v∗ c∗ ) = 0 ∂t∗

Dmol∗ grady∗ c∗ · n = 0

within Ω∗f

(5.7)

over Γ∗

(5.8)

c∗ = c∗(0) (x∗ , y∗ , t∗ ) + εc∗(1) (x∗ , y∗ , t∗ ) + ε2 c∗(2) (x∗ , y∗ , t∗ ) + · · ·

(5.9)

We look for c∗ and v∗ in the form:

v∗ = v∗(0) (x∗ , y∗ ) + εv∗(1) (x∗ , y∗ ) + ε2 v∗(2) (x∗ , y∗ ) + · · ·

(5.10)

where c∗(i) (x∗ , y∗ , t∗ ) and v∗(i) (x∗ , y∗ ) are y∗ -periodic and where x∗ = εy∗ . The method consists of substituting these expansions into the dimensionless system (5.75.8) and identifying the powers of ε. We note that due to the two spatial variables and choice of microscopic viewpoint, the spatial derivative takes the following form: grady∗ −→ ε gradx∗ + grady∗ 5.3.1. Homogenization 5.3.1.1. Boundary value problem for c∗(0) After substituting in the expansions (5.9) and (5.10), we obtain at the lowest order in ε: divy∗ (Dmol∗ grady∗ c∗(0) ) = 0

in Ω∗f

(5.11)

Dmol∗ grady∗ c∗(0) · n = 0

over Γ∗

(5.12)

where c∗(0) is periodic in y∗ . Evidently the solution is not unique: c∗(0) is only determined up to an additive function of x∗ . We introduce the space C of regular functions α deﬁned over Ω∗f , with a mean of zero over Ω∗f : 1 α = ∗ |Ω |

1 α dΩ = φ ∗ ∗ |Ω Ωf f| ∗

Ω∗ f

α dΩ∗ = 0

Diffusion/Advection in Porous Media

147

This condition, which is not required by the stated problem, is introduced to ensure the Hilbert character of C and to avoid the indeterminate nature of c∗(0) . The bilinear, symmetric, positive deﬁnite form: ∂α ∂β Dij ∗ ∗ dΩ∗ = Dmol∗ grady∗ α · grady∗ β dΩ∗ (α, β)C = ∗ ∂y ∂y Ω∗ Ω i j f f deﬁnes a scalar product over C. We multiply the two members of (5.11) by α ∈ C, and integrate over Ω∗f . Integrating by parts, and using boundary conditions over Γ∗ and periodicity, we obtain the following equivalent weak formulation: Dmol∗ grady∗ α · grady∗ c∗(0) dΩ∗ = 0 (5.13) ∀α ∈ C, (α, c∗(0) )C = Ω∗ f

We can use the Lax-Milgram theorem to show that this formulation ensures the existence and uniqueness of c∗(0) ∈ C, in other words satisfying the condition of zero mean. As a result, the solution of (5.11)-(5.12) must be: c∗(0) (x∗ , y∗ , t∗ ) = c∗(0) (x∗ , t∗ )

(5.14)

The concentration c∗(0) is constant over the period. 5.3.1.2. Boundary value problem for c∗(1) The second order c∗(1) satisﬁes: divy∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) )) = 0

within Ω∗f

(5.15)

Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ) · n = 0

over Γ∗

(5.16)

where c∗(1) is periodic in y∗ and gradx∗ c∗(0) appears as a source term. After multiplication of the two members of (5.15) by α ∈ C, and integration over Ω∗f , we obtain the weak formulation: ∀α ∈ C, (α, c∗(1) )C = Dmol∗ grady∗ α · grady∗ c∗(1) dΩ∗ Ω∗ f

=

Ω∗ f

Dmol∗ grady∗ α · gradx∗ c∗(0) dΩ∗

(5.17)

As was the case for c∗(0) , this formulation shows the existence and uniqueness of c∗(1) ∈ C, which means we can write the solution to linear problem (5.15)-(5.16) in the form: c∗(1) (x∗ , y∗ , t∗ ) = χdif∗ (y∗ ) · gradx∗ c∗(0) + c¯∗(1) (x∗ , t∗ )

(5.18)

148

Homogenization of Coupled Phenomena

where χdif∗ is a periodic vector with a mean of zero over Ω∗f . The function c¯∗(1) is an arbitrary function of x∗ and t∗ introduced by the external condition χdif∗ = 0. The vector χdif∗ satisﬁes: divy∗ (Dmol∗ (grady∗ χdif∗ + I)) = 0 within Ω∗f Dmol∗ (grady∗ χdif∗ + I) · n = 0

over Γ∗

χdif∗ = 0 5.3.1.3. Boundary value problem for c∗(2) Finally, bearing in mind the results above, the boundary value problem for c∗(2) can be reduced to: ∂c∗(0) − divy∗ (Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ))− ∂t∗ divx∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) )) = 0 Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ) · n = 0

within Ω∗f (5.19)

over Γ∗

(5.20)

where c∗(2) is y∗ -periodic. In contrast to the boundary condition for c∗(0) and c∗(1) , the boundary value problem for c∗(2) introduces a compatibility condition obtained by integration of (5.19) over Ω∗f . Bearing in mind the boundary conditions over Γ∗ , the periodicity, and after division by |Ω∗ |, we ﬁnd: φ

∂c∗(0) − divx∗ (Ddif∗ gradx∗ c∗(0) ) = 0 ∂t∗

(5.21)

where φ is the porosity and the effective diffusion tensor Ddif∗ is deﬁned as: dif∗ = Dij

1 |Ω∗ |

Ω∗ f

mol∗ Dik (Ijk +

∂χdif∗ j ) dΩ∗ ∂yk∗

(5.22)

5.3.2. Macroscopic diffusion model Returning to physical quantities, we ﬁnd: φ

∂c ¯ − divX (Ddif gradX c) = O(ε), ∂t

Ddif = Dc Ddif∗

(5.23)

whose relative precision is O(ε). The tensor Ddif is purely diffusive. It is possible to show, starting from equation (5.17), that the tensor Ddif is positive and symmetric.

Diffusion/Advection in Porous Media

149

In macroscopic model (5.23), c is the mass of solute per unit volume of ﬂuid. Using the mass of solute per unit of volume of the porous medium c = φ c, the model becomes: ∂c ¯ − divX (φ−1 Ddif gradX c) = O(ε) ∂t 5.4. Comparable diffusion and advection on the macroscopic scale The characteristic time is now: adv tc = tdif L = tL

which implies

Pel = O(ε)

and Pl = O(ε2 )

In dimensionless form, equations (5.3) and (5.4) can be written: ε2

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + εv∗ c∗ ) = 0 ∂t∗

Dmol∗ grady∗ c∗ · n = 0

within Ω∗f

over Γ∗

(5.24) (5.25)

5.4.1. Homogenization 5.4.1.1. Boundary value problems for c∗(0) and c∗(1) We introduce expansions (5.9) and (5.10) into the system of equations (5.34-5.35). It is easy to see that the boundary value problems for c∗(0) and c∗(1) are unchanged. Solutions (5.14) and (5.18) obtained in the preceding section remain valid: c∗(0) (x∗ , y∗ , t∗ ) = c∗(0) (x∗ , t∗ ) c∗(1) (x∗ , y∗ , t∗ ) = χdif∗ (y∗ ) · gradx∗ c∗(0) + c¯∗(1) (x∗ , t∗ ) 5.4.1.2. Boundary value problem for c∗(2) In contrast, the boundary value problem for c∗(2) becomes: ∂c∗(0) − divy∗ (Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ))− ∂t∗ divx∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ))+ divy∗ (c∗(0) v∗(1) + c∗(1) v∗(0) ) + divx∗ (c∗(0) v∗(0) ) = 0 Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ) · n = 0

over Γ∗

within Ω∗f

(5.26) (5.27)

150

Homogenization of Coupled Phenomena

where c(2) is y∗ -periodic, and v∗(0) and v∗(1) are the ﬂuid velocities for the ﬁrst two orders of approximation. These velocities are given by independent consideration of slow, steady-state ﬂow of an incompressible Newtonian ﬂuid in a porous medium (see Chapter 7). This leads to the following results. Conservation of mass of the ﬂuid to the ﬁrst two orders can be written: divy v∗(0) = 0,

divy v∗(1) + divx v∗(0) = 0

(5.28)

The no-slip boundary condition at Γ∗ gives: v∗(0) = v∗(1) = 0

(5.29)

In addition, to ﬁrst order, the velocity of the ﬂuid on the microscopic scale is proportional to the pressure gradient (see Chapter 7), v∗(0) (x∗ , y∗ ) = −

k∗ (y∗ ) gradx∗ p∗(0) , η∗

with

p∗(0) = p∗(0) (x∗ )

(5.30)

At the macroscopic scale, these results lead to Darcy’s law: divx v∗(0) = 0,

v∗(0) = −

K∗ gradx∗ p∗(0) η∗

(5.31)

The compatibility condition is obtained by integration of (5.26) over Ω∗f . Using the divergence theorem, the boundary conditions over Γ∗ (5.27), the periodicity and ﬁnally the no-slip condition at the boundary Γ∗ (5.29), we ﬁnd: φ

∂c∗(0) − divx∗ (Ddif∗ gradx∗ c∗(0) − v∗(0) c∗(0) ) = 0 ∂t∗

(5.32)

where Ddif∗ is the effective diffusion tensor introduced in the previous case and v∗(0) is the Darcy velocity given by equation (5.31). 5.4.2. Macroscopic diffusion-advection model In dimensional variables, the model (5.32) becomes: φ

∂c ¯ − divX (Ddif gradX c − vc) = O(ε), ∂t

Ddif = Dc Ddif∗

(5.33)

with a relative precision of O(ε). The tensor Ddif , purely diffusive, is the same as that deﬁned for the case where diffusion dominates: it is not modiﬁed by advection. In terms of the concentration per unit volume of the porous material, this model can be written: ∂c ¯ − divX (φ−1 Ddif gradX c − φ−1 vc) = O(ε) ∂t

Diffusion/Advection in Porous Media

151

5.5. Advection dominant at the macroscopic scale The characteristic time is now: dif tc = tadv L = εtL

then Pel = O(1)

and Pl = O(ε)

Equations (5.3) and (5.4) take the dimensionless form: ε

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + v∗ c∗ ) = 0 ∂t∗

Dmol∗ grady∗ c∗ · n = 0

within Ω∗f

(5.34)

over Γ∗

(5.35)

5.5.1. Homogenization 5.5.1.1. Boundary value problem for c∗(0) The boundary value problem for c∗(0) is now: divy∗ (Dmol∗ .grady∗ c∗(0) − v∗(0) c∗(0) ) = 0 Dmol∗ grady∗ c∗(0) · n = 0

within Ω∗f

(5.36)

over Γ∗

(5.37)

where c∗(0) is a periodic function of y∗ . With the help of equations (5.28) and (5.29) to ﬁrst order, the associated weak formulation is similar to (5.13): ∗(0) ∀α ∈ C, (α, c )C = Dmol∗ grady∗ α · grady∗ c∗(0) dΩ∗ = 0 (5.38) Ω∗ f

As a consequence, equation (5.14) remains applicable: c∗(0) (x∗ , y∗ , t∗ ) = c∗(0) (x∗ , t∗ )

(5.39)

The concentration c∗(0) is constant across the period. 5.5.1.2. Boundary value problem for c∗(1) The boundary value problem for c∗(1) is strongly modiﬁed: ∂c∗(0) − divy∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) )) ∂t∗ +divy∗ (c∗(0) v∗(1) + c∗(1) v∗(0) ) + divx∗ (c∗(0) v∗(0) ) = 0 Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ) · n = 0

in Ω∗f

over Γ∗

(5.40) (5.41)

152

Homogenization of Coupled Phenomena

where c∗(1) is y∗ -periodic and gradx∗ c∗(0) appears as a source term. This system introduces a compatibility condition for the existence of c∗(1) , which can be obtained by taking the volume mean of (5.40) over Ω∗ . Using the divergence theorem, the boundary condition over Γ∗ (5.41), periodicity and ﬁnally no-slip condition at the wall Γ∗ (5.29), we ﬁnd: φ

∂c∗(0) + divx∗ (v∗(0) c∗(0) ) = 0 ∂t∗

(5.42)

where the velocity v∗(0) is the Darcy velocity, the mean of v∗(0) over Ω∗ . As might be expected, advection alone is present at the ﬁrst order of approximation. We need to ﬁnd the ﬁrst correction to this macroscopic model which brings diffusion into play. For that, substituting (5.42) into equation (5.40), the latter becomes: divy∗ (Dmol∗ (gradx∗ c∗(0) + gradx∗ c∗(1) )) − v∗(0) · grady∗ c∗(1) = (v∗(0) − φ−1 v∗(0) ) · gradx∗ c∗(0)

in Ω∗f

(5.43)

The solution to the boundary value problem (5.43)–(5.41) can be described using the space C. Linearity allows us to write: c∗(1) (x∗ , y∗ , t∗ ) = χdis∗ (y∗ ) · gradx∗ c∗(0) + c¯∗(1) (x∗ , t∗ )

(5.44)

where χdis∗ is a periodic vector with a mean of zero over Ω∗f . Its components χdis∗ k are the solutions to three independent problems (k = 1, 2, 3): dis∗ ∂χdis∗ ∂ ∗(0) ∂χk mol∗ k (Ijk + = ∗ (Dij ∗ )) − vi ∂yi ∂yj ∂yi∗ ∗(0)

vk mol∗ Dij (Ijk +

∂χdis∗ k )ni = 0 ∂yj∗

− φ−1 v ∗(0) k

within Ω∗f

over Γ∗

(5.46)

which can be written in the form: divy∗ (Dmol∗ (grady∗ χdis∗ + I)) − v∗(0) · grady∗ χdis∗ = v∗(0) − φ−1 v∗(0) Dmol∗ (grady∗ χdis∗ + I) · n = 0 χdis∗ = 0

over Γ∗

(5.45)

within Ω∗f

Diffusion/Advection in Porous Media

153

We note that χdis∗ = χdif∗ , since χdis∗ depends on the velocity ﬁeld to ﬁrst order, and hence on the pressure gradient gradx∗ p(0) . It is interesting to note that the above system is the same as that obtained by Brenner [BRE 80] using the method of moments. 5.5.1.3. Boundary value problem for c∗(2) The macroscopic relationship satisﬁed by the corrector c∗(1) is obtained through the compatibility condition associated with the boundary value problem for c∗(2) : ∂c∗(1) − divy∗ (Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ))− ∂t∗ divx∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ))+ divy∗ (c∗(0) v∗(2) + c∗(1) v∗(1) + c∗(2) v∗(0) )+ divx∗ (c∗(0) v∗(1) + c∗(1) v∗(0) ) = 0 Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ) · n = 0

in Ω∗f

(5.47) over Γ∗

(5.48)

where c∗(2) is y∗ -periodic. If we take the volume mean of (5.47), we ﬁnd: ∂c(1) − divx∗ (Dmol∗ (I + grady∗ χdis∗ ).gradx∗ c∗(0) − ∂t∗ v∗(1) c∗(0) − c∗(1) v∗(0) ) = 0

(5.49)

which represents the condition for the existence of c∗(2) . This equation can be slightly modiﬁed by introducing equations (5.44) for c∗(1) and (5.30) for v∗(0) : ∂c(1) − divx∗ (Ddis∗ gradx∗ c∗(0) − v∗(1) c∗(0) − c¯∗(1) v∗(0) ) = 0 ∂t∗

(5.50)

where Ddis∗ is the effective dispersion tensor: dis∗ = Dij

1 |Ω∗ |

Ω∗ f

mol∗ Dik (Ijk +

∂χdis∗ k ∗ dis∗ ∂p∗(0) j ) + ik χ dΩ∗ ∗ ∂yk η∗ j ∂x∗k

(5.51)

Equation (5.50) represents the macroscopic behavior of the ﬁrst-order correction.

154

Homogenization of Coupled Phenomena

5.5.2. Dispersion model To second order, the concentration and velocity can be written: c∗ ≈ c∗(0) + εc∗(1) = φ c∗(0) + εφ c¯∗(1)

(5.52)

v∗ ≈ v∗(0) + εv∗(1)

(5.53)

Adding equations (5.42) and (5.50) multiplied by ε, term by term, we ﬁnd: ∂c∗ − divx∗ (εφ−1 Ddis∗ .gradx∗ c∗ − φ−1 v∗ c∗ ) = O(ε2 ) ∂t∗

(5.54)

where we note that the precision of this model is O(ε2 ). The diffusion term is small compared to the advection term, PeL = ε−1 . Returning now to physical quantities, we ﬁnd: ∂c tc Lc −1 dis φ−1 ¯ 2) − divX Lc (ε φ D gradX c − vc) = O(ε ∂t cc Dc cc vc cc

(5.55)

We recall that tc = tadv = Lc /vc and vc Lc /Dc = PeL = ε−1 . We obtain: L ∂c ¯ 2) − divX (φ−1 Ddis gradX c − φ−1 vc) = O(ε ∂t

(5.56)

As shown by Auriault and Adler [AUR 95], the dispersion tensor Ddis is positive, but is not generally symmetrical. Using the concentration per unit volume of ﬂuid cΩf = c(0) + ε c¯(1) = c/φ, the model becomes: φ

∂cΩf ¯ 2) − divX (Ddis gradX cΩf − vcΩf ) = O(ε ∂t

One important question concerns the physical meaning of the volume average v, which contains the ﬁrst correction v(1) . It turns out that v(1) is not irrotational – see (7.11b). As a result v(1) is not generally a ﬂux (a similar reasoning can be found for v(0) in section 7.2). As a consequence, v itself is not generally a ﬂux. Another consequence of the presence of v(1) is that v does not generally satisfy Darcy’s law. (For more details, see [AUR 05b].) 5.6. Very strong advection The characteristic macroscopic advection time tadv L becomes very large compared to the macroscopic diffusion time tdif L . The characteristic time of the observer is now: 2 dif tc = tadv L = ε tL

which implies

Pel = O(ε−1 ) and Pl = O(1)

Diffusion/Advection in Porous Media

155

Equations (5.3) and (5.4) can be written in dimensionless form as: ∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + ε−1 v∗ c∗ ) = 0 ∂t∗ Dmol∗ grady∗ c∗ · n = 0

within Ω∗f

over Γ∗

(5.57) (5.58)

We need to ﬁnd a suitable macroscopic model. We substitute the asymptotic expansions of c∗ and v∗ into the above system. At the lowest order in c∗ we obtain: divy∗ (c∗(0) v∗(0) ) = 0

within Ω∗f

(5.59)

The incompressibility of the ﬂuid, divy∗ v∗(0) = 0, means we can write the preceding relationship in the form: v∗(0) · grady∗ c∗(0) = 0 Dmol∗ grady∗ c∗(0) · n = 0

within Ω∗f

(5.60)

over Γ∗

(5.61)

To make this clearer, consider a constant concentration c0 applied from the outset at the entrance to the macroscopic sample, and maintained constant from then onwards. Equation (5.60) shows that c∗(0) remains constant along a streamline across a given period: c∗(0) = c0 if the ﬂuid particle which was at the entrance to the sample at time t∗ = 0 has reached this period, or c∗(0) = 0 if not. This situation applies throughout the macroscopic sample, since over the time tc diffusion has only developed over a length O(lc ). At the next order we have: ∂c∗(0) + divy∗ (−Dmol∗ grady∗ c∗(0) + c∗(0) v∗(1) + c∗(1) v∗(0) ) = 0 ∂t∗ Which, through integration over Ω∗f gives the following compatibility condition: ∂c∗(0) =0 ∂t∗ This result is impossible because this dimensionless term is O(1) by construction. The situation Pl = O(1) is therefore not homogenizable: a macroscopically equivalent continuum does not exist. This last case illustrates the consistency of the method mentioned in section 2.4.1 of Chapter 2. 5.7. Example: parallel slits

porous medium consisting of a periodic lattice of narrow

In order to illustrate the above results, we will consider a porous medium consisting of a periodic lattice of narrow parallel slits, whose characteristic dimensions are given

156

Homogenization of Coupled Phenomena

e2 L2

e1

(a)

L1 e2

l2

f

h

e1

(b)

Figure 5.2. Porous medium consisting of a periodic lattice of narrow slits:(a) macroscopic structure, (b) dimensionless period Ω∗

in Figure 5.2. The molecular diffusion tensor is isotropic, Dmol∗ = D mol∗ I, and ﬂow is driven by a macroscopic pressure gradient in the direction e1 : gradx∗ p∗(0) =

dp∗(0) e1 dx∗1

5.7.1. Analysis of the ﬂow As we will see in Chapter 7, the velocity v∗(0) and the pressure p∗(1) , both y -periodic, are obtained by solving over Ω∗ the boundary value problem which takes the following dimensionless form: ∗

η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) = 0 divy∗ (v∗(0) ) = 0

within

v∗(0) = 0

Γ∗

over

within

Ω∗f

Ω∗f

The speciﬁc geometry of the porous medium necessarily implies that: ∗(0)

v∗(0) = v1

(y2∗ ) e1

and p∗(1) = p∗(1) (y2∗ )

Diffusion/Advection in Porous Media

157

Thus the preceding problem reduces to: η∗

∗(0)

dp∗(0) d2 v 1 − =0 dy2∗2 dx∗1

∗(0)

v1

within

Ω∗f

= 0 over Γ∗

Solving this system of equations leads us to: ∗(0) v1

1 dp∗(0) = ∗ 2η dx∗1

y2∗2

h∗2 − 4

whose mean can be written: ∗(0)

v1

=−

φh∗2 dp∗(0) 12η∗ dx∗1

5.7.2. Determination of the dispersion coefﬁcient The dispersion coefﬁcient is obtained by solving over Ω∗ the boundary value problem (5.45-5.46). In this particular case, the components χdis∗ only depend on k y2∗ , and this boundary value problem becomes: d dy2∗

dis∗ ∗(0) ∗(0) mol∗ dχ1 D = v1 − φ−1 v1 dy2∗

d dy2∗

dχdis∗ 2 Dmol∗ 1 + =0 dy2∗

d dy2∗

dis∗ mol∗ dχ3 D = 0 within Ω∗f dy2∗

Dmol∗

D

Dmol∗

within Ω∗f

dχdis∗ 1 .n2 = 0 over Γ∗ dy2∗

mol∗

dχdis∗ 1 1+ dy2∗

.n2 = 0

dχdis∗ 3 .n2 = 0 over Γ∗ dy2∗

dis∗ dis∗ χdis∗ 1 = χ2 = χ3 = 0

within Ω∗f

over Γ∗

158

Homogenization of Coupled Phenomena

Solving this system, we ﬁnd: 1 = 24η ∗ Dmol∗

χdis∗ 1

y2∗2

h∗2 1− 2

7h∗4 + 240

dp∗(0) dx∗1

χdis∗ = −y2∗ 2 χdis∗ =0 3 Using equation (5.51), the two non-zero components of the dispersion tensor are: dis∗ D11

= φD

φh∗6 + 30240η ∗2 Dmol∗

mol∗

dp∗(0) dx∗1

2

dis∗ D33 = φDmol∗ dis∗ dis∗ We note that D33 is purely diffusive and that D11 is the sum of a diffusive part and a dispersive part arising from the coupling between the concentration at the ﬁrst order c∗(1) and the advection (v∗(0) ). In addition, it can be shown that:

Pe∗2 dis∗ D11 = φDmol∗ 1 + 210 where Pe∗ is the Péclet number deﬁned by: Pe∗ =

∗(0)

v1 h∗ φDmol∗

Finally, in dimensional form, the dispersion tensor for the porous medium consisting of a periodic lattice of narrow parallel slits can be written: ⎛ dis ⎞ D11 0 0 0 0 ⎠ Ddis = ⎝ 0 dis 0 0 D33 with: dis D11

= φD

mol

Pe2 1+ , 210

dis D33 = φDmol

and

Pe =

v1 h φDmol

Here we recover the results established by Wooding [WOO 60]. When the geometries are more complex, the determination of the dispersion tensor requires the numerical solution of the boundary value problem (5.45-5.46).

Diffusion/Advection in Porous Media

159

For example, numerical estimates can be found in [EDW 91; SAL 93; COE 97; SOU 97] of the dispersion tensor for porous two-dimensional and three-dimensional media (both periodic and random).

5.8. Conclusion The different models obtained, as well as their domains of validity, are summarized in Table 5.1. There is a continuum, as the Péclet number decreases, from the dispersion model to the diffusion-advection model and then to the diffusion model. Pel Pl

Time tc of observation

Diffusion/dispersion tensor

Model

ε2 ε2

2 tdif L = Lc /Dc

Diffusion: Ddif

Diffusion

Diffusion: Ddif

Diffusion-advection

Dispersion: Ddis

Advection Diffusion: Dispersion

ε

2 adv ε2 tdif L = Lc /Dc = tL

ε

tadv L = Lc /vc

ε−1 1

tadv L = Lc /vc

1

Not homogenizable

Table 5.1. Macroscopic models of solute transport in a porous medium

The experimental measure of the dispersion coefﬁcient raises various difﬁculties due to the size L of the macroscopic lattice being used [AUR 97]. The macroscopic models obtained above are useful in regions of the macroscopic domain where there is a separation of length scales. This introduces two types of limitation on the range of validity of the models. First this condition must be fulﬁlled both by the geometry of the porous medium: εg =

lc 1 Lc

(5.62)

where Lc is the characteristic size of the sample, and also by the physical process under consideration. In the case of solute transport in a test column, we can write that

160

Homogenization of Coupled Phenomena

condition as: εph =

lc 1, Lph

Lph ≈ |

∂ c −1 ( )| ∂X c0

(5.63)

where c0 is the concentration (assumed constant) at the entrance to the column. These limitations are analyzed in [AUR 97] for column experiments. In addition, the condition of separation of length scales is not met close to macroscopic boundaries: in general a boundary layer must be introduced to connect the solution obtained by homogenization and the conditions at the boundaries of the lattice [BEN 78]. Determination of the effective coefﬁcients from the experimental parameters will be valid if the thickness of this boundary layer is small compared to the size of the lattice and the measurement is performed outside this boundary layer [LEW 98]. It has been observed that for elliptical or parabolic problems such as problems of elasticity or pure diffusion, the thickness of the boundary layer is approximately lc [SAN 87], with lc ≈ 5 to 10 grain sizes [BEA 72; ANG 94]. Thus in these cases the limitation on the domain of validity of the macroscopic models is given by the conditions εg 1 and εph 1. When advection is present at the macroscopic scale, the thickness of the boundary layer changes from lc to δc > lc . Now we will evaluate δc . The characteristic time tdif l is the time taken by the solute to diffuse over a distance lc : tdif l =

lc2 Dc

When advection is present, the solute particles are also convected at velocity vc . Over the time tdif l , the solute particles are transported over a distance: vc tdif l =

lc2 vc Dc

The thickness of the boundary layer in the presence of advection is therefore: δ c = lc +

lc2 vc Dc

Compare δc and lc : vc lc δc =1+ = 1 + Pel lc Dc The thickness of the boundary layer increases with the Péclet number. The length Lc of the macroscopic lattice must be increased in consequence of this, particularly in the presence of dispersion. The numerical investigations of Salles et al. [SAL 93] conﬁrm the above results.

Chapter 6

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

6.1. Introduction The two preceding chapters have highlighted different models of macroscopic transfer depending on the properties of the constituents and their interfaces on the microscopic scale. Each of these models – valid for various periodic geometries obeying certain connectivity conditions – has been elaborated in an analytical manner for bilaminate composites. This chapter will extend and complete these results, shifting the focus onto ﬁbrous, cellular or granular materials: – the numerical procedure of homogenization is illustrated through determination of the macroscopic transfer coefﬁcients for composite media with spherical inclusions (connected or disconnected) and ﬁbrous media; – for comparison, common self-consistent analytical estimates are recalled, with this approach also being extended to three-constituent materials and those with contact resistance; – ﬁnally, comparison with experimental results allows us to judge the suitability of these models for describing the properties of real materials. The aim is mostly illustrative. The main purpose is to present some basic results, numerical or analytical, for periodic or random microstructures. The calculation of effective properties has been the subject of a rich literature which explores complex random three-dimensional geometries, or geometries corresponding to real materials. On this subject, outside the scope of the present work, we refer the reader for example to: [PER 79; SAN 82a; THO 90; ADL 92a; b; COE 97; and SAH 03].

162

Homogenization of Coupled Phenomena

6.2. Effective thermal conductivity for some periodic media In this ﬁrst part, we determine effective thermal conductivities for periodic composite media with spherical inclusions, either connected or disconnected, as well as for ﬁbrous media, by solving the canonical boundary value problem resulting from the process of homogenization as presented in Chapter 4 (section 4.2.2) where the contact between the constituents is assumed to be perfect. In addition, on the microscopic scale, the thermal conductivities of each of the constituents are assumed to be constant, isotropic, and of the same order of magnitude as each other: λa = λa I,

λb = λb I,

λa = O(λb )

and

6.2.1. Media with spherical inclusions, connected or non-connected 6.2.1.1. Microstructures Here we consider composite media consisting of a periodic lattice of spherical inclusions (phase b), connected or non-connected, in a matrix (phase a). The volume fractions of the matrix and the inclusions are ca and cb = 1 − ca respectively. The radius of the inclusions is denoted R. The periodic lattices we will consider are simple cubic (SC) and body centered cubic (BCC) lattices. The characteristic dimensions of the period corresponding to each of these microstructures under study are shown in Figure 6.1. Phase b for the SC lattice and the BCC lattice is connected when the

l

l e2

e3

(a)

b R

e2

a

e1

e3

b R

a e1

(b)

Figure 6.1. Composite media consisting of a periodic lattice of inclusions, connected or non-connected. Characteristic dimensions of the period. (a) Simple cubic lattice, (b) body centered cubic lattice

volume fraction of the inclusion cb is greater than π/6 ≈ 0.523 and √ π 3/8 ≈ 0.68 respectively. Above this volume fraction, or percolation threshold, region Ωb consists of connected, truncated spheres. For the simple cubic lattice, the contact number between the spheres is 6, whereas for the centered cubic lattice the contact number is 8, and then 12 when cb > 0.93.

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

163

Given the symmetries of each of these microstructures, the effective thermal conductivity tensor is isotropic: λeﬀ = λeﬀ I

(6.1)

6.2.1.2. Solution to the boundary value problem over the period In dimensionless form, the effective thermal conductivity for the materials in question is deﬁned in equation (4.82): 1 ∂t∗ai ∂t∗bi eﬀ∗ ∗ ∗ ∗ ∗ λa (δij + ) dΩ + λb (δij + ∗ ) dΩ λij = ∗ (6.2) |Ω | ∂yj∗ ∂yj Ω∗ Ω∗ a b where the y∗ -periodic vector t∗ (y∗ ) = (t∗1 , t∗2 , t∗3 ) is the solution to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) over the period Ω∗ . This boundary value problem can be written in the following compact form: divy∗ (λ∗ (grady∗ t∗ + I)) = 0 in Ω∗ [t∗ ]Γ∗ = 0 over Γ∗ [λ∗ (grady∗ t∗ + I)]Γ∗ · n = 0

over Γ∗

t∗ = 0 The three elementary solutions t∗i (y∗ ) are associated with unit macroscopic gradients in the three directions gradx∗ T ∗(0) = ei (i = 1, 2, 3). Given the symmetries of the microstructures, all we need to determine is t∗1 (y∗ ). This boundary value problem has been solved by ﬁnite element analysis [COM 08]. Figures 6.2 (a) and (b) present respectively, for each of these microstructures under study, the magnitude of vector t∗ = t∗3 e3 solving the canonical problem P(λ∗a , λ∗b ) for λa /λb = 10 and ca = 0.6. The symmetries of the microstructures imply that t∗3 is symmetric with respect to the axis e3 and non-symmetric with respect to axes e1 and e2 . These comments are valid whatever the volume fraction ca and the ratio of the conductivities λa /λb . 6.2.1.3. Effective thermal conductivity Figure 6.3 presents, for two conductivity ratios λa /λb (=10 or 100), the evolution of effective thermal conductivity λeﬀ /λa obtained from numerical simulations as a function of the volume fraction of the matrix ca . These ﬁgures show that (i) for the two microstructures under consideration, and whatever the conductivity ratio λa /λb (with λa > λb ), λeﬀ is a monotonically increasing function of ca , and (ii) whatever the conductivity ratio λa /λb , λeﬀ becomes sensitive to the arrangement of the inclusions when the volume fraction cb of the inclusions is greater than 0.5.

164

Homogenization of Coupled Phenomena

e2

e2

e3

e1

(a)

e3

e1

(b)

Figure 6.2. Elementary t∗3 solution to the canonical problem P(λ∗a , λ∗b ), (ca = 0, 6, λa /λb = 10). (a) Simple cubic lattice (SC) (black = -0.09, white = 0.09), (b) body centered cubic lattice (black = -0.073, white = 0.073)

6.2.1.3.1. Comparison with the Voigt bounds (V), Reuss (R) and Hashin and Shtrikman (HS+ and HS-) The upper Voigt bounds [VOI 87] (V) and lower Reuss bounds [REU 29] (R) are, for λeﬀ : λR =

λa λb λeﬀ ca λa + (1 − ca )λb = λV (1 − ca )λa + ca λb

(6.3)

If we make the additional assumption that the material is isotropic on the macroscopic scale, improved bounds have been established by Hashin and Shtrikman [HAS 63] (Chapter 1, equation 6.4). These bounds, denoted HS+ and HS-, are for λa > λb : λHS− = λb +

ca +

1 λa −λb

cb 3λb

λeﬀ λa +

cb +

1 λb −λa

ca 3λa

= λHS+

(6.4)

Figure 6.3 shows that numerical estimates of the effective conductivity of the two microstructures under study lie between bounds whatever the conductivity ratio and the volume fraction of the matrix ca . We can also note that the numerical results are very close to the upper Hashin and Shtrickman bound (6.4) over a large range of volume fraction of the matrix ca . The differences are considerably more pronounced when both phases are connected. 6.2.1.3.2. Comparison with the self-consistent estimate with simple inclusions The original self-consistent scheme involves considering simple homogenous spherical inclusions (Chapter 1, section 1.4). In this particular case, the effective conductivity is given implicitly by equation (1.30). As Figure 6.3 shows, this selfconsistent estimate (SCE) underestimates the conductivity of the two periodic media

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

165

1

a / b = 10

e& / a

0.8

V

0.6

HS+ 0.4

HS-

SCE

R 0.2

0 0

0.2

0.4

ca

0.6

0.8

1

1

a / b = 100

e& / a

0.8

V

0.6

HS+ 0.4

0.2

HS-

SCE

R 0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.3. Evolution of the effective thermal conductivity λeﬀ /λa of simple cubic () and centered cubic () lattices with spherical inclusions, as a function of the volume fraction of the matrix, ca . Comparison, for two conductivity ratios λa /λb (=10 or 100), of the numerical results with the Voigt (V) and Reuss (R) bounds (equation (6.3)), of Hashin and Shtrickman (HS+ and HS-) (equation (6.4)) and the self-consistent estimate (SCE) with simple inclusions (equation (1.30))

in question, whatever the volume fraction of the inclusions. The differences are more marked when the volume fraction ca of the matrix is low, i.e. when the inclusions are connected. As has already been emphasized, the traditional self-consistent scheme

166

Homogenization of Coupled Phenomena

with simple spherical inclusions does not allow us to capture the connected or dispersed nature of the phases independently of their concentration. We will see later (section 6.3.1) that this can be achieved by applying the self-consistent scheme with bi-composite substructures, for example. 6.2.1.3.3. Effects of the thermal conductivity contrast λa /λb Evolution of the effective thermal conductivity λeﬀ /λa , as derived from numerical simulations as a function of the thermal conductivity contrast λa /λb , for a ﬁxed volume fraction of the matrix ca = 0.5, is shown in Figure 6.4. 6

HS+ V

ca = 0.5

5

e& / a

4

HS-

3

2

R

1

HS0 10-4

HS+

V

R 10-2

100

102

104

a / b Figure 6.4. Evolution of the effective thermal conductivity λeﬀ /λa of simple cubic lattices () et and centered cubic lattices () with spherical inclusions as a function of the conductivity ratio λa /λb (ca = 0.5).Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.4))

This ﬁgure emphasizes the high sensitivity of conductivity λeﬀ to the conductivity contrast λa /λb , when 10−3 < λa /λb < 103 . For weaker or stronger conductivity ratios, effective conductivity becomes nearly constant. In addition, we ﬁnd a good agreement between the numerical results for the centered cubic lattice and the upper (if λa /λb > 1) and lower (if λa /λb < 1) Hashin and Shtrickman bounds (1.24). 6.2.1.3.4. High conductivity contrast When the conductivity ratio λa /λb = ∞ (or λb = 0), the effective thermal conductivity reduces to that of matrix a on its own: λeﬀ a (Chapter 4, section 4.2.3.2).

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

167

It is obtained by solving the canonical problem P(λ∗a , 0). The evolution of λeﬀ a /λa as a function of the volume fraction ca is shown in Figure 6.5. It can again be seen that the upper Hashin and Shtrikman bound [HAS 63] is still a good approximation to λeﬀ a across a wide range of volume fractions of the matrix ca . These results clearly apply to all diffusive phenomena, such as solute diffusion, of weak concentration in a ﬂuid saturating the porous skeleton consisting of inert dif mol spherical inclusions. By analogy, λeﬀ , where Ddif is a (ca )/λa becomes D (φ)/D mol is the molecular diffusion coefﬁcient and φ is the the effective diffusion tensor, D porosity (Chapter 5, section 5.3). 1

a / b =

e&/ a

0.8

0.6

V HS+

0.4

0.2

HS-

R

0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.5. Evolution of the effective thermal conductivity λeﬀ /λa of simple cubic () and centered cubic () lattices with spherical inclusions when λb = 0 (ca = 0.5). Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.4))

When the conductivity ratio λa /λb tends to zero, and the spherical inclusions are non-connected, this situation corresponds to that of highly conductive inclusions embedded in a matrix (Chapter 4, section 4.2.4). The effective conductivity λ+eﬀ is obtained by solving the speciﬁc boundary value problem deﬁned by equations (4.63– 4.64). When the inclusions are connected, the effective thermal conductivity tends towards that of the network of inclusions (phase b) λeﬀ b , given by solving the canonical problem P(0, λ∗b ). Figure 6.6 shows the evolution of λeﬀ /λa when λa /λb = 10−3 . This ﬁgure underlines the importance of the connectivity of the phases on the effective

168

Homogenization of Coupled Phenomena

1000

a / b = 0 . 0 01 HS+

e& / a

100

10

HS-

1 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.6. Evolution of the effective thermal conductivity λeﬀ /λa of simple cubic () and body centered cubic () lattices of spherical inclusions when λa /λb = 0, 001. Comparison of numerical results with the Hashin and Shtrikman (HS+ and HS-) bounds (equation 6.4)

conductivity. When the volume fraction of the inclusions is below the connectivity threshold (cb = 0.48 for the simple cubic lattice and cb = 0.32 for the body centered cubic lattice), the transfer and effective coefﬁcient (λ+eﬀ ) are controlled by the weakly-conducting matrix. However, the inﬂuence of highly conducting inclusions on the effective conductivity is not negligible. For a volume fraction of the matrix ca = 0.6, the effective conductivity is typically three times higher than that of the matrix. Finally, we observe that in this case the effective conductivities of the two microstructures are very close to the lower Hashin and Shtrikman bound. On the other hand, above the connectivity threshold the transfers are dictated by the connected network of highly conducting inclusions (≈ λeﬀ b ). In this case, the effective conductivities of the two materials are very close to the upper bound of Hashin and Shtrikman. The differences, very obviously close to the connectivity threshold, reﬂect the inﬂuence of the geometry of the contacts between spherical particles on the transfer processes. 6.2.2. Fibrous media consisting of parallel ﬁbers 6.2.2.1. Microstructures We will consider ﬁbrous media consisting of a periodic square or triangular lattice of parallel ﬁbers (phase b) buried in a matrix (phase a). The ﬁbers have a circular cross-section, and their radius is denoted R (Figure 6.7).

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

l

l 2R b

169

l

a

b 2R

$ l 3

a

e2 (a)

e1

(b)

Figure 6.7. Fibrous media consisting of (a) square and (b) triangular arrangements of parallel ﬁbers with a circular cross section, embedded in a matrix. Characteristic dimensions of the period

The ﬁbrous networks we will consider are not connected. For the square network, the volume fraction of the ﬁbers cb = πR2 /l2 , varies between 0 and cbmax = π/4 ≈ 0.785. For of the ﬁbers is given by cb = √ the triangular lattice, the volume fraction √ 2πR2 /( 3l2 ) and varies between 0 and cbmax = π/(2 3) ≈ 0.907. Given the symmetries of each of these microstructures, the effective thermal conductivity tensor is transversely isotropic and can be written: eﬀ λeﬀ = λeﬀ T (e1 ⊗ e1 + e2 ⊗ e2 ) + λL e3 ⊗ e3

(6.5)

eﬀ eﬀ eﬀ eﬀ where λeﬀ T (= λ11 = λ22 ) and λL (= λ33 ) are the transverse and longitudinal effective thermal conductivities respectively.

6.2.2.2. Solution to the boundary value problem over the period As for composites with spherical inclusions, the effective thermal conductivity of the materials under consideration, deﬁned by equation (6.2), is obtained by determining the y∗ -periodic vector t∗ (y∗ ) = (t∗1 , t∗2 , t∗3 ) which solves the canonical problem P(λ∗a , λ∗b ) (4.26–4.30) over the period. For the microstructures we are considering, all components t∗i are independent of y3∗ and it must be the case that t∗3 = 0. As a result, determination of the vector t∗ reduces to solving the canonical problem P(λ∗a , λ∗b ) in the plane (e1 , e2 ). This problem has been solved by ﬁnite element analysis [COM 08]. Figures 6.8 and 6.9 show respectively, for each of the microstructures studied, the mesh used for the numerical simulations as well as the solutions t∗1 and t∗2 . The symmetries of the microstructure imply that t∗1 (t2 ) is symmetric and symmetric with respect to the axes e1 and e2 (e2 and e1 ).

170

Homogenization of Coupled Phenomena

e2

e1

(a)

(b)

(c)

Figure 6.8. Fibrous media consisting of a square lattice of parallel ﬁbers (ca = 0.5, λa /λb = 100): (a) example of the mesh used for ﬁnite element numerical simulation, (b) t∗1 and (c) t∗2 (black = -0.125, white = 0.125), solutions to the canonical problem P(λ∗a , λ∗b )

e2

(a)

e1

(b)

(c)

Figure 6.9. Fibrous media consisting of a triangular lattice of parallel ﬁbers (ca = 0.5, λa /λb = 10) (a) example of the mesh used for ﬁnite element numerical simulation, (b) t∗1 and (c) t∗2 (black = -0.277, white = 0.277), solutions to the canonical problem P(λ∗a , λ∗b )

6.2.2.3. Effective thermal conductivity Since t∗3 = 0, equation (6.2) implies that the longitudinal conductivity λeﬀ L (= ) of the two microstructures under consideration is the arithmetic mean of the λeﬀ 33 conductivities of constituents a and b: λeﬀ L = ca λa + (1 − ca )λb

(6.6)

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

171

This result can easily be seen from the fact that the transfer parallel to the axis of the ﬁbers occurs at the same constant temperature gradient. This does not apply for transfer perpendicular to the axis of the ﬁbers. Figure 6.10 shows, for two conductivity ratios λa /λb (=10 or 100), evolution of the effective transverse thermal conductivity λeﬀ T /λa determined from numerical simulations as a function of the volume fraction of the matrix ca . As for composites with spherical inclusions, these ﬁgures show that (i) for the two microstructures under consideration, and whatever the conductivity ratio λa /λb (with λa > λb ), λeﬀ T is a monotonically increasing function of ca , and (ii) whenever the conductivity ratio λa /λb , λeﬀ T becomes sensitive to the arrangement of the ﬁbers when the volume fraction of the matrix ca falls below 0.4, i.e. when the volume fraction of the ﬁbers cb > 0.6. 6.2.2.3.1. Comparison with the Voigt (V), Reuss (R) and Hashin and Shtrikman (HS+ and HS-) bounds eﬀ The upper Voigt (V) bounds and lower Reuss (R) bounds for λeﬀ T and λL are given by equation (6.3). These bounds do not include the macroscopic anisotropy induced by the ﬁber arrangement. As we have already emphasized, when the transfer occurs parallel to the axis of the ﬁbers, each phase is subject to the same constant temperature V gradient, and for this reason the Voigt bound is reached: λeﬀ L =λ .

Despite the anisotropy caused by the arrangement of the ﬁbers, it is interesting to compare the numerical values of λeﬀ T with the bounds established by Hashin and Shtrikman [HAS 63] for a two-dimensional isotropic material. For λa > λb these bounds become: λHS− 2D = λb +

ca +

1 λa −λb

cb 2λb

λeﬀ T λa +

cb +

1 λb −λa

ca 2λa

= λHS+ 2D

(6.7)

As shown in Figure 6.10, the effective conductivities of the two microstructures lie between these bounds whatever the conductivity ratio and the volume fraction of the matrix ca . As for composites with spherical inclusions, it can be seen that the numerical results are very close to the upper bound of Hashin and Shtrikman (6.7) across a wide range of volume fractions. The differences become more pronounced close to the percolation threshold of the ﬁbers. 6.2.2.3.2. Inﬂuence of the thermal conductivity contrast λa /λb In the same way as for composites with spherical inclusions, Figure 6.11 emphasizes the high sensitivity of conductivity λeﬀ T to the thermal conductivity contrast λa /λb , when 10−2 < λa /λb < 102 . A good agreement can also be seen between the numerical results and the upper bounds (if λa /λb > 1) and lower bounds (if λa /λb < 1) of Hashin and Shtrikman.

172

Homogenization of Coupled Phenomena 1

a / b = 10 0.8

e& / T a

0.6

V HS+

0.4

HS-

R 0.2

0 0

0.2

0.4

ca

0.6

0.8

1

1

a / b = 100

e& / T a

0.8

0.6

V HS+

0.4

0.2

HS-

R 0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.10. Evolution of the effective transverse thermal conductivity λeﬀ T /λa of ﬁbrous media consisting of a square () or triangular () arrangement of parallel ﬁbers as a function of the volume fraction of the matrix ca . Comparison, for two conductivity ratios λa /λb (=10 or 100), of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.7))

6.2.2.3.3. Exchange of properties between the phases Keller [KEL 64] showed that the effective transverse conductivities of any parallel ﬁber network which has two orthogonal axes of symmetry e1 and e2 , have the following property: eﬀ λeﬀ 11 (λa , λb ) λ22 (λb , λa ) = λa λb

(6.8)

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

3

HS+ V

HS-

2

173

ca = 0.5

e& / T a

R

1

V R 0 10-4

10-2

100

HS-

102

HS+

104

a / b Figure 6.11. Evolution of the effective transverse thermal conductivity λeﬀ T /λa of ﬁbrous media consisting of a square () or triangular () lattice of parallel ﬁbers as a function of the conductivity ratio λa /λb (ca = 0.5). Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.7))

eﬀ where λeﬀ ii (., .) = λii (λmatrix , λﬁber ) (without summation). This property was then generalized by Mendelson [MEN 75] to any microstructure whose axes e1 and e2 are principal axes of the effective conductivity tensor. For the square and triangular lattices we are considering here, axes e1 and e2 are axes of symmetry and, conveniently, are also principal axes of λeﬀ . Because of this, property (6.8) becomes:

λeﬀ T (λa , λb ) = λa

λeﬀ T (λb , λa ) λb

−1 (6.9)

This property is satisﬁed by the numerical results presented in Figure 6.11. If we know the transverse conductivity λeﬀ T (λa , λb ) of a given microstructure, this equation means we can easily determine the effective transverse conductivity of this same microstructure if the properties of the phases are interchanged – in other words λeﬀ T (λb , λa ). 6.2.2.3.4. High conductivity contrast When the conductivity ratio λa /λb = ∞ (or λb = 0), the effective thermal conductivity tensor reduces to that of the matrix a alone: λeﬀ = λeﬀ a (Chapter 4,

174

Homogenization of Coupled Phenomena 1

a / b =

e& / Ta a

0.8

0.6

V HS+ 0.4

0.2

HS-

R

0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.12. Evolution of the effective transverse thermal conductivity λeﬀ Ta /λa of ﬁbrous media consisting of a square () or triangular () lattice of parallel non-conducting ﬁbers as a function of the volume fraction of the matrix (λa /λb = ∞). Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.7))

section 4.2.3.2). The effective longitudinal conductivity for the two microstructures considered reduces to: eﬀ λeﬀ L = λLa = ca λa eﬀ The transverse conductivity λeﬀ T = λTa can be obtained as before by numerical solution of the canonical boundary value problem P(λ∗a , 0) (4.26–4.30) over the period. The evolution of λeﬀ Ta /λa as a function of the volume fraction ca is shown in Figure 6.12.

The trends observed above for a ﬁnite conductivity ratio λa /λb are still valid. The upper bound of Hashin and Shtrikman [HAS 63] remains a good approximation to λeﬀ Ta across a wide range of concentration ca . Finally we note again that these results can be applied to all types of diffusion phenomena, in particular to solute diffusion, diluted in a ﬂuid saturating the ﬁbrous medium. By analogy, λeﬀ a (ca )/λa becomes Ddif (φ)/Dmol (Chapter 5, section 5.3). Finally, if the conductivity ratio λa /λb tends to zero, the effective longitudinal conductivity (6.6) reduces to: eﬀ λeﬀ L = λLb = (1 − ca )λb

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

175

In the transverse direction, this situation corresponds to highly conductive ﬁbers (inclusions) embedded in a matrix. In this particular case, the effective thermal conductivity tensor λ+eﬀ is obtained by solving the boundary value problem across the period (4.63–4.64) (Chapter 4, section 4.2.4). For the microstructures considered, the property (6.8) implies that: λ+eﬀ T = λa

λeﬀ Ta λa

−1

The ratio λ+eﬀ T /λa is clearly very close to the lower bound of Hashin and Shtrikman [HAS 63] across a wide range of the volume fraction ca (in contrast with the ratio λeﬀ Ta /λa , Figure 6.11). 6.3. Study of various self-consistent schemes The principle of self-consistent schemes was introduced in Chapter 1, taking as an example of application the traditional scheme where each constituent is treated independently of the others using a spherical substructure. We emphasized the morphological restrictions associated with this scheme (which leads to the fact that any phase with a concentration below one-third is dispersed). Here we will consider both procedures which can be applied to generic composite substructures, and also combinations of self-consistent schemes. These approaches, which do not require the periodicity of the medium, and lead to simple analytical formulations, can be justiﬁed in the context of random media and are often used for the representation of materials. 6.3.1. Self-consistent scheme for bi-composite inclusions 6.3.1.1. Granular or cellular media Consider a bi-composite medium where the a constituent, with concentration ca , is connected and the b constituent, with concentration cb = 1 − ca , is dispersed. Following the concept developed by [BRU 35; KER 56b; LAN 52; HAS 63], a single substructure Ω – consisting of a sphere Ωb of radius Rb of the b constituent surrounded by a concentric sphere Ωa of external radius Ra of medium a, following the same volume proportions as the real material (Figure 6.13) – is sufﬁcient to describe the material. We recall the reasoning developed in Chapter 1 (section 1.4) when embedding the composite sphere in a medium of conductivity λeﬀ . Subjecting this system to a uniform temperature gradient G = |G|ej = Gej at inﬁnity, the macroscopic (isotropic) conductivity satisﬁes general relation (1.29): λeﬀ =

N α=1

cα βα

176

Homogenization of Coupled Phenomena

G = G ez

ez

er

M

r Ra ( b

ey

O Rb a

ex

Homogeneous medium

Figure 6.13. Generic spherical composite substructure Ω of a bi-composite granular medium where the a constituent is connected

which for a single substructure reduces to: λeﬀ = β(λeﬀ , λa , λb )

(6.10)

where we recall that: β(λeﬀ , λa , λb )G =

1 ej · Ω

λb gradTb dΩ

λa gradTa dΩ + Ωa

Ωb

is the projection in the direction of G of the mean ﬂux crossing the substructure. We note that since there is only one substructure, the imposed gradient G is also the mean gradient over the substructure (for proof see section 1.4) so that: 1 G= Ω

λb gradTb dΩ = gradT

λa gradTa dΩ + Ωa

Ωb

We can easily express the problem of the bi-composite spherical substructure in equation form if we use spherical coordinates (r, θ, ϕ), with the origin at the center of the substructure Ω deﬁned by r < Ra , and the θ = 0 axis given by G. First of all, to encapsulate the isotropy of the space, the temperatures Tα in each medium (α = a, b), and the temperature T in the homogenous medium, must necessarily take the form: Tα = fα (r · r, r · G, G · G),

T = f (r · r, r · G, G · G)

Taking account of the linear dependence on G, Tα becomes: Tα = r · GFα (r) = Gcos(θ)fα (r)

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

177

whence: gradTα = G(fα cos(θ)er −

fα sin(θ)eθ ) r

Additionally, the Fourier equation can be written: λα (Tα ) = 0,

so that

r−2 ((r2 fα ) − 2fα ) = 0

and, integrating: fα (r) = Aα

Ra2 r + Bα 2 r Ra

Since the temperature takes a ﬁnite value at the center of the substructure, Ab = 0. This means that the ﬂux is uniform in the internal sphere. Following the same methodology, we have in the homogenous medium: f (r) = A

Ra2 r +B 2 r Ra

Also, since G is the uniform gradient applied at inﬁnity, we require that B = G. From continuity of temperature: gradTa dΩ + gradTb dΩ = Ta ndS = T ndS = Ω(A + B)ej Ωa

Ωb

∂Ω

∂Ω

and since G = gradT it follows that A = 0. Hence the ﬂux is also uniform in the medium outside the substructure, so that at its boundary the substructure is subject to homogenous conditions of temperature gradient and ﬂux. Because of this, we know that (for proof see Chapter 1, Reuss bound): 1 λa gradTa dΩ + λb gradTb dΩ = λeﬀ gradT = λeﬀ G Ω Ωa Ωb an equality which is consistent with (6.10). We still need to express the conditions of continuity of normal ﬂux and of temperature at the two interfaces (r = Rb , r = Ra ). Thus we obtain the following system of four equations in four unknowns: Bb

=

λb Bb

=

Aa + Ba R3 λa −2Aa a3 + Ba Rb

(6.11) (6.12)

178

Homogenization of Coupled Phenomena

Aa + Ba

=

G

(6.13)

λa (−2Aa + Ba ) = λeﬀ G

(6.14)

which only has non-trivial solutions for: λeﬀ = λa (1 +

1 − ca ) 1 ca − λb 3 1− λa

(6.15)

6.3.1.2. Fibrous media For ﬁbrous media we use a generic substructure consisting of a bi-composite cylinder (Figure 6.14), to which we apply the same calculation, bearing in mind that this geometry requires us to treat the transfers both along the axis and perpendicular to the axis of the ﬁbers. The solution for transfer parallel to the axis of the ﬁbers is obvious since they are subjected to a constant gradient: the macroscopic longitudinal conductivity is thus the arithmetic mean of the conductivities of the constituents. λeﬀ L = ca λa + (1 − ca ).λb The treatment of transfer perpendicular to the axis of the cylindrical substructure closely follows that for the spherical substructure. In the plane, r and θ designate the polar coordinates, Tα is the temperature in media α = a, b and T is the temperature in the homogenous medium. These take the form: Tα = Gcos(θ)gα (r),

T = Gcos(θ)g(r),

gα (r) = Aα

with

with

g(r) = A

Ra r + Bα r Ra

r Ra +B r Ra

The system to be solved is thus: Bb

=

Aa + Ba

λ b Bb

=

R2 λa (−Aa a2 Rb

Aa + Ba

=

G

(6.18)

λeﬀ T G

(6.19)

λa (−Aa + Ba ) =

(6.16) + Ba )

(6.17)

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

G = G ey

ey

179

er M r

Ra ( b

O

ex

Rb a Homogeneous medium

Figure 6.14. Generic composite cylindrical substructure Ω for a bi-composite ﬁbrous medium where the a constituent is connected

which gives: λeﬀ T = λa (1 +

1 − ca ) 1 ca − λb 2 1− λa

(6.20)

6.3.1.3. General remarks on bi-composite models We have seen that the bi-composite spherical (or cylindrical) substructure is neutral with respect to the equivalent medium, in the sense that replacing sphere (or a cylinder) in the equivalent medium with a substructure in no way disturbs the exterior ﬁeld, which remains uniform. This operation can therefore be repeated until the point where the whole space is occupied by homothetic substructures. As a result, the value λeﬀ is an exact value for this particular class of materials, whether the organization of the substructures is periodic, random, or anything else, and is independent of their size distribution (see Figure 6.15). As we have already mentioned several times, Hashin and Shtrikman [HAS 63] showed that for any bi-composite (random or otherwise) which has macroscopic isotropy, the value (6.15) determined using the more conductive connected constituent (λa > λb ) deﬁnes an upper bound (HS+), and conversely, (6.15) with the less conductive part (λa < λb ) deﬁning a lower bound (HS-). It can easily be veriﬁed that, in these models, the external phase of the substructure is necessarily connected: whether the conductivity of the internal phase is zero or inﬁnite, the effective conductivity remains ﬁnite. Thus, in contrast to traditional schemes, we can specify the connected or disperse nature of the phases, independent

180

Homogenization of Coupled Phenomena

Figure 6.15. Class of media consisting of homothetic composite spherical substructures for which the self-consistent result (6.15) is exact

of their concentration. It can also be observed that in the bi-composite inclusions the interaction between phases is explicit, whereas in the traditional scheme it is only expressed by way of the effective medium. These results apply to all phenomena governed by a diffusion process. For example, for diffusion of a solute with a diffusion coefﬁcient Dmol , diluted in a ﬂuid saturating a porous, granular, isotropic and inert medium, of porosity φ, the effective diffusion Ddif would be: 1 − φ 2φ (6.21) Ddif = Dmol 1 + φ = Dmol 3 −φ 3 −1 As for pressure diffusion, the permeability of mixtures consisting of clay with permeability ka and concentration ca , associated either with impermeable sand grains or with highly permeable pockets of water, would be respectively: 1 − ca eﬀ = ka 1 + ca kas , 3 −1

1 − ca eﬀ = ka 1 + 3 kaw ca

(6.22)

For macroscopically isotropic media, with the help of (6.15) we obtain estimates of the effective conductivities using models developed in Chapter 4: – bi-composites with conductivities of the same order of magnitude: 1 − ca ) λeﬀ = λa (1 + 1 ca − λb 3 1− λa

(6.23)

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

181

– highly conducting inclusions in a matrix: if we take the limit λb → ∞ we obtain: 1 − ca ) (6.24) λ+eﬀ = λa (1 + 3 ca – weakly conducting inclusions: in the limit λb → 0 we obtain: 1 − ca ) (6.25) λeﬀ a = λa (1 + ca 3 −1 We note that the term expressing the local non-equilibrium situation is calculated over the internal sphere only, and in the harmonic regime gives [AUR 83]: λb 1 3ωt iω iω (1 − τ (ω)dΩ = cb 1 + coth ) with ωt = 2 Ω Ωb iω ωt ωt Rb ρb Cb which concludes the formulation of the macroscopic parameters for this model. In Figures 6.3 and 6.10, the analytical values (6.15) and (6.20) (denoted HS+) are compared to those obtained numerically for periodic assemblies of spheres or cylinders, as a function of their concentration ca . We note the very good agreement of the two self-consistent and numerical results for the bi-composite substructure (HS+) as long as the inclusions are dispersed in the matrix. When this is not the case, the values diverge. In ﬁbrous media, contact between ﬁbers leads to an inversion of connectivity of the phases, which explains the observed differences. In a granular medium, contact between the spheres implies connectivity of both phases, and thus only a small difference. Approximations for other substructures close to their connectivity thresholds have been established by Andrianov et al. [AND 99]. Since in addition we know that the self-consistent values are exact regardless of the assembly (random or otherwise) of homothetic substructures, it is reasonable to assume that, other than for very speciﬁc morphologies, the self-consistent values (for a bi-composite spherical or cylindrical substructure) offer an acceptable estimate for bi-composites (cellular or ﬁbrous) where the matrix is connected and the inclusions dispersed. 6.3.2. Self-consistent scheme with multi-composite substructures 6.3.2.1. n-composite substructure The same principles can be used to treat n-composite substructures formed of concentric embedded spheres (or cylinders). Such a structure represents a medium formed of composite inclusions dispersed in a connected matrix, represented by the external surface of the substructure (Figure 6.16). For a medium with n constituents, numbered starting with constituent 1 at the center to constituent n on the outside of the substructure, the effective conductivity

182

Homogenization of Coupled Phenomena G = G ez

ez

er

( Rn

M

r ey

O 1

2 n1 n Homogeneous medium

ex

Figure 6.16. n substructure-composite

can be found by solving a 2nx2n system of equations. However, since boundary conditions at the interfaces are all of the same nature we can proceed by recurrence [BOU 96b]. Thus for an n-composite medium, deﬁning: γj = 1 − j

cj

j = 1, n

k=1 ck

the expression for the effective conductivity takes for form of a ﬁnite continued fraction: λeﬀ = λn (1+

γn 1 − γn − 3

1−

γn−1

λn−1 (1 + λn 1 − γn−1 − 3

where the term associated with layer i is: )

1 1−

..

. λi

)

1 .. 1−

γi λi (1 + .. 1 − γi . − 3

)(6.26)

1

.

..

.

γ2 λ2 (1 + λ3 1 − γ2 − 3

)

1 1−

λ1 λ2

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

183

For example, for a tri-composite medium, the effective conductivity would be given by: λeﬀ = λ3 (1 +

γ3 1 − γ3 − 3

)

1 1−

γ2 λ2 (1 + λ3 1 − γ2 − 3

)

1 1−

(6.27)

λ1 λ2

6.3.2.2. Treatment of a contact resistance In a multi-composite substructure, suppose one of the layers i has inﬁnitesimal thickness d = Ri+1 − Ri−1 and surface area Γ, so that the concentration ci of this constituent is linked to its speciﬁc surface area σ = Γ/Ω by ci = σd. How can we express the conditions between constituents i − 1 and i + 1? In layer i the normal components of the temperature gradient and ﬂux are effectively constant around the values: 1 (Ti−1 (Ri−1 ) − Ti+1 (Ri+1 )) , d

and

Q=

λi (Ti−1 (Ri−1 ) − Ti+1 (Ri+1 )) d

Thus, to a ﬁrst approximation, we can deduce from the two continuity equations for the ﬂux on the two surfaces of layer i: λi−1 grad(Ti−1 ) · er = Q

and

Q = λi+1 grad(Ti+1 ) · er

ﬁrstly the ﬂux continuity between constituents i − 1 and i + 1: λi−1 grad(Ti−1 ) · er = λi+1 grad(Ti+1 ) · er

(6.28)

and secondly, setting h = λi /d, the relationship between the ﬂux and the temperature differential between these two constituents: λi−1 grad(Ti−1 ) · er = h(Ti−1 (Ri−1 ) − Ti+1 (Ri+1 ))

(6.29)

Equations (6.28) and (6.29) deﬁne the boundary conditions encapsulating the presence of a contact resistance h−1 : when the resistance h−1 → ∞ the ﬂux is zero, and the surface introduces a perfect insulation; when h−1 → 0, the temperatures are continuous and the contact is perfect. Thus we can make direct use of the results for the multi-composite medium to estimate the effect of a contact resistance between the two constituents i − 1 and i + 1,

184

Homogenization of Coupled Phenomena

i by replacing λi with hd, and γi − 1 with σd/ 1 ck . Since d ≈ 0 and γi ≈ 1 the term associated with the contact resistance takes the form: λi h γi 1 1 1 ). )= (1+ (d+ ( )≈ .. .. .. 1 1 1 1 − γi σ σ . . . − + − i i .. .. 3 .. 3 1 ck 3h 1 ck . . . 1− d− λi h Thus for bi-composite media that are macroscopically isotropic, consisting of inclusions (medium b) dispersed in a matrix (medium a), with a speciﬁc surface area of contact σ and a resistance h−1 , the values of effective conductivity for the models developed in Chapter 4 (section 4.3) can be estimated as follows: – media with very low contact resistance (h−1 → 0) (model I), we recover the value obtained with perfect contact: 1 − ca ) (6.30) λIeﬀ = λeﬀ = λa (1 + 1 ca − λb 3 1− λa – media with moderate contact resistance (model II), in other words: h = O(λb /(3(1 − ca ))) σ we obtain an effective conductivity which depends on the contact resistance: 1 − ca λIIeﬀ = λa (1 + ) (6.31) 1 ca − λb 1 3 1− ( ) λb σ λa 1 + 3h(1−c a) λeﬀ a

– media with high contact resistance (h−1 → ∞) (model III), we recover the value where only the matrix contributes to the conduction: 1 − ca ) ; λIIIeﬀ = λeﬀ =0 (6.32) λIIIeﬀ = λIIIeﬀ a a = λa (1 + ca b − 1 3

As established through periodic homogenization, the dispersed inclusions, insulated by the contact resistance, do not participate in the transfer; – models with two temperature ﬁelds (coupled and decoupled): the same values (6.32) apply and H = hσ. 6.3.3. Combined self-consistent schemes Starting from the two traditional self-consistent schemes for composite substructures, it is possible to construct other schemes which are adapted to the morphology of the material.

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

185

6.3.3.1. Mixed self-consistent schemes A ﬁrst possibility involves applying a traditional scheme with composite inclusions. For example, for a medium with three constituents a, b and c, media b and c both being dispersed in matrix a with concentration ca < 1/3, it is clear that neither the traditional scheme nor the tri-composite sphere scheme are acceptable: in the traditional scheme phase a – with a concentration less than 1/3 – will be dispersed, and in the tri-composite scheme one of the constituents b or c will not be in contact with matrix a. If on the other hand we consider two bi-composite schemes {a, b} and {a, c}, and if we apply the traditional scheme to these, we are then able to respect the morphological charateristics of connectivity. We note however that the model is not unique since the respective concentration of the two substructures is an adjustable parameter. 6.3.3.2. Multiple self-consistent schemes Another possibility is to apply a self-consistent scheme several times. Consider for example a medium with three constituents, consisting of inclusions c in a matrix α, which itself consists of inclusions b in a matrix a. If a signiﬁcant difference in size exists between constituents a, b and inclusions c – the condition of separation of length scales – the heterogenous medium a, b behaves like a homogenous medium on the scale of the inclusions c. We can then proceed with a double homogenization, determining ﬁrst of all the effective conductivity λα of the matrix α with the help of a bi-composite inclusion {a, b}, and then the effective properties of the whole system using a substructure {α, c}. In this way the model we have constructed is unique, and its morphological structure is explicit (Figure 6.17), with phase a being connected, and phases b and c dispersed, but with very different sizes.

Figure 6.17. Morphology associated with a double homogenization procedure with a bi-composite inclusion

186

Homogenization of Coupled Phenomena

Applying result (6.15), and observing that in substructure {a, b} the concentrations of constituents a, b are respectively ca /(1 − cc ) and cb /(1 − cc ), and that in the substructure {α, c} the concentrations are cα = 1−cc and cc , we obtain in succession: λα = λa (1 +

cb ca − 3

1−

and λeﬀ = λα (1 +

),

1 λb λa

cc 1 − cc − 3

)

1 λc λα

1−

With the same concentrations ca , cb , cc we could also propose another model where the phase a remains connected, swapping the role of phases b and c, which then gives: λeﬀ = λβ (1 +

cb 1 − cb − 3

),

1 1−

with

λb λβ

λβ = λa (1 +

cc ca − 3

)

1 1−

λc λa

By way of example, we will consider the case of the permeability of a soil consisting of saturated clay (ca , ka ), grains of sand (cs , ks = 0) and pockets of water (cw , kw = ∞) [IBR 02]. The model where the sand/clay mixture surrounds the water pockets gives: λeﬀ = λa (1 + 3 ca 3

cs −1

)(1 + 3

cw ) 1 − cw

whereas the model where the clay and the water pockets surround the impermeable grains gives: λeﬀ = λa (1 + 3

cw cs )(1 + ) 1 − cs ca −1 3

Figure 6.18 illustrates the signiﬁcant difference between the two models and demonstrates the impact of the choice of morphology. Still on the subject of multiple combined schemes, we mention the recursive (or differential) model which proceeds by successive homogenizations with the traditional scheme [HAS 88; BER 02]. Its morphological interpretation is difﬁcult because of the fact that, at each stage, interactions between substructures occur through the intermediary of an effective medium, which assumes a very wide range of scales for the heterogenities. These examples should serve to demonstrate that the self-consistent approach with a composite substructure makes it fairly easy to describe media with inclusions dispersed in a matrix. On the other hand, media where several constituents are connected are clearly less easy to model.

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

187

cw

cs cw ez

cs

Figure 6.18. Impact of the morphology on the effective permeability of clay-sand-water mixtures. Isovalues normalized by the permeability of clay as a function of the concentration of sand cs and free water cw . (a) pockets of water dispersed in a sand-clay mixture (b) inclusions of sand dispersed in a macroporous saturated clay

Furthermore, we see that by altering the combination of self-consistent schemes used, many different models are possible. However, the difﬁculty here is not in establishing the value of the effective coefﬁcient, but rather in establishing what type of material it is reasonable to apply the model to. This is one of the important limitations of the self-consistent approach as compared to periodic homogenization.

188

Homogenization of Coupled Phenomena

6.4. Comparison with experimental results for the thermal conductivity of cellular concrete In this section we will compare self-consistent estimates with experimental results in order to judge the appropriateness of these models for describing the properties of real materials. We will concentrate here on the thermal properties of dry and damp cellular concrete, as studied in Boutin [BOU 96a]. Autoclaved cellular concrete is a construction material known for its insulating properties. Its weak thermal conductivity – lying between 0.08 and 0.3 W/mK – is of course higher than that of air (λair = 0.026 W/mK), but considerably lower than that of the minerals which it is made from (λs = 0.894 W/mK). This is a result of its high porosity – 0.65 < φ < 0.85 – which also leads to a low density of between 350 kg/m3 and 800 kg/m3 , clearly lower than that of the minerals it is made from ρs = 2650 kg/m3 . Its insulating properties depend strongly on its density, and we also know that for a given density the thermal conductivity increases signiﬁcantly with the presence of water in the material (λw = 0.602 W/mK). Based on experimental studies, empirical relationships have been established to link the conductivity to these two parameters, density and water level. We will compare these trends with selfconsistent estimates.

Figure 6.19. Macro-porosity of aerated concrete

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

189

The self-consistent approach with a composite substructure seems well suited to cellular concrete. This consists of a combination of porous structures on a series of very different length scales. We distinguish: – quasi-spherical macropores (with a diameter of the order of a millimeter) surrounded by a solid membrane (Figure 6.19); – a mesoporosity of the solid membrane formed of pores (with a size of around 10 μm) separating the crystalline aggregates; – ﬁnally, the microporosity of the crystalline aggregates (≈ 0.1μm). This scheme is not however entirely adequate, because although the unconnected spherical pores give a good description of the macroporosity, these only give a partial approximation to the porosity of the crystalline aggregates. This material, whose constituents are in perfect contact, is described by the canonical model (see Chapter 4, section 4.2.2). Although the air is a poor conductor compared to the solid, the memory effect behavior is never achieved, because the heat capacity of the air (ρCair = 10−3 J/Km3 ) is clearly smaller than that of the solid (ρCs = 1.6 J/Km3 ). To convince ourselves of this, we calculate in the harmonic regime the frequency ωnel at which the dynamic regime (local non-equilibrium) applies in the pores. It is enough to express that the thermal boundary layer thickness is of the same order as the size of the macropores lpore = 1 mm so that: λair = lpores , ρCair ωnel

numerically

ωnel ≈ 2.5 × 107 Hz

The macroscopic conduction model is only correct if the separation of length scales is respected. Let ωse be the frequency fulﬁlling this condition. Given the properties of the material, this condition leads us to: λbeton lpores , (1 − φ)ρCs ωse

numerically

ωse 1.2 × 104 Hz ωnel

As a consequence, when the problem is homogenizable, the air in the pores is still in the quasi-static regime, in other words in local equilibrium.

6.4.1. Dry cellular concrete To treat dry cellular concrete, we make direct use of result (6.15), considering that the material consists of bubbles of air surrounded by a solid membrane. Given the weak conductivity of the air with respect to that of the solid particles (λair λs ) and

190

Homogenization of Coupled Phenomena

expressing total porosity φ as a function of densities of the solid constituent ρs and the material ρ: ρ = ρs (1 − φ) we obtain, starting from equation (6.15): λeﬀ (ρ) = λs (1 +

φ 1−φ 3

−1

) = λs

2 2 = λs 3ρs −1 ρ −1

3 1−φ

(6.33)

Given the values of ρs and λs , in the normal range of densities the curvature of the function λeﬀ (ρ) is very slight, which means we can replace it by its gradient at the point ρ = 500 kg/m3 : λeﬀ (ρ) ≈ (−8.05 + 0.261ρ)10−3

(6.34)

Figure 6.20 shows the very good agreement between this theoretical expression, experimental results of Frey [FRE 92], and the empirical formula proposed by Millard [MIL 92]: λeﬀ (ρ) ≈ (−10.87 + 0.266ρ)10−3

Figure 6.20. Model-experiment comparison for the conductivity of dry cellular concrete as a function of its density. The continuous lines, which are almost coincidental, represent the empirical correlation of Millard [MIL 92] and the bi-composite inclusion model (equation 6.34). The squares () are the experimental values of Frey [FRE 92]

6.4.2. Damp cellular concrete Cellular concrete can take up water by condensation of water vapor in the smallest pores. Following the conventions established by experimental results, conductivity in

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

191

the damp state λeﬀ (ρ, u) follows from conductivity in the dry state λeﬀ (ρ), and from the content of water u by mass, with the empirical relationship: λeﬀ (ρ, u) ≈ λeﬀ (ρ)(1 + 4, 00u) For this damp material, we will examine two models. The ﬁrst model uses the self-consistent scheme with tri-composite inclusions with a layer of water, whose thickness depends on u, lying between the air and the solid membrane. In this description, the concentration of the water is the same whatever the size of the pores. The concentration of water in the pores cw can be expressed as a function of the content by mass of water u: u = cw φ

ρw ρ

where:

cw =

u ρw

1 1 − ρ ρs

−1

Due to the weak conductivity of the air compared to those of the two other constituents, the conductivity of cellular concrete with dry density ρ and a water content u can be written with the help of (6.27 ) as: λeﬀ (ρ, u) = λs (1 +

φ 1−φ − 3

1− 1−

cw 3

)

cw λw (1 + 2 ) 3 λs

and, in linearized form close to u = 0: λw ρs 1 eﬀ eﬀ λ (ρ, u) ≈ λ (ρ) 1 + u 1 − 3ρρs λs ρw a form which allows us to compare these results to those in the literature. For example, for ρ = 500 kg/m3 , we ﬁnd: λeﬀ (ρ, u) ≈ λeﬀ (ρ) (1 + 1.91u) Compared to the empirical relation, it is clear that the tri-composite sphere approach agrees qualitatively with the experiment, but underestimates the effect of the water content. This can be attributed to a poor description of the distribution of water in the pores. In the second approach, we will proceed by “double homogenization”. In order to better describe the distribution of water, which mostly occupies the small pores, we will consider that the air bubbles are surrounded by a porous medium saturated with water. We will then proceed by ﬁrst determining the effective conductivity of the

192

Homogenization of Coupled Phenomena

saturated medium, and then the conductivity of the ensemble. This approach assumes implicitly that the saturated pores are much smaller than the dry pores, so that on the length scale of the latter the saturated membrane can be considered as a homogenous medium. Following this description, the density of damp cellular concrete is that of the saturated medium. The volume density of water cw in this medium is thus directly linked to the water content by mass by the relationship: cw = u

ρs ρw

With a bi-composite structure where water is trapped in the solid sphere – which assures the connectivity of the solid phase – conductivity of the saturated membranes can be written: λsw = λs (1 +

cw

1 − cw 1 − 3 1 − λλws

)

and, with the presence of saturated pores of air with concentration cair , we obtain (neglecting λa ): λeﬀ (ρ, u) = λsw (1 +

cair 1−cair − 3

1

)

In order that this description with double homogenization should also reﬂect the properties of the dry material (u = 0), we must have: (1 +

cair cw φ ) = (1 + )(1 + ) 1 − cair 1−φ 1 − cw −1 −1 −1 3 3 3

from which it follows that: λeﬀ (ρ, u) = λeﬀ (ρ)(1 +

cw

1 − cw 1 − 3 1 − λλws

) (1 +

cw )−1 1 − cw −1 3

For water contents u < 0.2, approximating the curves by their tangent at the origin, we obtain the following approximation: λeﬀ (ρ, u) ≈ λeﬀ (ρ)(1 + u

9 2(2 +

λw λs )

λw ρ s ) λs ρw

Numerical and Analytical Estimates for the Effective Diffusion Coefﬁcient

193

which gives numerically: λeﬀ (ρ, u) ≈ λeﬀ (ρ)(1 + 3.45u) This equation is in good agreement with experimental results, particularly since the coefﬁcient obtained from the slope at the origin underestimates the dependence on u. Thus, for normal water contents, the approach by “double homogenization” appears to better describe the microstructure and gives a satisfactory description of the thermal properties of damp cellular concrete. In conclusion, despite simplifying the assumptions, we have obtained acceptable estimates for the conductivity of dry or damp cellular concrete in the form of simple analytical expressions which clearly show the essential parameters. This result is not speciﬁc to cellular concrete; other examples can be found which have a good agreement between theory and experiment, particularly for materials based on particles of vegetable matter [ARN 00; ARN 04], and for the permeability of heterogenous soils [IBR 02; KAC 04].

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PART THREE

Newtonian Fluid Flow Through Rigid Porous Media

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Chapter 7

Incompressible Newtonian Fluid Flow Through a Rigid Porous Medium

7.1. Introduction The slow ﬂow of a Newtonian ﬂuid in an isotropic porous medium is traditionally described using Darcy’s law [DAR 56], Darcy having carried out the ﬁrst experimental study of ﬂow through porous media: v = −

K gradX p η

(7.1)

where v is the ﬂuid ﬂux across a surface of the porous medium, K the permeability, η the dynamic viscosity and p the pressure. This is an example of the laws which had been discovered some time previously to model the effect of a viscous ﬂuid on an obstacle. Indeed, the physics described by (7.1) is that of the interaction of a Newtonian ﬂuid with an obstacle: a porous medium is an obstacle just like any other! Newton [NEW 87; NEW 26] analyzed the resistance R of a ﬂuid on a moving body and modeled it with: R = Av + Cv 2

(7.2)

where A and C are constants. The v 2 term is recognized as having an inertial origin, and Newton indicated that it is only an approximation – something with which all his successors would concur (Newton also proposed R = Av + Bv 3/2 + Cv 2 ). The laws of (7.1) and (7.2) describe a one-dimensional ﬂow. Clearly Darcy’s law (7.1) is equivalent to that of Newton (7.2) when the velocity is small. de Coulomb

198

Homogenization of Coupled Phenomena

gradP/hvi

[deC 01] also obtained equation (7.2) based on meticulous experiments. Equation (7.2) would then be rediscovered by many scientists [deP 04; WEI 45; DAR57 (DarcyWeisbach law); FOR 01 (Forchheimer law)]. Reynolds [REY 83] was the ﬁrst to have indicated the ranges of validity of the v and v 2 behavior. Chézy’s law [CHÉ 75; MOU 21], which can be written as R = Cv 2 , also appears as a special case of (7.2). It is hard to believe that none of these authors were aware of Newton’s work. Subsequently, many generalizations of Darcy’s law have been applied to more complex

hvi Figure 7.1. Evolution of the resistance to ﬂow as a function of velocity. The points correspond to the experimental results of Rasoloarijoana and Auriault [RAS 94]. Darcy’s law, a constant resistance obtained for very low velocities, can be corrected by weak inertial effects up to a velocity of 0.06 m/s (parabolic variation in resistance), beyond which inertial effects dominate (linear variation of resistance)

situations. In theoretical terms, we refer the reader to Marˇusi´c-Paloka and Mikeli´c [MAR 00] for the treatment of non-linearities which are neglected in the laminar regime. Equation (7.2) is only an approximation. We will not concern ourselves with investigation of this here. In this chapter, we intend to deﬁne the laws describing the ﬂow of an incompressible viscous Newtonian ﬂuid, using the method of multiple scale expansions, in a rigid porous medium in various different situations: – slow steady-state ﬂow of an incompressible Newtonian ﬂuid in a rigid matrix: Darcy’s law; – linear dynamics of an incompressible Newtonian ﬂuid in a matrix; – steady-state ﬂow of an incompressible Newtonian ﬂuid in a rigid matrix: the appearance of inertial non-linearities.

Incompressible Newtonian Fluid Flow

199

The work of Matheron [MAT 67] offers an interesting and different approach to that presented here, one which relies on probabilities. 7.2. Steady-state ﬂow of an incompressible Newtonian ﬂuid in a porous medium: Darcy’s law Here we will determine the macroscopic description of an isothermal ﬂow of a Newtonian ﬂuid in a rigid porous matrix. This topic has been the subject of much research in the context of the homogenization of periodic structures, initially by Ene and Sanchez-Palencia [ENE 75]. Other macroscopization techniques can also be applied [BEA 72; WHI 86]. A good review of early works can be found in Scheidegger [SCH 74]. We will consider a rigid porous matrix which is periodic with period Ω. The ﬂuid occupies the pores Ωf , and Γ represents the surface of the solid (Figure 7.2). To

Lc

lc

f

s n

(a)

(b)

Figure 7.2. (a) Porous medium, (b) period

simplify things, we will assume that the viscosity of the ﬂuid is constant. A variable viscosity, i.e. one which depends on the fast variable y∗ and which is periodic, would not introduce any further complications. The temperature is constant. The equations governing velocity v, pressure p and density ρ of an incompressible viscous Newtonian ﬂuid of viscosity η are in general terms the following, for a ﬂow in a matrix: – dynamics equation (Navier-Stokes): ∂v + (v grad ηΔX v − gradX p = ρ X )v ∂t – conservation of mass: divX (v) = 0

(7.3)

(7.4)

200

Homogenization of Coupled Phenomena

– no-slip condition: v|Γ = 0

(7.5)

Gravity is included here in the pressure term. Equations (7.3) and (7.4) introduce three dimensionless numbers: – the Reynolds number Re: |ρ(v gradX )v| Re = |ηΔX v| – the transient Reynolds number Rt : ∂v |ρ | ∂t Rt = |ηΔX v| – the number Q deﬁned by: |gradX p| Q= |ηΔX v| Since the ﬂow here is steady-state, Rt is zero. We also assume the movements to be slow: the local Reynolds number is small so that non-linearities do not appear on the macroscopic scale, at least to the ﬁrst order of approximation. Because of this, we can consider that: Rel = O

ρc vc lc ηc

= O(ε),

or

ReL = O

ρc vc Lc ηc

= O(1)

Thus the dominant terms in the Navier-Stokes equation are the viscous term and the pressure term. We will now evaluate ratio QL between these two quantities. In a ﬁltration experiment, the ﬂow is driven by a macroscopic pressure gradient: gradX p = O

pc Lc

At the same time, v varies inside the pores with a characteristic size lc : ηΔX v = O

ηc vc lc2

It follows that: pc =O Lc

ηc vc lc2

Incompressible Newtonian Fluid Flow

201

The dimensionless number QL is therefore: QL = O

pc L2c Lc ηc vc

= O(ε−2 )

The dimensionless equations describing the local situation for an incompressible ﬂuid can be written from the macroscopic viewpoint as: – dynamics equation: ε2 η ∗ Δx∗ v∗ − gradx∗ p∗ = ε2 ρ∗ (v∗ gradx∗ )v∗

(7.6)

– the conservation of mass reads: divx∗ v∗ = 0

(7.7)

– the no-slip condition: v∗ |Γ∗ = 0

(7.8)

7.2.1. Darcy’s law Having adopted the macroscopic viewpoint, we look for the unknowns v∗ and p∗ in the form: v∗ (x∗ , y∗ ) = v∗(0) (x∗ , y∗ ) + εv∗(1) (x∗ , y∗ ) + ε2 v∗(2) (x∗ , y∗ ) + · · · p∗ (x∗ , y∗ ) = p∗(0) (x∗ , y∗ ) + εp∗(1) (x∗ , y∗ ) + ε2 p∗(2) (x∗ , y∗ ) + · · ·

(7.9)

with y∗ = ε−1 x∗ , v∗(i) and p∗(i) being Ω∗ -periodic in y∗ . We will introduce these expansions into equations (7.6, 7.7, 7.8). Observing that the derivatives appearing in this system must be considered with respect to x∗ (macroscopic viewpoint), and that due to the double scale, the derivative operator becomes: ∂ ∂ + ε−1 ∗ ∂x∗ ∂y we obtain, identifying the powers of ε: grady∗ p∗(0) = 0

(7.10a)

η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) = 0 ———————-

(7.10b)

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Homogenization of Coupled Phenomena

divy∗ v∗(0) = 0

(7.11a)

divy∗ v∗(1) + divx∗ v∗(0) = 0

(7.11b) ———————-

v∗(0) = 0 v∗(1) = 0

(7.12a) over Γ∗

(7.12b) ———————-

Equation (7.10a) gives p∗(0) = p∗(0) (x∗ ). Equations (7.10b), (7.11a) and (7.12a) then represent the fundamental problem which must be solved over the period, where the unknowns v∗(0) and p∗(1) are Ω∗ -periodic in y∗ . We can see that v∗(0) and p∗(1) are linear functions of gradx∗ p∗(0) : v∗(0) = −

k∗ gradx∗ p∗(0) η∗

p∗(1) = a∗ · gradx∗ p∗(0) + p∗(1) (x∗ ) where the tensor k∗ is a function of the variable y∗ . Since the pressure p∗(1) is only deﬁned up to a constant value, we take the volume mean of the vector a∗ to be zero. Now consider (7.11b). This involves the local volume conservation of v∗(1) where divx∗ v∗(0) appears as a source term. Also, v∗(1) is Ω∗ -periodic and (7.12b) shows that it is zero over Γ∗ . The source term divx∗ v∗(0) satisﬁes a compatibility condition: its volume mean must be zero. This can be shown by integrating (7.11b) over Ω∗f . Using: 1 . dΩ∗ . = ∗ |Ω | Ω∗f we obtain: divx∗ v∗(0) = divx∗ v∗(0) = −

1 |Ω∗ |

=−

1 |Ω∗ |

Ω∗ f

divy∗ v∗(1) dΩ∗

∂Ω∗ f

v∗(1) · n dS ∗ = 0

(7.13)

where n is the unit normal to Γ∗ exterior to Ω∗f . Finally we have: divx∗ v∗(0) = 0, v∗(0) = −

K∗ gradx∗ p∗(0) , K∗ = k∗ η∗

(7.14)

Incompressible Newtonian Fluid Flow

203

which represents a macroscopic volume conservation, but also a conservation of momentum. Tensor K∗ is the permeability tensor. Equation (7.14b) is Darcy’s law (as long as v∗(0) is a ﬂux, i.e. a surface mean – see later). For a medium with a random structure, locally stationary, Ω∗ is the representative elementary volume, and the line of reasoning which leads to (7.14) is still valid. If we then include the uniqueness of the problem (7.10b, 7.11a, 7.12a) in the steady state case, all representations lead to K∗ and (7.14) follows. When viscosity varies across the period, the ﬂow law (7.14b) becomes v∗(0) = −Λ∗ gradx∗ p∗(0) , where Λ∗ is a hydraulic conductivity. Returning to dimensional variables with: x∗ =

X , Lc

v∗(0) =

η∗ =

η ηc

v(0) v ¯ = + O(ε), vc vc

p∗(0) =

p(0) p ¯ = + O(ε) pc pc

it follows that: ¯ divX v = O(ε),

v = −

ηc Lc vc 1 ∗ ¯ K gradX p + O(ε) pc η

But: μc Lc vc ∗ 2 ∗ 2 ∗ K = Q−1 L Lc K = lc K pc Thus we ﬁnally obtain: ¯ divX v = O(ε),

v = −

K ¯ gradX p + O(ε), η

K = lc2 K∗

¯ where O(ε) is a small term, of order ε relative to the other terms in the equality. 7.2.2. Comments on macroscopic behavior 7.2.2.1. Physical meaning of the macroscopic quantities The pressure p∗(0) does not raise any issues because, since it is independent of the local variable, its macroscopic deﬁnition is the same as that introduced at the microscopic scale. This is not a priori the case for v∗(0) , deﬁned as a volume mean, whereas a Darcy velocity is a ﬂux, i.e. a surface mean. In fact the two means are indeed equal here. This follows from the solenoidal character of v∗(0) . We start from the identity: ∗(0)

∂v ∂ ∗(0) ∗(0) (vk yi∗ ) = k ∗ yi∗ + vk Iik ∗ ∂yk ∂yk

204

Homogenization of Coupled Phenomena

Integrating over Ω∗f , with the divergence theorem and the no-slip condition (7.12a), we ﬁnd: 1 ∗(0) ∗(0) ∗ vi = ∗ v yi nk dS ∗ |Ω | ∂Ω∗f ∩∂Ω∗ k where ∂Ω∗f and ∂Ω∗ are the boundaries of Ω∗f and Ω∗ , and n is the unit normal vector exterior to Ω∗f . Let li∗ be the dimensionless length of the period along the yi∗ axis and Σ∗i the cross-section of the period at yi∗ = li∗ (see Figure 7.3).

l2 s1

f1

n

f s

y l1 Figure 7.3. Period Ω∗ of a porous medium (two-dimensional case) ∗(0)

Σ∗fi is the ﬂuid part of Σ∗i . Because vk yi∗ is Ω∗ -periodic in the yj∗ , j = i direction, and is zero for yi∗ = 0, we are left with: 1 ∗(0) ∗(0) ∗(0) vi = ∗ vi li∗ dS ∗ = |Σ∗i |−1 vi dS ∗ |Ω | Σ∗f Σ∗ f i

i

(without summation over i). As a result, v∗(0) is indeed a ﬂux. 7.2.2.2. Structure of the macroscopic law The mathematical structure of Darcy’s law is different to that of the Navier-Stokes equations from the outset. It still nevertheless involves the basic principles of mechanics, which can more easily be seen when Darcy’s law is written in the form: gradx∗ p∗(0) = −H∗ v∗(0) ,

H∗ = η ∗ K∗−1

which is permissible given the invertibility of K∗ (see section 7.2.2.4). The value of dimensionless number Q deﬁned above, or equation (7.6), gives an estimate of v(0) and thus of k and thence K: |K| = O(|k|) = O(lc2 )

Incompressible Newtonian Fluid Flow

205

This estimate is very approximate, since the geometry of the pores can alter the value of K to a very signiﬁcant extent, as we will see in Chapters 9 and 10. The equation above is often used to deﬁne lc starting from the permeability. Finally, in order to explicitly include gravity, all we need to do is to replace gradx∗ p∗(0) with gradx∗ p∗(0) + ρ∗ g∗ , where g∗ is the gravitational acceleration. 7.2.2.3. Study of the underlying problem In order to study the boundary-value problem (7.10b, 7.11a, 7.12a) where the unknowns v∗(0) and p∗(1) are Ω∗ -periodic in y∗ , we introduce the Hilbert space V of Ω∗ -periodic vectors with zero divergence, deﬁned over Ω∗f , with a value of zero on Γ∗ and possessing the scalar product: (α, v)V =

Ω∗ f

η∗

∂αi ∂vi ∗ dΩ ∂yj∗ ∂yj∗

Multiplying the two members of (7.11b) by α ∈ V and integrating them, we ﬁnd: Ω∗ f

∗

η αΔy∗ v

∗(0)

∗

dΩ =

∗(0)

Ω∗ f

αgradx∗ p

∗

dΩ +

Ω∗ f

αgrady∗ p∗(1) dΩ∗

Taking account of the equations: αΔv = div(α grad v) − grad α grad v α grad p = div(α p) − p divα of (7.11a), of the divergence theorem, and of (7.12a), it follows that: ∀α ∈ V,

∂αi η∗ ∗ ∗ ∂y Ωf j

∗(0)

∂vi dΩ∗ = − ∂yj∗

Ω∗ f

αi

∂p∗(0) ∗ dΩ ∂x∗i

(7.15)

Similarly, it is possible to show [SAN 80] that this formulation is equivalent to (7.10b), (7.11a) and (7.12a). The existence and uniqueness of the solution then follows from the Lax-Milgram theorem [NEC 67]. 7.2.2.4. Properties of K∗ Permeability tensor K∗ is positive and symmetric when the pores are connected in all three spatial directions. Let k∗j /η ∗ be the speciﬁc solution to (7.15) for velocity v∗(0) when: 1 if i = j, gradx∗i p∗(0) = 0 otherwise.

206

Homogenization of Coupled Phenomena

Consider the form (7.15) with on one hand v∗(0) = k∗q /η ∗ and α = k∗p /η ∗ , and on the other hand v∗(0) = k∗p /η∗ and α = k∗q /η. It follows, given the symmetry of the scalar product, that: ∗ ∗ ∂kpi ∂kqi ∗ ∗ ∗ ∗ dΩ = − kqp dΩ = − kpq dΩ∗ ∂yj∗ ∂yj∗ Ω∗ Ω∗ Ω∗ f f f The tensor K∗ is thus symmetric. We note that this symmetry follows from the conservation equations on the scale of the pores. Now consider (7.15) with α = v∗(0) : 1 |Ω∗ |

Ω∗ f

η∗

∗(0)

∗(0)

∂vi ∂vi dΩ∗ = ∂yj∗ ∂yj∗

−

1 |Ω∗ |

Ω∗ f

∗(0) ∗(0) ∂p dΩ∗ ∂x∗i

vi

=

∗ ∂p∗(0) ∂p∗(0) Kij ∂x∗i η ∗ ∂x∗j

(7.16)

The ﬁrst expression is positive. The same applies to the ﬁnal expression, and so K∗ is positive. 7.2.2.5. Energetic consistency The ﬁrst part of (7.16) represents the mean of the local dissipation density. We have: η ∗ Δy∗ v∗(0) = 2η ∗ divy∗ Dy∗ (v∗(0) ) where D is the strain rate tensor. The symmetry of D means we can write the weak formulation (7.15) in the form: ∀α ∈ V, 2η ∗ Dy∗ (α) : Dy∗ (v∗(0) )dΩ∗ = − α · gradx∗ p∗(0) dΩ∗ Ω∗ f

Ω∗ f

and (7.16) becomes: K 1 2η ∗ Dy∗ (v∗(0) ) : Dy∗ (v∗(0) )dΩ∗ = gradx∗ p∗(0) · ∗ gradx∗ p∗(0) ∗ |Ω | Ω∗f η Thus the mean of the local dissipation density equals the macroscopic dissipation density. 7.2.3. Non-homogenizable situations Now consider situations where QL = O(ε−2 ), with all other things remaining unchanged. It is sufﬁcient to study QL = O(ε−1 ) and QL = O(ε−3 ), with the other cases following immediately.

Incompressible Newtonian Fluid Flow

207

7.2.3.1. Case where QL = O(ε−1 ). The normalized Navier-Stokes equation can now be written: εη ∗ Δx∗ v∗ − gradx∗ p∗ = ερ∗ (v∗ gradx∗ )v∗ and we obtain for the different orders: η ∗ Δy∗ v∗(0) − grady∗ p∗(0) = 0

(7.17a)

η ∗ Δy∗ v∗(1) − gradx∗ p∗(0) − grady∗ p∗(1) = ρ∗ (v∗(0) grady∗ )v∗(0) ———————-

(7.17b)

divy∗ v∗(0) = 0

(7.18a)

divy∗ v∗(1) + divx∗ v∗(0) = 0

(7.18b)

divy∗ v∗(2) + divx∗ v∗(1) = 0

(7.18c) ———————-

v∗(0) = 0

(7.19a)

v∗(1) = 0

(7.19b)

v(∗2) = 0

over Γ∗

(7.19c) ———————-

Equations (7.17a, 7.18a, 7.19a), with v∗(0) and p∗(0) Ω∗ -periodic, represents a homogenous boundary-value problem. It is easy to show the existence and uniqueness of the solution. This follows directly from the discussion in 7.2.2.3. We obtain: v∗(0) = 0,

p∗(0) = p∗(0) (x∗ )

which is an acceptable because v∗(0) = O(1). The following problem involves Ω∗ -periodic v∗(1) and p∗(1) . It can now be written: ηΔy∗ v∗(1) − grady∗ p∗(1) − gradx∗ p∗(0) = 0 divy∗ v∗(1) = 0 v∗(1) = 0

over Γ∗

208

Homogenization of Coupled Phenomena

This is the boundary-value problem (the underlying problem over the unit cell) which we studied in 7.2.1., where v∗(0) has been replaced with v∗(1) . Thus: v∗(1) =

k∗ gradx∗ p∗(0) η∗

Finally (7.18c), along with (7.19c) and Ω∗ -periodicity, gives the compatibility condition: divx∗ v∗(1) = 0,

v∗(1) = −

K∗ gradx∗ p∗(0) η∗

As we will see, this is again an equivalent macroscopic description, but the ﬁrst nonzero term of v∗ is now εv∗(1) , so that the value of QL is ipso facto reduced to O(ε−2 ). Thus we have established by contradiction that a situation with QL = O(ε−1 ) does not exist in practice. 7.2.3.2. Case where QL = O(ε−3 ) The normalized Navier-Stokes equation is now written as: ε3 η ∗ Δx∗ v∗ − gradx∗ p = ε3 ρ∗ (v∗ gradx∗ )v∗ and we obtain for the different orders: grady∗ p∗(0) = 0

(7.20a)

gradx∗ p∗(0) + grady∗ p∗(1) = 0

(7.20b)

η∗ Δy∗ v∗(0) − grady∗ p∗(2) − gradx∗ p∗(1) = 0

(7.20c)

Equation (7.20a) again gives p∗(0) = p∗(0) (x∗ ). But (7.20b), with Ω∗ -periodic p∗(1) , introduces the compatibility condition gradx∗ p∗(0) = 0, which is not acceptable because gradx∗ p∗(0) = O(1). Thus p∗(0) is independent of x∗ and (7.20b) leads to p∗(1) = p∗(1) (x∗ ). Finally, (7.20c) gives v∗(0) in the form of a linear vectorial function of gradx∗ p∗(1) . Combining p∗(0) and p∗(1) into p∗(0) + εp∗(1) with: gradx∗ (p∗(0) + εp∗(1) ) = ε gradx∗ p∗(1) we again obtain a Darcy’s law with a smaller pressure gradient. But the value of QL is therefore reduced to the value O(ε−2 ) used in 7.2.1. A situation where QL = O(ε−3 ) is not homogenizable: an intrinsic macroscopic description does not exist. Here we have again arrived at a situation of non-homogenizability which is similar to the one we examined for the dynamics of elastic composites in section 3.3.2.3 of Chapter 3.

Incompressible Newtonian Fluid Flow

209

Thus the method of multiple scale expansions can be described as consistent with itself in that it only gives a result when homogenization is possible. To be rigorous, it is clear that a number QL = O(ε−3 ) indicates a strong pressure gradient, which implies a local non-linear description, with a different evaluation of the Reynolds number to that adopted here.

7.3. Dynamics of an incompressible ﬂuid in a rigid porous medium The force exerted on a rigid body with velocity v in slow transient motion in a Newtonian liquid is classically written as [NAV 22; LAN 71]: R=

t −∞

M (t − τ ) v(τ ) dτ

(7.21)

where M is a memory function which depends on the density and viscosity of the ﬂuid. (7.21) is a dynamic equation with memory, and the conservation of body momentum leads to an increased mass: its apparent mass is greater than its actual mass. The transient ﬂows in porous media are of the same nature [BIO 56b]. With respect to change of length scale studies, the results in this section can be found in Levy [LEV 79] and Auriault [AUR 80]. The study of dynamic behavior of a ﬂuid in a rigid porous matrix can at ﬁrst glance seem purely academic, since few applications can be envisaged where the ﬂuid can vibrate but the skeleton remains immobile, unless the ﬂuid density is much lower than that of the solid. Nevertheless it will become clear later (Chapters 8, 12 and 14) that this study is the cornerstone of treatment of the acoustics of deformable saturated porous media. We will therefore consider a rigid porous matrix. The notations are the same as those used previously, and we will adopt the microscopic viewpoint. The threedimensional period (the cell) is referred to as Ω∗ , with local variable y∗ . The period consists of a rigid solid part Ω∗s and of the pores Ω∗f . The boundary common to Ω∗s and Ω∗f will still be denoted Γ∗ . The pores are saturated by an incompressible ﬂuid of viscosity η. Such a period is again illustrated in Figure 7.2 for a two-dimensional problem.

7.3.1. Local description and estimates The local description is given by the Navier-Stokes equation, the incompressibility condition, and no-slip condition over Γ: ηΔX v − gradX p = ρ

∂v + (v gradX )v ∂t

(7.22)

210

Homogenization of Coupled Phenomena

divX v = 0

(7.23)

v|Γ = 0

(7.24)

The conservation of momentum (7.22) introduces three dimensionless numbers, and the macroscopic description clearly depends on the values of these numbers. As we have an acoustic vibration in mind, the perturbations are small and the same applies to the local Reynolds number, which we will take to be O(ε): Rel =

|ρv gradX )v| = O(ε) |ηΔX v|

The dynamics bring into play the transient inertial term on the local scale. The local transient Reynolds number is therefore O(1): ∂v | ∂t = O(1) Rtl = |ηΔX v| |ρ

This number can be expressed in different ways: ρc lc2 ωc Rtl = = ηc

lc δb

2 = (ωc τd )

2

where: δb =

ηc ωc ρc

gives the thickness of the boundary layer which appears along the ﬂuid-solid interface Γ, and: τd = lc

ρc ωc ηc

is the time taken by a diffusive shear wave to cross the period. A local transient Reynolds number O(1) corresponds to a boundary layer thickness of the same order of magnitude as the size of the pores, and to a vibration whose period is of the same order of magnitude as the time taken by the diffusive shear wave to cross the period. Finally, the estimate made in section 7.2 for the number Q still applies: Ql =

pc lc |gradX p| = = O(ε−1 ) |ηΔX v| ηc vc

Incompressible Newtonian Fluid Flow

211

This value of Ql is, as in the other situations considered up to now, the only one which leads to a macroscopic description. Thus, adopting the microscopic viewpoint, (7.22) can be written: ∗ ∂v ∗ ∗ η ∗ Δy∗ v∗ − ε−1 grady∗ p∗ = ρ∗ (7.25) + ε(v grad )v ∗ y ∂t∗ 7.3.2. Macroscopic behavior: generalized Darcy’s law We want to ﬁnd v∗ and p∗ in the form: v∗ (x∗ , y∗ , t∗ ) = v∗(0) (x∗ , y∗ , t∗ ) + εv∗(1) (x∗ , y∗ , t∗ ) + · · · p∗ (x∗ , y∗ , t∗ ) = p∗(0) (x∗ , y∗ , t∗ ) + εp∗(1) (x∗ , y∗ , t∗ ) + · · · x∗ = εy∗ , v∗(i) and p∗(i) are Ω∗ -periodic. Substituting these expansions into (7.25), (7.23) and (7.24), we obtain for the successive orders: grady∗ p∗(0) = 0 ρ∗

(7.26a)

∂v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) ∂t∗ ———————-

(7.26b)

divy∗ v∗(0) = 0

(7.27a)

divy∗ v∗(1) + divx∗ v∗(0) = 0

(7.27b) ———————-

v∗(0) = 0 v∗(1) = 0

(7.28a) over Γ∗

(7.28b) ———————-

Thus a series of boundary-value problems are deﬁned, which make it possible to obtain the successive terms v∗(i) and p∗(i) . The ﬁrst problem is deﬁned by (7.26a) and gives p∗(0) in the form: p∗(0) = p∗(0) (x∗ , t∗ ) The pressure is, to ﬁrst order, uniform in the pore. The second and fundamental problem of interested involves unknowns v∗(0) and p∗(1) . The pressure p∗(0) is at this

212

Homogenization of Coupled Phenomena

stage considered to be an arbitrary function of x∗ . With (7.26b), (7.26b) and (7.27a), it follows that: ρ∗

∂v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) ∂t∗

(7.29)

divy∗ v∗(0) = 0

(7.30)

v∗(0) = 0

(7.31)

where v∗(0) and p∗(1) are Ω∗ -periodic. It is convenient to study this problem in Fourier space. If ω ∗ is the frequency, (7.29) can be written in the form: ρ∗ iω ∗ v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0)

(7.32)

Problem (7.30), (7.31) and (7.32) is linear. The existence and uniqueness of v∗(0) and p∗(1) are investigated later in section 7.3.3. Unknowns v∗(0) and p∗(1) are linear functions of gradx∗ p∗(0) . In what follows, we will only require v∗(0) : ∗(0)

vi

= −λ∗ij (x∗ , ω∗ )

∂p∗(0) ∂x∗j

(7.33)

The λ∗ij components here are functions of ω ∗ , and are complex because of the inertial term contained in (7.32). The viscosity η ∗ and density ρ∗ are taken into consideration in λ∗ , but the separation of viscous and inertial effects in the macroscopic description turns out to be very difﬁcult (see section 7.3.3). Now we will look for the macroscopic description. This is obtained from (7.27b) and (7.28b): divy∗ v∗(1) + divx∗ v∗(0) = 0 v∗(1) = 0

over Γ∗

where v∗(1) is Ω∗ -periodic. As we have already seen several times, this volume conservation implies a compatibility equation: divx∗ v∗(0) = 0 From this it follows that: divx∗ v∗(0) = 0

(7.34)

Incompressible Newtonian Fluid Flow

213

with: v∗(0) = −Λ∗ (ω ∗ )gradx∗ p∗(0) ,

Λ∗ = λ∗

(7.35)

where Λ∗ is the dynamic hydraulic conductivity. System (7.33-7.34) represents the macroscopic description we are looking for. Relation (7.35) is a generalized Darcy’s law. The dynamic permeability K∗ can be introduced by: Λ∗ =

K∗ η∗

where K∗ depends on η ∗ ω ∗ /ρ∗ . In dimensional variables, the macroscopic model becomes: ¯ divX v = O(ε)

¯ v = −Λ(ω)gradX p + O(ε)

or alternatively: ¯ divX v = O(ε)

v = −

K(ω) ¯ gradX p + O(ε) η

7.3.3. Discussion of the macroscopic description Here we will restrict ourselves to the main points. (The reader is referred to [AUR 80; AVA 81; BOR 83; AUR 85b] for more details.) 7.3.3.1. Physical meaning of macroscopic quantities The physical meaning of macroscopic quantities v∗(0) and p∗(0) is the same as discussed in 7.2.2.1. We will not return to that here. 7.3.3.2. Energetic consistency We are now in a position to tackle energetic consistency. This involves showing that the macroscopic densities of viscous dissipation and kinetic energy are equal to the volume means of the microscopic densities of these same respective quantities. First of all, with φ representing porosity, and H∗ the inverse of Λ∗ (which exists, see below): H∗ = HR∗ + iHI∗ = Λ∗−1 we rewrite the generalized Darcy’s law in the form: gradx∗ φp∗(0) = −φHR∗ v∗(0) − φHI∗ ω ∗−1 v˙ ∗(0)

(7.36)

214

Homogenization of Coupled Phenomena

where v˙ ∗(0) is the mean acceleration and represents the time derivative of the velocity at constant frequency. Equation (7.36) is a generalized Darcy’s law when the frequency is constant. It is written in the form of a conservation of momentum equation. It introduces a dissipative term φHR∗ v∗(0) and an inertial term φHI∗ ω ∗−1 v˙ ∗(0) . We will now look for an variational form equivalent to the system (7.29a,b-7.32). Let W be the space of vectors with zero divergence, which are Ω∗ -periodic and take complex values, deﬁned over Ω∗f , zero over Γ∗ , and satisfying the scalar product: (α, β)W =

Ω∗ f

(

∂αi ∂ β˜i + αi β˜i ) dΩ∗ ∂yj∗ ∂yj∗

˜ is the complex conjugate of β. Thus the equivalent variational formulation where β can be written, proceeding as in section 7.2.2.3, as: ∀α ∈ W,

Ω∗ f

η∗(

∗(0)

∂vi ∂ α ˜i ∗(0) +iω ∗ ρ∗ vi α ˜ i ) dΩ∗ = − ∂yj∗ ∂yj∗

Ω∗ f

∂p∗(0) α ˜ i dΩ∗ (7.37) ∂x∗i

This formulation ensures the existence and uniqueness of v∗(0) [LEV 79]. We will take α = v∗(0) as real in (7.37). It follows for the right-hand side that:

∂p∗(0) ∗(0) vi dΩ∗ = |Ω∗ |gradx∗ p∗(0) · v∗(0) = −|Ω∗ |H∗ v∗(0) · v∗(0) ∂x∗i

Ω∗ f

Taking the real and imaginary parts of (7.37) after transformation in this way, we obtain: HR∗ v∗(0) · v∗(0) =

1 |Ω∗ |

HI∗ ω ∗−1 v∗(0) · v∗(0) =

Ω∗ f

1 |Ω∗ |

η∗

∗(0)

∗(0)

∂vi ∂vi dΩ∗ ∂yj∗ ∂yj∗

Ω∗ f

ρ∗ v∗(0) · v∗(0) dΩ∗

(7.38)

(7.39)

Introducing strain rate tensor D and using (7.29a), the right-hand side of (7.38) can be easily transformed into: 1 |Ω∗ |

∗(0)

∗(0)

∂vi ∂vi 1 ∗ η ∗ ∗ dΩ = |Ω∗ | ∗ ∂y ∂y Ωf j j ∗

Ω∗ f

2η ∗ Dy∗ (v∗(0) ) : Dy∗ (v∗(0) ) dΩ∗

so that (7.38) becomes: H

R∗

v

∗(0)

· v

∗(0)

1 = ∗ |Ω |

Ω∗ f

2η ∗ Dy∗ (v∗(0) ) : Dy∗ (v∗(0) ) dΩ∗

(7.40)

Incompressible Newtonian Fluid Flow

215

Equations (7.39) and (7.40) show the energetic consistency for kinetic and dissipated energy respectively. 7.3.3.3. The tensors H∗ and Λ∗ are symmetric Consider two real velocity ﬁelds v∗p and v∗q , corresponding respectively to = δip and vi∗q = δiq , given p and q, p = q. We will write the expression (7.37) for v∗(0) = v∗p , α = v∗q and then subsequently for v∗(0) = v∗q , α = v∗p . We then obtain: ∂v ∗q ∂vi∗p ∂p∗(0) ∗ η ∗ ( i∗ + iω ∗ ρ∗ vi∗q vi∗p ) dΩ∗ = − |Ω |Iip = |Ω∗ |H∗pq ∗ ∂yj ∂yj ∂x∗i Ω∗ f vi∗p

and: Ω∗ f

η∗ (

∂vi∗p ∂vi∗q ∂p∗(0) ∗ + iω ∗ ρ∗ vi∗p vi∗q ) dΩ∗ = − |Ω |Iiq = |Ω∗ |H∗qp ∗ ∗ ∂yj ∂yj ∂x∗i

The symmetry of the scalar product in the left-hand sides of these two above equations implies the equality of the right-hand sides: H∗pq = H∗qp 7.3.3.4. Low-frequency behavior The inertial term in (7.32) tends to zero as the frequency tends to zero. The generalized Darcy’s law then approaches the classical Darcy’s law: K∗ (0) K∗ = η∗ η∗

Λ∗ (ω ∗ ) → Λ∗ (0) =

when

ω∗ → 0

where K∗ is the steady state or intrinsic permeability tensor. 7.3.3.5. Additional mass effect Inertial density φHI∗ ω ∗−1 introduced in the generalized Darcy’s law takes the form of a tensor. It is easy to demonstrate an additional mass effect, familiar in ﬂuid mechanics, particularly in the isotropic case. We will start from (7.39) with HI∗ ij = I∗ ∗q H Iij and we will consider the velocity ﬁeld v introduced above: I∗

H ω

∗−1

vi∗q

1 =ρ |Ω∗ |

1 = ∗ |Ω |

∗

Ω∗ f

Ω∗ f

ρ∗ vi∗q vi∗q dΩ∗

vi∗q dΩ∗ = δiq

216

Homogenization of Coupled Phenomena

The Schwarz inequality applied to the integral of velocity v∗q leads to: 1 1 |Ω∗ | = | v∗q dΩ∗ | |Ω∗ | 2 | vi∗q vi∗q dΩ∗ | 2 Ω∗ f

Ω∗ f

Thus: φHI∗ ω ∗−1 ρ∗ We refer to Chapter 10 section 10.3.1.2 for a proof of the following properties [BOU 08]: dHR∗ 0 dω ∗

;

d(HI∗ /ω ∗ ) 0 dω ∗

7.3.3.6. Transient excitation: Dynamics with memory effects When the porous medium is subject to a transient excitation, the generalized Darcy’s law (for example (7.36)) must be replaced with: gradx∗ φp∗(0) = −F ∗−1 (φH∗ ) ∗ v∗(0)

(7.41)

where F ∗−1 (φH∗ ) is the inverse Fourier transform of φH∗ , and ∗ indicates a convolution product. Thus the generalized Darcy’s law for transient motion describes dynamics with memory effects, as in (7.21). 7.3.3.7. Quasi-periodicity When periodicity varies slowly, in other words in the case of quasi-periodicity, where the geometry depends on macroscopic variable x∗ , all the results obtained above remain valid. During the process of homogenization, x∗ then plays the role of an independent parameter. Thus macroscopic coefﬁcients Λ∗ and H∗ become dependent on x∗ . 7.3.4. Circular cylindrical pores The mean behavior of a viscous incompressible Newtonian ﬂuid through a porous matrix whose pores are circular cylinders has been known for a long time (see for example [BIO 56a; 56b]), in any case well before the formulation of the method of asymptotic expansions. This is because in this particular case, the periodicity is arbitrary along the pore axis. Thus the fundamental cellular boundary-value problem to be solved when homogenization is used is identical to the problem posed directly on the macroscopic scale: due to the arbitrary periodicity, the different terms in the asymptotic expansions are independent of the fast variable along the pore axis. This property of course applies regardless of the cross-section of the cylindrical pores.

Incompressible Newtonian Fluid Flow

217

This explains why this particular type of pore can be taken as a reference, and the results obtained can be generalized to other porous media [BIO 56a; b; AVA 81]. Signiﬁcant differences exist in the size of H(ω) however, notably for the behavior of the inertial coefﬁcient at high frequencies [BOR 83; AUR 85b]. Here we will present

e2

e2

a

e1

e3

a

Figure 7.4. Porous medium consisting of circular cylindrical pores: macroscopic sample and representative elementary volume

the derivation of the generalized Darcy’s law for a porous medium with parallel circular cylindrical pores, in the context of the method of asymptotic expansions. The medium is shown in Figure 7.4. The pores have radius a∗ in terms of the slow variable. The dimensionless problem to be solved is given by (7.29), (7.30) and (7.31): ρ∗

∂v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) ∂t∗

divy∗ v∗(0) = 0 v∗(0) = 0

over

within

within

Ω∗f

Ω∗f

Γ∗

The calculation is carried out in dimensional cylindrical variables (X, r, θ), with X pointing along the pore axis (e3 ). Bearing in mind pore geometry and incompressibility, v(0) points along the pore axis and is independent of X and θ, and p(1) is independent of θ. Also, the arbitrary periodicity in X direction implies that p(1) does not depend on X. Since the problem is macroscopically one-dimensional, the macroscopic pressure gradient is taken along the axis of the pore. The system to be solved can thus be simpliﬁed signiﬁcantly. The equations recalled above are written in dimensionless form. We will now return to the physical quantities (remember that the same notations were used in both cases in order to simplify matters). By projection onto the axes, we obtain, for constant pulsation: iω 1 dv (0) 1 dp(0) d2 v (0) − v (0) = + 2 dr r dr νf η dX

218

Homogenization of Coupled Phenomena

1 dp(1) =0 η dr where νf is the kinematic viscosity, νf = η/ρ. Over the pore boundary, the velocity satisﬁes: v (0) (a) = 0 It follows from this that p(1) = p(1) (X), and also that the solution v (0) can be written: 1

1

v (0) = AJ0 (i(iωνf−1 ) 2 r) + BN0 (i(iωνf−1 ) 2 r) −

1 dp(0) iωρ dX

where A and B are two constants of integration which must be determined, and J0 and N0 are Bessel functions. Since v (0) is ﬁnite at the origin, B = 0, and A can be obtained using the no-slip condition at r = a. It follows that: A=

1 1 iωρJ0 (i(iωνf−1 ) 2 a)

dp(0) dX

and the velocity distribution in the pore is given by: v (0) =

1

−1 +

J0 (i(iωνf−1 ) 2 r) 1 2

J0 (i(iωνf−1 ) a)

1 dp(0) iωρ dX

Finally, taking the mean over the cross-section S = πa2 of the pore, we obtain the generalized Darcy’s law for constant pulsation: v

(0)

φ = S

a

2πrv (0) dr 0

To ﬁrst order we have: v = −Λ

dp , dX 1

φ J2 (i(iωνf−1 ) 2 a) Λ=− iωρ J0 (i(iωνf−1 ) 12 a) where J2 is the Bessel function. The values of Λ and H = Λ−1 are shown in Figures 7.5 and 7.6 as a function of the dimensionless pulsation.

Incompressible Newtonian Fluid Flow

2/*a2 1

2

Log(a2 /2") Figure 7.5. Variation of Λ1 and Λ2 as a function of frequency

*HI /(')

0.1*HR /2 Log(a2 /2") Figure 7.6. Variation of the real (HR ) and imaginary (HI ) parts of H as a function of frequency

In the case of transient motion, the dynamic Darcy’s law becomes: φ v = − ρ

t −∞

G(t − τ )

dp (τ ) dτ dX

where memory function G is obtained via the inverse Fourier transform of Λρ/φ: G=4

−2 νf t λ−2 exp −λ n n a2 n=1 ∞

for

t>0

219

220

Homogenization of Coupled Phenomena

G=0

t<0

for

with λn being the zeroes of J0 . In the form of a dynamics equation, the law can then be written: t ∞ 8η dp −2 = − 2 v − ρ 1 + 4 φ βn M (t − τ ) ¨ v (τ ) dτ (7.42) v ˙ +ρ dX a −∞ n=1 with memory function M deﬁned by: M (t) = 4

∞

βn−2 exp

−βn−2

n=1

M =0

νf t a2

for

t>0

t<0

for

where βn are the zeroes of J2 . Functions G and M are shown in Figure 7.7. A dynamics equation with memory effects can be recognized in (7.42).

G M "t/a2 Figure 7.7. Memory functions M and G

7.4. Appearance of inertial non-linearities At low speeds the steady-state ﬂow of a ﬂuid in a Galilean porous medium is described by Darcy’s law. As the speed increases, the non-linear inertial terms grow, and the ﬂow law itself also becomes non-linear. The ﬂow law currently used is equation (7.2), known as the Darcy-Weisbach law or the Forchheimer law. Here we will study the appearance of non-linearities in Darcy’s law which appear for low local Reynolds numbers Rel . A study can be found [MEI 91] which was carried out for

Incompressible Newtonian Fluid Flow

221

Rel = O(ε1/2 ), and [WOD 91] an analysis can be found which includes all small Rel such that O(ε) < Rel < O(ε1/2 ). Here we will revisit the work of Rasoloarijaona [RAS 93] for Rel = O(ε). The viewpoint is macroscopic and the ﬂuid is still assumed to be incompressible. 7.4.1. Macroscopic model Consider a ﬂow with Reynolds number Rel = O(ε), i.e. ReL = O(1). The local dimensionless description can be written, as in equations (7.6-7.8): ε2 η ∗ Δx∗ v∗ − gradx∗ p∗ = ε2 ρ∗ (v∗ gradx∗ )v∗ divx∗ v∗ = 0 v∗ |Γ∗ = 0 In order to study the appearance of the non-linearities, we need to follow the expansions carried out for Darcy’s law to higher orders. Introducing the expansions of v∗ and p∗ into the above equations, we obtain successively: ∂p∗(0) =0 ∂yi∗ η∗

∗(0)

∂p∗(0) ∂p∗(1) ∂ 2 vi − − =0 ∂yk∗ ∂yk∗ ∂x∗i ∂yi∗

η

∗

η

∗

∗(0)

∗(1)

∗(0)

∂v ∂ ∂v ∂ ∂vi ( + i∗ )+ ∗ i∗ yk∗ ∂x∗k ∂yk xk ∂yk

∗(0)

−

∂p∗(2) ∂p∗(1) ∗(0) ∂vi − = ρ ∗ vk ∂x∗i ∂yi∗ ∂yk∗

∗(1) ∗(2) ∗(0) ∗(1) ∂ ∂vi ∂vi ∂ ∂vi ∂vi ∂p∗(2) ∂p∗(3) ( + )+ ∗( + ) − − = ∗ ∗ ∗ ∗ ∗ ∗ yk ∂xk ∂yk xk ∂xk ∂yk ∂xi ∂yi∗ ∗

ρ

∗(0) ∗(1) ∂vi vk ∂yk∗

+

∗(0) ∗(0) ∂v vk ( i ∗ ∂xk

———————-

∗(0)

∂vi =0 ∂yk∗ ∗(1)

∗(0)

∂vi ∂v + i∗ =0 ∂yk∗ ∂xk

∗(1) ∂vi + ) ∂yk∗

(7.43)

222

Homogenization of Coupled Phenomena ∗(2)

∗(1)

∗(3)

∗(2)

∂vi ∂vi =0 ∗ + ∂yk ∂x∗k ∂vi ∂v + i∗ =0 ∂yk∗ ∂xk ∗(0)

vi

∗(1)

vi

∗(2)

vi

(7.44) ———————-

=0 =0 over Γ∗

=0

(7.45) ———————-

Equation (7.43a) gives p∗(0) = p∗(0) (x∗ ). Then (7.43b), (7.44a) and (7.45a) represent the underlying problem for Darcy’s law (see section 7.2). This gives v∗(0) and p∗(1) as linear functions of gradx∗ p∗(0) : ∗(0)

vi

=−

∗ kij ∂p∗(0) η ∗ ∂x∗j

p∗(1) = a∗i

∂p∗(0) + p∗(1) (x∗ ) ∂x∗i

where tensor k∗ satisﬁes in particular: ∗ ∂kij =0 ∂yi∗

Since p∗(1) is only deﬁned up to a constant term, the volume mean of the a∗ vector can be taken to be zero. The existence of v∗(1) then leads to: divx∗ v∗(0) = 0,

v∗(0) = −

K∗ gradx∗ p∗(0) , η∗

K∗ = k∗

(7.46)

The problem which follows is given by (7.43c) with (7.44b) and (7.45b). Unknowns v∗(1) and p∗(1) appear as linear functions of: ∂p∗(0) , ∂x∗j

∂p∗(1) , ∂x∗j

∂p∗(0) ∂p∗(0) , ∂x∗j ∂x∗k

∂ 2 p∗(0) ∂x∗j ∂x∗k

so that: ∗(1)

η ∗ vi

∗ = −kij

∗(0) 2 ∗(0) ∗(0) ∂p∗(1) ∂p∗(0) ∗ ∂p ∗ ∂ p ∗ ∂p − l − m − n ijk ijk ij ∂x∗j ∂x∗j ∂x∗k ∂x∗j ∂x∗k ∂x∗j

Incompressible Newtonian Fluid Flow

p∗(2) = a∗i

223

∗(0) 2 ∗(0) ∗(0) ∂p∗(1) ∂p∗(0) ∗ ∂p ∗ ∂ p ∗ ∂p + b + c + d + p∗(2) (x∗ ) ij ij i ∂x∗i ∂x∗i ∂x∗j ∂x∗i ∂x∗j ∂x∗i

where we again ﬁnd tensors k∗ and a∗ that were introduced above. Since pressure p∗(2) is only deﬁned up to a constant term, volume means of the a∗ , b∗ , c∗ and d∗ coefﬁcients can be taken to be zero. The existence of v∗(2) implies, through integration of (7.44c): divx∗ v∗(1) = 0

(7.47)

Finally (7.43d) with (7.44c) and (7.45c) give v∗(2) and p∗(3) as linear functions of: ∂p∗(0) ∂p∗(1) ∂p∗(2) ∂p∗(0) ∂p∗(0) ∂ 2 p∗(0) ∂p∗(0) ∂p∗(1) ∂ 2 p∗(1) , , , , , , ∂x∗i ∂x∗i ∂x∗i ∂x∗j ∂x∗k ∂x∗j ∂x∗k ∂x∗i ∂x∗k ∂x∗j ∂x∗k ∂p∗(0) ∂p∗(0) ∂p∗(0) ∂p∗(0) ∂ 2 p∗(0) ∂ 3 p∗(0) , , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∂xi ∂xj ∂xk ∂xi ∂xj ∂xk ∂xj ∂xk ∂xl Here we only need the expression for v∗(2) : ∗(2)

η ∗ vi

= −n∗ ij

∗(2) ∗(0) ∂p∗(1) ∗ ∂p ∗ ∂p − k − r − ij ij ∂x∗j ∂x∗j ∂x∗j

∗ lijk (

∂p∗(0) ∂p∗(1) ∂p∗(0) ∂p∗(1) + )− ∗ ∗ ∂xj ∂xk ∂x∗k ∂x∗j

m∗ ijk

∗(0) 2 ∗(0) ∂ 2 p∗(1) ∂p∗(0) ∗ ∂p ∗ ∂ p − ∗ ∗ − qijk ∗ ∗ − qijk ∂xj ∂xk ∂xj ∂xk ∂x∗j ∂x∗k

s∗ijkl

∗(0) 2 ∗(0) ∂p∗(0) ∂p∗(0) ∂p∗(0) ∂ p ∂ 3 p∗(0) ∗ ∂p ∗ ∗ ∗ ∗ − sijkl ∗ ∗ ∗ − sijkl ∂xj ∂xk ∂xl ∂xj ∂xk ∂xl ∂x∗j ∂x∗k ∂x∗l

Tensors k∗ and a∗ , deﬁned above, are again found in these expressions. The existence of v∗(3) implies, by integration of (7.44d): divx∗ v∗(2) = 0

(7.48)

Ultimately, we ﬁnd that order n introduces derivatives of order n of the pressure into Darcy’s law on the macroscopic scale, and derivatives of order n + 1 into the

224

Homogenization of Coupled Phenomena

volume conservation equations – see (7.47) and (7.48). The corrector terms given by these last two equations reveal the inﬂuence of non-local mechanics. They may be non-negligible when the separation of scales is poor.

7.4.2. Macroscopically isotropic and homogenous medium In the case of macroscopic isotropy, all effective tensors of odd order are zero. Since the medium is homogenous, the even-order, isotropic tensors are independent of x∗ . Thus: ∗ Kij = K∗ Iij

v∗(0) = −

K∗ gradx∗ p∗(0) η∗

p∗(1) = φp∗(1) (x∗ ) and (7.46) becomes: ∂ 2 p∗(0) =0 ∂x∗k ∂x∗k

(7.49)

At the next order, bearing in mind homogenity, the expressions for v∗(1) and p∗(2) reduce to: ∗(1)

η ∗ vi

∗ = −kij

p∗(2) = a∗i

∗(0) ∂p∗(1) ∂p∗(0) ∂ 2 p∗(0) ∗ ∂p − lijk − m∗ijk ∗ ∗ ∗ ∗ ∗ ∂xj ∂xj ∂xk ∂xj ∂xk

∂p∗(1) ∂p∗(0) ∂p∗(0) ∂ 2 p∗(0) + b∗ij + c∗ij ∗ ∗ + p∗(2) (x∗ ) ∗ ∗ ∗ ∂xi ∂xi ∂xj ∂xi ∂xj

Equation (7.47) then becomes, with isotropy: divx∗ v∗(1) = 0,

v∗(1) = −

and p∗(2) = φp∗(2) (x∗ ) From this we deduce that: ∂ 2 p∗(1) =0 ∂x∗k ∂x∗k

K∗ gradx∗ p∗(1) η∗

Incompressible Newtonian Fluid Flow

225

Finally, homogenity simpliﬁes the expression for v∗(2) to: ∗(2)

η ∗ vi

∗ = −kij

∗(0) ∂p∗(2) ∂p∗(1) ∂p∗(0) ∂p∗(1) ∗ ∂p − l ( + ) ijk ∂x∗j ∂x∗j ∂x∗k ∂x∗k ∂x∗j

−m∗ ijk

∗(0) 2 ∗(0) ∂ 2 p∗(1) ∂p∗(0) ∗ ∂p ∗ ∂ p ∗ ∗ − qijk ∗ ∗ − qijk ∂xj ∂xk ∂xj ∂xk ∂x∗j ∂x∗k

−s∗ijkl

∗(0) 2 ∗(0) ∂p∗(0) ∂p∗(0) ∂p∗(0) ∂ p ∂ 3 p∗(0) ∗ ∂p ∗ ∗ ∗ ∗ − sijkl ∗ ∗ ∗ − sijkl ∂xj ∂xk ∂xl ∂xj ∂xk ∂xl ∂x∗j ∂x∗k ∂x∗l

With isotropy the means of l∗ , m∗ , q∗ and q∗ are zero. All fourth-order isotropic tensors π become: πijkl = λIij Ikl + α(Iik Ijl + Iil Ijk ) + α (Iik Ijl − Iil Ijk ) Bearing in mind (7.49), it follows that: ∗(2)

η ∗ vi

= −K∗

∗(0) ∗(0) 2 ∗(0) ∂p∗(2) ∂p∗(0) ∂p∗(0) ∂ p ∗ ∗ ∂p ∗ ∂p ∗ − (λ + 2α ) ∗ ∗ ∗ − 2α ∂xi ∂xi ∂xk ∂xk ∂x∗k ∂x∗i ∂x∗k

Summarizing the results obtained for the different orders of velocity, we obtain: ∗(0)

η ∗ vi∗ = η ∗ (vi = −K∗

∗(1)

+ εvi

∗(2)

+ ε2 vi

) + O(ε3 )

∗(1) ∗(2) ∂p∗(0) ∗ ∂p 2 ∗ ∂p − εK − ε K − ∂x∗i ∂x∗i ∂x∗i

ε2 (λ∗ + 2α∗ )

∗(0) 2 ∗(0) ∂p∗(0) ∂p∗(0) ∂p∗(0) ∂ p 2 ∗ ∂p − 2ε α + O(ε3 ) ∗ ∗ ∗ ∗ ∂xi ∂xk ∂xk ∂xk ∂x∗i ∂x∗k

Similarly: p∗ = p∗(0) + εp∗(1) + ε2 p∗(2) + O(ε3 ) = φp∗(0) + εφp∗(1) + ε2 φp∗(2) + O(ε3 ) With: vi∗ vk∗ vk∗ =

K∗3 ∂p∗(0) ∂p∗(0) ∂p∗(0) + O(ε) η ∗3 ∂x∗i ∂x∗k ∂x∗k

226

Homogenization of Coupled Phenomena

vk∗

∂vi∗ K∗2 ∂p∗(0) ∂ 2 p∗(0) = ∗2 + O(ε) ∗ ∂xk η ∂x∗k ∂x∗i ∂x∗k

and: p∗ = φ−1 p∗ ﬁnally, we obtain: vi∗ (1 − ε2 η ∗3 K∗−3 (λ∗ + 2α∗ )vk∗ vk∗ )+ 2ε2 η ∗2 K∗−2 α∗ vk∗

∂vi∗ K∗ ∂p∗ =− ∗ ∗ ∂xk η ∂x∗i

(7.50)

It can be shown [MEI 91] that λ∗ + 2α∗ 0. In dimensional variables, the model can be written: vi (1 − η 3 K−3 (λ + 2α)vk vk ) + 2η 2 K−2 α vk

∂vi K ∂p ¯ 3) =− + O(ε ∂Xk η ∂Xi

7.4.3. Conclusion In the case of an isotropic and homogenous medium with Rel = O(ε), macroscopic description (7.50) introduces two corrective terms to Darcy’s law. The ﬁrst is a velocity cubed term – and not a squared term as is often claimed, see for example Scheidegger [SCH 74]. The second corrective term is a standard convective term. This disappears in one-dimensional problems. It does not appear for Rel = O(ε1/2 ) [MEI 91], or for O(ε) < Rel < O(ε1/2 ) [WOD 91]. In general anisotropic or inhomogenous cases, it does not appear that the result can be written in the form of a simple generalized Darcy’s law.

7.5. Summary The different models of incompressible ﬂuid ﬂow in rigid porous media that we have obtained, along with their ranges of validity, are summarized below: – in the steady-state regime, when: p c lc = O(ε−1 ), ηc vc or equivalently: Ql =

QL = O(ε−2 ),

Rel =

ReL O(1),

ρc vc lc O(ε), ηc and RtL = 0

and Rtl =

ρc ωc lc2 =0 ηc

Incompressible Newtonian Fluid Flow

227

the laminar ﬂow within the rigid porous medium is described by Darcy’s law: K divX v = 0, v = − gradX p η where K [m2 ] is the intrinsic steady state permeability tensor, which is positive and symmetric. – in the dynamic regime, when: Ql = O(ε−1 ),

Rel O(ε),

and Rtl = O(1)

or equivalently: QL = O(ε−2 ),

ReL O(1),

and RtL = O(ε−2 )

the laminar ﬂow within the rigid porous medium is described by generalized Darcy’s law: K(ω) divX v = 0, v = −Λ(ω) gradX p = − gradX p η where Λ(ω) = ΛR + iΛI and K(ω) = KR + iKI are the dynamic hydraulic conductivity and dynamic permeability respectively. These tensors are symmetrical, frequency-dependent and take complex values. They satisfy the following equations: Λ−1 = H = HR + iHI ,

K−1 = H = HR + iHI

Finally, we observe that in the two above cases: – values of Ql = O(ε−1 ) (or QL = O(ε−2 )) lead to non-homogenizable situations. We will see in Chapter 11 that the deformability of the medium makes such situations homogenizable; – values of O(ε) < Rel < O(ε1/2 ) (or O(1) < ReL < O(ε−1/2 )) give rise to inertial non-linearities in the ﬂow law (see section 7.4).

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Chapter 8

Compressible Newtonian Fluid Flow Though a Rigid Porous Medium

8.1. Introduction Even though studies of ﬂow in porous media very often involve media saturated with water or (quasi)-incompressible ﬂuids (see previous chapter), the fact remains that in many ﬁelds of engineering (mining, pharmaceutical powders, acoustic insulation, aeronautic materials, etc.), the porous media are saturated with a gas, which is by nature very compressible. In this chapter therefore, we intend to deﬁne the ﬂow laws for a compressible Newtonian ﬂuid in a rigid porous medium, by using multiple scale expansions. Three ﬂow regimes will be examined: – ﬂow of a compressible Newtonian ﬂuid in a rigid matrix under pressure ﬂuctuations of the order of the equilibrium pressure. This regime of strong perturbation is governed by a non-linear macroscopic behavior (section 8.2); – slow steady-state ﬂow of a compressible Newtonian ﬂuid, a rareﬁed gas, in a rigid matrix. This situation involves slipping ﬂuid along the pore walls, and leads to Klinkenberg’s law (section 8.3); – dynamic linear ﬂow of a compressible Newtonian ﬂuid in the small perturbation regime in a rigid matrix. This case requires coupling with thermal effects to be included (section 8.4).

8.2. Slow isothermal ﬂow of a highly compressible ﬂuid Here we rederive the results from Auriault et al. [AUR 90b]. The equations describing the isothermal ﬂow of a compressible ﬂuid in a porous medium can be

230

Homogenization of Coupled Phenomena

written: – conservation of momentum: ηΔX v + (λf + 2η)gradX divX v − gradX p = ρ – conservation of mass: ∂ρ + divX (ρv) = 0 ∂t – no-slip condition: v|Γ = 0

∂v + (v gradX )v (8.1) ∂t (8.2)

(8.3)

– equation of state of the ﬂuid, assumed barotropic: f (p, ρ) = 0

(8.4)

In (8.1), the viscosities are functions of the pressure p, which is assumed to be known: λf = λf (p)

η = η(p)

We refer to Figure 7.3 for the notations. 8.2.1. Estimates Equations (8.1) and (8.2) introduce three independent dimensionless numbers: – the Reynolds number Re: |ρv gradX )v| Re = |ηΔX v| – the transient Reynolds number Rt : ∂v |ρ | ∂t Rt = |ηΔX v| – the number Q deﬁned by: |gradX p| Q= |ηΔX v| We will also make use of the Strouhal number S: ∂v ∂ρ |ρ | | Rt ∂t ∂t = = S= |divX (ρv)| |ρv gradX )v| Re |

These dimensionless numbers are subject to certain restrictions if an equivalent macroscopic description is to exist. These restrictions, which will be analyzed in

Compressible Newtonian Fluid Flow

231

section 8.2.3, can be written, using lc (the microscopic viewpoint) for normalization: Sl =

ω c lc O(ε), vc

Rel =

ρc vc lc O(1), ηc

Ql =

p c lc O(ε−1 ) ηc vc

where ωc is a characteristic angular frequency. With Lc as a characteristic length (in other words from the macroscopic viewpoint) these restrictions become: SL =

ωc Lc O(1), vc

QL =

pc Lc O(ε−2 ) ηc vc

ReL =

ρc vc Lc O(ε−1 ) ηc

There are restrictions on the Strouhal number and Reynolds number because we have chosen to study large-amplitude movements while remaining in the laminar regime. In the context of this assumption, a local Reynolds number Rtl of order O(1) is not possible. We will see in section 8.4 that this limitation disappears if we cover small acoustic perturbations. The ﬁnal restriction, which applies to Q, was analyzed earlier. We can identify various homogenizable situations worthy of interest: – Steady-state ﬂow: Rt = 0 and S = 0. Also, (i) if Rel = (ρc vc lc /ηc ) O(ε) so that the macroscopic ﬂow is described by standard Darcy’s law and (ii) if Rel = O(1), the non-linearities appear on the macroscopic scale. In both cases the conservation of macroscopic mass is non-linear. – Transient conservation of mass: SL = O(1). Also, if Rtl = O(ε2 ) and Rel = O(ε), then SL = O(1). The transient term in conservation of mass will be taken into account on the macroscopic scale. However, since Rtl = O(ε2 ), the transient term in the equation of motion is negligible. The ﬂow will be described macroscopically by the standard Darcy’s law, and the non-linear conservation of mass will contain a transient term. Finally, if Rtl = O(ε) and Rel = O(1), i.e. SL = O(1), nonlinearities appear in the ﬂow law.

8.2.2. Steady-state ﬂow With the above estimates and a macroscopic Reynolds number O(1), the equations (8.1-8.4) can be formally written as: ε2 η ∗ Δx∗ v∗ + ε2 (λ∗f + 2η ∗ )gradx∗ divx∗ v∗ − gradx∗ p∗ = ε2 ρ∗ (v∗ gradx∗ )v∗

(8.5)

232

Homogenization of Coupled Phenomena

divx∗ (ρ∗ v∗ ) = 0

(8.6)

v∗ |∗Γ = 0

(8.7)

f ∗ (p∗ , ρ∗ ) = 0

(8.8)

where we have used Lc to non-dimensionalize the lengths (i.e. we have chosen the macroscopic viewpoint). We will introduce the double coordinate system x∗i and yi∗ = ε−1 x∗i and the following expansions: v∗ (x∗ , y∗ ) = v∗(0) (x∗ , y∗ ) + εv∗(1) (x∗ , y∗ ) + ε2 v∗(2) (x∗ , y∗ ) + · · · p∗ (x∗ , y∗ ) = p∗(0) (x∗ , y∗ ) + εp∗(1) (x∗ , y∗ ) + ε2 p∗(2) (x∗ , y∗ ) + · · · ρ∗ (x∗ , y∗ ) = ρ∗(0) (x∗ , y∗ ) + ερ∗(1) (x∗ , y∗ ) + ε2 ρ∗(2) (x∗ , y∗ ) + · · · where v∗(i) , p∗(i) , ρ∗(i) are functions of x∗i and yi∗ that are Ω∗ -periodic in y∗ . From this we can deduce the expansions of λ∗f and η ∗ . We only require the ﬁrst term of these expansions: ∗(0)

λ∗f = λf

(p∗(0) ) + O(ε),

η ∗ = η ∗(0) (p∗(0) ) + O(ε)

Equation (8.5) gives for orders O(ε−1 ) and O(1): grady∗ p∗(0) = 0

(8.9a) ∗(0)

η ∗(0) Δy∗ v∗(0) + (λf

+ 2η ∗(0) )grady∗ divy∗ v∗(0) −gradx∗ p∗(0) − grady∗ p∗(1) = 0 ———————-

(8.9b)

In the same way (8.6), (8.7) and (8.8) give: divy∗ (ρ∗(0) v∗(0) ) = 0

(8.10a)

divy∗ (ρ∗(0) v∗(1) + ρ∗(1) v∗(0) ) + divx∗ (ρ∗(0) v∗(0) ) = 0 ———————-

(8.10b)

∗(0)

vi

∗(1)

vi

=0 =0

(8.11a) over Γ∗

(8.11b) ———————-

Compressible Newtonian Fluid Flow

f ∗ (p∗(0) , ρ∗(0) ) = 0 p∗(1) (

233

(8.12a)

∂f ∗ ∗(0) ∂f ∗ ) + ρ∗(1) ( ∗ )∗(0) = 0 ∗ ∂p ∂ρ ———————-

(8.12b)

where the subscripts y∗ and x∗ indicate the variable of derivation. With (8.9a) and (8.12a) it follows that: ∗(0)

p∗(0) = p∗(0) (x∗ ), ρ∗(0) = ρ∗(0) (x∗ ), λf

∗(0)

= λf

(x∗ ), η∗(0) = η ∗(0) (x∗ )

and (8.10a) reduces to: divy∗ v∗(0) = 0 Thus the boundary-value problems for v∗(0) and p∗(1) can be written, starting from (8.9b), (8.10a) and (8.11a) in the form: η ∗(0) Δy∗ v∗(0) − gradx∗ p∗(0) − grady∗ p∗(1) = 0 divy∗ v∗(0) = 0,

v∗(0) = 0

over Γ∗

(8.13)

where v∗(0) and p∗(1) are Ω∗ -periodic. (8.13) is the standard system already obtained for the ﬂow of an incompressible ﬂuid in a porous medium. The solution can be written: v∗(0) = −

k∗ gradx∗ p∗(0) η ∗(0)

p∗(1) = a∗ (y∗ ) · gradx∗ p∗(0) + p∗(1) (x∗ ) where k∗ and a∗ (already deﬁned in Chapter 7) are a second-order tensor and a vector. p∗(1) (x∗ ) is an arbitrary function of x∗ at this stage. The macroscopic description can be obtained from (8.10b), which we will write in the form: divy∗ (ρ∗(0) v∗(1) + ρ∗(1) v∗(0) ) = −divx∗ (ρ∗(0) v∗(0) )

(8.14)

This equation represents the conservation of the periodic quantity ρ∗(0) v∗(1) + ρ v , with the source term −divx∗ (ρ∗(0) v∗(0) ). The existence of v∗(1) and ρ∗(1) require that the source volume mean should be zero. It follows that: ∗(1) ∗(0)

divx∗ (ρ∗(0) v∗(0) ) = 0

234

Homogenization of Coupled Phenomena

where we recall that: 1 . = ∗ . dΩ∗ |Ω | Ω∗f Finally the macroscopic behavior is given to ﬁrst order of approximation by: divx∗ (ρ∗(0) v∗(0) ) = 0 v∗(0) = −

K∗ gradx∗ p∗(0) , η ∗(0) (p∗(0) )

(8.15) K∗ = k∗

f ∗ (p∗(0) , ρ∗(0) ) = 0

(8.16) (8.17)

Thus the ﬂow is governed by standard Darcy’s law. The non-linearities due to compressibility of the ﬂuid appear on the macroscopic scale in the conservation of mass (8.15). In dimensional variables, we have: ¯ divX (ρv) = O(ε) v = −

K ¯ gradX p + O(ε), η(p)

K = lc2 K∗

¯ f (p, ρ) = O(ε) When the equation of state is linear – in other words when it has the form ρ = Ap, where A is a constant – (8.15) becomes: ∗ K ∗(0) 2 divx∗ gradx∗ (p ) = 0 η ∗(0) or in dimensional variables: K ¯ divX gradX p2 = O(ε) η(p) Finally, we will consider a local Reynolds number O(1), with all other things remaining the same. The new normalization of equation (8.1) is: ε2 η ∗ Δx∗ v∗ + ε2 (λ∗f + 2η ∗ )gradx∗ divx∗ v∗ − gradx∗ p∗ = ερ∗ (v∗ gradx∗ )v∗ It can easily be seen that nothing has changed except the equation giving v∗(0) and p : ∗(1)

η ∗(0) Δy∗ v∗(0) − gradx∗ p∗(0) − grady∗ p∗(1) = ρ∗(0) (v∗(0) grady∗ )v∗(0)

Compressible Newtonian Fluid Flow

235

As a consequence, v∗(0) appears as a non-linear vectorial function of gradx∗ p∗(0) and ρ∗(0) . The ﬂow law is non-linear. 8.2.3. Transient conservation of mass Based on section 8.2.1, we will adopt the following values for the dimensionless numbers: SL = O(1),

RtL = O(1),

ReL = O(1)

System (8.1-8.4) can therefore now be formally written in the form: ε2 η ∗ Δx∗ v∗ + ε2 (λ∗f + 2η ∗ )gradx∗ divx∗ v∗ − gradx∗ p∗ = ε2 ρ ∗ (

∂v∗ + (v∗ gradx∗ )v∗ ) ∂t∗

∂ρ∗ + divx∗ (ρ∗ v∗ ) = 0 ∂t∗ v∗ |Γ∗ = 0 f ∗ (p∗ , ρ∗ ) = 0 The introduction of the asymptotic expansions of v∗ , p∗ and ρ∗ lead successively to: grady∗ p∗(0) = 0

(8.18a) ∗(0)

η ∗(0) Δy∗ v∗(0) + (λf

+ 2η ∗(0) )grady∗ divy∗ v∗(0) −gradx∗ p∗(0) − grady∗ p∗(1) = 0 ———————-

(8.18b)

In the same way (8.6), (8.7) and (8.8) give: divy∗ (ρ∗(0) v∗(0) ) = 0

(8.19a)

∂ρ∗(0) + divy∗ (ρ∗(0) v∗(1) + ρ∗(1) v∗(0) ) + divx∗ (ρ∗(0) v∗(0) ) = 0 ∂t∗ ———————-

(8.19b)

236

Homogenization of Coupled Phenomena ∗(0)

vi

∗(1)

vi

=0

(8.20a) over Γ∗

=0

(8.20b) ———————-

f ∗ (p∗(0) , ρ∗(0) ) = 0 p∗(1) (

(8.21a)

∗ ∂f ∗ ∗(0) ∗(1) ∂f ) + ρ ( )∗(0) = 0 ∂p∗ ∂ρ∗ ———————-

(8.21b)

We therefore obtain the same results as above, but now with a time-dependence: ρ∗(0) = ρ∗(0) (x∗ , t∗ ), ∗(0)

λf

∗(0)

= λf

p∗(0) = p∗(0) (x∗ , t∗ ),

(x∗ , t∗ ),

divy∗ v∗(0) = 0

η ∗(0) = η ∗(0) (x∗ , t∗ )

The boundary-value problem for v∗(0) and p∗(1) is the same as that obtained for an incompressible ﬂuid. Thus: v∗(0) = −

k∗ gradx∗ p∗(0) η ∗(0)

∂ρ∗(0) Equation (8.19b) however includes the additional term in the source term. The ∂t∗ corresponding compatibility condition therefore becomes: φ

∂ρ∗(0) + divx∗ (ρ∗(0) v∗(0) ) = 0 ∂t∗

(8.22)

where φ is the porosity. Thus the macroscopic behavior is given by: φ

∂ρ∗(0) + divx∗ (ρ∗(0) v∗(0) ) = 0 ∂t∗

v∗(0) = −

K∗ gradx∗ p∗(0) , η ∗(0) (p∗(0) )

f ∗ (p∗(0) , ρ∗(0) ) = 0

or in dimensional variables: φ

∂ρ ¯ + divX (ρv) = O(ε) ∂t

v = −

K ¯ gradX p + O(ε), η(p)

(8.23) ¯ f (p, ρ) = O(ε)

(8.24)

Compressible Newtonian Fluid Flow

237

This is a standard result [BEA 72]. As in the preceding section, a macroscopic Reynolds number O(ε−1 ) leads to a non-linear ﬂow law. It is now time to clarify the restrictions introduced at the start of the chapter on S, Re and Q. First of all we will consider a macroscopic Strouhal number SL of order O(ε−1 ). The conservation of mass must be rewritten in the following dimensionless form: ε−1

∂ρ∗ + divx∗ (ρ∗ v∗ ) = 0 ∂t∗

This normalization corresponds to that introduced by Keller [KEL 80]. The ﬁrst order term (8.19a) becomes: ∂ρ∗(0) + divy∗ (ρ∗(0) v∗(0) ) = 0 ∂t∗

(8.25)

However, contrary to (8.19a), equation (8.25) introduces a compatibility condition:

∂ρ∗(0) = 0, ∂t∗

so that

∂ρ∗(0) =0 ∂t∗

This result is impossible since by construction (∂ρ∗(0) /∂t∗ ) = O(1). The macroscopic Strouhal number is thus reduced to O(1)! From this we conclude that ﬂows characterized by a Strouhal number SL of order O(ε−1 ) are not homogenizable. This result can easily be extended to larger Strouhal numbers. A macroscopic Reynolds number of order O(ε−2 ) also leads to a normalization which causes it to be reduced by an order of magnitude. In order to simplify things, we will consider a steady-state ﬂow. Equation (8.9a) is now written: −grady∗ p∗(0) = ρ∗(0) (v∗(0) grady∗ )v∗(0) This equation gives the boundary-value problem for v∗(0) , p∗(0) and ρ∗(0) when we supplement it with (8.10a), (8.11a) and (8.12a): divy∗ (ρ∗(0) v∗(0) ) = 0,

v∗(0) = 0

over Γ∗ ,

f ∗ (p∗(0) , ρ∗(0) ) = 0

If we assume the solution to be unique, it takes the following form: ρ∗(0) = ρ∗(0) (x∗ , t∗ ),

p∗(0) = p∗(0) (x∗ , t∗ ),

v∗(0) = 0

Once again we get an impossible result because v∗(0) = O(1). The Reynolds number is thus reduced. A ﬂow with a macroscopic Reynolds number of order O(ε−2 ) (or greater) cannot be homogenized. Finally, a QL of an order other than O(ε−2 ) will automatically be increased or decreased to order O(ε−2 ) by cancellation of the ﬁrst relevant terms in the asymptotic expansions. A ﬂow with QL = O(ε−2 ) is not homogenizable.

238

Homogenization of Coupled Phenomena

8.3. Wall slip: Klinkenberg’s law Take the slow isothermal steady-state ﬂow of a gas in a rigid porous medium [SKJ 99b; CHA 04a; CHA 04b]. If the pressure is sufﬁciently high and the pores of characteristic size l are not too small, the mean free path λ of the molecules is very small compared to l, λ l. The gas ﬂow can be described on the scale of the pores by the Navier-Stokes equations and the no-slip condition on the surface of the pores: the results from section 8.2 apply and the ﬂow is described by Darcy’s law. For lower pressures, the mean free path increases and Darcy’s law is no longer valid. Wall slip comes into play on the surface of the pores, following Navier law [NAV 22]. Darcy’s law becomes Klinkenberg’s law [ADZ 37a; b; KLI 41]: v=−

Kk gradX p, η

K k = K(1 +

b ) pm

(8.26)

where b depends on the ﬂuid and the porous medium, and pm is the mean pressure on the sample. In what follows, we will look at a periodic porous medium, and we will assume a separation of scale between the pores and macroscopic scale of l/L = ε 1. The mean free path λ is sufﬁciently small compared to l that the ﬂow can be described by the Navier-Stokes equations, and sufﬁciently large that the condition on the walls of the pores is the Navier slip condition. We will assume: ε

λ 1 l

(8.27)

8.3.1. Pore scale description and estimates Here we have, the period Ω, the ﬂuid part Ωf , and the solid/ﬂuid interface Γ. At all points on Γ we will introduce a set of orthonormal axes deﬁned by two tangential vectors t1 and t2 and a normal vector n exterior to Ωf , and where t1 is collinear to the wall slip velocity of the ﬂuid (Figure 8.1). The description of ﬂow on the pore scale is the same as in section 8.2, where the no-slip boundary condition over Γ is replaced by the Navier slip condition (8.30). The ﬂow is slow and steady, so the non-linearities in the Navier-Stokes equation and transient terms can be ignored: ηΔX v + (λf + 2η)gradX divX v − gradX p = 0 within Ωf

(8.28)

divX ρv = 0 within Ωf

(8.29)

v = −cλ (t1 · gradX v · n) t1 f (p, ρ) = 0

over Γ

(8.30) (8.31)

Compressible Newtonian Fluid Flow

239

lc

t1 s n f

slip boundary condition

Figure 8.1. Period

where v is the velocity vector and ρ is the density. The viscosities η and λf are functions of pressure: η = η(p),

λf = λf (p)

(8.32)

In the wall slip equation (8.30), [CER 88; 90] c is: c = 1.1466

(8.33)

and the mean free path is given by: πRT /2 λ= p η

(8.34)

In this expression T is temperature and R is deﬁned by: R = R0 /M

(8.35)

where R0 is the gas constant and M is the molecular weight. When p is sufﬁciently large, λ becomes negligible and (8.30) becomes the no-slip condition: v=0

sur Γ

(8.36)

The shear velocity over Γ is given by: τn = −t1 · gradX v · n = t1i

∂vi nj ∂Xj

(8.37)

240

Homogenization of Coupled Phenomena

We will use lc as the characteristic length to non-dimensionalize the local description. This introduces two dimensionless numbers: number Q already introduced for our study of Darcy’s law; and Knudsen number Kn: Ql =

pc lc , ηc vc

Kn =

cλc |cλ(t1 · gradX v · n) t1 | = |v| lc

(8.38)

Looking at the situation described by Darcy’s law, with the approximation (8.27) the two above numbers can be estimated as: Ql = O(ε−1 ),

Kn = O(ε0 )

(8.39)

Thus the dimensionless description becomes: εη∗ Δy∗ v∗ + ε(λ∗f + η ∗ )grady∗ divy∗ v∗ − grady∗ p∗ = 0

(8.40)

divy∗ (ρ∗ v∗ ) = 0 within Ω∗f

(8.41)

v∗ = −Kn (t1 · grady∗ v∗ · n) t1 over Γ∗

(8.42)

f (p∗ , ρ∗ ) = 0

(8.43)

We have left Kn in (8.42) since we will later make use of the fact that Kn, although O(1), is small. 8.3.2. Klinkenberg’s law In order to study macroscopic behavior, we will proceed in two stages. Firstly we will set out the underlying problem over the period using asymptotic expansions in powers of ε. Second we will study this fundamental problem, making use of the fact that Kn is small. We will introduce the following expansions: v∗ (x∗ , y∗ ) = v∗(0) (x∗ , y∗ ) + εv∗(1) (x∗ , y∗ ) + ε2 v∗(2) (x∗ , y∗ ) + · · · p∗ (x∗ , y∗ ) = p∗(0) (x∗ , y∗ ) + εp∗(1) (x∗ , y∗ ) + ε2 p∗(2) (x∗ , y∗ ) + · · · ρ∗ (x∗ , y∗ ) = ρ∗(0) (x∗ , y∗ ) + ερ∗(1) (x∗ , y∗ ) + ε2 ρ∗(2) (x∗ , y∗ ) + · · · where v∗(i) , p∗(i) , ρ∗(i) are functions of x∗i and yi∗ , Ω∗ -periodic in y∗ , and x∗i = εyi∗ . The tangent vector t1 depends on velocity. Consequently t1 takes the form: t1 = t1 (x∗ , y∗ ) + ε t1 (x∗ , y∗ ) + ε2 t1 (x∗ , y∗ ) + . . . (0)

(1)

(2)

(8.44)

Compressible Newtonian Fluid Flow

241

(0)

where t1 is the unit vector of v∗(0) . The same applies for Kn, η ∗ and λ∗f , which depend on the pressure: ϕ∗ = ϕ∗(0) (x∗ , y∗ ) + ε ϕ∗(1) (x∗ , y∗ ) + ε2 ϕ∗(2) (x∗ , y∗ ) + . . .

(8.45)

ϕ∗ = Kn, η ∗ , λ∗f ,

(8.46)

ϕ∗(0) = ϕ∗ (p∗(0) )

After substitution into the dimensionless description, and identiﬁcation of the powers of ε, the lowest order gives the familiar result (see section 8.2): grady∗ p∗(0) = 0, p∗(0) = p∗(0) (x∗ ), ρ∗(0) = ρ∗(0) (x∗ )

(8.47)

The fundamental problem, involving unknowns v∗(0) and p∗(1) , appears when: η ∗ Δy∗ v∗(0) − G∗ − grady∗ p∗(1) = 0 within Ω∗f

(8.48)

divy∗ v∗(0) = 0 within Ω∗f

(8.49)

v∗(0) = −Kn(0) (t1 · grady∗ v∗(0) · n) t1 (0)

(0)

over Γ∗

(8.50)

where v∗(0) and p∗(1) are Ω-periodic. The vector G∗ is the macroscopic pressure gradient responsible for the ﬂow: G∗ = gradx∗ p∗(0)

(8.51)

Evidently v∗(0) is a linear function of G∗ : v∗(0) = −λk∗ G∗ ,

λk∗ =

k∗ η∗

(8.52)

We will investigate λ∗k below.

8.3.3. Small Knudsen numbers Knudsen number Kn(0) is assumed to be small. We therefore look for v∗(0) and in the form: p ∗(1)

v∗(0) = v∗0 + Kn(0) v∗1 + (Kn(0) )2 v∗2 + . . .

(8.53)

p∗(1) = p∗0 + Kn(0) p∗1 + (Kn(0) )2 p∗2 + . . .

(8.54)

242

Homogenization of Coupled Phenomena

Substitution of these expansions into the fundamental problem shows that they are n valid up to the Kn(0) term on the condition that: ε1/n Kn 1

(8.55)

After substitution and identiﬁcation of the powers of Kn(0) , we get the basic Darcy’s law problem (see section 7.2): η ∗ Δy∗ v∗0 − G∗ − grady∗ p∗0 = 0, within Ω∗f

(8.56)

divy∗ v∗0 = 0 within Ω∗f

(8.57)

v∗0 = 0 over Γ∗

(8.58)

where v∗0 and p∗0 are Ω∗ -periodic. The solution for velocity takes the form: v∗0 = −

k∗ ∗ G η∗

(8.59)

Taking the mean volume, we get Darcy’s law: v∗0 = −

K∗ ∗ G η∗

1 K = k = ∗ Ω ∗

∗

(8.60) Ω∗ f

k∗ dΩ∗

(8.61)

The permeability tensor K∗ is positive and symmetrical. The Kn(0) corrector satisﬁes: η ∗ Δy∗ v∗1 − grady∗ p∗1 = 0 within Ω∗f

(8.62)

divy∗ · v∗1 = 0 within Ω∗f

(8.63)

v∗1 = − (t1 · grady∗ v∗0 · n) t1 (0)

(0)

over Γ

(8.64)

where v∗1 and p∗1 are Ω∗ -periodic. We will construct an Ω∗ -periodic solenoidal (0) (0) vector vs∗ within Ω∗f which takes the value vs∗ = −(t1 · grady∗ v∗0 · n) t1 over ∗ ∗0 ∗ ∗ ∗ Γ [LAD 69]. Since v is linear in G , vs is also linear in G . We will also set v∗1 = w∗ + vs∗ and look for w∗ ∈ V, where V is the Hilbert space deﬁned in section 7.2. After multiplication of (8.62) by u ∈ V and integrating over Ω∗f , we obtain the following weak formulation: ∀ u ∈ V, (u, w∗ + vs∗ )V = 0

(8.65)

Compressible Newtonian Fluid Flow

243

Lax-Milgram theorem assures the existence of a unique w∗ . Since vs∗ is linear in G , w∗ is also linear in G∗ . The corrector v∗1 takes the form: ∗

v∗1 = −

k∗1 ∗ G η∗

v∗1 = −

(8.66)

K∗1 ∗ G η∗

(8.67)

where: K∗1 = k∗1

(8.68)

The two ﬁrst terms of (8.53) give: v∗(0) = v∗0 + Kn(0) v∗1 = −

K∗1 K∗ + Kn(0) ∗ ∗ η η

G∗

(8.69)

In dimensional variables, and neglecting the error of order O(ε), the ﬂow law becomes: β K I + Hk gradX p v = − (8.70) η(p) p where K is the permeability tensor, and Klinkenberg tensor Hk is deﬁned by: Hk = K−1 K1 and β is given by: cη(p) πRT /2 β= l

(8.71)

(8.72)

Equation (8.70) is the tensor form of Klinkenberg’s law. This is a local law. It is valid for: ε Kn 1

(8.73)

8.3.4. Properties of the Klinkenberg tensor Hk 8.3.4.1. Hk is positive Consider the tensor K1 . Multiplying (8.56) by v∗1 using: v∗1 · Δy∗ v∗0 = divy∗ (v∗1 · grady∗ v∗0 ) − grady∗ v∗1 : grady∗ v∗0

(8.74)

244

Homogenization of Coupled Phenomena

and integrating over Ω∗f , we obtain: (v∗1 , v∗0 )V + Ω∗ v∗1 · G∗ = =

Γ∗

Ω∗ f

η ∗ divy∗ (v∗1 · grady∗ v∗0 )dS ∗

η ∗ (v∗1 · grady∗ v∗0 ) · n dS ∗ =

=−

Γ∗

η ∗ τn∗(0) t1 · grady∗ v∗0 · n dS ∗ (0)

Γ∗

η ∗ (τn∗(0) )2 dS ∗ < 0

(8.75)

The ﬁrst term on the left-hand side is zero, as can be seen by setting u = v∗0 in (8.65). Finally we ﬁnd that: −v∗1 · G∗ = G∗ ·

K∗1 ∗ G >0 η∗

(8.76)

Consequently K1 is positive. From this we infer that Hk is positive. 8.3.4.2. Symmetries Consider K1 . For this, we examine two pairs of solutions va∗0 , va∗1 and vb∗0 , vb∗1 from equations (8.56-8.58) and (8.62-8.64), for G∗ = G∗a and G∗ = G∗b respectively. We introduce va∗0 and vb∗1 into (8.75). After the use of (8.65), where we set u = va∗0 , we ﬁnd: (0) (0) ∗ ∗1 ∗ η ∗ t1b · grady∗ vb∗0 · n t1b · grady∗ va∗0 · n dS ∗ (8.77) −Ω vb · Ga = Γ∗

Using (8.67) on the left-hand side of (8.77), we obtain: −vb∗1 · G∗a =

K∗1 ∗ G η∗ b

· G∗a = G∗b ·

t

K∗1 (K∗1 ) · G∗a = G∗a · ∗ · G∗b (8.78) ∗ η η

where t (K∗1 ) is the reverse of K∗1 . Introducing the right-hand side of (8.78) into (8.77) gives: Ω∗ G∗a ·

K∗1 · G∗b = η∗

Γ∗

η ∗ t1b · grady∗ vb∗0 · n t1b · grady∗ va∗0 · n dS ∗ (8.79) (0)

(0)

For vb∗0 and va∗1 , we get (8.79), with A and B swapped around: Ω∗ G∗b ·

K∗1 · G∗a = η∗

Γ∗

η ∗ t1a · grady∗ va∗0 · n t1a · grady∗ vb∗0 · n dS ∗ (8.80) (0)

(0)

Compressible Newtonian Fluid Flow

245

Subtracting (8.80) from (8.79) member by member, we obtain: ∗1 t K∗1 (K ) ∗ ∗ Ω Ga · − · G∗b η∗ η∗ =

Γ∗

η ∗ (t1b · grady∗ vb∗0 · n t1b · grady∗ va∗0 · n− (0)

(0)

tta · grady∗ va∗0 · n t1a · grady∗ vb∗0 · n) dS ∗ (0)

(0)

(8.81)

In general the right-hand side of (8.81) is not zero. Thus K1 is not generally symmetrical. The same applies for Hk . 8.4. Acoustics in a rigid porous medium saturated with a gas Here we will consider the acoustics of porous media saturated with a gas. These media are often used for their acoustic absorption abilities [ALL 93]. This property stems from the free-space wave encountering the porous medium, pushing air into the pores. Because of this the wave is only partially reﬂected. Additionally, the wave transmitted is damped by viscous dissipation and by thermal exchange with the porous skeleton [ZWI 49; ATT 83]. In order to describe these phenomena we will analyze gas ﬂow in the pores starting from the following assumptions: – In the context of acoustics, levels of pressure are weak (20 Pa corresponding to the pain threshold) and only introduce small deformations. In consequence, the effects of advection and convection are negligible and the physics is linear. – It is assumed the porous skeleton cannot be deformed. This assumption is acceptable for the moment due to the high density and high rigidity of the porous matrix relative to the gas. Nevertheless, it may become relevant again for highly porous media which then come into the regime of Biot theory, see Chapter 12. – The air is assumed to be an ideal gas in terms of its response to volume variations associated with pressure, and as a Newtonian ﬂuid in terms of its response to distortions created by the ﬂow. – Thermal exchanges within the gas and at the walls result from conductive heat diffusion. In order to proceed with the homogenization we will assume the medium to be periodic (see Figure 8.2 for the notations related to the period). We will consider sufﬁciently low frequencies to be large compared to the period for the acoustic

246

Homogenization of Coupled Phenomena

wavelength, so that the separation of scales condition is respected (see Chapter 3). We will follow the analysis presented in Boutin et al. [BOU 98] but limit ourselves to media with simple porosity.

8.4.1. Harmonic perturbation of a gas in a porous medium Gas saturating pores of the rigid skeleton (of porosity φ) is subject to small harmonic perturbations (of frequency f = ω/2π) around its equilibrium state (where pressure, temperature and density take the values P e , T e and ρe ). Variables describing perturbations are the variations in pressure p, in temperature T , in density ρ, and velocity of the gas v. D(v) is the tensor measuring strain rate. The parameters

Lc

lc

f

s n

(a)

(b)

Figure 8.2. (a) porous medium, (b) period

governing motion and transfer of heat are viscosity η, thermal conductivity λ, ratio of speciﬁc heats γ, and heat capacity at constant pressure, Cp . Assuming the air to be an ideal gas, the equilibrium values are linked by the following equation: 1 Pe = Cp (1 − ) T e ρe γ The linearized equations describing harmonic oscillations are the following, where the exp(iωt) term is omitted and the variables are complex. In the pores (Ωf ) of the periodic cell, we have (Figure 8.2): – conservation of mass: ρ divX (v) + iω e = 0 ρ

(8.82)

Compressible Newtonian Fluid Flow

247

– the Navier-Stokes equation: divX (2ηDX (v)) − gradX p − iωρe v = 0

(8.83)

– the heat equation: divX (λgradX T ) − iω(ρe Cp T − p) = 0 – the equation of state for the gas: ρ T p = e+ e Pe ρ T

(8.84)

(8.85)

At the gas-solid interface Γ, the boundary conditions are: – the no-slip condition for gas at the wall: v/Γ = 0

(8.86)

– the isothermal condition which we will justify in the following section: T/Γ = 0

(8.87)

8.4.2. Analysis of local physics We need to specify the physics of the gas in the pores. We will start by justifying the isothermal wall condition by analyzing thermal exchanges in the system. In the harmonic regime, diffusive heat transfer occurs over the thickness δt of the thermal boundary layer deﬁned by: λ δt = ρCp ω where conduction parameter λ and heat capacity ρCp are those of the medium (gas or solid) in which we are considering the transfer. Since the range of acoustic frequencies extends from 50Hz to 20kHz, and given the properties: – of the air, λ = 0.026 W/mK, ρe Cp = 1230 J/m3 K – of common solids for which λ 1.5 W/mKρs Cps 2 106 J/m3 K, the thicknesses of the thermal boundary layers in the air δt , and solid δts , vary between: 10μm > δt > 250μm 2μm > δts > 50μm which are values comparable to or smaller than the size of the pores in contemporary materials. Because of this, characteristic temperature variations Tc in the gas and Tcs

248

Homogenization of Coupled Phenomena

solid matrix have gradients which are of the order of Tcs /δts and Tc /δt respectively. Also, the equality of thermal ﬂuxes at the interface Γ requires that: Tc Tcs λs =O λ δts δt from which we can determine the relative order of magnitude of temperature charge in the solid matrix relative to that of the gas: O

Tcs Tc

=

λρe Cp λs ρs Cps

But the values given above show that the thermal impedance of the air is considerably weaker (105 times) than that of the solid. As a consequence: Ts O 1 T It is therefore reasonable to consider that the skeleton is isothermal and that thermal effects are only signiﬁcant in gas. (This condition can nevertheless be called into question for certain mousses where the skeleton consists of walls which are thinner than the solid thermal boundary layer.) Having established this condition, we will return to the description of wave propagation. On one hand, mass and heat transfer with adherence and isothermality at the walls require that velocity and temperature must vary with pore scale. On the other hand, during wave propagation the pressure and volume variation oscillates with the wavelength λ, which can be linked to the macroscopic length scale through Lc = λ/(2π). (If we assume one variable changes on a local scale, it is equivalent to considering wavelengths and pores on the same scale which is incompatible with the separation of length scales.) We therefore have: O(gradX p) =

pc Lc

O(divX v) =

vc Lc

(8.88)

Looking at the pore scale physics obtained when: – The three terms in the Navier-Stokes equation are of the same order, indicating that the pressure gradient is matched by the viscous and inertial terms: 2ηvc pc e = O (ωρ (8.89) O vc ) = O lc2 Lc The ﬁrst equality indicates that the viscous layer is of the same order as pore scale: η δv = = O(lc ) e ρ ω

Compressible Newtonian Fluid Flow

249

These three estimates are represented by the following orders of two dimensionless numbers introduced in the preceding sections: |gradX p| RtL = = O(ε−2 ) |divX (2ηDX (v))| L |ωρe v| QL = = O(ε−2 ) |divX (2ηDX (v))| L – The three terms in the Fourier equation are of the same order, meaning that the heat source due to pressure is distributed between the conduction and thermal inertial terms, so that: λTc (8.90) = O (ωρe Cp Tc ) = O(ωpc ) O lc2 We note that, in accordance with earlier analysis, the ﬁrst equation indicates that the thermal boundary layer is of the pore scale order in other words: λ δt = = O(lc ) ρe Cp ω which is also consistent with the previous assumption since the air (and more generally in gases), thermal and viscous boundary layers are of the same order. We therefore have: |ωp| |ωρe Cp T | −2 = O(ε ); = O(ε−2 ) |divX (λgradX T )| L |divX (λgradX T )| L – Since the gas is not in an isothermal state, the relative variations in pressure and temperature are of the same order as the density variations. These variations balance out volume variations, so that: p ρc vc ρc Tc c (8.91) = O ; O =ω e = O O e e e P T ρ Lc ρ 8.4.3. Non-dimensionalization and renormalization For this problem we will apply the homogenization technique from the macroscopic viewpoint, formulated in terms of variables x and y = ε−1 x, where we recall that x is the measurement in the chosen system of units (metric) of distance X (see Chapter 3). In dimensionless form in terms of variable x∗ , the three conservation equations take the following form (it is clear that the boundary conditions and equation of state do not require normalization): ρc vc divx∗ (v∗ ) + iω e ρ∗ = 0 Lc ρ ηvc pc divx∗ (2Dx∗ (v∗ )) − gradx∗ p∗ − iωρe vc v∗ = 0 L2c Lc

250

Homogenization of Coupled Phenomena

λTc divx∗ (gradx∗ T ∗ ) − iωρe Cp Tc T ∗ + iωpc p∗ = 0 L2c So that the physics described by the estimates (8.89, 8.90, 8.91) is followed, conduction terms in the Fourier equation and viscous terms in the Navier-Stokes equation must be renormalized by a factor of (lc /Lc )2 = ε2 , with the conservation of mass remaining unchanged. Returning to physical variables we obtain: ρ 1 divx∗ (v) + iω e = 0 Lc ρ ε2

1 1 divx∗ (2ηDx∗ (v)) − gradx∗ p − iωρe v = 0 L2c Lc

ε2

1 divx∗ λ(gradx∗ T ) − iω(ρe Cp T − p) = 0 L2c

Which, using the metric variable x = Lc x∗ , or Lc = Lc 1m , is: ρ 1 divx (v) + iω e = 0 1m ρ ε2

1 1 divx (2ηDx (v)) − gradx p − iωρe v = 0 2 1m 1m

ε2

1 divx λ(gradx T ) − iω(ρe Cp T − p) = 0 12m

For the homogenization treatment, we will abstract ourselves from the unit of 1m , which is mathematically neutral. Finally the equation of state means that we can eliminate density from the heat equation and that, using conservation of mass: divX (2ηDX (v)) = η(X (v)+graddivX (v)) = η(X (v)−iωgradX (

p T − )) Pe Te

Thus, in addition to the wall conditions and equation of state, the gas is governed by the three following renormalized differential equations for variables p, v and T : T p − e) = 0 Pe T p T e 2 N (p, v, T ) = −gradp − iρ ωv + ε η (v) − iωgrad( e − e ) = 0 P T

G(p, v, T ) = div(v) + iω(

F (p, T ) = iω(p − ρe Cp T ) + ε2 div(λgradT ) = 0

Compressible Newtonian Fluid Flow

251

8.4.4. Homogenization We now proceed to homogenization using spatial variables x and y = ε−1 x. The modiﬁed spatial derivatives in ε−1 ∂y + ∂x are introduced in differential operators G, N, F , and so we look for pressure p, temperature T , and velocity v in the form of expansions in powers of ε: p(x, y) =

∞

εi p(i) (x, y);

T (x, y) =

∞

0

v(x, y) =

∞

εi T (i) (x, y)

(8.92)

0

εi v(i) (x, y)

(8.93)

0

where p(i) , T (i) and v(i) are Ω-period with respect to variable y. The solution to the problems, leading to the macroscopic description accurate to ﬁrst order, is detailed below. 8.4.4.1. Pressure and temperature The Navier-Stokes equation at order ε−1 reduces simply to −grady p(0) = 0, giving a constant pressure in the pores: p(0) (x, y) = p(0) (x) When this result is introduced into the Fourier equation of order ε0 , this leads to the system governing T (0) within Ωf [SAN 80]: iωp(0) (x) − iωρe Cp T (0) + λy T (0) = 0 (0)

T/Γ = 0 ;

T (0)

Ω − periodic

The solution to this transfer problem in the harmonic regime, where p(0) (x) acts as a forcing term, takes the form: λT

(0)

(x, y) = θ

y δt

iωp(0) (x)

(8.94)

The temperature distribution characterized by θ (solution for p(0) (x) = λ/iω) has complex values and depends on the local variable and frequency through the intermediary of dimensionless variable y/δt . The system deﬁning θ also shows that: O(θ) = (size of Ωf )2 = (lc )2

252

Homogenization of Coupled Phenomena

At low frequencies, the forcing and inertial terms tend to zero, and since the wall is isothermal T (0) → 0: the gas tends to an isothermal regime. At high frequencies, the conduction term is negligible compared to the forcing and inertial terms. Thus within pores the temperature tends to be uniform T (0) → p(0) /(ρe Cp ) corresponding to the adiabatic regime. The thermal boundary layer of thickness δt → 0 keeps this consistent with boundary conditions at the wall. 8.4.4.2. Velocity ﬁeld It is easy to show that the problem deﬁning v(0) , where gradx p(0) acts as a forcing term, is identical to that found in the dynamics of incompressible ﬂuids, which was treated in detail in the previous chapter [AUR 80]: −grady p(1) − gradx p(0) − iω

ρe (0) ηv + y ηv(0) = 0 η

divy v(0) = 0 (0)

v/Γ = 0 ;

v(0)

p(1)

Ω − periodic

As a consequence we ﬁnd solution: ηv(0) (x, y) = k

y δv

gradx p(0)

(8.95)

where k has complex values, depends on the local variable and frequency through the intermediary of dimensionless variable (y/δv ), and O(k) = (lc )2 . 8.4.4.3. Macroscopic conservation of mass At order ε the conservation of mass can be written: (0) T (0) p − divy v(1) + divx v(0) + iω =0 Pe Te and by integration over the cell pores:

divy v(0) + divx v(0) + iω(

Ωf

p(0) T (0) − e ) dΩ = 0 e P T

But, using the divergence theorem, the periodicity and no-slip condition over Γ:

v(1) · n dS = 0

divy v(1) dΩ = Ωf

∂Ωf

Compressible Newtonian Fluid Flow

253

As a result, swapping integration with respect to y and derivation with respect to x:

divx

v

(1)

dΩ + iω

Ωf

Ωf

p(0) dΩ − Pe

Ωf

T (0) dΩ Te

=0

and, introducing the macroscopic physical variables: 1 φ 1 (0) (0) (0) (0) v = v dΩ = v dΩ ; T Ωf = T (0) dΩ Ω Ωf Ωf Ω f Ωf Ω f we obtain the equation of macroscopic conservation: divx v(0) + iωφ

T (0) Ωf Te

p(0) − Pe

=0

(8.96)

8.4.5. Biot-Allard model The macroscopic description given by the three equations below corresponds to the phenomenological descriptions established by Attenborough [ATT 83] and Allard [ALL 93]. It has been validated experimentally by many measurements on materials with simple microstructures: divX v(0) + iωφ v

(0)

T (0) Ωf Te

p(0) − Pe

K(ω) =− gradX p(0) ; η

T (0) ρe Cp = iωΘ Te λ

1 1− γ

=0

φ K= Ωf p(0) ; Pe

kdΩ Ωf

1 Θ= Ωf

θdΩ Ωf

The equivalent medium is characterized by the dynamic permeability tensor K/η and effective compressibility: ρe Cp 1 φ 1− Θ (1 − ) /P e λ γ Θ is sometimes known as the “thermal permeability” because of strong analogies between the problem of visco-inertial ﬂow and diffuso-inertial heat transfer [LAF 97]. The two following chapters provide calculations of the dynamic permeability for three-dimensional periodic structures, and self-consistent estimates of K and Θ. In an isotropic medium, we recall simply that at low frequencies viscous effects dominate

254

Homogenization of Coupled Phenomena

and that K tends towards intrinsic permeability (with a real value) K. At high frequencies inertia dominates and K tends towards a pure imaginary value, (φη)/(iωρe τ∞ ) where τ∞ is the high-frequency limit of the tortuosity. The transition between these two domains takes place around the critical frequency determined by setting the inertial and viscous effects as equal in the macroscopic ﬂow. ωc =

φη Kρe τ∞

In an anisotropic medium, we ﬁnd three critical frequencies deﬁned in the same way, but using the principal values of intrinsic permeability tensor K and tortuosity tensory τ ∞ . As far as the complex effective compressibility is concerned, at low frequencies it tends towards the true isothermal compressibility φ/P e , and true adiabatic compressibility φ/γP e at high frequencies. The critical thermal frequency separating these two regions of behavior takes the form: ωt =

λ Λ2t ρe Cp

where length Λt is linked to the ratio of pore volume to surface area of the interface [CHA 91]. If we only retain pressure as a variable, the macroscopic description leads to the following wave equation: ρe Cp K 1 p(0) (0) =0 − gradX p ) + iωφ 1 − Θ (1 − ) η λ γ Pe

divX

from which we deduce the expression for complex velocity (in an isotropic medium): c=

iω

K Pe e η φ[1 − Θ ρ Cp (1 − 1 )] λ γ

Bearing in mind the frequency characteristics of the macroscopic parameters, the wave tends towards a diffusion wave at low frequencies, and towards a wave propagating with attenuation at high frequencies. These properties are the signature of a P2 type wave in the Biot model (see Chapters 12 and 14), with the difference that here the skeleton is rigid and complex compressibility is a function of frequency. Finally, we emphasize in conclusion to this ﬁrst-order description that: – since the gas compressibility is only apparent at the macroscopic scale, the dynamic permeability is identical to that obtained for an incompressible ﬂuid; – the two dissipative effects, thermal and viscous, are decoupled;

Compressible Newtonian Fluid Flow

255

– the thermal non-equilibrium of gas on the pore scale leads to a complex compressibility. We note that homogenization makes it possible to treat these situations of local non-equilibrium when induced by a uniform forcing term over the cell (here the zero-order pressure); – this model must be modiﬁed for media with double porosity (i.e. when the skeleton is itself microporous), because a pressure diffusion phenomenon takes place in the micropores and introduces an additional attenuation mechanism [BOU 98; OLN 03]. This underlines the importance of accurate microstructure morphology for the macroscopic description.

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Chapter 9

Numerical Estimation of the Permeability of Some Periodic Porous Media

9.1. Introduction In preceding chapters, we have seen that the slow ﬂow of a compressible or incompressible Newtonian ﬂuid in a rigid porous medium is described on the macroscopic scale by Darcy’s law [DAR 56]: v = −

K gradX p η

(9.1)

where the intrinsic or steady state permeability tensor K [m2 ] is positive and symmetric, vector v [m s−1 ] represents the Darcy velocity, gradp [Pa m−1 ] is the pressure gradient, and η [Pa s] is the dynamic ﬂuid viscosity. The permeability tensor is intimately linked to the microstructure of the porous medium and can vary by several powers of 10 depending on the material. Since Darcy’s investigations, many experimental, theoretical and numerical works have been carried out with the intention of demonstrating the role of microstructure parameters (porosity, particle arrangement, etc.) on permeability, and hence to establish predictive expressions. The ﬁrst well-known works in this context are those of Kozeny [KOZ 27] and Carman [CAR 37] who likened the complex geometry of the porous medium to a network of capillary tubes. Following that, many works have been carried out with the aim of characterizing ﬂow within ideal microstructures consisting mainly of periodic or random lattices: – parallel cylinders with circular or elliptical cross sections, [HAP 59; KUW 59; HAS 59; HOW 74; JAC 82; SAN 82a; DRU 84; SAN 88; EDW 90; BER 93; GHA 95; HIG 96; CLA 97; BOU 00; IDR 04]; – spheres [HAS 59; SAN 82b; ZIC 82; BOU 00].

258

Homogenization of Coupled Phenomena

These idealized microstructures of ﬁbrous or granular media clarify the inﬂuence of certain microstructural parameters on permeability, such as porosity, diameter of grains or cylinders, their arrangement, etc. Since the start of the 1990s, it has been possible to numerically calculate the transport properties of three-dimensional random microstructures, be they reconstructed or actual structures obtained through for example X-ray microtomography [ADL 92a; ADL 92b; SPA 94; KOP 98; HIL 01; MAN 02; ZHA 00; BER 05]. The permeability tensor is thus determined by numerically solving the Stokes equations over a representative elementary volume (REV) of the porous material while applying either homogenous conditions in terms of pressure (or velocity) gradient, or periodicity conditions to its boundaries, despite the fact that the microstructure may not be periodic. In the dynamic case, in Chapter 7 we saw that generalized Darcy’s law can be written in the form: K(ω) gradX p (9.2) v = − η where K(ω) is dynamic permeability. These Kij components depend on the pulsation ω and have complex values. Clearly, when ω −→ 0, we recover the steady state permeability deﬁned earlier (9.1). In the same way as for steady state permeability, theoretical investigations (cell model, self-consistent scheme) [UMN 00; THI 02; BOU 08] and numerical investigations [BOR 83; CHA 92] have been carried out in order to quantify the inﬂuence of microstructure and pulsation ω on permeability. These works mostly involve idealized porous media consisting of periodic or random assemblies of cylinders or spheres. Very recently the dynamic permeability of reconstructed uni- or bi-modal three-dimensional porous media, as well as that of real materials viewed through X-ray microtomography, has been determined numerically [MAL 08]. This chapter makes no pretense of providing an exhaustive review of numerical or analytical estimates of the permeability of porous media. Various studies can be found in the works cited above which offer the reader a more complete appreciation of this. It is however clear that a few numerical illustrations would be useful to complement the theoretical approaches in Chapter 7. Consequently, in this chapter we will present various well-known results regarding estimates of the steady state and/or dynamic permeability of various idealized ﬁbrous and granular media consisting of a periodic lattice of parallel cylinders or spheres. These estimates were obtained by solving the boundary-value problems given by periodic homogenization across a period which were presented in Chapter 7. Here we solve these boundary-value problems using the ﬁnite element method [COM 08]. The results which we present will emphasize the well known inﬂuence of various microstructural parameters (porosity, arrangement) on the permeability of these types of media. They will also be compared to other numerical or analytical results available in the literature. Finally, in Chapter 10 all these results will be compared to self-consistent estimates of permeability.

Numerical Estimation of Permeability

259

9.2. Permeability tensor: recap of results from periodic homogenization With the help of periodic homogenization, it has been shown that dynamic permeability tensor K(ω), which is positive and symmetric, is deﬁned by: Kij = lc2 K∗ij = lc2 k∗ij (ω)

(9.3)

where lc is a characteristic microscopic length and k∗ (ω ∗ ) is a second-order tensor such that ﬂuid velocity in dimensionless form: v∗(0) = −

k∗ (ω ∗ ) gradx∗ p∗(0) η∗

(9.4)

is a solution to the following boundary value problem deﬁned over the period: η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) = iω ∗ ρ∗ v∗(0) divy∗ v∗(0) = 0

within Ω∗f

within Ω∗f

v∗(0) = 0 over Γ∗

(9.5) (9.6) (9.7)

Macroscopic pressure gradient gradx p∗(0) is a source term, and the ﬁrst order pressure p∗(1) can be taken to have a mean of zero across the period: p∗(1) = 0. ∗(0) ∗(0) ∗(0) The unknowns v∗(0) = (v1 , v2 , v3 ) and p∗(1) are Ω∗ -periodic. Steady state permeability is worked out by solving the same boundary-value problem with ω ∗ = 0: k∗ = k∗ (0) and then: K∗ = k∗ = k∗ (0) = K∗ (0) In what follows, all the numerical results presented have been obtained by numerical solution of the boundary-value problem (9.5-9.7) using the ﬁnite element method, with a P2-P1 velocity-pressure formulation [COM 08]. 9.3. Steady state permeability of ﬁbrous media In this section we will only discuss steady state permeability: ω ∗ = 0. 9.3.1. Microstructures We will ﬁrst consider ﬂow in a ﬁbrous medium consisting of a square or triangular periodic lattice of parallel cylinders (ﬁbers). The cross-section of the cylinders is

260

Homogenization of Coupled Phenomena

circular, and their radius is Rs (Figure 9.1). Characteristic dimensions of the period for each of the microstructures are shown in Figure 9.1. For the square lattice, the l

l 2Rs 2Rs

e2

$ l 3

l

e2

e1

e1 (b)

(a)

Figure 9.1. Characteristic dimensions of the period, and examples of two-dimensional meshes used in the numerical simulations (a) square lattice (b) triangular lattice

solid volume fraction c = πRs2 /l2 varies between 0 and cmax √ =2 π/4 ≈ 0.785. For 2 the triangular lattice, the solid volume fraction c = 2πR /( 3l ) varies between 0 s √ and cmax = π/(2 3) ≈ 0.907. Given the symmetries of each of these microstructures, their permeability tensor is transverse isotropic and is written: K = KT (e1 ⊗ e1 + e2 ⊗ e2 ) + KL e3 ⊗ e3

(9.8)

where KT (= K11 = K22 ) and KL (= K33 ) are the transverse and longitudinal permeability respectively. 9.3.2. Transverse permeability The study of ﬂow perpendicular to the cylinders can be reduced to a two-dimensional problem in the plane (e1 , e2 ). The transverse permeability KT of the ﬁbrous media under consideration is obtained by solving boundary-value problem (9.5-9.7) while imposing for example a macroscopic pressure gradient gradx∗ p(0) = −e1 . Using equation (9.4), we have: ∗(0)

v∗(0) = v1

∗(0)

e1 + v2

e2 =

∗ k11 k∗ e1 + 12 e2 ∗ η η∗

Numerical Estimation of Permeability

e2 e1

(a)

261

e2 e1

(b)

e2 e1

e2 e1

(d)

(c)

Figure 9.2. Triangular arrangement of parallel ﬁbers (c = 0.3). Velocity ﬁeld v∗(0) and p∗(1) solutions to the boundary-value problem (9.3 − 9.5) when ∗(0) ∗(0) ∗(0) ∗(1) gradx∗ p∗(0) = −e1 . (a) v∗(0) /vmax (b) v1 /v1max , (c) p(1) /pmax , ∗(0) ∗(0) (d) v2 /v2max (after [SAW 04])

then with (9.3), we arrive at: ∗(0)

∗ ∗ K11 = KT = η ∗ v1

∗ = k11 ,

∗(0)

∗ K12 = η ∗ v2

∗ = k12 =0

9.3.2.1. Mesh, velocity ﬁelds and microscopic pressure ﬁelds Figure 9.1 shows an example of a mesh used to carry out simulations when the solid volume fraction is c = 0.3. By exploiting the symmetries of the cell, it would have been possible to carry out the computations over only one quarter of the cell.

262

Homogenization of Coupled Phenomena

Figure 9.2 shows, for a solid volume fraction of c = 0.3 and a pressure gradient gradx∗ p(0) = −e1 , the velocity ﬁeld v∗(0) and pressure p∗(1) which solve equations (9.5-9.7). These different quantities are normalized with respect to their maximum value. Because of the symmetries of microstructure and imposed pressure gradient, this ﬁgure shows that: ∗(0)

– v1 is symmetric with respect to the e1 and e2 axes, and that its intensity is greatest in the gap between the cylinders; ∗(0)

– v2 is antisymmetric with respect to the e1 and e2 axes, which implies that ∗(0) ∗ v2 = 0 and as a result we do indeed have K12 = 0; – p∗(1) is symmetric with respect to e1 and antisymmetric with respect to e2 , and because of this the condition p∗(1) = 0 is indeed satisﬁed. 9.3.2.2. Transverse permeability KT Figures 9.3 (a) and (b) show the dimensionless transverse permeability KT /Rs2 as a function of solid volume fraction for (a) a square and (b) triangular lattice of parallel cylinders. The transverse permeability KT /Rs2 of the media we are considering is a monotonically decreasing function of solid volume fraction, which varies over several factors of 10 and tends to 0 for a solid volume fraction close to cmax , in other words when the cylinders are in contact. We can also see that the inﬂuence of cylinder arrangement on KT /Rs2 is on a level of less than 10% for solid volume fraction below 0.35, i.e. when the cylinders are very spaced out or dilute. The numerical results of Berdichevsky and Cai [BER 93] and predictions of analytic expressions for the transverse permability put forward by Drummond and Tahir [DRU 84] are also reported in these ﬁgures. These expressions are, for the square and triangular lattices respectively: 1 KT −ln(c) − 1, 476 + 2c − 1, 774c2 + O(c3 ) = (square) Rs2 8c 1 KT c2 = (−ln(c) − 1, 497 + 2c − − 0, 739c4 + O(c5 )) Rs2 8c 2

(triangular)

It can be seen that (i) there is a good agreement between the different numerical results across the whole range of solid volume fraction values [c − cmax ] and (ii) the analytical expressions put forward by Drummond and Tahir [DRU 84] are suitable for describing the permeability of this type of medium when the solid volume fraction is below 0.3 for the square lattice, and 0.4 for the triangular lattice. Bruschke and Advani [BRU 93] proposed a hybrid analytical expression KT /Rs2 , which is a combination of the cell model method [HAP 59; KUW 59] and the lubrication model of Keller [KEL 64]. These two models are valid for low and high solid volume fraction values respectively. The hybrid model is: Klub K cell Khyb = ξ1 2 + ξ2 2 Rs2 Rs Rs

Numerical Estimation of Permeability

3

10

Dimensionless permeability

101

-1

10

-3

10

-5

10

-7

10

0

0.2

0.4

0.6

0.8

1

0.8

1

Solid volume fraction (c)

(a) 3

10

Dimensionless permeability

101

-1

10

10-3

10-5

10-7 0

(b)

0.2

0.4

0.6

Solid volume fraction (c)

Figure 9.3. Evolution of transverse permeability KT /Rs2 as a function of the solid volume fraction for (a) a square and (b) triangular arrangement of parallel ﬁbers. () numerical results, () numerical results of Berdichevsky and Cai [BER 93], (- - -) Approximation [DRU 84], (—) hybrid model with τ = 0.5

263

264

Homogenization of Coupled Phenomena

with: ξ1 = 1 − eτ (Φ/(Φ−1)) ,

ξ2 = 1 − eτ ((Φ−1)/Φ)

Kcell 3 1 1 = − (ln c + − 2c + c2 ) Rs2 8c 2 2 ⎛

Klub Rs2

⎞−1 √ 1+√Φ arctan √ 1 (1 − Φ) ⎜ 1 1− Φ ⎟ √ = + Φ + 1⎠ √ 3 ⎝3 Φ 3A 2 1 − Φ Φ 2

√ where Φ = c/cmax , A = 1 and A = 3 for the square and triangular lattices of parallel ﬁbers respectively. The τ parameter, adjusted to ﬁt the numerical results, is equal to 0.5 for both microstructures [IDR 04]. As Figures (a) and (b) show, the hybrid model of Bruschke and Advani [BRU 93] gives a good description of the numerical results across the whole range of solid volume fraction values [0 − cmax ]. 9.3.3. Longitudinal permeability Longitudinal permeability KL of the porous media considered is obtained by solving boundary-value problem (9.5-9.7) with a macroscopic pressure gradient gradx∗ p(0) = −e3 . Equation (9.4) gives: ∗(0)

v∗(0) = v1

∗(0)

e1 + v2

∗(0)

e2 , +v3

e3 =

∗ ∗ ∗ k13 k23 k33 e + e , + e3 1 2 η∗ η∗ η∗

then with equation (9.3) we deduce that: ∗(0)

∗ K13 = η ∗ v1

∗ = k13 , ∗(0)

∗ = KL∗ = η ∗ v3 K33

∗(0)

∗ K23 = η ∗ v2

∗ = k23

∗ = k33

9.3.3.1. Mesh, velocity ﬁelds Figures 9.4 (a) and (b) show an example of the three-dimensional mesh used for the simulations when solid volume fraction is c = 0.3. Figures 9.4 (c) and (d) show (0) the normalized velocity ﬁeld v3 solving (9.5-9.7) when gradx∗ p∗(0) = −e3 . The ∗(0) ∗(0) other velocity components, v1 and v2 , as well as the pressure p∗(1) , are zero. 9.3.3.2. Longitudinal permeability KL The evolution of dimensionless longitudinal permeability KL /Rs2 of the two microstructures as a function of solid volume fraction is shown in Figure 9.5. As

Numerical Estimation of Permeability

e3

265

e2 e1

e

$ l 3

h

h l (a)

(c)

(b)

(d)

l

Figure 9.4. Flow parallel to the ﬁbers. (a,b) Examples of 3D meshes used for the simulations. (c,d) Velocity ﬁeld v∗(0) solving the boundary-value problem ∗(0) (9.3 − 9.5) when gradx∗ p∗(0) = −e3 . Visualization of v∗(0) /vmax (after [SAW 04])

far as the transverse permeability is concerned, KL /Rs2 decreases as solid volume fraction increases, but tends to a ﬁnite value when c = cmax . This is very clear due to the fact that beyond cmax , ﬂow parallel to the ﬁbers is still possible, something which is not the case for transverse ﬂow. Once again we can see that the inﬂuence of the cylinder arrangement on KL /Rs2 is very low, less than 10% for solid volume fraction values below 0.35. Finally, we observe that for a given microstructure the ratio KL /KT is always greater than 1 regardless of solid volume fraction value [0 − cmax ]. This ratio typically varies between 2 and 4 for solid volume fraction values below 0.5. It then grows very rapidly as c approaches cmax . These media are more permeable in the longitudinal direction than in the transverse direction. Finally, the numerical results of Berdichevsky and Cai [BER 93] and analytic expressions for longitudinal permeability put forward by Drummond and Tahir [DRU 84] are again shown in these ﬁgures. These expressions are, for the square and triangular lattices respectively: 1 KL = Rs2 4c

−ln(c) − 1, 476 + 2c −

c2 + O(c4 ) 2

(square)

Homogenization of Coupled Phenomena

3

10

2

Dimensionless permeability

10

1

10

0

10

-1

10

10-2 10-3 10-4 10-5 0

0.2

(a)

0.4

0.6

0.8

1

0.8

1

Solid volume fraction (c)

3

10

101 Dimensionless permeability

266

-1

10

-3

10

10-5

-7

10

0 (b)

0.2

0.4

0.6

Solid volume fraction (c)

Figure 9.5. Evolution of the longitudinal permeability KL /Rs2 as a function of the solid volume fraction for (a) a square and (b) triangular arrangement of parallel ﬁbers. () numerical results, () numerical results of Berdichevsky and Cai [BER 93], (- - -) approximation from Drummond and Tahir [DRU 84]

Numerical Estimation of Permeability

1 KL = Rs2 4c

267

−ln(c) − 1, 498 + 2c −

c2 + O(c6 ) 2

(triangular)

The numerical results agree with the analytic expressions of Drummond and Tahir [DRU 84] across the whole range of solid volume fraction values [0 − cmax ]. 9.4. Steady state and dynamic permeability of granular media 9.4.1. Microstructures We will now consider ﬂow in a granular medium whose microstructure consists of a periodic lattice of spheres, all of the same radius Rs . The periodic lattices we will consider are simple cubic (SC), body centered cubic (BCC) and face centered cubic (FCC), with porosities varying between 0 and 1. The solid phases for the SC, BCC and the FCC lattices are connected for porosities which are below 0.47, 0.32 and 0.26 respectively. In these cases, the microstructures consist of truncated spheres. 9.4.2. Methodology The symmetries of each of these microstructures imply that their steady state permeability tensor K is isotropic: K = K(e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 )

(9.9)

where K = K11 = K22 = K33 . Permeability K of the porous media considered can be obtained by solving the boundary-value problem (9.5-9.7) with a macroscopic pressure gradient gradx∗ p∗(0) = −e3 , for example. Equation (9.4) gives us: ∗(0)

v∗(0) = v1

∗(0)

e1 + v2

∗(0)

e2 + v3

e3 =

∗ ∗ k13 k23 k33∗ e + e2 + ∗ e3 1 η∗ η∗ η

and then with equation (9.3) and the symmetries, we have: ∗(0)

∗ K13 = η ∗ v1

∗ = k13 = 0, ∗(0)

∗ = η ∗ v3 K∗ = K33

∗(0)

∗ K23 = η ∗ v2

∗ = k23 =0

∗ = k33

Given the symmetries of each of these microstructures, the boundary-value problem (9.5-9.7) with ω ∗ = 0, has been solved over just 1/16th of the period (Figure 9.6). In the same way, dynamic permeability K(ω) = K(ω)I is obtained by solving the boundary-value problem (9.5-9.7) for different values of ω ∗ .

268

Homogenization of Coupled Phenomena

e3

e2 e1

(a)

(b)

(c)

Figure 9.6. (a) simple cubic, (b) body centered cubic, (c) face centered cubic lattices of spheres. Mesh for the ﬂuid part over 1/16th of the period ∗(0)

Figure 9.7 shows for example the evolution of v3 as a function of the ratio ω/ωc for the case of a SC lattice, when the porosity is equal to 0.3. ωc , deﬁned by equation (9.13), represents the critical frequency which deﬁnes the transition between the high

e3 e2 e1 (a)

(b)

(c)

(d) ∗(0)

Figure 9.7. Inﬂuence of the ratio ω/ωc on the velocity v3 in a simple cubic lattice (φ = 0.3): (a) ω/ωc = 0, (b) ω/ωc = 1, (c) ω/ωc = 10 and (d) ω/ωc = 100

Numerical Estimation of Permeability

269

and low frequency regimes. It can be seen that the viscous effects which dominate at low frequencies are mostly conﬁned to the boundary layer at very high frequencies. 9.4.3. Steady state permeability 1

10

K/R2s

10-1

10-3

-5

10

0

0.2

0.4

0.6

0.8

1

* Figure 9.8. Evolution of steady state permeability K/Rs2 as a function of the porosity. Numerical results: (◦) simple cubic, () body centered cubic and () face centered cubic

The evolution of the steady state permeability K/Rs2 of different spherical lattices, as determined from numerical simulations with ω ∗ = 0, is presented in Figure 9.8. This ﬁgure again underlines the strong variations (several factors of 10) in permeability as a function of porosity of the porous medium. As far as ﬁbrous media are concerned, it can be seen that in the dilute regime, in other words for high porosities ( 0.6), the permeability is not very sensitive to the arrangement of particles. In the highly concentrated regime, in other words when the solid phase is connected, the inﬂuence of the arrangement of grains on permeability is more marked. For a porosity of φ = 0.2, permeability of the FCC lattice is about 40% weaker than that of the SC and BCC lattices. These differences are clearly intimately linked to the geometric characteristics (diameter, tortuosity, etc.) of the pores of the different periodic lattices, which are very different at low porosity. 9.4.4. Dynamic permeability 9.4.4.1. Effect of frequency The generalized Darcy’s law (9.2) can be written in inverse form: ηHv = gradp

270

Homogenization of Coupled Phenomena

with: H = K−1 = HR + iHI where HR and HI are the real and imaginary parts of H. As for K, the H tensor is positive symmetric (see Chapter 7) and can be written in the isotropic case as: H = HI = (HR + iHI )I The real and imaginary parts of H respectively characterize the viscous dissipation and inertial effects or additional mass effects (see Chapter 7, section 7.3.3). Figure 9.9 shows the typical evolution of the real and imaginary parts of H as a function of ω/ωc for different periodic lattices of spheres and for two values of the porosity. The real part HR and imaginary part HI /ω are increasing and decreasing functions of ω respectively, as shown in Chapter 10, section 10.3.1.2. The imaginary part has two horizontal asymptotes at low and high frequencies. As for steady state permeability, it can again be observed that in the dilute regime (Figure 9.9 (a) and (b)), the dynamic permeability of the porous media considered is not very sensitive to the arrangement of spheres. In the concentrated regime (Figure 9.9 (c) and (d)), the differences are more pronounced. 9.4.4.2. Low-frequency approximation When ω = 0, we of course recover steady state permeability: H(ω = 0) = HR I =

1 I K

At low frequencies or in the quasi-static regime, when ω −→ 0, viscous effects dominate (Figure 9.7) and permeability can be approximated by [BOR 83; AUR 85b]:

iωρ Kτ0 K(ω) K 1 − η φ

(9.10)

where K is steady state permeability and τ0 is steady state or low-frequency tortuosity. The tortuosity τ0 φηHI /(ρω) characterizes the horizontal asymptote at low frequencies which is visible in Figures 9.9 (b) and (d). Using periodic homogenization, we can show that steady state tortuosity can be written in tensorial form: τ 0 = φk∗ k∗ (k∗ k∗ )−1

(9.11)

Numerical Estimation of Permeability

271

3 10

KHR

* HI/(')

2.5

2

1.5

1 10-2

10-1

(a)

100

101

102

1 10-2

103

10-1

100

(b)

/c

101

102

103

102

103

/c

1.6

KHR

* HI /(')

10

1 10-2

(c)

10-1

100

101

/c

102

103

1.4

1.2

1 10-2

(d)

10-1

100

101

/c

Figure 9.9. Evolution of the real and imaginary parts of H(ω) as a function of ω/ωc . (◦), () and () represent the numerical results for simple cubic, body centered cubic and face centered cubic lattices respsectively. (a) and (b) φ = 0.3, (c) and (d) φ = 0.7

where tensor k∗ is the solution to the boundary-value problem (9.5-9.7) with ω ∗ = 0. For the cubic arrangements considered, this tensor is isotropic: τ 0 = τ0 I. The evolution of steady state tortuosity τ0 as a function of the porosity, as obtained from numerical simulations, is shown in Figure 9.10. The steady state tortuosity τ0 for each of the lattices of spheres decreases as porosity increases, and varies between 4 and 1 across the range of porosities studied. Once again, the inﬂuence of the spheres’ arrangement is very pronounced in the concentrated regime, in other words for very low porosities.

272

Homogenization of Coupled Phenomena

4

+0

3

2

1 0

0.2

0.4

0.6

0.8

1

* Figure 9.10. Evolution of steady state tortuosity τ0 as a function of porosity. Numerical results: (◦) simple cubic, () body centered cubic and () face centered cubic

9.4.4.3. High-frequency approximation At high frequencies, inertial effects dominate and viscous effects are conﬁned to the boundary layer (Figure 9.7). In this case, dynamic permeability can be approximated by: ηφ H(ω) τ∞ iωρ

1+

M ωc 2 iω

(9.12)

where τ∞ is the tortuosity and M a form factor. This time tortuosity τ∞ φηHI /(ρω) characterizes the horizontal asymptote at high frequencies as seen in Figures 9.9 (b) and (d). The critical frequency ωc is deﬁned by: ωc =

φη Kρτ∞

The form factor M is deﬁned by [JOH 86]: M=

8τ∞ K Λ2v φ

(9.13)

Numerical Estimation of Permeability

273

where Λv is the viscous length of the medium. The tortuosity τ∞ , and viscous length Λv deﬁned by [JOH 86; CHA 92]: 2 v dS 2 (9.14) = Γ 2 Λv v dΩ Ωf can be deduced through the study of an ideal ﬂuid in a porous medium: iρωv = −gradp

within Ωf

div(v) = 0 within Ωf Using periodic homogenization, Auriault et al. [AUR 85b] showed that, to a ﬁrst order of approximation, the solution to this problem has the form: v∗(0) = −

1 iρ∗ ω ∗

(grady∗ χ∗ + I) · gradx∗ p∗(0)

(9.15)

where the Ω∗ -periodic vector χ∗ is the solution to the following boundary-value problem: divy∗ (grady∗ χ∗ + I) = 0 within Ω∗f (grady∗ χ∗ + I) · n = 0

over Γ∗

χ∗ = 0 This problem is similar to the one which we established for the diffusion of a solute in an inert porous medium (see Chapter 5). The form factor M is given by equations (9.14) and (9.15). The tortuosity τ∞ follows from equations (9.12) and (9.15), and has the tensorial form: τ ∞ = φ grady∗ χ∗ + I−1 For the cubic arrangements considered, this tensor is isotropic: τ ∞ = τ∞ I. Figure 9.11 shows evolution of the inﬁnite-frequency tortuosity τ∞ and form factor M as a function of porosity, as determined from numerical simulations. Like steady state tortuosity, tortuosity τ∞ for the cubic lattices considered decreases as the porosity increases. For a given lattice, and whatever the porosity, we have τ0 > τ∞ . The form factor M for cubic lattices systematically displays a maximum at the percolation threshold. Once again, these parameters are not very sensitive to the arrangement of spheres in the dilute regime.

Homogenization of Coupled Phenomena

3.5

3

+

2.5

2

1.5

1 0

0.2

0.4

(a)

0.6

0.8

1

0.6

0.8

1

*

2.5

2

1.5

M

274

1

0.5

0 0

(b)

0.2

0.4

*

Figure 9.11. Evolution of inﬁnite-frequency tortuosity τ∞ and the form factor M as a function of porosity. Numerical results: (◦) simple cubic, () body centered cubic and () face centered cubic

Chapter 10

Self-consistent Estimates and Bounds for Permeability

10.1. Introduction

The physical origin of steady state or dynamic permeability has been clearly established, both by phenomenological or thermodynamic approaches – developed by [BIO 41] for quasi-static consolidation, [ZWI 49] for the acoustics with rigid skeletons, [BIO 56a] for the dynamics of deformable media, and continued by others [JOH 86; ALL 93] – and also by micro-macro approaches using homogenization of periodic media (HPM) [SAN 80; AUR 80] as described in the preceding chapters.

However even if the physics is well known, the question of an acceptable estimate, or bounding, for this coefﬁcient, is still an open one. Dimensional analysis indicates that the intrinsic permeability is of the order of φl2 , where l is the characteristic pore size. This estimate cannot be used in practice however because it requires a quantiﬁcation of l which takes into account the geometric pore complexity. Also, in dynamics although it is the viscous (or thermal) boundary layer which provides the characteristic length, the effects of morphology remain central to the problem, as shown by Zhou and Sheng [ZHO 89]. This same situation is found in the context of rareﬁed gases where, in the Klinkenberg effect, it is the mean free path which is comparable with the pore dimensions. Furthermore, aside from the exact analytical expressions for thin slits or circular cylinders (see Chapter 7), more realistic morphologies are quite rarely studied. Here the reader is directed to the works of [SPA 94; ADL 92a; ADL 92b] which discuss reconstructions of actual porous media.

276

Homogenization of Coupled Phenomena

For steady state (or intrinsic) permeability, parametric studies have been carried out through numerical simulations, [SAN 82a] for lattices of spheres, using the selfconsistent method (SCM) (see [BER 93] for ﬁbrous media, [BOU 00] for granular media), or by “cell models” of ﬁbrous media for the intrinsic permeability and thermal permeability [TAR 96]. In the dynamic regime, were studied [AUR 85b] high-tortuosity two-dimensional channels, and Chapman and Higdon [CHA 92] numerically simulated lattices of spheres. Analytical estimates were proposed by Umnova et al. [UMN 00], making use of a “cell model”, using the self-consistent method for assemblies of cylinders [THI 02] and by [BOU 08] for assemblies of spheres. These considerations illustrate the complementarity of homogenization of periodic media and the self-consistent approach: – The homogenization of periodic media makes it possible, i) to rigorously determine the macroscopic laws starting from phenomena on pore scale and ii) to give theoretical expressions as well as properties of the coefﬁcients, whatever the morphology of the pores in the periodic structure. It is therefore an indispensable guide for rigorous expression of the physics on micro- and macroscopic scales. However any kind of quantiﬁcation requires numerical simulations, and (very) many simulations would be necessary to identify correlations between permeability and microstructure. – The aim of the self-consistent method is to come up with plausible estimates of the effective coefﬁcients, based on analytical solutions in simple conﬁgurations which are assumed to represent the essential characteristics of the medium. In other words, in the case of permeability, the connectivity of the ﬂuid phase, pore size and porosity. We should emphasize that the underlying solution is exact, but that its application to actual media is generally a conjecture since, in contrast to the homogenization of periodic media, the actual microstructure is not explicitly treated. In addition, the different nature of the Navier-Stokes equation on microscopic scale and Darcy’s law on macroscopic scale (steady state, dynamic, etc.) introduces an additional difﬁculty. Thus self-consistent approaches and those using “cell models”, which do not make the same assumptions, will lead to different estimates. The self-consistent method has for a long time been used for homogenization problems where the local conservation equation takes the symbolic form: div[A(y)[grady (variable) + gradx (variable)]] = 0 This type of problem is considered in the contexts of heat transfer (Chapter 4), diffusion (Chapter 5) and elasticity (Chapter 13). In those cases the micro and macro descriptions have the same form, and the effective parameters are independent of any local length scale. In contrast, all the problems treated in this chapter belong to another class, which is characterized by a local conservation of the form: Δy (variable) + Source + (Optional inertial term) = 0

Self-consistent Estimates and Bounds for Permeability

277

The macroscopic description then has two unusual characteristics: – it has a different nature to the local description; – it implies parameters which are intrinsically linked to a local length scale. However, we will show in this chapter that by combining the results of periodic homogenization and the self-consistent approach it remains possible to establish estimates of the macroscopic parameters, to deduce bounds for them, and to specify the area within which they can be applied. We will strictly limit ourselves to physical situations where the effective coefﬁcients have been established by homogenization in Chapters 7 and 8: we will consider porous media formed from assemblies of spheres (or cylinders), either ordered or disordered, of variable size. The problems we will consider involve determining: – the (steady state) intrinsic permeablity of granular or ﬁbrous media in section 10.2. – the dynamic permeability of granular media in section 10.3. – the permeability for rareﬁed ﬂuids, and the Klinkenberg effect in section 10.4. – the thermal permeability involved in the dynamic compressibility of gases as well as trapping in the transient regime, in section 10.5. These problems have equivalents in the ﬁeld of solute diffusion-absorption. The analogies will be drawn in section 10.6 by studying the following parameters of granular media: – the trapping constant for media having an inﬁnite absorptivity; – the trapping constant in the transient regime; – the trapping constant in media with a ﬁnite absorptivity. Currently, the averaging of sophisticated calculations make it possible to quantify these parameters for complex three-dimensional microstructures. Self-consistent models are still required to provide analytical expressions which encapsulate the essential parts of the physics and which are easily parameterizable. The method also makes it possible to compare the parameters which are produced by different underlying physics. 10.1.1. Notation and glossary Here we will specify the main notations which we will use in this chapter: – the representative elementary volume (REV) consists of ﬂuid volume and solid volume, i.e. Ω = Ωf ∪ Ωs , Γ is the interface between the two phases, Γf = ∂Ω ∩ ∂Ωf is the ﬂuid boundary of the REV. We will denote porosity as φ = Ωf /Ω;

278

Homogenization of Coupled Phenomena

– ρ represents the density of the saturating ﬂuid (kg/m3 ), η its dynamic viscosity (Pas), λ its thermal conductivity (J/(msK)), Cp its heat capacity at constant pressure (J/kg); – we will represent the pressure in the pores by P , and V will be the Darcy velocity, so that, referring to the preceding chapters: 1 v(0) dΩ, P = p(0) (x) V= Ω Ωf – when we return to local problems obtained through homogenization, we will systematically express them with the help of variables x and y = ε−1 x, with x being the measurement of distance X expressed in the chosen length scale (meters for convenience) – see Chapter 3. Depending on the ﬁelds of application, different deﬁnitions (and units) can be found in the literature for permeability. In order to avoid any confusion, here we will use: – (hydraulic) conductivity in units of m2 (Pas)−1 in tensor form – represented by Λ(ω) in the dynamic case and K/η in the steady state case – linking ﬂux to pressure gradient; – permeability with coefﬁcients in units of m2 , i.e. dynamic permeability K(ω) = ηΛ(ω) in the harmonic regime and intrinsic permeability K = K(ω = 0) in the steady state regime, denoted K in an isotropic medium; – dynamic tortuosity τ (ω) in tensor form, linked to the inverse of the dynamic conductivity, or alternatively the dynamic resistance, H(ω) = ηH = Λ−1 (ω) by the relation: τ (ω) = φH(ω)/(iωρ) = φηH(ω)/(iωρ). Units In geophysics, the traditional unit of intrinsic permeability is the Darcy = 0.97 × 10−12 m2 . The hydraulic conductivity Λ, in units of ms−1 used in soil mechanics is linked to intrinsic permeability by Λ = (ρ.g/η)K ≈ 107 K where ηwater ≈ 10−3 Pas at 20˚C and g = 9.81 m/s2 is the acceleration due to gravity. Finally the resistivity of air σ, in units of Pas/m2 , used in acoustics, is the inverse of conductivity – i.e. σ = ηair /K.

10.2. Intrinsic (or steady state) permeability of granular and ﬁbrous media Among the various coefﬁcients – heat transfer, mass, solubility, etc. – of porous media, permeability in the steady state or dynamic regime is one of the most important physical parameters in ﬁelds connected with soil mechanics, with the ultrasonic

Self-consistent Estimates and Bounds for Permeability

279

Figure 10.1. Representative elementary volume of a periodic porous medium

ausculation of materials, with the construction of ﬁber-reinforced composites, etc. In industrial applications, the enormous variability of this parameter depending on pore size, porosity (and frequency) makes its estimation often more important than that of the other less sensitive coefﬁcients. For example, the intrinsic permeability of soils extends over a very wide range as a function of the granulometry: from 10−7 to 10−10 m2 for alluvia varying from gravels to coarse sands, from 10−10 to 10−12 m2 for true sands, from coarse to ﬁne, from 10−12 to 10−16 m2 for argilaceous sands, loams and silts (with rocks containing hydrocarbon reservoirs lying in the intermediate range 10−10 to 10−15 m2 ), and from 10−16 to 10−19 m2 for clays from soft to highly consolidated states. Drawing on homogenization results established in Chapter 7, in section 10.2.2 we will develop the self-consistent method for granular media. We will show in section 10.2.4 that the two estimates obtained in section 10.2.3 show good agreement with those obtained from numerical simulations on periodic lattices of spheres (Chapter 9) and those obtained from empirical laws for natural media. In addition, bounds on permeability are established for different classes of ordered or disordered porous media. The extension to ﬁbrous media is treated in section 10.2.5. These results are from Boutin [BOU 00]. 10.2.1. Summary of results obtained through periodic homogenization For what follows, we need to return brieﬂy to the main results established through homogenization of periodic media (Figure 10.1) which we will present here in terms of the variables x and y = ε−1 x.

280

Homogenization of Coupled Phenomena

10.2.1.1. Global and local descriptions – energetic consistency Darcy’s law can be written (the conservation of mass divx (V) = 0 will not be used in the self-consistent approach): V=−

K gradx P η

(10.1)

– P is ﬂuid pressure – in fact the zero-order stress in the ﬂuid – which is constant (0) in the pores, i.e. σf (x, y) = P (x)I; – V = Ω1 Ωf v(0) dΩ is the Darcy velocity, with local velocity v(0) being governed by the following system of equations (expressed here in terms of variables x and y), where gradx P acts as a forcing term [ENE 75; AUR 80]: ⎧ (1) (0) ⎪ ⎨ −grady p − gradx P + divy (2ηDy (v )) = 0 (0) 0 divy (v ) = 0 Sv ⎪ (0) ⎩ v/Γ = 0 ; v(0) & p(1) Ω − periodic – K is the positive symmetric tensor for intrinsic permeability: 1 K= ei ⊗ ki dΩ Ω Ωf

(10.2)

The three velocity ﬁelds ki /η, associated with the pressure ﬁelds pi(1) , are solutions for unit macroscopic pressure gradients in the three directions, gradx P = −ei , i = 1, 2, 3. As a consequence, with incompressibility and the no-slip condition, ki and pi(1) satisfy: −grady pi(1) + ei + y (ki ) = 0 The problem this describes is purely geometric since it is independent of ﬂuid viscosity. The ki ﬁelds are therefore intrinsic to the pore structure. The variational formulation associated with the local problem, − = |Ω|−1 Ωf −dΩ, is as follows: ∀w ∈ W

;

2ηDy (v(0) ) : Dy (w) = −gradx P · w

W is the vectorial space of real ﬁelds: % W = w / w Ω − periodic; w/Γ = 0;

where

(10.3)

& divy (v) = 0

Taking v(0) as a test ﬁeld, we obtain: 2ηDy (v(0) ) : Dy (v(0) ) = −gradx P · v(0) = gradx P ·

K gradx P η

(10.4)

Self-consistent Estimates and Bounds for Permeability

281

which conﬁrms energetic consistency: in each of the principal directions ei , Kii /η corresponds to the power dissipated by viscosity under a unit macroscopic pressure gradient. 10.2.1.2. Connections between the micro- and macroscopic descriptions The quantities appearing in the macroscopic description are linked to local quantities: – macroscopic velocity V is the mean of the local velocity; – pressure P is the zero-order stress in the pores. Due to periodicity, the higherorder stresses are self-equilibrating at the cell boundary, in other words: (1) σ f n dS = 0 Γf

– intrinsic permeability tensor K expresses energetic consistency, and so is the identity between viscous dissipation taking place in the representative porous volume and in the same volume of the equivalent Darcy medium. In the following section these homogenization results are incorporated into a selfconsistent approach. 10.2.2. Self-consistent estimate of the permeability of granular media The self-consistent approach, which allows estimation of the macroscopic coefﬁcients of a heterogenous medium, consists of [HIL 65; HAS 68]: – making an assumption of the macroscopic behavior – the coefﬁcients of which are to be determined – and considering one or more generic substructures, representing the local physics; – solving the problem over substructure subject to a macroscopic forcing term, including the energetic equivalence between the substructure and equivalent medium; – deducing from this relationship the macroscopic coefﬁcient(s) we are looking for. This method has been applied to ﬁbrous media [BER 93] and to granular media [BOU 00]. Tarnow [TAR 96] uses a similar philosophy with a “cell model”, but replaces the energetic condition by a condition of zero vorticity on the substructure boundary. 10.2.2.1. Formulation of the self-consistent problem In accordance with the results of homogenization, the equivalent medium is assumed to follow Darcy’s law as recalled earlier. For convenience, the analysis is presented in an isotropic medium (K = KI), although the same reasoning can be directly transposed to the principal permeability values in an anisotropic medium.

282

Homogenization of Coupled Phenomena

The idea is to compare the inclusion to an identical volume of an equivalent medium whose intrinsic permeability K is to be determined, when the inclusion and equivalent medium are subject to a uniformly by applied pressure gradient G of amplitude G and direction E, in other words (the uppercase variables refer to the Darcy medium): gradP = G = GE

V=−

so

K GE η

For a granular medium, the simplest generic inclusion I which expresses the connectivity of ﬂuid, pore size and porosity is a composite of spheres with volume ΩI and boundary Γf = ∂ΩI consisting of a solid sphere (of radius βR, 0 < β < 1, and volume Ωs ) surrounded by a concentric spherical shell ﬁlled with ﬂuid, whose external radius is R and whose volume is Ωf = ΩI − Ωs , so that the porosity is φ = 1 − β 3 (Figure 10.2). The spherical symmetry and preferred direction introduced by the pressure gradient make it natural to use spherical coordinates (r, θ, ϕ) orientated as shown in Figure 10.2 (θ = 0 corresponding to er = ez = E). nP = G ez ez

er

M

r R

( I ey

O

s #R

f ex Darcy medium

Figure 10.2. Generic spherical ﬂuid-solid inclusion I (BR) with volume ΩI , boundary ∂ΩI = Γf , along with its associated spherical coordinate system. Solid sphere: radius Rs = βR, volume Ωs . Concentric spherical shell ﬁlled with ﬂuid: external radius R, volume Ωf = ΩI − Ωs . Solid concentration cs = β 3 , porosity φ = Ωf /ΩI = 1 − β 3

The ﬂuid in shell βR < r < R is governed by the following system of equations: the Navier-Stokes equation, incompressibility, no-slip condition, total pressure p decomposed into a pressure induced by the uniform macroscopic gradient r · gradP and an additional pressure π. They are expressed using a single system of spatial

Self-consistent Estimates and Bounds for Permeability

283

variables (the lowercase variables referring to the ﬂuid in the inclusion): −gradp + (ηv) = 0

(10.5)

div(v) = 0

(10.6)

v(βR) = 0

(10.7)

p=π+r·G

(10.8)

Aside from the periodicity condition, this system is identical to that established by homogenization. 10.2.2.2. General expression for the ﬁelds in the inclusion In order to respect the isotropy of the space, ﬁelds in inclusion must be isotropic function of the pair {position vector r, imposed gradient G}. The theory of tensorial representation [BOE 87], implies that, with Fp , Fr , FG being arbitrary functions of the three scalars r · r, G · G and r · G: p = Fp (r · r, G.G, r · G) ηv = rFr (r · r, G · G, r · G) + GFG (r · r, G · G, r · G) In addition, since the solution depends linearly on G, the general form of the solution only involves three independent functions of r: p = (r · G)h1 (r);

ηv = r(r · G)f1 (r) + Gg1 (r)

which, using the spherical harmonics technique [LAM 32; HOW 74], can be transformed into: p = G · gradh(r);

ηv = G[grad ⊗ gradf (r) + g(r)I]

(10.9)

The functions f , g, h must satisfy the Navier-Stokes equation and the incompressibility condition. Taking the divergence of (10.5), and making use of incompressibility: (p) = 0, which implies (here and in what follows, the derivatives d/dr are indicated by a prime ): (G · gradh) = G · er (h) = 0

thus

h = c0

from which it follows that: h(r) = c0

r2 1 − c1 + c 2 6 r

(10.10)

284

Homogenization of Coupled Phenomena

In what follows, we will take the constant of integration c2 , which has no physical signiﬁcance, to be zero. In addition, incompressibility gives us: div(G[grad ⊗ gradf + gI]) = G · grad[f + g] = G · er (f + g) = 0 whence: g = −f + a0 ;

grad ⊗ gradf + gI = grad ⊗ gradf − f I + a0 I

But since grad ⊗ grad(r 2 ) − (r2 )I = −4I, the constant a0 can be cancelled by including a term −(a0 /4)r2 in f . Thus v can be expressed using a single potential function f : ηv = G[grad ⊗ gradf − If ]

(10.11)

Substituting expression (10.11) for v into the Navier-Stokes equation leads to: G [−Ih + [grad ⊗ grad − I][−h + f ]] = 0 With (10.10), and setting A = −h + f : (A /r) (rA ) )+r⊗r =0 G [−c0 I + (grad ⊗ grad − I)A] = G −I(c0 + r r which gives the two equations: integrated to give: A = −h + f = −c0

(A /r) (rA ) = 0 and = −c0 which can be r r

r2 + c2 4

r2 1 Finally, f satisﬁes the following equation: f = −c0 − c1 + c2 whose 12 r solution is: f = −c0

r4 r r2 1 − c1 + c2 − c3 + c4 240 2 6 r

(10.12)

As for h, the constant of integration c4 is taken to be zero as it has no physical signiﬁcance. To summarize, the pressure and the velocity deﬁned by (10.9, 10.10, 10.11, 10.12) involve four constants of integration. The next step consists of deﬁning the correct boundary conditions in order to determine the permeability of the equivalent Darcy medium.

Self-consistent Estimates and Bounds for Permeability

285

10.2.2.3. Boundary conditions In order to express the boundary conditions, we will use the following expressions: ηv ηvr ηvθ 2ηD(v)er

= G[f /r − f ] + er (G · r)(f /r)

(10.13)

= (G · er )[f /r − f + r(f /r) ] = −2(f /r)(G · er )

(10.14)

= (G · eθ )[f /r − f ] = −[f /r + f ](G · eθ )

(10.15)

= −Gf + er (G · er )[f − 4(f /r) ]

(10.16)

2ηDrr

= −4(f /r) (G · er )

(10.17)

2ηDrθ

= −f (G · eθ )

(10.18)

ηcurl(v)

= −(G ∧ r)

(Δf ) r

(10.19)

10.2.2.3.1. Conditions on velocity At the ﬂuid-solid interface (r = βR) the no-slip condition v(βR) = 0 gives, by combining equations (10.14) and (10.15): −2

f (βR) =0 βR

2 − f (βR) = 0 3

(10.20)

(10.21)

Furthermore, the mean velocity in the inclusion I equals that in the equivalent medium: K 1 1 f (R) G =− G V= vdΩ = r(v · er ) dS + 0 = −2 Ω Ωf Ω ∂ΩI R η η and so: −2

f (R) +K =0 R

(10.22)

This equation also indicates that normal velocities in the ﬂuid and in the equivalent medium are identical on the inclusion boundary, in other words vr (R) = Vr (R) (see equation 10.14.). 10.2.2.3.2. Conditions on stress at the inclusion boundary This condition is less obvious than the preceding ones, because the transfer of viscous stress to the skeleton does not a priori lead to any requirements on the values

286

Homogenization of Coupled Phenomena

of pressure or stress on the boundary. On the other hand, it is physically reasonable (1) to transpose the result from homogenization: Γf σ f .n dS = 0 which stems from the periodicity, and thus represents the stationarity of the representative volume. Translated to the current situation, this equality indicates that the force exerted by the residual stress σr on the inclusion boundary ∂ΩI – in other words the difference between the stress in the ﬂuid and pressure in the equivalent Darcy medium – is zero. This indicates that the equilibrium of the inclusion is satisﬁed on average by the zeroorder stress. We can therefore write: σ r n dS = 0 with : σ r = −(p − P )I + 2ηD(v) (10.23) ∂ΩI

Introducing these expressions for the pressure and the shear stress, and integrating, we obtain: (G · r)(−h + 1)er + (−Gf + er (G · er )[f − 4(f /r) ])er dS 0 = ∂ΩI

=

4πR2 G (−h + R + f − 4(f /r) − 3f ) 3

so that, after simpliﬁcation, the global equilibrium can be expressed as: −h (R) + R − 2(f ) (R) = 0

(10.24)

10.2.2.3.3. Energetic consistency We still need to express the energetic equivalence between the inclusion and the same volume of the equivalent Darcy medium. According to equation (10.4.) obtained by homogenization, the conservation of energy during upscaling can be written as:

2ηDy (v) : Dy (v)dΩ = −gradP · Ωf

vdΩ

(10.25)

Ωf

Similarly, taking the scalar product of the Navier-Stokes equation (10.5) with ﬁeld v, and integrating by parts over pore volume, we obtain (bearing in mind the no-slip condition):

[−per + 2ηD(v)er ] · v dS

2ηD(v) : D(v)dΩ = Ωf

∂ΩI

By comparison with the previous equality, the following relation must be satisﬁed:

[−per + 2ηD(v)er ] · v dS = −gradP · ∂ΩI

vdΩ Ωf

Self-consistent Estimates and Bounds for Permeability

287

We note that the right-hand side can be transformed into:

−gradP ·

vdΩ = −gradP · Ωf

r(v · er ) dS = − ∂ΩI

P (v · er ) dS ∂ΩI

so that, associated with the global equilibrium condition (10.23) multiplied by the mean velocity V, energetic consistency requires that: [(P − p)er + 2ηD(v)er ] · [v − V] dS = 0 ∂ΩI

and since the normal velocities are identical over ∂ΩI (c.f. (10.22)), all that remains is: 2ηDrθ [vθ − Vθ ] dS = 0 ∂ΩI

Given the expressions for the ﬁelds, we deduce that on the boundary (r = R): – either the shear stress ηDrθ is zero: f (R) = 0

(10.26)

In this case the stress on the face normal to the boundary is equal to the pressure in the equivalent Darcy medium, in other words σ(R).er = P (R)er ; – or the tangential velocities are equal, in other words vθ = Vθ : 2 (10.27) − f (R) + K = 0 3 In this case, the velocities in the ﬂuid and equivalent Darcy medium are identical on the inclusion boundary, in other words v(R) = V(R), and additionally Drr (R) = 0. To summarize, ﬁve parameters (c0 , c1 , c2 , c3 , K) can be determined from the two systems of ﬁve equations, (10.20, 10.21, 10.22, 10.24, 10.26), or (10.20, 10.21, 10.22, 10.24, 10.27), with both respecting all the conditions for micro-macro transformation identiﬁed through homogenization. Two possibilities can therefore be envisaged: – Either over ∂ΩI the ﬂuid stress is continuous with the pressure in the equivalent Darcy medium (this is the assumption used by [BER 93]), and so we will refer to the pressure approach (denoted by P). The tangential velocities in the inclusion and equivalent medium are not continuous over ∂ΩI . – Or over ∂ΩI the velocity in the inclusion is continuous with that of the equivalent medium, and so we will refer to the velocity approach (denoted by V). Over ∂ΩI , neither the pressure nor the stress in the inclusion are continuous with the pressure in the equivalent Darcy medium.

288

Homogenization of Coupled Phenomena

Although these two assumptions are consistent, we can assign a greater reliability to the pressure approach because, referring back to homogenization, the boundary (1) condition on the pressure is exact to zero-order (since σf = P + εσf + . . . ), which is not the case for the condition on the velocity. 10.2.2.3.4. Comment on the zero-vorticity condition We note that the self-consistent assumptions do not lead to the condition of zero vorticity on the inclusion boundary as used by Tarnow [TAR 96] and Umnova et al. [UMN 00]. This condition therefore violates energetic consistency. Despite this issue, consideration of the kinematics of the ﬂow at the boundary of two identical inclusions argues in favor of this zero-vorticity condition. In fact the vorticity is strong in regions of constriction, but becomes zero when the velocity reaches its maximum at the middle of the constriction, i.e. the boundary of the inclusion. Following (10.19), the assumption of zero vorticity can be expressed by (f ) (R) = 0, which when substituted into the global equilibrium equation (10.24) gives: −h (R) + R = 0

(10.28)

which indicates that pressure (and not stress) on the ﬂuid boundary equals the pressure in the Darcy medium. This condition (which will be denoted by C) is thus very close to the P- condition, with the two being identical up to zero order.

10.2.3. Solution and self-consistent estimates 10.2.3.1. Pressure approach: p ﬁeld After introducing the expressions (10.10-10.12) for the h and f functions into the ﬁve conditions, this linear system (10.20, 10.21, 10.22, 10.24, 10.26) becomes: −

2 (βR)3 c3

−

−

2 R3 c3

−

2 c R4 3

+

− Kp

−

2 3 c2 2 c 3 2 2 3 c2

+

0

+

+ +

1 βR c1 2 c 3βR 1 1 c R 1 3 R3 c1

0

(βR)2 30 c0 (βR)2 c0 18 R2 c 30 0

+ + + + −

0 R c 10 0

=0 =0 =0 =1 =0

The solution gives the P- estimate denoted Kp : Kp = ψp (β)R2 =

ψp (β) 2 Rs β2

with

ψp (β) =

1 3

−1 +

2 + 3β 5 β(3 + 2β 5 )

(10.29)

Self-consistent Estimates and Bounds for Permeability

289

as well as the four ci coefﬁcients. The velocity ﬁeld p deﬁned by these coefﬁcients is the exact solution for ﬂuid ﬂow under the pressure imposed by a unit macroscopic pressure gradient at the inclusion boundary (r = R): σn = −P n, with gradP = ez . 10.2.3.2. Velocity approach: v ﬁeld In this case the system to be solved (10.20-10.21-10.22-10.27-10.24) is the following:

Kv Kv

−

2 (βR)3 c3

−

− +

2 c R3 3

− − −

0

2 3 c2 2 3 c2 2 c 3 2 2 c 3 2

+ + + +

1 βR c1 2 3βR c1 1 c R 1 2 c 3R 1 3 c R3 1

+ + + + +

(βR)2 30 c0 (βR)2 18 c0 R2 c 30 0 R2 c 18 0

0

=0 =0 =0 =0 =1

From this we can deduce the V-estimate denoted Kv : ψv (β) 2 Kv = ψv (β)R = Rs , β2 2

1 ψv (β) = 18

1−β (1 − β 2 )2 4 −5 β 1 − β5

(10.30)

Starting from the velocity ﬁeld v deﬁned by the coefﬁcients ci which are solutions to the equation system, we deduce the exact ﬁeld v ηHv for ﬂuid ﬂow under a unit macroscopic ﬂux imposed at the boundary of the inclusion (r = R): v ηHv = −ez Inserted into a medium of intrinsic permeability Kp – or alternatively Kv – inclusion I{β, R} is quasi-neutral, since its presence modiﬁes neither the ﬁeld in the Darcy medium nor the volume density of dissipated energy, and the continuity of stress – or alternatively velocity – is respected. 10.2.3.3. Comparison of estimates ψp /β 2 and ψv /β 2 presented as a function of porosity in Figure 10.3 show the extreme sensitivity of permeability to this parameter. In view of the inﬁnite extent of the permeabilities range (with with constant external radius of the inclusion), the two estimates are fairly close since for a ﬁxed inclusion: 1/4 Kv /Kp 1

(10.31)

The estimate Kc which corresponds to the assumption of zero vorticity, lies between Kp and Kv , and takes the following value: Kv < Kc = ψc (β)R2 < Kp ;

ψc (β) =

2 (1 − β)3 5 + 6β + 3β 2 + β 3 (10.32) 9 β 5

290

Homogenization of Coupled Phenomena

2

10

0

10

-2

K/Rs2

10

-4

10

-6

10

-8

10

0

0.2

0.4

(a)

0.6

0.8

1

0.6

0.8

1

*

1

0.75

0.5

0.25

0 0

0.2

0.4

(b)

*

Figure 10.3. (a) Logarithmic plot of estimates, using the spherical inclusion, of the normalized intrinsic permeability, i.e. K/Rs2 , as a function of porosity: P-estimate (continuous line); V-estimate (dotted line), C-estimate (mixed line). (b) Ratio Kv /Kp (continuous line) and Kc /Kp (mixed line) as a function of porosity

The discrepancies between these different estimates are weak for dilute concentrations of solids (φ ≈ 1) but are signiﬁcant at high concentrations (φ ≈ 0) as the following bounds on the behavior show: φ → 1;

ψp

1 2 − ; 9β 3

φ → 0,

ψp

2(1 − β)3 ; 3

ψv

1 2 − ; 9β 2

ψv

ψc

(1 − β)3 ; 6

2 2 − 9β 5

ψc

2(1 − β)3 3

Self-consistent Estimates and Bounds for Permeability

291

101

K/Rs2

10-1

10-3

10-5

10-7 0

0.2

0.4

0.6

0.8

1

* Figure 10.4. Comparison of the numerical results obtained by ﬁnite element simulations on simple cubic lattices (◦), body centered cubic lattices (), face centered cubic lattices () and the estimates: P estimate (continuous line); V estimate (dotted line), C estimate (mixed line)

10.2.4. From spherical substructure to granular materials The self-consistent approach leads on one hand to two values which appear physically acceptable, and on the other hand to a coefﬁcient which depends not only on the concentration of constituents – as in problems of elasticity, conductivity, etc. – but also on the size of inclusions. How can these estimates be applied and in which situations? We will tackle this question in this section. 10.2.4.1. Cubic lattices of spheres The permeability of cubic lattices of identical spheres (simple, body centered and face centered) has been determined numerically in the previous chapter, to which the reader is referred for a bibliography and calculations. In Figure 10.4 we compare these numerical results to the analytical estimates for an inclusion where the solid sphere has a radius identical to that in the lattice and has the same solid concentration, β 3 . Very good agreement is seen between numerical results and estimates across the full range of porosities. Values for the periodic medium are very close to the P and C estimates, and lie between the two estimates across almost the entire range of porosity. Thus the geometric approximation which consists of comparing the spherical inclusion to the cubic period cell, with the pressure approach, leads to an excellent approximation. The low level inﬂuence of the type of arrangement used leads us to think that the P and C estimates are appropriate for media which present a narrow granulometry.

292

Homogenization of Coupled Phenomena

Figure 10.5. Granular media consisting of ordered or disordered generic inclusions

10.2.4.2. Bounds on the permeability of ordered or disordered granular media Consider the class of granular media consisting of ordered or disordered generic spherical inclusions, I{βα , Rα } of volume Ωα , where the solid spheres are polydisperse, with the spheres ﬁlling all space (Figure 10.5). In what follows, we will show that the self-consistent approaches in pressure and in velocity make it possible to establish bounds for the permeability of these media. and representative Let us subject one of these granular media, with permability K volume: = ∪Iα {βα , Rα } Ω

;

= |Ω|

Ωα

α

where the subscript α denotes the inclusion Iα {βα , Rα }, and the hat . the medium, to a unit macroscopic pressure gradient gradP = ez , inducing a macroscopic ﬂux V = −K/ηe z . From among velocity ﬁelds in the ﬂuid which respect the NavierStokes equation, incompressibility and the no-slip condition, we will distinguish (here [-] expresses the discontinuity across the boundaries of the inclusions ∂ΩI ): – ﬁelds with continuous stresses, denoted vp , so that on the boundary ∂ΩI of every inclusion: [σ p n] = 0; – ﬁelds continuous in velocity, denoted vv , so that on the boundary ∂ΩI of every inclusion: [vv ] = 0; which is the only ﬁeld for which the stresses and velocity – the exact solution v are continuous throughout.

Self-consistent Estimates and Bounds for Permeability

293

For every ﬁeld pair (vp ,vv ) we will introduce the function of dissipated power E: E(vp , vv ) =

%

2η

& D(vp − vv ) : D(vp − vv )dΩ

(10.33)

Ωα

α

. By construction E is positive, and reaches its minimum (zero) for vp = vv = v Expanding: % 2ηD(vp ) : D(vp ) + 2ηD(vv ) : D(vv ) E(vp , vv ) = α

Ωα

& −4ηD(vp ) : D(vv ) dΩ But, using the incompressibility condition (div(vv ) = 0), and then integrating by parts, the two last terms become: 2ηD(vp ) : D(vv )dΩ = σ p : D(vv )dΩ Ωα

Ωα

=−

div(σ p ) · vv dΩ +

(σp n) · vv dS

Ωα

∂Ωα

Thus, since div(σ p ) = 0: E(vp , vv ) = E(vp , 0) + E(0, vv ) − 2

α

(σ p n) · vv dS

∂Ωα

10.2.4.2.1. Upper bound on K p by dividing up We construct a ﬁeld continuous in stress with velocity vector v p = Pα , where the medium into individual inclusions, and setting within each Iα , v p is the self-consistent P ﬁeld, deﬁned in section 10.3.4.1. By proceeding in this p n = P n is satisﬁed on the boundary of each inclusion. We then way, the condition σ ). On as a ﬁeld continuous in velocity, and calculate E( choose the solution v vp , v one hand: p n) · v dS = dS = − (σ Pv gradP · r ( v · n) dS α

∂Ωα

∂Ωα

α

=

α

α

∂Ωα

K dS = ΩgradP gradP · v · V = −Ω η Ωα

294

Homogenization of Coupled Phenomena

On the other hand, using (10.4-10.25): E( vp , 0) =

Ωα

α

Kpα , η

) = Ω and E(0, v

K η

from which we ﬁnally arrive at: ' Ωα Ω ) = E( vp , v K −K pα η α Ω

(10.34)

) 0 we deduce that: Since E( vp , v K

Ωα α

Ω

Kpα

10.2.4.2.2. Lower bound on K In a similar way, we can assume a unit macroscopic ﬂux V = −ez , corresponding which induces the exact local to the macroscopic pressure gradient gradP = ez η/K v = η/K. We construct a continuous ﬁeld in velocity, v v by setting v ﬂow ﬁeld v vα η/Kvα in each inclusion Iα , where v is the self-consistent V ﬁeld, deﬁned in v = V. We will take section 10.3.4.2, so that on the boundary of each inclusion v as a continuous ﬁeld strain, and calculate E( v is the strain η/K v ) where σ v vη/K, tensor solution: σ n) · V dS v dS = − − (η/K)( σ n) · v (η/K)( α

∂Ωα

=

α

α

∂Ωα

Ω P n · V dS = ΩgradP · V = −η K ∂Ωα

As earlier, using (10.4-10.25), we obtain: ' Ωα η η v ) = Ω >0 − E( vη/K, v Kvα K α Ω In the end, then, we have the following bounds for intrinsic permeability of any ordered or disordered granular medium involving poly-disperse solid spheres: ( )−1 Ωα Ωα 1 <
α

These general bounds, which depend on morphology, can be reﬁned for speciﬁc media.

Self-consistent Estimates and Bounds for Permeability

295

10.2.4.2.3. “Optimized” bounds for granular materials “P-optimized” granular media are media for which all the inclusions I{βα , Rα } give the same P estimate, which we will label Kp , i.e.: ∀I{βα , Rα } ∈ Ω

;

2 Kpα = ψp (βα )Rα = Kp

of these media is thus “optimally” described by: The actual permeability K ( Kp

Ωα Kpα α

)−1

Kvα Ω

<
Ωα Ω

α

Kp = Kp

which, making use of (10.31), gives the deﬁnitive result: Kp < Kp
;

2 Kvα = ψv (βα )Rα = Kv

we establish the following restriction: < 4Kv Kv < K Since these bounds apply for any ﬂux in the principal directions, the maximum anisotropy ratio of these P- or V-optimized granular media is 4. The restrictions can be improved if the solid concentration of all the inclusions is below a value of cmax < 1. In order to ﬁt the optimization condition, these media must meet a morphological constraint which requires that the solid concentration decreases with inclusion radius. More precisely, it can be shown, by inverting the functions ψ using the Padé approximation, that geometric parameters βα and Rα of the inclusions must satisfy: – for P-optimized granular media (Kp ), with ξ = β(R) ∼ =ξ

1 + 94 ξ 1 + 92 ξ(1 + 12 ξ(1 +

3

4 3 ξ))

,

1 + 32 ξ 1 + 3ξ(1 + 12 ξ(1 +

3

ξ 3 ))

,

and ζ =

β(Rs ) ∼ =ζ

– for V-optimized granular media (Kv ), with ξ = β(R) ∼ =ξ

2R2 9Kp

2R2 9Kv

β(Rs ) ∼ =ζ

3

2Rs2 9Kp

1 + ζ3 1+

ζ 2

ζ2 4

+

and ζ =

3

1 + 3ζ +

ζ3 √ 3 3

+ ζ4

2Rs2 9Kv

9ζ 4 9ζ 2 4

1+

+

+ ζ3 ζ3 + √ + ζ4 3 4 3

296

Homogenization of Coupled Phenomena

10.2.4.2.4. Bounds for granular media with homothetic inclusions We will consider β0 -homothetic granular media where the I(β0 , Rα ) inclusions have the same solid concentration β03 , which is identical in this case to the medium. of these media is therefore limited by: The actual permeability K (

Ωα 1 ψv (β0 ) 2 Rα α Ω

)−1 < ψp (β0 )
Ωα α

Ω

2 Rα

or alternatively, in terms of the solid sphere’s radii: 3 Rsα R5 −2 −2 α < K < ψp (β0 )β0 α sα ψv (β0 )β0 3 α Rsα α Rsα Using Hölder inequality, it can be shown that the equivalent medium radius, associated with its speciﬁc surface, is bound by two characteristic granulometry terms: 3 3 5 R R α Rsα α sα < Re = α sα < 2 3 α Rsα α Rsα α Rsα For dense granulometries the bounds are close together, but they become very wide apart for very widely spaced granulometries. Contrary to P- and V-optimized media, the morphology of homothetic media is not suited to a close bounding of their permeability. 10.2.4.3. Empirical laws We conclude this section by comparing the self-consistent estimates to well-known empirical laws. 10.2.4.3.1. Kozeny-Carman formula By comparing a porous medium to a lattice of channels which has the same porosity φ and speciﬁc surface (or equivalent radius Re ), Kozeny [KOZ 27] put forward an expression for permeability whose empirical form factor has been measured by Carman [CAR 37] for granular media:

KKC

Re = 2γ 3

2

φ3 1 with 2γ = 2 (1 − φ) 5

We will compare this expression to the self-consistent estimates for an inclusion I{β, R} such that φ = 1 − β 3 and R = β −1 Re . At high solid concentrations, the function: (1 − β 3 )3 KKC = 2 Re 45β 4

Self-consistent Estimates and Bounds for Permeability

297

behaves like (1 − β)3 3/5, which places it between the behavior of ψp and ψv (see section 10.2.3.3), with the coefﬁcient 3/5 being close to the coefﬁcient 2/3 of ψp . Across a wider range, the Kozeny-Carman formula clearly diverges at high porosities (i.e. above 50%), but correctly describes moderate (and low) porosities. 10.2.4.3.2. Casagrande formula Based on experimental observations, Casagrande put forward the following empirical law which gives a linear relationship between the relative variations in the φ : permeability of soil and that of the void ratio e = 1−φ ΔK Δe =b with b = 2 for e0 = 0.85 Ke=e0 e0 For standard soils, 0.5 < e < 1, i.e. 0.8 < β < 0.9, and the permeability estimates can be linearized over this interval. For the P- (V-) estimates we obtain b = 2.43 (2.58) whereas Kozeny-Carman gives b = 2.54, values which are reasonably consistent with that suggested by Casagrande. 10.2.4.3.3. Hazen formula Finally we mention that for the same reason (0.5 < e < 1), the permeability given by Hazen for media with a granulometry which is not very spread out: KH = 10−3 (D10 )2 ;

D10 (m)

(D10 being such that 10% of the mass of material consists of particles of diameter less than D10 ) is also consistent with the order of magnitude of the P and V estimates. This expression also indicates that the characteristic pore size for these granulometries is determined by the small particles, and is of the order of D10 /10. 10.2.5. Intrinsic permeability of ﬁbrous media Fibrous media can be modeled using the same method by considering cylindrical inclusions J (β, R). These inclusions, of porosity φ = 1 − β 2 , consisting of a solid cylinder (of radius Rs = βR with 0 < β < 1) surrounded by a co-axial cylindrical ﬂuid shell of external radius R, are subjected to ﬂows parallel or perpendicular to the cylinders. Thus we obtain P and V estimates for the transverse (T) and longitudinal (L) intrinsic permeability. The reader is referred to [BER 93] and [BOU 00] for details of the analytical calculations. The values obtained in this way are given below, and are shown in Figure 10.6. – P estimate: KpL = ΦpL (β)R2 ;

1 1 ΦpL (β) = − [log(β) + (1 − β 2 )(3 − β 2 )] 2 4

298

Homogenization of Coupled Phenomena

KpT = ΦpT (β)R2 ;

1 1 − β4 ΦpT (β) = − [log(β) + ] 4 2(1 + β 4 )

– V estimate: KvT = ΦvT (β)R2 ;

KvT = ΦvT (β)R2

1 1 − β2 ΦvL (β) = 2ΦvL (β) = − [log(β) + ] 2 1 + β2 We again observe that the P estimate is greater than the V estimate, with a maximum ratio of 4, which is reached at high solid concentrations. Also, whatever the solid concentration, the permeability is twice as large in the longitudinal than transverse direction – exactly so for the V estimate, and approximately so for the P estimate. 10.2.5.1. Periodic arrangements of identical cylinders Comparisons of the estimates for a J (β, R) inclusion and the numerical results presented in the previous chapter for square and triangular lattices of cylinders with the same solid radius βR and concentration c = β 2 (given in Figure 10.6) show: – A very good agreement for longitudinal ﬂows up to concentrations close to the maximum concentration of the lattice. – For transverse ﬂows, the self-consistent and periodic estimates diverge when the solid concentration approaches the maximum concentration. These discrepancies can be explained by the fact that at the maximum concentration ﬂow remains possible across the inclusions, while the lattice of cylinders in contact is impermeable (in this direction). Close to this threshold, the ﬂow is governed by a lubrication mechanism in the constrictions between the cylinders, an effect which does not describe the transverse ﬂow in the inclusion. 10.2.5.2. Permeability bounds for ideal ordered and disordered ﬁbrous media formed Consider ordered or disordered ﬁbrous media with representative volume Ω by the arrangement of parallel cylinders with radius Jα (βα , Rα ) and volume Ωα , where the solid cylinders may have different radii and where the cylinders ﬁll the space in the same way as for the spherical inclusions. Applying the same method of partition of local ﬁelds into the various inclusions, we establish the longitudinal and transverse permeability bounds, in the same form as (10.94) where the Kpα and Kvα are replaced either by KpLα and KvLα or by KpTα and KvTα . Since the anisotropy ratio is quasi-independent of the solid concentration, optimization of the morphology in order to obtain “optimized” bounds in a given direction also makes it possible to obtain bounds which are almost as close in the other direction.

Self-consistent Estimates and Bounds for Permeability

299

2

Dimensionless permeability

10

0

10

-2

10

-4

10

-6

10

-8

10

0

0.2

(a)

0.4

0.6

0.8

1

0.8

1

Porosity

2

Dimensionless permeability

10

0

10

-2

10

-4

10

-6

10

-8

10

0

(b)

0.2

0.4

0.6

Porosity

Figure 10.6. Logarithmic plot of P (continuous line) and V (dotted line) estimates for a cylindrical inclusion of the normalized intrinsic permeability in the (a) transverse and (b) longitudinal directions, i.e. K/Rs2 , as a function of porosity. Comparisons with numerical results obtained by ﬁnite element simulations of square () and triangular () lattices of parallel cylinders

10.3. Dynamic permeability In this section, we will extend the self-consistent analysis of steady-state ﬂows to the harmonic regime. The ﬁrst estimates of permeability where made by Umnova et al. [UMN 00] based on a spherical cell model with zero vorticity on the boundary, and by Thiery and Boutin [THI 02] who used a self-consistent approach for arrangements

300

Homogenization of Coupled Phenomena

of cylinders. Here we present the expansions established in Boutin and Geindreau [BOU 08]. Again the self-consistent method leads to two estimates for granular media, which are in good agreement with numerical simulations on periodic lattices, and which make it possible to establish bounds on the dynamic permeability of ideal ordered or disordered porous media. In order to treat the dynamic regime, the porous medium is subjected to small harmonic perturbations with frequency f = ω/2π. Due to the linearity, all the variables take the form Aexp(iωt) where A is the complex amplitude. In what follows, exp(iωt) will be systematically omitted.

10.3.1. Summary of homogenization results To make use of homogenization results in the self-consistent approach, we must again revisit certain homogenization results demonstrated in Chapter 4. 10.3.1.1. Global and local description – energetic consistency Dynamic Darcy’s law (the macroscopic conservation of mass is not used in the self-consistent approach) can be written: V=−

K(ω) gradx P η

(10.36)

where we recall that: – P is pressure – in fact the zero-order stress in the ﬂuid – which is constant in the (0) pores, i.e. σ f (x, y) = P (x)I; – V is the Darcy velocity, where V = Ω1 Ωf v(0) dΩ, for the local velocity v(0) , which is governed by the system of equations describing the ﬂuid volume Ωf (expressed here in dimensional variables), with gradx P acting as a forcing term, [AUR 80]: ⎧ (1) (0) (0) ⎪ ⎨ −grady p − gradx P − iωρv + divy [2ηDy (v )] = 0 (0) 0 divy (v ) = 0 Sv ⎪ (0) ⎩ v/Γ = 0 ; v(0) & p(1) Ω − periodic – K(ω) is the complex dynamic permeability tensor whose real part is KR and whose imaginary part is KI : 1 ei ⊗ ki dΩ = KR + iKI (10.37) K(ω) = Ω Ωf The three complex velocity ﬁelds ki /η associated with pressure ﬁelds pi(1) are the solutions under unit macroscopic pressure gradients in the three directions, in other

Self-consistent Estimates and Bounds for Permeability

301

words gradx P = −ei . As a consequence, with the incompressibility and no-slip condition, the local ﬁelds ki satisfy: 1 −grady pi(1) + ei − 2 ki + y (ki ) = 0 δv where: δv = η/iωρ is the thickness (complexity) of the viscous boundary layer. ki therefore only depends on the local spatial variable and the frequency via the dimensionless variable y/δv . In inverse form, generalized Darcy’s law becomes: ηH V = −gradx P

;

H = K−1 = HR + iHI

A proof of symmetry of tensors K(ω) and H(ω) can be found in Chapter 4, along with their properties of positivity, which in the isotropic case where K = KI = (KR + iKI )I and H = HI = (HR + iHI )I can be written (exp(−iωt) variations would give opposite signs to the imaginary parts): KR > 0,

KI < 0

and

HR > 0,

HI > ωρ/(ηφ)

These properties can be established with the help of the variational formulation associated with local problem (− = |Ω|−1 Ωf −dΩ expresses the mean over the representative volume of the medium): ∀w ∈ W,

2ηDy (v(0) ) : Dy (w) + iωρv(0) · w = −gradx P · w (10.38)

where W is the vectorial space of complex ﬁelds: % W = w/

w

Ω − periodic;

w/Γ = 0;

& divy (v) = 0

Taking v(0) as a test ﬁeld (the variables with a line over them denote complex conjugates), we obtain: 2ηDy (v(0) ) : Dy (v(0) ) + iωρv(0) · v(0) = −gradx P · v(0) = gradx P · K gradx P = V · ηHV η

(10.39)

which shows the energetic consistency: the real and imaginary parts of Kii /η – (or ηHii ) are the effective power dissipated through viscosity and the effective kinetic power produced per cycle under a unit macroscopic pressure (or ﬂux) gradient in each of the principal directions ei .

302

Homogenization of Coupled Phenomena

10.3.1.2. Frequency characteristics of dynamic permeability Recall in the isotropic case the properties established through homogenization by Auriault et al. [AUR 85b]: In the quasi-static regime – (low frequencies), i.e. when l/δv → 0, viscous effects dominate and: iωρ Kτ0 K(ω) K 1 − (10.40) η φ where K = K(ω = 0) is the intrinsic permeability and τ0 is the low frequency tortuosity coefﬁcient. At high frequencies –, i.e. when l/δv → ∞, inertial effects dominate and viscous effects are conﬁned to a boundary layer, which implies that: M ωc ηφ H(ω) τ∞ 1 + (10.41) iωρ 2 iω where τ∞ is tortuosity, M a form factor, which is generally close to 1, and ωc critical frequency dividing the low- and high-frequency domains. The latter can be obtained by equating viscous effects (i.e. the real part estimated using the low-frequency approximation) and inertial effects (i.e. the imaginary part estimated using the highfrequency approximation): ωc =

φη Kρτ∞

After Johnson et al. [JOH 86] (see section 9.4.4.3), M ωc 2δv = 2 iω Λv where Λv is the viscous length characterizing the medium. Also recall that the ηφ dimensionless functions K/K and H iωρτ = HKωc /ω only depend on the ∞ dimensionless frequency ω/ωc . The approximate formula for H(ω) [JOH 87] offers the advantage of being a causal function which connects the low-frequency behaviors and the high-frequency behavior: M iω H(ω)ηφ ωc 1+ (10.42) τ∞ 1 + iωρ iω 2 ωc To conclude this section we will now show (in the isotropic case, but the result also applies to principal values of an anisotropic medium) that the real part of

Self-consistent Estimates and Bounds for Permeability

303

H – the viscous dissipation for unit velocity – grows with frequency. We also show that contrary to this, the imaginary part of H divided by the pulsation – the inertial effect of added mass or tortuosity – decreases with frequency: dHR 0 ; dω

d(HI /ω) 0 dω

(10.43)

In order to analyze the variations of H with frequency, let us calculate the virtual power of a velocity ﬁeld – the solution at a given frequency – under a velocity ﬁeld which is the solution at another frequency, and vice-versa. In order to do this, we introduce solutions u and u corresponding to real mean unit velocities at frequencies ω and ω , so that: u = u = e

;

|e| = 1

arg(e) = 0

The imposed pressure gradients associated with u and u are −ηHe and −ηH e respectively, and the variational ω-formulation (10.87) for u (and similarly for u , ω , H = H(ω )) is: ∀w ∈ W

;

2ηDy (u) : Dy (w) + iωρu · w = +He · w

Choosing test ﬁeld w = H u (w = Hu) for the variational ω- (ω -) formulation, we ﬁnd: 2ηDy (u) : Dy (u )H + iωρu.u H = ηHH 2ηDy (u ) : Dy (u)H + iω ρu · uH = ηH H By subtracting these we obtain the following equality which is valid for all frequencies ω and ω : 2ηDy (u) : Dy (u )(H − H) + iρu · u (ωH − ω H) = 0 From this it follows, taking the limit where ω approaches ω, that: 2ηDy (u) : Dy (u)

dH d(H/ω) + iρu · uω 2 =0 dω dω

(10.44)

In addition, choosing w = u in the variational ω-formulation: 2ηDy (u) : Dy (u) + iρu · uω = ηH and solving equations (10.44-10.45) gives by elimination of H: dH iρ = u.u dω η

;

ω2

d(H/ω) = −2Dy (u) : Dy (u) dω

(10.45)

304

Homogenization of Coupled Phenomena

so that, returning to the real and imaginary parts: 2

iρ dHR = (u + u) · (u − u) dω η

2iω 2

d(HI /ω) = −2[Dy (u) + Dy (u)] : [Dy (u) − Dy (u)] dω

In order to establish the sign of these derivatives, we use w = u − u as a test ﬁeld in the variational ω-formulation in original and then conjugated form. In this way we obtain: 2ηDy (u) : [Dy (u) − Dy (u)] + iωρu.(u − u) = 0 2ηDy (u) : [Dy (u) − Dy (u)] − iωρu · (u − u) = 0 which, by summation and subtraction leads to: η[Dy (u) + Dy (u)] : [Dy (u) − Dy (u)]

=

2iωρuI · uI

iωρ(u + u) · (u − u)

=

8ηDy (uI ) : Dy (uI )

and consequently: 4η dHR = Dy (uI ) : Dy (uI ) dω ω

;

2ρ d(HI /ω) = − uI · uI dω ω

Since the terms on the right-hand side are positive and negative, respectively, we arrive at the result mentioned earlier.

10.3.2. Self-consistent estimates of dynamic permeability In this section we will return to the self-consistent approach adopted for intrinsic permeability, transposing it to the dynamic permeability [BOU 08]. We will work with the same type of generic spherical inclusion I and retain the same notations (see Figure 10.2). Since the general principles of the upscaling discussed in section 10.2.1.2 still apply, we will stick to emphasizing the differences introduced by inertial effects.

10.3.3. Formulation of the problem in the inclusion In accordance with the results of periodic homogenization, the equivalent medium is assumed to follow the dynamic Darcy’s law recalled above. For convenience,

Self-consistent Estimates and Bounds for Permeability

305

we will consider isotropic media, but the reasoning can be transferred to the principal values of an anisotropic medium. As before, we need to compare the inclusion to an identical volume of an equivalent medium whose dynamic permeability K(ω) is to be determined, when the two are subjected to a uniform forcing pressure gradient G of amplitude G and direction E, i.e. (with the uppercase variables referring to the dynamic Darcy medium): gradP = G = GE

where

V=−

K(ω) GE η

The ﬂuid in the shell βR < r < R is governed by the following system of equations, identical to that established by homogenization (aside from the periodicity condition): the dynamic Navier-Stokes equation, incompressibility, no-slip condition, total pressure p decomposed into a pressure induced by the uniform macroscopic gradient, i.e. r.G, and an additional pressure π. The equations are expressed using a single system of spatial variables (with the lowercase variables referring to the ﬂuid in the inclusion): −gradp − iωρv + (ηv) = 0

(10.46)

div(v) = 0

(10.47)

v(βR) = 0

(10.48)

p=π+r·G

(10.49)

10.3.3.1. Expressions for the ﬁelds We can establish that, as in the steady state case, pressure and velocity take the form: p = G · gradh(r)

with

h(r) = c0

r2 1 − c1 6 r

ηv = G[grad ⊗ gradf − If ]

(10.50) (10.51)

so that after introducing the (complex) thickness of the viscous boundary layer δv = η/iωρ the dynamic Navier-Stokes equation becomes: 1 0 = G −Ih + [grad ⊗ grad − I][−h + ( − 2 )f ] δv which, similarly to the steady state case, results in: −h + ( −

r2 1 )f = −c0 + c2 2 δv 4

(10.52)

306

Homogenization of Coupled Phenomena

Making use of the expression for h, f satisﬁes a Helmholtz equation with forcing terms: ( −

1 r2 1 − c1 + c2 )f = −c0 2 δv 12 r

which solution is the sum of speciﬁc solutions and spherical wavefunctions of viscous diffusion (spherical harmonics [LAM 32]). These ones introduce two additional constants, c and c (the constant c2 which is of no physical signiﬁcance is taken to be zero): r/δv −r/δv r2 1 δv2 2 2 e e +c ) (10.53) f = δv c0 ( + ) + c1 + δv (c 12 2 r r/δv −r/δv 10.3.3.2. Boundary conditions From a formal point of view, the expressions for pressure and velocity are identical to those obtained in the steady state case (but of course the function f and the h coefﬁcients are different). Because of this, the expressions for the velocities and stresses given in section 10.2.2.3 remain valid, and boundary conditions retain the same form as in the steady state case, speciﬁcally: – no-slip condition at the ﬂuid-solid interface (v(βR) = 0): −2

f (βR) =0 βR

2 − f (βR) = 0 3

(10.54) (10.55)

– the identity of the mean velocity in the inclusion I and equivalent Darcy medium (which in fact leads to the identity of the normal velocities vr (R) = Vr (R)): f (R) +K=0 (10.56) R – the condition of mean equilibrium of inclusion under the macroscopic pressure gradient, i.e.: (−(p − P )I + 2ηD(v))n dS = 0 (10.57) −2

∂ΩI

which gives: −h (R) + R − 2(f ) (R) = 0

(10.58)

– energetic consistency, which again leads to two possibilities on the boundary (r = R): – either the shear stress ηDrθ goes to zero (and stress on the face normal to the boundary equals pressure in the equivalent Darcy medium, i.e. σ(R)er = P (R)er ). We then have: f (R) = 0 (10.59)

Self-consistent Estimates and Bounds for Permeability

307

– or the tangential velocities are equal, i.e. vθ = Vθ (and consequently the ﬂuid velocities and equivalent Darcy medium are identical on the inclusion boundary, i.e. v(R) = V(R) and also Drr (R) = 0). In this case: 2 (10.60) − f (R) + K = 0 3 Justiﬁcation of the energetic consistency conditions is similar to that in the steady state case. Homogenization shows that the micro/macro energetic consistency can be expressed as (10.39): [ηDy (v) : Dy (v) − iωρv · v] dΩ = −gradP · vdΩ (10.61) Ωf

Ωf

At the same time, in the inclusion, taking the scalar product of the Navier-Stokes equation (10.46) with v ¯, and integrating over the ﬂuid volume we obtain: [ηD(v) : D(v) − iωρv · v] dΩ = [−per + 2ηD(v)er ] · v ¯ dS Ωf

∂ΩI

By comparison with the previous equality we deduce the condition: [−per + 2ηD(v)er ] · v ¯ dS = −gradP · vdΩ = − P (¯ v · er ) dS ∂ΩI

Ωf

∂ΩI

If we substitute in the condition of global equilibrium (10.57) multiplied by the ¯ this equality takes the form: conjugate of the mean velocity V, ¯ dS = 0 [(P − p)er + 2ηD(v)er ] · [¯ v − V] ∂ΩI

which, due to the normal velocities being equal, can be simpliﬁed to: 2ηDrθ [v¯θ − V¯θ ] dS = 0 ∂ΩI

from which we can deduce the two conditions mentioned above. 10.3.4. Solution and self-consistent estimates In order to determine the ﬁve constants in the problem (c0 , c1 , c, c , K(ω)) two choices of ﬁve conditions are therefore possible, only differing in the energetic consistency condition chosen. They lead to two estimates: – system (10.54, 10.55, 10.56, 10.58, 10.59) which – at the inclusion/equivalent medium interface – chooses continuity of stress over continuity of tangential velocity, leading to the P-estimate;

308

Homogenization of Coupled Phenomena

– system (10.54, 10.55, 10.56, 10.58, 10.60) which instead chooses continuity of velocity over continuity of stress, leading to the V-estimate. For the same reasons as in the steady state case, the P choice appears more reliable since it uses a condition which is exact to zero order, in contrast to the V choice. 10.3.4.1. P estimate: p ﬁeld With x being the complex number x = R/δv , linear system (10.54, 10.55, 10.59, 10.56, 10.58) can be written: βx

c eβx

+ βx

1 1 e c βx (1 − βx ) βx x e cN x x −c x1 (1 − x1 ) ex

− − +

−βx

c e−βx

−βx

1 c βx (1 *

1 e + βx ) −βx 6 c (1 + x2 )(x + 1) −x c x1 (1 + x1 ) e−x

+ −x + 2 e−x

=0

+

c0 2 c0 6

+ −

0 c0 6

+

0

=0 = 12 K = δ2p

+

0

+

−

c1 (βR)3 6c1 R3 c1 2R3 3c1 R3

− −

=0

v

where: 6 N = (1 + 2 )(x − 1) − 2 x Solving this leads to the P estimate of dynamic permeability, denoted Kp (ω): 3 Ap + Bp tanh[x(β − 1)]/x 2 Kp = δv 1 − ; Dp = 2 (10.62) 1 + Dp ap + bp tanh[x(β − 1)]/x where: Ap = −

2x2 x2 (6 + x2 ) + [3 + (βx)2 ] − 3β + cosh[x(β − 1)] 6β

Bp = [3 + (βx)2 ] − 3βx2 +

x2 (2 + x2 ) 2β

ap =

(3 + (βx)2 )(6 + x2 ) 2 + x2 − 3β 6 2

bp =

(3 + (βx)2 )(2 + x2 ) 6 + x2 − βx2 2 2

The inverse Hp of Kp is given by: 1 3 Hp = 2 1 + δv Dp − 2

Self-consistent Estimates and Bounds for Permeability

309

The velocity ﬁeld p deﬁned by coefﬁcients c, c , c0 , c1 is the exact solution for ﬂuid ﬂow in the inclusion I{β, R} under the pressure imposed by a unit macroscopic pressure gradient on its boundary, i.e. at r = R: σn = −P n, with gradP = ez . 10.3.4.2. V estimate: v ﬁeld In this case the system (10.54, 10.55, 10.60, 10.56, 10.58) to be solved is: βx

c eβx

+ βx

1 1 e c βx (1 − βx ) βx x e cx x −c x1 (1 − x1 ) ex

− + +

−βx

c e−βx

−βx

1 1 e c βx (1 + βx ) −βx −x e c −x −x c x1 (1 + x1 ) e−x

+

0

+

−

c1 (βR)3

+

+

0

+

−

c1 2R3 3c1 R3

−

c0 2 c0 6 c0 2 c0 6

+

0

=0 =0 Kv = − 32δ 2 =

v

1 2

v =K δ2 v

This leads to the V estimate of the dynamic permeability, denoted Kv (ω); Av + Bv tanh[x(β − 1)]/x 3 2 (10.63) K v = δv 1 − ; Dv = 2 1 + Dv av + bv tanh[x(β − 1)]/x where: Av = −

2x2 x2 + [3 + (βx)2 ] − 3β + cosh[x(β − 1)] β

Bv = [3 + (βx)2 ] − 3βx2 +

x2 (3 + x2 ) 3β

av = [3 + (βx)2 ] − β(3 + x2 ) bv =

(3 + (βx)2 )(3 + x2 ) − 3βx2 3

The inverse Hv of Kv is: 1 3 Hv = 2 1 + δv Dv − 2 Starting from velocity ﬁeld v deﬁned by c, c , c0 , c1 coefﬁcients that solve the system, we deduce the exact ﬁeld v ηHv for ﬂuid ﬂow in I{β, R} under a unit macroscopic ﬂux imposed on its boundary r = R: v ηHv = −ez . As in the steady state case, when inserted into a medium of intrinsic permeability Kp (ω) (or Kv (ω)), the inclusion I{β, R} is quasi-neutral, since its presence modiﬁes

310

Homogenization of Coupled Phenomena

neither the ﬁeld in the Darcy medium nor the volume density of kinetic or dissipated energy, and the stress (or velocity) continuity is respected. We note that the assumption of zero vorticity on the inclusion boundary contravenes energetic consistency, although it gives values which are fairly close. The analytical expression for these is as follows: 1 2 (10.64) Kc = 2δv 1 − 1 − Ac Ac =

(1 − β 3 ) β 2 1 − 1/β + (x2 − 1/β)tanh(x(β − 1))/x + 2 3 x 1 + tanh(x(β − 1))/x

10.3.4.3. Commentary and comparisons with numerical results for periodic lattices The variations of Kp (ω), Kv (ω) and Kc (ω) with frequency are shown in Figure 10.7, which shows that these estimates are consistent with the dynamic permeability properties detailed in section 10.3.1.2. We also compare these estimates to numerical results obtained using periodic lattices of spheres (simple, centered and face centered cubic) presented in the previous chapter. 10.3.4.3.1. Low frequencies Expanding our results at low frequencies, we can show that the estimates follow the behavior in (10.40). We recover the estimates for intrinsic permeability Kp , Kv and Kc given in 10.2.3. The low-frequency tortuosity coefﬁcients lie between 1 and 9/5 with τ0p > τ0c > τ0v . 10.3.4.3.2. High frequencies High-frequency behavior obtained by expansion of the estimates also follows the behavior in (10.41). From this we can obtain the P, V and C values for tortuoisty τ∞ and form factor M . For tortuosity, the three estimates are identical: τ∞ = 1 +

3−φ β3 = < τ0v < τ0c < τ0p 2 2

The reason for this is that at high frequencies viscous effects become negligible compared to inertia. Outside a boundary layer at the solid surface, the ﬂow behaves as a perfect ﬂuid governed by conditions which involve neither tangential ﬂux nor viscous stress. Thus the three approaches lead to the same solution for τ∞ . This is identical to a diffusion problem where the inverse of iωρ plays the role of conduction: the immobile solid can be considered to have inﬁnite density, in other words zero conduction (see Chapter 6, section 6.3.1, equation (6.25)). The results of Hashin [HAS 68] applied to the tortuosity lead to the above values.

Self-consistent Estimates and Bounds for Permeability

311

100

1 % & )

0.8

-1

||K||/Kp

KR /Kp

10 0.6

0.4

-2

10 0.2

0 -2 10

-1

0

10

10

1

2

10

10-3 10-2

3

10

/cp

(a)

10-1

0

0

-0.1

-0.1

-0.2

-0.3

-0.4

100

101

102

103

/cp

(b)

arg(K)/$

KI/Kp

10

-0.2

-0.3

-0.4 % & )

-0.5 -2 10

-1

0

10

10

10

1

2

10

-0.5 -2 10

3

10

/cp

(c)

-1

10

0

10

10

1

2

10

3

10

/cp

(d)

Figure 10.7. P, V and C estimates of the dynamic permeability as a function of the logarithm of frequency normalized by the P critical frequency: (a) real parts, (b) modulus (c) imaginary parts and (d) phase. P estimate (continuous line); V estimate (dotted line); C estimate (mixed line). Calculations carried out for a porosity of 1/3, shown with the amplitudes normalized by the P intrinsic permeability Kp (0) = Kp . The P, V and C curves are shifted because of the differences in the critical frequencies

With the high- and low-frequency limits, we can deduce the critical frequencies: ωci =

ηφ , Ki ρτ∞

i =p ,v ,c

The form factors are given by: Mp = 2

9β 2 2φ2

2

φKp , τ∞ R2

Mv = 2

9(β 2 + β 6 ) 2φ2

2

φKv τ∞ R 2

312

Homogenization of Coupled Phenomena 1

4

10

-1

10

+0

K/R2s

3

10-3

2 10-5

10-7

1 0

0.2

0.4

(a)

0.6

0.8

1

0

0.2

0.4

(b)

*

2.5

3

2

2.5

1.5

0.8

1

0.6

0.8

1

+

M

3.5

0.6

*

2

1

1.5

0.5

0

1 0

0.2

0.4

(c)

0.6

0.8

0

1

*

0.2

0.4

*

(d)

Figure 10.8. Evolution of the dynamic permeability parameters at low frequencies as a function of the porosity: (a) steady state permeability. K/Rs2 , (b) steady state tortuosity τ0 , (c) tortuosity τ∞ , (d) form factor M . P estimate (continuous line); C estimate (dotted line); V estimate (mixed line). Comparison with the numerical results obtained for cubic lattices of spheres

Mc = 2

9β 3 2φ2

2

φKc τ∞ R 2

As Figure 10.8 (d) shows, these coefﬁcients vary from 0 at high porosity to 2/3 at low porosity, and 1 < Mp /Mv < 2. In terms of characteristic viscous length compared to size of the solid spheres, we obtain: Λc 4φ 2φ(3 − φ) Λp τ∞ = = = Rs Rs 9(1 − φ) 9(1 − φ)

;

Λv =

Λp 1 + β4

(10.65)

Self-consistent Estimates and Bounds for Permeability

313

1.6

KHR

*HI /(')

10

1 -2

10

-1

10

0

10

(a)

1

10

10

2

1.4

1.2

1 -2 10

3

10

-1

10

0

10

(b)

/c

1

10

1

10

10

2

10

3

2

10

/c 3

10

KHR

*HI /(')

2.5

2

1.5

1 -2

10

(c)

-1

10

0

10

1

10

2

10

/c

3

10

1 -2 10

(d)

-1

10

0

10

10

3

/c

Figure 10.9. Real and imaginary parts of H as a function of frequency, displayed in dimensionless form. Comparison of P estimates (continuous line); C estimates (dotted line); V estimates (mixed line) with the numerical results obtained for cubic lattices of spheres. Figures (a) and (b): φ = 0.3. Figures (c) and (d): φ = 0.7

10.3.4.3.3. Full frequency range In Figure 10.9, the frequency variations of the real and imaginary parts of Hp , Hc and Hv , given in dimensionless form, reveal the inequalities: Kc R ω Kv R ω Kp R ω Hp ( Hc ( H ( ) ) ) η ωcp η ωcc η v ωcv φ I ω φ I ω φ I ω H ( H ( H ( ) ) ) ωρ p ωcp ωρ c ωcc ωρ v ωcv

314

Homogenization of Coupled Phenomena

These functions display the same characteristics and, in accordance with (10.43), HR (HI /ω) grows (decreases) monotonically with frequency. The difference between the critical P and V frequencies increases the differences between the dimensional parameters. 10.3.4.3.4. Comparison with calculations for periodic lattices of spheres The three analytical estimates give a correct qualitative description of the permeability of the lattices (simple cubic, centered cubic and face-centered cubic) of spheres, whether they are separate or interpenetrating. Quantitatively the P and C estimates are better across the whole frequency range. There is good accuracy for the four parameters K, τ0 , τ∞ , M when porosity is greater than 0.5. In contrast, for lower porosities the differences between the estimates and calculations can be as much as a factor of two. 10.3.4.4. Bounds on the dynamic permeability of granular media Consider again the class of granular media formed from spherical inclusions I{βα , Rα } of volume Ωα , where the solid spheres are poly-dispersed, with the ensemble ﬁlling the whole space, and where the volume is represented as follows (with the subscript α designating the Iα {βα , Rα } inclusion, and the hat . designating the medium): = ∪Iα {βα , Rα } Ω

;

= |Ω|

Ωα

α

Through energetic analysis and partition of the local ﬁeld between individual inclusions, we have established boundaries on steady state permeability. This same approach, repeated in the harmonic regime, leads to the following boundaries for real and its inverse H, valid across all and imaginary parts of dynamic permeability K frequencies and for all ordered and disordered distributions of inclusions [BOU 08]: , Ωα R KR K pα α Ω

;

Ωα I , |KI | |K | pα α Ω

Ωα R , H HR vα Ω

;

, Ωα I HI H vα Ω

α

α

In order to establish these results, we will subject a granular medium of dynamic permeability K(ω) to a unit harmonic macroscopic pressure gradient of amplitude gradP = ez . This leads to a homogenous macroscopic ﬂux of (complex) amplitude V = −K(ω)e z . From among ﬁelds satisfying the dynamic Navier-Stokes equation in all the inclusions, as well as the incompressibility and no-slip conditions, we will

Self-consistent Estimates and Bounds for Permeability

315

distinguish the following (where [-] indicates discontinuity across the interface between the inclusions): – continuous stress vp , so that on the boundary of every inclusion: [σ p n] = 0; – continuous velocity vv , so that on the boundary of every inclusion: [vv ] = 0; , the only ﬁeld which is continuous in both strain and – the exact solution v velocity. For every (vp , vv ) pair we will introduce the complex power function E: % & 2ηD(vp −vv ) : D(vp − vv )+iωρ(vp −vv )·(vp − vv ) dΩ E(vp , vv ) = α

Ωα

By construction, the real and imaginary parts of E are positive, and their minima . Expanding E we obtain: (zero) is reached when vp = vv = v E(vp , vv ) =

[2ηD(vp ) : D(vp ) + iωρvp .vp ] + [2ηD(vv ) : D(vv ) + iωρvv .vv ] Ωα

α

' −[2ηD(vp ) : D(vv ) + iωρvp .vv ] − [2ηD(vp ) : D(vv ) + iωρvp .vv ] dΩ

Integrating by parts and using the properties of the ﬁelds (div(vv ) = 0, div(σp ) = iωρvp ), the last two integrals can be transformed into surface integrals and: E(vp , vv ) = E(vp , 0) + E(0, vv ) − [(σp .n).¯ vv + (σ p .n).vv ] dS α

∂Ωα

(ω) 10.3.4.5. Bounds on the real and imaginary parts of K p by dividing up the medium We will construct a ﬁeld with continuous strain v p = pα , where p is the into its individual inclusions, and setting in each Iα , v p n = P n is P-consistent ﬁeld deﬁned in section 10.3.4.1. In this way condition σ for satisﬁed on the boundary of each inclusion. We now choose the exact solution v ) it can be observed that: vp , v our velocity-continuous ﬁeld vv . To calculate E( . * + ¯ ¯ + Pn · v + (σ dS = dS p n) · v p n) · v Pn · v − (σ α

=

α

∂Ωα

α

∂Ωα

, + * Ω ¯ dS = Ω[gradP + gradP · v ·V+gradP ·V] = −2 KR gradP · v η Ωα

316

Homogenization of Coupled Phenomena

also, with (10.39-10.61): E( vp , 0) =

Ωα

α

Kpα η

;

K ) = Ω E(0, v η

and ﬁnally: ( ) ' ' Ωα 1 Ω , R ) = E( vp , v K − K (10.66) Ωα Kpα + Ωα K − 2Ω K = pα η α η α Ω v leads to consideration of E( vp , i v). Observing In a similar way, choosing vv = i ), and: that E(0, i v) = E(0, v −

-

. p n) · i p n) · i (σ v + (σ v dS

∂Ωα

α

= Ω[gradP · iV + gradP · iV] = 2i

,I Ω K η

we ﬁnd that: ( ) ' ' Ωα 1 Ω ,I E( vp , i v) = K + K (10.67) Ωα Kpα + Ωα K + 2iΩ K = pα η α η α Ω ) and imaginary part of From the positive nature of the real part of E( vp , v E( vp , i v), we can deduce the following bounds (recall that KI < 0): , Ωα R K KR < pα α Ω

;

Ωα I , |KI | < |K | pα α Ω

(ω) 10.3.4.6. Bounds on the real and imaginary parts of H Conversely, suppose we have a unit macroscopic ﬂux V = −ez , corresponding to a macroscopic pressure gradient gradP = H(ω)e z associated with the exact local H(ω). We can construct a velocity-continuous ﬁeld v v by setting velocity solution v v = vα Hvα in each inclusion Iα , where v was deﬁned in section 10.3.4.2. If v v = V on the boundary of each inclusion, and taking v H(ω) we do this, v as a stress v v ), it can be observed that: continuous ﬁeld. To calculate E( vH, −

α

-

. σ n) · v σ n) · v v dS v + (H (H

∂Ωα

Self-consistent Estimates and Bounds for Permeability

=−

. σ n) · V dS σ n) · V + (H (H

∂Ωα

α

=

α

317

,R [gradP · V + gradP · V] = −2Ω H (P n · V + P n · V) dS = Ω

∂Ωα

As a consequence, using (10.39-10.61): v v ) = 2E( vH,

' ' ,R Ωα + Ωα (Hvα Hvα )Kvα − 2Ω H Ωα H =Ω Hvα − H Ω

α

α

We also establish, as earlier, that: ' ' ,I Ωα v v ) = 2E(−i vH, (H H )K = Ω + H H Ωα H+Ω −2 Ω i H α vα vα vα vα α α Ω v v ) and the imaginary part of The positive nature of the real part of E( vH, v ) gives: E(i vH, v Ωα R , H HR < vα α Ω

, Ωα I H HI < vα α Ω

;

which forms the second part of the aforementioned result. 10.3.4.7. Low-frequency bounds follow directly The bounds already established for the intrinsic permeability K from inequalities that apply to the real and imaginary parts of K(0) and H(0): (

Ωα 1 Kvα α Ω

)−1 K

Ωα α

Ω

Kpα

For the low-frequency tortuosity coefﬁcient, two upper bounds can be established, with the simplest one stemming from the inequality which applies to the imaginary part of H(0): Ωα τ0vα τ0 φα Ω φ α

318

Homogenization of Coupled Phenomena

10.3.4.8. High-frequency bounds for tortuosity At high frequencies, expressions for bounds on the form factor are unfortunately rather complex. On the other hand for tortuosity τ∞ we obtain: (

Ωα φα τ α Ω ∞α

)−1 <

Ωα τ∞α τ, ∞ < φα Ω φ α

If all the inclusions have the same solid concentration (so that it is a β0 homothetic medium as described in section 10.2.4.2), the two bounds are identical and the estimates give the exact value of the high-frequency tortuosity (τ, ∞ = τ∞α ), in agreement with Hashin’s result. On the other hand, as we mentioned in section 10.2.4.2, this type of morphology does not allow the bounds on the intrinsic permeability to be optimized. Again, the whole approach presented here for granular media can be applied equally to ﬁbrous media (with the base solutions for the cylindrical inclusions being expressed with the help of Bessel functions, see [THI 02]). 10.4. Klinkenberg correction to intrinsic permeability By construction, Darcy’s law incorporates the no-slip condition on the walls of the pores. When ﬂuid slips at the wall, the ﬂow law is modiﬁed by the Klinkenberg effect. This is the case for ﬂow of rareﬁed ﬂuids in which the length of mean free path of the molecules is of the same order as pore size. These situations can be encountered either for low-pressure gases or in porous media with very small pores. In order to estimate the effective coefﬁcients for Klinkenberg’s law, we will use the self-consistent method to treat local problems established through periodic homogenization, as described in Chapter 8 [SKJ 99a]. 10.4.1. Local and global descriptions obtained through homogenization The mean free path λ is linked to the state of the gas in terms of pressure P and temperature T , and to its properties of viscosity η and molar mass M (with R = 8.37 J/mole ˚K being the ideal gas constant) through: η πRT λ(P, T ) = P 2M When λ is of pore size order, Klinkenberg’s law can be written (macroscopic conservation of mass will not be used here): V=−

Kk (P ) gradx P η

(10.68)

Self-consistent Estimates and Bounds for Permeability

319

– P is the pressure – in fact the zero-order stress – which is constant in the pores, (0) i.e. σ f (x, y) = P (x)I. – V = Ω1 Ωf v(0) dΩ is the mean of the local velocity v(0) . This is governed by the same system of equations as in Darcy’s law, except the no-slip condition is replaced by the wall-slip condition. The latter gives velocity at the wall as a function of local velocity gradient and the mean free path corrected by a dimensionless constant c, which is close to 1. In dimensional variables, and designating normal to the wall by n and the tangent in the direction of the ﬂow by t, we have [SKJ 99a]: ⎧ −grady p(1) − gradx P + divy [2ηDy (v(0) )] = 0 ⎪ ⎪ ⎨ divy (v(0) ) = 0 S 0 v/Γ = −cλ(tgrady v(0) .n)t ⎪ ⎪ ⎩ v(0) & p(1) Ω − periodic As before, gradx P acts as a forcing term, but as well as this, through the wallslip condition which involves λ(P, T ), pressure P becomes a parameter in the local problem. Because of this, Kk depends on the pressure. – Kk (P ) = Ω1 Ωf ei ⊗ kki dΩ is the positive symmetric Klinkenberg permeability tensor, constructed starting from the three velocity ﬁelds kki /η which are the solutions under unit macroscopic pressure gradients in the three directions, gradx P = −ei . The kki ﬁelds depend on position through the dimensionless variable y/cλ. If λ is of a smaller order than the pore size, the wall-slip condition can be treated as a distribution of the Darcy ﬂow, and the pressure dependence – formulated here in the isotropic case – is explicit: b k ; b>0 K (P ) = K 1 + P 10.4.2. Self-consistent estimates of Klinkenberg permeability Since the general principles which apply to upscaling, given in section 10.2.1.2, are the same, self-consistent treatment is carried out in the same way as the calculation for intrinsic permeability in section 10.2.2.1, where only the boundary condition at the wall is altered [BOU 07]. We will use the same type of generic spherical inclusion I and retain the same notations (see Figure 10.2). In accordance with the results of periodic homogenization, the equivalent medium is assumed to follow Klinkenberg’s law. For convenience, we will consider isotropic media, but the reasoning can easily be transferred to the principal values of an anisotropic medium. Under a uniform macroscopic gradient gradP = G, ﬂuid in the shell βR < r < R is governed by the following system of equations (Navier-Stokes equation, incompressibility, wall-slip condition, total pressure p decomposed into a pressure

320

Homogenization of Coupled Phenomena

induced by the uniform macroscopic gradient r·gradP and an additional pressure π), expressed with a single system of spatial variables (with the lowercase variables referring to the ﬂuid in the inclusion):

v(βR) · er = v(βR) · eϕ = 0

;

−gradp + (ηv) = 0

(10.69)

div(v) = 0

(10.70)

v(βR) · eθ = cλDrθ

(10.71)

p=π+r·G

(10.72)

Physically this system is correct as long as λ is smaller than, or of the same order as, pore size (beyond this, we enter the Knudsen diffusion regime where the continuum description on the scale of the pores is lost). Introducing the ratio of the mean free path to the solid sphere radius, using coefﬁcient χ, we can express our assumption as: χ=

cλ = O(1) βR

The wall slip does not modify the general form of pressure and velocity solutions given by (10.9, 10.10, 10.11, 10.12). Boundary conditions on normal velocities (10.20, 10.22), global equilibrium (10.24) and energetic consistency (10.26 or 10.27), are also retained. On the other hand, the wall-slip condition means that we must replace (10.21) with: −f (βR) = −

cλ f (βR) 2

and if we replace f with its expression (10.12), we deduce that: −

1 1 (βR)2 (1 − χ)c0 = 0 (2 + 3χ)c3 + c2 − 3 (βR) 3 20

(10.73)

The two equation systems (10.20, 10.73, 10.22, 10.24, 10.26) and (10.20, 10.73, 10.22, 10.24, 10.27) lead to two estimates Kpk and Kvk for the Klinkenberg permeability. Here we will only discuss the P estimate which, as we have already mentioned, follows from boundary conditions which are closer to actual physical conditions. By solving the linear system we obtain the following expression, where we again ﬁnd the dimensionless function ψp for steady state permeability (10.29): 5

Kpk (P )

=

ψpk (β, χ)R2

with

3ψpk (β, χ)

+ 1 = [3ψp (β) + 1]

1−β 1 + 3χ 3β 5 +2 5

1−β 1 + 3χ 2β 5 +3

Self-consistent Estimates and Bounds for Permeability

which can also be written as: ( ) 5 2 3χ ) (1 − β 1 Kpk (P ) = Kp 1 + ) (1 + 1−β 5 3ψp (β) (2β 5 + 3)(3β 5 + 2) 1 + 3χ 2β 5 +3 Recalling that: η c cλ = χ= βR P βR

321

(10.74)

πRT 2M

the P estimate above gives dependence of the Klinkenberg permeability on the pressure, as a function of the solid concentration β 3 of the inclusions. We note that if λ is of the same order of magnitude as the pore size, i.e. χ = O(1), Darcy’s law is not simply corrected by a b/P term; for example, in porous media with weak or strong solid concentration we have: φ→1 ;

(β → 0)

;

Kpk (P ) Kp (1 +

3 χ ) 51+χ

φ→0 ;

(β → 1)

;

Kpk (P ) Kp (1 +

3 χ ) 1 + 3χ(1 − β) 2(1 − β)

This result can be compared with Klinkenberg’s experimental observations (described by [CHA 04a]) which show that the b coefﬁcient decreases steadily as the pressure drops. With a weak Klinkenberg effect, i.e. χ 1, we recover the b/P correction, with the b coefﬁcient being expressed as: bp = ηc

(1 − β 5 )2 πRT 1 πRT 1 (1 − β 5 )2 R = ηc 2M Kp (2β 5 + 3)2 2M Kp (2β 5 + 3)2 ψp (β)

This estimate is interesting for two reasons. Firstly, for an actual material whose porosity and intrinsic permeability characteristics are known, it allows us to evaluate the level of pressure above which the Klinkenberg effect should be taken into account, and to estimate the correction. It also helps us to understand the signiﬁcant variation in this parameter with different materials and experimental conditions (the values reported by Chastanet [CHA 04a] vary from 10−2 to 102 M P a). For natural materials such as compacted rock whose porosity (and hence β) varies across a relatively small range, we in fact observe that when the permeability is weaker b is larger. However the power law bp ∼ Kp−0.5 is only valid at constant porosity, and cannot be applied to general cases. In conclusion, in both steady state and dynamic regimes it is possible to establish bounds on the Klinkenberg permeability based on self-consistent P and V solutions

322

Homogenization of Coupled Phenomena

using the same energetic technique, with partition of the ﬁelds into the two selfconsistent ﬁelds, and assuming, based on the separation of scales hypothesis, that each inclusion in the representative volume is subjected to the same mean level of pressure. Because of the pressure dependence, it is only possible to construct “optimized” media which offer “optimally” improved bounds for a ﬁxed value of pressure, with optimization being lost if the pressure is changed. 10.5. Thermal permeability – compressibility of a gas in a porous medium In their study of acoustically absorbing materials, Zwikker and Kosten [ZWI 49] showed that in the harmonic regime the apparent compressibility of a gas in a porous medium involves local heat transfer effects between the gas and matrix. This local effect is described macroscopically by the thermal permeability, as discussed in Chapter 8. Here we will examine this coefﬁcient from the self-consistent angle for granular media; ﬁbrous media are discussed in Tarnow [TAR 96]. 10.5.1. Dynamic compressibility obtained by homogenization Homogenization (among various approaches) leads to a macroscopic pressure law for a gas subjected to a harmonic acoustic pressure P which has the following form (equilibrium values are indicated with the superscript e , λ is the thermal conduction coefﬁcient of the gas, ρe Cp is its heat capacity at constant pressure, and γ is the adiabatic constant): div(V ) + φC(ω)[iωφP ] = 0

with C(ω) =

1 1 iωρe Cp (1 − )Θ(ω)] [1 − e P λ γ

The coefﬁcient Θ(ω) which appears in dynamic compressibility C(ω) is known as the thermal permeability [LAF 97]. It originates from the heat transfer in the gas in contact with the isothermal porous matrix (a reasonable assumption except for materials with a very low solid concentration). Denoting the gas-solid temperature difference as T0 , the local problem can be expressed in the following manner [SAN 80; BOU 98]: iωP − iωρe Cp T(0) + λy (T(0) ) = 0

(10.75)

T(0) Ω − periodic

(10.76)

(0) T/Γ = 0

;

This problem resembles that of the trapping constant in terms of its scalar formulation, and that of the dynamic permeability through the presence of inertial terms (see section 8.4). By analogy, the solution is thus expressed in the form: θ 1 T(0) = iωP and Θ = θdΩ λ Ωf Ω f

Self-consistent Estimates and Bounds for Permeability

323

θ being the temperature distribution obtained when iωP/λ is unitary and which therefore satisﬁes: 1−

1 θ + y (θ) = 0 δt2

where: δt =

λ/(iωρe Cp )

is the (complex) thickness of the thermal boundary layer. Since thermal equilibrium is not achieved in the pores, θ takes complex values and depends on the local variable and on the frequency through the single dimensionless variable y/δt . The local problem can be written in the variational form (with complex conjugate variables being shown by a bar over): [λgrady (T(0) ) · grady (a) ∀a ∈ A Ωf

+iωρe Cp T(0) a]dΩ = iωP

adΩ

(10.77)

Ωf

where A is the vectorial space of scalar Ω-periodic ﬁelds which are zero on the ﬂuid/solid boundary Γ. Taking T(0) as a test ﬁeld: 1 [λgrady (T(0) ) · grady (T(0) ) + iωρe Cp T(0) T(0) ]dΩ = Ωf Ωf Θ 1 iωP (10.78) T(0) dΩ = iωP iωP Ωf Ω f λ This equality establishes the consistency of the micro and macro descriptions: the real and imaginary parts of Θω2 /λ are respectively the effective power dissipated by diffusion and the effective thermal kinetic power produced per cycle under a unit macroscopic pressure. 10.5.2. Self-consistent estimate of the thermal permeability of granular media In order to determine the self-consistent estimate of Θ(ω), we will again work on the same generic spherical inclusion I(β, R), retaining the notations in Figure 10.2. Field θ in the gas shell βR < r < R is governed by: 1−

iωρe Cp θ + (θ) = 0 λ

(10.79)

324

Homogenization of Coupled Phenomena

θ/Γ = 0

(10.80)

Due to the spherical symmetry of the problem, the solution only depends on r. Integration of the Helmholtz equation with a constant forcing term gives the general form of the solution, which is the sum of a speciﬁc constant solution and of thermal diffusion spherical wavefunctions: θ=

δt2

e−r/δt er/δt + c 1+c r/δt −r/δt

The determination of the two constants of integration c and c requires two boundary conditions, which are the isothermal condition on the solid sphere, i.e. θ(βR) = 0, and the micro-macro energetic consistency condition. In order to establish the latter, we will take the scalar produce of the Fourier equation with the θ¯ ﬁeld. Integrating by parts over the pore volume, and taking account of the isothermal wall condition, we obtain: e [λgrad(θ) · grad(θ) + iωρ Cp θθ]dΩ = θdΩ + grad(θ) · er θ dS Ωf

Ωf

∂Ω

In order to respect the micro-macro consistency condition (10.78) (divided by iωP/λ2 ), the surface integral must have a value of zero. As a consequence, the thermal gradient must be zero on the boundary of the inclusion (the alternative to zero temperature would require insulation of the inclusion at its boundary, using an isothermal condition which is not consistent with the local physics). We will therefore write θ (R) = 0. If we substitute the expression for θ into the two boundary conditions θ(βR) = 0 and θ (R) = 0, we obtain a 2x2 system of equations, the solution to which allows us to determine θ. Integration of this ﬁeld over the pore volume gives the self-consistent estimate of the thermal permeability Θsc (ω): 1 + x tanh[x(β − 1)] 3β Θsc (ω) = δt2 1 − β 3 + 2 βx −1 (10.81) x x + tanh[x(β − 1)] where x = R/δt 10.5.3. Properties of thermal permeability The properties of Θsc (ω) are very similar to those of the dynamic permeability which were discussed in section 10.3.1.2. We also note the identity of the analytical functions involved in Θsc and Kc . This result is no coincidence. It stems from the close relationship between the scalar problem of thermal permeability and the

Self-consistent Estimates and Bounds for Permeability

325

vectorial problem of dynamic permeability. This will be examined in general in section 10.6. For the inclusion under study, the condition of zero vorticity has an exact analog in the condition of zero thermal gradient, which allows us to establish an exact analytical connection between the velocity and temperature ﬁelds. If we abstract ourselves from the analytical values of the self-consistent estimates, the results given below for Θsc (ω) also apply to the thermal permeability Θ(ω) of all porous media. At low frequencies – i.e. when l/δt → 0, conduction dominates, and the gas is in the quasi-isothermal regime. The low frequency expansion gives: iωρe Cp Θsc (0)ζ0 Θsc (ω) Θsc (0) 1 − λ where Θsc (0) is the steady state thermal permeability whose estimate is directly linked to that of the intrinsic permeability Kc (given by (10.32)): Θsc (0) =

3Kc 3ψc (β) = R2 2φ 2(1 − β 3 )

;

3ψc (β) 5 + 6β + 3β 2 + β 3 = 2(1 − β 3 ) 15β

and ζ0 is the corrector coefﬁcient for the low-frequency heat capacity. At high frequencies – i.e. when l/δt → ∞, inertia dominates and the gas is in the quasi-adiabatic regime. Conductive effects are conﬁned to the thermal boundary layer, implying that: λ 1+ e iωρ Cp Θsc (ω)

Mt ω t 2 iω

– ωt is the critical frequency separating low- and high-frequency domains. This is obtained by equating conductive effects (i.e. the low-frequency real part) and inertial effects (i.e. the high-frequency imaginary part) in Θ(ω): λ ωt = Θsc (0)ρe Cp – Mt is a form factor whose self-consistent estimate is: 6(1 − β)3 β 3 (5 + 6β + 3β 2 + β 3 ) Mtac = 5(1 − β 3 )3 From Champoux and Allard [CHA 91]:

2δt Mt ωt = 2 iω Λt

326

Homogenization of Coupled Phenomena

where Λt is a characteristic thermal length in the medium which, relative to the size of the solid sphere, has a self-consistent estimate of: Λtac 2φ = Rs 3(1 − φ) Thus we can deduce the isothermal compressibility corrections at low frequencies, and the adiabatic corrections at high frequencies: ω 1, ωt

C(ω)

ω 1, ωt

C(ω)

iω 1 1 − (1 − ) ωt γ 1 Mt ωt 1 1− (1 − ) γP e 2 iω γ 1 Pe

Across the whole frequency range – Θ(ω)/Θ(0) only depend on the dimensionless frequency ω/ωt , and can be approximated by the following causal function, the same as [JOH 87], which links the low- and high-frequency behavior (but without ζ0 ): ωt 1 1+ e iωρ Cp Θsc (ω) iω

1+

Mt iω 2 ωt

(10.82)

Also, following the same line of reasoning as for dynamic permeability, but with the variational formulation (10.77), we can show that: – the real part of Θ−1 – i.e. resistivity due to conduction – grows with frequency, but on the contrary the imaginary part of Θ−1 divided by the pulsation – i.e. additional heat capacity coefﬁcient – decreases with frequency: d(Θ−1 )R 0 dω

;

d((Θ−1 )I /ω) 0 dω

– the thermal permeability of granular media respects the above upper bounds (see equation (10.94) for notations), which means we can give an upper bound to the imaginary part of the compressibility, and a lower bound to its real part: Ωα Ωα ,I | ,R ; |Θ ΘR |ΘIα | Θ α Ω Ω α

α

10.5.4. Signiﬁcance of connectivity of phases From a formal point of view, the local problem (10.75) is identical to that which governs thermal conduction in a high-contrast medium as described in Auriault [AUR 83] (see also [BOU 93] for the effective compressibility of gas in a ﬂuid with

Self-consistent Estimates and Bounds for Permeability

327

bubbles and section 4.2.3). From a physical point of view these two problems differ from that of the thermal permeability for the following reasons: – In thermal permeability the source term comes from pressure in the pores which requires their connectivity: the phase which is out of local equilibrium must be therefore be connected. – In high-contrast conduction it is the least conducting phase which is out of equilibrium, with the source being the temperature imposed at the walls by the more conductive phase. In order for this mechanism to develop, we therefore require that the phase in local equilibrium must be connected (if not, the overall conductivity is reduced to that of the least conductive phase, and both phases are in local equilibrium). This is also the case in ﬂuids with bubbles where the gas, which is out of equilibrium, is surrounded by ﬂuid. In order to express these morphological requirements in a model that uses composite spherical inclusions I(β, R), the positioning of the phases must be reversed: for thermal permeability the region which is out of equilibrium must be the external part of the inclusion; for high-contrast conduction, the region out of equilibrium must be the internal part of the inclusion. The models which follow from this cannot therefore give the same answers, as demonstrated by the mean temperature in an isolated sphere of radius Ri [AUR 83; ATT 83]:

Θ (ω) =

δt2

3ω 1 + t (1 − iω

iω coth ωt

iω ) ωt

with

ωt =

λ Ri2 ρe Cp

which is independent of the porosity (Θ (0) = Ri2 /15), and only equals Θ(ω) at high frequencies. 10.5.5. Critical thermal and viscous frequencies Since the dynamic compressibility always has a real part, transition from the isothermal to adiabatic regime has much less signiﬁcant consequences than transition from the viscous to inertial regimes in dynamic permeability. The transition frequencies of these two phenomena are connected by: ωc φΘ(0) ηCp = ωt Kτ∞ λ According to estimates obtained using the spherical solid-ﬂuid inclusion, these two frequencies are close since, for air ηCp /λ 0.77, and 1 Θsc (0)/Kp 1.5, and 1 τ∞ 1.5. We know that in reality they can be fairly different, and that normally ωc > ωt because permeability is mostly determined by the throat zones whereas thermal transfers take effect across the whole pore. These aspects, which

328

Homogenization of Coupled Phenomena

depend on the morphological complexity of the medium, are not described by the ﬂow in the inclusion.

10.6. Analogy between the trapping constant and permeability In this section we intend to compare different macroscopic transfer coefﬁcients. The aim is to make use of the analogies that exist between local problems, all governed by diffusion mechanisms, whether they involve viscous, thermal or solute transport effects. We will examine in turn: – the links between the description of steady-state diffusion-trapping regime, steady state thermal permeability and intrinsic permeability; – the transient description of the diffusion-trapping regime and dynamic thermal permeability; – the steady-state diffusion-trapping regime with a ﬁnite absorptivity and the Klinkenberg permeability.

10.6.1. Trapping constant Torquato [TOR 90] drew attention to the analogy between the following local problems: – steady-state ﬂow of a viscous ﬂuid under an imposed pressure gradient; – steady-state diffusive transport of a solute generated by a homogenous source in the pores and which is absorbed instantaneously when it comes into contact with the solid matrix. The steady-state regime is attained when the level of production of the reactant exactly matches the rate of capture at the walls. This last problem, which leads to the deﬁnition of trapping constant Υ, is also a direct transposition of the problem which deﬁnes the steady state thermal permeability. By comparison of these two problems, Torquato [TOR 90] established that: K

φ Υ

The question is whether this bound given by the trapping constant provides an acceptable estimate of intrinsic permeability. This question is all the more reasonable since their equality is observed for uni-directional geometries. We will examine this point in an indirect manner by calculating the self-consistent estimate of the trapping constant and by comparing it to estimates of intrinsic permeability. If we assume that the self-consistent values correctly describe periodic lattices of spheres, we have a way of comparing these two parameters for these media.

Self-consistent Estimates and Bounds for Permeability

329

Through periodic homogenization, Rubinstein and Torquato [RUB 88] gave the following macroscopic description of the trapping, for a volume source of solute whose homogenous production rate in the pores is S, by a perfectly absorbing medium (D is the diffusion coefﬁcient for the solute): 1 1 1 C= S ; = gdΩ (10.83) ΥD Υ Ωf Ω f – C = Ω1f Ωf c(0) dΩ is the mean concentration in the volume of the pores, with local solute concentration c(0) being governed by i) the conservation of solute, taking into account the source and diffusive transfer following Fick’s law, ii) the condition of total absorption at the walls, and iii) the periodicity conditions. Expressed in dimensional variables, the local problem is therefore as follows: S + divy [Dgrady (c(0) )] = 0

c(0)

(10.84)

c/Γ = 0

(0)

(10.85)

Ω − periodic

(10.86)

– Υ is the trapping constant, i.e. the mean of concentration ﬁeld g/D that is the solution for a unit source S = 1, which therefore satisﬁes over Ωf : 1 + y (g) = 0 In this form, it is clear that g depends only on the pore geometry. The local problem can be written in the following variational form: (0) grady (c ) · grady (a)dΩ = S adΩ ∀a ∈ A ; D Ωf

(10.87)

Ωf

where A is the vectorial space of Ω-periodic scalar ﬁelds which are zero on the ﬂuidsolid boundary Γ. Taking c(0) as a test ﬁeld: 1 1 D grady (c(0) ) · grady (c(0) )dΩ = S c(0) dΩ = S S (10.88) Ωf Ωf Ωf Ωf ΥD This equality establishes the micro-macro consistency, i.e. the identity between the diffusive dissipation which occurs in the representative pore volume and in the same volume of the equivalent medium. It is clear that this problem is identical to that of the thermal permeability in the steady-state regime ((10.79) with ω = 0), and that as a result g = θ(ω=0) , so that: 1/Υ = Θsc (0)

330

Homogenization of Coupled Phenomena

How does this relate to the intrinsic permeability? 10.6.1.1. Comparison between the trapping constant and intrinsic permeability Here we will reproduce the reasoning in Torquato [TOR 90], for an isotropic medium for simplicity. Starting from the solution in terms of concentration g for a unit source, i.e. S = 1, we construct the vectorial ﬁeld g such that g = ge1 . Because of this, it satisﬁes: e1 + y (g) = 0

(10.89)

g · n/Γ = 0

;

g

Ω − periodic

Also recall that (−k1 /η; −p1 ) – the solution to the local Darcy problem under a unit pressure gradient e1 – satisﬁes (see 10.2.1.1): grady p1 + e1 + y (k1 ) = 0

(10.90)

1

divy (k ) = 0 k1/Γ = 0

;

k1

& p1

Ω − periodic

The ﬁeld w = k1 − g is thus the solution to: grady p1 + y (w) = 0

(10.91)

divy (w) = −divy (g) w · n/Γ = 0

;

w

Ω − periodic

Looking at the scalar product of (10.91) with w and integrating parts, we ﬁnd, bearing in mind periodicity and the zero value at the ﬂuid-solid interface: 0< 2Dy (w) : Dy (w)dΩ = grady p1 · wdΩ Ωf

Ωf

The integral on the right-hand side can be integrated by parts to give: p1 divy (w)dΩ = p1 divy (g)dΩ = − grady p1 · gdΩ − Ωf

Ωf

Ωf

But, calculating the scalar product of (10.90) by g: − grady p1 · gdΩ = e1 · gdΩ + y (k1 ) · gdΩ Ωf

Ωf

Ωf

Self-consistent Estimates and Bounds for Permeability

=

gdΩ +

Ωf

331

Ωf

y (k11 )gdΩ

and with the periodicity, the Dirichlet condition over Γ, and the conservation equation over g, the last integral can be transformed into: − grady (k11 ) · grady gdΩ = y (g)k11 dΩ = − k11 dΩ Ωf

Ωf

Ωf

In summary, we obtain the result mentioned previously in the form: 1 1 φ 1 −K 2Dy (w) : Dy (w)dΩ = gdΩ − k 1 dΩ = 0 Ω Ωf Ω Ωf Ω Ωf 1 Υ 10.6.1.2. Self-consistent estimate of the trapping constant for granular media In order to estimate the trapping constant for granular media we will work with the same generic spherical inclusion I(β, R), retaining the same notations (see Figure 10.2). The solute concentration c in the ﬂuid shell βR < r < R is governed by the system of equations (10.6.1; 10.85) without any periodicity condition. Due to the spherical symmetry of the problem, the solution only depends on r. This result is already known, since by analogy we have already shown that 1/Υ = Θsc (0). However, in order to illustrate the simplicity of the method we will show it directly. By integration – recall that in spherical coordinates c(r) = r−2 [r 2 c ] – we can deduce the general form of the solution, which contains two constants of integrations a and b: c=

a S r2 [ + + b] D 6 r

The determination of these two constants requires two boundary conditions, speciﬁcally total absorption condition at the walls, c(βR) = 0, and the micro-macro consistency condition. In order to establish the latter, we will calculate the scalar product of the solute conservation equation (10.6.1) with ﬁeld c. Integrating by parts over the volume of the pores, and taking account of the total absorption, we obtain: D grad(c) · grad(c)dΩ = S cdΩ + grad(c) · er c dS Ωf

Ωf

∂Ω

In order to respect the micro-macro consistency condition (10.88), this last integral must be zero, which means a concentration gradient of zero at the inclusion boundary is needed. There is no alternative to zero since it is equivalent to isolating the inclusion using a condition of total absorption at the boundary, which is not consistent with local physics when the ﬂuid is connected. As a result, we set c (R) = 0.

332

Homogenization of Coupled Phenomena

The solution is clear, and after taking the mean over the volume of the pores, we obtain the self-consistent estimate of trapping constant Υsc : 1 = R2 F(β) Υsc

F(β) =

with

5 − 9β + 5β 3 − β 6 15β(1 − β 3 )

(10.92)

Based on the inequality we established theoretically, we have effectively obtained a value greater than that of the permeability estimates: Kv < Kp <

φ Υsc

;

φ 3 = Kc Υsc 2

Finally we note that the values for a dilute solid concentration (which lead to the same result as Smoluchowski), or a concentrated solution, are similar to the values of the permeability estimates, up to a factor of 3/2 (see (10.2.3.3)): φ→0

φF

1 3β

− 35 ;

φ→1

φF (1 − β)3

(10.93)

Over the entire range of solid concentrations, φ/Υsc only overestimates the intrinsic permeability Kp by a factor of between 1.2 and 1.5. This factor can be attributed to the absence of kinematic limitation of incompressibility in the diffusion problem. For these ideal geometries, the reduction of the vectorial problem to a scalar problem does in fact lead to acceptable estimates – through a much simpler calculation. It appears however that for more complex geometries, although the values are still of the same magnitude, the disagreements can be greater. We ﬁnish by returning to the granular media introduced in section 10.2.4.2, to which we refer for the notations used. Using the same method of partitioning ﬁelds, and making use of the positivity of the function: % & D c) = grad(gsc − c) · grad(gsc − c)dΩ E(gsc , α

Ωα

we can establish an upper bound on the inverse of the trapping constant in these media: Ωα 1 1 < Υsc Υ Ω

(10.94)

α

10.6.2. Diffusion-trapping in the transient regime The diffusion-trapping problem in the transient regime is described by the equation: S(t) −

∂c(t) + D(c(t)) = 0 ∂t

(10.95)

Self-consistent Estimates and Bounds for Permeability

333

In order to examine this we will consider an impulse source S(t) = δ(t) and return to the frequency space by Fourier transform. The equation becomes: 1 − iωc + D(c) = 0

(10.96)

which is the exact analog of the thermal permeability equation (10.79), where D plays the role of λ/(ρe Cp ) and c = θ/D. Using the solution already established in the thermal case, and returning to the initial time-dependent problem by the inverse Fourier transformation, we obtain a self-consistent estimate for the impulse response of the generic spherical inclusion I(β, R) which is: 1 ∞ −iωt e Ξsc (ω)dΩ Cimp (t) = D −∞ where: 1 + x tanh[x(β − 1)] 3β −1 Ξsc (ω) = δd2 1 − β 3 + 2 βx x x + tanh[x(β − 1)] with: ;

x = R/δd

δd =

√ iωD

The function Ξsc (ω) has the characteristic frequency: ωd = D/Θsc (0). When the rate of solute generation S is not constant, the mean concentration response becomes: C(t) =

t

Cimp (t)S(t − τ )dτ

0

which shows a memory effect of order O(2πωd ) which has the same order of magnitude as the trapping constant (or inverse of the intrinsic permeability) divided by the molecular diffusion coefﬁcient. Also, it can easily be shown that Cimp (t) exp(−at) is the impulse response of: δ(t) − ac(t) −

∂c(t) + D(c(t)) = 0 ∂t

so that the diffusion-trapping problem in the presence of an additional absorption in the pores (characterized by a) is also solved. 10.6.3. Steady-state diffusion-trapping regime in media with a ﬁnite absorptivity We will brieﬂy consider the steady-state diffusion-trapping regime for media with a ﬁnite absorptivity. This problem is the scalar equivalent to that of the Klinkenberg

334

Homogenization of Coupled Phenomena

permeability, because the Dirichlet condition representing total absorption is then replaced by a mixed condition which relates ﬂux to the local concentration. If we designate the rate of absorption per unit of surface area as α, the condition at the interface becomes: Dgrady (c(0) ) · n = αc(0)

over

Γ

and introduces a new characteristic length which, when compared to the solid sphere radius, leads to the dimensionless parameter D/(αβR). The determination of the macroscopic model and its self-consistent estimate using the inclusion I(β, R) closely follows that for the trapping constant and Klinkenberg permeability. We obtain: C=

1 S Υα D

;

1 = R2 Fα (β)dΩ Υαsc

(10.97)

With the following estimate (F being given by the trapping constant for inﬁnite absorption – see (10.92)): Fα (β) = F(β) +

D 1 1 − β3 α βR 3β

(10.98)

At high rates of absorption, it is diffusion which regulates the steady-state regime equilibrium (with a corrector effect due to absorption). On the other hand, at weak absorption rates it is the reaction mechanism at the wall which controls the equilibrium. In this case it is the speciﬁc surface of the wall which becomes the relevant geometric parameter since then: C

α S Υ0

;

1 1 − β3 =R Υ0ac 3β 2

(10.99)

10.7. Conclusion We have seen that self-consistent estimates give a fairly good description of periodic lattices. They can also be applied to ordered and disordered media formed from elements of variable size with speciﬁc morphologies, where the solid concentration varies as a function of the size of the element. On the other hand, their use for poly-disperse morphologies with homogenous solid concentration can lead to erroneous values. For ﬂow problems, two self-consistent estimates are possible, but the pressure approach is more reliable. The interest in the velocity approach lies mostly in the framework which it offers for the permeability of optimized media. In addition, the method put forward for partition of the ﬁelds also allows the determination of bounds for mixed media consisting of spheres and of cylinders, or of non-coaxial cylinders. We also note that saturating linear viscoelastic ﬂuids can be treated in the same way by replacing the viscosity by a complex modulus. In the

Self-consistent Estimates and Bounds for Permeability

335

steady-state regime, a rheological memory effect then appears in Darcy’s law, which in the dynamic case is combined with memory effects of inertial origin; the reader is referred to Boutin [BOU 89a] on this subject. When written in a general (and approximate) form, all the estimates obtained using the spherical inclusion I(β, R) follow the same trends: Steady state permeability

(1 − β 3 ) [a0 + (a1 − a0 )β]R2 β

where the a0 and a1 coefﬁcients arising from the behavior in the limit of weak and strong solid concentrations depend on the scalar or vectorial problem being treated, and the P- or V- approach which is used. Ratio βc = (a1 − a0 )/a0 is an indication of the solid concentration cs = βc3 at which a transition occurs between the behavior in the dilute and concentrated regimes. Dynamic effects can be expressed through a Johnson-type multiplicative function (see section 10.3.1.2). The effects of mixed boundary conditions at the ﬂuid-solid interface introduce a dimensionless parameter χ ∼ 1/βR, and a correction which is proportional to this value as long as it remains small. In order to express the connectivity of the pores, we have used a composite sphere. This approach is forced upon us because the use of a simple self-consistent approach which uses pure ﬂuid and pure solid spheres fails due to two problems: – the no-slip boundary condition, crucial to the physics of the ﬂow, can only be implicitly expressed on average at the inclusions’ interface, but how accurate is that? – it is difﬁcult to imagine the morphology of a medium (even an ideal one) which could correspond to this type of model These difﬁculties are inherent to the simple self-consistent modeling – whatever the local physics being examined – but their repercussions on the model become particularly apparent for permeability problems. Although the geometry used for the structure is an extreme simpliﬁcation of the pore geometry, these results underline the high sensitivity of the macroscopic parameters to the material morphology. Because of this, the estimates given here can be applied to materials whose pores are fairly regular, but no claim is made that they are applicable in general. More complex morphologies which introduce dead regions in the ﬂow – for example cellular materials where the cells are only connected by small oriﬁces in the walls – lead to high tortuosities even at very low solid concentrations [BOR 83], and are not correctly described by a simple spherical bi-composite inclusion.

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PART FOUR

Saturated Deformable Porous Media

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Chapter 11

Quasi-statics of Saturated Deformable Porous Media

The interesting behavior of saturated porous media is fundamentally due to the presence, on ﬁne scale of the pores, of two materials which are fundamentally different in nature: a solid which forms the porous matrix and a liquid which occupies the pores. The sharp local discontinuities lead to macroscopic descriptions which are very unfamiliar compared to the behavior of other materials. The distinction is generally made between two main classes of macroscopic descriptions: – traditional monophasic descriptions, with a constitutive law which involves a particular velocity associated with the presence of interstitial ﬂuid (see for example [TSY 69]); – diphasic descriptions with two displacement ﬁelds, one for the porous matrix and the other for the ﬂuid, such as those introduced by Terzaghi [TER23; 43], and by Biot [BIO41; 55; 62].

The studies listed above were carried out directly on the macroscopic scale using phenomenological and experimental approaches. Here we will use the method of asymptotic expansions in order to specify the domains of validity of these descriptions and the properties of effective coefﬁcients. For the reasons invoked in the preceding chapters, the porous medium will be assumed to be periodic. And for further simplicity we will assume that the material forming the porous matrix is linear elastic and that the deformations are small. The interstitial ﬂuid will be treated as an incompressible viscous Newtonian ﬂuid.

340

Homogenization of Coupled Phenomena

Despite this we will focus on linear materials. Many practical applications have required the non-linearity of the skeleton to be taken into account, mainly through phenomenological or thermodynamic approaches [COU 95; ELH 02] or micromechanical approaches [DOR 06]. In the ﬁrst part we will investigate the behavior of an empty porous matrix. This study, along with that already carried out on ﬂows in rigid porous media, will act as the basis for our diphasic macroscopic behavior, where it exists. The quasi-static behavior of the saturated porous medium is then treated in the second part. Three types of behavior are found, and their domains of validity are speciﬁed: diphasic behavior, elastic monophasic behavior and viscoelastic monophasic behavior. 11.1. Empty porous matrix Here we will investigate the behavior of a deformable porous matrix when the pores are empty [AUR 90b]. The local structure is periodic with period Ω, and the solid part is Ωs (Figure 11.1).

Lc

lc

f

s n

(a)

(b)

Figure 11.1. (a) Porous medium; (b) period

11.1.1. Local description On the microscopic scale, the linear elastic solid satisﬁes the Navier equations and standard condition of zero normal stress at the interface Γ: divX (σ s ) = ρs σs n = 0

∂ 2 us ∂t

over Γ

with

σ s = a : eX (us )

within Ωs

Quasi-statics of Saturated Deformable Porous Media

341

where σ s , us and ρs are respectively the microscopic stress, displacement and density of the solid. The eX tensor is the strain tensor, with components: eXij

1 = 2

∂usi ∂usj + ∂Xj ∂Xi

and a is the elasticity tensor, periodic with period O(lc ), with possible discontinuities at which the standard equations of continuity of displacement and of normal stress are satisﬁed. The tensor a satisﬁed the symmetries: aijkh = akhij = ajikh = aijhk and the ellipticity condition: ∀eij

aijkh eij ekh γ eij eij

γ>0

Taking the macroscopic viewpoint, and introducing the following quantities into the above description: X = Lc x∗ ,

a = a c a∗ ,

us = usc u∗s ,

ρs = ρsc ρ∗s

the dimensionless microscopic description becomes: divx∗ (σ ∗s ) = PL ρ∗s

∂ 2 u∗s ∂t∗2

with

σ ∗s = a∗ : ex∗ (u∗s )

σ ∗s n = 0 over Γ∗

within Ω∗s

(11.1) (11.2)

where the dimensionless number PL is deﬁned by: 2

PL =

|ρs ∂∂tu2s | =O |divX (σ s )|

ρsc L2c ac t2c

In this chapter the motion is assumed to be slow. The inertial terms are assumed to be small, and will not appear in the macroscopic model. Because of this, we are able to assume here that (see Chapter 3, section 3.8): PL = O(ε)

342

Homogenization of Coupled Phenomena

11.1.2. Equivalent macroscopic behavior Now we will homogenize the system of equations (11.1-11.2). For that we will look for u∗s in the form: u∗s = u∗(0) (x∗ , y∗ ) + εu∗(1) (x∗ , y∗ ) + ε2 u∗(2) (x∗ , y∗ ) + · · · with y∗ = x∗ /ε. We recall that, because of the two spatial variables and the choice of macroscopic viewpoint, the spatial derivative takes the following form: gradx∗ −→ gradx∗ + ε−1 grady∗ The expansion of σ ∗s can then be written in the following form: + σ ∗(0) + εσ ∗(1) + ... σ ∗s = ε−1 σ ∗(−1) s s s with: σ ∗(−1) = a∗ : ey∗ (u∗(0) ) s σ ∗(0) = a∗ : (ey∗ (u∗(1) ) + ey∗ (u∗(0) )) s σ ∗(1) = a∗ : (ey∗ (u∗(2) ) + ey∗ (u∗(1) )) s After substituting the asymptotic expansion into the dimensionless system (11.1-11.2), we obtain the various successive boundary value problems as follows. 11.1.2.1. Boundary-value problem for u∗(0) The lowest order gives a boundary-value problem for u∗(0) : =0 divy∗ (a∗ : ey∗ (u∗(0) )) = divy∗ σ ∗(−1) s a∗ : ey∗ (u∗(0) ) n = σ ∗(−1) n=0 s

over Γ∗

within Ω∗s

(11.3) (11.4)

In order to investigate this boundary-value problem, as well as subsequent ones, we will introduce the space E of vectors u∗ , deﬁned over Ω∗s , periodic, with zero mean over Ω∗s : u∗ dΩ∗ = 0 (11.5) Ω∗ s

and satisfying the following scalar product: ∗ ∗ (u , α )E = a∗ijkl ey∗ ij (u∗ ) ey∗ kl (α∗ ) dΩ∗ Ω∗ s

(11.6)

Quasi-statics of Saturated Deformable Porous Media

343

In particular, it can be shown without difﬁculty (i) that with conditions of zero mean and periodicity, the only ﬁeld in space E with zero deformation is a ﬁeld which is zero over all space, (ii) the positivity of (11.6) due to the ellipticity of a∗ . If we multiply both sides of (11.3) by α∗ ∈ E and integrate over Ω∗s by parts, we can show with the help of the divergence theorem that: 0=

∗(−1)

∂σsij

Ω∗ s

=

∂Ω∗ s

(u∗(0) ) ∗ αi dΩ∗ ∂yj∗

∗(−1)

σsij

(u∗(0) )αi∗ nj dS ∗ −

∗(−1)

Ω∗ s

σsij

(u∗(0) )

∂αi∗ dΩ∗ ∂yj∗

The surface integral is zero over Γ∗ due to (11.4), and also over the rest of ∂Ω∗s by periodicity. Finally the symmetry of σ∗s means that we can write: ∗ ∗(−1) ∂αi σsij ∗ ∂yj

=

∗ ∗(−1) ∂αj σsji ∂yi∗

=

∗ ∗(−1) ∂αj σsij ∂yi∗

1 ∗(−1) = σsij 2

∂αj∗ ∂αi∗ + ∂yj∗ ∂yi∗

Thus we obtain the following variational form: (u∗(0) , α∗ )E = a∗ijkh ey∗ kh (u∗(0) ) ey∗ ij (α∗ ) dΩ∗ = 0 ∀α∗ ∈ E,

(11.7)

Ω∗ s

This formulation is equivalent to the system (11.3-11.4) with u∗(0) having a mean of zero (see [AUR 77]). In fact, equation (11.5) is separate to the problem represented by equations (11.3-11.4). It is introduced here so that (11.6) represents a scalar product. Using Lax-Milgram theorem, we can show that equation (11.7) ensures the existence and uniqueness of the solution u∗(0) which belongs to E, i.e. satisfying the condition of zero mean. It is clear that this solution is also zero. The general solution follows by the addition of a constant ﬁeld independent of y∗ . So in summary, u∗(0) takes the form: u∗(0) = u∗(0) (x∗ ) 11.1.2.2. Boundary-value problem for u∗(1) At the next order, i.e. taking (11.1) to order ε−1 and (11.2) to order ε0 , we obtain the following cellular problem which deﬁnes u∗(1) : = 0 within Ω∗s divy∗ (a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))) = divy∗ σ ∗(0) s

(11.8)

n = 0 over Γ∗ a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n = σ ∗(0) s

(11.9)

344

Homogenization of Coupled Phenomena

where u∗(1) is Ω-periodic in y∗ . The equivalent weak formulation (up to an arbitrary vector u∗(1) (x∗ )) can be obtained as for u∗(0) by multiplying both sides of (11.8) by α∗ ∈ E and integrating over Ω∗s . The calculations are identical to those in -a- but with ey∗ (u∗(0) ) replaced by ey∗ (u∗(1) ) + ex∗ (u∗(0) ), so that the variational formulation becomes: a∗ijkh (ey∗ kh (u∗(1) ) + ex∗ kh (u∗(0) )) ey∗ ij (α∗ ) dΩ∗ = 0 (11.10) ∀α∗ ∈ E, Ω∗ s

Here again the properties of elastic tensor a∗ assure the existence and uniqueness of the solution to (11.10) which appears as a linear vectorial function of ex∗ (u∗(0) ). If then the vector ξ ∗lm is the speciﬁc solution corresponding to ex∗ ij (u∗(0) ) = (δil δjm + δjl δim )/2, with l and m ﬁxed, the solution to (11.8-11.9) can be written: ∗(1)

ui

∗(1)

= ξi∗lm ex∗ lm (u∗(0) ) + ui

(x∗ )

where u∗(1) (x∗ ) is arbitrary and ξ ∗ (y∗ ) belongs to E, so that: ξ ∗ dΩ∗ = 0 Ω∗ s

11.1.2.3. Boundary-value problem for u∗(2) The cellular problem deﬁning u∗(2) can be obtained from (11.1) at order ε0 and (11.2) at order ε: + divx∗ σ ∗(0) =0 divy∗ σ ∗(1) s s σ∗(1) n=0 s ∗(0)

within Ω∗s

(11.11)

over Γ∗

(11.12)

∗(1)

where σ s and σ s are the zero- and ﬁrst-order terms in the expansion of σ ∗s in powers of ε. Equation (11.11) is a conservation equation of the periodic quantity ∗(1) ∗(0) σ s , satisfying condition (11.12) over Γ∗ , with the source term −divx∗ σ s . There therefore exists a compatibility condition (the existence of u∗(2) ), which can be obtained by integrating (11.11) over Ω∗s . Let us deﬁne the total stress σ T∗ by: ∗ within Ω∗s σs σ T∗ = 0 within Ω∗f ∗(1)

The compatibility condition can be written, with (11.12) and the periodicity of σ s divx∗ σ ∗(0) = divx∗ σ ∗(0) = divx∗ σ T∗(0) s s 1 =− ∗ |Ω |

Ω∗ s

divy∗ σ ∗(1) s

1 dΩ = − ∗ |Ω | ∗

∂Ω∗ s

σ ∗(1) n dS ∗ = 0 s

:

Quasi-statics of Saturated Deformable Porous Media ∗(0)

And using the deﬁnition of σ s T∗(0)

σij

345

it follows that:

= a∗ijkh (ey∗ kh (u∗(1) ) + ex∗ kh (u∗(0) )) = a∗ijkh (ey∗ kh (ξ∗lm ) ex∗ lm (u∗(0) ) + ex∗ kh (u∗(0) )) = c∗ijkh ex∗ kh (u∗(0) )

with: c∗ijkh = a∗ijkh + a∗ijlm ey∗ lm (ξ ∗kh )

(11.13)

In conclusion, the macroscopic dimensional description can be written to ﬁrst ¯ order of approximation, with a relative error O(ε), as: ¯ divX σ T = O(ε)

(11.14)

¯ σ T = c : eX (us ) + O(ε)

(11.15)

11.1.3. Investigation of the equivalent macroscopic behavior The structure of the system of equations (11.14-11.15) which describes the equivalent macroscopic behavior is identical to that of system (11.1-11.2) for local behavior. Macroscopic distribution does however introduce quantities whose physical meaning needs to be interpreted, and coefﬁcients whose properties need to be speciﬁed. 11.1.3.1. Physical meaning of quantities involved in macroscopic description The quantity σ T∗(0) is the volume mean of a stress. We will show that it can also be interpreted as a surface mean, and consequently as a stress. We will begin with the following identity, which holds for any second-order tensor: ∗(0)

∗(0)

∂σsij ∗ ∂(σsij yk∗ ) ∗(0) = y + σsik ∗ ∂yj ∂yj∗ k ∗(0)

We integrate over Ω∗s . Using (11.8), the divergence of σ s becomes zero and it follows, with the divergence theorem and condition over Γ∗ (see Figure 11.1) that: ∗(0) σsik

1 = ∗ |Ω |

∗(0)

∗ ∂Ω∗ s ∩∂Ω

σsij yk∗ nj ds

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Homogenization of Coupled Phenomena

The surface integral can be interpreted as a stress. We will assume the parallelepipedal period to be described by axes y1∗ , y2∗ , y3∗ and to occupy volume yi∗ li∗ , i = 1, 2, 3, where li∗ represents the dimensionless length of the period along yi∗ axis. Then the integrals over yj∗ = 0 and yj∗ = li∗ , j = k faces are zero through the ∗(0)

periodicity, as are the integrals over yk∗ = 0. Taking σ s left with: ∗(0) ∗ ∗(0) ∗ ∗ σsij yk nj dS = lk σsij nj dS ∗

= 0 within Ω∗f , we are then

∗ =l∗ yk k

∗ ∂Ω∗ s ∩∂Ω

But for yk∗ = lk∗ , nj = δjk and: ∗(0) σsik

=

lk∗

1 |Ω∗ |

∗ =l∗ yk k

∗(0) σsij

1 dS = ∗ |Σk | ∗

∗(0)

∗ =l∗ yk k

σsij dS ∗

where Σ∗k is the surface of Ω∗ perpendicular to yk∗ . The ﬁnal term on the right-hand side represents a surface mean, and hence a stress. We note that the result follows ∗(0) from the solenoidal nature of σ s : ∗(0)

∂σsij =0 ∂yj∗ ∗(1)

Conversely, the mean volume of the second term σ s of σ ∗s does not in general represent a stress since: ∗(1)

in the asymptotic expansion

∗(0)

∂σsij ∂σsij

= 0 ∗ =− ∂yj ∂x∗j This does not raise any issues in terms of the signiﬁcance of u∗(0) though, because it involves the same physical quantity on both scales, since u∗(0) is a function of x∗ alone. 11.1.3.2. Properties of the effective elastic tensor Firstly, c∗ possesses the symmetries of an elastic tensor. by construction:

On one hand,

c∗ijkh = c∗jikh = c∗ijhk On the other hand, c∗ijkh = c∗khij . Consider the formulation in (11.10) where we set ﬁrstly: ex∗ ij (u∗(0) ) = (δip δjq + δjp δiq )/2

and

α∗ = ξ ∗rs

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347

and then: ex∗ ij (u∗(0) ) = (δir δjs + δjr δis )/2

α∗ = ξ ∗pq

and

The symmetry of the scalar product means that we can write: a∗ijkh ey ∗ ij (ξ ∗pq ) ey∗ kh (ξ∗rs ) dΩ∗ = − a∗pqkh ey∗ kh (ξ ∗rs ) dΩ∗ Ω∗ s

Ω∗ s

=−

Ω∗ s

a∗rskh ey∗ kh (ξ ∗pq ) dΩ∗

The last equality shows the symmetry of c∗ (cf. deﬁnition of c∗ (11.13)). Then c∗ satisﬁes an ellipticity property similar to that satisﬁed by a∗ : ∀ex∗ ij ,

c∗ijkh ex∗ ij ex∗ kh β ∗ ex∗ ij ex∗ ij ,

β∗ > 0

The left-hand side of this inequality can be written: ∗(0)

c∗ijkh ex∗ ij (u∗(0) ) ex∗ kh (u∗(0) ) = σsij ex∗ ij (u∗(0) ) =

1 ex∗ ij (u∗(0) ) |Ω∗ |

Ω∗ s

∗

a∗ijkh (ey∗ kh (u

(1)

) + ex∗ kh (u∗(0) )) dΩ∗

Also the formulation in (11.10), with α∗ = u∗(1) can be written: a∗ijkh (ey∗ kh (u∗(1) ) + ex∗ kh (u∗(0) )) ey∗ ij (u∗(1) ) dΩ∗ = 0 Ω∗ s

Combining these two last inequalities term by term we obtain: 1 a∗ (ey∗ ij (u∗(1) ) c∗ijkh ex∗ ij (u∗(0) ) ex∗ kh (u∗(0) ) = ∗ |Ω | Ω∗s ijkh +ex∗ ij (u∗(0) )) (ey∗ kh (u∗(1) ) + ex∗ kh (u∗(0) )) dΩ∗

(11.16)

and recalling the ellipticity properties of a∗ , it follows that: 1 (ey∗ ij (u∗(1) )) c∗ijkh ex∗ ij (u∗(0) ) ex∗ kh (u∗(0) ) γ ∗ ∗ |Ω | Ω∗s +ex∗ ij (u∗(0) )) (ey∗ ij (u∗(1) )) + ex∗ ij (u∗(0) )) dΩ∗ > 0 Indeed the right-hand side cannot be zero because then the periodicity of u∗(1) would lead to ex∗ ij (u∗(0) ) = 0, which is incompatible with our assumption that it is of O(1).

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Homogenization of Coupled Phenomena

11.1.3.3. Energetic consistency We need to show that the mean of the local elastic energy density equals the macroscopic elastic energy density. That follows directly from (11.16) when we observe that this equation can be written: 1 ∗(0) ∗(0) ∗(0) ∗(0) ∗ σ e dΩ∗ σsij ex ij (u ) = ∗ |Ω | Ω∗s sij ij ∗(0)

where eij

= ey∗ ij (u∗(1) ) + ex∗ ij (u∗(0) ) is the zero-order strain.

11.1.4. Calculation of the effective coefﬁcients If we accept the existence of the equivalent macroscopic description and its structure, calculation of the equivalent effective coefﬁcients can be abstracted from the formalism of the two-scale expansions. We will consider the problem of the empty porous matrix discussed previously. We need to determine the effective elastic tensor over a given period Ω. As we have seen, this tensor is deﬁned starting from this single period. The method involves considering the medium to be inﬁnitely periodic, with a constant period given by the speciﬁc period Ω we have chosen. Such a medium is macroscopically homogenous. In addition, since the effective coefﬁcients are independent of excitation we can only consider a state of the material where the macroscopic stress is also homogenous. This will lead to a macroscopic strain which is also homogenous. Thus the local strain is Ω-periodic. Of course, the local displacement itself is not strictly periodic, but the gradient Sanchez-Palencia [SAN 74] showed that it can be written in the general form: ˜i + Aij Xj ui = u ˜ is an Ω-periodic vector and it can be shown [AUR 77], after continuation where u ˜ into the pores, that A represents the macroscopic strain. If we substitute this of u expression for displacement into the local equations (recall that in this treatment there is only a single spatial variable): ∂σsij = 0, ∂Xj σsij nj = 0

σsij = aijkh eXkh (us )

within Ωs

over Γ

˜: we obtain the following boundary-value problem for u ∂ (aijkh (eXkh (˜ u) + Akh )) = 0 within Ωs ∂Xj (aijkh (eXkh (˜ u) + Akh )nj = 0

over Γ

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349

˜ Ω-periodic. This problem is simply the boundary-value problem (11.8-11.9) with u which deﬁnes u∗(1) when the asymptotic expansion method is used, with: ˜ ∗ = u∗(1) u A∗ij = ex∗ ij (u∗(0) ) Thus: u ˜∗i = ξi∗jk A∗jk + u∗i (x∗ ) The effective elastic tensor ceﬀ is deﬁned here by: σij = ceﬀ ijkl Akl It follows that: ∗kl ∗ ∗ ) ceﬀ∗ ijkl = aijkl + aijpq ey ∗ pq (ξ

and: ceﬀ∗ = c∗ This method is more accessible than the asymptotic expansion method. For this reason it is often used for the calculation of effective coefﬁcients in simple macroscopic models. It should again be pointed out that this method starts from the initial assumption that the structure of the macroscopic description is correct. 11.2. Deformable saturated porous medium In this section we will investigate the macroscopic description of a deformable porous matrix ﬁlled with a liquid, when the movements are slow: the velocities are small and accelerations are negligible [AUR 77]. We will assume that the solid part is connected, as is the ﬂuid part. However, the assumption of connectivity of the liquid is only required in the case of diphasic macroscopic behavior discussed in section 11.2.2. With the aim of simpliﬁcation, we will again adopt the assumptions of previous chapters regarding rheology of the constituents: – the material forming the porous matrix is linear elastic and the deformations are small. The matrix has a periodic structure (a material with a random structure will lead to the same macroscopic description, as we saw earlier, if we assume local stationarity);

350

Homogenization of Coupled Phenomena

– the interstitial liquid is viscous Newtonian and incompressible; – certain alterations to these assumptions are possible without signiﬁcantly changing the structure of the results. These will be pointed out where appropriate. The development that follows is based on the work of Auriault and SanchezPalencia [AUR 77] for the diphasic behavior, and the monophasic macroscopic behavior is also presented in Sanchez-Palencia [SAN 80] [Chapter 8] for dynamics and in Auriault [AUR87b; 90a] for the quasi-static case. 11.2.1. Local description and estimates On the microscopic scale, the medium satisﬁes the Navier equation in the solid part Ωs , and the Navier-Stokes equation in the ﬂuid part Ωf , with standard conditions on the interface Γ (with normal n), i.e. continuity of normal stress and of displacement: divX (σ s ) = ρs

∂ 2 us ∂t2

divX (σ f ) = ρf (

with

σ s = a : eX (us )

(11.17)

∂v + (vgrad).v) ∂t with

divX v = 0

within Ωs

σ f = 2ηD(v) − pI

within Ωf

within Ωf

(11.18) (11.19)

(σs − σ f )n = 0 over Γ

(11.20)

u˙ s − v = 0 over Γ

(11.21)

In these equations, the subscripts s and f refer to the solid and ﬂuid respectively. We will use the same notations as before for the solid. Note that u˙ s represents the time derivative of solid displacement. In general terms, a dot above a quantity indicates the time derivative. In what follows, we will assume that stresses on the macroscopic scale, densities and displacements of the solid and ﬂuid are all of the same order of magnitude as each other: = O(σ macro ), σ macro s f

ρs = O(ρf ),

us = O(uf )

(11.22)

For this investigation we will carry out analysis from the macroscopic viewpoint. The inertial terms are again assumed to be small, so that they do not appear on the macroscopic scale. This can in particular be expressed as: 2

PL =

|ρs ∂∂tu2s | =O |divX (σ s )|

ρsc L2c ac t2c

= O(ε)

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351

In the Navier-Stokes equation, we assume that the Reynolds number is sufﬁciently small that it does not intervene in the ﬁrst order of the macroscopic description. Because of this we can take: ReL = O(1) Similarly the ratio of the inertial term to pressure term is also assumed to be very small on the macroscopic scale: ∂v | ∂t = O(ε) TL = |gradX p| |ρf

Under these conditions, and so as not to overburden our discussion, we will not include these inertial terms in the Navier-Stokes equations from now on. System (11.17-11.21) only involves a single dimensionless number, then the ratio between pressure and viscosity terms in the Stokes equation. This is number Q introduced in Chapter 7: Q = |gradX p|/|ηΔX v| Choosing Lc for non-dimensionalization, we have: QL = O(

pc L2c ac uc Lc a c tc )= = Lc ηc vc L c η c ω c uc ηc

We saw in Chapter 7 that when the porous matrix is rigid, the only homogenizable situation is: QL = O(ε−2 ) and this leads to a macroscopic ﬂuid ﬂow relative to the solid – Darcy’s law. Due to the fact that in the present case the skeleton is deformable, other situations are homogenizable. It is sufﬁcient to consider the following cases: – QL = O(ε−2 ): this case leads to a diphasic macroscopic description. This is the subject of sections 11.2.2 and 11.2.3; – QL = O(ε−1 ): the corresponding macroscopic description is monophasic elastic. This is studied in section 11.2.4; – QL = O(1): the macroscopic description becomes monophasic viscoelastic. Section 11.2.5 analyzes this third situation.

352

Homogenization of Coupled Phenomena

11.2.2. Diphasic macroscopic behavior: Biot model We have QL = O(ε−2 ). Equations (11.17-11.21) are now considered as being written in dimensionless form, with (11.18) replaced with: ε2 η ∗ Δx∗ v∗ − gradx∗ p∗ = O(ε)

(11.23)

σ ∗f = 2η ∗ ε2 D∗x∗ (v∗ ) − p∗ I

(11.24)

Having taken the macroscopic viewpoint for non-dimensionalization, we will look for unknowns u∗s , v∗ and p∗ in the form: u∗s (x∗ , y∗ , t∗ ) = u∗(0) (x∗ , y∗ , t∗ ) + εu∗(1) (x∗ , y∗ , t∗ ) + · · · v∗ (x∗ , y∗ , t∗ ) = v∗(0) (x∗ , y∗ , t∗ ) + εv∗(1) (x∗ , y∗ , t∗ ) + · · · p∗ (x∗ , y∗ , t∗ ) = p∗(0) (x∗ , y∗ , t∗ ) + εp∗(1) (x∗ , y∗ , t∗ ) + · · ·

(11.25)

with y∗ = ε−1 x∗ , u∗(i) , v∗(i) and p∗(i) Ω∗ -periodic in y∗ . Since the motion is quasistatic, time is not explicitly involved. 11.2.2.1. Boundary-value problem for u∗(0) At the lowest order, (11.17) and (11.20) lead to the problem for u∗(0) that we have already encountered for the empty porous matrix: divy∗ (a∗ : ey∗ (u∗(0) )) = divy∗ σ ∗(−1) =0 s a∗ : ey∗ (u∗(0) ) n = σ ∗(−1) n=0 s

within Ω∗s

over Γ∗

where u∗(0) is Ω∗ -periodic in y∗ . We therefore have: u∗(0) = u∗(0) (x∗ , t∗ ) 11.2.2.2. Boundary-value problem for p∗(0) and v∗(0) Also, at the lowest order (11.23) leads, as in Chapter 7, to: grady∗ p∗(0) = 0 and thus we again ﬁnd: p∗(0) = p∗(0) (x∗ , t∗ )

within Ω∗f

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353

At the following order (11.23), (11.19) and (11.21) form a system of equations for unknown v∗(0) : η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) = 0 divy∗ v∗(0) = 0 v∗(0) = u˙ ∗(0)

within Ω∗f

within Ω∗f over Γ∗

where v∗(0) and p∗(1) are Ω∗ -periodic in y∗ . Introducing the relative velocity w∗ = v∗(0) − u˙ ∗(0) , we arrive at the same system of equations studied in Chapter 7 where v∗(0) is replaced with w∗ : η ∗ Δy∗ w∗ − grady∗ p∗(1) − gradx∗ p∗(0) = 0 divy∗ w∗ = 0 w∗ = 0

within Ω∗f

within Ω∗f over Γ∗

where w∗ and p∗(1) are Ω∗ -periodic in y∗ . It follows that: wi∗ = −

∗ kij ∂p∗(0) ∗ η ∂x∗j

Thus we obtain a Darcy’s law for the relative motion: w∗ = v∗(0) − φu˙ ∗(0) = −

K∗ gradx∗ p∗(0) , K∗ = k∗ η∗

(11.26)

K∗ is the permeability tensor of the rigid skeleton. 11.2.2.3. Boundary-value problem for u∗(1) We will now return to system (11.17) and (11.20). The next order is a problem for the unknown u∗(1) : = 0 within Ω∗s divy∗ (a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))) = divy∗ σ ∗(0) s a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n = σ ∗(0) n = −p∗(0) n s

over Γ∗

with u∗(1) Ω∗ -periodic in y∗ . This system, with p∗(0) = 0, is identical to the one we studied for the empty porous matrix. The unknown u∗(1) appears here as a linear

354

Homogenization of Coupled Phenomena

function of ex∗ (u∗(0) ) and of p∗(0) , up to an arbitrary vector u∗(1) that is a function of x∗ and t∗ : ∗(1)

ui

∗(1)

= ξi∗lm ex∗ lm (u∗(0) ) − ηi∗ p∗(0) + ui

(x∗ , t∗ )

where ξ ∗ (y∗ ) is the third-order tensor with zero mean which has already been introduced in section 11.1 and η ∗ is a vector with zero mean which is obtained by solving the above system with ex∗ (u∗(0) ) = 0 and p∗(0) = −1. 11.2.2.4. First compatibility equation Consider the conservation of momentum of the solid and ﬂuid at order ε0 , with the condition of continuity of normal stresses over Γ∗ at order ε: + divx∗ σ ∗(0) =0 divy∗ σ ∗(1) s s ∗(1)

divy∗ σ f

within Ω∗s

− gradx∗ p∗(0) = 0 ∗(1)

(σ ∗(1) − σf s

)n = 0

within Ω∗f

over Γ∗

These conservation equations lead to a compatibility equation which can be obtained by integration of the ﬁrst two equations over Ω∗s and Ω∗f respectively. We will deﬁne the total stress σ T∗ by: σ T∗ =

σ ∗s σ ∗f

within Ω∗s within Ω∗f

It follows that: divx∗ σ T∗(0) = 0

(11.27)

σ T∗(0) = c∗ : ex∗ (u∗(0) ) − α∗ p∗(0)

(11.28)

where: c∗ijkh = a∗ijkh + a∗ijlm ey∗ lm (ξ ∗kh ) is the effective elastic tensor introduced for the macroscopic behavior of the empty porous matrix, and: ∗ αij = φIij + a∗ijlm ey∗ lm (η ∗ )

is a new elastic tensor.

Quasi-statics of Saturated Deformable Porous Media

355

11.2.2.5. Second compatibility equation A second compatibility equation can be obtained starting from the conservation of volume (11.19) at order ε0 : divx∗ v∗(0) + divy∗ v∗(1) = 0 Integrating over Ω∗f , and using boundary condition (11.21) at order ε, we ﬁnd: divx∗ (v∗(0) − φu˙ ∗(0) ) = −γ ∗ : e˙ x∗ (u∗(0) ) − β ∗ p˙∗(0)

(11.29)

with two new elastic coefﬁcients: ∗ ∗ij γij = φIij − ξp,p ∗ β ∗ = ηp,p

11.2.2.6. Macroscopic description In summary, the system describing the dimensionless macroscopic behavior of the deformable saturated porous medium is, to ﬁrst order of approximation, (11.27), (11.28), (11.29) and (11.26). In physical variables, the behavior can be written: ¯ divX σ T = O(ε)

(11.30)

¯ σ T = c : eX (us ) − α p + O(ε)

(11.31)

¯ divX (v − φu˙ s ) = −γ : e˙ X (us ) − β p˙ + O(ε)

(11.32)

v − φu˙ s = −

K ¯ gradX p + O(ε) η

(11.33)

¯ is a giving seven scalar equations in seven scalar unknowns usi , vi and p. O(ε) small term of order ε relative to the other terms in the equation. The above system of equations is equivalent to the Biot model. The medium is described by two displacement (or velocity) ﬁelds: those of the solid and of the liquid. The process this describes is known as consolidation in soil mechanics. In practice, the number of unknowns can be reduced, as will be discussed below. 11.2.3. Properties of the macroscopic diphasic description 11.2.3.1. Properties of macroscopic quantities and effective coefﬁcients We have already studied the macroscopic quantities involved in (11.30-11.33), apart from σ T , for which the logic for σs can be identically applied. Thus σ T and v are indeed ﬂuxes, and represent a stress and a Darcy velocity respectively. The tensors c and K have also already been analyzed. We will therefore consider the tensors a, γ and b.

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Homogenization of Coupled Phenomena

11.2.3.2. The coupling between (11.31) and (11.32) is symmetric, α = γ This property follows from the symmetry of the scalar product in the space E introduced earlier and with variational form giving u∗(1) . This can be obtained as in the case of the empty porous matrix, since only the boundary condition on Γ∗ is altered. It follows that: ∀α∗ ∈ E,

Ω∗ s

a∗ijkh (ey∗ (u∗(1) ) + ex∗ (u∗(0) )) ey∗ kh (α∗ ) dΩ∗

=−

Γ∗

p∗(0) αi∗ ni dS ∗

where n∗ is the exterior normal to Ω∗s . Consider this form with ﬁrst u∗(1) = η ∗ and α∗ = ξ∗lm and second with u∗(1) = ξ ∗lm and α∗ = η ∗ . The symmetry of the scalar product means that we can write: Ω∗ s

a∗ijkh

ey ∗ ij (ξ

∗lm

∗

∗

) ey∗ kh (η ) dΩ =

Γ∗

ξi∗lm ni dS ∗

=−

Ω∗ s

a∗ijlm ey∗ ij (η ∗ ) dΩ∗

The periodicity of the vector ξ∗lm , and the divergence theorem, then lead to the result: ∗lm ∗ ∗lm ∗ ∗lm ∗ ξi ni dS = ξi ni dS = ξi,i dΩ = − a∗ijlm ey∗ ij (η ∗ ) dΩ∗ Γ∗

∂Ω∗ s

Ω∗ s

Ω∗ s

Then, using the deﬁnitions of α∗ and γ ∗ , the last equality implies: α∗ = γ ∗ 11.2.3.3. The tensor α∗ is symmetric This follows immediately from the deﬁnition of α∗ and the properties of a∗ ∗ αij = φIij + a∗ijlm ey∗ lm (η ∗ )

The tensor α∗ is symmetric and real; hence it is diagonalizable.

Quasi-statics of Saturated Deformable Porous Media

357

11.2.3.4. The coefﬁcient β ∗ is positive, β ∗ > 0 We will now consider the β ∗ coefﬁcient. For this we will again look at the variational form giving u∗(1) , this time with u∗(1) = α∗ = η ∗ : Ω∗ s

a∗ijkh ey ∗ ij (η ∗ ) eykh (η) dΩ∗ =

Γ∗

ηi∗ ni dS ∗ =

Ω∗ s

∗ ηi,i dΩ∗

(11.34)

where we have made use of the periodicity of η ∗ . The property of ellipticity of the tensor a∗ , which corresponds to the positivity of the local strain energy, implies that the left-hand side of these equalities is strictly positive, which leads to the result. 11.2.3.5. Speciﬁc cases 11.2.3.6. Homogenious matrix material We will assume that a∗ is independent of the variable y∗ : the material forming the porous matrix is homogenous. The expression for c∗ becomes: c∗ijkh = a∗ijkh + a∗ijlm ey∗ lm (ξ ∗kh ) = (1 − φ)a∗ijkh + a∗ijlm ey∗ lm (ξ ∗kh ) We will introduce the inverse d∗ of a∗ through: a∗ijkh d∗αβkh =

1 (Iαi Iβj + Iαj Iβi ) 2

It then follows that: ey∗ lm (ξ∗kh ) = (c∗ijkh − (1 − φ) a∗ijkh ) d∗ijlm from which we deduce that: ∗kh = (c∗ijkh − (1 − φ) a∗ijkh ) d∗ijtt = c∗ijkh d∗ijtt − (1 − φ) Ikh ξi,i

so that α∗ and γ ∗ are given by: ∗ij ∗ ∗ = γij = φIij − ξl,l = Iij − c∗khij d∗khtt αij

We also deduce from the deﬁnition of α∗ that: ∗ ey∗ lm (η ∗ ) = (αij − φIij ) d∗ijlm = ((1 − φ)Iij − c∗khij d∗khtt ) d∗ijlm

and: ∗ = (1 − φ)d∗iill − c∗khij d∗khtt d∗ijll β ∗ = ηl,l

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Homogenization of Coupled Phenomena

In certain applications, the material forming the porous matrix is only slightly compressible, so that the approximation of incompressibility is often adopted, particularly in soil mechanics. In this case: d∗ijll = 0 and: ∗ = Iij , αij

β∗ = 0

We then recover the standard system of equations for soil mechanics. 11.2.3.7. Homogenous and isotropic matrix material and macroscopically isotropic matrix In this case: a∗ijtr = λ∗ Iij Itr + μ∗ (Iit Ijr + Iir Ijt ) c∗ijtr = λ∗p Iij Itr + μ∗p (Iit Ijr + Iir Ijt ) where λ∗ , μ∗ and λ∗p , μ∗p are respectively the Lamé coefﬁcients of the matrix material and of the macroscopic porous structure. Then the expression for α∗ can be simpliﬁed: ∗ αij = Iij −

c∗ijll 3λ∗p + 2μ∗p = (1 − )Iij = α∗ Iij 3λ∗ + 2μ∗ 3λ∗ + 2μ∗

Given that: 3λ∗p + 2μ∗p 0 3λ∗ + 2μ∗ it is clear that: α∗ 1 Also, in this case: α∗ = φ + (λ∗ +

2μ∗ ∗ )ηi,i 3

and (11.34) leads to α∗ > φ and hence: φ α∗ 1

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359

Furthermore we still have: 0

3λ∗p + 2μ∗p )1−φ 3λ∗ + 2μ∗

Finally: 3 β = ∗ 3λ + 2μ∗ ∗

3λ∗p + 2μ∗p (1 − φ) − ∗ 3λ + 2μ∗

and we recover the fact that for an incompressible material, α∗ = 1 et β ∗ = 0. 11.2.3.8. Diphasic consolidation equations: Biot model Making use of the fact that α∗ = γ ∗ , the system describing diphasic consolidation on the macroscopic scale can be written, omitting terms of relative order O(ε): divX σ T = divX (c : eX (us ) − α p) = 0

(11.35)

K gradX p η

(11.36)

v − φu˙ s = −

divX (v − φu˙ s ) = −α : e˙ X (us ) − β p˙

(11.37)

In practice, attempts are made to reduce the number of unknowns and of equations. This can be achieved in two ways. The ﬁrst method involves eliminating v from the last two equations (11.36) and (11.37). Thus we obtain: divX (c : eX (us ) − α p) = 0 divX

K gradX p η

= α : e˙ X (us ) + β p˙

which represents the standard system used for soil consolidation, giving four scalar equations in four unknowns usi and p. The second method involves symmetrizing the system. For that we integrate with respect to time the volume conservation (11.37) and set v = φu˙ f where uf is the (mean) displacement of the ﬂuid. It follows for β = 0 that: p = −β −1 α : eX (us ) − φβ −1 divX (uf − us ) + f (X)

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Homogenization of Coupled Phenomena

Since in general terms p, usi and ufi are changes relative to a given initial state, the constant of integration f (X) can be taken to be zero. We now introduce the value of p into expression (11.36) for Darcy’s law: divX (φp) = −divX −φβ −1 α : eX (us ) − φ2 β −1 divX (uf − us ) = −φ2 ηK−1 (u˙ f − u˙ s ) Thus we recover the expression which was introduced by Biot [BIO 55] using a phenomenological approach. With those notations: divX (φp) = divX (Q : eX (us ) + Rθ) = b(u˙ f − u˙ s )

(11.38)

φp = −Q : eX (us ) − Rθ with: Q = φβ −1 (α − φI), b = φ2 ηK−1 , θ = divX uf ,

or

or

Qij = φβ −1 (αij − φIij )

−1 bij = φ2 ηKij

and R = φ2 β −1

Now we will introduce the partial stress on the skeleton through: σ s = σ T + φpI Subtracting equation (11.38) from equation (11.35), term by term, we ﬁnd: divX σ s = −b(u˙ f − u˙ s )

(11.39)

with: σ s = π : eX (us ) + Q θ πijkh = cijkh + β −1 (αij − φIij )(αkh − φIkh ) Equations (11.38) and (11.39) were introduced by Biot [BIO 55] following a direct analysis on the macroscopic scale. Their advantage is that they give a good representation of the symmetry of ﬂuid-solid coupling, with the presence of the same tensor Q in both equations. Their generalization to dynamic perturbations is very well suited to the study of wave propagation, which will be discussed in the following chapter.

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361

11.2.3.9. Effective stress We will introduce the effective stress through: σ eﬀ = σ T + pI Since the ﬁrst order of p∗ (p∗(0) ) is constant over the period, the concept of effective stress that we have introduced is consistent with the meaning it is given in soil mechanics. This concept, well deﬁned in the context of linear elasticity, can under certain conditions be extended to skeletons which display a non-linear behavior [deB 96; GEI 99]. 11.2.3.10. Compressible interstitial ﬂuid When the ﬂuid is highly compressible, the results from Chapter 8 apply, since we have seen that, as far as the ﬂow law is concerned, the problem with the deformable skeleton can be reduced to that of the rigid skeleton by replacing the Darcy velocity with a relative velocity. When ﬂuid is compressible, it is possible to retain the linearity of the description. In this case the equation of ﬂuid state can in fact be written: p˙ = −Kf divX v where Kf is the modulus of ﬂuid compressibility. This equation gives, to ﬁrst order: divy∗ v∗(0) = 0 so that the results involving Darcy’s law remain valid. Only the conservation of volume is again modiﬁed, and it can be obtained from the second order: p˙∗(0) = −Kf∗ (divx∗ v∗(0) + divy∗ v∗(1) ) A calculation analogous to that carried out earlier then leads to the same structure of the macroscopic conservation equation which in (5.38) (with a relative ¯ approximation of O(ε)): ¯ divX (v − φu˙ s ) = −α : e˙ X (us ) − β p˙ + O(ε) but the coefﬁcient β is now deﬁned by: β = ηl,l +

φ Kf

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Homogenization of Coupled Phenomena

11.2.4. Monophasic macroscopic elastic behavior: Gassman model We will now analyze the case where: QL = O(ε−1 ) System (11.17-11.21) is again considered as being written in dimensionless form, with (11.18) replaced by: εη ∗ Δx∗ v∗ − gradx∗ p∗ = O(ε)

(11.40)

σ ∗f = 2η ∗ εDx∗ (v∗ ) − p∗ I

(11.41)

As before, system (11.17) and (11.20) at the lowest order, is unchanged and leads to u∗(0) : u∗(0) = u∗(0) (x∗ ) Since time is not explicitly involved here, we will not include it in our variables. Equations (11.40) and (11.19) at order ε−1 and (11.21) at order ε0 determine v∗(0) and p∗(0) : η ∗ Δy∗ v∗(0) − grady∗ p∗(0) = 0 divy∗ v∗(0) = 0 v∗(0) = u˙ ∗(0)

within Ω∗f

within Ω∗f over Γ∗

where v∗(0) and p∗(0) are Ω∗ -periodic in y∗ . The solution is evident: v∗(0) = u˙ ∗(0) (x∗ ),

p∗(0) = p∗(0) (x∗ )

Thus we again ﬁnd that to ﬁrst order there is no displacement of the ﬂuid with respect to the matrix. The equations giving u∗(1) are unchanged, and lead to: ∗(1)

ui

∗(1)

= ξi∗lm ex∗ lm (u∗(0) ) − ηi∗ p∗(0) + ui

(x∗ )

The compatibility equation stemming from the conservation of momentum is also unchanged: divx∗ σ T∗(0) = 0 σ T∗(0) = c∗ : ex∗ (u∗(0) ) − α∗ p∗(0)

Quasi-statics of Saturated Deformable Porous Media

363

On the other hand, the conservation of mass (11.19) at order ε0 leads to a compatibility equation which is entirely different. Because: v∗(0) = φ u˙ ∗(0) equation (11.29) becomes: 0 = α∗ : e˙ x∗ (u∗(0) ) + β ∗ p˙ ∗(0) where we have made use of the result α∗ = γ ∗ , which is still valid. Integrating this equation with respect to time, and making use of the fact that the constant of integration can be taken to be zero, as above, because p∗(0) and ex∗ (u∗(0) ) are changes relative to a given initial state, we obtain: p∗(0) = −β ∗−1 α∗ : ex∗ (u∗(0) ) Thus the macroscopically equivalent medium is described in physical variables by: ¯ divX σ T = O(ε) ¯ σ T = c+ : eX (us ) + O(ε) with: −1 αij αkh c+ ijkh = cijkh + β

¯ where O(ε) is a small term of order ε relative to the other terms in the equality. The macroscopic behavior is that of an elastic solid. This is the quasi-static model of Gassman [GAS 51].

11.2.5. Monophasic viscoelastic macroscopic behavior Now the viscous stress and pressure are of the same order of magnitude on the macroscopic scale: QL = O(1) and (11.18) and (11.19) can be written: η ∗ Δx∗ v∗ − gradx∗ p∗ = O(ε)

(11.42)

σ ∗f = 2η ∗ Dx∗ (v∗ ) − p∗ I

(11.43)

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Homogenization of Coupled Phenomena

Let u∗ be the displacement deﬁned by u∗s within Ω∗s and u∗f within Ω∗f . We will ﬁrst of all consider motion at constant pulsation ω ∗ (ω ∗ is small because the motion ∗ ∗ is quasi-static). We will look for all the quantities with a time dependence of eiω t , 2 with i = −1. Thus: v∗ = iω ∗ u∗f Equations (11.17) and (11.42) at order ε−2 , (11.19) and (11.20) at order ε−1 and (11.21) at order ε0 give u∗(0) as a solution to: divy∗ (a∗ : ey∗ (u∗(0) )) = 0 η ∗ Δy∗ (iω ∗ u∗(0) ) = 0 divy∗ u∗(0) = 0

within Ω∗s within Ω∗f

within Ω∗f

a∗ : ey∗ (u∗(0) ) n = 2η ∗ ey∗ (iω ∗ u∗(0) ) n [u∗(0) ] = 0

over Γ∗

over Γ∗

where u∗(0) is Ω∗ -periodic in y∗ . It follows that: u∗(0) = u∗(0) (x∗ , ω ∗ ) The same equations at the next order lead to u∗(1) and p∗(0) : divy∗ (a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))) = 0 η ∗ Δy∗ (iω ∗ u∗(1) ) − grady∗ p∗(0) = 0 divy∗ u∗(1) + divx∗ u∗(0) = 0

within Ω∗s within Ω∗f

within Ω∗f

a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n = (−p∗(0) I + 2η ∗ iω ∗ (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n -

. u∗(1) = 0

over Γ∗

Making use of the linearity, the solution is: ∗(1)

ui

∗(1)

= χ∗lm ex∗ lm (u∗(0) ) + ui i

p∗(0) = ζ ∗lm ex∗ lm (u∗(0) )

(x∗ , ω ∗ )

over Γ∗

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365

where u∗(1) (x∗ , ω ∗ ) is arbitrary and independent of y∗ . The tensors χ∗ and ζ ∗ take complex values and depend on the variables y∗ and ω ∗ . We observe that no arbitrary additive function of x∗ appears in the expression for p∗(0) . This results from the continuity of normal stresses over Γ∗ written at order ε0 (see above). Finally the next order leads to the compatibility condition which, as usual, plays the role of macroscopic description. It takes the same form as in the previous section. Thus we obtain in dimensional variables a behavior of the form: ¯ divX σ T = O(ε) ¯ σ T = c++ : eX (us ) + O(ε) c++ ijkh =

1 |Ω|

1 + |Ω|

(aijkh + aijlm eXlm (χkh )) dΩ Ωs

(−ζ ij Ikh + 2iωηeXij (χkh ) + 2iωηIij Ikh ) dΩ Ωf

¯ where O(ε) is a small term of order ε relative to the other terms in the equality. But here, c++ is complex and depends on ω. For an arbitrary transient motion the constitutive law becomes: T ++ ¯ σij = TF−1 ) ∗ eXkh (us ) + O(ε) ijkh (c ++ ) is the inverse Fourier transform of c++ , and ∗ here indicates where TF−1 ijkh (c the convolution product. The equivalent macroscopic behavior is that of a linear viscoelastic solid.

11.2.6. Relationships between the three macroscopic models Auriault and Sanchez-Palencia [AUR 77] showed that Biot’s diphasic model tends to the monophasic elastic model when the characteristic time tc of excitation decreases. Nevertheless, when tc decreases, the inertial terms increase and the macroscopic behavior becomes dynamic, which happens in the example discussed in Chapter 12. In order to study the relationships between the three quasi-static macroscopic models more precisely, we will try and ﬁnd a dimensionless number which characterizes these models, which is independent of the excitation applied to the porous medium. The number PL can be estimated as: PL =

ρsc L2c ρsc lc2 = ε−2 = O(ε) 2 ac tc ac t2c

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Homogenization of Coupled Phenomena

Substituting the value of tc drawn from this last expression into QL we ﬁnd: t2c =

ρsc lc2 −3 ε , ac

QL =

3 ac tc = Rε− 2 , ηc

R=

lc √ ac ρsc ηc

where R is a dimensionless number which is independent of excitation. The domains of validity of the three macroscopic models obtained above can be expressed as a 3 function of R. In addition, for R O(ε− 2 ), the characteristic time of excitation is very large and the ﬂuid does not play a part in the interaction. Conditions (11.221 ) and (11.223 ) are relaxed. We obtain the following classiﬁcation: 3 – R O(ε− 2 ): the saturated porous medium behaves like an empty porous matrix. On the corresponding characteristic timescale, the ﬂuid does not play any relevant role; 1

– R = O(ε− 2 ): Biot’s diphasic model; 1

– R = O(ε 2 ): monophasic elastic model, or Gassman model; 3

– R = O(ε 2 ): monophasic viscoelastic model; 5

– R = O(ε 2 ): the separation of scales no longer exists. macroscopically equivalent medium.

There is no

From this we can deduce that porous media subjected to a quasi-static excitation such as PL = O(ε) and TL = O(ε) can be split into two distinct categories based on their quasi-static macroscopic behavior: – media where R 1, for which only a O(ε−p ), p > 0 estimate is possible. Such media can only display diphasic behavior or the behavior of an empty porous matrix. Monophasic elastic behavior is seen in the dynamic regime (see Chapter 12); 1

3

– media where R 1, for which only the O(ε 2 ) and O(ε 2 ) estimates are possible. Such media can only display monophasic elastic or viscoelastic behavior. We will again ﬁnd a similar classiﬁcation for the dynamic behavior (PL = O(1)) of saturated porous media in Chapter 12. In general terms, for PL = O(εp ), p 1, we have the following classiﬁcation: p – R O(ε 2 −2 ): empty porous matrix, p

– R = O(ε 2 −1 ): Biot’s diphasic model, p

– R = O(ε 2 ): Gassman model, p

– R = O(ε 2 +1 ): monophasic viscoelastic model, p

– R = O(ε 2 +2 ): non-homogenizable medium. For PL < O(ε2 ), the diphasic behavior is no longer possible when R 1.

Chapter 12

Dynamics of Saturated Deformable Porous Media

12.1. Introduction Theoretical modeling of the acoustics of porous saturated deformable media was ﬁrst achieved using phenomenological approaches: Frenkel [FRE 44] showed the existence of two volume waves, but without additional mass, and then Biot [BIO 56a; 56b] gave the complete diphasic model. Experimental demonstration of the slow ¯ ¯ volume wave was achieved by Oura [OUR 52] and Plona [PLO 80]. Here we will apply the method of asymptotic expansions in order to determine the macroscopic description of porous saturated deformable media subject to lowamplitude mechanical vibrations. In order to keep the presentation simple, the behavior of the medium will be assumed to be isothermal. As in the previous chapter, the matrix is linear elastic and the ﬂuid incompressible viscous Newtonian. The generalization to less “simple” materials can lead to difﬁculties, but the presentation which follows will include the main facts involved in the equivalent macroscopic behavior. We will begin as usual with estimates for the local phenomena. This involves gathering together the individual behaviors of the constituents that we have already studied, in terms of the dynamics of the ﬁltering ﬂuid (Chapter 7) and the dynamics of an elastic composite (Chapter 3). As in Chapter 11 for the quasi-static behavior of the saturated porous medium, three main types of macroscopic behavior can be distinguished, based on the values of the dimensionless numbers: – The diphasic macroscopic behavior will be investigated in sections 12.3 and 12.4. In these the description takes the same structure as that introduced by Biot [BIO56a; 56b]. The presentation here makes use of Levy [LEV 79] and Auriault

368

Homogenization of Coupled Phenomena

[AUR 80]. We use it to demonstrate the role played by the dynamic permeability tensor, which was introduced in Chapter 7 and which contains all the information about dynamic and viscous coupling between the two phases on the macroscopic scale. The thickness of the viscous boundary layer at boundary Γ has a size which is of the same order as that of the pores. – The monophasic elastic macroscopic behavior is obtained in section 12.5 when conditions are such that relative motion of the ﬂuid with respect to the skeleton is impossible, and ﬂuid viscosity is too low for it to survive the micro-macro transition (at least to ﬁrst order of approximation). The thickness of the viscous boundary layer at boundary Γ of the pores is very large compared to the pore size. This leads to the model described in Gassman [GAS 51]. – Finally, the monophasic viscoelastic behavior is the subject of section 12.6. This time the viscosity of the ﬂuid is high enough for it to be involved in the ﬁrst order of macroscopic behavior. The relative motion of the ﬂuid and solid appears in the second order of approximation. This distinction between three types of macroscopic behavior was introduced by Sanchez-Palencia [SAN 80] for the dynamics of suspensions. For porous media, the reader is referred to Boutin and Auriault [BOU 90] for a more detailed and extensive discussion of viscoelastic ﬂuids.

12.2. Local description and estimates The medium satisﬁed the Navier equation in the solid part and the Navier-Stokes equation in the ﬂuid part, with standard conditions on the interface Γ, i.e. the continuity of normal stress and of displacement: divX (σ s ) = ρs

∂ 2 us ∂t

σ s = a : eX (us ) divX (σ f ) = ρf (

within Ωs

∂v + (vgrad).v) ∂t

(12.1) (12.2) (12.3)

σ f = 2ηD(v) − pI

(12.4)

divX v = 0

(12.5)

within Ωf

(σ s − σ f )n = 0 over Γ

(12.6)

u˙ s − v = 0 over Γ

(12.7)

Dynamics of Saturated Deformable Porous Media

369

We will again assume in what follows that the stresses on the macroscopic scale and displacements are both of the same order of magnitude. The same will be the case for densities: σ macro = O(σ macro ), s f

us = O(uf )

and

ρs = O(ρf )

(12.8)

In terms of the solid, we saw in Chapter 3 that homogenizable dynamics implies that: 2

|ρs ∂∂tu2s | PL = = O(1) |divX (σ s )| so that, using the macroscopic viewpoint, equation (12.1) can be considered to be written in dimensionless variables. In the Navier-Stokes equation, the Reynolds number is very small due to the low amplitude of vibrations. We will assume that it is sufﬁciently small that it does not affect the ﬁrst order of the macroscopic description. Because of this, we can take: Rel = O(ε)

or

ReL = O(1)

Under these conditions, and so as not to overburden the presentation, we will not include the convective inertial terms in the equations. On the other hand, we are studying the richest behavior, the dynamic regime, which means that we should take the ratio of the inertial term to pressure term to be of order O(1) macroscopically: ∂v | ∂t = O(1) TL = |gradX p| |ρf

We still need to analyze the number Q, the ratio between pressure and viscosity terms: Q = |gradX p|/|ηΔX v| As for the quasi-static behavior (see Chapter 11), we only need to consider the following three cases: – QL = O(ε−2 ): this case leads to a diphasic macroscopic description. This is the subject of sections 12.3 and 12.4. This corresponds to a macroscopic transient Reynolds number of O(ε−2 ): ∂v |ρf | ∂t = O(ε−2 ) RtL = |ηΔX v|

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Homogenization of Coupled Phenomena

so that, on the scale of the pores, just as in Chapter 8: Rtl = O(1) We recall that this number can be expressed in different ways: Rtl = where:

ρfc lc2 ωc lc = ( )2 = (ωc τd )2 ηc δb

ηc ωc ρfc is the thickness of the boundary layer which appears along the ﬂuid-solid interface Γ, and: ρfc τd = lc ηc ωc is the time taken for the diffusive shear wave to cross the period. Here: δb =

δb = O(l)

,

τd = O(ωc−1 )

– QL = O(ε−1 ): as investigated in Chapter 7 where the matrix is rigid, there is no movement of the ﬂuid with respect to the matrix (to ﬁrst order). Here the corresponding macroscopic description is monophasic elastic. This is investigated in section 12.5. The macroscopic transient Reynolds number is now O(ε−1 ), and δb = O(ε−1 lc ). – QL = O(1): there is no movement of the ﬂuid with respect to the matrix (to ﬁrst order). The macroscopic description becomes monophasic viscoelastic. Section 12.6 treats this third situation. The macroscopic transient Reynolds number is now O(1), and δb = O(ε−2 lc ). 12.3. Diphasic macroscopic behavior: Biot model We will carry out an analysis at constant frequency ω. We now assume system (12.1-12.7) to be written in dimensionless form, with (12.3) and (12.4) replaced with: ε2 η ∗ Δx∗ v∗ − gradx∗ p∗ = ρ∗f iω ∗ v∗ + O(ε) σ∗f = 2η ∗ ε2 Dx∗ (v∗ ) − p∗ I

(12.9) (12.10)

and (12.1) is rewritten in the form: divx∗ (σ s ) = −ω ∗2 ρ∗s u∗s Having adopted the macroscopic viewpoint for the non-dimensionalization, we will look for unknowns u∗s , v∗ and p∗ in the form: u∗s (x∗ , y∗ , ω ∗ ) = u∗(0) (x∗ , y∗ , ω ∗ ) + εu∗(1) (x∗ , y∗ , ω ∗ ) + · · ·

Dynamics of Saturated Deformable Porous Media

371

v∗ (x∗ , y∗ , ω ∗ ) = v∗(0) (x∗ , y∗ , ω∗ ) + εv∗(1) (x∗ , y∗ , ω ∗ ) + · · · p∗ (x∗ , y∗ , ω ∗ ) = p∗(0) (x∗ , y∗ , ω∗ ) + εp∗(1) (x∗ , y∗ , ω ∗ ) + · · · with y∗ = ε−1 x∗ , u∗(i) , v∗(i) and p∗(i) Ω∗ -periodic in y∗ . It is easy to see that for u∗(0) , p∗(0) , and u∗(1) , the lowest orders lead to the same results as those obtained in the quasi-static case: u∗(0) = u∗(0) (x∗ , ω ∗ ) p∗(0) = p∗(0) (x∗ , ω ∗ ) ∗(1)

ui

∗(1)

= ξi∗lm ex∗ lm (u∗(0) ) − ηi∗ p∗(0) + ui

(x∗ , ω ∗ )

The problem giving v∗(0) is that of the quasi-static case with the addition of the dynamic term: η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) = ρ∗f iω ∗ v∗(0) divy∗ v∗(0) = 0

within Ω∗f

within Ω∗f

v∗(0) = iω ∗ u∗(0)

over Γ∗

with v∗(0) and p∗(1) Ω∗ -periodic in y∗ . Introducing the relative velocity w∗ = v∗(0) − iω ∗ u∗(0) , we obtain a system which is similar to that studied in section 7.5 where v∗(0) is replaced with w∗ : η ∗ Δy∗ w∗ − grady∗ p∗(1) − gradx∗ p∗(0) = ρ∗f iω ∗ w∗ − ρ∗f ω ∗2 u∗(0) divy∗ w∗ = 0 w∗ = 0

within Ω∗f

within Ω∗f over Γ∗

where w∗ and p∗(1) are Ω∗ -periodic in y∗ . It follows that the system giving w∗ is identical to that giving v∗(0) when the skeleton is assumed to be rigid, with gradx∗ p∗(0) replaced with gradx∗ p∗(0) − ρ∗f ω ∗2 u∗(0) : wi∗

=

−λ∗ij

∂p∗(0) ∗(0) − ρ∗f ω ∗2 uj ∂x∗j

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Homogenization of Coupled Phenomena

Thus we obtain a generalized Darcy’s law for the relative motion: w∗ = v∗(0) − φiω ∗ u∗(0) = −Λ∗ (gradx∗ p∗(0) − ρ∗f ω ∗2 u∗(0) ) Λ∗ = λ∗

(12.11)

The tensor Λ∗ is the dynamic hydraulic conductivity tensor for the rigid skeleton. Recall that here the components Λ∗ij are functions of ω ∗ and are complex. Consider the conservation of momentum for the solid and ﬂuid at order ε0 , with the condition of normal strain continuity over Γ∗ at order ε: divy∗ σ ∗(1) + divx∗ σ ∗(0) = −ρ∗s ω ∗2 u∗(0) s s ∗(1)

divy∗ σ f

− gradx∗ p∗(0) = ρ∗f iω ∗ v∗(0) ∗(1)

(σ ∗(1) − σf s

)n = 0

within Ω∗s within Ω∗f

over Γ∗

These conservation equations lead to a compatibility equation obtained by integration of the two ﬁrst equations over Ω∗s and Ω∗f respectively. We will introduce the total stress σ T∗ deﬁned by: σ T∗ =

σ ∗s σ ∗f

within Ω∗s within Ω∗f

It follows that: divx∗ σ T ∗(0) = −ρ∗s ω ∗2 u∗(0) + ρ∗f iω ∗ v∗(0)

(12.12)

σT∗(0) = c∗ : ex∗ (u∗(0) ) − α∗ p∗(0)

(12.13)

where: c∗ijkh = a∗ijkh + a∗ijlm ey∗ lm (ξ ∗kh ) and: ∗ = φIij + a∗ijlm ey∗ lm (η ∗ ) αij

are the elastic tensors introduced for macroscopic behavior of the empty porous matrix and the quasi-static behavior of the saturated deformable porous medium.

Dynamics of Saturated Deformable Porous Media

373

A second compatibility equation can be obtained starting from the conservation of volume (12.5) at order ε0 : divx∗ v∗(0) + divy∗ v∗(1) = 0 Integrating over Ω∗f , and using boundary condition (12.6) at order ε, we ﬁnd: divx∗ (v∗(0) − φiω∗ u∗(0) ) = −iω ∗ γ ∗ : ex∗ (u∗(0) ) − iω ∗ β ∗ p∗(0)

(12.14)

with the two elastic coefﬁcients already introduced in Chapter 11: ∗ ∗ij γij = φIij − ξp,p ∗ β ∗ = ηp,p

The conservation of mass equation (12.14) is identical to the one obtained in the quasi-static regime. In summary, the system of equations describing, to ﬁrst order, the macroscopic behavior of the saturated deformable porous medium at constant frequency ω ∗ is in this case (12.12), (12.13), (12.14) and (12.11). This is the model given in Biot [BIO56a; 56b]. It can be written in dimensionless variables, making use of the symmetry α = γ: ¯ divX σ T = −ρs ω 2 us + ρf iωv + O(ε)

(12.15)

¯ σ T = c : eX (us ) − α p + O(ε)

(12.16)

¯ divX (v − φiωus ) = −iωα : eX (us ) − iωβ p + O(ε)

(12.17)

¯ v − iωφus = −Λ (gradX p − ρf ω 2 us ) + O(ε)

(12.18)

so that, in the quasi-static case, we have seven scalar equations in seven unknowns usi , vi and p. Tensors c, α and β are the elastic tensors introduced in the steady state case. Only the dynamic hydraulic conductivity Λ is complex, and frequencydependent. It is the time derivatives that have given rise to the factors of iω. Chapter 14 will focus on the study of wave propagation in these media.

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Homogenization of Coupled Phenomena

12.4. Study of diphasic macroscopic behavior 12.4.1. Equations for the diphasic dynamics of a saturated deformable porous medium In order to obtain a more symmetrical description, we will follow the same strategy as in Chapter 11 for the quasi-static case, and again introduce the partial stresses −φp for the ﬂuid and σ s for the solid–through: T σsij = σij + φpIij

and the (mean) displacement uf of the ﬂuid through: v = φiωuf We will also introduce the inverse H of Λ: −1 I Hij (ω) = HR ij (ω) + iHij (ω) = Λij (ω)

Equation (12.18) can now be written (in what follows we will ignore the ¯ terms O(ε)): −

∂φp = φ2 Hij iω(ufj − usj ) − φ ω 2 ρf usi ∂Xi

2 −1 I = φ2 HR Hij )ω 2 usj − φ2 ω −1 HIij ω 2 ufj ij iω(ufj − usi ) − (φρf Iij − φ ω

Deducting this equality term by term from equation (12.15), we then ﬁnd: ∂σsij = −φ2 HR ij iω(ufj − usi ) ∂Xj −((ρs − φρf )Iij + φ2 ω −1 HIij )ω 2 usj − (φρf Iij − φ2 ω −1 HIij )ω 2 ufj In addition, the conservation of volume (12.17) can be written (as in the quasistatic case), when β = 0: p = −αij β −1 eXij (us ) − φβ −1

∂ (ufi − usi ) ∂Xi

Finally, introducing the notation from Biot [BIO56a; 56b] and returning to time derivatives, the system of equations describing the macroscopic diphasic monochromatic behavior can be written in the following form: ∂σsij = −bij (u˙ fj − u˙ sj ) + ρ11ij u ¨sj + ρ12ij u ¨fj ∂Xj

Dynamics of Saturated Deformable Porous Media

375

∂(−φp) = bij (u˙ fj − u˙ sj ) + ρ21ij u ¨sj + ρ22ij u ¨fj ∂Xi σsij = πijkh eXkh (us ) + Qij θ −φp = Qjk eXjk (us ) + Rθ where: bij = φ2 HR−1 ij where π, Q, R and θ have already been deﬁned in Chapter 11. πijkh = cijkh + β −1 (αij − φIij )(αkh − φIkh ) Qij = φβ −1 (αij − φIij ) R = φ2 β −1 θ=

∂ufi ∂Xi

and where the tensorial densities ρAB are deﬁned by: ρ11ij = (ρs − φρf )Iij + φ2 ω −1 HIij = ρs Iij − ρ12ij ρ12ij = ρ21ij = φρf Iij − φ2 ω −1 HIij ρ22ij = φ2 ω −1 HIij The structure of this description is identical to that of Biot [BIO56a; 56b]. The method used here provides the means of calculating all the effective coefﬁcients. As can easily be seen, coefﬁcients π, Q and R can be obtained from the elastic properties of the material which forms the matrix. The viscous friction and apparent densities ρAB can be determined from the densities of the constituents and from tensor H, which then contains all the information about viscous and inertial couplings. It is worth noting that H is obtained by solving a ﬂow problem for the ﬂuid in the rigid porous matrix. 12.4.2. Rheology and dynamics The stresses σ T , σ s and pI are deﬁned in the same way as for a quasi-static motion, which means we can apply the standard macroscopic constitutive law of the matrix to the dynamic case. For the matrix there is a clear separation between rheology and dynamics. The same is not true for the ﬂuid, since the viscosity η is involved in the calculation of the inertial terms.

376

Homogenization of Coupled Phenomena

12.4.3. Additional mass The tensorial densities ρAB depend on pulsation ω. They characterize the additional densities and inertial couplings between the two phases. By deﬁnition, these tensors are symmetrical. Finally, in the isotropic case, we have: φHI ω −1 ρf ρ12 = ρ21 0,

ρ11 ρs ,

ρ22 φρf

12.4.4. Transient motion When the medium is subjected to some transient motion, memory effects appear in the inertial and dissipative parts. The partial dynamics of the two constituents depend on their past history, an effect which we observed previously in Chapter 7 for the dynamics of a ﬂuid ﬁltering through a rigid matrix. In this case Λ and H must be replaced by convolution operators involving FT−1 (Λ ) and FT−1 (K) respectively – the inverse Fourier transforms of the tensors Λ and H. 12.4.5. Small pulsation ω We will take the limit where the pulsation ω tends to zero. Then, when ω → 0: ρ12ij = ρ21ij → φρf Iij − φ2 lim ω −1 HIij = 0 ω→0

ρ11ij (0) = ρs Iij ,

ρ22ij (0) = φρf Iij

In the isotropic case where φHI ω −1 ρf : ρ12 (0) = ρ21 (0) < 0 ρ11 (0) > ρs ,

ρ22 (0) > φρf

So in general, the tensorial character of the densities, as well as dynamic coupling between the two constituents, is retained when the pulsation tends to zero. The memory effect disappears when the duration of memory tends to zero. From this point of view the partial dynamics revert to the standard type. 12.4.6. Dispersive waves The diphasic medium is dispersive: the velocity of wave propagation depends on pulsation ω. As shown in Biot [BIO 56a; 56b], such a medium generates a shear wave

Dynamics of Saturated Deformable Porous Media

377

(S wave) and two volume compression waves: a P1 wave where the motions of the ﬂuid and solid are almost in phase with each other, and a P2 wave, which is highly damped, where their motions are very much out of phase. These two waves were ¯ ¯ demonstrated experimentally by Oura [OUR 52] and Plona [PLO 80]. We recall that a monophasic medium only generates a single compression wave. Chapter 14 will focus on the study of wave propagation in these media.

12.5. Macroscopic monophasic elastic behavior: Gassman model We will now analyze the case where: QL = O(ε−1 ) The system (12.1-12.7) is again considered as being written in dimensionless form, with (12.3) and (12.4) replaced with: εη ∗ Δx∗ v∗ − gradx∗ p∗ = ρ∗f iω ∗ v∗

(12.19)

σ∗f = 2η ∗ εDx∗ (v∗ ) − p∗ I

(12.20)

As before, system (12.1), (12.3) and (12.6), at the lowest order, is unchanged and leads to u∗(0) : u∗(0) = u∗(0) (x∗ , ω ∗ ) Equations (12.19) and (12.5) at order ε−1 and (12.7) at order ε0 determine v∗(0) and p∗(0) : η ∗ Δy∗ v∗(0) − grady∗ p∗(0) = 0 divy∗ v∗(0) = 0 v∗(0) = iω ∗ u∗(0)

within Ω∗f

within Ω∗f over Γ∗

where v∗(0) and p∗(0) are Ω∗ -periodic in y∗ . The solution is clear: v∗(0) = iω ∗ u∗(0) (x∗ , ω ∗ ),

p∗(0) = p∗(0) (x∗ , ω ∗ )

We ﬁnd, that to ﬁrst order there is no displacement of the ﬂuid with respect to the matrix. The equations giving u∗(1) are unchanged, and lead to: ∗(1)

ui

∗(1)

= ξi∗lm ex∗ lm (u∗(0) ) − ηi∗ p∗(0) + ui

(x∗ , ω ∗ )

378

Homogenization of Coupled Phenomena

The compatibility equation arising from the conservation of momentum is also unchanged: divx∗ σ T ∗(0) = −ρ∗s ω ∗2 u∗(0) + ρ∗f iω ∗ v∗(0) σ T∗(0) = c∗ : ex∗ (u∗(0) ) − α∗ p∗(0) On the other hand, the conservation of mass (12.5) at order ε0 leads to a compatibility equation which is entirely different. Because: v∗(0) = iω ∗ u∗(0) equation (12.14) becomes: 0 = −iω ∗ α∗ : ex∗ (u∗(0) ) − iω ∗ β ∗ p∗(0) where we have made use of the result α∗ =γ ∗ which is still valid. From this equation we can obtain the value of p∗(0) : p∗(0) = −β ∗−1 α∗ : ex∗ (u∗(0) ) s Thus the macroscopically equivalent medium is described in dimensional variables, and to the ﬁrst order of approximation, by: ¯ divX σ T = −(ρs + φρf )ω 2 u + O(ε)

(12.21)

¯ σT = c+ : eX (u) + O(ε)

(12.22)

with: −1 c+ ijkh = cijkh + αij αkh β

The macroscopic behavior is that of an elastic solid: the Gassman model [GAS 51]. This is the dynamic equivalent of the quasi-static elastic behavior studied in Chapter 11, with the elastic tensor c+ being the one deﬁned in this section.

12.6. Monophasic viscoelastic macroscopic behavior Now the viscous stress and pressure have the same order of magnitude on the macroscopic scale: QL = O(1)

Dynamics of Saturated Deformable Porous Media

379

and (12.3) and (12.4) can be written: η ∗ Δx∗ v∗ − gradx∗ p∗ = ρ∗f iω ∗ v∗

(12.23)

σ∗f = 2η ∗ Dx∗ (v∗ ) − p∗ I

(12.24)

Let u∗ be the displacement deﬁned by u∗s within Ω∗s , and u∗f within Ω∗f . Equations (12.1) and (12.24) at order ε−2 , (12.5) and (12.6) at order ε−1 , and ﬁnally (12.7) at order ε0 , give u∗(0) as the solution to: divy∗ (a∗ : ey∗ (u∗(0) )) = 0

within Ω∗s

η ∗ Δy∗ (iω ∗ u∗(0) ) = 0 divy∗ u∗(0) = 0

within Ω∗f within Ω∗f

a∗ : ey∗ (u∗(0) ) n = 2η ∗ ey∗ (iω ∗ u∗(0) ) n iω ∗ u∗(0) − v∗(0) = 0

over Γ∗

over Γ∗

where u∗(0) is Ω∗ -periodic in y∗ . It follows that: u∗(0) = u∗(0) (x∗ , ω ∗ ) The same equations at the following order lead to u∗(1) and p∗(0) : divy∗ (a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))) = 0 η ∗ Δy∗ (iω ∗ u∗(1) ) − grady∗ p∗(0) = 0 divy∗ u∗(1) + divx∗ u∗(0) = 0

within Ω∗s within Ω∗f

within Ω∗f

a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n = (−p∗(0) I + 2η ∗ iω ∗ (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n iω ∗ u∗(1) − v∗(1) = 0

over Γ∗

Making use of the linearity, the solution can be written: ∗(1)

ui

∗(1)

= χ∗lm ex∗ lm (u∗(0) ) + ui i

p∗(0) = ζ ∗lm ex∗ lm (u∗(0) )

(x∗ , ω ∗ )

over Γ∗

380

Homogenization of Coupled Phenomena

The vector u∗(1) (x∗ , ω ∗ ) is arbitrary and independent of y∗ . The tensors ξ∗ and χ have complex values and depend on the variables y∗ and ω ∗ . We note that no arbitrary additive function of x∗ appears in the expression for p∗(0) . This is a result of the continuity of normal stress over Γ∗ written to order ε0 (see above). ∗

Finally, the next order leads to the compatibility condition which, as usual, plays the role of macroscopic description. In physical variables, and neglecting terms of relative order O(ε), this takes the same form as in the previous section, so that to ﬁrst order of approximation: ¯ divX σ T = −(ρs + φρf )ω 2 u + O(ε)

(12.25)

¯ σT = c++ : eX (u) + O(ε)

(12.26)

c++ ijkh =

1 |Ω|

+

1 |Ω|

(aijkh + aijlm eXlm (χkh )) dΩ Ωs

(−ζij Ikh + 2iωηeyij (χkh ) + 2iωηIij Ikh ) dΩ Ωf

But here, c++ takes complex values and depends on ω. For an arbitrary transient motion, the constitutive law becomes: T ++ σij = TF−1 ) ∗ eXkh (us ) ijkh (c ++ where TF−1 ) is the inverse Fourier transform of c++ , and here ∗ indicates ijkh (c the convolution product. The equivalent macroscopic behavior is that of a linear viscoelastic solid.

12.7. Choice of macroscopic behavior associated with a given material and disturbance In order to investigate the relationship between the three dynamic macroscopic models, we will proceed as in section 11.2.6 by introducing the dimensionless number R, which is independent of the disturbance applied to the porous medium. The number PL can be estimated here as: PL =

2 ρsc L2c −2 ρsc lc = ε = O(1) ac t2c ac t2c

Substituting the value of tc obtained form this last expression into QL we ﬁnd: t2c =

ρsc lc2 −2 ε , ac

QL =

ac tc = Rε−1 , ηc

R=

lc √ ac ρsc ηc

Dynamics of Saturated Deformable Porous Media

381

The domains of validity of the three macroscopic models obtained earlier can be expressed as a function of R. They can be divided into the following categories: – R O(ε−2 ): the saturated porous medium behaves like an empty porous matrix. The characteristic time of disturbance is very large, a time at which the ﬂuid is no longer being disturbed. Conditions (12.81 ) and (12.82 ) are relaxed; – R = O(ε−1 ): Biot’s diphasic model; – R = O(1): Gassman’s monophasic elastic model; – R = O(ε): monophasic viscoelastic model; – R O(ε2 ): the separation of scales is poor, and there is no macroscopically equivalent model. From this we ﬁnd that porous media can be divided into two distinct categories based on their macroscopic dynamic behavior: – media A such that R 1, for which only estimates of O(ε−p ), p > 1, O(ε−1 ) and O(1) are possible. Such media can only display the behavior of an empty lattice, the Biot diphasic model or the Gassman monophasic elastic model. – media B such that R 1 for which only estimates of O(1) and O(ε) are possible. Such media can only display the behavior of the Gassman monophasic elastic model or the monophasic viscoelastic model. The different behaviors are shown in Figure 12.1. In Figure 12.2 we have gathered Nonhomogenizable (a)

(b)

2

Monophasic viscoelastic

Gassman 1

1

1

Material B

Biot

2

1

Empty pores

3

2

QL

R

Material A

Figure 12.1. (a) Classiﬁcation of the dynamic behavior of porous saturated media as a function of the dimensionless number QL ; (b) as a function of R

together the different quasi-static models, PL O(ε), and dynamic models, PL = O(1), as well as the values of PL and R corresponding to non-homogenizable porous media.

382

Homogenization of Coupled Phenomena

PL 1 NH

NH

NH

NH

3

5/2

2

3/2

NH

NH

NH

NH

NH

1/2

VE

NH 1 G

NH

1/2

NH

NH

NH

1

3/2

2

B

R

EP

NH

G

VE

NH

EP

EP

EP

EP

EP

EP

EP

EP

B

2 NH

B

G

VE

VE

G

B

Material B

3

EP

Material A

Figure 12.2. Quasi-static and dynamic models of saturated porous media. B: Biot model, G: Gassman model, NH: non-homogenizable, EP: empty pores, VE: monophasic viscoelastic model

12.7.1. Effects of viscosity Consider a given porous medium saturated by a ﬂuid of variable viscosity. We obtain the following classiﬁcation: √ – η η0 = lc ρsc ac . The saturated porous medium is of type A. Such media can only display Biot diphasic behavior or Gassman monophasic elastic behavior. √ – η η0 = lc ρsc ac . The saturated porous medium is of type B. Such media can only display Gassman monophasic elastic or monophasic viscoelastic behavior. We will now show that there is a continuous transition, both from the diphasic model to the elastic model, and also from the viscoelastic model to the elastic model. 12.7.1.1. Transition from diphasic behavior to elastic behavior We will assume that the viscosity η = η1 = O(εη0 ), and we will increase this towards η0 . For η = η1 the behavior is diphasic. The generalized Darcy’s law is: uf − φus = −

Λ (gradX p − ω 2 ρf us ) iω

As η increases, the magnitude of inertial terms in the Navier-Stokes equation decreases relative to the viscous terms, and the norm of the conductivity Λ tends

Dynamics of Saturated Deformable Porous Media

383

towards that of the quasi-static permeability: Λ→

K l2 = O( c ) η η0

The tensor Λ is multiplied by a factor of O(ε) and: ufi − ηusi → O(ε) To ﬁrst order it follows that: uf − φus = 0 Thus, by substituting this value into (12.17), we recover the conservation of volume obtained in section 12.5 for the macroscopically monophasic elastic medium: 0 = iωα : eX (us ) + β iωp The diphasic behavior degenerates gradually into monophasic elastic behavior. 12.7.1.2. Transition from viscoelastic behavior to elastic behavior Viscosity is now η = η2 = O(ε−1 η0 ), and we will decrease it towards η0 . For η = η2 the behavior is monophasic viscoelastic. The problem giving u∗(1) and p∗(0) can be written (see section 12.6): divy∗ (a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))) = 0 η2∗ Δy∗ (iω0∗ u∗(1) ) − grady∗ p∗(0) = 0 divy∗ u∗(1) + divx∗ u∗(0) = 0

within Ω∗s within Ω∗f

(12.27)

within Ω∗f

a∗ : (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n = (−p∗(0) I + 2η2∗ iω0∗ (ey∗ (u∗(1) ) + ex∗ (u∗(0) ))n u˙ ∗(1) − v∗(1) = 0

over Γ∗

Making use of the linearity, the solution can be written: ∗(1)

ui

∗(1)

= χ∗lm ex∗ lm (u∗(0) ) + ui i

p∗(0) = ζ ∗lm ex∗ lm (u∗(0) )

(x∗ , ω0∗ )

over Γ∗ (12.28)

384

Homogenization of Coupled Phenomena

If the viscosity decreases from η2 to η0 , the terms in (12.27) and (12.28) associated with η2 become O(ε), and can be relegated to the next order of approximation. The above system then becomes identical to the corresponding system from the elastic case (see section 12.5). The rest of the homogenization calculation is then identical to that carried out in the elastic case. The monophasic viscoelastic behavior gradually degenerates into monophasic elastic behavior. 12.7.2. Effect of rigidity of the porous skeleton Now we consider that only the rigidity of the porous skeleton is variable. We obtain the following classiﬁcation: – ac a0 = ηc2 /lc2 ρsc . The saturated porous medium is of type A. Such media can only display Biot diphasic behavior or Gassman monophasic elastic behavior. – ac a0 = ηc2 /lc2 ρsc . The saturated porous medium is of type B. Such media can only display Gassman monophasic elastic or monophasic viscoelastic behavior. We can show, as earlier, that there is a continuous transition both from the diphasic model to the elastic model, and also from the viscoelastic model to the elastic model. 12.7.3. Effect of frequency We will now investigate how the model of the porous medium is affected by the frequency. The characteristics of the porous medium are assumed to be constant. In contrast to the earlier discussions, ε is no longer constant. We will examine the case of low-dispersion waves separately, the P1 and S waves (which exist for all three behaviors), and then the case of the dispersive P2 wave (which only exists for diphasic behavior). 12.7.3.1. Low-dispersion P1 and S waves The characteristic macroscopic length can be linked to the wavelength λ through: Lc =

λc 2π

For low-dispersion waves, velocity c is independent of frequency. We have: Lc =

cc ωc

From this we can deduce the value of the separation of scales parameter: ε=

l c ωc cc

Dynamics of Saturated Deformable Porous Media

385

This expression allows the characterization of evolution of behaviors of materials A and B as a function of frequency: – Material A. As the frequency decreases, ε−1 increases. In the limit, R = O(ε−1 ) becomes O(1). The diphasic behavior tends towards Gassman monophasic elastic behavior at low frequency. – Material B. As the frequency decreases, ε decreases. In the limit, R = O(ε) becomes O(1). The monophasic viscoelastic behavior tends towards Gassman monophasic elastic behavior at low frequency. This is indeed what is observed in practice: the relative motion of ﬂuid with respect to the solid is not excited at low frequencies by the P1 and S waves. 12.7.3.2. Dispersive P2 wave This wave only exists when the behavior is diphasic – in material A. As seen in section 11.2.6, the diphasic behavior of material A is only retained at low frequencies for PL = O(ε), and so this is the only possible behavior. As ω decreases, with PL changing from O(1) to O(ε), the P2 wave degenerates into a diffusive wave as described in the previous chapter. In summary, the behavior becomes diphasic at low frequencies, even if the P1 and S waves have an elastic behavior. The diphasic character disappears for smaller frequencies such that PL O(ε2 ). 12.7.4. Effect of pore size Consider a given saturated porous medium subject to a disturbance at constant frequency. We will modify pore size homothetically. The mechanical properties of the medium are the same, and Lc remains unchanged. We obtain the following classiﬁcation: √ – lc l0 = ηc / ρsc ac . The saturated porous medium is of type A. Such media can only display Biot diphasic behavior or Gassman monophasic elastic behavior. If we consider a material with lc l0 displaying a diphasic behavior, R = O(ε−1 ), and decrease lc . The number R decreases and ε−1 grows. The number R becomes O(1) and the model becomes the Gassman model. √ – lc l0 = ηc / ρsc ac . The saturated porous medium is of type A. Such media can only display Gassman monophasic elastic or monophasic viscoelastic behavior. If we consider a material with lc l0 displaying a monophasic viscoelastic behavior, R = O(ε), and modify lc the number R, and also ε, are modiﬁed by the same proportions. The scale of R remains unchanged, as does the model. 12.7.5. Application example: bituminous concretes The above results can be translated to media saturated with a viscoelastic ﬂuid [BOU 90], such as bituminous concretes consisting of grains of characteristic

386

Homogenization of Coupled Phenomena

size l ≈ 1cm surrounded by bitumen with a volume fraction of the order of 15% (Figure 12.3).

Figure 12.3. Microstructure of a bituminous concrete

Figure 12.4. Domains of validity of the different models of bituminous concretes, as a function of frequency and temperature

Dynamics of Saturated Deformable Porous Media

387

The bitumen, in the harmonic regime and for small deformations, satisﬁes the time-temperature equivalence principle (Williams-Landel-Ferry law), and its linear viscoelastic modulus M (ω, T ) evolves over several factors of 10 as the pressure varies between -15˚C and 150˚C. Experimentally, this material exhibits three types of behavior. At low temperature and high frequency, the behavior is elastic, with a high elastic modulus. At high temperature and low frequency, the behavior is elastic with a low elastic modulus. In the intermediate temperature and frequency range, the behavior is viscoelastic. These behaviors match those that we established by homogenization. Following the classiﬁcation set out earlier, the type of behavior at ﬁxed temperature and frequency is determined by the value of the exponent x in: |M (ω, T )| = εx |C|

with

ε=

ωl cs

where C is the macroscopic modulus of elasticity of the skeleton and cs is the velocity of the shear waves. For x < 1, the behavior is viscoelastic (and the time-temperature equivalence law for bitumen also applies to bituminous concretes), for x = 1 the behavior is elastic, and for x > 1 the behavior is biphasic (Figure 12.4).

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Chapter 13

Estimates and Bounds for Effective Poroelastic Coefﬁcients

13.1. Introduction This chapter is dedicated to determining the effective poroelastic parameters (c, α and β) which appear in the Biot [BIO 41] model, through the use of the results of periodic homogenization. These results are brieﬂy discussed in section 13.2. Section 13.3 involves the determination of the effective parameters of a cohesive periodic granular medium consisting of a cubic arrangement of spheres all with the same diameter. The effect of solid volume fraction and Poisson ratio of the constitutive material on the effective parameters is demonstrated. These numerical results are then compared ﬁrst with the classical bounds (Voigt and Reuss, Hashin and Strickman), second with self-consistent estimates (section 13.4), and third with experimental results (section 13.5). These comparisons will allow us to draw a connection between these models and the properties of real materials. 13.2. Recap of the results of periodic homogenization On the macroscopic scale, the quasi-static behavior of a saturated porous elastic medium is described by the Biot model [BIO 41] (11.30-11.33) which takes the form: divX σ T = 0,

σ T = c : eX (us ) − α p

(13.1)

divX (v − φu˙ s ) = −α : e˙ X (us ) − β p˙

(13.2)

1 v − φu˙ s = − K gradX p η

(13.3)

390

Homogenization of Coupled Phenomena

where c, α and β are the effective poroelastic parameters which depend on the material microstructure and elastic properties of the constitutive material, K is the permeability tensor, which depends purely on the microstructure of the porous medium, and: 1 . = Ω

dΩ Ω

When the constitutive material is heterogenous on the scale of the period Ω, these effective parameters can be written, in dimensionless form: c∗ijkh = (1 − φ)a∗ijkh + a∗ijlm ey∗ lm (ξkh∗ )

(13.4)

∗ ij∗ αij = φ δij − ξp,p = φ δij + a∗ijkh ey∗ kh (η ∗ )

(13.5)

β ∗ = ey∗ ii (η ∗ )

(13.6)

The tensor of the effective moduli of the porous matrix c∗ is symmetrical and positive, as is the Biot tensor α∗ , and the coefﬁcient β ∗ > 0 (see Chapter 11, section 11.2.3). ξ∗ (y∗ ) is a third-order periodic tensor, with a mean of zero, ξ∗ = 0, which is the solution to the following boundary-value problem: divy∗ (a∗ : ey∗ (ξ ∗ E∗ )) = 0 within Ω∗s

(13.7)

(a∗ : E∗ + a∗ : ey∗ (ξ∗ E∗ ))n = 0 over Γ∗

(13.8)

where E∗ = e∗x (u∗(0) ) corresponds to an imposed macroscopic strain. ζikh∗ represents the ith component of microscopic displacement u∗(1) of the solid skeleton when E∗ = ek ⊗ eh and p∗(0) = 0. The vector η ∗ (y∗ ) is periodic, and is the solution to: divy∗ (a∗ : ey∗ (η ∗ )) = 0 within Ω∗s (a∗ : ey∗ (η ∗ ) + I)n = 0 over Γ∗

(13.9) (13.10)

with η ∗ = 0 and where I is the identity tensor. ηi∗ represents the ith component of the microscopic displacement u∗(1) of the skeleton when a pressure p∗(0) = 1 is applied and E∗ = 0. We note that in the dynamic case the poroelastic parameters c, α and β are unchanged and, as a result, the above deﬁnitions remain valid. Only permeability K is altered, and this depends on pulsation ω. This permeability is identical to that

Effective Poroelastic Coefﬁcients

391

determined in Chapters 9 and 10 when the porous medium is rigid. In what follows, we will therefore only be interested in the poroelastic parameters c, α and β. 13.3. Periodic granular medium Here we will restrict ourselves to the numerical estimation of the effective poroelastic properties of a periodic granular medium consisting of a simple-cubic arrangement of spheres. Numerical estimates can also be found for the poroelastic properties of granular, ﬁssured or cellular materials in [CHA 94; POU 96a; POU 96b; LYD 00; ARN 02; ROB 02; MOU 08]. 13.3.1. Microstructure and material Here we consider a cohesive saturated granular medium consisting of a periodic cubic arrangement of spheres which all have the same diameter. Figure 13.1 shows the solid (connected) and ﬂuid parts of the period of the medium, considered for two values of solid volume fraction: c = 0.7 and 0.9. The period dimension, denoted l, l

e3

2a (a)

solid

e3

e2 e1

fluid

(b)

solid

e2 e1

fluid

Figure 13.1. Characteristic dimensions of the period of a saturated cohesive granular material consisting of a simple cubic arrangement of identical spheres (a) solid volume fraction c = 0.7 (b) solid volume fraction c = 0.9

is assumed to be constant. The radius of the spheres is R. The solid volume fraction of the porous medium, c = Ωs /Ω, varies between cmin = π/6 0.523 and 1. This particular granular medium can be considered to be a connected assembly of truncated spheres, each of which is linked to six others by identical junctions. Finally, we assume that the elastic properties of the constitutive material on the microscopic scale are heterogenous and isotropic: aijkl = λδij δkl + μ(δik δjl + δil δjk )

(13.11)

where λ and μ are the Lamé coefﬁcients at a given temperature, with λ = Eν/((1 + ν)(1 − 2ν)) and μ = E/(2(1 + ν)), where E and ν are the Young’s modulus and the Poisson’s ratio. The coefﬁcient of compressibility is K = λ + (2/3)μ.

392

Homogenization of Coupled Phenomena

13.3.2. Effective elastic tensor c 13.3.2.1. Methodology The microstructure we are considering displays cubic symmetry. As a result, the elastic tensor of the effective moduli can be written in the following form (see for example [NUN 84]): cijkl = λ(1 + A)δij δkl + μ(1 + B)(δik δjl + δil δjk ) + 2μ(C − B)δijkl (13.12) where δijkl = 1 if i = j = k = l, or δijkl = 0 otherwise. The functions A, B and C depend on the microstructure and on Poisson’s ratio ν. The material is heterogenous on the period scale, and as a result functions A, B and C are independent of Young’s modulus E. Equation (13.12) gives: c1111 = c2222 = c3333 = λ(1 + A) + 2μ(1 + C) = λp + 2μ2p

(13.13)

c1122 = c1133 = c2233 = λ(1 + A) = λp

(13.14)

c1212 = c1313 = c2323 = 2μ(1 + B) = 2μ1p

(13.15)

c1122 = c1133 = c2233 = c2211 = c3311 = c3322 = 0

(13.16)

c1213 = c1223 = c1323 = c1312 = c2312 = c2313 = 0

(13.17)

where λp = λ(1+A), μ2p = μ(1+C) and μ1p = μ(1+B) are the three elastic moduli of the porous medium. On the macroscopic scale, the porous medium is orthotropic, with cubic symmetry. It exhibits two shear coefﬁcients: μ1p and μ2p characterize the shears in the planes (ei , ek ) and (ei + ej , ek ), respectively. In order to determine these coefﬁcients, problem (13.7-13.8) must be solved over the period, successively assuming E∗ = ei ⊗ ei and E∗ = ei ⊗ ej . These boundary-value problems have been solved using ﬁnite element analysis, using quadratic Lagrange elements [COM 08]. Figure 13.1 shows the lattices used to carry out the simulations. Given the symmetries of the period, the domain of the calculation can be reduced to 1/8 of the period. The solutions to problem (13.7-13.8) for E∗ = ei ⊗ ei and E∗ = ei ⊗ ej are ζ ii∗ and ζ ij∗ respectively. As a result, equations (13.4) and (13.11) lead to the following (without summation over the repeated indices): c∗iiii = (λ∗ +2μ∗ )(1−φ+ey∗ ii (ζ ii∗ ))+λ(ey∗ jj (ζ ii∗ )+ey∗ kk (ζ ii∗ )) ∗

(13.18)

c∗iijj = (λ∗ + 2μ∗ )ey∗ jj (ζ ii∗ ) + λ(ey∗ ii (ζ ii ) + ey∗ kk (ζ ii ))

(13.19)

c∗ijij = 2μ∗ (1 − φ + ey∗ ij (ζ ij ))

(13.20)

Effective Poroelastic Coefﬁcients

393

0

-0.1

-0.2

-0.3

-0.4

f g h

-0.5

-0.6 0.5

0.6

0.7 0.8 solid volume fraction (c)

0.9

1

Figure 13.2. Evolution of the functions f , g and h as a function of the solid volume fraction c when ν = 0.3

Finally, setting (without summation over the repeated indices): f = ey∗ ii (ζ ii∗ ), g = ey∗ jj (ζ ii∗ ) = ey∗ kk (ζ ii∗ ), h = ey∗ ij (ζ ij∗ ) we can write equations (13.13)-(13.20) as: λp /λ = 1 + A = 1 − φ + f + 2g(1 + (μ/λ))

(13.21)

μ1p /μ = 1 + B = 1 − φ + h

(13.22)

μ2p /μ = 1 + C = 1 − φ + f − g

(13.23)

Functions f , g and h depend only on the microstructure of the porous medium and on Poisson’s ratio ν, since μ/λ = 2ν/(1 − 2ν). Figure 13.2 shows the typical evolution of the functions f , g and h as a function of solid volume fraction c when ν = 0.3. When c = 1, we have: f =g=h=0

so that

λp /λ = μ1p /μ = μ2p /μ = 1

When c = cmin , we have: f = h = −cmin , g = 0,

so that

λp /λ = μ1p /μ = μ2p /μ = 0

394

Homogenization of Coupled Phenomena

13.3.2.2. Compressibility and shear moduli Figures 13.3 and 13.4 show the evolution of the moduli λp /λ, μ1p /μ and μ2p /μ as a function of solid volume fraction, for different values of the Poisson’s ratio ν between 0.1 and 0.49. The modulus of compressibility, deﬁned by: 2 Kp = λp + μ2p 3 1 Q= 0.1 Q= 0.2 Q= 0.3 Q= 0.4 Q 0.49

0.8

Op/O

0.6

0.4

0.2

(a)

0 0.5

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

0.9

1

1 Q= 0.1 Q= 0.2 Q= 0.3 Q= 0.4 Q= 0.49

0.8

Kp / K

0.6

0.4

0.2

0 0.5

(b)

0.6

0.7 0.8 Solid volume fraction (c)

Figure 13.3. Evolution of the poroelastic parameters λp /λ, and Kp /K as a function of the solid volume fraction, and for different values of the Poisson’s ratio ν

Effective Poroelastic Coefﬁcients

395

1 Q= 0.1 Q= 0.2 Q= 0.3 Q= 0.4 Q= 0.49

0.8

P1p/P

0.6

0.4

0.2

(a)

0 0.5

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

0.9

1

1 Q = 0.1 Q= 0.2 Q= 0.3 Q= 0.4 Q= 0.49

0.8

P2p/P

0.6

0.4

0.2

(b)

0 0.5

0.6

0.7 0.8 Solid volume fraction (c)

Figure 13.4. Evolution of the poroelastic parameters μ1p /μ et μ2p /μ as a function of the solid volume fraction, and for different values of the Poisson’s ratio ν

is also shown in Figure 13.3 (b) Figure 13.3 shows that λp /λ, depends strongly on solid volume fraction and on Poisson’s ratio ν, as does Kp /K. On the other hand, Figure 13.4 emphasizes the fact that the two shear moduli of the porous matrix are very insensitive to variations in ν. The variations of μ1p /μ and μ2p /μ are mostly non-linear for solid volume fraction values close to cmin , i.e. when the junctions between the spheres are small. These non-linearities are similar to those observed experimentally [GRE 90].

396

Homogenization of Coupled Phenomena 1.8 Q= 0.1 Q= 0.2 Q= 0.3 Q= 0.4 Q= 0.49

1.6

1+A / 1+B P 2p/P 1p

1.4

1.2

1

0.8

0.6 0.5 (a)

0.6

0.7 0.8 solid volume fraction (c)

0.9

1

Figure 13.5. Evolution of the ratio μ2p /μ1p as a function of solid volume fraction, for different values of Poisson’s ratio ν

13.3.2.3. Degree of anisotropy Figure 13.5 shows the evolution of the ratio μ2p /μ1p as a function of solid volume fraction for different values of ν. This ratio μ2p /μ1p characterizes the anisotropy of the tensor of the effective moduli c. For a given value of ν, the ratio μ2p /μ1p increases as the solid volume fraction decreases. It has a maximum of the order of 1.6 for low solid volume fraction values when ν = 0.49. This ratio decreases as the contact number between the particles increases. It has a maximum of the order of 1.1 for a body centered cubic lattice of spheres (8 contacts) [MOU 08]. 13.3.2.4. Young’s modulus and Poisson’s ratio The anisotropy of the microstructure means we cannot identify a unique Young’s modulus or Poisson’s ratio. In what follows, we will deﬁne the Young’s modulus Ep and the Poisson’s ratio νp of a granular cohesive material, relative to the principal axes of the microstructure, by: Ep =

μ2p (3λp + 2μ2p ) , μ2p + λp

νp =

λp 2(μ2p + λp )

(13.24)

When c = 1, we have λp /λ = μ1p /μ = μ2p /μ = 1, and as a result, with equation (13.24), it follows that Ep /E = νp /ν = 1. When c = cmin , λp /λ = μ1p /μ = μ2p /μ = 0, and with equation (13.24) we have Ep /E = 0. Poisson’s ratio νp is undetermined.

Effective Poroelastic Coefﬁcients

397

1 Q= 0.1 Q= 0.2 Q= 0.3 Q= 0.4 Q= 0.49

0.8

Ep / E

0.6

0.4

0.2

(a)

0 0.5

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

0.9

1

0.5 Q= 0.1 Q= 0.2 Q= 0.3 Q= 0.4 Q= 0.49

0.4

Qp

0.3

0.2

0.1

(b)

0 0.5

0.6

0.7 0.8 Solid volume fraction (c)

Figure 13.6. Evolution of the poroelastic parameters Ep /E and νp as a function of the solid volume fraction, and for different values of the Poisson’s ratio ν

Figures 13.6 (a) and (b) show the evolution of Ep /E and of Poisson’s ratio νp for different values of ν as a function of solid volume fraction. It can be observed that: (i) Young’s modulus Ep depends strongly on Poisson’s ratio ν. As with μ2p /μ, the ratio Ep /E varies non-linearly for solid volume fraction values close to cmin : and (ii) νp is not necessarily an increasing function of solid volume fraction. When ν = 0.1 and c < 1, Figure 13.6 (b) shows that the effective Poisson’s ratio νp for the microstructure

398

Homogenization of Coupled Phenomena

under consideration can be greater than that of the constitutive material. As we have already emphasized for c = cmin , equation (13.24) does not allow an estimate of Poisson’s ratio νp . Walton [WAL 87] showed that Poisson’s ratio νp of an aggregate of grains whose compactness is close to c = cmin , i.e. with small inter-grain junctions, is νp (cmin ) = ν/(10 − 6ν). Typically, this Poisson’s ratio νp (cmin ) is equal to 0.01 and 0.069 for ν = 0.1 and ν = 0.49 respectively. These values are consistent with the numerical results shown in Figure 13.6 (b).

13.3.3. Biot tensor The symmetries of the microstructure we are considering imply that the Biot tensor is isotropic: α = αI. As a consequence, equation (13.5) becomes: ij = φ − (f + 2g) α = φ − ζp,p

(13.25)

Using equations (13.21), (13.22) and (13.24), equation (13.25) can then be written in the traditional form (Chapter 11, section 11.2.3.2): α=1−

Kp 3λp + 2μ2p =1− 3λ + 2μ K

(13.26)

Then, equations (13.5), (13.9), (13.25) and (13.26) lead to the following expression for β: β=

−(f + 2g) 1 (α − φ) = = K K K

c−

Kp K

(13.27)

When the material is incompressible on the microscopic scale, K −→ ∞ and ν −→ 0.5, equations (13.26) and (13.27) lead to the well-known results α = 1 and β = 0. Figure 13.7 shows the evolution of α as a function of Poisson’s ratio ν for different values of solid volume fraction. It can be seen that the coefﬁcient α: (i) grows as ν increases, and decreases as solid volume fraction increases, and (ii) is very sensitive to variations in Poisson’s ratio once ν > 0.4.

13.4. Inﬂuence of microstructure: bounds and self-consistent estimates In this section, the numerical results obtained for the simple cubic microstructure will be compared ﬁrst with those obtained for a body centered cubic microstructure and second with the standard Voigt-Reuss, Hashin and Shtrikman bounds and selfconsistent estimates with simple inclusions or for a bi-composite (see Chapter 1, sections 1.3 and 1.4).

Effective Poroelastic Coefﬁcients

399

1

0.8

D

0.6

0.4 c=0.9 c=0.8 c=0.7 c=0.6 c=0.55

0.2

0 0

0.1

0.2 0.3 Poisson coefficient Q

0.4

0.5

Figure 13.7. Evolution of α as a function of Poisson’s ratio ν for different values of solid volume fraction

13.4.1. Voigt and Reuss bounds In the case of a porous medium, the lower Voigt bound [VOI 87] (V) and upper Reuss bound [REU 29] (R), can be reduced to: μR = 0 < μp c μ = μV ,

K R = 0 Kp c K = K V

(13.28)

As a result, using equations (13.28) and (13.26)-(13.27), the lower and upper bounds for Young’s modulus Ep , Poisson’s ratio νp , and Biot coefﬁcient α can be written: E R = 0 Ep c E = E V ,

ν R = −1 νp ν V = 1/2

αR = 1 − c α 1 = αV

(13.29) (13.30)

These bounds do not take into account the macroscopic anisotropy induced by the arrangement of the spherical grains. 13.4.2. Hashin and Shtrikman bounds Using a variational method, Hashin and Shtrikman [HAS 63] proposed improved bounds for the compressibility and shear moduli of an isotropic macroscopically elastic composite. For the case of a porous medium, these bounds become: K HS− = 0 Kp

4c μK = K HS+ 4μ + 3(1 − c)K

(13.31)

400

Homogenization of Coupled Phenomena

μHS− = 0 μp

c μ(9K + μ) = μHS+ (9K + μ) + 6(1 − c)(K + 2μ)

(13.32)

From this we can deduce the lower and upper bounds of Young’s modulus Ep , Poisson’s ratio νp , the Biot coefﬁcient α: E HS− = 0 Ep

9K HS+ μHS+ = E HS+ 3K HS+ + μHS+

ν HS− = −1 νp 1/2 = ν HS+ αHS− =

(1 − c)(4μ + 3K) α 1 = αHS+ 4μ + 3(1 − c)K

(13.33) (13.34) (13.35)

Although these bounds are valid for a macroscopically isotropic material, we will compare them in rough terms to the numerical results obtained for the cubic arrangement of truncated spheres, whose macroscopic behavior displays cubic symmetry. 13.4.3. Self-consistent estimates The self-consistent estimation scheme [HIL 65; BUD 65], involves considering each heterogenity, i.e. an inclusion with a given shape and elastic properties, to be embedded in an inﬁnite elastic matrix whose properties are the effective properties of the material being considered. Solving this problem leads to an implicit equation which is a function of the effective properties we are looking for (see Chapter 1, section 1.4). If we assume that the inclusions are spherical and that the porous medium is macroscopically heterogenous and isotropic, the results of Hill [HIL 65] show that the effective modulus of compressibility K SCE and effective shear modulus μSCE of porous medium satisfy the following coupled equations: 1 1−c c + SCE = SCE K SCE − K K K + 4/3μSCE

(13.36)

1−c c 3 K SCE + = − μSCE − μ μSCE 5μSCE μSCE (K SCE + 4/3μSCE )

(13.37)

The solution of equations (13.36)-(13.37) gives: μSCE = K SCE =

√ 3 −K(2 + c) − μ(7/3 − 5c) + Δ 16 4μSCE K c 3(1 − c)K + 4μSCE

(13.38)

(13.39)

Effective Poroelastic Coefﬁcients

401

where Δ = (K(2+c)+μ(7/3−5c))2 −32μK(1−2c). We note that K SCE and μSCE are only real and positive if c > 0.5. Using equations (13.38)-(13.39), the Young’s modulus E SCE , Poisson’s ratio ν SCE and Biot coefﬁcient αSCE can all be written: E SCE =

9K SCE μSCE , 3K SCE + μSCE

αSCE =

(1 − c)(3K + 4μSCE ) 3(1 − c)K + 4μSCE

ν SCE =

3K SCE − 2μSCE 6K SCE + 2μSCE (13.40)

Christensen and Lo [CHR 79] put forward a self-consistent method where the medium is viewed as an assembly of composite spheres. In the case of porous media, the spheres are hollowed out and the pores are not connected with each other. Their results show that in this case the effective modulus of compressibility of the porous medium is K CH = K HS+ , and that the shear modulus μCH satisﬁes a quadratic equation of the form: CH 2 CH μ μ + a3 = 0 a1 + 2a2 (13.41) μ μ where a1 , a2 and a3 are three parameters which depend on the geometry and on ν. We do not reproduce the expressions for a1 , a2 and a3 here. 13.4.4. Comparison: numerical results, bounds and self-consistent estimates Figures 13.8 (a,b,c and d) show the evolution of λp /λ, μ1p /μ, Kp /K, Ep /E and νp as a function of solid volume fraction for ν = 0.4 for simple cubic (SC) and body centered cubic (BCC) arrangements of spheres, as well as the bounds of Voigt (V), Reuss (R), Hashin and Shtrikman (HS) and the two-phase SCE and threephase Christensen and Lo (CH) self-consistent estimates. Figure 13.9 (e) shows the evolution of α as a function of ν and for c = 0.9. It can be seen that: – All the numerical results and self-consistent estimates respect the various bounds. – For a given solid volume fraction, the porous medium consisting of a BCC arrangement of spheres is softer than the porous medium consisting of a SC arrangement. This is directly linked to the dimensions of the contacts between the spheres, which are smaller in the BCC microstructure. The numerical results for a BCC arrangement display a “jump” at a solid volume fraction of the order of 0.93. This “jump” corresponds to an increase, from 8 to 12, in the contact number between the spheres. – The self-consistent estimate and the numerical results for the SCE arrangement are very close to each other over a wide range of solid volume fraction values.

402

Homogenization of Coupled Phenomena

1

1

(SC) (BCC)

(SC) (BCC)

0.8

0.8

0.6

0.6 μ1p / μ

Kp / K

V

0.4

0.4

V HS+ CH

HS+ 0.2

0.2 SCE R, HS-

SCE 0 0.5 (a)

0.6

1

0.7 0.8 Solid volume fraction (c)

0.9

R, HS0 0.5 (b)

1

0.6

0.5

(SC) (BCC)

0.8

0.7 0.8 Solid volume fraction (c) (SC) (BCC)

0.9

1

V, HS+

0.4

0.6

0.3

Qp

Ep / E

V

HS+

0.4

SCE 0.2

0.2

0.1 Q/(10-6Q) [Walton, 1987] R, HS-

SCE 0 0.5 (c)

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

0 0.5

(d)

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

Figure 13.8. Evolution of λp /λ, μ1p /μ, Kp /K, Ep /E and νp as a function of solid volume fraction for ν = 0.4. Comparison of numerical results for simple cubic (SC) and body centered cubic (BCC) arrangements of spheres, along with the bounds of Voigt (V), Reuss (R), Hashin and Shtrikman (HS) and the self-consistent estimates (SCE) and Christensen and Lo (CH)

– The self-consistent estimate for bi-composite inclusions CH lies further from the numerical results. This estimate, which corresponds to a microstructure consisting of spherical voids, does not allow the connectivity of both phases to be expressed simultaneously. This estimate would be more suited to describing the behavior of a type of foam material, where the pores are not necessarily interconnected. – Finally, Figure 13.8 (d) shows that the value of Poisson’s ratio given by Walton [WAL 87] for c = cmin is consistent with the numerical results for both arrangements. These observations are valid whatever the value of ν considered.

Effective Poroelastic Coefﬁcients

403

1 (SC) (BCC)

V, HS+

0.8

D

0.6

0.4

0.2

SCE

HS-

R 0 0

0.1

0.2 0.3 Poisson coefficient Q

0.4

0.5

Figure 13.9. Evolution of α as a function of ν for c = 0.9. Comparison of numerical results for simple cubic (SC) and body centered cubic (BCC) arrangements of spheres, along with the bounds of Voigt (V), Reuss (R), Hashin and Shtrikman (HS) and the self-consistent estimates (SCE)

13.5. Comparison with experimental data We will ﬁrst compare the numerical results for both the SC and BCC microstructures, as well as the bounds and self-consistent estimates, with the experimental results of Fate [FAT 75] and Green et al. [GRE 90]. Using an ultrasonic technique, Fate [FAT 75] measured Young’s modulus Ep and the shear modulus μp of industrial samples of polycrystalline silicone nitride obtained by hot compaction. These measurements were carried out on samples whose solid volume fraction varied between 0.745 < c < 1. Green et al. [GRE 90], using an acoustic technique separately measured, the Young’s modulus Ep and shear modulus μp of a sample of partially sintered alumina. The solid volume fraction of samples varied between 0.58 < c < 1. Based on measurements of Ep and of μp , and assuming the material to be isotropic, we can estimate the modulus of compressibility Kp and the Poisson’s ratio νp . The intrinsic elastic parameters (K, μ, E and ν) of both materials are listed in Table 13.1. Figures 13.10 (a, b, c, and e) show the evolution of Kp /K, Ep /E, μp /μ and νp measured by Fate [FAT 75] and Green et al. [GRE 90]. The Poisson’s ratio of both materials is of the order of 0.2. The bounds, self-consistent estimates and numerical values obtained for both the SC and BCC arrangements are also shown in these ﬁgures. It can be seen that the numerical results for a SC arrangement are consistent with the

404

Homogenization of Coupled Phenomena

Material K (GPa) μ (GPa) E(GPa) ν Polycristalline silicone nitride [FAT 75] 289 118.2 173.6 0.22 Sintered alumina [GRE 90] 221.7 167 399 0.2 Rock [ARN 02] 37 44 94.5 0.07 Table 13.1. Elastic properties of the materials

1

1 Fate (1975) Green (1990) (SC) (BCC)

0.8

0.6

0.8

0.6 Ep / E

Kp / K

V

HS+

0.4

Fate (1975) Green (1990) (SC) (BCC)

0.4

0.2

0.2

V

HS+

SCE

SCE R, HS-

R, HS0 0.5 (a)

0.6

1

0.7 0.8 Solid volume fraction (c)

0.9

0 0.5 (b)

1

0.5

Fate (1975) Green (1990) (SC) (BCC)

0.8

0.6

0.9

1

V, HS+

0.3

Qp

μp / μ

0.7 0.8 Solid volume fraction (c) Fate (1975) Green (1990) (SC) (BCC)

0.4

V

HS+

0.4

0.6

0.2

SCE

CH 0.2

0.1

SCE R, HS-

0 0.5

(c)

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

0 0.5

(d)

0.6

0.7 0.8 Solid volume fraction (c)

0.9

Figure 13.10. Evolution of Kp /K, Ep /E, μp /μ and of Poisson’s ratio νp as a function of solid volume fraction. Comparison of the experimental data of [FAT 75; GRE 90] with the bounds of Voigt (V), Reuss (R), Hashin and Shtrikman (HS), the two-phase (SCE) and composite sphere (CH) self-consistent estimates, and numerical results for simple cubic (SC) and body centered cubic (BCC) arrangements of spheres (ν = 0.2)

1

Effective Poroelastic Coefﬁcients 1

1 Arns (2002) (SC) (BCC)

Arns (2002) (SC) (BCC) 0.8

0.8

0.6

V

Ep / E

Kp / K

0.6

HS+ 0.4

0.2

0.4

0.2

SCE

0 0.5 (a)

0.6

1

0.7 0.8 Solid volume fraction (c)

0.9

HS+

SCE

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

0.9

1

0.5

Arns (2002) (SC) (BCC)

Arns (2002) (SC) (BCC) 0.4

0.3

V

Qp

μp / μ 0.4

V

0 0.5 (b)

1

0.8

0.6

405

HS+

0.2

SCE

CH 0.2

0.1 SCE

0 0.5

(c)

0.6

0.7 0.8 Solid volume fraction (c)

0.9

1

0 0.5

(d)

0.6

0.7 0.8 Solid volume fraction (c)

Figure 13.11. Comparison of the experimental results of Han [HAN 86] (after [ARN 02]), with numerical results for simple cubic (SC) and body centered cubic (BCC) arrangements of spheres, as well as the bounds of Voigt (V), Reuss (R), Hashin and Shtrikman (HS) and the two-phase SCE and composite sphere (CH) self-consistent estimates (ν = 0.1)

experimental results of Fate [FAT 75] and Green et al. [GRE 90] for solid volume fraction values greater than 0.7. For low solid volume fraction values (0.58 < c < 0.65), the results of Green et al. [GRE 90] clearly show the non-linear evolution of the effective moduli. As we have already emphasized, the numerical estimates obtained for both CS and BCC microstructures display similar non-linear evolutions for solid volume fraction values close to cmin . Observed differences between the numerical results and the experimental data are mostly due to the fact that junctions

406

Homogenization of Coupled Phenomena

(arrangement, number and dimensions) between the particles in real samples and in the idealized microstructures are different. Finally, we will compare the numerical results for SC and BCC arrangements, the bounds and self-consistent estimates to the experimental results of Han [HAN 86] for the rocks of Fontainebleau with solid volume fraction values between 0.78 and 0.925 [ARN 02]. The modulus of compressibility Kp and shear modulus μp were again measured using acoustic techniques. These experimental results, as well as the numerical results for SC and BCC arrangements, bounds and self-consistent estimates, are shown in Figures 13.11 (a, b, c, and e) which show the evolution of experimental values of Kp /K, Ep /E, μp /μ and νp . A good agreement can be seen between the experimental results and numerical results for a BCC arrangement of spheres.

Chapter 14

Wave Propagation in Isotropic Saturated Poroelastic Media

14.1. Introduction This chapter focuses on the propagation of mechanical waves in deformable saturated porous media. We will emphasize both the similarities and differences introduced by isotropic poroelastic dynamics as compared to normal elastodynamics. For these developments we will work directly on the macroscopic scale by using (following [BOU 87a; 89a; 87b]) diphasic behavior as established by homogenization in Chapter 12. Since the works of Biot [BIO 56a], which predicted the existence of two compression modes, and their experimental demonstration by Plona [PLO 80], wave propagation in poroelastic media has been the subject of much research, among which we draw particular attention to the early work of Deresiewicz [DER 62] and Geertsma and Smit [GEE 61]. A good review on this subject can be found in Corapcioglu [COR 91], and a bibliographic selection of recent work can be found in Mavko et al. [MAV 03]. Initially targeted at geophysics, the ﬁeld of application has since expanded hugely, to take in the ﬁelds of biomechanics, acoustics, ultrasonic ausculation, tomography, and more. Here we will revisit this now well-known analysis while highlighting the different behavior of the three modes of propagation in an isotropic poroelastic diphasic medium (sections 14.2 and 14.3). The refraction conditions for plane waves at interfaces between a porous medium and another medium (porous, elastic or ﬂuid) are then discussed in section 14.4. The characteristics of the Rayleigh wave are also discussed (section 14.5).

408

Homogenization of Coupled Phenomena

We then present the standard tools for solving dynamic elasticity problems [AKI 80; KAU 06], transposed to porous media. First we establish (section 14.6) the analytical expression for the fundamental solutions (Green’s functions) associated with natural point excitations [BOU 87b]. We then construct the integral representation adapted to such media for two- and three-dimensional problems in the harmonic regime (section 14.7). Finally, these results are accompanied by an analysis of the wave ﬁelds radiated by point sources and dislocations (section 14.8). Due to linear behavior with memory effects, it is much simpler to consider problems in the established harmonic regime, and potentially then to return to the time domain using inverse Fourier transform, rather than working directly in the time domain. In what follows we will systematically work in the harmonic regime, with each variable having the form Ae+iωt , where A is the complex amplitude. For convenience, we will restrict ourselves to macroscopically isotropic materials, both in terms of the properties of the skeleton and of the ﬂow. The generalization to anisotropic materials does not pose any difﬁculties in principle but, as for elastic media, makes the analytical expressions very unwieldy (if indeed they exist). In this chapter, thermal exchanges between the two phases are not taken into account. Given the materials forming normal porous matrices (in geophysics, biomechanics, etc.), this simpliﬁcation is reasonable in most cases, in particular when the pores are (quasi-)saturated by a liquid. For highly deformable materials saturated with a gas – such as certain acoustic absorbers, foams, etc. – the effects of thermal exchange can easily be integrated by introducing a complex effective compressibility for the gaseous phase (see Chapter 8). We also emphasize that wave propagation is investigated on the macroscopic scale, which implicity assumes that the conditions for validity of our description of the ensemble behavior are respected, particularly in terms of separation of scales. Because of this, at high frequencies, when the wavelengths approach the size of the heterogenities, the diffractive effects which appear cannot be described using this analysis. Although we will not explicitly state it again, this restriction must always be borne in mind. 14.2. Basics 14.2.1. Notation To simplify the presentation, we will use the following notations in this chapter: – φ is porosity, ρs , ρf and ρm = (1 − φ)ρs + φρf are densities of the materials forming the porous skeleton, the saturating ﬂuid and the saturated porous medium.

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409

– With the assumption of isotropy, the elastic tensor C which characterizes the porous skeleton on the macroscopic scale can be expressed with the help of the two Lamé coefﬁcients λ and μ; the second-order coupling tensor α reduces to the scalar α = 1 − Kb /Ks ; the scalar modulus of compressibility β = (α − φ)/Ks + φ/Kf remains unchanged. In these expressions Kb , Ks and Kf are the moduli of incompressibility of the skeleton structure of the material forming the skeleton and of the ﬂuid. – The dimensionless coefﬁcient ζ = α2 /[β(λ + 2μ)] measures the contrast between compressibility of the skeleton alone (in oedometric deformation) and of the ﬂuid and solid constituents. – The dynamic permeability tensor (in units of m2 ) is simply deﬁned in an isotropic medium by K(ω)I, where I is the identity tensor. K(ω) is linked to the dynamic conductivity Λ(ω) (in m2 /(Pas)) and to the dynamic viscosity η (Pas) by: K(ω) = ηΛ(ω) We will also use the dynamic tortuosity (complex, dimensionless), whose real part expresses the inertial effects of additional mass (i.e. the real coefﬁcient ρ22 (ω) introduced in Chapter 12): φη τ (ω) = iωρf K(ω) – Us and Uf are the solid and ﬂuid displacements (i.e. the mean displacement over pore volume). We note that the Darcy velocity v mentioned above (i.e. the mean velocity over the whole solid and ﬂuid volumes) is iωφUf . Finally e(Us ) is the strain tensor of the solid matrix. – σ and P represent the tensor of total stress (positive for tension) and interstitial pressure (positive for compression). In this framework, the system describing the macroscopic diphasic behavior of a saturated deformable isotropic porous medium under driven harmonic vibrations with pulsation ω takes the following form: σ divσ

=

λdivUs I + 2μe(Us ) − αP I

(14.1)

=

−ω [(1 − φ)ρs Us + φρf Uf ]

(14.2)

K(ω) 2 [ω ρf Us − gradP ] η

(14.3)

−αdivUs − βP

(14.4)

iωφ(Uf − Us ) = φdiv(Uf − Us ) =

2

We recall that the problem linearity has meant that we can ignore the time-dependent term eiωt in all the equations.

410

Homogenization of Coupled Phenomena

In these equations we can recognize both the macroscopic constitutive laws (i.e. the poroelasticity (14.1) and dynamic Darcy’s law (14.3)), and also the conservation equations (vectorial conservation of momentum (14.2) and scalar conservation of ﬂuid mass (14.4)). 14.2.2. Comments on the parameters 14.2.2.1. Elastic coefﬁcients Many granular materials such as soils have macroscopic properties which are very different to that of their constituents. The grains of the skeleton are extremely rigid; typically the modulus of incompressibility of quartz grains Ks is of the order of 4 × 1010 Pa, while the skeleton itself is much more deformable, with moduli 100 to 1000 times smaller depending on the solid volume fraction and the state of ambient stress (a modulus of incompressibility Kb = (3λ + 2μ)/3 of 8 × 107 Pa is an appropriate value for a soil). For porous rocks, the contrast in properties is less pronounced, but remains relevant (the material/skeleton factor may vary between 10 and 50). If the saturating ﬂuid is water – or hydrocarbons in oil reservoirs – then it is not very compressible (for pure water, Kl ≈ 2 × 109 Pa). However when the saturation is not perfect, the presence of gas considerably increases ﬂuid compressibility: if cg is the concentration of gas in the ﬂuid, and Pg is its equilibrium pressure (for example, at atmospheric pressure Pg ≈ 105 P a Kl ), the modulus of ﬂuid incompressibility becomes: Kf = [

1 − cg cg −1 Pg + ] ≈ Kl Pg cg

As a consequence of these orders of magnitude, in most cases the coefﬁcient α = 1 − Kb /Ks takes values close to 1 (to within a few percent). Also, the modulus of compressibility: β=

φ α−φ + Ks Kf

can be estimated for perfectly saturated media by φ/Kf (except in the case of very low porosities where the compressibility of the grains becomes crucial). This approximation becomes very good for imperfect saturation. Finally, depending on the medium, the contrast coefﬁcient ζ can take values from ζ 1 (ﬂuid very compressible compared to the skeleton) to ζ 1 in the opposite case. 14.2.2.2. Dynamic permeability Dynamic permeability K(ω) is the main parameter which controls the ﬂuid-solid interactions, and as a result the biphasic behavior of the medium. The reader is referred

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411

to Chapters 7 to 10 for a detailed analysis of these properties. We recall brieﬂy that at low frequencies viscous effects dominate and: K(ω) → K(0) = K where K(0) = K is the intrinsic permeability (m2 ), and that at high frequencies inertial forces dominate: τ (ω) =

φη → τ∞ iωρf K(ω)

and lead to an effect of additional mass which is quantiﬁed by the real parameter τ∞ 1, which is often termed the medium tortuosity. Naturally, viscous and additional mass effects are present whatever the frequency. The transition between the low- and high-frequency domains occurs around the characteristic medium frequency, fc = ωc /2π where the viscous terms in the dynamic permeability (estimated using the low-frequency approximation) are of the same order as the inertial terms (using the high-frequency approximation). In this way we obtain: ωc =

φη φη = K(0)ρf τ∞ Kρf τ∞

which introduces the dimensionless frequency: ω ∗ = ω/ωc . 14.2.3. Degrees of freedom and dimensionless parameters The system of differential equations (14.1-14.4) governing the porous medium links two steady state ﬁelds, those of total stress and the interstitial pressure, to the two kinematic ﬁelds of the solid and ﬂuid displacements. However, the knowledge of the solid displacement ﬁeld Us and pressure ﬁeld P is enough to describe the porous medium (see section 12.4). To convince ourselves of this, all that is needed is to recognize that the total stress is then completely determined by constitutive law, and ﬂuid displacement by dynamic ﬁltration law. Thus the porous medium is a system described by the four independent scalar variables made up (for example) of the solid displacement and pressure,which are the ones normally used in consolidation theory. Similarly, four scalar conservation equations (three in force, one in mass) describe the porous medium. We note that, in the isotropic case, poroelastic behavior depends on nine independent parameters: two dimensionless – porosity φ and coefﬁcient α – and seven dimensional – densities of the two phases, Lamé coefﬁcients of the skeleton β and also the frequency f and dynamic conductivity K(ω)/η. But, if we assume that the dimensionless frequency dependence displayed by the conductivity (or dynamic

412

Homogenization of Coupled Phenomena

tortuosity) is known, the only new parameter introduced is characteristic frequency fc , because K(ω)/η = (φ/iωρf )τ −1 (ω ∗ ). Thus, since the seven dimensional parameters only involve three independent units, they can be reformulated in terms of 7 − 3 = 4 dimensionless parameters – the density ratio ρs /ρf , Poisson’s ratio of the skeleton ν, “compressibility” ratio ζ = α2 /[β(λ + 2μ)], and dimensionless frequency, ω ∗ = ω/ωc = f /fc . As a result any quantity associated with the medium only depends on the six dimensionless parameters (the above four plus α and φ) we have just identiﬁed.

14.3. Three modes of propagation in a saturated porous medium Using a method analogous to the approach in an elastic medium [ACH 73], the wave equations can be obtained by keeping the kinematic variables Us and Uf unknow, and eliminating steady state variables σ and P from the system. The procedure involves substituting the expression for the stress and pressure gradient, given by constitutive law (14.1) and generalized Darcy’s law (14.3), into the momentum balance (14.2) and mass (14.4), (of which we take the gradient). The equations that are obtained are: (λ + μ)graddivUs + μΔUs = −ω 2 {[(1 − φ)ρs − α(1 − τ )ρf ]Us + (φ − ατ )ρf Uf } 1 graddiv[(α − φ)Us + φUf ] = −ω 2 ρf [(1 − τ )Us + τ Uf ] β where we recall that τ is the dynamic tortuosity: τ (ω) =

φη τ∞ K(0) = ∗ iωρf K(ω) iω K(ω)

(14.5)

We will introduce the Helmholtz decomposition in order to solve these propagation equations. Any sufﬁciently regular vector ﬁeld U can be decomposed uniquely (aside from an additive constant) into the sum of two ﬁelds which are the derivatives of potentials: U = gradΦ + curlΨ

with

divΨ = 0

Φ and Ψ are the scalar and vector potentials of the ﬁeld U. This decomposition makes it possible to distinguish the volume-conserving motions (div curl ≡ 0) associated with vector potential, and irrotational motions (curl grad ≡ 0) associated with the scalar potential. We will therefore set: Us = gradΦ + curlΨ

with

divΨ = 0

Wave Propagation in Poroelastic Media

Uf = gradΦf + curlΨf

413

divΨf = 0

with

Substituting these expressions into the propagation equations, we obtain: % & grad (λ + 2μ)ΔΦ + ω 2 [((1 − φ)ρs − α(1 − τ )ρf )Φ + (φ − ατ )ρf Φf ] % & +curl μΔΨ + ω 2 [((1 − φ)ρs − α(1 − τ )ρf )Ψ + (φ − ατ )ρf Ψf ] = 0 grad

' 1 [(α − φ)ΔΦ + φΔΦf ] + ω 2 ρf [(1 − τ )Φ + τ Φf ] + β % & curl ω 2 ρf [(1 − τ )Ψ + τ Ψf ] = 0

Observing that div ΔΨ = div ΔΨf = 0, and that a particular consequence of the Helmholtz theorem is that any zero ﬁeld has potentials of zero, we can deduce that each of the terms under the differential operators grad and curl is exactly zero. This leads us to four equations grouped into two independent systems, one involving the scalar potentials of the two phases, and the other their vector potentials. −ω 2 [((1 − φ)ρs − α(1 − τ )ρf )Ψ

+

(φ − ατ )ρf Ψf ] = μΔΨ

−ω 2 ρf [(1 − τ )Ψ + τ Ψf ] = 0 −ω [((1 − φ)ρs − α(1 − τ )ρf )Φ 2

(14.7)

+ (φ − ατ )ρf Φf ] = (λ + 2μ)ΔΦ

−ω 2 ρf [(1 − τ )Φ + τ Φf ] =

(14.6)

1 [(α − φ)ΔΦ + φΔΦf ] β

(14.8) (14.9)

We emphasize that this decoupling of potentials is only possible for isotropic materials. In anisotropic materials, the three modes generally involve variations in volume and rotational motion. The wave characteristics then depends on the direction of propagation, and can be determined using a Christoffel tensor adapted for use in poroelasticity. 14.3.1. Wave equations For the vector potentials, system (14.6-14.7) leads to: ΔΨ + δ3 2 Ψ = 0

Ψf = μ3 Ψ

(14.10)

with: δ3 2 = ω 2

φρf ρm [1 − ] μ ρm τ

μ3 = 1 −

1 τ

(14.11)

414

Homogenization of Coupled Phenomena

Thus the vector potentials for both phases are linked linearly through μ3 and are solutions to a Helmholtz equation with wavenumber δ3 . Since the variations in volume of this type of ﬁeld are zero, the equation for conservation of mass (14.4) shows that the pressure also remains zero. For scalar potentials, the elimination of Φf from the system (14.8)-(14.9) leads to the following differential equation:

λ + 2μ ΔΔΦ + ω 2 ρf β

' τ α2 1 ρm (λ + 2μ + )+ ( − 2α) ΔΦ φ β β ρf +ω 4 (ρm τ − φρf )

ρf Φ=0 φ

(14.12)

or alternatively, (Δ + δ1 2 )(Δ + δ2 2 )Φ = 0 δ1 2 and δ2 2 are the complex roots of the associated second-order dispersion equation: λ + 2μ 2 X − ω 2 ρf β

' τ α2 1 ρm (λ + 2μ + )+ ( − 2α) X φ β β ρf +ω 4 (ρm τ − φρf )

ρf =0 φ

(14.13)

The general solution is the sum of any two solutions of the two independent Helmholtz equations: Φ = Φ1 + Φ2

Φj

solution to:

(Δ + δj 2 )Φj = 0

j = 1, 2

The scalar potential of the ﬂuid displacements then follows from system (14.8-14.9): Φf = μ1 Φ1 + μ1 Φ2 with: μj = 1 −

1 φ τ−α

(1 +

ρm δj 2 λ + 2μ [ − 1]) αρf ω 2 ρm

j = 1, 2

(14.14)

From equation (14.4) for the conservation of mass, we can determine the pressure: P =

−1 Δ[φ(Φf − Φ) + αΦ] β

Wave Propagation in Poroelastic Media

=

415

& 1% [φ(μ1 − 1) + α]δ1 2 Φ1 + [φ(μ2 − 1) + α]δ2 2 Φ2 β

Finally the solid and ﬂuid displacement ﬁelds, and the pressure, can be expressed in the following form (with divΨ = 0): Us

=

gradΦ1 + gradΦ2 + curlΨ

(14.15)

Uf

=

(14.16)

P

=

μ1 gradΦ1 + μ2 gradΦ2 + μ3 curlΨ & 1% [n(μ1 − 1) + α]δ1 2 Φ1 + [n(μ2 − 1) + α]δ2 2 Φ2 β

(14.17)

which makes clear the three modes of propagation in a porous saturated medium: two distinct compression modes – known as P1 and P2 – and a shear mode, known as S. The characteristics of each of these three modes, which we will subsequently refer to as 1 (for P1), 2 (for P2) and 3 (for S), are completely determined by the parameters δj and μj , which represent the wavenumber and amplitude ratio (as well as the phase difference) of ﬂuid displacements relative to the solid. Because of the dynamic permeability, these parameters are complex and depend on frequency. Since they have been obtained from the same mechanical system, there are links between them. We will point out two interesting identities here between the characteristics of the modes and the elasticity, viscosity and dynamic permeability parameters of the medium. These identities can be obtained from (14.11), (14.13) and (14.14): −iωηβ μ δ 1 2 δ2 2 = 2 K(ω) λ + 2μ δ3

(14.18)

φ(μ1 − 1) + α][φ(μ2 − 1) + α] = −(λ + 2μ)β

(14.19)

The second equation demonstrates the vital role that the compressibility ratio ζ (which is real and frequency-independent) plays in the relative ﬂuid/solid displacement (which is complex and frequency-dependent) for the two compression modes. We also observe that the four degrees of freedom of the porous medium can be found in the potential-based description. In fact, the knowledge of the two scalar potentials and of two components of the vector potential (with the third being imposed, up to a constant value, by the condition of zero divergence) is enough to completely determine all the variables. In conclusion, the three main differences between propagation in elastic and poroelastic isotropic media are: – the presence of two compressive modes; – the existence of differential motion between the ﬂuid and solid phases; – the frequency-dependent and the complex component of the modes.

416

Homogenization of Coupled Phenomena

14.3.2. Elementary wave ﬁelds: plane waves Returning to temporal variables which have a harmonic time-dependence, each potential φ (scalar, or component of a vector potential) is determined by a wave equation of the form: ω 2 Δφ − δ 2 ∂tt φ = 0

φ = eiωt Φ

with

whose solution is usually decomposed into a basis of elementary solutions: φu = ei(ωt−δu.x) 2

where u is a vector with components uj , (j = 1, 2, 3) such that | u | = u.u = 3 2 j=1 uj = 1. When this vector is purely real, the φu functions describe homogenous plane waves; when it includes a complex component it refers to inhomogenous plane waves. Since these two types of ﬁeld form the basis of many models and numerical developments, we will brieﬂy recall their main properties. 14.3.2.1. Homogeneous plane waves Since the wavenumbers are complex, we will separate their real part (which we will take to be positive by convention) and imaginary part by setting: δ = δ + iδ Considering a potential of the form:

φu (t, x) = ei(ωt−δu.x) = ei(ωt−δ u.x) eδ

u.x

we observe that: φu (t + t , x) = φu (t, x − x )eδ

u.x

with

x =

ωt u δ

This indicates that the signal at point M (x) at an instant t + t corresponds to the signal which exists at the instant t at point M (x − x ), multiplied by eδ u.x . It therefore refers to a phenomenon which propagates in the u direction, at a velocity u.x /t = ω/δ , with a decay rate determined by δ . Physically the waves can only decay along their direction of propagation, which requires that δ < 0. The polarization of both types of waves can easily be determined from the following equations (where A is a given constant vector): gradφ = iδuφ

curlAφ = iδu ∧ Aφ

Wave Propagation in Poroelastic Media

417

which indicates that for compressive waves (P1 and P2) the solid and ﬂuid displacements are parallel to the direction of propagation, whereas for the shear waves they are perpendicular to that direction. To summarize, we have recovered the standard equations for an absorbing dispersive isotropic medium (as in the linear isotropic viscoelastic case) which deﬁne the characteristics of the waves: – phase velocity: cj = ω/δj ; – wavelength: λj = 2π/δj ; – attenuation per wavelength: 2πχj = 2π | −2πχj

e

δj δj

| (over λj the signal decreases by

);

– spatial decay: δj (over a distance d the signal decreases by e−|δj |d ); to which can be added: – the amplitude ratio between ﬂuid and solid displacements, | μj |, and the value by which the ﬂuid phase leads the solid phase, arg μj ; – the pressure/potential ratio (P1 and P2 waves):

1 β [φ(μj

− 1) + α]δj 2 .

14.3.2.2. Inhomogenous plane waves We will now examine the characteristics of inhomogenous waves by considering the case where the vector u is complex, in other words: u = u + iu We note that the condition | u | = 1 requires both the orthogonality of u and u , 2 2 (u .u = 0) and also | u | = 1 + | u | > 1. Separating the real and imaginary parts in the argument of the potential, we ﬁnd: 2

φu (t, x) = ei(ωt−[δ u −δ

u ].x) [δ u +δ u ].x

e

As in a viscoelastic medium [BOR 73], two preferred directions appear: – the direction of propagation, given by: d = √

u −χu 1+(1+χ2 )|u |2

, based on which

we deﬁne the velocity of propagation cinh and the wavelength λinh analogously to the values c and λ which we determined above for homogenous waves: 1 1 cinh = c λinh = λ 2 2 1 + (1 + χ2 )| u | 1 + (1 + χ2 )| u | – the direction of maximum attenuation given by: a = √

χu +u . χ2 +(1+χ2 )|u |2

418

Homogenization of Coupled Phenomena

Displac ement

nt ceme Displa

We note that d and a are not perpendicular (except in the case where χ = 0, i.e. for elastic waves) so that, contrary to the case for homogenous plane waves, the planes of constant phase and constant amplitude are not the same (see Figure 14.1). To make it easier to visualize this inhomogenous wave we can compare it with two homogenous

F Displac ement

nt ceme Displa

N

F

N S

l1

Figure 14.1. Inhomogenous wave ﬁelds. Above: weakly attenuated waves (P1 or S type); below: strongly attenuated waves (P2 type)

waves. It can easily be shown that: – the homogenous plane wave with the same direction of propagation as the inhomogenous wave propagates more rapidly in any direction; – the homogenous plane wave with the same attenuation direction as the inhomogenous wave is less attenuated in any direction. Finally for the polarization, at any instant: in P mode: Ust = gradΦ = iδuΦ = u (iδΦ) − u (iδΦ) in S mode: Ust = curlΨ = iδu ∧ Ψ = u ∧ (iδΨ) − u ∧ (iδΨ) As a result, the motion of a particle is no longer longitudinal (in the P mode) nor transverse (in the S mode), but ellipsoidal, with the axes of the ellipse deﬁned by u and u for a P wave, and perpendicular to these for the S wave.

Wave Propagation in Poroelastic Media

419

14.3.3. Physical characteristics of the modes Now we will return to the physical characteristics of the modes of propagation. To complement the numerical calculation of δj and μj , it is useful to examine the speciﬁc cases of low and high frequencies, and high compressibility contrasts between the ﬂuid and solid, in order to gain a better understanding of the phemonenon [BOU 87a]. 14.3.3.1. Low frequencies, f fc In Chapters 7 and 10 it was shown that at low frequencies viscous effects dominate, and the dynamic permeability behaves in the following way (like any causal transfer function, the real part is even and the imaginary part is odd): ' τ0 + O(ω ∗ 2 (1 + iω ∗ )) K(ω) ≈ K(0) 1 − iω ∗ τ∞

with

ω∗ =

ω f = ωc fc

which gives for the dynamic tortuosity τ : τ (ω) =

τ∞ K(0) τ∞ φη = ∗ ≈ ∗ + τ0 iωρf K(ω) iω K(ω) iω

Using this expression in equations (14.11), (14.13) and (14.14), we arrive at the low-frequency characeristics which are listed in Table 14.1 below. In general terms it can be seen that at low frequencies the inertial effects of Darcy’s law, characterized by the real part of the dynamic tortuosity τ0 , are not involved in the values in this limit. We also recall that since ωc τ∞ = φη/K(0)ρf , the dimensionless frequency ω ∗ /τ∞ only involves steady state parameters. Properties c = ω/δ

P1 wave (λ+2μ)(1+ζ) ρm φρf ρm [1 + αρf (1+1/ζ) ]2 2ρm

c1 = ∗

χ = δ /δ χ1 = τω∞ |μ| | μ1 |= 1 ρm ω∗ arg(μ) ϕ1 = − τ∞ [1 − αρf (1+1/ζ) ]

P2wave 2ω K(0) c2 = ηβ(1+ζ)

S wave c3 =

μ ρm ω ∗ φρf τ∞ 2ρm

χ2 = 1 χ3 = 1 | μ2 |= −1 + α (1 + ) | μ3 |= 1 φ ζ ∗ ϕ2 = π ϕ3 = − τω∞

Table 14.1. Velocity, attenuation, amplitude ratio and ﬂuid-solid phase α2 ) difference for the three low-frequency modes (ζ = (λ+2μ)β

For P1 and P2 waves, the complex velocities (Cj = ω/δj , j = 1, 2) are ﬁrst order solutions to the equation: β ρf τ ∞ ρm τ∞ ρm − 2αρf ρf [ ∗ − ρf ]C 4 − [ + ∗ β(1 + ζ)]C 2 + 1 = 0 λ + 2μ iω φ λ + 2μ iω φ

420

Homogenization of Coupled Phenomena

As for the complex velocity of the S wave, that is given directly to ﬁrst order by: C32 =

μ φρf −1 [1 − iω ∗ ] ρm ρm τ∞

14.3.3.1.1. Low-frequency P1 and S waves To second order, the velocities of these waves are the same as in an elastic medium whose characteristics, independent of the permeability, are: – the mean density of the saturated medium: ρm = (1 − φ)ρs + φρf ; – the rigidity of the skeleton: μ; – the Lamé coefﬁcient λ of the skeleton, to which the modulus α2 /β should be added, which is associated with compressibility of the ﬂuid and grains in the skeleton. Thus in switching from a dry to a saturated medium, the S-wave velocity is only affected by the density correction, and a correction to the modulus is required for the P1 waves. For low-rigidity materials (such as, for example, soils with a poorlyconsolidated surface) this correction may be very signiﬁcant, with the low compressibility of the ﬂuid completely masking the skeleton’s elastic properties. Of course for highly rigid or imperfectly saturated materials, this effect disappears, as the interstitial ﬂuid appears to be very compressible. In contrast to elastic media, P1 and S waves display an attenuation per wavelength which varies linearly with ω ∗ /τ∞ . The attenuation is thus proportional to the intrinsic permeability and to the ﬂuid viscosity, growing linearly with frequency, and it is seen to be systematically greater in the P1 mode than in the S mode. It is important to emphasize that these two modes produce almost identical displacements of both phases (the same amplitude, with a slight phase retardance in the ﬂuid compared to solid). 14.3.3.1.2. Low-frequency P2 wave Since δ2 2 approaches the purely imaginary value −iωηβ(1 + ζ)/K(0), the P2 wave does not propagate, but tends towards a diffusive pressure wave. The effect is similar to that in quasi-static consolidation in a medium whose characteristics would be the intrinsic permeability and an equivalent elastic modulus deﬁned by the series combination of the modulus of incompressibility β −1 and oedometric modulus of the skeleton (corrected by the√ multiplicative factor α−2 ). The velocity, which grows with permeability, varies with ω, and the attenuation per wavelength is strong, χ2 ≈ 1. The ﬂuid and solid displacements have different amplitudes which are are half a period out of phase from one another, which explains the very high attenuation compared to other modes. A consequence of the diffusive character of the P2 wave is that when it appears it only has a signiﬁcant effect over a distance of the order of its wavelength. This observation was put to good use by Mei and Foda [MEI 81] in order to treat

Wave Propagation in Poroelastic Media

421

the diffusive pressure zone in a simpliﬁed manner as a boundary layer (of thickness 2K(0) λ2 ≈ 2π ). ωηβ(1 + ζ) We note that at low frequencies the P1 and P2 modes result from very different physical mechanisms which inhibit or favor differential ﬂuid/solid displacement. This is made very clear in the case of a high compressibility contrast between the skeleton and the ﬂuid (ζ 1 or 1), where it is the more rigid phase and the medium density which determines the velocity of the P1 waves, whereas it is the more compressible phase and steady state permeability which determines the velocity of the P2 waves. 14.3.3.2. High frequencies: f fc At high frequencies, intertial effects dominate, and the dynamic tortuosity τ behaves in the following manner (where M is a form factor for the material which is generally close to 1): φη ≈ τ∞ [1 + τ= iωρf K(ω)

M ] 2iω∗

Making use of this expansion, we can see that all three modes propagate and display common characteristics (see Tables 14.2 and 14.3). Their velocities stabilize on √ ∗ constant values and their attenuation per wavelength, which varies as 1/ ω , tends to zero with a linear dependence on the√ real, dissipative part of H(ω). However, since ∗ the attenuation per unit length varies as ω , the attenuation remains a very important characteristic of the three modes. Finally, all three modes produce differential ﬂuidsolid displacements (in phase opposition for the P2 wave). Properties P1 wave (ζ 1) P2 wave (ζ 1) c = ω/δ

c1 =

λ+2μ

φ βρf τ∞

c2 = φρ ρm − τ∞f √ √ χ = δ /δ O( ω ∗−1 ) O( ω ∗−1 ) α 1 −1 |μ| | μ1 |= 1 − τ∞ | μ2 |= φζ φ −1 ατ ∞ √ arg(μ) ϕ1 = O( ω ∗−1 ) ϕ2 = π

S wave μ c3 = φρ ρm − τ∞f √ O( ω ∗−1 ) −1 | μ3 |= 1 − τ∞ √ ϕ3 = O( ω ∗−1 )

Table 14.2. Mode properties at high frequencies – skeleton that is very rigid compared to the ﬂuid, ζ 1

14.3.3.2.1. High-frequency S wave For the S wave, the calculation leads directly to: C32 =

μ φρf −1 [1 − ] ρm ρm τ∞

μ3 ≈ 1 − 1/τ∞

422

Homogenization of Coupled Phenomena

P1 wave (ζ 1)

Properties c = ω/δ c1 = χ = δ /δ |μ| arg(μ)

P2 wave (ζ 1)

ρ

φ βρf τ∞

f ∞] 1− ρm [2− τφ ρ

f 1− ρm √ ∗−1 O( ω )

| μ1 |= ϕ1 =

φ τ∞

ρ /ρf −1 1 + mτ∞ −φ √ O( ω ∗−1 )

λ+2μ ρm

c2 =

√

1 ρf ∞] 1− ρm [2− τφ ∗−1 ω )

O( | μ2 |= −1 + 1/φ ϕ2 = π

Properties

S wave μ c = ω/δ c3 = φρ ρm − τ∞f √ χ = δ /δ O( ω ∗−1 ) −1 |μ| | μ3 |= 1√− τ∞ arg(μ) ϕ3 = O( ω ∗−1 )

Table 14.3. Mode properties at high frequencies – ﬂuid very incompressible compared to the skeleton, ζ 1

In comparison to the low-frequency velocity, the term for skeleton rigidity is retained, but the apparent density of the ensemble is reduced by inertial coupling, leading to a higher velocity. Bearing in mind the normal values of the porosity and tortuosity (τ∞ ), this correction is generally small. As a result the velocity dispersion of S waves is quite low across the whole range of frequencies. The ﬂuid and solid motion tends to be in phase, but their amplitudes will be different. We observe that in the very speciﬁc case where τ∞ = 1, we recover the S-wave velocity in a dry medium, and the ﬂuid tends to be stationary. All these properties are independent of the contrast in compressibility between the ﬂuid and solid, ζ. 14.3.3.2.2. P1 and P2 waves at high frequencies The velocities of these waves, in terms of their real part and to ﬁrst order, can be obtained from the roots of the following equation: ρf τ ∞ β ρf τ∞ φρf 4 ρm − 2αρf [ρm − + β(1 + ζ)]C 2 + 1 = 0 ]C − [ λ + 2μ n τ∞ λ + 2μ n It can be shown analytically that the velocity of the P1 mode at high frequency remains fairly close to its low-frequency value (this follows from the fact that across the whole frequency range 0 <| τ |−1 < 1). As a result the dispersion of the P1 mode is also low across the whole frequency range (but more pronounced than for the S wave). Each of the compressive modes creates differential ﬂuid/solid displacements, which are in phase for P1 and in opposition for P2. By way of example, in Table 14.2 we give the parameters for the simple case of a skeleton which is highly rigid relative to the ﬂuid. We note that if in addition τ∞ = 1, the velocity of the P1 wave is the same

Wave Propagation in Poroelastic Media

423

as that of P waves in a dry medium (with the ﬂuid tending to be stationary) and that of the P2 wave is the same as the velocity of compression waves in the ﬂuid (with the solid tending to be stationary). This very speciﬁc decoupling disappears in the general case. The characteristics in the case of a skeleton which is very deformable compared to the ﬂuid (Table 14.3) illustrate this. 14.3.3.3. Full spectrum For an arbitrary frequency, we need to proceed numerically, using models of dynamic permeability, such as those given by Johnson et al. [JOH 87] or ones obtained through the self-consistent approach (see Chapter 10). The values shown in Figures 14.2 and 14.3 are for a sand whose characteristics are given in Table 14.4. The transition between the low- and high-frequency properties occurs in the frequency range 10−1 fc < f < 10fc . For the P1 and S modes, this is the region where dispersion is concentrated and where the attenuation per wavelength reaches its maximum. The three models of dynamic permeability emphasize that the closer the tortuosity τ∞ is to 1 (and consequently the weaker the inertial coupling is), the more the biphasic effects at medium and high frequencies become signiﬁcant, leading to an increase in attenuation, dispersion and amplitude of the differential displacements.

14.4. Transmission at an elastic-poroelastic interface The analysis of transmission-refraction conditions for a plane wave incident on a planar interface between two semi-inﬁnite media, one elastic and the other poroelastic, gives a simple illustration of the biphasic effects of a porous medium, and its frequency dispersion [DUT 83]. The existence of two compressive modes, one highly dispersive, leads us to expect that signiﬁcant differences will appear compared to the transmission between two elastic media. On the other hand, since S waves are not very dispersive, when only these modes are involved, as is the case for the transmission of SH-waves (i.e. waves polarized in the plane of the interface), the effects of poroelasticity will be small. We will now consider the problem of P or SV-plane wave (i.e. a wave which is polarized in the plane deﬁned by the normal to the interface and direction of propagation) refraction at a planar interface (with normal ez and lying at z = 0). We will assume that the incident wave (indexed by I) coming from one or the other medium is of oblique incidence in the (x, z) plane (Figure 14.4). We will determine the plane waves of types P1, P2 and SV (indexed 1, 2 and 3 respectively) reﬂected and refracted in the poroelastic medium (z > 0), and the P and SV-waves (indexed 4 and 5) in the elastic medium (z < 0). The potential of each wave has the form: – compressive waves:

j

Φj = Aj e−iδj p

.x

j = 1, 2, 4

– shear waves polarized in the (x, z) plane: Ψj = Aj ey e−iδj p

j

.x

j = 3, 5

424

Homogenization of Coupled Phenomena

Figure 14.2. Characteristics of the three P1, P2 and S modes as a function of the dimensionless frequency determined for characteristic values of a sand with three different frequency models of the dynamic permeability, corresponding to τ∞ = 1 for the ﬁrst and τ∞ = 1.8 for the last two; above: phase and group velocities; below: attenuation per unit length

With Aj , pj , δj being respectively the complex amplitude, the normalized 2 propagation vector (complex or real), | pj | = 1, and the wavenumber of wave j.

Wave Propagation in Poroelastic Media

UF /US

2

1

0

f /fc 104

0.01

1

100

$

0

f /fc 4

10

0.01

1

100

Figure 14.3. Characteristics of the three P1, P2 and S modes as a function of the dimensionless frequency determined for characteristic values of a sand with three different frequency models of the dynamic permeability, corresponding to τ∞ = 1 for the ﬁrst and τ∞ = 1.8 for the last two; above: relative amplitude of the ﬂuid motion compared to the solid motion; below: phase

425

426

Homogenization of Coupled Phenomena

ρs (kg/m3 ) Ks (Pa) λ(Pa) μ(Pa) ρf (kg/m3 ) Kf (Pa) φ Sand 2700 3.61010 5.8107 3.8107 1000 2109 0.3 10 8 9 Sandstone 2700 3.610 8.310 1.2510 1000 2109 0.2 Table 14.4. Representative mechanical parameters for a sand and a sandstone

S

P

P

P1 P2

S

Figure 14.4. System of waves refracted at a planar interface (z = 0) between a poroelastic (lower) medium and an elastic medium (upper). Case of a P wave incident in the (x, z) plane

In the elastic medium characterized by the Lamé coefﬁcients λe and μe and the density ρe , we have: δ4 = ω/cP

δ5 = ω/cS

cP =

λe + 2μe ρe

cS =

μe ρe

14.4.1. Expression for the conditions at the interface We will now express the conditions at the interface z = 0. The wave ﬁelds must satisfy: – continuity of solid displacement: ±UIs +

3

Ujs −

j=1

5

Uj = 0

j=4

– continuity of total stress: ⎡ ⎤ 3 5 ⎣±σ I + σj − σ j ⎦ .ez = 0 j=1

j=4

(± takes the + sign if the incident wave is in the porous medium or the − sign if it comes from the elastic medium). In addition to these normal elastic conditions we must add a condition associated with the ﬂuid phase, which depends on the nature of

Wave Propagation in Poroelastic Media

427

the interface and only involves the modes of the porous medium. This is: – Either the condition of zero pressure in the case of a “free surface” contact. Depending on whether the incident wave is in the porous medium or the elastic medium, this can be written: PI + P1 + P2 = 0

P1 + P2 = 0

– Or the condition of zero ﬂux in the case of an impermeable “closed surface” contact. Depending on whether the incident wave is in the porous medium or the elastic medium, this can be written: ⎤ ⎡ ⎤ ⎡ 3 3 ⎣ (Ujs − Ujs )⎦ .ez = 0 ⎣(UIs − UIs ) + (Ujs − Ujs )⎦ .ez = 0 j=1

j=1

Since all these conditions must be simultaneously satisﬁed at all points M (x, y, 0) on the interface, the arguments of the potentials for each wave must be identical. This ﬁrstly requires that the propagation vectors pj should all lie in the same plane (x, z) as that of the incident wave, and also leads to Snell-Descartes’ law: δI pIx = δj pjx = wx

j = 1, ..., 5

These equations express the conservation of wavenumber in the interface plane, and make it possible to determine the components of propagation vectors, making use 2 of the fact that pjy = 0 and | pj | = 1: pjx =

δI I p δj x

pjz = ±[1 − (pjx )2 ]1/2

with the conventions, for all complex a, that (a1/2 ) > 0; ± takes the + sign if the wave propagates towards z < 0, and the − sign otherwise. Thus for the wavenumbers normal to the interface: wzj = δj pjz = −[δj2 − wx2 ]1/2 wzj = δj pjz = [δj2 − wx2 ]1/2

j = 1, 2, 3 j = 4, 5

Outside the case of normal incidence (pjx = pIx = 0, and pjz = pIz = 1), the complex nature of wavenumbers has an effect on some of the propagation vectors, making the waves concerned slightly inhomogenous. In addition, when | pjx |> 1, pjz has a strong imaginary component, implying a high inhomogenity in the wave. This is the effect of critical refraction which is familiar in the context of elasticity. The velocity dispersion causes the critical angle of refraction to vary with frequency.

428

Homogenization of Coupled Phenomena

We still need to specify the boundary conditions for the displacements, and with the help of constitutive laws, the stresses and pressure as a function of the amplitudes of the potentials of each wave. Thus we obtain a linear system in the following general matrix form: [M C].[AP ] + [M I] = 0 [M C] and [M I] are the (5x5) matrices for the boundary conditions and incident conditions respectively. Each column of [M C] (or [M I]) corresponds to a different refracted (or incident) wave, and each line to one of the conditions on the interface. [AP ] is the matrix of the transmission coefﬁcients for the potentials, with APij representing the potential amplitude of wave i under incidence of wave j. Below, the expression for [M C] can be found, with the lines organized in the following order: continuity of Ux , of Uz , of σzz , of σxz , and ﬁnally the two boundary conditions corresponding to either the free surface ( βP = 0) or closed surface (Uf z − Usz = 0). The matrix [M I] can be deduced from [M C] by replacing the directions of propagation of the refracted waves by those of the incident waves, i.e., according to Snell-Descartes’ law, by changing the wzj to −wzj : [M C] =

⎡

⎤

wx wx −wz3 −wx wz5 w w w −w −w x x z1 z2 z4 ⎥ ⎢ 2 + λδ 2 + 2 + λδ 2 + 2 + −2{μwz2 −2μwx wz3 2μe wz4 2μe wx wz5 ⎥ ⎢ −{2μwz1 1 2 ⎥ ⎢δ1 2 [φ(μ1 − 1) + α] α } δ2 2 [φ(μ2 − 1) + α] α } 2 λe δ4 ⎥ ⎢ β β 2 − w 2 ) 2μ w w 2 − w 2 )⎥ ⎢ −2μw w −2μw w −μ(w μ (w x z1 x z2 e x z4 e x x z3 z5 ⎥ ⎢ 2 ⎥ ⎢ δ1 [φ(μ1 − 1) + α] δ2 2 [φ(μ2 − 1) + α] 0 0 0 ⎥ ⎢ − − − − − ⎦ ⎣ wz2 (μ2 − 1) wx (μ3 − 1) 0 0 wz1 (μ1 − 1) P1 P2 S P S

This system makes it possible to treat any type of planar poroelastic-elastic interface (free surface, contact with a rigid substrate, with a ﬂuid, etc.) by adapting the Lamé coefﬁcients of the elastic medium. Solving the system allows us to determine the matrix terms [AP ]. Since these are dimensionless, they only depend (in frequency terms) on the dimensionless frequency f /fc . From a physical point of view, it is more informative to consider the displacement amplitudes AD of the incident and refracted waves. These follow naturally from the values of AP through the following equation: ADij =

δi APij δj

14.4.2. Transmission of compression waves The cases of refraction of normally-incident compression waves at the interface between a dry and a saturated medium demonstrate the main effects. In the examples shown in Figure 14.5, parameters for the saturated medium are those for sand as given

Wave Propagation in Poroelastic Media

Figure 14.5. Amplitude of the reﬂection-transmission coefﬁcients for a normally-incident compression wave. Variation as a function of the dimensionless frquency. Inﬂuence of the interface conditions, of the incident wave and of the dynamic permeability model τ∞ = 1, 2, 3. Left: P1 and P2 waves in the porous medium, reﬂected if P1 incident, transmitted if P incident; right: P wave in the elastic medium, transmitted if P1 incident, reﬂected if P incident

429

430

Homogenization of Coupled Phenomena

in Table 14.4, and parameters for the elastic medium correspond to the same sand, but dry. The results are shown as a function of dimensionless frequency for various dynamic permeability models corresponding to τ∞ = 1, 2, 3 (with the same value of the intrinsic permeability) and permeable or impermeable interface conditions. In general terms, the strong dispersive effect of the porous medium can be seen in the transmission-reﬂection coefﬁcients, as well as the fact that a weak inertial coupling (τ∞ close to 1) favors the generation of the P2 wave. The ﬁrst two examples describe the transmission of a normally-incident P1 wave in the saturated porous medium towards the dry medium with either free-surface or closed-surface boundary conditions. The effect of this condition is most signiﬁcant at medium and high frequencies. As would intuitively be expected, the appearance of the P2 wave is restricted in the case of an impermeable interface [RAS 84]. In addition, the impermeability condition acts to increase the impedance of the porous medium (facilitating transmission to the dry medium), whereas the free-surface condition has the opposite effect and reduces transmission. The next example describes transmission of a normally-incident P wave from dry medium to saturated medium, for the impermeable boundary condition. At low frequencies (f fc ) the P2 wave has a low amplitude, and the situation is close to that of two elastic media, where the velocities would be those of the P and P1 waves, and the densities those of the two media. At moderate and high frequencies the division between the two waves is entirely reversed, with a notable drop in reﬂection of the P wave at the expense of transmission of the P2 wave. In fact, when it is propagative the P2 wave displays very little impedance, which facilitates the transmission of the P wave. Finally, the biphasic effects are weakest for an incident P wave and an impermeable interface (they are barely visible for the reﬂected P wave), and they are concentrated mostly at high frequencies.

14.5. Rayleigh waves Rayleigh waves are a mode which develops at the free surface of a homogenous half-space in the absence of an incident ﬁeld [ACH 73]. In porous media [DER 62] as in elastic media, only the P-SV waves can lead to guided surface waves. We should therefore identify the combinations of planar P1, P2 and SV-waves which satisfy the free-surface conditions – the zero stresses σzz , σxz and zero pressure. SnellDescantes’ law requires the same horizontal wavenumber wx for all three modes. This wavenumber, which characterizes the surface oscillations, deﬁnes the wavenumber of the Rayleigh mode, wx = δR . Reusing results from the previous section in this

Wave Propagation in Poroelastic Media

431

speciﬁc case (where the upper medium has zero characteristics and no incident wave), we end up solving the following system: [M L].[A] = 0 where [A] is the vector formed by potential amplitudes of the P1, P2 and S waves forming the Rayleigh wave, and [M L] is the (3x3) matrix of the free-surface conditions (with the lines expressing the zero values of σzz , σxz and βP respectively): ⎡

2 −{2μwz1 + λδ12 + ⎢ δ1 2 [φ(μ1 − 1) + α] α } β [M L] = ⎢ ⎣ −2μwx wz1 δ1 2 [φ(μ1 − 1) + α]

⎤ 2 −2{μwz2 + λδ22 + −2μwx wz3 ⎥ δ2 2 [φ(μ2 − 1) + α] α β} ⎥ 2 −2μwx wz2 −μ(wx2 − wz3 ) ⎦ δ2 2 [φ(μ2 − 1) + α] 0

This system only has non-zero solutions when the determinant of [M L] is zero, which gives the dispersion equation. Observing that: μ

δ 2 − δ32 λ+2μ (μj − 1 + α/φ)δj2 i i = (−1) (μ1 − 1 + α/φ)δ12 − (μ2 − 1 + α/φ)δ22 δi2 − δj2

for i, j = 1, 2

The dispersion equation for Rayleigh waves in a poroelastic medium takes the following form: [2 − (

δ3 2 2 ) ] = δR

4

δ3 1 − ( )2 δR

5

μ δ22 − δ32 λ+2μ

δ22 − δ12

μ δ12 − δ32 λ+2μ δ1 1 − ( )2 + δR δ22 − δ12

δ2 1 − ( )2 δR

6

As well as the presence of two compressive modes, two notable differences can be seen compared to the elastic case: – ﬁrstly, since the coefﬁcients are complex, the solution δR is also complex, which indicates a decay of the surface Rayleigh wave in its direction of propagation; – also, the frequency dependence of the three volume modes imposes a velocity dispersion for the Rayleigh wave. At low frequencies we simply have, to ﬁrst approximation, the normal dispersion equation for Rayleigh waves in an elastic medium, with velocities that are those of the P1 and S modes at low frequencies (cP 10 et cS0 ), shown in Table 14.1: [2 − (cR0 /cS0 )2 ]2 = 4 1 − (cR0 /cS0 )2 1 − (cR0 /cP 10 )2

432

Homogenization of Coupled Phenomena

√ It can also be shown that in this case the attenuation per wavelength is of O( ω ∗ ). At medium and high frequencies, the numerical solution to the dispersion equation always leads to a solution whose velocity is lower than that of the S waves. This type of analysis can be applied to other localized propagation modes close to an interface, such as Stoneley waves (cf. for example [SCH 88]) or layer modes (such as Love waves [DER 62]). 14.6. Green’s functions The determination of Green’s functions in an isotropic deformable saturated porous medium was examined by Gringarten and Ramey [GRI 73], who considered the diffusion of a ﬂuid source in a porous medium, by Burridge and Vargas [BUR 79] who determined in the time domain the far ﬁeld emitted by a point force, and by Norris [NOR 85] who introduced forces in the ﬂuid phase without real physical signiﬁcance, which makes interpretation difﬁcult. The ﬁrst satisfactory solution [BON 87], was obtained by drawing an analogy with thermoelasticity. Here we will show the solution which was established directly from the poroelasticity equations in the harmonic regime [BOU 87b]. This offers the advantage of giving physical signiﬁcance to the source terms, and being valid at all frequencies and at all distances. 14.6.1. Source terms Green’s functions are the responses of an inﬁnite medium to point excitations. Physically, these source terms can only appear in the conservation equations, i.e. the dynamics equation and the continuity equation. Since the elasticity and the ﬁltration law are (macroscopic) constitutive laws, they cannot be modiﬁed by the addition of a source. As a result, in order to ﬁnd these fundamental solutions in the harmonic regime, we are led to introduce the following point harmonic excitations: – into the dynamics equation, a vectorial force density Fδ|x| eiωt , which corresponds to a point harmonic force applied at the origin (δ|x| is the Dirac distribution); – into the continuity equation, a distribution V δ|x| eiωt , which corresponds to a volume injected (V > 0) harmonically at the point of the origin. We therefore need to solve the following problem (the time-dependence in eiωt can again be omitted through linearity): σ

=

λdivUs I + 2μe(Us ) − αP I

divσ

=

−ω 2 [(1 − φ)ρs Us + φρf Uf ] − Fδ|x|

iωφ(Uf − Us ) =

K(ω) 2 [ω ρf Us − gradP ] η

Wave Propagation in Poroelastic Media

φdiv(Uf − Us ) =

433

−αdivUs − βP + V δ|x|

In order to do this, we will rewrite this differential system in terms of the four independent variables Us and P , eliminating Uf and σ. Thus we obtain, introducing the dynamic tortuosity τ (ω): (λ+μ)graddivUs +μΔUs +ω 2 [ρm −

φρf φ ]Us −[α− ]gradP = −Fδ|x| (14.20) τ τ

φ K ΔP − βP − [α − ]divUs = −V δ|x| iωη τ

(14.21)

Deﬁning as R the response vector, with four components (Us , P ), and by S the impulse vector, with four components (F, V ), the system can be written in the following condensed form: ⎤ Usx ⎢ Usy ⎥ ⎥ R=⎢ ⎣ Usz ⎦ P ⎡

B.R + Sδ|x| = 0

with

⎡

⎤ Fx ⎢ Fy ⎥ ⎥ S=⎢ ⎣ Fz ⎦ V

where B is the symmetric differential operator (4x4) for the harmonic poroelasticity: Bii = (λ + μ)∂ii + μΔ + ω 2 [ρm −

φρf ], τ

Bij = (λ + μ)∂ij ,

i, j = 1, 2, 3;

i = j

B4i = Bi4 = −[α −

φ ]∂i , τ

B44 =

i = 1, 2, 3

i = 1, 2, 3

K Δ−β iωη

One very important property of B is that the differential operators which couple between displacements and pressure are only ﬁrst order, whereas the strongly coupled variables are linked by second order differential operators. 14.6.2. Determination of the fundamental solutions In order to determine the Green’s functions we need to identify displacementpressure responses gk to the four unit point impulses Sk (the three force components

434

Homogenization of Coupled Phenomena

and the injected volume). For that we need to solve the four following differential systems (δjk is the Kronecker delta function): B.gk + Sk = 0

S k j = δjk δ|x|

j, k = 1, ..., 4

Combining these responses gk into a single matrix G (4x4) such that Gki = (gk )i , the problem takes the form: B.G + δ|x| I = 0 so that, transposed into the space of distributions [ROD 71]: [B.δ|x| I] ∗ G + δ|x| I = 0 This last equation shows that the Green’s matrix G is the inverse B, according to the spatial convolution product. The solution is obtained with the help of the Kupradze method [KUP 79], which makes it possible to reduce the matrix problem to an essentially scalar problem. Here we will discuss the principles without going into full detail of the calculations, which can be found in Boutin et al. [BOU 87b]. Expressed in the space of distributions, the method of solving the differential system is strongly analogous to solving a linear system. Consider distribution matrix B which is constructed from the cofactors of the matrix B. The cofactors, in the sense of distributions, are obtained using normal matrix calculus techniques, but with the convolution product replacing the standard product. In the same way as for linear systems, we then have by construction: [B.δ|x| I] ∗ [B .δ|x| I] = Det{B}δ|x| I where Det{B}δ|x| is the distribution corresponding to the determinant (in terms of the convolution of distributions) of B. We will call θ the scalar solution to the equation: Det{B}δ|x| ∗ θ + δ|x| = 0 Multiplying this equation by the unit matrix I, θI is then naturally the solution to the differential system: [Det{B}δ|x| I] ∗ θI + δ|x| I = 0 and, making use of the expression for the determinant: [B.δ|x| I] ∗ [B .δ|x| I] ∗ θI + δ|x| I = 0 from which we deduce by simple comparison with the initial problem that: G = [B .δ|x| I] ∗ θI = B (θI)

Wave Propagation in Poroelastic Media

435

Thus, once the solution θ has been determined, the Green’s matrix can be obtained by applying the differential operator B to θI. In the case of the poroelastic operator, the calculation leads to the following expressions for the cofactors and for determinant [BOU 87a], expressions which involve the wavenumbers of the three P1, P2 and S modes: Det{B} = μ(Δ + δ32 )D

with D = μ(λ + μ)

K [Δ + δ1 2 ][Δ + δ2 2 ][Δ + δ3 2 ] iωη

B = μ(Δ + δ32 )B The terms of B have the following expressions: for i = 1, 2, 3: Bii = −[(λ + μ)(

for i, j = 1, 2, 3;

φ K K Δ − β) − (α − )2 ]∂ii + [Δ + δ12 ][Δ + δ22 ](λ + 2μ) iωη τ iωη

i = j:

Bij = −[(λ + μ)(

φ K Δ − β) − (α − )2 ]∂ij iωη τ

for i = 1, 2, 3: B4i = Bi4 = (α −

φ )μ[Δ + δ32 ]∂i τ

and ﬁnally: = μ[Δ + δ32 ][(λ + 2μ)Δ + μδ32 ] B44

The presence of the μ(Δ + δ32 ) term which is common to B and Det{B} means that we can reduce the scalar problem to that of determining the function θ such that: [B.δ|x| I] ∗ [B .δ|x| I] = [Dδ|x| ] ∗ θ I + δ|x| I with Green’s functions then being obtained by applying B to θ I. For convenience, K θ so that: we will write ϕ = μ(λ + μ) iω [Δ + δ12 ][Δ + δ22 ][Δ + δ32 ]ϕ + δ|x| = 0 It is natural to look for ϕ in the form of a linear combination of Green’s functions of the Helmholtz equations associated with the three modes of propagation, i.e.: ϕ=

3 j=1

aj ϕj

with

[Δ + δj2 ]ϕj + δ|x| = 0

436

Homogenization of Coupled Phenomena

2 By identiﬁcation we can deduce the coefﬁcients (with the convention δj+3 = δj2 ):

aj =

[δ12

−

2 δj+1 2 δ3 ][δ32

2 − δj+2 − δ22 ][δ22 − δ12 ]

j = 1, 2, 3

Since we know that, for the Helmholtz equation, the Green’s functions radiating from the source to inﬁnity (when the time dependence is eiωt ) are: ϕj =

e−iδj |x| 4π | x |

(14.22)

the function ϕ is then entirely determined. iωη leads, after a number of μ(λ + μ)K(ω) manipulations, to the Green’s matrix whose terms are given below. The symmetry of the poroelastic operator is reﬂected in the cofactors, and consequently in the Green’s matrix. The coefﬁcients appearing in the various terms only depend on the wavenumbers and coefﬁcients μi of the three modes of propagation. Finally, applying the operator B to Iϕ

For a unit force F (| F |= 1) in direction en , n = 1, 2, 3, the components of solid displacement and pressure are given by the following (to avoid any confusion with the wavenumbers, the Kronecker delta is represented by δ˜jn here): n

(Us )j =

Gnj

P n = Gn4

=

=

1 − 4π −

1 4π

5

∂j

3

∂n

ξ∂n

k=1

e−iδk |x| αk |x|

−iδ1 |x|

e

|x|

−

e

1 e−iδ3 |x| ˜n − δ μ |x| j ' −iδ2 |x|

6

|x|

For a unit injected volume V (V = 1), the components of solid displacement and pressure are given by: 4

G4j

=

P 4 = G44

=

(Us )j =

' e−iδ1 |x| e−iδ2 |x| ξ∂j − |x| |x| ' −iδ1 |x| e e−iδ2 |x| iωη 1 1 α2 + α1 − 4π K(ω) α3 |x| |x| 1 − 4π

with the following coefﬁcients: α1

=

1 μ [δ 2 − δ2] μδ32 [δ22 − δ12 ] 2 λ + 2μ 3

(14.23)

Wave Propagation in Poroelastic Media

1

[δ 2 δ22 ] 1

−

μ δ2] λ + 2μ 3

437

α2

=

α3

=

−

1 μδ32

(14.25)

ξ

=

1 μ1 − μ2

(14.26)

μδ32 [δ12

−

(14.24)

Due to the isotropy of the medium, the response to a force in an arbitrary direction can be determined from the response to a force of a known orientation by rotating the displacement ﬁeld in the same manner. The expressions obtained correspond to sources applied at the origin of an inﬁnite homogenous medium. In the case of a source applied at a point MS , distance to the origin, | x |, simply needs to be replaced by distance from the point of measurement MR to the source location MS , | xMR − xMS |. These Green’s functions established in the harmonic regime allow us to determine the response to a transient point source S(t), using the inverse Fourier transform. The temporal response then takes the form of a convolution product: ˆ ˆ ˆ R(t) = G ∗ S = G(τ )S(t − τ )dτ with G(τ ) = G(ω)e−iωτ dω In general, the frequency dependence of terms in the Green’s matrix is too complex to get an analytical expression for temporal responses, and numerical methods must be used. If on the other hand the frequency spectrum of the source is low enough to use the quasi-static approximation of Darcy’s law, it is then possible to obtain timedependent expressions for Green’s functions. The reader is referred to Cheng and Detournay [CHE 98] on this topic.

14.6.3. Fundamental solutions in plane geometry This method can also be applied to plane geometry problems in the x1 − x2 plane. The response vector then reduces to two displacement components, along with the pressure, and the impulse vector has two force components and the injected volume component. In this case the source terms represent line sources perpendicular to the plane. The solution is very similar to the three-dimensional case, with the main difference arising from the two-dimensional Helmholtz equation, whose Green’s function (radiating from the source to inﬁnity with a eiωt time dependence) is the zeroth order Hankel function: ϕj =

H02 (δj | x |) 4i

| x |=

x21 + x22

(14.27)

438

Homogenization of Coupled Phenomena

(when time dependence is e−iωt , the solution radiating from the origin is: H01 [δj | x |] ). 4i The various terms of the (3x3) Green’s matrix take the following form for planar problems, with αi and ξ coefﬁcients being exactly the same in two- and three-dimensions. For a unit force F (| F |= 1) in the en , n = 1, 2 direction, the components of solid displacement and pressure are given by: 6 5 3 i 1 2 n n 2 n αk H0 (δj | x |) − H0 (δ3 | x |)δ˜j (Us )j = Gj = ∂j ∂n 4 μ k=1

P n = Gn3

& i% ξ∂n H02 (δ1 | x |) − H02 (δ2 | x |) 4

=

For a unit injected volume V (V = 1), the components of the solid displacement and pressure are given by: (Us 3 )j = G3j

=

P 3 = G33

=

& i% ξ∂j H02 (δ1 | x |) − H02 (δ2 | x |) 4 ' iωη 1 i 2 2 α2 H0 (δ1 | x |) + α1 H0 (δ2 | x |) 4 K(ω) α3

14.6.4. Symmetry of the Green’s matrix, and reciprocity theorem The symmetry of the Green’s matrix leads to a reciprocity theorem, which has a simple expression in the Fourier domain. Consider two different loads, on an inﬁnite porous medium: Case a: a point source SaA at point source A. Case b: a point source Sb B at point source B. In case (a) the response at B is Ra B = G(B−A) .Sa A , and in case (b) the response at A is Rb A = G(A−B) .Sb B . In both cases, the source-detector distance is the same, but the roles are reversed. The derivatives in terms of the Green’s matrix are taken with respect to the components of the source-receiver vector, and so swapping the source and receiver results in a change of sign of each derivative. Thus the terms which only involve a single derivative (the pressure under a force, and the displacements under an injected volume) are opposite in the two cases, so that: i

j

Gji(B−A) = (−1)δ4 +δ4 Gji(A−B)

Wave Propagation in Poroelastic Media

439

We will now calculate the complex work of response 1 under impulse 2, including the four variables and the four types of sources (we specify complex because the actual work is determined using only the real parts of the variables). Respecting the conventions of positive pressure under compression, and positive volume source under injection, the work is of the form: j

Wab = Uas B Fb B − PBa VBb = Ra jB Sb jB (−1)δ4 so that: i

Wab = Gij(B−A) S(1) iA S(2) jB (−1)δ4 or alternatively, reversing the source and receiver and then using the symmetry of the Green’s matrix: Wab = Gij(A−B) Sa iA Sb jB = Gji(A−B) Sb jB Sa iA which ﬁnally gives: Wab = Uas B .Fb B − PBa VBb = Ubs A .Fa A − PAb VAa = Wba This equality gives a direct demonstration of the reciprocity in an inﬁnite poroelastic medium. We note that is makes it possible to link the responses to sources of different types, in particular the pressure response to a forceﬁeld and the displacement response to a ﬂuid injection. The transposition of this result into the time domain requires the introduction of convolution products of the temporal variables, which shows that this energetic equality does not have a signiﬁcance at any one instant in time.

14.6.5. Properties of radiated ﬁelds Starting from solid displacements, it is easy to identify potentials and to obtain a more manageable representation for the ﬁeld radiated by each source. As explained earlier, ﬂuid displacements can be deduced by simple multiplication of the potentials by the μj coefﬁcients, and pressure given by Green’s functions can be expressed in an equivalent manner using the potentials of the compression waves. For a unit force acting an an arbitrary direction e, the potentials of the P1, P2 and S waves are given by: Φ1 = −α1 div(ϕ1 e)

Φ2 = −α2 div(ϕ2 e)

Ψ = −α3 curl(ϕ3 e)

440

Homogenization of Coupled Phenomena

For a unit source of ﬂuid volume, only the compression waves are created, in keeping with the isotropic character of the source: Φ1 = −ξϕ1

Φ2 = +ξϕ2

Ψ=0

−iδj |x| H 2 (δj | x |) with ϕj = e in the in the three-dimensional model, and ϕj = 0 4i 4π | x | two-dimensional model, with the coefﬁcients given by (14.23-14.26).

Calculation of the successive derivatives of ϕj leads to an expression describing the geometry of the ﬁelds radiated by a force F e: Us =

−

2

F 2

4π| x |

4π| x |

2

αj

j=1

e−iδ3 |x| +α3 |x|

P =

j=1

2

F

αj

e−iδj |x| e{iδj | x | +1} |x|

e−iδj |x| x(x.e) 2 {(iδj | x |) + 3iδj | x | +3} | x | | x |2

5

F x.e ξ 4π | x |2

−e{iδ3 | x | +1} + [e −

x(x.e) 2

|x|

6 2

]{(iδ3 | x |) + 3iδ3 | x | +3}

' e−iδ1 |x| e−iδ2 |x| (iδ1 | x | +1) − (iδ2 | x | +1) |x| |x|

and similarly for a volume V : Us P

' e−iδ1 |x| e−iδ2 |x| (iδ1 | x | +1) − (iδ2 | x | +1) = |x| |x| ' e−iδ1 |x| e−iδ2 |x| V iω 1 α2 + α1 = − 4π K α3 |x| |x| V x ξ 4π | x |2

The symmetries of the ﬁelds can be seen: cylindrical symmetry around the axis of the force and spherical symmetry centered on the volume source. In planar geometry, similar expressions can be established by using the recurrence relations:

1 d z dz

p

2 [z −ν Hν2 (z)] = (−1)p z −ν−p Hν+p (z)

Wave Propagation in Poroelastic Media

441

14.6.5.1. Far-ﬁeld – near-ﬁeld – quasi-static regime In the far-ﬁeld (i.e. at distances from the source which are large compared to the −δj |x| wavelengths, so that | δj x | 1), each of the wavetrains decreases with e 2 : the |x| exponential decrease due to attenuation is added to the normal geometric decrease in an elastic medium. In the near-ﬁeld (i.e. at distances from the source which are small compared to the wavelengths, so that | δj x | 1), expansion of the ϕj functions about zero shows the nature of singularities at the point source. Combinations of the three modes systematically reduces by one order the singularity that each mode creates separately. For the (three-dimensional) near-ﬁeld of a point force, we obtain: F Us ≈ 8π | x | F Uf ≈ 8π | x | P ≈

5

5

6

x(x.e) x(x.e) 1 1 {e − 2 } + μ {e + 2 } λ + 2μ |x| |x|

6

x(x.e) x(x.e) 1 − α/φ μ3 {e − 2 } + μ {e + 2 } λ + 2μ |x| |x|

F x.e δ12 − δ22 8π | x | φ(μ1 − μ2 )

and for a volume source: Us ≈

δ12 − δ22 V x 8π | x | φ(μ1 − μ2 )

Uf ≈

V x 1 4π | x |3 φ

P ≈−

V iω 1 4π K | x |

A remarkable property of the near ﬁelds is that they reintroduce a partial decoupling of the elastic and ﬂow processes. Thus for a force, the near ﬁeld of the solid displacement is independent of frequency, exactly as would be observed for the elastic skeleton: the presence of the ﬂuid and dynamic ﬁltration law has no effect. The H 2 (δj | x |) −1 ≈ singularity has the form | x | (and Log| x | in two dimensions since 0 4i −Log(| x |)/2π) for the ﬂuid and solid displacements (which are different), whereas the pressure is simply discontinuous (continuous in | x | .Log| x | in two dimension). In the same way, for a volume source the near-ﬁeld pressure is that of diffusion in a porous medium characterized by the dynamic permeability, independent of the −1 skeleton’s properties. The singularity has the form | x | for pressure and ﬂuid displacements, and only solid displacements are discontinuous. This reduction in order of the singularities for the coupled terms is a direct consequence of the low degree of differentiation of these terms in the poroelastic operator B. This is found consistently in all near ﬁelds, whatever the nature of the point source.

442

Homogenization of Coupled Phenomena

In the quasi-static regime, the wavelengths tend to inﬁnity. The low-frequency Green’s functions can then be directly deduced from the near-ﬁeld expressions, and there are several simpliﬁcations in the coefﬁcients, which take their values for the δ12 − δ22 −iωη α low-frequency limit (in particular ). ≈ φ(μ1 − μ2 ) K(0) λ + 2μ 14.6.5.2. Decomposition into elementary waves It is often useful to decompose the radiated ﬁeld into elementary waves whose properties are known. In the three-dimensional case, the Sommerﬁeld integral leads to a decomposition into cylindrical waves with axis Oxj (J0 is the zero-order Bessel function): ∞ e−iδ|x| 1 dk 2 = h = [δ 2 − k 2 ]1/2 kJ0 (kr)e−ihxj | x | = x2j + r 2 4π | x | 4π 0 ih We can alternatively use a (homogenous and inhomogenous) plane wave decomposition, obtained using a Fourier transform: ∞ ∞ dk1 dk2 1 e−iδ|x| = h = [δ 2 − k12 − k22 ]1/2 e−ik1 x1 e−ik2 x2 eihx3 | |x| 2π −∞ −∞ ih Similarly, in the two-dimensional case: H02 (δ | x |) 1 = 4i 4π

∞

−∞

e−ikx1 e−ihx2

dk ih

h = [δ 2 − k2 ]1/2

In order to get from these expressions to those in the Green’s matrix, all that is required is to take the derivative of the term under the integral sign. These decompositions, often used in elasticity [AKI 80] for the calculation of synthetic seismograms, and in acoustics [ALL 93], form the basis for the method of discrete wavenumbers [BOU 80] which can be applied directly to poroelastic media [BOU 87a]. 14.6.6. Energy and moment sources: explosion and injection In geophysics, explosions are often used as dynamic sources. Since these involve isotropic sources which cannot be modeled by a force, they are represented in an elastic medium [AKI 80; KAU 06] by a point excitation of the form Dgradδ|x| , which respects the spherical symmetry. Since this source is homogenous with a volumetric force density, D represents an energy. In accordance with its expression, this source corresponds to force dipoles in all three spatial directions. These dipoles (or doublets) are formed of two forces pointing along the same axis, of the same

Wave Propagation in Poroelastic Media

443

amplitude, in opposite directions, and whose points of application, which are inﬁnitely close, lie on the axis. This representation makes it possible to obtain the potentials for the radiated ﬁeld by simple derivation (with respect to the coordinates of the point source) of the potentials associated with a force along each axis, and summation over the three directions. This reasoning, based on the superposition principle, is justiﬁed by the linearity of the behavior. Applying this source to the poroelastic medium skeleton (i.e. in equation (14.20) in place of the point force), we obtain the spherically symmetric potentials (or radial in two dimensional models) of the two compression waves (the S wave is not generated): Φ1 = −α1 (δ12 ϕ1 − δ|x| )D

Φ2 = −α2 (δ22 ϕ2 − δ|x| )D

Ψ=0

Thus, in the same way as two compression modes exist, we have identiﬁed two isotropic sources of different natures. The comparison between a volume injection source and a source formed of three dipoles shows that the relative amplitude ratio of the radiated P2 and P1 waves is multiplied in this last case by the coefﬁcient: C=

φ(μ1 − 1) + α φ(μ2 − 1) + α

Since this term only depends weakly on frequency, it can be approximated by α2 . Consequently for a (λ + 2μ)β ﬂuid which is more compressible than the skeleton, a volume injection favors the appearance of the P2 wave, which the triple dipole does not, but the opposite effect is seen for a less compressible ﬂuid.

its value in the low-frequency limit, C ≈ ζ =

The near ﬁeld of the triple dipole is given below: Us ≈

D x 1 3 4π | x | λ + 2μ

Uf ≈

D x 1 − α/φ 4π | x |3 λ + 2μ

P ≈

D 1 δ22 − δ12 4π | x | φ(μ1 − μ2 )

By comparison with the near ﬁeld of an injected volume, we can deduce the (D, V ) combination of the two isotropic sources, which cancels out the differential ﬂuid-solid displacements close to the point of application: V /D = α/(λ + 2μ) It can be seen that both sources are in phase, and their amplitude ratio is independent of frequency. Considering, following Auriault and Sanchez-Palencia [AUR 77], that under a brief impulse (i.e. with a characteristic time less than 1/fc ), the medium reacts instantaneously as a monophasic medium, this combination of sources allows us to simulate an explosion in a poroelastic medium. Figure 14.6 shows an application of these results to seismic exploration in porous media.

444

Homogenization of Coupled Phenomena

time

receivers

groundwater level

sand

time

k = 5.1010 m2

k = 1010 m2

time

k = 5.109 m2

Figure 14.6. Simulation of seismic exploration using explosions. For sufﬁciently permeable soils, the effect of the P2 wave generated at the source is visible on the synthetic seismograms calculated at the surface [BOU 87b]

Wave Propagation in Poroelastic Media

445

Analogously, we can introduce a source which corresponds to a pair of torques (oriented in the same direction) along each axis. For example the double torque Mz along ez is formed of two forces along ex (or ey ) with the same amplitude, opposite directions, and whose points of application, inﬁnitely close, lie on the Oy (or Ox) axis. The expression for the potentials can be obtained by derivation (with respect to the coordinates of the point source) of the potentials radiated by the forces. Thus for a moment Mz we obtain: Φ1 = Φ2 = 0

Ψ=

Mz x ∧ ez ϕ3 μ |x|

A pair of torques with a given orientation is thus the source of a single shear wave, polarized perpendicular to the axis and with cylindrical symmetry around that axis. 14.7. Integral representation Knowledge of Green’s functions makes it possible to develop an integral representation. This formulation offers the possibility of numerically solving dynamic boundary-value problems in porous media using the boundary equations method [PRE 84; BOU 89a]. Consider a porous medium occupying a volume Ω bounded by the closed surface Γ, and suppose that the response vector and its gradient are known at all points on Γ, which has the exterior normal n (this point will be elaborated on later). To begin with, we will also assume that the medium is free from any sources. The aim is to ﬁnd the response R, consistent with the boundary conditions on boundary Γ, and such that inside Ω: B.R + s = 0

with

s=0

In order to ﬁnd R, we will introduce the distribution R which is deﬁned over all space by: R =

R within the volume

R =

0

Ω

outside the volume Ω

Expressed in the space of distributions, the differential system becoms: B∗R+S =0

B = B.δ|x| I

where S is a distribution which has a value of zero outside Γ since by construction: B.R = B.R =

0

within the volume

Ω

B.R = B.0 = 0 outside the volume Ω

446

Homogenization of Coupled Phenomena

On the surface, distribution associated with the discontinuities in R and in its gradient can be calculated by differentiation of distributions. At all points on Γ, the derivative of R is: ∂j R = ∂j R + [R](n.ej δΓ ) where [R] is the step change in R across Γ along n, so that [R] = 0 − R,and δΓ is the Dirac distribution attached to the surface Γ (deﬁned by: < δΓ , g >= Γ gds for all functions g). In the same way: ∂jk R = ∂jk R + [∂j R](n.ek δΓ ) + [R](n.ek ∂j δΓ ) = ∂jk R − ∂j R(n.ek δΓ ) − R(n.ek ∂j δΓ ) Using these identities, we obtain the distribution over the boundary in condensed form: B ∗ R = −(Cn R)δΓ −t (Cn δΓ )R = −S where Cn is the (4x4) differential operator of order 1, which is not symmetrical and is deﬁned by: Cnii = λni ∂i + μni ∂i + μ∂n , Cnij = λni ∂j + μnj ∂i , , Cni4 = −αni , Cn4i = −iωρf

Cn44 = −

i = 1, 2, 3

i, j = 1, 2, 3;

i = j

i = 1, 2, 3 K(ω) ni , η

i = 1, 2, 3

K(ω) ∂n iωη

The physical signiﬁcance of this operator becomes clear when it is applied to a response vector R: – for the three ﬁrst components (j = 1, 2, 3), we obtain: (Cn .R)j = (σ.n)j – the fourth component making use of the dynamic Darcy’s law (14.3) gives: (Cn .R)4 = n(Uf − Us ).n

Wave Propagation in Poroelastic Media

447

Thus we recover the stress vector exerted on all surface elements dsn of boundary Γ and the ﬂuid ﬂux across it. In other words, Cn .Rds expresses the source (in terms of force and injected ﬂuid) which results from the stress and ﬂux values on the boundary. If the distribution S is known, inversion of the initial system can be achieved by convolution with Green’s matrix: G ∗ (B ∗ R) = −G ∗ S = −G ∗ {(Cn .R)δΓ +t (Cn δΓ ).R} But, by construction G is the inverse convolution of B, and using the commutativity and associativity of the convolution, we obtain: G ∗ (B ∗ R) = (G ∗ B) ∗ R = −δ|x| I ∗ R = −R Before we reveal the integral formulation (in function space), recall that the convolutions with derivatives of δΓ introduce a change of sign (for example < ∂1 δΓ , g >= − Γ ∂1 gds). Because of this, we are led to introduce the operator Cn obtained from Cn by changing the signs of the derivative-free terms, so that: i

j

Cn i = (−1)δ4 +δ4 Cn ji j

Finally, at all points M inside Ω: G(M −Q) .Cn Q RQ ds − RM = RM = Γ

and at all points M outside Ω: RM = 0 = G(M −Q) .Cn Q RQ ds − Γ

Γ

t

t Γ

(Cn Q G(M −Q) ).RQ ds t

(Cn Q G(M −Q) ).RQ ds t

According to the Huygens-Fresnel principle, this result can be interpreted as meaning that the response inside any closed surface can always be considered as being produced by a suitable distribution of sources lying on the surface, whose amplitude is deﬁned by the static and kinematic conditions on that surface. Also to be found in this formulation are the boundary conditions that must be taken into account for wellposed problems, either in terms of displacement and pressure, or in terms of stress and ﬂux. For points on the boundary, this expression is not valid, since the matrices G and Cn .t G display singularities when the receiver and source are merged. This issue can be addressed by approaching this limit using the near-ﬁeld expressions (and their derivatives). The order of singularities ensures the convergence of the integral over G. Conversely, since the singularities of Cn .t G are of a higher order (since the operator Cn introduces an additional derivation of the terms of G), the associated integral t

448

Homogenization of Coupled Phenomena

becomes inﬁnite. As in the case of elasticity, it can be decomposed into its principal Cauchy value and a (self-)inﬂuence tensor A (4x4) applied to the response, such that if Q0 ∈ Γ: t t t t (Cn Q G(Q0 −Q) ).RQ ds = (Cn Q G(Q0 −Q) ).RQ ds + AQ0 .RQ0 Γ

v.p.Γ

The inﬂuence tensor at a given point depends on the geometry of the surface at this point and can be calculated analytically, but when a tangent plane exists it can be shown by consideration of the symmetry of the near ﬁelds that: AQ0 = − 12 I. We therefore obtain at all points Q0 on a regular surface: 1 t t RQ0 = G(Q0 −Q) .Cn Q RQ ds − (Cn Q G(Q0 −Q) ).RQ ds 2 Γ v.p.Γ Finally, if the medium Ω also contains a source distribution s, we simply need to add them to S. A complementary integral then appears which is due to the radiation of these internal sources, and ﬁnally the formulation becomes: Ω δM RM = G(M −Q) .(Cn Q .RQ )ds Γ

−

t v.p.Γ

(Cn Q G(M −Q) ).RQ ds + t

Ω

G(M −P ) .sP dv

Ω Ω Ω = 1 if M ∈ Ω; δM = 1/2 if M ∈ Γ; or δM = 0 otherwise with: δM

This formulation can be directly used for the development of methods of solution by boundary elements in porous media. We should emphasize again that this integral representation is only valid in the harmonic regime. The return to the time domain for transient regimes can be obtained through use of the inverse Fourier transform, which substitutes convolution products of temporal variables in place of the products of the corresponding complex variables in the frequency domain. 14.8. Dislocations in porous media In this section we repeat the results presented in Boutin [BOU 89a]. The integral representation indicates that, just as the term G.(Cn .R)ds expresses the ﬁeld radiated by discontinuities in stress and the ﬂux over elementary surface dsn lying at the origin, t similarly the term t (Cn G).Rds corresponds to the ﬁeld radiated by discontinuities in displacement and pressure across this same surface, oscillating harmonically in time. Thus the displacement-pressure response due to: – an extending ﬁssure (at the origin, and abrupt), in other words a discontinuity in the component of Us perpendicular to the surface dsn (Us .n discontinuous);

Wave Propagation in Poroelastic Media

449

– a slip (discontinuity in the displacement component parallel to the surface); – a pressure discontinuity across the surface; t are given by the matrix t (Cn G) which we will denote Tn . By analogy with elasticity, we will refer to these types of sources as dislocations. As we are considering isotropic media, it is possible to ﬁx the orientation of the surface element without losing the generality of the results. In what follows we will take n = e1 . The displacement/pressure ﬁeld t1 created by an extension dislocation [Us1 ] ([.] indicates the step change in the variable) across dse1 is given by the ﬁrst column of Te 1 , so that: t1 i = (Te 1 )i1 = {t (Ce 1 G)}i1 [Us1 ]ds t

In the same way the displacement/pressure ﬁelds created by a slip dislocation [Us2 ] orientated along e2 along the surface dse1 (or a slip dislocation [Us3 ] along dse3 ) are given by t2 and t3 respectively, the second and third column of Te 1 , so that: t2 i = (Te 1 )i2 = {t (Ce 1 G)}i2 [Us2 ]ds t

t3 i = (Te 1 )i3 = {t (Ce 1 G)}i3 [Us3 ]ds t

Finally, a step change in pressure [P] across dse1 generates the t4 ﬁeld: t4 i = (Te 1 )i4 = {t (Ce 1 G)}i4 [P ]ds t

The expression for operator Cn (with n = e1 ) makes it possible to express the different components of these ﬁelds, remembering that the derivatives are taken with respect to the coordinates of the point source: t1 i

=

λ

3

∂j Gij + 2μ∂1 Gi1 − αGi4

(14.28)

j=1

t2 i

=

μ(∂2 Gi1 + ∂1 Gi2 )

(14.29)

t3 i

=

μ(∂3 Gi1 + ∂1 Gi3 )

(14.30)

t4 i

=

K(ω) (ρf ω 2 Gi1 − ∂1 Gi4 ) iωη

(14.31)

Knowing the potentials associated with Green’s functions, it is easy to determine the potentials of ﬁelds radiated by the dislocations, whose expressions can be found summarized in Table 14.5.

450

Homogenization of Coupled Phenomena

Sources F ek

P1 wave Φ1 −α1 ∂k ϕ1

P2 wave Φ2 −α2 ∂k ϕ2

S wave Ψ −α3 curl(ϕ3 ek )

V

−ξϕ1

+ξϕ2

0

D

−α1 δ12 ϕ1

−α2 δ22 ϕ2

0

[Usj ]ek

[P ]ek

2μα1 ∂jk ϕ1 2μα2 ∂jk ϕ2 −μα3 curl(∂j ϕ3 ek + ∂k ϕ3 ej ) −δkj (λα1 δ12 + αξ)ϕ1 −δkj (λα2 δ22 + αξ)ϕ2 iω K(ω) ρf α1 ∂k ϕ1 η (ω) K + ξϕ iωη

1

iω K(ω) ρf α2 ∂k ϕ2 η (ω) K + ξϕ iωη

iω K(ω) ρf α3 curl(ϕ3 ek ) η

2

Table 14.5. Potentials (undeﬁned at the origin) of the ﬁelds radiated by abrupt unit sources. The coefﬁcients are given in (14.23-14.26) and the ϕi functions in (14.22) (three-dimensional case) and (14.27) (two-dimensional case)

Additionally, in these expressions each derivative can be interpreted as the response due to a source dipole (for example a dipole along e1 for Ri = ∂1 Gi1 ) or a torque (for example a torque along e2 ∧ e3 for Ri = ∂3 Gi2 ). Consequently, as in the case of elasticity [BUR 64] each dislocation is associated with an equivalent distribution of volume forces. For an extension (or normal) dislocation [Us1 ] along dse1 the equivalent source system is given by: SN f = {λgradδ|x| + 2μe1 ∂1 δ|x| }[Us1 ]ds

SvN = −αδ|x| [Us1 ]ds

(14.32)

which is a combination of triple force dipoles, an additional force dipole along e1 , and a ﬂuid injection. For a slip (or tangential) dislocation [Us2 ] along dse1 the equivalent source system is a double torque, in other words an abrupt distribution with a net force and torque of zero, formed from two opposing torques ±e1 ∧ e2 : ST f = μ{e2 ∂1 δ|x| + e1 ∂2 δ|x| }[Us2 ]ds

SvT = O

(14.33)

For a step change in pressure across dse1 , we obtain the distribution: SP f =−

K(ω) iωρf e1 δ|x| [P ]ds η

SvP = −

K(ω) ∂1 δ|x| [P ]ds iωη

which includes a force and a dipole of injected ﬂuid.

(14.34)

Wave Propagation in Poroelastic Media

451

If we compare the equivalent volume forces at dislocations in poroelasticity and elasticity, it can be seen that they are the same for a slip (which is consistent with the absence of resistance to ﬂuid shear), but in normal dislocation an additional source appears due to the presence of ﬂuid. In the case of a step change in pressure we note that the skeleton is also disturbed. In order to better understand these equivalent sources, it is interesting to identify the variables associated with each of them via the expression for complex work. We will consider an inﬁnite medium and calculate the complex work for an abrupt ˜ s , P˜ ). dislocation on an arbitrary test ﬁeld of the displacement-pressure response (U As with the reciprocity theorem, we have: W =

˜ s Sf − P˜ .Sv )dv (U

In the case of a normal dislocation: W

N

= [Us1 ]ds

˜ s {λgradδ|x| + 2μe1 ∂1 δ|x| } − P˜ .αδ|x| dv U

so that: ˜ s + 2μ∂1 U ˜ s − αP˜ }[Us ]ds = σ ˜11 [Us1 ]ds W N = {λdivU 1 1 We also obtain by similar calculations for a tangential dislocation and a step change in pressure: WT = σ ˜12 [Us2 ]ds

˜f −U ˜ s }[P ]ds W P = n{U 1 1

Thus in every case the dual variable is recovered, either in stress or in ﬂux. Following the same procedure as for elementary sources, it is possible to determine the near ﬁelds radiated by the dislocations (they can also be obtained by derivation and linear combination of the near ﬁelds of elementary sources, in accordance with expressions (14.33-14.34) for the equivalent volume forces at the dislocations). It can again be shown that for dislocations of the solid matrix (discontinuity in displacement), the near ﬁeld of solid displacement is identical to that which is observed for the elastic −2 skeleton’s alone. The singularity has the form | x | for ﬂuid and solid displacements −1 (which are different) and is one order lower for the pressure, | x | . For a step change in pressure, the near pressure ﬁeld is that of the porous medium characterized by dynamic permeability, independent of the skeleton’s properties. The singularities −2 −1 go as | x | for the pressure and ﬂuid displacements, and as | x | for the solid

452

Homogenization of Coupled Phenomena

displacements. The expressions for the near ﬁelds are given below: Near ﬁeld associated with [Us1 ]e1 ds 5 5 6 6 [Us1 ]ds μ μ μ x21 x − + 2x1 e1 + 3(1 − ) Us ≈ 3 λ + 2μ λ + 2μ | x |2 λ + 2μ 4π| x | [Us1 ]ds P ≈ 4π| x |

5

6

x2 δ 2 − δ22 ω ρf − μ 1 (1 − 1 2 ) φ(μ1 − μ2 ) |x| 2

Near ﬁeld associated with [Us2 ]e1 ds: 5 6 [Us2 ]ds μ μ x 1 x2 + (x1 e1 − x2 e2 ) 3x(1 − ) Us ≈ 3 λ + 2μ | x |2 λ + 2μ 4π| x | P ≈−

δ 2 − δ22 x21 [Us2 ]ds 2μ 1 4π| x | φ(μ1 − μ2 ) | x |2

Near ﬁeld associated with [P ]e1 ds: 5 6 α [P ]ds xx21 ω 2 ρf K(ω) 1 xx21 (e1 − (e1 + Us ≈ − 2) − 2) 8π| x | λ + 2μ η μ |x| |x| P ≈−

[P ]ds x1 4π| x | | x |2

For the integral representation, these results can be used to show that at an angular point Q0 the coupled terms (between displacement and pressure) of the inﬂuence tensor AQ0 are zero, with the other terms being those for the elastic medium formed by the skeleton and for diffusion. To conclude, we emphasize that for Green’s functions, the general expressions (14.28-14.31) for ﬁelds radiated by dislocations are valid at all frequencies and at all distances, with the transformation to the time domain being achieved by a Fourier transform. These results ﬁnd practical application in inversion problems in acoustic emission, hydraulic fracturing and in seismology.

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Index

A acoustics in a rigid porous medium saturated with a gas, 253 additional mass, 386 advection, 155, 157 asymptotic expansion, 68 B Biot model, 362, 380 Biot tensor, 408 Biot-Allard model, 261 bounds Hashin and Shtrikman, 49, 172, 179, 410 on dynamic permeability, 324 on static permeability, 301, 308 Reuss, 46, 172, 179, 409 Voigt, 44, 172, 179, 409

composite bilaminate, 34, 123, 129, 145 elastic, 108 compression simple, 107 concrete aerated, 196 bituminous, 396 conditions homogenizability, 66 connectivity of phases, 56, 336 consistency energetic, 214 contact resistance, 191 continuum equivalent, 59 convection, 149 coupling, 142, 163, 366 D

C Casagrande formula, 307 characteristic length, 39, 66, 104 characteristic time of advection, 151 of diffusion, 151 choice of model, 146, 165, 234, 375, 391 coefﬁcient effective, 93 effective poroelastic numerical estimates, 401

Darcy’s law, 205, 207 generalized, 219, 382 Darcy-Weisbach equation, 205 deformable saturated porous medium quasi-static behavior, 359 description macroscopic, 70 diffusion, 149, 152 diffusion-advection, 155 dispersion, 160 doubly porous, 77

474

Homogenization of Coupled Phenomena

E effect additional mass, 223 memory, 125 effective coefﬁcient, 93 effective poroelastic coefﬁcients deﬁnition, 362 properties, 365 elasto-dynamics, 95 elasto-statics, 89 elementary volume, 68 representative, 72, 74 energetic consistency, 214, 221, 358 equivalent continuum, 59 estimate self-consistent, 55 F ﬁnite element, 171, 177, 270 ﬂuid compressible Newtonian, 237 force point, 107 Forchheimer law, 205 form factor, 280, 312 Fredholm alternative, 70

spherical bi-composite, 183 strongly conducting, 174 weakly conducting, 125, 174 interface elastic-poroelastic, 434 K Klinkenberg effect, 246, 328 Klinkenberg tensor, 251 Klinkenberg’s law, 246 Knudsen number, 248 Kozeny-Carman formula, 306 L lattice centered cubic, 170, 275 face centered cubic, 275 of cylindrical pores, 224 of narrow parallel slits, 162 of parallel ﬁbers, 176, 268 simple cubic, 170, 275 length characteristic, 39, 66, 104 length scales separation of, 165 M

G Gassman model, 371, 387 Green’s functions, 442 H Hazen formula, 307 Hill macro-homogeneity, 73 Hill principle, 43 homogenizability conditions, 66 homogenization, 33, 61 I inclusion connected, 170 dispersed, 170 highly conductive, 130 simple spherical, 53, 170 spherical, 292

macroscopic behavior diphasic, 362, 380 monophasic elastic, 371, 387 monophasic viscoelastic, 373, 389 macroscopic description, 70 mass additional, 223 material composite, 113, 170 doubly porous, 77 periodic structure, 72 random structure, 72 with periodic structure, 170 medium aerated, 183, 196 cellular, 183, 196 deformable porous saturated dynamic behavior, 377 quasi-static behavior, 349

Index

ﬁbrous static permeability, 268, 288, 307 thermal conductivity, 176, 186 granular dynamic permeability, 277, 314 Klinkenberg permeability, 328 poroelastic coefﬁcients, 401 static permeability, 277, 288, 291 thermal conductivity, 170, 183 thermal permeability, 331 trapping constant, 340 periodic static permeability, 268 rigid porous comparable diffusion and advection, 155 diffusion dominated, 152 diffusion-convection, 149 dispersion model, 160 dynamics of an incompressible ﬂuid, 217 saturated by a gas: acoustics, 253 memory effect, 386 mesh periodic, 82 method homogenization, 33, 61 multiple scale expansion, 65, 73, 79 self-consistent, 50, 53 statistical, 73 upscaling, 33 model Biot, 362, 380 Biot-Allard, 261 choice of, 104, 146, 165, 234, 375, 391 Gassman, 371, 387 statistical, 64 two-ﬁeld, 142 multiple scales, 79, 104

475

numerical simulation, 171, 177, 270 P Péclet number, 151 perforated plate, 85 periodicity, 68 quasi-, 94 permeability dynamic bounds, 324 numerical estimates, 265 of granular media, 277, 314 of periodic media, 277 self-consistent estimates, 285, 314 tensor properties, 223 Klinkenberg (estimate), 328 static bounds, 301, 308 empirical laws, 306 numerical estimates, 265 of ﬁbrous media, 268, 288, 307 of granular media, 277, 288, 291 of periodic media, 268 self-consistent estimates, 285 tensor properties, 213 tensor dynamic, 221 static, 210 thermal (estimate), 331 phases connectivity, 56 plate perforated, 85 point force, 107 poroelasticity, 362 propagation wave, 108 R

N Navier-Stokes, 207 Newtonian ﬂuid incompressible, 205 number Biot, 136 Péclet, 151

radiated ﬁelds, 450 resistance contact, 134, 191 Reynolds number, 208, 229, 238, 361 rigid porous medium dynamics of an incompressible ﬂuid, 217

476

Homogenization of Coupled Phenomena

inertial non-linearities of a ﬂow, 228 saturated by a gas: acoustics, 253 slow ﬂow of a Newtonian ﬂuid compressible, 237 S scales multiple, 65, 79, 104 separation of, 60, 66, 73, 79, 104, 165 self-consistent combined methods, 192 estimate, 55, 172, 410 method, 50, 53, 183 separation of scales, 60, 66, 73, 79, 104 simple compression, 107 slits narrow parallel, 162 sources, 453 spatial variables, 100 macroscopic, 66 microscopic, 66 stationarity, 68 statistical modeling, 64 stress effective, 370 partial, 370, 384 total, 354, 364, 382 Strouhal number, 238 structure periodic, 72 random, 72 substructure n-composite, 189 bi-composite, 183

T thermal conductivity contrast, 125, 130, 174, 181 effective, 122 ﬁbrous media, 176, 186 granular media, 170, 183 numerical estimates, 170 periodic media, 170 self-consistent estimates, 183 tortuosity high frequency, 280, 312 low frequency, 278, 311 transfer diffusion-convection, 149 diffusive, 149 heat, 34 transport thermal, 113 trapping constant, 338 for a granular medium, 340 U unit cell, 68 upscaling, 33 W wave diffusion, 109 dispersive, 387 modes of propagation, 422 plane, 108, 426 propagation in a porous saturated medium, 417 Rayleigh, 440 wave equations, 423 wave propagation, 108