The Rock Physics Handbook - PDF Free Download

This page intentionally left blank The Rock Physics Handbook, Second Edition Tools for Seismic Analysis of Porous Med...

Author: Gary Mavko | Tapan Mukerji | Jack Dvorkin

1005 downloads 4048 Views 4MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

This page intentionally left blank

The Rock Physics Handbook, Second Edition Tools for Seismic Analysis of Porous Media

The science of rock physics addresses the relationships between geophysical observations and the underlying physical properties of rocks, such as composition, porosity, and pore fluid content. The Rock Physics Handbook distills a vast quantity of background theory and laboratory results into a series of concise, self-contained chapters, which can be quickly accessed by those seeking practical solutions to problems in geophysical data interpretation. In addition to the wide range of topics presented in the First Edition (including wave propagation, effective media, elasticity, electrical properties, and pore fluid flow and diffusion), this Second Edition also presents major new chapters on granular material and velocity–porosity–clay models for clastic sediments. Other new and expanded topics include anisotropic seismic signatures, nonlinear elasticity, wave propagation in thin layers, borehole waves, models for fractured media, poroelastic models, attenuation models, and cross-property relations between seismic and electrical parameters. This new edition also provides an enhanced set of appendices with key empirical results, data tables, and an atlas of reservoir rock properties expanded to include carbonates, clays, and gas hydrates. Supported by a website hosting MATLAB routines for implementing the various rock physics formulas presented in the book, the Second Edition of The Rock Physics Handbook is a vital resource for advanced students and university faculty, as well as in-house geophysicists and engineers working in the petroleum industry. It will also be of interest to practitioners of environmental geophysics, geomechanics, and energy resources engineering interested in quantitative subsurface characterization and modeling of sediment properties. Gary Mavko received his Ph.D. in Geophysics from Stanford University in 1977

where he is now Professor (Research) of Geophysics. Professor Mavko co-directs the Stanford Rock Physics and Borehole Geophysics Project (SRB), a group of approximately 25 researchers working on problems related to wave propagation in earth materials. Professor Mavko is also a co-author of Quantitative Seismic Interpretation (Cambridge University Press, 2005), and has been an invited instructor for numerous industry courses on rock physics for seismic reservoir characterization. He received the Honorary Membership award from the Society of Exploration Geophysicists (SEG) in 2001, and was the SEG Distinguished Lecturer in 2006. Tapan Mukerji received his Ph.D. in Geophysics from Stanford University in 1995 and

is now an Associate Professor (Research) in Energy Resources Engineering and

a member of the Stanford Rock Physics Project at Stanford University. Professor Mukerji co-directs the Stanford Center for Reservoir Forecasting (SCRF) focusing on problems related to uncertainty and data integration for reservoir modeling. His research interests include wave propagation and statistical rock physics, and he specializes in applied rock physics and geostatistical methods for seismic reservoir characterization, fracture detection, time-lapse monitoring, and shallow subsurface environmental applications. Professor Mukerji is also a co-author of Quantitative Seismic Interpretation, and has taught numerous industry courses. He received the Karcher award from the Society of Exploration Geophysicists in 2000. Jack Dvorkin received his Ph.D. in Continuum Mechanics in 1980 from Moscow

University in the USSR. He has worked in the Petroleum Industry in the USSR and USA, and is currently a Senior Research Scientist with the Stanford Rock Physics Project at Stanford University. Dr Dvorkin has been an invited instructor for numerous industry courses throughout the world, on rock physics and quantitative seismic interpretation. He is a member of American Geophysical Union, Society of Exploration Geophysicists, American Association of Petroleum Geologists, and the Society of Petroleum Engineers.

The Rock Physics Handbook, Second Edition Tools for Seismic Analysis of Porous Media

Gary Mavko Stanford University, USA

Tapan Mukerji Stanford University, USA

Jack Dvorkin Stanford University, USA

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521861366 © G. Mavko, T. Mukerji, and J. Dvorkin 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13

978-0-511-65062-8

eBook (NetLibrary)

ISBN-13

978-0-521-86136-6

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page xi

1

Basic tools

1.1 1.2 1.3 1.4

The Fourier transform The Hilbert transform and analytic signal Statistics and probability Coordinate transformations

1 6 9 18

2

Elasticity and Hooke’s law

21

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

21 23 35 39 40 43 47

2.9 2.10 2.11 2.12

Elastic moduli: isotropic form of Hooke’s law Anisotropic form of Hooke’s law Thomsen’s notation for weak elastic anisotropy Tsvankin’s extended Thomsen parameters for orthorhombic media Third-order nonlinear elasticity Effective stress properties of rocks Stress-induced anisotropy in rocks Strain components and equations of motion in cylindrical and spherical coordinate systems Deformation of inclusions and cavities in elastic solids Deformation of a circular hole: borehole stresses Mohr’s circles Static and dynamic moduli

3

Seismic wave propagation

81

3.1 3.2

Seismic velocities Phase, group, and energy velocities

81 83

v

1

54 56 68 74 76

vi

Contents

3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

NMO in isotropic and anisotropic media Impedance, reflectivity, and transmissivity Reflectivity and amplitude variations with offset (AVO) in isotropic media Plane-wave reflectivity in anisotropic media Elastic impedance Viscoelasticity and Q Kramers–Kronig relations between velocity dispersion and Q Waves in layered media: full-waveform synthetic seismograms Waves in layered media: stratigraphic filtering and velocity dispersion Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation Scale-dependent seismic velocities in heterogeneous media Scattering attenuation Waves in cylindrical rods: the resonant bar Waves in boreholes

138 146 150 155 160

4

Effective elastic media: bounds and mixing laws

169

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17

Hashin–Shtrikman–Walpole bounds Voigt and Reuss bounds Wood’s formula Voigt–Reuss–Hill average moduli estimate Composite with uniform shear modulus Rock and pore compressibilities and some pitfalls Kuster and Tokso¨z formulation for effective moduli Self-consistent approximations of effective moduli Differential effective medium model Hudson’s model for cracked media Eshelby–Cheng model for cracked anisotropic media T-matrix inclusion models for effective moduli Elastic constants in finely layered media: Backus average Elastic constants in finely layered media: general layer anisotropy Poroelastic Backus average Seismic response to fractures Bound-filling models

169 174 175 177 178 179 183 185 190 194 203 205 210 215 216 219 224

5

Granular media

229

5.1 5.2

Packing and sorting of spheres Thomas–Stieber model for sand–shale systems

229 237

86 93 96 105 115 121 127 129 134

vii

Contents

5.3 5.4 5.5

Particle size and sorting Random spherical grain packings: contact models and effective moduli Ordered spherical grain packings: effective moduli

245 264

6

Fluid effects on wave propagation

266

6.1 6.2 6.3 6.4

266 272 273

6.20 6.21

Biot’s velocity relations Geertsma–Smit approximations of Biot’s relations Gassmann’s relations: isotropic form Brown and Korringa’s generalized Gassmann equations for mixed mineralogy Fluid substitution in anisotropic rocks Generalized Gassmann’s equations for composite porous media Generalized Gassmann equations for solid pore-filling material Fluid substitution in thinly laminated reservoirs BAM: Marion’s bounding average method Mavko–Jizba squirt relations Extension of Mavko–Jizba squirt relations for all frequencies Biot–squirt model Chapman et al. squirt model Anisotropic squirt Common features of fluid-related velocity dispersion mechanisms Dvorkin–Mavko attenuation model Partial and multiphase saturations Partial saturation: White and Dutta–Ode´ model for velocity dispersion and attenuation Velocity dispersion, attenuation, and dynamic permeability in heterogeneous poroelastic media Waves in a pure viscous fluid Physical properties of gases and fluids

331 338 339

7

Empirical relations

347

7.1

Velocity–porosity models: critical porosity and Nur’s modified Voigt average Velocity–porosity models: Geertsma’s empirical relations for compressibility Velocity–porosity models: Wyllie’s time-average equation Velocity–porosity models: Raymer–Hunt–Gardner relations

6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19

7.2 7.3 7.4

242

282 284 287 290 292 295 297 298 302 304 306 310 315 320 326

347 350 350 353

viii

Contents

7.5

7.8 7.9 7.10 7.11 7.12 7.13 7.14

Velocity–porosity–clay models: Han’s empirical relations for shaley sandstones Velocity–porosity–clay models: Tosaya’s empirical relations for shaley sandstones Velocity–porosity–clay models: Castagna’s empirical relations for velocities VP–VS–density models: Brocher’s compilation VP–VS relations Velocity–density relations Eaton and Bowers pore-pressure relations Kan and Swan pore-pressure relations Attenuation and quality factor relations Velocity–porosity–strength relations

358 359 363 380 383 383 384 386

8

Flow and diffusion

389

8.1 8.2 8.3 8.4 8.5 8.6 8.7

Darcy’s law Viscous flow Capillary forces Kozeny–Carman relation for flow Permeability relations with Swi Permeability of fractured formations Diffusion and filtration: special cases

389 394 396 401 407 410 411

9

Electrical properties

414

9.1 9.2 9.3 9.4 9.5

Bounds and effective medium models Velocity dispersion and attenuation Empirical relations Electrical conductivity in porous rocks Cross-property bounds and relations between elastic and electrical parameters

414 418 421 424 429

Appendices

437

Typical rock properties Conversions Physical constants Moduli and density of common minerals Velocities and moduli of ice and methane hydrate

437 452 456 457 457

7.6 7.7

A.1 A.2 A.3 A.4 A.5

355 357

ix

Contents

A.6 A.7 A.8

Physical properties of common gases Velocity, moduli, and density of carbon dioxide Standard temperature and pressure

468 474 474

References Index

479 503

Preface to the Second Edition

In the decade since publication of the Rock Physics Handbook, research and use of rock physics has thrived. We hope that the First Edition has played a useful role in this era by making the scattered and eclectic mass of rock physics knowledge more accessible to experts and nonexperts, alike. While preparing this Second Edition, our objective was still to summarize in a convenient form many of the commonly needed theoretical and empirical relations of rock physics. Our approach was to present results, with a few of the key assumptions and limitations, and almost never any derivations. Our intention was to create a quick reference and not a textbook. Hence, we chose to encapsulate a broad range of topics rather than to give in-depth coverage of a few. Even so, there are many topics that we have not addressed. While we have summarized the assumptions and limitations of each result, we hope that the brevity of our discussions does not give the impression that application of any rock physics result to real rocks is free of pitfalls. We assume that the reader will be generally aware of the various topics, and, if not, we provide a few references to the more complete descriptions in books and journals. The handbook contains 101 sections on basic mathematical tools, elasticity theory, wave propagation, effective media, elasticity and poroelasticity, granular media, and pore-fluid flow and diffusion, plus overviews of dispersion mechanisms, fluid substitution, and VP–VS relations. The book also presents empirical results derived from reservoir rocks, sediments, and granular media, as well as tables of mineral data and an atlas of reservoir rock properties. The emphasis still focuses on elastic and seismic topics, though the discussion of electrical and cross seismic-electrical relations has grown. An associated website (http://srb.stanford.edu/books) offers MATLAB codes for many of the models and results described in the Second Edition. In this Second Edition, Chapter 2 has been expanded to include new discussions on elastic anisotropy including the Kelvin notation and eigenvalues for stiffnesses, effective stress behavior of rocks, and stress-induced elasticity anisotropy. Chapter 3 includes new material on anisotropic normal moveout (NMO) and reflectivity, amplitude variation with offset (AVO) relations, plus a new section on elastic impedance (including anisotropic forms), and updates on wave propagation in stratified media, and borehole waves. Chapter 4 includes updates of inclusion-based effective media models, thinly layered media, and fractured rocks. Chapter 5 contains xi

xii

Preface

extensive new sections on granular media, including packing, particle size, sorting, sand–clay mixture models, and elastic effective medium models for granular materials. Chapter 6 expands the discussion of fluid effects on elastic properties, including fluid substitution in laminated media, and models for fluid-related velocity dispersion in heterogeneous poroelastic media. Chapter 7 contains new sections on empirical velocity–porosity–mineralogy relations, VP –VS relations, pore-pressure relations, static and dynamic moduli, and velocity–strength relations. Chapter 8 has new discussions on capillary effects, irreducible water saturation, permeability, and flow in fractures. Chapter 9 includes new relations between electrical and seismic properties. The Appendices has new tables of physical constants and properties for common gases, ice, and methane hydrate. This Handbook is complementary to a number of other excellent books. For indepth discussions of specific rock physics topics, we recommend Fundamentals of Rock Mechanics, 4th Edition, by Jaeger, Cook, and Zimmerman; Compressibility of Sandstones, by Zimmerman; Physical Properties of Rocks: Fundamentals and Principles of Petrophysics, by Schon; Acoustics of Porous Media, by Bourbie´, Coussy, and Zinszner; Introduction to the Physics of Rocks, by Gue´guen and Palciauskas; A Geoscientist’s Guide to Petrophysics, by Zinszner and Pellerin; Theory of Linear Poroelasticity, by Wang; Underground Sound, by White; Mechanics of Composite Materials, by Christensen; The Theory of Composites, by Milton; Random Heterogeneous Materials, by Torquato; Rock Physics and Phase Relations, edited by Ahrens; and Offset Dependent Reflectivity – Theory and Practice of AVO Analysis, edited by Castagna and Backus. For excellent collections and discussions of classic rock physics papers we recommend Seismic and Acoustic Velocities in Reservoir Rocks, Volumes 1, 2 and 3, edited by Wang and Nur; Elastic Properties and Equations of State, edited by Shankland and Bass; Seismic Wave Attenuation, by Tokso¨z and Johnston; and Classics of Elastic Wave Theory, edited by Pelissier et al. We wish to thank the students, scientific staff, and industrial affiliates of the Stanford Rock Physics and Borehole Geophysics (SRB) project for many valuable comments and insights. While preparing the Second Edition we found discussions with Tiziana Vanorio, Kaushik Bandyopadhyay, Ezequiel Gonzalez, Youngseuk Keehm, Robert Zimmermann, Boris Gurevich, Juan-Mauricio Florez, Anyela Marcote-Rios, Mike Payne, Mike Batzle, Jim Berryman, Pratap Sahay, and Tor Arne Johansen, to be extremely helpful. Li Teng contributed to the chapter on anisotropic AVOZ, and Ran Bachrach contributed to the chapter on dielectric properties. Dawn Burgess helped tremendously with editing, graphics, and content. We also wish to thank the readers of the First Edition who helped us to track down and fix errata. And as always, we are indebted to Amos Nur, whose work, past and present, has helped to make the field of rock physics what it is today. Gary Mavko, Tapan Mukerji, and Jack Dvorkin.

1

Basic tools

1.1

The Fourier transform Synopsis The Fourier transform of f(x) is defined as Z 1 FðsÞ ¼ f ðxÞei2pxs dx 1

The inverse Fourier transform is given by Z 1 f ðxÞ ¼ FðsÞeþi2pxs ds 1

Evenness and oddness A function E(x) is even if E(x) ¼ E(–x). A function O(x) is odd if O(x) ¼ –O(–x). The Fourier transform has the following properties for even and odd functions: Even functions. The Fourier transform of an even function is even. A real even function transforms to a real even function. An imaginary even function transforms to an imaginary even function. Odd functions. The Fourier transform of an odd function is odd. A real odd function transforms to an imaginary odd function. An imaginary odd function transforms to a real odd function (i.e., the “realness” flips when the Fourier transform of an odd function is taken). real even (RE) ! real even (RE) imaginary even (IE) ! imaginary even (IE) real odd (RO) ! imaginary odd (IO) imaginary odd (IO) ! real odd (RO) Any function can be expressed in terms of its even and odd parts: f ðxÞ ¼ EðxÞ þ OðxÞ where EðxÞ ¼ 12½ f ðxÞ þ f ðxÞ OðxÞ ¼ 12½ f ðxÞ f ðxÞ 1

2

Basic tools

Then, for an arbitrary complex function we can summarize these relations as (Bracewell, 1965) f ðxÞ ¼ reðxÞ þ i ieðxÞ þ roðxÞ þ i ioðxÞ

FðxÞ ¼ REðsÞ þ i IEðsÞ þ ROðsÞ þ i IOðsÞ As a consequence, a real function f(x) has a Fourier transform that is hermitian, F(s) ¼ F*(–s), where * refers to the complex conjugate. For a more general complex function, f(x), we can tabulate some additional properties (Bracewell, 1965): f ðxÞ , FðsÞ f ðxÞ , F ðsÞ f ðxÞ , F ðsÞ f ðxÞ , FðsÞ 2 Re f ðxÞ , FðsÞ þ F ðsÞ 2 Im f ðxÞ , FðsÞ F ðsÞ f ðxÞ þ f ðxÞ , 2 ReFðsÞ f ðxÞ f ðxÞ , 2 ImFðsÞ The convolution of two functions f(x) and g(x) is Z þ1 Z þ1 f ðzÞ gðx zÞ dz ¼ f ðx zÞ gðzÞ dz f ðxÞ gðxÞ ¼ 1

1

Convolution theorem If f(x) has the Fourier transform F(s), and g(x) has the Fourier transform G(s), then the Fourier transform of the convolution f(x) * g(x) is the product F(s) G(s). The cross-correlation of two functions f(x) and g(x) is Z þ1 Z þ1 f ðxÞ ? gðxÞ ¼ f ðz xÞ gðzÞ dz ¼ f ðzÞ gðz þ xÞ dz 1

1

where f* refers to the complex conjugate of f. When the two functions are the same, f*(x) ★ f(x) is called the autocorrelation of f(x).

Energy spectrum The modulus squared of the Fourier transform jF(s)j2 ¼ F(s) F*(s) is sometimes called the energy spectrum or simply the spectrum. If f(x) has the Fourier transform F(s), then the autocorrelation of f(x) has the Fourier transform jF(s)j2.

3

1.1 The Fourier transform

Phase spectrum The Fourier transform F(s) is most generally a complex function, which can be written as FðsÞ ¼ jFjei’ ¼ Re FðsÞ þ i Im FðsÞ where jFj is the modulus and ’ is the phase, given by ’ ¼ tan1 ½Im FðsÞ=Re FðsÞ The function ’(s) is sometimes also called the phase spectrum. Obviously, both the modulus and phase must be known to completely specify the Fourier transform F(s) or its transform pair in the other domain, f(x). Consequently, an infinite number of functions f(x) , F(s) are consistent with a given spectrum jF(s)j2. The zero-phase equivalent function (or zero-phase equivalent wavelet) corresponding to a given spectrum is FðsÞ ¼ jFðsÞj Z 1 f ðxÞ ¼ jFðsÞj eþi2pxs ds 1

which implies that F(s) is real and f(x) is hermitian. In the case of zero-phase real wavelets, then, both F(s) and f(x) are real even functions. The minimum-phase equivalent function or wavelet corresponding to a spectrum is the unique one that is both causal and invertible. A simple way to compute the minimumphase equivalent of a spectrum jF(s)j2 is to perform the following steps (Claerbout, 1992): (1) Take the logarithm, B(s) ¼ ln jF(s)j. (2) Take the Fourier transform, B(s) ) b(x). (3) Multiply b(x) by zero for x < 0 and by 2 for x > 0. If done numerically, leave the values of b at zero and the Nyquist frequency unchanged. (4) Transform back, giving B(s) + i’(s), where ’ is the desired phase spectrum. (5) Take the complex exponential to yield the minimum-phase function: Fmp(s) = exp[B(s) þ i’(s)] ¼ jF(s)jei’(s). (6) The causal minimum-phase wavelet is the Fourier transform of Fmp(s) ) fmp(x). Another way of saying this is that the phase spectrum of the minimum-phase equivalent function is the Hilbert transform (see Section 1.2 on the Hilbert transform) of the log of the energy spectrum.

Sampling theorem A function f(x) is said to be band limited if its Fourier transform is nonzero only within a finite range of frequencies, jsj < sc, where sc is sometimes called the cut-off frequency. The function f(x) is fully specified if sampled at equal spacing not exceeding Dx ¼ 1/(2sc). Equivalently, a time series sampled at interval Dt adequately describes the frequency components out to the Nyquist frequency fN ¼ 1/(2Dt).

Basic tools

2

2 Π(s) Boxcar(s)

sinc(x)

Sinc(x)

4

0

−2 −3

−2

−1

0 x

1

2

3

0

−2

−2

−1

0 s

1

2

Figure 1.1.1 Plots of the function sinc(x) and its Fourier transform (s).

The numerical process to recover the intermediate points between samples is to convolve with the sinc function: 2sc sincð2sc xÞ ¼ 2sc sinðp2sc xÞ=p2sc x where sincðxÞ

sinðpxÞ px

which has the properties: sincð0Þ ¼ 1 n ¼ nonzero integer sincðnÞ ¼ 0 The Fourier transform of sinc(x) is the boxcar function (s): 8 1 > < 0 jsj > 2 ðsÞ ¼

> :

12

/

jsj ¼ 12

1

jsj < 12

Plots of the function sinc(x) and its Fourier transform (s) are shown in Figure 1.1.1. One can see from the convolution and similarity theorems below that convolving with 2sc sinc(2scx) is equivalent to multiplying by (s/2sc) in the frequency domain (i.e., zeroing out all frequencies jsj > sc and passing all frequencies jsj < sc.

Numerical details Consider a band-limited function g(t) sampled at N points at equal intervals: g(0), g(Dt), g(2Dt), . . . , g((N – 1)Dt). A typical fast Fourier transform (FFT) routine will yield N equally spaced values of the Fourier transform, G( f ), often arranged as N N þ1 þ2 ðN 1Þ N 1 2 3 2 2 Gð0Þ Gðf Þ Gð2f Þ Gð fN Þ GðfN þ f Þ Gð2f Þ Gðf Þ

5

1.1 The Fourier transform

time domain sample rate Dt Nyquist frequency fN ¼ 1/(2Dt) frequency domain sample rate Df ¼ 1/(NDt) Note that, because of “wraparound,” the sample at (N/2 þ 1) represents both fN.

Spectral estimation and windowing It is often desirable in rock physics and seismic analysis to estimate the spectrum of a wavelet or seismic trace. The most common, easiest, and, in some ways, the worst way is simply to chop out a piece of the data, take the Fourier transform, and find its magnitude. The problem is related to sample length. If the true data function is f(t), a small sample of the data can be thought of as f ðtÞ; atb fsample ðtÞ ¼ 0; elsewhere or 1 t 2ða þ bÞ fsample ðtÞ ¼ f ðtÞ ba where (t) is the boxcar function discussed above. Taking the Fourier transform of the data sample gives Fsample ðsÞ ¼ FðsÞ ½jb aj sincððb aÞsÞeipðaþbÞs More generally, we can “window” the sample with some other function o(t): fsample ðtÞ ¼ f ðtÞ oðtÞ yielding Fsample ðsÞ ¼ FðsÞ WðsÞ Thus, the estimated spectrum can be highly contaminated by the Fourier transform of the window, often with the effect of smoothing and distorting the spectrum due to the convolution with the window spectrum W(s). This can be particularly severe in the analysis of ultrasonic waveforms in the laboratory, where often only the first 1 to 112 cycles are included in the window. The solution to the problem is not easy, and there is an extensive literature (e.g., Jenkins and Watts, 1968; Marple, 1987) on spectral estimation. Our advice is to be aware of the artifacts of windowing and to experiment to determine the sensitivity of the results, such as the spectral ratio or the phase velocity, to the choice of window size and shape.

Fourier transform theorems Tables 1.1.1 and 1.1.2 summarize some useful theorems (Bracewell, 1965). If f(x) has the Fourier transform F(s), and g(x) has the Fourier transform G(s), then the Fourier

6

Basic tools

Table 1.1.1

Fourier transform theorems.

Theorem

x-domain

s-domain

Similarity

f(ax)

,

1 s F jaj a

Addition

f(x) þ g(x)

,

F(s) þ G(s)

Shift

f(x – a)

,

Modulation

f(x) cos ox

,

Convolution

f(x) * g(x)

,

e–i2pasF(s) 1 ! 1 ! F s þ F sþ 2 2p 2 2p F(s) G(s)

Autocorrelation

f(x) * f (–x)

,

jF(s)j2

Derivative

f 0 (x)

,

i2psF(s)

Table 1.1.2

*

Some additional theorems.

Derivative of convolution Rayleigh Power ( f and g real)

d ½ f ðxÞ gðxÞ ¼ f 0 ðxÞ gðxÞ ¼ f ðxÞ g0 ðxÞ dx Z 1 Z 1 j f ðxÞj2 dx ¼ jFðsÞj2 ds 1 1 Z 1 Z 1 f ðxÞ g ðxÞ dx ¼ FðsÞ G ðsÞ ds 1 1 Z 1 Z 1 f ðxÞ gðxÞ dx ¼ FðsÞ GðsÞ ds 1

1

transform pairs in the x-domain and the s-domain are as shown in the tables. Table 1.1.3 lists some useful Fourier transform pairs.

1.2

The Hilbert transform and analytic signal Synopsis The Hilbert transform of f(x) is defined as Z 1 1 f ðx0 Þ dx0 FHi ðxÞ ¼ p 1 x0 x which can be expressed as a convolution of f(x) with (–1/px) by FHi ¼

1 f ðxÞ px

The Fourier transform of (–1/px) is (i sgn(s)), that is, þi for positive s and –i for negative s. Hence, applying the Hilbert transform keeps the Fourier amplitudes or spectrum the same but changes the phase. Under the Hilbert transform, sin(kx) is converted to cos(kx), and cos(kx) is converted to –sin(kx). Similarly, the Hilbert transforms of even functions are odd functions and vice versa.

7

1.2 The Hilbert transform and analytic signal

Table 1.1.3

Some Fourier transform pairs. sin px

i 1 1 d sþ d s 2 2 2

cos px

1 1 1 d sþ þd s 2 2 2

d(x)

1

sinc(x)

(s)

sinc2(x)

L(s)

e–px

2

e–ps

2

–1/px

i sgn(s)

x0 x20 þ x2

p exp(–2px0jsj)

e–jxj

jxj–1/2

2 1 þ ð2psÞ2 jsj–1/2

The inverse of the Hilbert transform is itself the Hilbert transform with a change of sign: Z 1 1 FHi ðx0 Þ dx0 f ðxÞ ¼ p 1 x0 x or 1 f ðxÞ ¼ FHi px The analytic signal associated with a real function, f(t), is the complex function SðtÞ ¼ f ðtÞ i FHi ðtÞ As discussed below, the Fourier transform of S(t) is zero for negative frequencies.

8

Basic tools

The instantaneous envelope of the analytic signal is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ EðtÞ ¼ f 2 ðtÞ þ F2Hi ðtÞ The instantaneous phase of the analytic signal is ’ðtÞ ¼ tan1 ½FHi ðtÞ=f ðtÞ ¼ Im½lnðSðtÞÞ The instantaneous frequency of the analytic signal is d’ d l dS o¼ ¼ Im lnðSÞ ¼ Im dt dt S dt Claerbout (1992) has suggested that o can be numerically more stable if the denominator is rationalized and the functions are locally smoothed, as in the following equation: E3 2D S ðtÞ dSðtÞ dt 5 ¼ Im4 o hS ðtÞ SðtÞi where 〈·〉 indicates some form of running average or smoothing.

Causality The impulse response, I(t), of a real physical system must be causal, that is, IðtÞ ¼ 0;

for t < 0

The Fourier transform T( f ) of the impulse response of a causal system is sometimes called the transfer function: Z 1 IðtÞ ei2pft dt Tð f Þ ¼ 1

T( f ) must have the property that the real and imaginary parts are Hilbert transform pairs, that is, T( f ) will have the form Tð f Þ ¼ Gð f Þ þ iBð f Þ where B( f ) is the Hilbert transform of G( f ): Z

1

Gð f 0 Þ df 0 0 1 f f Z 1 1 Bð f 0 Þ df 0 Gð f Þ ¼ p 1 f 0 f 1 Bð f Þ ¼ p

9

1.3 Statistics and probability

Similarly, if we reverse the domains, an analytic signal of the form SðtÞ ¼ f ðtÞ iFHi ðtÞ must have a Fourier transform that is zero for negative frequencies. In fact, one convenient way to implement the Hilbert transform of a real function is by performing the following steps: (1) Take the Fourier transform. (2) Multiply the Fourier transform by zero for f < 0. (3) Multiply the Fourier transform by 2 for f > 0. (4) If done numerically, leave the samples at f ¼ 0 and the Nyquist frequency unchanged. (5) Take the inverse Fourier transform. The imaginary part of the result will be the negative Hilbert transform of the real part.

1.3

Statistics and probability Synopsis The sample mean, m, of a set of n data points, xi, is the arithmetic average of the data values: n 1X xi n i¼1

m¼

The median is the midpoint of the observed values if they are arranged in increasing order. The sample variance, s2, is the average squared difference of the observed values from the mean: 2 ¼

n 1X ðxi mÞ2 n i¼1

(An unbiased estimate of the population variance is often found by dividing the sum given above by (n – 1) instead of by n.) The standard deviation, s, is the square root of the variance, while the coefficient of variation is s/m. The mean deviation, a, is a¼

n 1X jxi mj n i¼1

Regression When trying to determine whether two different data variables, x and y, are related, we often estimate the correlation coefficient, r, given by (e.g., Young, 1962) 1 Pn i¼1 ðxi mx Þðyi my Þ ; ¼n x y

where jj 1

10

Basic tools

where sx and sy are the standard deviations of the two distributions and mx and my are their means. The correlation coefficient gives a measure of how close the points come to falling along a straight line in a scatter plot of x versus y. jrj ¼ 1 if the points lie perfectly along a line, and jrj < 1 if there is scatter about the line. The numerator of this expression is the sample covariance, Cxy, which is defined as Cxy ¼

n 1X ðxi mx Þðyi my Þ n i¼1

It is important to remember that the correlation coefficient is a measure of the linear relation between x and y. If they are related in a nonlinear way, the correlation coefficient will be misleadingly small. The simplest recipe for estimating the linear relation between two variables, x and y, is linear regression, in which we assume a relation of the form: y ¼ ax þ b The coefficients that provide the best fit to the measured values of y, in the leastsquares sense, are y ; x

a¼

b ¼ my amx

More explicitly, a¼

n

P

P P xi yi ð xi Þð yi Þ ; P P n x2i ð xi Þ2

slope

P P 2 P P ð yi Þ xi ð xi yi Þð xi Þ ; b¼ P P n x2i ð xi Þ2

intercept

The scatter or variation of y-values around the regression line can be described by the sum of the squared errors as E2 ¼

n X

ðyi y^i Þ2

i¼1

where y^i is the value predicted from the regression line. This can be expressed as a variance around the regression line as ^y2 ¼

n 1X ðyi y^i Þ2 n i¼1

The square of the correlation coefficient r is the coefficient of determination, often denoted by r2, which is a measure of the regression variance relative to the total variance in the variable y, expressed as

11

1.3 Statistics and probability

variance of y around the linear regression total variance of y Pn ^y2 ðyi y^i Þ2 ¼ 1 ¼ 1 Pni¼1 2 y2 i¼1 ðyi my Þ

r 2 ¼ 2 ¼ 1

The inverse relation is ^y2 ¼ y2 ð1 r 2 Þ Often, when doing a linear regression the choice of dependent and independent variables is arbitrary. The form above treats x as independent and exact and assigns errors to y. It often makes just as much sense to reverse their roles, and we can find a regression of the form x ¼ a0 y þ b0 Generally a 6¼ 1/a0 unless the data are perfectly correlated. In fact, the correlation pﬃﬃﬃﬃﬃﬃ coefficient, r, can be written as ¼ aa0 . The coefficients of the linear regression among three variables of the form z ¼ a þ bx þ cy are given by Cxz Cyy Cxy Cyz 2 Cxx Cyy Cxy Cxx Cyz Cxy Cxz c¼ 2 Cxx Cyy Cxy b¼

a ¼ mz mx b my c The coefficients of the n-dimensional linear regression of the form z ¼ c0 þ c1 x1 þ c2 x2 þ þ cn xn are given by 2 3 2 3 c0 ð1Þ z 6 c1 7 6 7

ð2Þ 7 1 6 6 c2 7 6z 7 6 7 ¼ MT M MT 6 .. 7 6 .. 7 4 . 5 4 . 5 zðkÞ cn where the k sets 2 ð1Þ 1 x1 6 6 1 xð2Þ 1 6 M¼6 .. 6 .. 4. . 1

ðkÞ

x1

of independent variables form columns 2:(n þ 1) in the matrix M: 3 ð1 Þ x2 xðn1Þ 7 ð2 Þ x2 xðn2Þ 7 7 .. .. 7 7 . . 5 ðk Þ

x2

xðnkÞ

12

Basic tools

Variogram and covariance function In geostatistics, variables are modeled as random fields, X(u), where u is the spatial position vector. Spatial correlation between two random fields X(u) and Y(u) is described by the cross-covariance function CXY(h), defined by CXY ðhÞ ¼ Ef½XðuÞ mX ðuÞ½Yðu þ hÞ mY ðu þ hÞg where E{} denotes the expectation operator, mX and mY are the means of X and Y, and h is called the lag vector. For stationary fields, mX and mY are independent of position. When X and Y are the same function, the equation represents the auto-covariance function CXX(h). A closely related measure of two-point spatial variability is the semivariogram, g(h). For stationary random fields X(u) and Y(u), the cross-variogram 2gXY(h) is defined as 2gXY ðhÞ ¼ Ef½Xðu þ hÞ XðuÞ½Yðu þ hÞ YðuÞg When X and Y are the same, the equation represents the variogram of X(h). For a stationary random field, the variogram and covariance function are related by gXX ðhÞ ¼ CXX ð0Þ CXX ðhÞ where CXX(0) is the stationary variance of X.

Distributions

A population of n elements possesses nr (pronounced “n choose r”) different subpopulations of size r n, where ðnÞr nðn 1Þ ðn r þ 1Þ n! n ¼ ¼ ¼ r r! 1 2 ðr 1Þr r!ðn rÞ! Expressions of this kind are called binomial coefficients. Another way to say this is

that a subset of r elements can be chosen in nr different ways from the original set. The binomial distribution gives the probability of n successes in N independent trials, if p is the probability of success in any one trial. The binomial distribution is given by N n fN; p ðnÞ ¼ p ð1 pÞNn n The mean of the binomial distribution is given by mb ¼ Np and the variance of the binomial distribution is given by 2b ¼ Npð1 pÞ

13

1.3 Statistics and probability

The Poisson distribution is the limit of the binomial distribution as N ! 1 and p ! 0 so that l ¼ Np remains finite. The Poisson distribution is given by fl ðnÞ ¼

ln el n!

The Poisson distribution is a discrete probability distribution and expresses the probability of n events occurring during a given interval of time if the events have an average (positive real) rate l, and the events are independent of the time since the previous event. n is a non-negative integer. The mean of the Poisson distribution is given by mP ¼ l and the variance of the Poisson distribution is given by 2P ¼ l The uniform distribution is given by 8 > < 1 ; axb f ðxÞ ¼ b a > : 0; elsewhere The mean of the uniform distribution is m¼

ða þ bÞ 2

and the standard deviation of the uniform distribution is jb aj ¼ pﬃﬃﬃﬃﬃ 12 The Gaussian or normal distribution is given by 2 1 2 f ðxÞ ¼ pﬃﬃﬃﬃﬃﬃ eðxmÞ =2 2p

where s is the standard deviation and m is the mean. The mean deviation for the Gaussian distribution is rﬃﬃﬃ 2 a¼ p When m measurements are made of n quantities, the situation is described by the n-dimensional multivariate Gaussian probability density function (pdf): 1 1 T 1 exp C ðx mÞ ðx mÞ fn ðxÞ ¼ 2 ð2pÞn=2 jCj1=2

14

Basic tools

where xT ¼ (x1, x2, . . . , xn) is the vector of observations, mT ¼ (m1, m2, . . . , mn) is the vector of means of the individual distributions, and C is the covariance matrix: C ¼ ½Cij where the individual covariances, Cij, are as defined above. Notice that this reduces to the single variable normal distribution when n ¼ 1. When the natural logarithm of a variable, x ¼ ln(y), is normally distributed, it belongs to a lognormal distribution expressed as " # 1 1 lnðyÞ a 2 f ðyÞ ¼ pﬃﬃﬃﬃﬃﬃ exp 2 b 2p by where a is the mean and b2 is the variance. The relations among the arithmetic and logarithmic parameters are m ¼ eaþb =2 ; 2

2

2 ¼ m2 ðeb 1Þ;

a ¼ lnðmÞ b2=2

b2 ¼ ln 1 þ 2=m2

The truncated exponential distribution is given by (1 x 0 X expðx=XÞ; PðxÞ ¼ 0; x<0 The mean m and variance s2 of the exponential distribution are given by m¼X 2 ¼ X 2 The truncated exponential distribution and the Poisson distribution are closely related. Many random events in nature are described by the Poisson distribution. For example, in a well log, consider that the occurrence of a flooding surface is a random event. Then in every interval of log of length D, the probability of finding exactly n flooding surfaces is Pl D ðnÞ ¼

ðl DÞn expðl DÞ n!

The mean number of occurrences is lD, where l is the mean number of occurrences per unit length. The mean thickness between events is D=ðlDÞ ¼ 1=l. The interval thicknesses d between flooding events are governed by the truncated exponential l expðldÞ; x 0 PðdÞ ¼ 0; x<0 the mean of which is 1/l.

15

1.3 Statistics and probability

The logistic distribution is a continuous distribution with probability density function given by Pð xÞ ¼

eðxmÞ=s s½1 þ eðxmÞ=s

2

¼

x m 1 sech2 4s 2s

The mean m and variance s2 of the logistic distribution are given by m ¼m 2 ¼

p2 s2 3

The Weibull distribution is a continuous distribution with probability density function ( k1 k k x eðx=lÞ ; x 0 l l PðxÞ ¼ 0; x<0 where k > 0 is the shape parameter and l > 0 is the scale parameter for the distribution. The mean m and variance s2 of the logistic distribution are given by 1 m ¼ l 1 þ k 2 2 2 ¼l 1 þ m2 k where G is the gamma function. The Weibull distribution is often used to describe failure rates. If the failure rate decreases over time, k < 1; if the failure rate is constant k ¼ 1; and if the failure rate increases over time, k > 1. When k ¼ 3, the Weibull distribution is a good approximation to the normal distribution, and when k ¼ 1, the Weibull distribution reduces to the exponential distribution.

Monte Carlo simulations Statistical simulation is a powerful numerical method for tackling many probabilistic problems. One of the steps is to draw samples Xi from a desired probability distribution function F(x). This procedure is often called Monte Carlo simulation, a term made popular by physicists working on the bomb during the Second World War. In general, Monte Carlo simulation can be a very difficult problem, especially when X is multivariate with correlated components, and F(x) is a complicated function. For the simple case of a univariate X and a completely known F(x) (either analytically or numerically), drawing Xi amounts to first drawing uniform random variates Ui between 0 and 1, and then evaluating the inverse of the desired cumulative distribution function (CDF) at these Ui: Xi ¼ F –1(Ui). The inverse of the CDF is called the quantile function. When F1(X) is not known analytically, the inversion can be easily done by

Basic tools

1

Uniform [0 1] random number

0.8 CDF F(X)

16

0.6 0.4 Simulated X 0.2 0 0

5

10 X

15

20

Figure 1.3.1 Schematic of a univariate Monte Carlo simulation.

table-lookup and interpolation from the numerically evaluated or nonparametric CDF derived from data. A graphical description of univariate Monte Carlo simulation is shown in Figure 1.3.1. Many modern computer packages have random number generators not only for uniform and normal (Gaussian) distributions, but also for a large number of well-known, analytically defined statistical distributions. Often Monte Carlo simulations require simulating correlated random variables (e.g., VP, VS). Correlated random variables may be simulated sequentially, making use of the chain rule of probability, which expresses the joint probability density in terms of the conditional and marginal densities: P(VP, VS) ¼ P(VSjVP) P(VP). A simple procedure for correlated Monte Carlo draws is as follows: draw a VP sample from the VP distribution; compute a VS from the drawn VP and the VP–VS regression; add to the computed VS a random Gaussian error with zero mean and variance equal to the variance of the residuals from the VP–VS regression. This gives a random, correlated (VP, VS) sample. A better approach is to draw VS from the conditional distributions of VS for each given VP value, instead of using a simple VP–VS regression. Given sufficient VP–VS training data, the conditional distributions of VS for different VP can be computed.

Bootstrap “Bootstrap” is a very powerful computational statistical method for assigning measures of accuracy to statistical estimates (e.g., Efron and Tibshirani, 1993). The general idea is to make multiple replicates of the data by drawing from the original data with replacement. Each of the bootstrap data replicates has the same number of samples as the original data set, but since they are drawn with replacement, some of the data may be represented more than once in the replicate data sets, while others might be

17

1.3 Statistics and probability

missing. Drawing with replacement from the data is equivalent to Monte Carlo realizations from the empirical CDF. The statistic of interest is computed on all of the replicate bootstrap data sets. The distribution of the bootstrap replicates of the statistic is a measure of uncertainty of the statistic. Drawing bootstrap replicates from the empirical CDF in this way is sometimes termed nonparametric bootstrap. In parametric bootstrap the data are first modeled by a parametric CDF (e.g., a multivariate Gaussian), and then bootstrap data replicates are drawn from the modeled CDF. Both simple bootstrap techniques described above assume the data are independent and identically distributed. More sophisticated bootstrap techniques exist that can account for data dependence.

Statistical classification The goal in statistical classification problems is to predict the class of an unknown sample based on observed attributes or features of the sample. For example, the observed attributes could be P and S impedances, and the classes could be lithofacies, such as sand and shale. The classes are sometimes also called states, outcomes, or responses, while the observed features are called the predictors. Discussions concerning many modern classification methods may be found in Fukunaga (1990), Duda et al. (2000), Hastie et al. (2001), and Bishop (2006). There are two general types of statistical classification: supervised classification, which uses a training data set of samples for which both the attributes and classes have been observed; and unsupervised learning, for which only the observed attributes are included in the data. Supervised classification uses the training data to devise a classification rule, which is then used to predict the classes for new data, where the attributes are observed but the outcomes are unknown. Unsupervised learning tries to cluster the data into groups that are statistically different from each other based on the observed attributes. A fundamental approach to the supervised classification problem is provided by Bayesian decision theory. Let x denote the univariate or multivariate input attributes, and let cj, j ¼ 1, . . . , N denote the N different states or classes. The Bayes formula expresses the probability of a particular class given an observed x as Pðcj j xÞ ¼

Pðx; cj Þ Pðx j cj Þ Pðcj Þ ¼ PðxÞ PðxÞ

where P(x, cj) denotes the joint probability of x and cj; P(x j cj) denotes the conditional probability of x given cj; and P(cj) is the prior probability of a particular class. Finally, P(x) is the marginal or unconditional pdf of the attribute values across all N states. It can be written as PðXÞ ¼

N X j¼1

PðX j cj Þ Pðcj Þ

18

Basic tools

and serves as a normalization constant. The class-conditional pdf, P(x j cj), is estimated from the training data or from a combination of training data and forward models. The Bayes classification rule says: classify as class ck if Pðck j xÞ > Pðcj j xÞ for all j 6¼ k: This is equivalent to choosing ck when P(x j ck)P(ck) > P(x j cj)P(cj) for all j 6¼ k. The Bayes classification rule is the optimal one that minimizes the misclassification error and maximizes the posterior probability. Bayes classification requires estimating the complete set of class-conditional pdfs P(x j cj). With a large number of attributes, getting a good estimate of the highly multivariate pdf becomes difficult. Classification based on traditional discriminant analysis uses only the means and covariances of the training data, which are easier to estimate than the complete pdfs. When the input features follow a multivariate Gaussian distribution, discriminant classification is equivalent to Bayes classification, but with other data distribution patterns, the discriminant classification is not guaranteed to maximize the posterior probability. Discriminant analysis classifies new samples according to the minimum Mahalanobis distance to each class cluster in the training data. The Mahalanobis distance is defined as follows: T M2 ¼ x mj 1 x mj where x is the sample feature vector (measured attribute), mj are the vectors of the attribute means for the different categories or classes, and S is the training data covariance matrix. The Mahalanobis distance can be interpreted as the usual Euclidean distance scaled by the covariance, which decorrelates and normalizes the components of the feature vector. When the covariance matrices for all the classes are taken to be identical, the classification gives rise to linear discriminant surfaces in the feature space. More generally, with different covariance matrices for each category, the discriminant surfaces are quadratic. If the classes have unequal prior probabilities, the term ln[P(class)j] is added to the right-hand side of the equation for the Mahalanobis distance, where P(class)j is the prior probability for the jth class. Linear and quadratic discriminant classifiers are simple, robust classifiers and often produce good results, performing among the top few classifier algorithms.

1.4

Coordinate transformations Synopsis It is often necessary to transform vector and tensor quantities in one coordinate system to another more suited to a particular problem. Consider two right-hand rectangular Cartesian coordinates (x, y, z) and (x0 , y0 , z0 ) with the same origin, but

19

1.4 Coordinate transformations

with their axes rotated arbitrarily with respect to each other. The relative orientation of the two sets of axes is given by the direction cosines bij, where each element is defined as the cosine of the angle between the new i0 -axis and the original j-axis. The variables bij constitute the elements of a 3 3 rotation matrix [b]. Thus, b23 is the cosine of the angle between the 2-axis of the primed coordinate system and the 3-axis of the unprimed coordinate system. The general transformation law for tensors is 0 MABCD... ¼ bAa bBb bCc bDd . . . Mabcd...

where summation over repeated indices is implied. The left-hand subscripts (A, B, C, D, . . .) on the bs match the subscripts of the transformed tensor M0 on the left, and the right-hand subscripts (a, b, c, d, . . .) match the subscripts of M on the right. Thus vectors, which are first-rank tensors, transform as u0i ¼ bij uj or, in matrix notation, as 0 01 0 10 1 u1 b11 b12 b13 u1 B 0C B CB C B u2 C ¼ B b21 b22 b23 CB u2 C @ A @ A@ A 0 u3 b31 b32 b33 u3 whereas second-rank tensors, such as stresses and strains, obey 0ij ¼ bik bjl kl or ½0 ¼ ½b½½bT in matrix notation. Elastic stiffnesses and compliances are, in general, fourth-order tensors and hence transform according to c0ijkl ¼ bip bjq bkr bls cpqrs Often cijkl and sijkl are expressed as the 6 6 matrices Cij and Sij, using the abbreviated 2-index notation, as defined in Section 2.2 on anisotropic elasticity. In this case, the usual tensor transformation law is no longer valid, and the change of coordinates is more efficiently performed with the 6 6 Bond transformation matrices M and N, as explained below (Auld, 1990): ½C0 ¼ ½M½C½MT ½S0 ¼ ½N½S½NT

20

Basic tools

The elements of the 6 6 M and N matrices are given in terms of the direction cosines as follows: 2

b211

6 2 6 b 6 21 6 2 6 b 6 31 M¼6 6b b 6 21 31 6 6b b 4 31 11 b11 b21

3

b212

b213

2b12 b13

2b13 b11

b222

b223

2b22 b23

2b23 b21

b232

b233

2b32 b33

2b33 b31

b22 b32

b23 b33

b22 b33 þ b23 b32

b21 b33 þ b23 b31

b32 b12

b33 b13

b12 b33 þ b13 b32

b11 b33 þ b13 b31

7 7 7 7 7 2b31 b32 7 7 b22 b31 þ b21 b32 7 7 7 b11 b32 þ b12 b31 7 5

b12 b22

b13 b23

b22 b13 þ b12 b23

b11 b23 þ b13 b21

b22 b11 þ b12 b21

2b11 b12 2b21 b22

and 2

b211

6 6 b2 21 6 6 6 b2 31 6 N¼6 6 2b b 6 21 31 6 6 2b b 4 31 11 2b11 b21

3

b212

b213

b12 b13

b13 b11

b222

b223

b22 b23

b23 b21

b232

b233

b32 b33

b33 b31

2b22 b32

2b23 b33

b22 b33 þ b23 b32

b21 b33 þ b23 b31

2b32 b12

2b33 b13

b12 b33 þ b13 b32

b11 b33 þ b13 b31

7 7 7 7 7 b31 b32 7 7 b22 b31 þ b21 b32 7 7 7 b11 b32 þ b12 b31 7 5

2b12 b22

2b13 b23

b22 b13 þ b12 b23

b11 b23 þ b13 b21

b22 b11 þ b12 b21

b11 b12 b21 b22

The advantage of the Bond method for transforming stiffnesses and compliances is that it can be applied directly to the elastic constants given in 2-index notation, as they almost always are in handbooks and tables.

Assumptions and limitations Coordinate transformations presuppose right-handed rectangular coordinate systems.

2

Elasticity and Hooke’s law

2.1

Elastic moduli: isotropic form of Hooke’s law Synopsis In an isotropic, linear elastic material, the stress and strain are related by Hooke’s law as follows (e.g., Timoshenko and Goodier, 1934): ij ¼ ldij eaa þ 2meij or eij ¼

1 ½ð1 þ nÞij ndij aa E

where eij ¼ elements of the strain tensor ij ¼ elements of the stress tensor eaa ¼ volumetric strain (sum over repeated index) aa ¼ mean stress times 3 (sum over repeated index) dij ¼ 0 if i 6¼ j and dij ¼ 1 if i ¼ j In an isotropic, linear elastic medium, only two constants are needed to specify the stress–strain relation completely (for example, [l, m] in the first equation or [E, v], which can be derived from [l, m], in the second equation). Other useful and convenient moduli can be defined, but they are always relatable to just two constants. The three moduli that follow are examples. The Bulk modulus, K, is defined as the ratio of the hydrostatic stress, s0, to the volumetric strain: 0 ¼ 13 aa ¼ Keaa The bulk modulus is the reciprocal of the compressibility, b, which is widely used to describe the volumetric compliance of a liquid, solid, or gas: b¼

21

1 K

22

Elasticity and Hooke’s law

Caution

Occasionally in the literature authors have used the term incompressibility as an alternate name for Lame´’s constant, l, even though l is not the reciprocal of the compressibility. The shear modulus, m, is defined as the ratio of the shear stress to the shear strain: ij ¼ 2meij ;

i 6¼ j

Young’s modulus, E, is defined as the ratio of the extensional stress to the extensional strain in a uniaxial stress state: zz ¼ Eezz ;

xx ¼ yy ¼ xy ¼ xz ¼ yz ¼ 0

Poisson’s ratio, which is defined as minus the ratio of the lateral strain to the axial strain in a uniaxial stress state: v¼

exx ; ezz

xx ¼ yy ¼ xy ¼ xz ¼ yz ¼ 0

P-wave modulus, M ¼ VP2 , defined as the ratio of the axial stress to the axial strain in a uniaxial strain state: zz ¼ Mezz ;

exx ¼ eyy ¼ exy ¼ exz ¼ eyz ¼ 0

Note that the moduli (l, m, K, E, M) all have the same units as stress (force/area), whereas Poisson’s ratio is dimensionless. Energy considerations require that the following relations always hold. If they do not, one should suspect experimental errors or that the material is not isotropic: K ¼lþ

2m 0; 3

m 0

or 1 < n 12;

E 0

In rocks, we seldom, if ever, observe a Poisson’s ratio of less than 0. Although permitted by this equation, a negative measured value is usually treated with suspicion. A Poisson’s ratio of 0.5 can mean an infinitely incompressible rock (not possible) or a liquid. A suspension of particles in fluid, or extremely soft, watersaturated sediments under essentially zero effective stress, such as pelagic ooze, can have a Poisson’s ratio approaching 0.5. Although any one of the isotropic constants (l, m, K, M, E, and n) can be derived in terms of the others, m and K have a special significance as eigenelastic constants (Mehrabadi and Cowin, 1989) or principal elasticities of the material (Kelvin, 1856). The stress and strain eigentensors associated with m and K are orthogonal, as discussed in Section 2.2. Such an orthogonal significance does not hold for the pair l and m.

23

2.2 Anisotropic form of Hooke’s law

Relationships among elastic constants in an isotropic material (after Birch, 1961).

Table 2.1.1

K

E

l

v

M

m

l þ 2m/3

m 3lþ2m lþm

—

l 2ðlþmÞ

l þ 2m

—

—

l 3Kl

3K – 2l

3(K – l)/2

3K2m 2ð3KþmÞ

K þ 4m/3

— —

Kl 3Kl

—

9K

—

9Km 3Kþm

K – 2m/3

Em 3ð3mEÞ

—

E2m m ð3mEÞ

E/(2m) – 1

m 4mE 3mE

—

—

3K 3KE 9KE

3KE 6K

3K 3KþE 9KE

3KE 9KE

l 1þn 3n

l ð1þnÞð12nÞ n

—

—

l 1n n

l 12n 2n

2ð1þnÞ m 3ð12nÞ

2m(1 þ v)

2n m 12n

—

m 22n 12n

—

n 1þn

—

3K 1n 1þn

3K 12n 2þ2n

—

3K(1 – 2v)

3K

E 3ð12nÞ

—

En ð1þnÞð12nÞ

—

Eð1nÞ ð1þnÞð12nÞ

E 2þ2n

M 43 m

—

M 2m

M2m 2ðMmÞ

—

—

Table 2.1.1 summarizes useful relations among the constants of linear isotropic elastic media.

Assumptions and limitations The preceding equations assume isotropic, linear elastic media.

2.2

Anisotropic form of Hooke’s law Synopsis Hooke’s law for a general anisotropic, linear, elastic solid states that the stress sij is linearly proportional to the strain eij, as expressed by ij ¼ cijkl ekl in which summation (over 1, 2, 3) is implied over the repeated subscripts k and l. The elastic stiffness tensor, with elements cijkl, is a fourth-rank tensor obeying the laws of tensor transformation and has a total of 81 components. However, not all 81 components are independent. The symmetry of stresses and strains implies that cijkl ¼ cjikl ¼ cijlk ¼ cjilk

24

Elasticity and Hooke’s law

reducing the number of independent constants to 36. In addition, the existence of a unique strain energy potential requires that cijkl ¼ cklij further reducing the number of independent constants to 21. This is the maximum number of independent elastic constants that any homogeneous linear elastic medium can have. Additional restrictions imposed by symmetry considerations reduce the number much further. Isotropic, linear elastic materials, which have maximum symmetry, are completely characterized by two independent constants, whereas materials with triclinic symmetry (the minimum symmetry) require all 21 constants. Alternatively, the strains may be expressed as a linear combination of the stresses by the following expression: eij ¼ sijkl kl In this case sijkl are elements of the elastic compliance tensor which has the same symmetry as the corresponding stiffness tensor. The compliance and stiffness are tensor inverses, denoted by cijkl sklmn ¼ Iijmn ¼ 12ðdim djn þ din djm Þ The stiffness and compliance tensors must always be positive definite. One way to express this requirement is that all of the eigenvalues of the elasticity tensor (described below) must be positive.

Voigt notation It is a standard practice in elasticity to use an abbreviated Voigt notation for the stresses, strains, and stiffness and compliance tensors, for doing so simplifies some of the key equations (Auld, 1990). In this abbreviated notation, the stresses and strains are written as six-element column vectors rather than as nine-element square matrices: 2 3 2 3 1 ¼ 11 e1 ¼ e11 6 2 ¼ 22 7 6 e2 ¼ e22 7 6 7 6 7 6 3 ¼ 33 7 6 e3 ¼ e33 7 7 6 7 T¼6 6 4 ¼ 23 7 E ¼ 6 e4 ¼ 2e23 7 6 7 6 7 4 5 ¼ 13 5 4 e5 ¼ 2e13 5 6 ¼ 12 e6 ¼ 2e12 Note the factor of 2 in the definitions of strains, but not in the definition of stresses. With the Voigt notation, four subscripts of the stiffness and compliance tensors are reduced to two. Each pair of indices ij(kl) is replaced by one index I(J) using the following convention:

25

2.2 Anisotropic form of Hooke’s law

ijðklÞ IðJÞ 11 1 22 2 33 3 23; 32 4 13; 31 5 12; 21 6 The relation, therefore, is cIJ ¼ cijkl and sIJ ¼ sijkl N, where 8 > < 1 for I and J ¼ 1; 2; 3 N ¼ 2 for I or J ¼ 4; 5; 6 > : 4 for I and J ¼ 4; 5; 6 Note how the definition of sIJ differs from that of cIJ. This results from the factors of 2 introduced in the definition of strains in the abbreviated notation. Hence the Voigt matrix representation of the elastic stiffness is 0

c11 B c12 B B c13 B B c14 B @ c15 c16

c12 c22 c23 c24 c25 c26

c13 c23 c33 c34 c35 c36

c14 c24 c34 c44 c45 c46

c15 c25 c35 c45 c55 c56

1 c16 c26 C C c36 C C c46 C C c56 A c66

and similarly, the Voigt matrix representation of the elastic compliance is 0

s11 B s12 B B s13 B B s14 B @ s15 s16

s12 s22 s23 s24 s25 s26

s13 s23 s33 s34 s35 s36

s14 s24 s34 s44 s45 s46

s15 s25 s35 s45 s55 s56

1 s16 s26 C C s36 C C s46 C C s56 A s66

The Voigt stiffness and compliance matrices are symmetric. The upper triangle contains 21 constants, enough to contain the maximum number of independent constants that would be required for the least symmetric linear elastic material. Using the Voigt notation, we can write Hooke’s law as 0 1 0 10 1 1 c11 c12 c13 c14 c15 c16 e1 B 2 C B c12 c22 c23 c24 c25 c26 CB e2 C B C B CB C B 3 C B c13 c23 c33 c34 c35 c36 CB e3 C B C¼B CB C B 4 C B c14 c24 c34 c44 c45 c46 CB e4 C B C B CB C @ 5 A @ c15 c25 c35 c45 c55 c56 A@ e5 A 6 c16 c26 c36 c46 c56 c66 e6

26

Elasticity and Hooke’s law

It is very important to note that the stress (strain) vector and stiffness (compliance) matrix in Voigt notation are not tensors. Caution

Some forms of the abbreviated notation adopt different definitions of strains, moving the factors of 2 and 4 from the compliances to the stiffnesses. However, the form given above is the more common convention. In the two-index notation, cIJ and sIJ can conveniently be represented as 6 6 matrices. However, these matrices no longer follow the laws of tensor transformation. Care must be taken when transforming from one coordinate system to another. One way is to go back to the four-index notation and then use the ordinary laws of coordinate transformation. A more efficient method is to use the Bond transformation matrices, which are explained in Section 1.4 on coordinate transformations.

Voigt stiffness matrix structure for common anisotropy classes The nonzero components of the more symmetric anisotropy classes commonly used in modeling rock properties are given below in Voigt notation.

Isotropic: two independent constants The structure of the Voigt elastic stiffness matrix for an isotropic linear elastic material has the following form: 2 3 c11 c12 c12 0 0 0 6 c12 c11 c12 0 0 0 7 6 7 6 c12 c12 c11 0 0 0 7 6 7; c12 ¼ c11 2c44 6 0 0 0 c44 0 0 7 6 7 4 0 0 0 0 c44 0 5 0 0 0 0 0 c44 The relations between the elements c and Lame´’s parameters l and m of isotropic linear elasticity are c11 ¼ l þ 2m;

c12 ¼ l;

c44 ¼ m

The corresponding nonzero isotropic compliance tensor elements can be written in terms of the stiffnesses: s11 ¼

c11 þ c12 ; ðc11 c12 Þðc11 þ 2c12 Þ

s44 ¼

1 c44

Energy considerations require that for an isotropic linear elastic material the following conditions must hold: K ¼ c11 43 c44 > 0;

m ¼ c44 > 0

27

2.2 Anisotropic form of Hooke’s law

Cubic: three independent constants When each Cartesian coordinate plane is aligned with a symmetry plane of a material with cubic symmetry, the Voigt elastic stiffness matrix has the following form: 2

c11 6 c12 6 6 c12 6 6 0 6 4 0 0

c12 c11 c12 0 0 0

c12 c12 c11 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

3 0 0 7 7 0 7 7 0 7 7 0 5 c44

The corresponding nonzero cubic compliance tensor elements can be written in terms of the stiffnesses: s11 ¼

c11 þ c12 ; ðc11 c12 Þðc11 þ 2c12 Þ

s12 ¼

c12 ; ðc11 c12 Þðc11 þ 2c12 Þ

s44 ¼

1 c44

Energy considerations require that for a linear elastic material with cubic symmetry the following conditions must hold: c44 0;

c11 > jc12 j;

c11 þ 2c12 > 0

Hexagonal or transversely isotropic: five independent constants When the axis of symmetry of a transversely isotropic material lies along the x3-axis, the Voigt stiffness matrix has the form: 2

c11 6 c12 6 6 c13 6 6 0 6 4 0 0

c12 c11 c13 0 0 0

c13 c13 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

3 0 0 7 7 0 7 7; 0 7 7 0 5 c66

c66 ¼ 12ðc11 c12 Þ

The corresponding nonzero hexagonal compliance tensor elements can be written in terms of the stiffnesses: c33 1 ; s11 s12 ¼ 2 c11 c12 c33 ðc11 þ c12 Þ 2c13 c13 c11 þ c12 ¼ ; s33 ¼ ; c33 ðc11 þ c12 Þ 2c213 c33 ðc11 þ c12 Þ 2c213

s11 þ s12 ¼ s13

s44 ¼

1 c44

Energy considerations require that for a linear elastic material with transversely isotropic symmetry the following conditions must hold: c44 0;

c11 > jc12 j;

ðc11 þ c12 Þc33 > 2c213

28

Elasticity and Hooke’s law

Orthorhombic: nine independent constants When each Cartesian coordinate plane is aligned with a symmetry plane of a material with orthorhombic symmetry, the Voigt elastic stiffness matrix has the following form: 2 3 c11 c12 c13 0 0 0 6 c12 c22 c23 0 0 0 7 6 7 6 c13 c23 c33 0 7 0 0 6 7 6 0 0 0 c44 0 0 7 6 7 4 0 0 0 0 c55 0 5 0 0 0 0 0 c66 Monoclinic: 13 independent constants When the symmetry plane of a monoclinic medium is orthogonal to the x3-axis, the Voigt elastic stiffness matrix has the following form: 2 3 c11 c12 c13 0 c15 0 6 c12 c22 c23 0 c25 0 7 6 7 6 c13 c23 c33 0 c35 0 7 6 7 6 0 7 0 c 0 0 c 44 46 6 7 4 c15 c25 c35 0 c55 0 5 0 0 0 c46 0 c66

Phase velocities for several elastic symmetry classes For isotropic symmetry, the phase velocity of wave propagation is given by rﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃ c11 c44 VS ¼ VP ¼ where r is the density. In anisotropic media there are, in general, three modes of propagation (quasilongitudinal, quasi-shear, and pure shear) with mutually orthogonal polarizations. For a medium with transversely isotropic (hexagonal) symmetry, the wave slowness surface is always rotationally symmetric about the axis of symmetry. The phase velocities of the three modes in any plane containing the symmetry axis are given as quasi-longitudinal mode (transversely isotropic): pﬃﬃﬃﬃﬃ VP ¼ ðc11 sin2 y þ c33 cos2 y þ c44 þ MÞ1=2 ð2Þ1=2 quasi-shear mode (transversely isotropic): pﬃﬃﬃﬃﬃ VSV ¼ ðc11 sin2 y þ c33 cos2 y þ c44 MÞ1=2 ð2Þ1=2

29

2.2 Anisotropic form of Hooke’s law

pure shear mode (transversely isotropic): 1=2 c66 sin2 y þ c44 cos2 y VSH ¼ where M ¼ ½ðc11 c44 Þ sin2 y ðc33 c44 Þ cos2 y2 þ ðc13 þ c44 Þ2 sin2 2y and y is the angle between the wave vector and the x3-axis of symmetry (y ¼ 0 for propagation along the x3-axis). The five components of the stiffness tensor for a transversely isotropic material are obtained from five velocity measurements: VP(0 ), VP(90 ), VP(45 ), VSH(90 ), and VSH(0 ) ¼ VSV(0 ): c11 ¼ VP2 ð90 Þ 2 c12 ¼ c11 2VSH ð90 Þ

c33 ¼ VP2 ð0 Þ 2 c44 ¼ VSH ð0 Þ

and c13 ¼ c44 þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 2 42 VPð45 Þ 2VPð45 Þ ðc11 þ c33 þ 2c44 Þ þ ðc11 þ c44 Þðc33 þ c44 Þ

For the more general orthorhombic symmetry, the phase velocities of the three modes propagating in the three symmetry planes (XZ, YZ, and XY) are given as follows: quasi-longitudinal mode (orthorhombic, XZ plane): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=2 2 2 VP ¼ c55 þ c11 sin y þ c33 cos y þ ðc55 þ c11 sin2 y þ c33 cos2 yÞ2 4A ð2Þ1=2

quasi-shear mode (orthorhombic, XZ plane): VSV ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=2 c55 þ c11 sin y þ c33 cos y ðc55 þ c11 sin2 y þ c33 cos2 yÞ2 4A ð2Þ1=2 2

2

pure shear mode (orthorhombic, XZ plane): 1=2 c66 sin2 y þ c44 cos2 y VSH ¼ where A ¼ ðc11 sin2 y þ c55 cos2 yÞðc55 sin2 y þ c33 cos2 yÞ ðc13 þ c55 Þ2 sin2 y cos2 y quasi-longitudinal mode (orthorhombic, YZ plane): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=2 2 2 VP ¼ c44 þ c22 sin y þ c33 cos y þ ðc44 þ c22 sin2 y þ c33 cos2 yÞ2 4B ð2Þ1=2

30

Elasticity and Hooke’s law

quasi-shear mode (orthorhombic, YZ plane): VSV ¼

c44 þ c22 sin2 y þ c33 cos2 y

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=2 ðc44 þ c22 sin2 y þ c33 cos2 yÞ2 4B ð2Þ1=2

pure shear mode (orthorhombic, YZ plane): VSH

1=2 c66 sin2 y þ c55 cos2 y ¼

where B ¼ ðc22 sin2 y þ c44 cos2 yÞðc44 sin2 y þ c33 cos2 yÞ ðc23 þ c44 Þ2 sin2 y cos2 y quasi-longitudinal mode (orthorhombic, XY plane): VP ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=2 c66 þ c22 sin2 ’ þ c11 cos2 ’ þ ðc66 þ c22 sin2 ’ þ c11 cos2 ’Þ2 4C ð2Þ1=2

quasi-shear mode (orthorhombic, XY plane): VSH ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=2 c66 þ c22 sin ’ þ c11 cos ’ ðc66 þ c22 sin2 ’ þ c11 cos2 ’Þ2 4C ð2Þ1=2 2

2

pure shear mode (orthorhombic, XY plane): VSV

1=2 c55 cos2 ’ þ c44 sin2 ’ ¼

where C ¼ ðc66 sin2 ’ þ c11 cos2 ’Þðc22 sin2 ’ þ c66 cos2 ’Þ ðc12 þ c66 Þ2 sin2 ’ cos2 ’ and y and ’ are the angles of the wave vector relative to the x3 and x1 axes, respectively.

Kelvin notation In spite of its almost exclusive use in the geophysical literature, the abbreviated Voigt notation has several mathematical disadvantages. For example, certain norms of the fourth-rank stiffness tensor are not equal to the corresponding norms of the Voigt stiffness matrix (Thomson, 1878; Helbig, 1994; Dellinger et al., 1998), and the eigenvalues of the Voigt stiffness matrix are not the eigenvalues of the stress tensor. The lesser-known Kelvin notation is very similar to the Voigt notation, except that each element of the 6 6 matrix is weighted according to how many elements of the actual stiffness tensor it represents. Kelvin matrix elements having indices 1, 2, or 3 each map to only a single index pair in the fourth-rank notation, 11, 22, and 33,

31

2.2 Anisotropic form of Hooke’s law

respectively, so any matrix stiffness element with index 1, 2, or 3 is given a weight of 1. Kelvin elements with indices 4, 5, or 6 each represent two index pairs (23, 32), (13, 31), and pﬃﬃﬃ(12, 21), respectively, so each element containing 4, 5, or 6 receives a weight of 2. The weighting must be applied for each pﬃﬃﬃ Kelvin index. For example, the Kelvin notation would map c^11 ¼ c1111 , c^14 ¼ 2c1123 , c^66 ¼ 2c1212 . One way to convert a Voigt stiffness matrix into a Kelvin stiffness matrix is to pre- and post-multiply by the following weighting matrix (Dellinger et al., 1998): 2 3 1 0 0 0 0 0 60 1 0 0 0 0 7 7 6 60 0 1 0 0 0 7 6 pﬃﬃﬃ 7 60 0 0 7 2 0 0 6 7 pﬃﬃﬃ 40 0 0 0 2 p0ﬃﬃﬃ 5 0 0 0 0 0 2 yielding 0

c^11 B c^ B 12 B B c^13 B B c^ B 14 B @ c^15

c^12 c^22

c^13 c^23

c^14 c^24

c^15 c^25

c^23 c^24 c^25

c^33 c^34 c^35

c^34 c^44 c^45

c^35 c^45 c^55

1 c^16 c^26 C C C c^36 C C c^46 C C C c^56 A

c^16

c^26

c^36

c^46

c^56

c^66

0

c11

B c B 12 B B c13 ¼B B pﬃﬃ2ﬃc B 14 B pﬃﬃﬃ @ 2c15 pﬃﬃﬃ 2c16

c12

c13

c22

c23

c23 pﬃﬃﬃ 2c24 pﬃﬃﬃ 2c25 pﬃﬃﬃ 2c26

c33 pﬃﬃﬃ 2c34 pﬃﬃﬃ 2c35 pﬃﬃﬃ 2c36

pﬃﬃﬃ 2c14 pﬃﬃﬃ 2c24 pﬃﬃﬃ 2c34 2c44

pﬃﬃﬃ 2c15 pﬃﬃﬃ 2c25 pﬃﬃﬃ 2c35 2c45

2c45

2c55

2c46

2c56

pﬃﬃﬃ 1 2c16 pﬃﬃﬃ C 2c26 C pﬃﬃﬃ C 2c36 C C 2c46 C C C 2c56 A 2c66

where the c^ij are the Kelvin elastic elements and cij are the Voigt elements. Similarly, the stress and strain elements take on the following weights in the Kelvin notation: 2

^1 ¼ 11

6 ^ 6 2 6 6 ^3 T^ ¼ 6 6 ^ 6 4 6 4 ^5

3

7 7 7 ¼ 33 7 pﬃﬃﬃ 7 ; ¼ 223 7 7 pﬃﬃﬃ 7 ¼ 213 5 pﬃﬃﬃ ^6 ¼ 212 ¼ 22

2

^e1 6 ^e 6 2 6 6 ^e3 E^ ¼ 6 6 ^e 6 4 6 4 ^e5 ^e6

¼ e11

3

7 7 7 ¼ e33 7 pﬃﬃﬃ 7 ¼ 2e23 7 pﬃﬃﬃ 7 7 ¼ 2e13 5 pﬃﬃﬃ ¼ 2e12 ¼ e22

32

Elasticity and Hooke’s law

Hooke’s law again takes on 0 1 0 c^11 c^12 c^13 ^1 B ^2 C B c^12 c^22 c^23 B C B B ^3 C B c^13 c^23 c^33 B C¼B B ^4 C B c^14 c^24 c^34 B C B @ ^5 A @ c^15 c^25 c^35 c^16 c^26 c^36 ^6

the familiar form in the Kelvin notation: 10 1 ^e1 c^14 c^15 c^16 B ^e2 C c^24 c^25 c^26 C CB C B C c^34 c^35 c^36 C CB ^e3 C C C c^44 c^45 c^46 CB B ^e4 C A @ ^e5 A c^45 c^55 c^56 ^e6 c^46 c^56 c^66

The Kelvin notation for the stiffness matrix has a number of advantages. The Kelvin elastic matrix is a tensor. It preserves the norm of the 3 3 3 3 notation: X X c^2ij ¼ c2ijkl i;j¼1;...;6

i;j;k;l¼1;...;3

The eigenvalues and eigenvectors (or eigentensors) of the Kelvin 6 6 C-matrix are geometrically meaningful (Kelvin, 1856; Mehrabadi and Cowin, 1989; Dellinger et al., 1998). Each Kelvin eigentensor corresponds to a state of the medium where the stress and strain ellipsoids are aligned and have the same aspect ratios. Furthermore, a stiffness matrix is physically realizable if and only if all of the Kelvin eigenvalues are non-negative, which is useful for inferring elastic constants from laboratory data.

Elastic eigentensors and eigenvalues The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor are called eigentensors when projected into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is six (Kelvin, 1856; Mehrabadi and Cowin, 1989). In the case of isotropic linear elasticity, there are two unique eigentensors. The total stress tensor ~ and the total strain tensor ~e for the isotropic case can be decomposed in terms of the deviatoric second-rank tensors (~ dev and ~edev ) and ~ scaled unit tensors [1 trð~ ÞI~ and 1 trð~eÞI]: 3

3

ÞI~ ~ ¼ ~dev þ 13 trð~ ~e ¼ ~edev þ 1 trð~eÞI~ 3

For an isotropic material, Hooke’s law can be written as ~ ¼ l trð~eÞI~ þ 2m~e where l is Lame´’s constant and m is the shear modulus. However, if expressed in terms of the stress and strain eigentensors, Hooke’s law becomes two uncoupled equations: ~ tr ð~ÞI~ ¼ 3K tr ð~eÞI;

~dev ¼ 2m~edev

33

2.2 Anisotropic form of Hooke’s law

Table 2.2.1

Eigenelastic constants for several different symmetries.

Symmetry

Eigenvalue

Multiplicity of eigenvalue

Isotropic

c11 þ 2c12 ¼ 3K 2c44 ¼ 2m

1 5

Cubic

c11 þ 2c12 c11 c12 2c44 pﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ c33 þ 2c13 b þ b2 þ 1 ; b ¼ 4c132 ðc11 þ c12 c33 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ c33 þ 2c13 b b2 þ 1 c11 c12 2c44

1 2 3

Transverse Isotropy

1 1 2 2

where K is the bulk modulus. Similarly, the strain energy U for an isotropic material can be written as

2U ¼ 2m tr ~e2dev þ K ½tr ð~eÞ2 where m and K scale two energy modes that do not interact. Hence we can see that, although any two of the isotropic constants (l, m, K, Young’s modulus, and Poisson’s ratio) can be derived in terms of the others, m and K have a special significance as eigenelastic constants (Mehrabadi and Cowin, 1989) or principal elasticities of the material (Kelvin, 1856). The stress and strain eigentensors associated with m and K are orthogonal, as shown above. Such an orthogonal significance does not hold for the pair l and m. Eigenelastic constants for a few other symmetries are shown in Table 2.2.1, expressed in terms of the Voigt constants (Mehrabadi and Cowin, 1989).

Poisson’s ratio defined for anisotropic elastic materials Familiar isotropic definitions for elastic constants are sometimes extended to anisotropic materials. For example, consider a transversely isotropic (TI) material with uniaxial stress, applied along the symmetry (x3-) axis, such that 33 6¼ 0

11 ¼ 12 ¼ 13 ¼ 22 ¼ 23 ¼ 0

One can define a transversely isotropic Young’s modulus associated with this experiment as E33 ¼

33 2c231 ¼ c33 e33 c11 þ c12

34

Elasticity and Hooke’s law

where the cij are the elastic stiffnesses in Voigt notation. A pair of TI Poisson ratios can similarly be defined in terms of the same experiment: n31 ¼ n32 ¼

e11 c31 ¼ e33 c11 þ c12

If the uniaxial stress field is rotated normal to the symmetry axis, such that 11 6¼ 0

12 ¼ 13 ¼ 22 ¼ 23 ¼ 33 ¼ 0

then one can define another TI Young’s modulus and pair of Poisson ratios as E11 ¼

11 c2 ðc11 þ c12 Þ þ c12 ðc33 c12 þ c231 Þ ¼ c11 þ 31 e11 c33 c11 c231 e33 c31 ðc11 c12 Þ n13 ¼ n23 ¼ ¼ e11 c33 c11 c231 n12 ¼ n21 ¼

e22 c33 c12 c231 ¼ e11 c33 c11 c231

Caution

Just as with the isotropic case, definitions of Young’s modulus and Poisson ratio in terms of stresses and strains are only true for the uniaxial stress state. Definitions in terms of elastic stiffnesses are independent of stress state. In spite of their similarity to their isotropic analogs, these TI Poisson ratios have several distinct differences. For example, the bounds on the isotropic Poisson ratio require that 1 nisotropic

1 2

In contrast, the bounds on the TI Poisson ratios defined here are (Christensen, 2005) jn12 j 1;

jn 31 j

1=2 E33 ; E11

jn13 j

1=2 E11 E33

Another distinct difference is the relation of Poisson’s ratio to seismic velocities. In an isotropic material, Poisson’s ratio is directly related to the VP/VS ratio as follows: ðVP =VS Þ2 ¼

2ð1 nÞ ð1 2nÞ

or, equivalently, ðVP =VS Þ2 2 i n¼ h 2 ðVP =VS Þ2 1

35

2.3 Thomsen’s notation for weak elastic anisotropy

However, in the TI material, the ratio of velocities propagating along the symmetry (x3-) axis is 2 VP c33 ¼ VS c44 and is not simply related to the n31 Poisson ratio. Caution

The definition of Young’s modulus and Poisson’s ratio for anisotropic materials, while possible, can be misinterpreted, especially when compared with isotropic formulas. For orthorhombic and higher symmetries, the “engineering” parameters – Poisson ratios, nij, shear moduli, Gij, and Young’s moduli, Eij – can be conveniently defined in terms of elastic compliances (Jaeger et al., 2007): E11 ¼ 1=s11

E22 ¼ 1=s22

E33 ¼ 1=s33

n21 ¼ s12 E22

n31 ¼ s13 E33

n32 ¼ s23 E33

G12 ¼ 1=s66

G13 ¼ 1=s55

G23 ¼ 1=s44

Existence of a strain energy leads to the following constraints: G12 ; G13 ; G23 ; E11 ; E22 ; E33 > 0 E11 E11 ðn21 Þ2 < 1; ðn31 Þ2 < 1; E22 E33

E22 ðn32 Þ2 < 1 E33

Assumptions and limitations The preceding equations assume anisotropic, linear elastic media.

2.3

Thomsen’s notation for weak elastic anisotropy Synopsis A transversely isotropic elastic material is completely specified by five independent constants. In terms of the abreviated Voigt notation (see Section 2.2 on elastic anisotropy) the elastic constants can be represented as 2 3 c11 c12 c13 0 0 0 6 c12 c11 c13 0 0 0 7 6 7 6 c13 c13 c33 0 0 0 7 6 7; where c66 ¼ 1ðc11 c12 Þ 2 6 0 0 7 0 0 c44 0 6 7 4 0 0 0 0 c44 0 5 0 0 0 0 0 c66

36

Elasticity and Hooke’s law

and where the axis of symmetry lies along the x3-axis. Thomsen (1986) suggested the following convenient notation for a TI material that is only weakly anisotropic. His notation uses the P- and S-wave velocities (denoted by a and b, respectively) propagating along the symmetry axis, plus three additional constants: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c11 c33 a ¼ c33 =; b ¼ c44 =; e ¼ 2c33 g¼

c66 c44 ; 2c44

d¼

ðc13 þ c44 Þ2 ðc33 c44 Þ2 2c33 ðc33 c44 Þ

In terms of these constants, the three phase velocities can be approximated conveniently as VP ðyÞ að1 þ d sin2 y cos2 y þ e sin4 yÞ a2 2 2 VSV ðyÞ b 1 þ 2 ðe dÞ sin y cos y b VSH ðyÞ bð1 þ g sin2 yÞ where y is the angle of the wave vector relative to the x3-axis; VSH is the wavefront velocity of the pure shear wave, which has no component of polarization in the symmetry axis direction; VSV is the pseudo-shear wave polarized normal to the pure shear wave; and VP is the pseudo-longitudinal wave. Berryman (2008) extends the validity of Thomsen’s expressions for P- and quasiSV-wave velocities to wider ranges of angles and stronger anisotropy with the following expressions: 2 sin2 ym sin2 y cos2 y 2 VP ðyÞ a 1 þ e sin y ðe dÞ 1 cos 2ym cos 2y 2 a 2 sin2 ym sin2 y cos2 y ðe dÞ VSV ðyÞ b 1 þ 1 cos 2ym cos 2y b2 where tan2 ym ¼

c33 c44 c11 c44

Berryman’s formulas give more accurate velocities at larger angles. They are designed to give the angular location of the peak (or trough) of the quasi-SV-velocity closer to the correct location; hence, the quasi-SV-velocities are more accurate than those from Thomsen’s equations. Thomsen’s P-wave velocities will sometimes be more accurate than Berryman’s at small y and sometimes worse. For weak anisotropy, the constant e can be seen to describe the fractional difference between the P-wave velocities parallel and orthogonal to the symmetry axis (in the weak anisotropy approximation):

37

2.3 Thomsen’s notation for weak elastic anisotropy

e

VP ð90 Þ VP ð0 Þ VP ð0 Þ

Therefore, e best describes what is usually called the “P-wave anisotropy.” Similarly, for weak anisotropy the constant g can be seen to describe the fractional difference between the SH-wave velocities parallel and orthogonal to the symmetry axis, which is equivalent to the difference between the velocities of S-waves polarized parallel and normal to the symmetry axis, both propagating normal to the symmetry axis: g

VSH ð90 Þ VSV ð90 Þ VSH ð90 Þ VSH ð0 Þ ¼ VSV ð90 Þ VSH ð0 Þ

The small-offset normal moveout velocity is affected by VTI (transversely isotropic with vertical symmetry axis) anisotropy. In terms of the Thomsen parameters, NMO (normal moveout) velocities, VNMO;P , VNMO;SV , and VNMO;SH for P-, SV-, and SHmodes are (Tsvankin, 2001): pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;P ¼ a 1 þ 2d 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a VNMO;SV ¼ b 1 þ 2; ¼ ðe dÞ b pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;SH ¼ b 1 þ 2g An additional anellipticity parameter, , was introduced by Alkhalifah and Tsvankin (1995): ¼

ed 1 þ 2d

is important in quantifying the effects of anisotropy on nonhyperbolic moveout and P-wave time-processing steps (Tsvankin, 2001), including NMO, DMO (dip moveout), and time migration. The Thomsen parameters can be inverted for the elastic constants as follows: c33 ¼ a2 ;

c44 ¼ b2

c11 ¼ c33 ð1 þ 2eÞ; c66 ¼ c44 ð1 þ 2gÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c13 ¼ 2c33 ðc33 c44 Þd þ ðc33 c44 Þ2 c44 Note the nonuniqueness in c13 that results from uncertainty in the sign of (c13 þ c44). Tsvankin (2001, p. 19) argues that for most cases, it can be assumed that (c13 þ c44) > 0, and therefore, the þsign in the equation for c13 is usually appropriate. Tsvankin (2001) summarizes some bounds on the values of the Thomsen parameters:

the lower bound for d: d 1 b2 =a2 =2;

an approximate upper bound for d: d 2= a2 =b2 1 ; and in VTI materials resulting from thin layering, e > d and g > 0.

38

Elasticity and Hooke’s law

Ranges of Thomsen parameters expected for thin laminations of isotropic materials (Berryman et al., 1999).

Table 2.3.1

e d g e–d

m ¼ const

l ¼ const m 6¼ const

lþm ¼ const m 6¼ const

lþ2m ¼ const m 6¼ const

n ¼ const l, m 6¼ const

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

Transversely isotropic media consisting of thin isotropic layers always have Thomsen (1986) parameters, such that e – d 0 and g 0 (Tsvankin, 2001). Berryman et al. (1999) find the additional conditions summarized in Table 2.3.1, based on Backus (1962) average analysis and Monte Carlo simulations of thinly laminated media. Although all of the cases in the table have d 0, Berryman et al. find that d can be positive if the layers have significantly varying and positively correlated shear modulus, m, and Poisson ratio, n. Bakulin et al. (2000) studied the Thomsen parameters for anisotropic HTI (transversely isotropic with horizontal symmetry axis) media resulting from aligned vertical fractures with crack normals along the horizontal x1-axis in an isotropic background. For example, when the Hudson (1980) penny-shaped crack model (Section 4.10) is used to estimate weak anisotropy resulting from crack density, e, the dry-rock Thomsen parameters in the vertical plane containing the x1-axis can be approximated as ðVÞ

edry ¼ ðVÞ ddry ðVÞ

gdry

ðVÞ

dry

c11 c33 8 ¼ e 0; 2c33 3

ðc13 þ c55 Þ2 ðc33 c55 Þ2 8 gð1 2gÞ ¼ e 1þ ¼ 0; 2c33 ðc33 c55 Þ 3 ð3 2gÞð1 gÞ c66 c44 8e ¼ ¼ 0; 2c44 3ð3 2gÞ ðVÞ ðVÞ edry ddry 8 gð1 2gÞ ¼ ¼ e 0 ðVÞ 3 ð3 2gÞð1 gÞ 1 þ 2ddry

where g ¼ VS2 =VP2 is a property of the unfractured background rock. In the case of fluid-saturated penny-shaped cracks, such that the crack aspect, a, is much less than the ratio of the fluid bulk modulus to the mineral bulk modulus, Kfluid/Kmineral, then the weak-anisotropy Thomsen parameters can be approximated as ðVÞ

esaturated ¼ ðVÞ

dsaturated ¼

c11 c33 ¼0 2c33 ðc13 þ c55 Þ2 ðc33 c55 Þ2 32ge ¼ 0 2c33 ðc33 c55 Þ 3ð3 2gÞ

39

2.4 Tsvankin’s extended Thomsen parameters

ðVÞ

c66 c44 8e ¼ 0 2c44 3ð3 2gÞ 32ge ðVÞ ¼ dsaturated ¼ 0 3ð3 2gÞ

gsaturated ¼ ðVÞ

saturated

In intrinsically anisotropic shales, Sayers (2004) finds that d can be positive or negative, depending on the contact stiffness between microscopic clay domains and on the distribution of clay domain orientations. He shows, via modeling, that distributions of clays domains, each having d 0, can yield a composite with d > 0 overall if the domain orientations vary significantly from parallel. Rasolofosaon (1998) shows that under the assumptions of third-order elasticity an isotropic rock obtains ellipsoidal symmetry with respect to the propagation of qP waves. Hence e ¼ d in the symmetry planes (see Section 2.4) on Tsvankin’s extended Thomsen parameters.

Uses Thomsen’s notation for weak elastic anisotropy is useful for conveniently characterizing the elastic constants of a transversely isotropic linear elastic medium.

Assumptions and limitations The preceding equations are based on the following assumptions: material is linear, elastic, and transversely isotropic; anisotropy is weak, so that e, g, d 1.

2.4

Tsvankin’s extended Thomsen parameters for orthorhombic media The well-known Thomsen (1986) parameters for weak anisotropy are well suited for transversely isotropic media (see Section 2.3). They allow the five independent elastic constants c11, c33, c12, c13, and c44 to be expressed in terms of the more intuitive P-wave velocity, a, and S-wave velocity, b, along the symmetry axis, plus additional constants e, g, and d. For orthorhombic media, requiring nine independent elastic constants, the conventional Thomsen parameters are insufficient. Analogs of the Thomsen parameters suitable for orthorhombic media can be defined (Tsvankin, 1997), recognizing that wave propagation in the x1–x3 and x2–x3 symmetry planes (pseudo-P and pseudo-S polarized in each plane and SH polarized normal to each plane) is analogous to propagation in two different VTI media. We once again define vertically propagating (along the x3-axis) P- and S-wave velocities, a and b, respectively: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a ¼ c33 =; b ¼ c55 =

40

Elasticity and Hooke’s law

Unlike in a VTI medium, S-waves propagating along x3-axis in an orthorhombic pthe ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ medium can have two different velocities, bx2 ¼ c44 = and bx1 ¼ c55 =, for waves polarized in the x2 and x1 directions, respectively. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Either polarization can be chosen as a reference, though here we take b ¼ c55 = following the definitions of Tsvankin (1997). Some results shown in later sections will use redefined polarizations in the definition of b. For the seven constants, we can write eð2Þ ¼

c11 c33 2c33

ðc13 þ c55 Þ2 ðc33 c55 Þ2 2c33 ðc33 c55 Þ c66 c44 ¼ 2c44

eð1Þ ¼

ðc23 þ c44 Þ2 ðc33 c44 Þ2 2c33 ðc33 c44 Þ c66 c55 ¼ 2c55

dð2Þ ¼

dð1Þ ¼

gð2Þ

gð1Þ

dð3Þ ¼

c22 c33 2c33

ðc12 þ c66 Þ2 ðc11 c66 Þ2 2c11 ðc11 c66 Þ

Here, the superscripts (1), (2), and (3) refer to the TI-analog parameters in the symmetry planes normal to x1, x2, and x3, respectively. These definitions assume that one of the symmetry planes of the orthorhombic medium is horizontal and that the vertical symmetry axis is along the x3 direction. These Thomsen–Tsvankin parameters play a useful role in modeling wave propagation and reflectivity in anisotropic media.

Uses Tsvankin’s notation for weak elastic anisotropy is useful for conveniently characterizing the elastic constants of an orthorhombic elastic medium.

Assumptions and limitations The preceding equations are based on the following assumptions: material is linear, elastic, and has orthorhombic or higher symmetry; the constants are definitions. They sometimes appear in expressions for anisotropy of arbitrary strength, but at other times the applications assume that the anisotropy is weak, so that e, g, d 1.

2.5

Third-order nonlinear elasticity Synopsis Seismic velocities in crustal rocks are almost always sensitive to stress. Since so much of geophysics is based on linear elasticity, it is common to extend the familiar linear elastic terminology and refer to the “stress-dependent linear elastic moduli” – which can

41

2.5 Third-order nonlinear elasticity

have meaning for the local slope of the strain-curves at a given static state of stress. If the relation between stress and strain has no hysteresis and no dependence on rate, then it is more accurate to say that the rocks are nonlinearly elastic (e.g., Truesdell, 1965; Helbig, 1998; Rasolofosaon, 1998). Nonlinear elasticity (i.e., stress-dependent velocities) in rocks is due to the presence of compliant mechanical defects, such as cracks and grain contacts (e.g., Walsh, 1965; Jaeger and Cook, 1969; Bourbie´ et al., 1987). In a material with third-order nonlinear elasticity, the strain energy function E (for arbitrary anisotropy) can be expressed as (Helbig, 1998) E ¼ 12 cijkl eij ekl þ 16 cijklmn eij ekl emn where cijkl and cijklmn designate the components of the second- and third-order elastic tensors, respectively, and repeated indices in a term imply summation from 1 to 3. The components cijkl are the usual elastic constants in Hooke’s law, discussed earlier. Hence, linear elasticity is often referred to as second-order elasticity, because the strain energy in a linear elastic material is second order in strain. The linear elastic tensor (cijkl) is fourth rank, having a minimum of two independent constants for a material with the highest symmetry (isotropic) and a maximum of 21 independent constants for a material with the lowest symmetry (triclinic). The additional tensor of third-order elastic coefficients (cijklmn) is rank six, having a minimum of three independent constants (isotropic) and a maximum of 56 independent constants (triclinic) (Rasolofosaon, 1998). Third-order elasticity is sometimes used to describe the stress-sensitivity of seismic velocities and apparent elastic constants in rocks. The apparent fourth-rank stiffness tensor, c~eff , which determines the speeds of infinitesimal-amplitude waves in a rock under applied static stress can be written as c~eff ijkl ¼ cijkl þ cijklmn emn where emn are the principal strains associated with the applied static stress. Approximate expressions, in Voigt notation, for the effective elastic constants of a stressed VTI (transversely isotropic with a vertical axis of symmetry) solid can be written as (Rasolofosaon, 1998; Sarkar et al., 2003; Prioul et al., 2004) 0 ceff 11 c11 þ c111 e11 þ c112 ðe22 þ e33 Þ 0 ceff 22 c11 þ c111 e22 þ c112 ðe11 þ e33 Þ 0 ceff 33 c33 þ c111 e33 þ c112 ðe11 þ e22 Þ 0 ceff 12 c12 þ c112 ðe11 þ e22 Þ þ c123 e33 0 ceff 13 c13 þ c112 ðe11 þ e33 Þ þ c123 e22 0 ceff 23 c13 þ c112 ðe22 þ e33 Þ þ c123 e11 0 ceff 66 c66 þ c144 e33 þ c155 ðe11 þ e22 Þ 0 ceff 55 c44 þ c144 e22 þ c155 ðe11 þ e33 Þ 0 ceff 44 c44 þ c144 e11 þ c155 ðe22 þ e33 Þ

42

Elasticity and Hooke’s law

Table 2.5.1 Experimentally determined third-order elastic constants c111, c112, and c123 and derived constants c144, c155, and c456, determined by Prioul and Lebrat (2004), using laboratory data from Wang (2002). Six different sandstone and six different shale samples are shown. c111 (GPa)

c112 (GPa)

c123 (GPa)

10 245 9 482 6 288 8 580 8 460 12 440

966 1745 1744 527 1162 3469

966 1745 1744 527 1162 3094

6 903 4 329 7 034 4 160 1 294 1 203

976 2122 2147 2013 510 637

976 1019 296 940 119 354

c144 (GPa)

c155 (GPa)

c456 (GPa)

0 0 0 0 0 188

2320 1934 1136 2013 1825 2243

1160 967 568 1006 912 1027

0 552 1222 536 196 141

1482 552 1222 536 196 141

741 0 0 0 0 0

Sandstones

Shales

where the constants c011 , c033 , c013 , c044 , c066 are the VTI elastic constants at the unstressed reference state, with c012 ¼ c011 2c066 . e11, e22, and e33 are the principal strains, computed from the applied stress using the conventional Hooke’s law, eij ¼ sijkl kl . For these expressions, it is assumed that the direction of the applied principal stress is aligned with the VTI symmetry (x3-) axis. Furthermore, for these expressions it is assumed that the stress-sensitive third-order tensor is isotropic, defined by the three independent constants, c111, c112, and c123 with c144 ¼ ðc112 c123 Þ=2, c155 ¼ ðc111 c112 Þ=4, and c456 ¼ ðc111 3c112 þ 2c123 Þ=8. It is generally observed (Prioul et al., 2004) that c111 < c112 < c123, c155 < c144, c155 < 0, and c456 < 0. A sample of experimentally determined values from Prioul and Lebrat (2004) using laboratory data from Wang (2002) are shown in the Table 2.5.1. The third-order elasticity as used in most of geophysics and rock physics (Bakulin and Bakulin, 1999; Prioul et al., 2004) is called the Murnaghan (1951) formulation of finite deformations, and the third-order constants are also called the Murnaghan constants. Various representations of the third-order constants that can be found in the literature (Rasolofosaon, 1998) include the crystallographic set (c111, c112, c123) presented here, the Murnaghan (1951) constants (l, m, n), and Landau’s set (A, B, C) (Landau and Lifschitz, 1959). The relations among these are (Rasolofosaon, 1998)

43

2.6 Effective stress properties of rocks

c111 ¼ 2A þ 6B þ 2C ¼ 2l þ 4m c112 ¼ 2B þ 2C ¼ 2l c123 ¼ 2C ¼ 2l 2m þ n Chelam (1961) and Krishnamurty (1963) looked at fourth-order elastic coefficients, based on an extension of Murnaghan’s theory. It turns out from group theory that there will only be n independent nth-order coefficients for isotropic solids, and n2 – 2n þ 3 independent nth-order constants for cubic systems. Triclinic solids have 126 fourth-order elastic constants.

Uses Third-order elasticity provides a way to parameterize the stress dependence of seismic velocities. It also allows for a compact description of stress-induced anisotropy, which is discussed later.

Assumptions and limitations

2.6

The above equations assume that the material is hyperelastic, i.e., there is no hysteresis or rate dependence in the relation between stress and strain, and there exists a unique strain energy function. This formalism assumes that strains are infinitesimal. When strains become finite, an additional source of nonlinearity, called geometrical or kinetic nonlinearity, appears, related to the difference between Lagrangian and Eulerian descriptions of motion (Zarembo and Krasil’nikov, 1971; Johnson and Rasolofosaon, 1996). Third-order elasticity is often not general enough to describe the shapes of real stress–strain curves over large ranges of stress and strain. Third-order elasticity is most useful when describing stress–strain within a small range around a reference state of stress and strain.

Effective stress properties of rocks Synopsis Because rocks are deformable, many rock properties are sensitive to applied stresses and pore pressure. Stress-sensitive properties include porosity, permeability, electrical resistivity, sample volume, pore-space volume, and elastic moduli. Empirical evidence (Hicks and Berry, 1956; Wyllie et al., 1958; Todd and Simmons, 1972; Christensen and Wang, 1985; Prasad and Manghnani, 1997; Siggins and Dewhurst, 2003, Hoffman et al., 2005) and theory (Brandt, 1955; Nur and Byerlee, 1971; Zimmerman, 1991a; Gangi and Carlson, 1996; Berryman, 1992a, 1993; Gurevich, 2004) suggest that the pressure dependence of each of these rock properties, X, can be

44

Elasticity and Hooke’s law

represented as a function, fX, of a linear combination of the hydrostatic confining stress, PC, and the pore pressure, PP: X ¼ fX ðPC nPP Þ;

n1

The combination Peff ¼ PC nPP is called the effective pressure, or more generally, C the tensor eff ij ¼ ij nPP dij is the effective stress. The parameter n is called the “effective-stress coefficient,” which can itself be a function of stress. The negative sign on the pore pressure indicates that the pore pressure approximately counteracts the effect of the confining pressure. An expression such as X ¼ fX ðPC nPP Þ is sometimes called the effective-stress law for the property X. It is important to point out that each rock property might have a different function fi and a different value of ni (Zimmerman, 1991a; Berryman, 1992a, 1993; Gurevich, 2004). Extensive discussions on the effective-stress behavior of elastic moduli, permeability, resistivity, and thermoelastic properties can be found in Berryman (1992a, 1993). Zimmerman (1991a) gives a comprehensive discussion of effective-stress behavior for strain and elastic constants. Zimmerman (1991a) points out the distinction between the effective-stress behavior for finite pressure changes versus the effective-stress behavior for infinitesimal increments of pressure. For example, increments of the bulk-volume strain, eb, and the pore-volume strain, ep, can be written as 1 @VT eb ðPC ; PP Þ ¼ Cbc ðPC mb PP ÞðdPC nb dPP Þ; Cbc ¼ VT @PC PP 1 @VP ep ðPC ; PP Þ ¼ Cpc ðPC mp PP ÞðdPC np dPP Þ; Cpc ¼ VP @PC PP where the compressibilities Cbc and Cpc are functions of PC mPP . The coefficients mb and mp govern the way that the compressibilities vary with PC and PP. In contrast, the coefficients nb and np describe the relative increments of additional strain resulting from pressure increments dPC and dPP. For example, in a laboratory experiment, ultrasonic velocities will depend on the values of Cpc, the local slope of the stress–strain curve at the static values of PC and PP. On the other hand, the sample length change monitored within the pressure vessel is the total strain, obtained by integrating the strain over the entire stress path. The existence of an effective-stress law, i.e., that a rock property depends only on the state of stress, requires that the rock be elastic – possibly nonlinearly elastic. The deformation of an elastic material depends only on the state of stress, and is independent of the stress history and the rate of loading. Furthermore, the existence of an effective-stress law requires that there is no hysteresis in stress–strain cycles. Since no rock is perfectly elastic, all effective-stress laws for rocks are approximations. In fact, deviation from elasticity makes estimating the effective-stress coefficient from laboratory data sometimes ambiguous. Another condition required for the existence of an effective-pressure law is that the pore pressure is well defined and uniform throughout the pore space. Todd and Simmons (1972) show that the effect of pore

45

2.6 Effective stress properties of rocks

pressure on velocities varies with the rate of pore-pressure change and whether the pore pressure has enough time to equilibrate in thin cracks and poorly connected pores. Slow changes in pore pressure yield more stable results, describable with an effective-stress law, and with a larger value of n for velocity. Much discussion focuses on the value of the effective-stress constants, n (and m). Biot and Willis (1957) predicted theoretically that the pressure-induced volume increment, dVT, of a sample of linear poroelastic material depends on pressure increments ðdPC ndPP Þ. For this special case, n ¼ a ¼ 1 K=KS , where a is known as the Biot coefficient or Biot–Willis coefficient. K is the dry-rock (drained) bulk modulus and KS is the mineral bulk modulus (or some appropriate average of the moduli if there is mixed mineralogy), defined below. Explicitly, dVT 1 K dPP ¼ dPC 1 VT K KS where dVT, dPC, and dPP signify increments relative to a reference state.

Pitfall

A common error is to assume that the Biot–Willis effective-stress coefficient a for volume change also applies to other deformation-related rock properties. For example, although rock elastic moduli vary with crack and pore deformation, there is no theoretical justification for extrapolating a to elastic moduli and seismic velocities. Other factors determining the apparent effective-stress coefficient observed in the laboratory include the rate of change of pore pressure, the connectivity of the pore space, the presence or absence of hysteresis, heterogeneity of the rock mineralogy, and variation of pore-fluid compressibility with pore pressure. Table 2.6.1, compiled from Zimmerman (1984), Berryman (1992a, 1993), and H. F. Wang (2000), summarizes the theoretically expected effective incremental stress laws for a variety of rock properties. These depend on four defined rock moduli: 1 1 @VT ; K ¼ modulus of the drained porous frame; ¼ K VT @Pd PP 1 1 @VT ¼ ; KS ¼ unjacketed modulus; if monomineralic; KS VT @PP Pd KS ¼ Kmineral ; otherwise KS is a poorly understood average of the mixed mineral moduli 1 1 @V ¼ ; if monomineralic; KS ¼ K ¼ Kmineral K V @PP Pd 1 1 @V 1 1 1 ¼ ¼ KP V @Pd PP K KS

46

Elasticity and Hooke’s law

Theoretically predicted effective-stress laws for incremental changes in confining and pore pressures (from Berryman, 1992a).

Table 2.6.1

Property

General mineralogy

Sample volumea

dVT VT dV V

Pore volumeb Porosityc Solid volumed Permeabilitye Velocity/elastic modulif

¼ K1 ðdPC adPP Þ

¼ K1P ðdPC bdPP Þ d a ¼ K ðdPC dPP Þ 1 ¼ ð1ÞK ðdPC dPP Þ S h i a dk 2 k ¼ h K þ 3K ðdPC kdPP Þ dVS VS

dVP VP

¼ f ðdPC yPP Þ

Notes: VT is the total volume. a ¼ 1 K=KS , Biot coefficient; usually in the range a 1; if monomineralic, a ¼ 1 K=Kmineral . b V ¼ VT , pore volume. b ¼ 1 KP =K , usually, b 1, but it is possible that b > 1. c ¼ b a a; if monomineralic, ¼ 1.

2ð1aÞ d e ¼ a a 1a ða Þ; if monomineralic, ¼ . a b. k ¼ 1 3hðaÞþ2 1: a

S Þ=@PC h 2 þ m 4, where m is Archie’s cementation exponent. f y ¼ 1 @ð1=K @ð1=KÞ=@PC ; if monomineralic,

y ¼ 1.

where Pd ¼ PC PP is the differential pressure, VT is the sample bulk (i.e., total) volume, and Vf is the pore volume. The negative sign for each of these rock properties follows from defining pressures as being positive in compression and volumes positive in expansion. There is still a need to reconcile theoretical predictions of effective stress with certain laboratory data. For example, simple, yet rigorous, theoretical considerations (Zimmerman, 1991a; Berryman, 1992a; Gurevich, 2004) predict that nvelocity ¼ 1 for monomineralic, elastic rocks. Experimentally observed values for nvelocity are sometimes close to 1, and sometimes less than one. Speculations for the variations have included mineral heterogeneity, poorly connected pore space, pressure-related changes in pore-fluid properties, incomplete correction for fluid-related velocity dispersion in ultrasonic measurements, poorly equilibrated or characterized pore pressure, and inelastic deformation.

Uses Characterization of the stress sensitivity of rock properties makes it possible to invert for rock-property changes from changes in seismic or electrical measurements. It also provides a means of understanding how rock properties might change in response to tectonic stresses or pressure changes resulting from reservoir or aquifer production.

47

2.7 Stress-induced anisotropy in rocks

Assumptions and limitations

2.7

The existence of effective-pressure laws assumes that the rocks are hyperelastic, i.e., there is no hysteresis or rate dependence in the relation between stress and strain. Rocks are extremely variable, so effective-pressure behavior can likewise be variable.

Stress-induced anisotropy in rocks Synopsis The closing of cracks under compressive stress (or, equivalently, the stiffening of compliant grain contacts) tends to increase the effective elastic moduli of rocks (see also Section 2.5 on third-order elasticity). When the crack population is anisotropic, either in the original unstressed condition or as a result of the stress field, then this condition can impact the overall elastic anisotropy of the rock. Laboratory demonstrations of stress-induced anisotropy have been reported by numerous authors (Nur and Simmons, 1969a; Lockner et al., 1977; Zamora and Poirier, 1990; Sayers et al., 1990; Yin, 1992; Cruts et al., 1995). The simplest case to understand is a rock with a random (isotropic) distribution of cracks embedded in an isotropic mineral matrix. In the initial unstressed state, the rock is elastically isotropic. If a hydrostatic compressive stress is applied, cracks in all directions respond similarly, and the rock remains isotropic but becomes stiffer. However, if a uniaxial compressive stress is applied, cracks with normals parallel or nearly parallel to the applied-stress axis will tend to close preferentially, and the rock will take on an axial or transversely isotropic symmetry. An initially isotropic rock with arbitrary stress applied will have at least orthorhombic symmetry (Nur, 1971; Rasolofosaon, 1998), provided that the stress-induced changes in moduli are small relative to the absolute moduli. Figure 2.7.1 illustrates the effects of stress-induced crack alignment on seismicvelocity anisotropy discovered in the laboratory by Nur and Simmons (1969a). The crack porosity of the dry granite sample is essentially isotropic at low stress. As uniaxial stress is applied, crack anisotropy is induced. The velocities (compressional and two polarizations of shear) clearly vary with direction relative to the stressinduced crack alignment. Table 2.7.1 summarizes the elastic symmetries that result when various applied-stress fields interact with various initial crack symmetries (Paterson and Weiss, 1961; Nur, 1971). A rule of thumb is that a wave is most sensitive to cracks when its direction of propagation or direction of polarization is perpendicular (or nearly so) to the crack faces. The most common approach to modeling the stress-induced anisotropy is to assume angular distributions of idealized penny-shaped cracks (Nur, 1971; Sayers, 1988a, b; Gibson and Tokso¨z, 1990). The stress dependence is introduced by assuming or inferring distributions or spectra of crack aspect ratios with various orientations.

48

Elasticity and Hooke’s law

5.0 Stress (bars) 300 4.6 200 VP (km/s)

4.8

4.4 100 4.2 4.0 0 3.8 3.6

0

20 40 60 80 Angle from stress axis (deg) 3.2

3.2

300

2.9

200

2.8

100

2.7

Vs (SV) (km/s)

Vs (SH) (km/s)

400 3.0

2.6

Stress (bars) 3.1 400

Stress (bars)

3.1

200

2.9

2.8 100 2.7

0 0

300

3.0

20 40 60 80 Angle from stress axis (deg)

2.6

0 0

20 40 60 80 Angle from stress axis (deg)

Figure 2.7.1 The effects of stress-induced crack alignment on seismic-velocity anisotropy measured in the laboratory (Nur and Simmons, 1969a).

The assumption is that a crack will close when the component of applied compressive stress normal to the crack faces causes a normal displacement of the crack faces equal to the original half-width of the crack. This allows us to estimate the crack closing stress as follows: close ¼

3pð1 2nÞ p aK0 ¼ am 4ð1 n2 Þ 2ð1 nÞ 0

where a is the aspect ratio of the crack, and n, m0, and K0 are the Poisson ratio, shear modulus, and bulk modulus of the mineral, respectively (see Section 2.9 on the deformation of inclusions and cavities in elastic solids). Hence, the thinnest cracks will close first, followed by thicker ones. This allows one to estimate, for a given aspect-ratio distribution, how many cracks remain open in each direction for any applied stress field. These inferred crack distributions and their orientations can be put into one of the popular crack models (e.g., Hudson, 1981) to estimate the resulting effective elastic moduli of the rock. Although these penny-shaped crack models have been relatively successful and provide a useful physical interpretation, they are limited to low crack concentrations and may not effectively represent a broad range of crack geometries (see Section 4.10 on Hudson’s model for cracked media).

49

2.7 Stress-induced anisotropy in rocks

Dependence of symmetry of induced velocity anisotropy on initial crack distribution and applied stress and its orientation.

Table 2.7.1

Symmetry of initial crack distribution

Applied stress

Random

Hydrostatic Uniaxial Triaxiala

Axial

Hydrostatic Uniaxial Uniaxial Uniaxial Triaxiala Triaxiala

Orthorhombic

Hydrostatic Uniaxial Uniaxial Uniaxial Triaxiala Triaxiala Triaxiala

Orientation of applied stress

Parallel to axis of symmetry Normal to axis of symmetry Inclined Parallel to axis of symmetry Inclined Parallel to axis of symmetry Inclined in plane of symmetry Inclined Parallel to axis of symmetry Inclined in plane of symmetry Inclined

Symmetry of induced velocity anisotropy

Number of elastic constants

Isotropic Axial Orthorhombic

2 5 9

Axial Axial

5 5

Orthorhombic

9

Monoclinic Orthorhombic

13 9

Monoclinic

13

Orthorhombic Orthorhombic

9 9

Monoclinic

13

Triclinic Orthorhombic

21 9

Monoclinic

13

Triclinic

21

Note: a Three generally unequal principal stresses.

As an alternative, Mavko et al. (1995) presented a simple recipe for estimating stress-induced velocity anisotropy directly from measured values of isotropic VP and VS versus hydrostatic pressure. This method differs from the inclusion models, because it is relatively independent of any assumed crack geometry and is not limited to small crack densities. To invert for a particular crack distribution, one needs to assume crack shapes and aspect-ratio spectra. However, if rather than inverting for a crack distribution, we instead directly transform hydrostatic velocity–pressure data to stress-induced velocity anisotropy, we can avoid the need for parameterization in terms of ellipsoidal cracks and the resulting limitations to low crack densities. In this sense, the method of Mavko et al. (1995) provides not only a simpler but also a more general solution to this problem, for ellipsoidal cracks are just one particular case of the general formulation.

50

Elasticity and Hooke’s law

The procedure is to estimate the generalized pore-space compliance from the measurements of isotropic VP and VS. The physical assumption that the compliant part of the pore space is crack-like means that the pressure dependence of the generalized compliances is governed primarily by normal tractions resolved across cracks and defects. These defects can include grain boundaries and contact regions between clay platelets (Sayers, 1995). This assumption allows the measured pressure dependence to be mapped from the hydrostatic stress state to any applied nonhydrostatic stress. The method applies to rocks that are approximately isotropic under hydrostatic stress and in which the anisotropy is caused by crack closure under stress. Sayers (1988b) also found evidence for some stress-induced opening of cracks, which is ignored in this method. The potentially important problem of stress–strain hysteresis is also ignored. The anisotropic elastic compliance tensor Sijkl(s) at any given stress state s may be expressed as Sijkl ðÞ ¼ Sijkl ðÞ S0ijkl Z p=2 Z 2p 0

0 ^ T mÞ ^ 4W2323 ^ T mÞ ^ mi mj mk ml sin y dy d ¼ W3333 ðm ðm ¼0

y¼0

Z

þ

p=2 y¼0

Z

0 ^ T mÞ ^ dik mj ml þ dil mj mk þ djk mi ml W2323 ðm ¼0

þdjl mi mk sin y dy d 2p

where 1 Siso ðpÞ 2p jjkk 1 0 iso W2323 ðpÞ ¼ ½Siso abab ðpÞ Saabb ðpÞ 8p 0 W3333 ðpÞ ¼

The tensor S0ijkl denotes the reference compliance at some very large confining hydrostatic pressure when all of the compliant parts of the pore space are closed. The iso 0 expression Siso ijkl ðpÞ ¼ Sijkl ðpÞ Sijkl describes the difference between the compliance under a hydrostatic effective pressure p and the reference compliance at high pressure. These are determined from measured P- and S-wave velocities versus the 0 0 hydrostatic pressure. The tensor elements W3333 and W2323 are the measured normal and shear crack compliances and include all interactions with neighboring cracks and pores. These could be approximated, for example, by the compliances of idealized ellipsoidal cracks, interacting or not, but this would immediately reduce the general^ ðsin y cos ; sin y sin ; cos yÞT denotes the unit normal to the ity. The expression m crack face, where y and f are the polar and azimuthal angles in a spherical coordinate system, respectively.

51

2.7 Stress-induced anisotropy in rocks

An important physical assumption in the preceding equations is that, for thin 0 0 0 cracks, the crack compliance tensor Wijkl is sparse, and thus only W3333 , W1313 , and 0 W2323 are nonzero. This is a general property of planar crack formulations and reflects an approximate decoupling of normal and shear deformation of the crack and decoupling of the in-plane and out-of-plane deformations. This allows us to write 0 0 Wjjkk W3333 . Furthermore, it is assumed that the two unknown shear compliances 0 0 are approximately equal: W1313 W2323 . A second important physical assumption is that for a thin crack under any stress field, it is primarily the normal component of ^ T m, ^ resolved on the faces of a crack, that causes it to close and to have stress, ¼ m a stress-dependent compliance. Any open crack will have both normal and shear deformation under normal and shear loading, but it is only the normal stress that determines crack closure. For the case of uniaxial stress s0 applied along the 3-axis to an initially isotropic rock, the normal stress in any direction is sn ¼ s0 cos2y. The rock takes on a transversely isotropic symmetry, with five independent elastic constants. The five independent components of DSijkl become Z Suni 3333

p=2

¼ 2p 0

Z

p=2

þ 2p

0 p=2

Z Suni 1111 ¼ 2p

0

Z

0 p=2

Z Z

0

Z

0

Suni 1133 ¼ 2p Suni 2323

p=2

p=2

¼ 2p 0

Z

þ 2p þ 2p 0

0 cos2 yÞ 4W2323 ð0 cos2 yÞ sin4 y sin y dy

0 2W2323 ð0 cos2 yÞ sin2 y sin y dy

0 1 8 ½W3333 ð0

0 cos2 yÞ 4W2323 ð0 cos2 yÞ sin4 y sin y dy

0 1 2 ½W3333 ð0

0 cos2 yÞ 4W2323 ð0 cos2 yÞ sin2 y cos2 y sin y dy

0 1 2 ½W3333 ð0

0 cos2 yÞ 4W2323 ð0 cos2 yÞ sin2 y cos2 y sin y dy

p=2

0

Z

0 4W2323 ð0 cos2 yÞ cos2 y sin y dy

0 3 8 ½W3333 ð0

p=2

þ 2p Suni 1122 ¼ 2p

0 0 ½W3333 ð0 cos2 yÞ 4W2323 ð0 cos2 yÞ cos4 y sin y dy

p=2

0 1 2 W2323 ð0

cos2 yÞ sin2 y sin y dy

0 W2323 ð0 cos2 yÞ cos2 y sin y dy

0 Note that in the above equations, the terms in parentheses with W2323 ðÞ and 0 0 0 W3333 ðÞ are arguments to the W2323 and W3333 pressure functions, and not multiplicative factors. Sayers and Kachanov (1991, 1995) have presented an equivalent formalism for stress-induced anisotropy. The elastic compliance Sijkl is once again written in the form

52

Elasticity and Hooke’s law

Sijkl ¼ Sijkl ðÞ S0ijkl where S0ijkl is the compliance in the absence of compliant cracks and grain boundaries and DSijkl is the excess compliance due to the cracks. DSijkl can be written as Sijkl ¼ 14 ðdik ajl þ dil ajk þ djk ail þ djl aik Þ þ bijkl where aij is a second-rank tensor and bijkl is a fourth-rank tensor defined by 1 X ðrÞ ðrÞ ðrÞ ðrÞ B n n A V r T i j 1 X ðrÞ ðrÞ ðrÞ ðrÞ ðrÞ ðrÞ ¼ BN BT ni nj nk nl AðrÞ V r

aij ¼ bijkl

In these expressions, the summation is over all grain contacts and microcracks within ðrÞ ðrÞ the rock volume V. BN and BT are the normal and shear compliances of the rth discontinuity, which relate the displacement discontinuity across the crack to the ðrÞ applied traction across the crack faces; ni , is the ith component of the normal to the discontinuity, and A(r) is the area of the discontinuity. A completely different strategy for quantifying stress-induced anisotropy is to use the formalism of third-order elasticity, described in Section 2.5 (e.g., Helbig, 1994; Johnson and Rasolofosaon, 1996; Prioul et al., 2004). The third-order elasticity approach is phenomenological, avoiding the physical mechanisms of stress sensitivity, but providing a compact notation. For example, Prioul et al. (2004) found that for a stressed VTI (transversely isotropic with vertical symmetry axis) material, the effective elastic constants can be approximated in Voigt notation as 0 ceff 11 c11 þ c111 e11 þ c112 ðe22 þ e33 Þ 0 ceff 22 c11 þ c111 e22 þ c112 ðe11 þ e33 Þ 0 ceff 33 c33 þ c111 e33 þ c112 ðe11 þ e22 Þ 0 ceff 12 c12 þ c112 ðe11 þ e22 Þ þ c123 e33 0 ceff 13 c13 þ c112 ðe11 þ e33 Þ þ c123 e22 0 ceff 23 c13 þ c112 ðe22 þ e33 Þ þ c123 e11 0 ceff 66 c66 þ c144 e33 þ c155 ðe11 þ e22 Þ 0 ceff 55 c44 þ c144 e22 þ c155 ðe11 þ e33 Þ 0 ceff 44 c44 þ c144 e11 þ c155 ðe22 þ e33 Þ

where the constants c011 , c033 , c013 , c044 , c066 are the VTI elastic constants at the unstressed reference state, with c012 ¼ c011 2c066 . The quantities e11, e22, and e33 are the principal strains, computed from the applied stress using the conventional Hooke’s law, eij ¼ sijkl kl . For these expressions, it is assumed that the direction of

53

2.7 Stress-induced anisotropy in rocks

the applied principal stress is aligned with the VTI symmetry (x3-) axis. Furthermore, for these expressions it is assumed that the stress-sensitive third-order tensor is isotropic, defined by the three independent constants, c111, c112, and c123 with c144 ¼ ðc112 c123 Þ=2 and c155 ¼ ðc111 c112 Þ=4. In practice, the elastic constants (five for the unstressed VTI rock and three for the third-order elasticity) can be estimated from laboratory measurements, as illustrated by Sarkar et al. (2003) and Prioul et al. (2004). For an intrinsically VTI rock, hydrostatic loading experiments provide sufficient information to invert for all three of the third-order constants, provided that the anisotropy is not too weak. However, for an intrinsically isotropic rock, nonhydrostatic loading is required in order to provide enough independent information to determine the constants. Once the constants are determined, the stress-induced elastic constants corresponding to any (aligned) stress field can be determined. The full expressions for stress-induced anisotropy of an originally VTI rock are given by Sarkar et al. (2003) and Prioul et al. (2004). Sarkar et al. (2003) give expressions for stress-induced changes in Thomsen parameters of an originally VTI rock, provided that both the original and stressinduced anisotropies are weak: c155 c155 ð22 33 Þ; eð2Þ ¼ e0 þ 0 0 ð11 33 Þ 0 0 c33 c55 c33 c55 c155 c155 0 ð2Þ 0 d þ 0 0 ð22 33 Þ; d ¼ d þ 0 0 ð11 33 Þ c33 c55 c33 c55 c456 c456 g0 þ 0 0 ð22 33 Þ; gð2Þ ¼ g0 þ 0 0 ð11 33 Þ 2c55 c55 2c55 c55 c155 ð22 11 Þ c033 c055 1 1 c456 ¼ ðc111 3c112 þ 2c123 Þ ðc111 c112 Þ; 4 8

eð1Þ ¼ e0 þ dð1Þ ¼ gð1Þ ¼ dð3Þ ¼ c155 ¼

The results are shown in terms of Tsvankin’s extended Thomsen parameters (Section 2.4). Orthorhombic symmetry occurs when 11 6¼ 22 . (Strictly speaking, three different principal stresses applied to a VTI rock do not result in perfect orthorhombic symmetry. However, orthorhombic parameters are adequate in the case of weak anisotropy.)

Uses Understanding or, at least, empirically describing the stress dependence of velocities is useful for quantifying the change of velocities in seismic time-lapse data due to changes in reservoir pressure, as well as in certain types of naturally occurring overpressure. Since the state of stress in situ is seldom hydrostatic, quantifying the impact of stress on anisotropy can often improve on the usual isotropic analysis.

54

Elasticity and Hooke’s law

Assumptions and limitations

2.8

Most models for predicting or describing stress-induced anisotropy that are based on cracks and crack-like flaws assume an isotropic, linear, elastic solid mineral material. Methods based on ellipsoidal cracks or spherical contacts are limited to idealized geometries and to low crack densities. The method of Mavko et al. (1995) has been shown to sometimes under predict stress-induced anisotropy when the maximum stress difference is comparable to the mean stress magnitude. More generally, there is evidence (Johnson and Rasolofosaon, 1996) that classical elastic formulations (nonlinear elasticity) can fail to describe the behavior of rocks at low stresses. The methods presented here assume that the strains are infinitesimal. When strains become finite, an additional source of nonlinearity, called geometrical or kinetic nonlinearity, appears, related to the difference between Lagrangian and Eulerian descriptions of motion (Zarembo and Krasil’nikov, 1971; Johnson and Rasolofosaon, 1996).

Strain components and equations of motion in cylindrical and spherical coordinate systems Synopsis The equations of motion and the expressions for small-strain components in cylindrical and spherical coordinate systems differ from those in a rectangular coordinate system. Figure 2.8.1 shows the variables used in the equations that follow. In the cylindrical coordinate system (r, f, z), the coordinates are related to those in the rectangular coordinate system (x, y, z) as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y r ¼ x2 þ y2 ; tanðÞ ¼ x x ¼ r cosðÞ; y ¼ r sinðÞ; z ¼ z The small-strain components can be expressed through the displacements ur, uf, and uz (which are in the directions r, f, and z, respectively) as @ur 1 @u ur @uz ; e ¼ þ ; ezz ¼ @r r @z r @ 1 1 @ur @u u þ ¼ @r r 2 r @ 1 @u 1 @uz 1 @uz @ur ¼ þ ; ezr ¼ þ @z 2 @z r @ 2 @r

err ¼ er ez

The equations of motion are @rr 1 @r @zr rr @ 2 ur þ þ þ ¼ 2 @r @z r @t r @

55

2.8 Strain components and equations of motion

z

z r

Spherical coordinates

Cylindrical coordinates

θ

y

y

f

x

f

x

r

Figure 2.8.1 The variables used for converting between Cartesian, spherical, and cylindrical coordinates.

@r 1 @ @z 2r @ 2 u þ þ þ ¼ 2 @r @z r @t r @ @rz 1 @z @zz rz @ 2 uz þ þ þ ¼ 2 @r @z r @t r @ where r denotes density and t time. In the spherical coordinate system (r, f, y) the coordinates are related to those in the rectangular coordinate system (x, y, z) as r¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x2 þ y2 þ z2 ;

x ¼ r sinðyÞ cosðÞ;

tanðÞ ¼

y ; x

z cosðyÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 x þ y2 þ z2

y ¼ r sinðyÞ sinðÞ;

z ¼ r cosðyÞ

The small-strain components can be expressed through the displacements ur, uf, and uy (which are in the directions r, f, and y, respectively) as @ur 1 @u ur uy ; e ¼ þ þ ; @r r r tanðyÞ r sinðyÞ @ 1 1 @ur @u u þ er ¼ @r r 2 r sinðyÞ @ 1 1 @u u 1 @uy þ ey ¼ 2 r @ r tanðyÞ r sinðyÞ @ 1 @uy uy 1 @ur ery ¼ þ r 2 @r r @y err ¼

eyy ¼

1 @uy ur þ r r @y

The equations of motion are @rr 1 @r 1 @ry 2rr þ ry cotðyÞ yy @ 2 ur þ þ þ ¼ 2 @r r @t r sinðyÞ @ r @y @r 1 @ 1 @y 3r þ y cotðyÞ @ 2 u þ þ þ ¼ 2 @r @t r sinðyÞ @ r @y r @ry 1 @ 1 @y 3ry þ ðyy ÞcotðyÞ @ 2 uy þ þ þ ¼ 2 @r @t r sinðyÞ @ r @y r

56

Elasticity and Hooke’s law

Uses The foregoing equations are used to solve elasticity problems where cylindrical or spherical geometries are most natural.

Assumptions and limitations The equations presented assume that the strains are small.

2.9

Deformation of inclusions and cavities in elastic solids Synopsis Many problems in effective-medium theory and poroelasticity can be solved or estimated in terms of the elastic behavior of cavities and inclusions. Some static and quasistatic results for cavities are presented here. It should be remembered that often these are also valid for certain limiting cases of dynamic problems. Excellent treatments of cavity deformation and pore compressibility are given by Jaeger and Cook (1969) and by Zimmerman (1991a).

General pore deformation Effective dry compressibility Consider a homogeneous linear elastic solid that has an arbitrarily shaped pore space – either a single cavity or a collection of pores. The effective dry compressibility (i.e., the reciprocal of the dry bulk modulus) of the porous solid can be written as 1 1 @p ¼ þ Kdry K0 p @ dry where Kdry is the effective bulk modulus of the dry porous solid, K0 is the bulk modulus of the solid mineral material, is the porosity, p is the pore volume, @p =@jdry is the derivative of the pore volume with respect to the externally applied hydrostatic stress. We also assume that no inelastic effects such as friction or viscosity are present. This is strictly true, regardless of the pore geometry and the pore concentration. The preceding equation can be rewritten slightly as 1 1 ¼ þ Kdry K0 K where K ¼ p =ð@p =@Þjdry is defined as the dry pore-space stiffness. These equations state simply that the porous rock compressibility is equal to the intrinsic mineral compressibility plus an additional compressibility caused by the pore space.

57

2.9 Deformation of inclusions and cavities

Caution: “Dry rock” is not the same as gas-saturated rock

The dry-frame modulus refers to the incremental bulk deformation resulting from an increment of applied confining pressure with pore pressure held constant. This corresponds to a “drained” experiment in which pore fluids can flow freely in or out of the sample to ensure constant pore pressure. Alternatively, the dry-frame modulus can correspond to an undrained experiment in which the pore fluid has zero bulk modulus, and in which pore compressions therefore do not induce changes in pore pressure. This is approximately the case for an air-filled sample at standard temperature and pressure. However, at reservoir conditions, the gas takes on a non-negligible bulk modulus and should be treated as a saturating fluid. An equivalent expression for the dry rock compressibility or bulk modulus is Kdry ¼ K0 ð1 bÞ where b is sometimes called the Biot coefficient, which describes the ratio of porevolume change Dup to total bulk-volume change DV under dry or drained conditions: p Kdry b¼ ¼ V dry K

Stress-induced pore pressure: Skempton’s coefficient If this arbitrary pore space is filled with a pore fluid with bulk modulus, Kfl, the saturated solid is stiffer under compression than the dry solid, because an increment of pore-fluid pressure is induced that resists the volumetric strain. The ratio of the induced pore pressure, dP, to the applied compressive stress, ds, is sometimes called Skempton’s coefficient and can be written as B

dP 1 ¼ d 1 þ K ð1=Kfl 1=K0 Þ 1 ¼

1 1 þ ð1=Kfl 1=K0 Þ 1=Kdry 1=K0

where Kf is the dry pore-space stiffness defined earlier in this section. For this definition to be true, the pore pressure must be uniform throughout the pore space, as will be the case if: (1) there is only one pore, (2) all pores are well connected and the frequency and viscosity are low enough for any pressure differences to equilibrate, or (3) all pores have the same dry pore stiffness.

58

Elasticity and Hooke’s law

Given these conditions, there is no additional limitation on pore geometry or concentration. All of the necessary information concerning pore stiffness and geometry is contained in the parameter Kf.

Saturated stress-induced pore-volume change The corresponding change in fluid-saturated pore volume, up, caused by the remote stress is 1 dp 1 dP 1=Kfl ¼ ¼ p d sat Kfl d 1 þ K ð1=Kfl 1=K0 Þ

Low-frequency saturated compressibility The low-frequency saturated bulk modulus, Ksat, can be derived from Gassmann’s equation (see Section 6.3 on Gassmann). One equivalent form is 1 1 1 ¼ þ þ K K 0 fl Ksat K0 K þ K K K0 K þ Kfl 0

fl

where, again, all of the necessary information concerning pore stiffness and geometry is contained in the dry pore stiffness Kf, and we must ensure that the stress-induced pore pressure is uniform throughout the pore space.

Three-dimensional ellipsoidal cavities Many effective media models are based on ellipsoidal inclusions or cavities. These are mathematically convenient shapes and allow quantitative estimates of, for example, Kf, which was defined earlier in this section. Eshelby (1957) discovered that the strain, eij, inside an ellipsoidal inclusion is homogeneous when a homogeneous strain, e0ij , (or stress) is applied at infinity. Because the inclusion strain is homogeneous, operations such as determining the inclusion stress or integrating to obtain the displacement field are straightforward. It is very important to remember that the following results assume a single isolated cavity in an infinite medium. Therefore, substituting them directly into the preceding formulas for dry and saturated moduli gives estimates that are strictly valid only for low concentrations of pores (see also Section 4.8 on self-consistent theories).

Spherical cavity For a single spherical cavity with volume p ¼ 43 pR3 and a hydrostatic stress, ds, applied at infinity, the radial strain of the cavity is

59

2.9 Deformation of inclusions and cavities

dR 1 ð1 vÞ ¼ d R K0 2ð1 2vÞ where v and K0 are the Poisson ratio and bulk modulus of the solid material, respectively. The change of pore volume is dp ¼

1 3ð1 vÞ p d K0 2ð1 2vÞ

Then, the volumetric strain of the sphere is eii ¼

dp 1 3ð1 vÞ ¼ d p K0 2ð1 2vÞ

and the single-pore stiffness is 1 1 dp 1 3ð1 vÞ ¼ ¼ K p d K0 2ð1 2vÞ Remember that this estimate of Kf assumes a single isolated spherical cavity in an infinite medium. Under a remotely applied homogeneous shear stress, t0, corresponding to remote shear strain e0 ¼ t0/2m0, the effective shear strain in the spherical cavity is e¼

15ð1 vÞ t0 2m0 ð7 5vÞ

where m0 is the shear modulus of the solid. Note that this results in approximately twice the strain that would occur without the cavity.

Penny-shaped crack: oblate spheroid Consider a dry penny-shaped ellipsoidal cavity with semiaxes a b ¼ c. When a remote uniform tensional stress, ds, is applied normal to the plane of the crack, each crack face undergoes an outward displacement, U, normal to the plane of the crack, given by the radially symmetric distribution UðrÞ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4ð1 v2 Þc 1 ðr=cÞ2 3pK0 ð1 2vÞ

d

where r is the radial distance from the axis of the crack. For any arbitrary homogeneous remote stress, ds is the component of stress normal to the plane of the crack. Thus, ds can also be thought of as a remote hydrostatic stress field. This displacement function is also an ellipsoid with semiminor axis, U(r ¼ 0), and semimajor axis, c. Therefore the volume change, du, which is simply the

60

Elasticity and Hooke’s law

integral of U(r) over the faces of the crack, is just the volume of the displacement ellipsoid: 4 16c3 ð1 v2 Þ dp ¼ pUð0Þ c2 ¼ d 9K0 ð1 2vÞ 3 Then the volumetric strain of the cavity is eii ¼

dp 4ðc=aÞ ð1 v2 Þ ¼ d p 3pK0 ð1 2vÞ

and the pore stiffness is 1 1 dp 4ðc=aÞ ð1 v2 Þ ¼ ¼ K p d 3pK0 ð1 2vÞ An interesting case is an applied compressive stress causing a displacement, U(0), equal to the original half-width of the crack, thus closing the crack. Setting U(0) ¼ a allows one to compute the closing stress: close ¼

3pð1 2vÞ p aK0 ¼ am 4ð1 v2 Þ 2ð1 vÞ 0

where a ¼ (a/c) is the aspect ratio. Now, under a remotely applied homogeneous shear stress, t0, corresponding to remote shear strain e0 ¼ t0/2m0, the effective shear strain in the cavity is e ¼ t0

2ðc=aÞ ð1 vÞ pm0 ð2 vÞ

Needle-shaped pore: prolate spheroid Consider a dry needle-shaped ellipsoidal cavity with semiaxes a b ¼ c and with pore volume ¼ 43pac2 . When a remote hydrostatic stress, ds, is applied, the pore volume change is dp ¼

ð5 4vÞ p d 3K0 ð1 2vÞ

The volumetric strain of the cavity is then eii ¼

dp ð5 4vÞ d ¼ p 3K0 ð1 2vÞ

and the pore stiffness is 1 1 dp ð5 4vÞ ¼ ¼ K p d 3K0 ð1 2vÞ

61

2.9 Deformation of inclusions and cavities

Note that in the limit of very large a/c, these results are exactly the same as for a two-dimensional circular cylinder. Estimate the increment of pore pressure induced in a water-saturated rock when a 1 bar increment of hydrostatic confining pressure is applied. Assume that the rock consists of stiff, spherical pores in a quartz matrix. Compare this with a rock with thin, penny-shaped cracks (aspect ratio a ¼ 0.001) in a quartz matrix. The moduli of the individual constituents are Kquartz ¼ 36 GPa, Kwater ¼ 2.2 GPa, and vquartz ¼ 0.07. The pore-space stiffnesses are given by K-sphere ¼ Kquartz K-crack ¼

2ð1 2vquartz Þ ¼ 22:2 GPa 3ð1 vquartz Þ

3paKquartz ð1 2vquartz Þ ¼ 0:0733 GPa ð1 v2quartz Þ 4

The pore-pressure increment is computed from Skempton’s coefficient: Ppore 1 ¼B¼ 1 1 Þ Pconfining 1 þ K ðKwater Kquartz Bsphere ¼ Bcrack ¼

1 ¼ 0:095 1 þ 22:2½ð1=2:2Þ ð1=36Þ 1 ¼ 0:970 1 þ 0:0733½ð1=2:2Þ ð1=36Þ

Therefore, the pore pressure induced in the spherical pores is 0.095 bar and the pore pressure induced in the cracks is 0.98 bar.

Two-dimensional tubes A special two-dimensional case of long, tubular pores was treated by Mavko (1980) to describe melt or fluids arranged along the edges of grains. The cross-sectional shape is described by the equations 1 x ¼ R cos y þ cos 2y 2þg 1 sin 2y y ¼ R sin y þ 2þg where g is a parameter describing the roundness (Figure 2.9.1).

62

Elasticity and Hooke’s law

g =0

g =1

g =∞

2R

2R

2R

Figure 2.9.1 The cross-sectional shapes of various two-dimensional tubes.

Consider, in particular, the case on the left, g ¼ 0. The pore volume is 12 paR2 , where a R is the length of the tube. When a remote hydrostatic stress, ds, is applied, the pore volume change is dp ¼

ð13 4v 8v2 Þ p d 3K0 ð1 2vÞ

The volumetric strain of the cavity is then eii ¼

dp ð13 4v 8v2 Þ ¼ d 3K0 ð1 2vÞ p

and the pore stiffness is 1 1 dp ð13 4v 8v2 Þ ¼ ¼ K p d 3K0 ð1 2vÞ In the extreme, g ! 1, the shape becomes a circular cylinder, and the expression for pore stiffness, Kf, is exactly the same as that derived for the needle-shaped pores above. Note that the triangular cavity (g ¼ 0) has about half the pore stiffness of the circular one; that is, the triangular tube can give approximately the same effective modulus as the circular tube with about half the porosity. Caution

These expressions for Kf, dup, and eii include an estimate of tube shortening as well as a reduction in pore cross-sectional area under hydrostatic stress. Hence, the deformation is neither plane stress nor plane strain.

Plane strain The plane-strain compressibility in terms of the reduction in cross-sectional area A is given by ( 6ð1vÞ g!0 1 1 dA m0 ; ¼ ¼ 2ð1vÞ K0 A d ; g!1 m 0

63

2.9 Deformation of inclusions and cavities

The latter case (g ! 1) corresponds to a tube with a circular cross-section and agrees (as it should) with the expression given below for the limiting case of a tube with an elliptical cross-section with aspect ratio unity. A general method of determining K0 for nearly arbitrarily shaped two-dimensional cavities under plane-strain deformation was developed by Zimmerman (1986, 1991a) and involves conformal mapping of the tube shape into circular pores. For example, pores with cross-sectional shapes that are n-sided hypotrochoids given by 1 cosðn 1Þy ðn 1Þ 1 y ¼ sinðyÞ þ sinðn 1Þy ðn 1Þ

x ¼ cosðyÞ þ

(see the examples labeled (1) in Table 2.9.1) have plane-strain compressibilities 1 1 dA 1 1 þ ðn 1Þ1 ¼ ¼ K0 A d K0cir 1 ðn 1Þ1 where 1=K0cir is the plane-strain compressibility of a circular tube given by 1 2ð1 nÞ ¼ 0cir m0 K Table 2.9.1 summarizes a few plane-strain pore compressibilities.

Two-dimensional thin cracks A convenient description of very thin two-dimensional cracks is in terms of elastic line dislocations. Consider a crack lying along –c < x < c in the y ¼ 0 plane and very long in the z direction. The total relative displacement of the crack faces u(x), defined as the displacement of the negative face (y ¼ 0–) relative to the positive face (y ¼ 0þ), is related to the dislocation density function by BðxÞ ¼

@u @x

where B(x) dx represents the total length of the Burger vectors of the dislocations lying between x and x þ dx. The stress change in the plane of the crack that results from introduction of a dislocation line with unit Burger vector at the origin is ¼

m0 2pDx

where D ¼ 1 for screw dislocations and D ¼ (1 – v) for edge dislocations (v is Poisson’s ratio and m0 is the shear modulus). Edge dislocations can be used to describe mode I and mode II cracks; screw dislocations can be used to describe mode III cracks.

1

(1)

1.581

3

(1)

1.188

2

(1)

p/8a

Plane-strain compressibility normalized by the compressibility of a circular tube.

1/2a

(2)

2/3a

(3)

Notes: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i3=2 (1) 1 1 x ¼ cosðyÞ þ ðn1Þ cosðn 1Þy, where n ¼ number of sides; y ¼ sinðyÞ þ ðn1Þ sinðn 1Þy. (2) y ¼ 2b 1 ðx=cÞ2 ðellipseÞ. (3) y ¼ 2b 1 ðx=cÞ2 (nonelliptical, “tapered” crack).

m0 2ð1nÞK0

Table 2.9.1

65

2.9 Deformation of inclusions and cavities

The stress here is the component of traction in the crack plane parallel to the displacement: normal stress for mode I deformation, in-plane shear for mode II deformation, and out-of-plane shear for mode III deformation. Then the stress resulting from the distribution B(x) is given by the convolution Z c m Bðx0 Þ dx0 ðxÞ ¼ 0 2pD c x x0 The special case of interest for nonfrictional cavities is the deformation for stressfree crack faces under a remote uniform tensional stress, ds, acting normal to the plane of the crack. The outward displacement distribution of each crack face is given by (using edge dislocations) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cð1 vÞ 1 ðx=cÞ2 2cð1 v2 Þ 1 ðx=cÞ2 UðxÞ ¼ ¼ m0 3K0 ð1 2vÞ The volume change is then given by dp 2pc2 a ð1 v2 Þ ¼ pUð0Þca ¼ d 3K0 ð1 2vÞ It is important to note that these results for displacement and volume change apply to any two-dimensional crack of arbitrary cross-section as long as it is very thin and approximately planar. They are not limited to cracks of elliptical cross-section. For the special case in which the very thin crack is elliptical in cross-section with half-width b in the thin direction, the volume is u ¼ pabc, and the pore stiffness under plane-strain deformation is given by 1 1 dA ðc=bÞð1 vÞ 2ðc=bÞ ð1 v2 Þ ¼ ¼ ¼ K0 A d m0 3K0 ð1 2vÞ Another special case is a crack of nonelliptical form (Mavko and Nur, 1978) with initial shape given by "

2 #3=2 x U0 ðxÞ ¼ 2b 1 c0 where c0 is the crack half-length and 2b is the maximum crack width. This crack is plotted in Figure 2.9.2. Note that unlike elliptical cracks that have rounded or blunted ends, this crack has tapered tips where faces make a smooth, tangent contact. If we apply a pressure, P, the crack shortens as well as thins, and the pressure-dependent length is given by 1=2 2ð1 vÞ c ¼ c0 1 P 3m0 ðb=c0 Þ

66

Elasticity and Hooke’s law

b 0 −b

b 0 −b −c

0

−c

c

0

c

Figure 2.9.2 A nonelliptical crack shortens as well as narrows under compression. An elliptical crack only narrows.

Then the deformed shape is 3 x2 3=2 c Uðx; PÞ ¼ 2b 1 ; c0 c

jxj c

An important consequence of the smoothly tapered crack tips and the gradual crack shortening is that there is no stress singularity at the crack tips. In this case, crack closure occurs (i.e., U ! 0) as the crack length goes to zero (c ! 0). The closing stress is close ¼

3 3 a0 E0 a0 m0 ¼ 2ð1 vÞ 4ð1 v2 Þ

where a0 ¼ b/c0 is the original crack aspect ratio, and m0 and E0 are the shear and Young’s moduli of the solid material, respectively. This expression is consistent with the usual rule of thumb that the crack-closing stress is numerically ~a0E0. The exact factor depends on the details of the original crack shape. In comparison, the pressure required to close a two-dimensional elliptical crack of aspect ratio a0 is close ¼

1 a0 E0 2ð1 v2 Þ

Ellipsoidal cracks of finite thickness The pore compressibility under plane-strain deformation of a two-dimensional elliptical cavity of arbitrary aspect ratio a is given by (Zimmerman, 1991a) 1 1 dA 1 v 1 2ð1 v2 Þ 1 ¼ aþ ¼ ¼ aþ K0 A d m0 a 3K0 ð1 2vÞ a where m0, K0, and v are the shear modulus, bulk modulus, and Poisson’s ratio of the mineral material, respectively. Circular pores (tubes) correspond to aspect ratio a ¼ 1 and the pore compressibility is given as 1 1 dA 2ð1 vÞ 4ð1 v2 Þ ¼ ¼ ¼ 0 K A d m0 3K0 ð1 2vÞ

67

2.9 Deformation of inclusions and cavities

Rp

Ri

Rc

Rp

Rc

Figure 2.9.3 Deformation of a single- and double-layer spherical shell.

Deformation of spherical shells The radial displacement ur of a spherical elastic shell (Figure 2.9.3) with inner pore radius Rp and outer radius Rc under applied confining pressure Pc and pore pressure Pp is given by (Ciz et al., 2008) ur ¼

R3p r ðPp Pc Þ ðPp Pc Þ þ 2 4r mð1 Þ 3Kð1 Þ

where r is the radial distance from the center of the sphere, K is the bulk modulus of the solid elastic material, m is the shear modulus of the solid elastic material, and f is the porosity. The associated volume change of the inner spherical pore is dV ¼

3V 1

1 1 1 þ dPp þ dPc : 3K 4m 3K 4m

The volume change of the total sphere (shell plus pore) is dV ¼

3V 1 1 1 þ dPp þ dPc : 1 3K 4m 3K 4m

Similarly, the radial displacement of a double-layer sphere (Figure 2.9.3) is given by (Ciz et al., 2008) ur ¼

An r Bn þ 2 r 3

where Rp is the pore radius, Rc is the outer sphere radius, and Ri is the radius at the boundary between the inner and outer elastic shells. The elastic moduli of the inner solid shell are K1 and m1, and the moduli of the outer shell are K2 and m2:

1 n A1 ¼ ð4m1 m2 þ 3K2 m2 ÞR3i þ ð3K2 m1 3K2 m2 ÞR3c R3p Pp o ð3K2 m1 þ 4m1 m2 ÞR3c R3i Pc

68

Elasticity and Hooke’s law

B1 ¼

A2 ¼

B2 ¼

i 3 1 nh 3 3 3 3 K K þ K m þ ðK m þ K m ÞR R 2 2 1 2 2 2 i Rp Ri Pp c 4 1 2 o

34 K1 K2 þ K1 m2 R3c R3i R3p Pc i 1 nh ð3K1 m1 3K1 m2 ÞR3p ð3K1 m2 þ 4m1 m2 ÞR3i R3c Pc o þ ð3K1 m2 4m1 m2 ÞR3i R3p Pp

i 1 nh ðK1 m1 K2 m1 ÞR31 34K1 K2 þ K2 m1 R3p R3c R3i Pc o

þ 34 K1 K2 þ K1 m1 R3c R3i R3p Pp

¼ K2 m1 ð3K1 þ 4m2 ÞR3c R3i K1 m2 ð3K2 þ 4m1 ÞR3i R3p þ 4m1 m2 ðK1 K2 ÞR6i þ 3K1 K2 ðm2 m1 ÞR3c R3p

Uses The equations presented in this section are useful for computing deformation of cavities in elastic solids and estimating effective moduli of porous solids.

Assumptions and limitations The equations presented in this section are based on the following assumptions. Solid material must be homogeneous, isotropic, linear, and elastic. Results for specific geometries, such as spheres and ellipsoids, are derived for single isolated cavities. Therefore, estimates of effective moduli based on these are limited to relatively low pore concentrations where pore elastic interaction is small. Pore-pressure computations assume that the induced pore pressure is uniform throughout the pore space, which will be the case if (1) there is only one pore, (2) all pores are well connected and the frequency and viscosity are low enough for any pressure differences to equilibrate, or (3) all pores have the same dry pore stiffness.

2.10

Deformation of a circular hole: borehole stresses Synopsis Presented here are some solutions related to a circular hole in a stressed, linear, elastic, and poroelastic isotropic medium.

69

2.10 Deformation of a circular hole: borehole stresses

Hollow cylinder with internal and external pressures The cylinder’s internal radius is R1 and the external radius is R2. Hydrostatic stress p1 is applied at the interior surface at R1 and hydrostatic stress p2 is applied at the exterior surface at R2. The resulting (plane-strain) outward displacement U and radial and tangential stresses are U¼

ðp2 R22 p1 R21 Þ ðp2 p1 ÞR21 R22 1 rþ 2 2 2ðl þ mÞðR2 R1 Þ 2mðR22 R21 Þ r

rr ¼

ðp2 R22 p1 R21 Þ ðp2 p1 ÞR21 R22 1 ðR22 R21 Þ r 2 ðR22 R21 Þ

yy ¼

ðp2 R22 p1 R21 Þ ðp2 p1 ÞR21 R22 1 þ ðR22 R21 Þ ðR22 R21 Þ r2

where l and m are the Lame´ coefficient and shear modulus, respectively. If R1 ¼ 0, we have the case of a solid cylinder under external pressure, with displacement and stress denoted by the following: U¼

p2 r 2ðl þ mÞ

rr ¼ yy ¼ p2 If, instead, R2 ! 1, then p2 r ðp2 p1 ÞR21 þ 2mr 2ðl þ mÞ R21 p1 R2 rr ¼ p2 1 2 þ 2 1 r r 2 R p1 R2 yy ¼ p2 1 þ 21 2 1 r r U¼

These results for plane strain can be converted to plane stress by replacing v by v/(1 þ v), where v is the Poisson ratio.

Circular hole with principal stresses at infinity The circular hole with radius R lies along the z-axis. A principal stress, sxx, is applied at infinity. The stress solution is then xx R2 xx 4R2 3R4 rr ¼ 1 2 þ 1 2 þ 4 cos 2y 2 r 2 r r 2 4 xx R xx 3R 1þ 2 1 þ 4 cos 2y yy ¼ 2 r 2 r

70

Elasticity and Hooke’s law

xx 2R2 3R4 1 þ 2 4 sin 2y ry ¼ 2 r r 8mUr r 2R R2 ¼ ð 1 þ 2 cos 2yÞ þ 1 þ þ 1 2 cos 2y Rxx r R r 2 8mUy 2r 2R R sin 2y ¼ þ 1 2 Rxx r R r where y is measured from the x-axis, and ¼ 3 4v; 3v ¼ ; 1þv

for plane strain for plane stress

At the cavity surface, r ¼ R, rr ¼ ry ¼ 0 yy ¼ xx ð1 2 cos 2yÞ Thus, we see the well-known result that the borehole creates a stress concentration of syy ¼ 3sxx at y ¼ 90 .

Stress concentration around an elliptical hole If, instead, the borehole is elliptical in cross-section with a shape denoted by (Lawn and Wilshaw, 1975) x2 y2 þ ¼1 b2 c2 where b is the semiminor axis and c is the semimajor axis, and the principal stress sxx is applied at infinity, then the largest stress concentration occurs at the tip of the long axis (y ¼ c; x ¼ 0). This is the same location at y ¼ 90 as for the circular hole. The stress concentration is yy ¼ xx ½1 þ 2ðc=Þ1=2 where r is the radius of curvature at the tip given by ¼

b2 c

When b c, the stress concentration is approximately rﬃﬃﬃ yy 2c c ¼2 xx b

71

2.10 Deformation of a circular hole: borehole stresses

z⬘

i

sv

z

y y⬘ a

sh

θ

sH x⬘

x

Figure 2.10.1 The coordinate system for the remote stress field around an inclined, cylindrical borehole.

Stress around an inclined cylindrical hole We now consider the case of a cylindrical borehole of radius R inclined at an angle i to the vertical axis, in a linear, isotropic elastic medium with Poisson ratio n in a nonhydrostatic remote stress field (Jaeger and Cook, 1969; Bradley, 1979; Fjaer et al., 2008). The borehole coordinate system is denoted by (x, y, z) with the z-axis along the axis of the borehole. The remote principal stresses are denoted by n ; the vertical stress; H ; the maximum horizontal stress; and h ; the minimum horizontal stress The coordinate system for the remote stress field (shown in Figure 2.10.1) is denoted by (x0 , y0 , z0 ) with x0 along the direction of the maximum horizontal stress, and z0 along the vertical. The angle a represents the azimuth of the borehole x-axis with respect to the x0 -axis, while y is the azimuthal angle around the borehole measured from the x-axis. Assuming plane-strain conditions with no displacements along the z-axis, the stresses as a function of radial distance r and azimuthal angle y are given by (Fjaer et al., 2008): ! ! 0xx þ 0yy 0xx 0yy R2 R4 R2 rr ¼ 1 2 þ 1 þ 3 4 4 2 cos 2y 2 r 2 r r R4 R2 R2 þ 0xy 1 þ 3 4 4 2 sin 2y þ pw 2 r r r ! ! 0xx þ 0yy 0xx 0yy R2 R4 1þ 2 1 þ 3 4 cos 2y yy ¼ 2 r 2 r R4 R2 0xy 1 þ 3 4 sin 2y pw 2 r r

72

Elasticity and Hooke’s law

zz ¼ ry ¼ yz ¼ rz ¼

R2 2 0 0 0 R v 2 xx yy 2 cos 2y þ 4xy 2 sin 2y r r ! 0 0 4 2 yy xx R R R4 R2 0 1 3 4 þ 2 2 sin 2y þ xy 1 3 4 þ 2 2 cos 2y 2 r r r r 2 R 0xz sin y þ 0yz cos y 1 þ 2 r 2 R 0 0 xz cos y þ yz sin y 1 2 r 0zz

In the above equations pw represents the well-bore pressure and 0ij is the remote stress tensor expressed in the borehole coordinate system through the usual coordinate transformation (see Section 1.4 on coordinate transformations) involving the direction cosines of the angles between the (x0 , y0 , z0 ) axes and the (x, y, z) axes as follows: 0xx ¼ b2xx0 H þ b2xy0 h þ b2xz0 n 0yy ¼ b2yx0 H þ b2yy0 h þ b2yz0 n 0zz ¼ b2zx0 H þ b2zy0 h þ b2zz0 n 0xy ¼ bxx0 byx0 H þ bxy0 byy0 h þ bxz0 byz0 n 0yz ¼ byx0 bzx0 H þ byy0 bzy0 h þ byz0 bzz0 n 0zx ¼ bzx0 bxx0 H þ bzy0 bxy0 h þ bzz0 bxz0 n and bxx0 ¼ cos a cos i;

bxy0 ¼ sin a cos i;

bxz0 ¼ sin i

byx0 ¼ sin a;

byy0 ¼ cos a;

byz0 ¼ 0

bzx0 ¼ cos a sin i;

bzy0 ¼ sin a sin i;

bzz0 ¼ cos i

Stress around a vertical cylindrical hole in a poroelastic medium The circular hole with radius Ri lies along the z-axis. Pore pressure, pf, at the permeable borehole wall equals the pressure in the well bore, pw. At a remote boundary R0 Ri the stresses and pore pressure are zz ðR0 Þ ¼ n rr ðR0 Þ ¼ yy ðR0 Þ ¼ h pf ðR0 Þ ¼ pf0

73

2.10 Deformation of a circular hole: borehole stresses

The stresses as a function of radial distance r from the hole are given by (Risnes et al., 1982; Bratli et al., 1983; Fjaer et al., 2008) " 2 # R2i R0 1 2n ðpf0 pw Þ 1 rr ¼ h þ ðh pw Þ 2 2 r 2ð1 nÞ R0 Ri ( ) " # R2 R0 2 lnðR0 =rÞ

a 2 i 2 1 þ r lnðR0 =Ri Þ R0 Ri " # R2i R0 2 1 2n 1þ yy ¼ h þ ðh pw Þ 2 ðpf0 pw Þ 2 r 2ð1 nÞ R0 Ri ( ) " # 2 R2i R0 lnðR0 =rÞ 1 1þ

a 2 þ lnðR0 =Ri Þ r R0 R2i R2 1 2n zz ¼ n þ 2nðh pw Þ 2 i 2 ðpf0 pw Þ 2ð1 nÞ R0 Ri 2 2R 2 lnðR0 =rÞ n

a n 2 i 2þ lnðR0 =Ri Þ R0 Ri where n is the dry (drained) Poisson ratio of the poroelastic medium, a ¼ 1 Kdry =K0 is the Biot coefficient, Kdry is effective bulk modulus of dry porous solid, and K0 is the bulk modulus of solid mineral material. In the limit R0 =Ri ! 1 the expressions simplify to (Fjaer et al., 2008) " # 2 Ri 1 2n Ri 2 lnðR0 =rÞ þ ðpf0 pw Þ a rr ¼ h ðh pw Þ r r 2ð1 nÞ lnðR0 =Ri Þ " # 2 Ri 1 2n Ri 2 lnðR0 =rÞ yy ¼ h þ ðh pw Þ ðpf0 pw Þ þ a r r 2ð1 nÞ lnðR0 =Ri Þ zz ¼ n ðpf0 pw Þ

1 2n 2 lnðR0 =rÞ n a 2ð1 nÞ lnðR0 =Ri Þ

Uses The equations presented in this section can be used for the following: estimating the stresses around a borehole resulting from tectonic stresses; estimating the stresses and deformation of a borehole caused by changes in borehole fluid pressure.

Assumptions and limitations The equations presented in this section are based on the following assumptions: the material is linear, isotropic, and elastic or poroelastic.

74

Elasticity and Hooke’s law

Extensions More complicated remote stress fields can be constructed by superimposing the solutions for the corresponding principal stresses.

2.11

Mohr’s circles Synopsis Mohr’s circles provide a graphical representation of how the tractions on a plane depend on the angular orientation of the plane within a given stress field. Consider a stress state with principal stresses s1 s2 s3 and coordinate axes defined along the corresponding principal directions x1, x2, x3. The traction vector, T, acting on a plane with outward unit normal vector, n ¼ (n1, n2, n3), is given by Cauchy’s formula as T ¼ sn where s is the stress tensor. The components of n are the direction cosines of n relative to the coordinate axes and are denoted by n1 ¼ cos ;

n2 ¼ cos g;

n3 ¼ cos y

and n21 þ n22 þ n23 ¼ 1 where f, g, and y are the angles between n and the axes x1, x2, x3 (Figure 2.11.1).

x3

q

n T g

f

x2

x1

Figure 2.11.1 Angle and vector conventions for Mohr’s circles.

75

2.11 Mohr’s circles

P 45⬚ 60⬚

30⬚ 15⬚

45⬚

30⬚

75⬚ 60⬚ 75⬚

A

B

Figure 2.11.2 Three-dimensional Mohr’s circle.

The normal component of traction, s, and the magnitude of the shear component, t, acting on the plane are given by ¼ n21 1 þ n22 2 þ n23 3 t2 ¼ n21 21 þ n22 22 þ n23 23 2

Three-dimensional Mohr’s circle The numerical values of s and t can be read graphically from the three-dimensional Mohr’s circle shown in Figure 2.11.2. All permissible values of s and t must lie in the shaded area. To determine s and t from the orientation of the plane (f, g, and y), perform the following procedure. (1) Plot s1 s2 s3 on the horizontal axis and construct the three circles, as shown. The outer circle is centered at (s1 þ s3)/2 and has radius (s1 – s3)/2. The left-hand inner circle is centered at (s2 þ s3)/2 and has radius (s2 – s3)/2. The right-hand inner circle is centered at (s1 þ s2)/2 and has radius (s1 – s2)/2. (2) Mark angles 2y and 2f on the small circles centered at A and B. For example, f ¼ 60 plots at 2f ¼ 120 from the horizontal, and y ¼ 75 plots at 2y ¼ 150 from the horizontal, as shown. Be certain to include the factor of 2, and note the different directions defined for the positive angles. (3) Draw a circle centered at point A that intersects the right-hand small circle at the mark for f. (4) Draw another circle centered at point B that intersects the left-hand small circle at the point for y. (5) The intersection of the two constructed circles at point P gives the values of s and t. Reverse the procedure to determine the orientation of the plane having particular values of s and t.

76

Elasticity and Hooke’s law

P 45⬚ 60⬚ 75⬚

30⬚ f = 158

Figure 2.11.3 Two-dimensional Mohr’s circle.

Two-dimensional Mohr’s circle When the plane of interest contains one of the principal axes, the tractions on the plane depend only on the two remaining principal stresses, and using Mohr’s circle is therefore simplified. For example, when y ¼ 90 (i.e., the x3-axis lies in the plane of interest), all stress states lie on the circle centered at B in Figure 2.11.2. The stresses then depend only on s1 and s2 and on the angle f, and we need only draw the single circle, as shown in Figure 2.11.3.

Uses Mohr’s circle is used for graphical determination of normal and shear tractions acting on a plane of arbitrary orientation relative to the principal stresses.

2.12

Static and dynamic moduli In a uniaxial stress experiment (Figure 2.12.1), Young’s modulus E is defined as the ratio of the axial stress s to the axial strain ea, while Poisson’s ratio n is defined as the (negative) ratio of the radial strain er to the axial strain: E¼

; ea

n¼

er ea

It follows from these definitions that Poisson’s ratio is zero if the sample does not expand radially during axial loading and Poisson’s ratio is 0.5 if the radial strain is half the axial strain, which is the case for fluids and incompressible solids. Poisson’s ratio must lie within the range 1 < n 0:5. The speeds of elastic waves in the solid are linked to the elastic moduli and the bulk density r by the wave equation. The corresponding expressions for Poisson’s ratio and Young’s modulus are:

77

2.12 Static and dynamic moduli

s

ea

er

Figure 2.12.1 Uniaxial loading experiment. Dashed lines show undeformed sample, and solid lines show deformed sample.

n¼

1 ðVP =VS Þ2 2 ; 2 ðVP =VS Þ2 1

E ¼ 2VS2 ð1 þ nÞ

where VP and VS are the P- and S-wave velocities, respectively. The elastic moduli calculated from the elastic-wave velocities and density are the dynamic moduli. In contrast, the elastic moduli calculated from deformational experiments, such as the one shown in Figure 2.12.1, are the static moduli. In most cases the static moduli are different from the dynamic moduli for the same sample of rock. There are several reasons for this. One is that stress–strain relations for rocks are often nonlinear. As a result, the ratio of stress to strain over a large-strain measurement is different from the ratio of stress to strain over a very small-strain measurement. Another reason is that rocks are often inelastic, affected, for example, by frictional sliding across microcracks and grain boundaries. More internal deformation can occur over a large-strain experiment than over very small-strain cycles. The strain magnitude relevant to geomechanical processes, such as hydrofracturing, is of the order of 102, while the strain magnitude due to elastic-wave propagation is of the order of 107 or less. This large strain difference affects the difference between the static and dynamic moduli. Relations between the dynamic and static moduli are not simple and universal because: (a) the elastic-wave velocity in a sample and the resulting dynamic elastic moduli depend on the conditions of the measurement, specifically on the effective pressure and pore fluid; and (b) the static moduli depend on details of the loading experiment. Even for the same type of experiment – axial loading – the static Young’s modulus may be strongly

Elasticity and Hooke’s law

40 Wang hard 30 E static (GPa)

78

20 Eissa and Kazi

10

Wang soft Mese and Dvorkin

0

0

10

20

30

40

50

E dynamic (GPa)

Figure 2.12.2 Comparison of selected relations between dynamic and static Young’s moduli.

affected by the overall pressure applied to the sample, as well as by the axial deformation magnitude. Some results have been reviewed by Scho¨n (1996) and Wang and Nur (2000). Presented below and in Figure 2.12.2 are some of their equations, where the moduli are in GPa and the impedance is in km/s g/cc. In all the examples, Estat is the static Young’s modulus and Edyn is the dynamic Young’s modulus. Data on microcline-granite, by Belikov et al. (1970): Estat ¼ 1:137Edyn 9:685 Igneous and metamorphic rocks from the Canadian Shield, by King (1983): Estat ¼ 1:263Edyn 29:5 Granites and Jurassic sediments in the UK, by McCann and Entwisle (1992): Estat ¼ 0:69Edyn þ 6:4 A wide range of rock types, by Eissa and Kazi (1988): Estat ¼ 0:74Edyn 0:82 Shallow soil samples, by Gorjainov and Ljachowickij (1979) for clay: Estat ¼ 0:033Edyn þ 0:0065 and for sandy, wet soil: Estat ¼ 0:061Edyn þ 0:00285

79

2.12 Static and dynamic moduli

Empirical relations between static and dynamic bulk moduli in dry tight sandstones from the Travis Peak formation.

Table 2.12.1

Kstat ¼ a þ bKdyn (GPa) Pressure (MPa)

a (GPa)

b

5 20 40 125

0.98 3.16 1.85 1.85

0.490 0.567 0.822 1.13

Wang and Nur (2000) for soft rocks (defined as rocks with the static Young’s modulus < 15 GPa): Estat ¼ 0:41Edyn 1:06 Wang and Nur (2000) for hard rocks (defined as rocks with the static Young’s modulus > 15 GPa): Estat ¼ 1:153Edyn 15:2 Mese and Dvorkin (2000) related the static Young’s modulus and static Poisson’s ratio (nstat) to the dynamic shear modulus calculated from the shear-wave velocity in shales and shaley sands: Estat ¼ 0:59mdyn 0:34;

nstat ¼ 0:0208mdyn þ 0:37

where mdyn is the dynamic shear modulus mdyn ¼ VS2 . The same data were used to obtain relations between the static moduli and the dynamic S-wave impedance, IS dyn : Estat ¼ 1:99IS

dyn

3:84;

nstat ¼ 0:07IS

dyn

þ 0:5

as well as between the static moduli and the dynamic Young’s modulus: Estat ¼ 0:29Edyn 1:1;

nstat ¼ 0:00743Edyn þ 0:34

Jizba (1991) compared the static bulk modulus, Kstat and the dynamic bulk modulus, Kdyn, and found the following empirical relations as a function of confining pressure in dry tight sandstones from the Travis Peak formation (Table 2.12.1).

Assumptions and limitations There are only a handful of well-documented experimental data where large-strain deformational experiments have been conducted with simultaneous measurement of the dynamic P- and S-wave velocity. One of the main uncertainties in applying

80

Elasticity and Hooke’s law

laboratory static moduli data in situ comes from the fact that the in-situ loading conditions are often unknown. Moreover, in most cases, the in-situ conditions are so complex that they are virtually impossible to reproduce in the laboratory. In most cases, static data exhibit a strongly nonlinear stress dependence, so that it is never clear which data point to use as the static modulus at in-situ conditions. Because of the strongly nonlinear dependence of static moduli on the strain magnitude, the isotropic linear elasticity equations that relate various elastic moduli to each other might not be applicable to static moduli. Finally, this section refers only to isotropic descriptions of static and dynamic moduli.

3

Seismic wave propagation

3.1

Seismic velocities Synopsis The velocities of various types of seismic waves in homogeneous, isotropic, elastic media are given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K þ 43 l þ 2 VP ¼ ¼ rﬃﬃﬃ VS ¼ sﬃﬃﬃ E VE ¼ where VP is the P-wave velocity, VS is the S-wave velocity, and VE is the extensional wave velocity in a narrow bar. In addition, r is the density, K is the bulk modulus, m is the shear modulus, l is Lame´’s coefficient, E is Young’s modulus, and v is Poisson’s ratio. In terms of Poisson’s ratio one can also write VP2 2ð1 vÞ ¼ ð1 2vÞ VS2 VE2 ð1 þ vÞð1 2vÞ ¼ 2 ð1 vÞ VP VE2 ¼ 2ð1 þ vÞ VS2 V 2 2VS2 V 2 2V 2 ¼ E 2 S v ¼ P 2 2 2VS 2 VP VS

81

82

Seismic wave propagation

The various wave velocities are related by VP2 4 VE2 =VS2 ¼ VS2 3 VE2 =VS2 VE2 3V 2 =V 2 4 ¼ 2P 2S 2 VS VP =VS 1 The elastic moduli can be extracted from measurements of density and any two wave velocities. For example, ¼ VS2

K ¼ VP2 43 VS2 E ¼ VE2 V 2 2VS2 v ¼ P 2 2 VP VS2 The Rayleigh wave phase velocity VR at the surface of an isotropic homogeneous elastic half-space is given by the solution to the equation (White, 1983)

2 1=2 1=2 VR2 VR2 VR2 2 2 4 1 2 1 2 ¼0 VS VP VS

(Note that the equivalent equation given in Bourbie´ et al. (1987) is in error.) The wave speed is plotted in Figure 3.1.1 and is given by " 2 1=2 #1=3 " 2 1=2 #1=3 VR2 q q p3 q q p3 8 þ þ ¼ þ þ þ 2 4 27 4 27 2 2 3 VS 2 q p3 þ for >0 4 27 " # p1=2 VR2 cos1 ð27q2 =4p3 Þ1=2 8 ¼ 2 cos þ 2 3 3 3 VS for

2 q p3 þ <0 4 27

8 16V 2 p ¼ 2S ; 3 VP

q¼

272 80VS2 27 3VP2

The Rayleigh velocity at the surface of a homogeneous elastic half-space is nondispersive (i.e., independent of frequency).

3.2 Phase, group, and energy velocities

0.95

0.95

0.94

0.94

0.93

0.93 VR/VS

VR/VS

83

0.92

0.92

0.91

0.91

0.90

0.90

0.89

0

0.1

0.2 0.3 0.4 Poisson's ratio

0.5

0.89 0.15 0.2 0.25 0.3 0.35 0.4 0.45 (VS /VP)2

Figure 3.1.1 Rayleigh wave phase velocity normalized by shear velocity.

Assumptions and limitations These equations assume isotropic, linear, elastic media.

3.2

Phase, group, and energy velocities Synopsis In the physics of wave propagation we often talk about different velocities (the phase, group, and energy velocities: Vp, Vg, and Ve, respectively) associated with the wave phenomenon. In laboratory measurements of core sample velocities using finite bandwidth signals and finite-sized transducers, the velocity given by the first arrival does not always correspond to an easily identified velocity. A general time-harmonic wave may be defined as Uðx; tÞ ¼ U0 ðxÞ cos½ot pðxÞ where o is the angular frequency and U0 and p are functions of position x; U can be any field of interest such as pressure, stress, or electromagnetic fields. The surfaces given by p(x) ¼ constant are called cophasal or wave surfaces. In particular, for plane waves, p(x) ¼ k · x, where k is the wave vector, or the propagation vector, and is in the direction of propagation. For the phase to be the same at (x, t) and (x þ dx, t þ dt) we must have o dt ðgrad pÞ dx ¼ 0 from which the phase velocity is defined as Vp ¼

o jgrad pj

For plane waves grad p ¼ k, and hence Vp ¼ o/k. The reciprocal of the phase velocity is often called the slowness, and a polar plot of slowness versus the direction of propagation is termed the slowness surface. Phase velocity is the speed of advance

84

Seismic wave propagation

of the cophasal surfaces. Born and Wolf (1980) consider the phase velocity to be devoid of any physical significance because it does not correspond to the velocity of propagation of any signal and cannot be directly determined experimentally. Waves encountered in rock physics are rarely perfectly monochromatic but instead have a finite bandwidth, Do, centered around some mean frequency o. The wave may be regarded as a superposition of monochromatic waves of different frequencies, which then gives rise to the concept of wave packets or wave groups. Wave packets, or modulation on a wave containing a finite band of frequencies, propagate with the group velocity defined as 1 Vg ¼ gradð@p=@oÞo which for plane waves becomes @o Vg ¼ @k o The group velocity may be considered to be the velocity of propagation of the envelope of a modulated carrier wave. The group velocity can also be expressed in various equivalent ways as Vg ¼ Vp l

dVp dl

Vg ¼ Vp þ k

dVp dk

or 1 1 o dVp ¼ 2 Vg Vp Vp do where l is the wavelength. These equations show that the group velocity is different from the phase velocity when the phase velocity is frequency dependent, direction dependent, or both. When the phase velocity is frequency dependent (and hence different from the group velocity), the medium is said to be dispersive. Dispersion is termed normal if the group velocity decreases with frequency and anomalous or inverse if it increases with frequency (Elmore and Heald, 1985; Bourbie´ et al., 1987). In elastic, isotropic media, dispersion can arise as a result of geometric effects such as propagation along waveguides. As a rule such geometric dispersion (Rayleigh waves, waveguides) is normal (i.e., the group velocity decreases with frequency). In a homogeneous viscoelastic medium, on the other hand, dispersion is anomalous or inverse and arises owing to intrinsic dissipation. The energy velocity Ve represents the velocity at which energy propagates and may be defined as

85

3.2 Phase, group, and energy velocities

kz /w Ve = Vg

k

Vp = Ve cos kx /w

anisotropic slowness surface

Figure 3.2.1 In anisotropic media, energy propagates along Ve, which is always normal to the slowness surface and in general is deflected by the angle c away from Vp and the wave vector k.

Ve ¼

Pav Eav

where Pav is the average power flow density and Eav is the average total energy density. In isotropic, homogeneous, elastic media all three velocities are the same. In a lossless homogeneous medium (of arbitrary symmetry), Vg and Ve are identical, and energy propagates with the group velocity. In this case the energy velocity may be obtained from the group velocity, which is usually somewhat easier to compute. If the medium is not strongly dispersive and a wave group can travel a measurable distance without appreciable “smearing” out, the group velocity may be considered to represent the velocity at which the energy is propagated (though this is not strictly true in general). In anisotropic, homogeneous, elastic media, the phase velocity, in general, differs from the group velocity (which is equal to the energy velocity because the medium is elastic) except along certain symmetry directions, where they coincide. The direction in which Ve is deflected away from k (which is also the direction of Vp) is obtained from the slowness surface (shown in Figure 3.2.1), for Ve (¼ Vg in elastic media) must always be normal to the slowness surface (Auld, 1990). The group velocity in anisotropic media may be calculated by differentiation of the dispersion relation obtained in an implicit form from the Christoffel equation given by jk2 cijkl nj nl o2 ik j ¼ ðo; kx ; ky ; kz Þ ¼ 0 where cijkl is the stiffness tensor, ni are the direction cosines of k, r is the density, and dij is the Kronecker delta function. The group velocity is then evaluated as Vg ¼

rk @=@o

where the gradient is with respect to kx, ky, and kz.

86

Seismic wave propagation

The concept of group velocity is not strictly applicable to attenuating viscoelastic media, but the energy velocity is still well defined (White, 1983). The energy propagation velocity in a dissipative medium is neither the group velocity nor the phase velocity except when (1) the medium is infinite, homogeneous, linear, and viscoelastic, and (2) the wave is monochromatic and homogeneous, i.e., planes of equal phase are parallel to planes of equal amplitude, or, in other words, the real and imaginary parts of the complex wave vector point in the same direction (in general they do not), in which case the energy velocity is equal to the phase velocity (BenMenahem and Singh, 1981; Bourbie´ et al., 1987). For the special case of a Voigt solid (see Section 3.8 on viscoelasticity) the energy transport velocity is equal to the phase velocity at all frequencies. For wave propagation in dispersive, viscoelastic media, one sometimes defines the limit V1 ¼ lim Vp ðoÞ o!1

which describes the propagation of a well-defined wavefront and is referred to as the signal velocity (Beltzer, 1988). Sometimes it is not clear which velocities are represented by the recorded travel times in laboratory ultrasonic core sample measurements, especially when the sample is anisotropic. For elastic materials, there is no ambiguity for propagation along symmetry directions because the phase and group velocities are identical. For nonsymmetry directions, the energy does not necessarily propagate straight up the axis of the core from the transducer to the receiver. Numerical modeling of laboratory experiments (Dellinger and Vernik, 1992) indicates that, for typical transducer widths (10 mm), the recorded travel times correspond closely to the phase velocity. Accurate measurement of group velocity along nonsymmetry directions would essentially require point transducers of less than 2 mm width. According to Bourbie´ et al. (1987), the velocity measured by a resonant-bar standing-wave technique corresponds to the phase velocity.

Assumptions and limitations In general, phase, group, and energy velocities may differ from each other in both magnitude and direction. Under certain conditions two or more of them may become identical. For homogeneous, linear, isotropic, elastic media all three are the same.

3.3

NMO in isotropic and anisotropic media Synopsis The two-way seismic travel time, t, of a pure (nonconverted) mode from the surface, through a homogeneous, isotropic, elastic layer, to a horizontal reflector is hyperbolic

87

3.3 NMO in isotropic and anisotropic media

x3 x1 Surface

x2

e lin ke Dip line

ri St

θ

Azimuth

Zero-offset ray Dip f ctor

Refle

Figure 3.3.1 Schematic showing the geometry for NMO with a dipping reflector.

t2 ¼ t20 þ

x2 V2

where x is the offset between the source and the receiver, t0 is the zero-offset, two-way travel time, and V is the velocity of the layer. The increase in travel time with distance, specifically the extra travel time relative to t0, is called normal moveout (or NMO), and the velocity that determines the NMO, is called the NMO velocity, VNMO. In the case of a homogeneous layer above a horizontal reflector, VNMO ¼ V. When the reflector is dipping with angle f from the horizontal (Figure 3.3.1), the travel time equation in the vertical plane along dip becomes (Levin, 1971) t2 ¼ t20 þ

x2 ðV= cos Þ2

or VNMO ðÞ ¼ V= cos . Again, t0 is the zero-offset, two-way travel time. More generally, the azimuthally varying NMO velocity is (Levin, 1971) 2 VNMO ð ; Þ ¼

V2 1 cos2 sin2

where f is the reflector dip and z is the azimuth relative to a horizontal axis in the dip direction. Note that in the strike direction VNMO ¼ V.

88

Seismic wave propagation

When the Earth consists of horizontal, homogeneous, isotropic layers down to the reflector, the two-way travel time equation is approximately hyperbolic, with the approximation being best for small offsets: t2 t20 þ

x2 2 VNMO

At zero dip, the NMO velocity is often approximated as the root mean squared (RMS) velocity, VNMO VRMS , where 2 ¼ VRMS

N X i¼1

Vi2 ti =

N X

ti

i¼1

where Vi is the velocity of the ith layer and ti is the two-way, zero-offset travel time of the ith layer. The summations are over all layers from the surface to the reflector. The approximation VNMO VRMS is best at offsets that are small relative to the reflector depth. If the NMO velocity is estimated at two horizontal reflectors (e.g., VNMOðnÞ at the base of layer n and VNMOðn1Þ at the base of layer (n 1)), then the RMS equation can be inverted to yield the Dix equation (Dix, 1955) for the interval velocity, Vn, of the nth layer: ! n n1 X X 1 2 2 VNMO ti VNMO ti Vn2 ðn Þ ðn1Þ tn i¼1 i¼1 It is important to remember that this model assumes: (1) flat, homogeneous, isotropic layers; (2) the offsets are small enough for the RMS velocity to be a reasonable estimate of the moveout velocity; and (3) the estimated interval velocities are themselves averages over velocities of thinner layers lying below the resolution of the seismic data. The equivalent of the Dix equation for anisotropic media is shown below.

NMO in an anisotropic Earth For a heterogeneous elastic Earth with arbitrary anisotropy and arbitrary dip, the NMO velocity of pure modes (at offsets generally less than the depth) can be written as (Grechka and Tsvankin, 1998; Tsvankin, 2001) 1 2 VNMO ð Þ

¼ W11 cos2 þ 2W12 sin cos þ W22 sin2

where z is the azimuth relative to the x1-axis. The components Wij are elements of a

~ and are defined as Wij ¼ 0 @pi =@xj , where pi ¼ @=@xi are symmetric matrix W horizontal components of the slowness vector for rays between the zero-offset

reflection point and the surface location xi ; xj , ðx1 ; x2 Þ is the one-way travel time

89

3.3 NMO in isotropic and anisotropic media

from the zero-offset reflection point, and t0 is the one-way travel time to the CMP (common midpoint) location x1 ¼ x2 ¼ 0. The derivatives are evaluated at the CMP location. This result assumes a sufficiently smooth reflector and sufficiently smooth lateral velocity heterogeneity, such that the travel time field exists (e.g., no shadow zones) at the CMP point and the derivatives can be evaluated. This can be rewritten as 1 2 VNMO ð Þ

¼ l1 cos2 ð Þ þ l2 sin2 ð Þ

~ and is the rotation of the eigenvectors of where l1 and l2 are the eigenvalues of W, ~ W relative to the coordinate system. l1 and l2 are typically positive, in which case VNMO ð Þ is an ellipse in the horizontal plane. The elliptical form allows the exact expression for VNMO ð Þ to be determined by only three parameters, the values of the NMO velocity at the axes of the ellipse, and the orientation of the ellipse relative to the coordinate axes. The elliptical form simplifies modeling of azimuthally varying NMO for various geometries and anisotropies, since only VNMO ð Þ along the ellipse axes needs to be derived. For example, we can write 1 2 ð Þ VNMO

¼

cos2

sin2

þ 2 2 ð ¼ 0Þ VNMO ð ¼ =2Þ VNMO

where the axes of the ellipse are at ¼ 0 and ¼ =2. A simple example of the elliptical form is the azimuthally varying moveout velocity for a homogeneous, isotropic layer above a dipping reflector, given above, which can be written as (Grechka and Tsvankin, 1998) 1 sin2

cos2

þ ¼ 2 V2 VNMO ð ; Þ ðV= cos Þ2 where f is the dip and z is the azimuth relative to a horizontal axis in the dip direction.

VTI symmetry with horizontal reflector NMO velocities in a VTI medium (vertical symmetry axis) overlying a horizontal reflector are given for P, SV, and SH modes (Tsvankin, 2001) by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;P ð0Þ ¼ VP0 1 þ 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;SV ð0Þ ¼ VS0 1 þ 2; VNMO;SH ð0Þ ¼ VS0

2 VP0 ¼ ð" Þ VS0

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 2

where VP0 and VS0 are the vertical P- and S-wave velocities, and e, d, and g are the Thomsen parameters for VTI media. In this case, the NMO ellipse is a circle, giving

90

Seismic wave propagation

an azimuthally independent NMO velocity. These expressions for NMO velocity hold for anisotropy of arbitrary strength (Tsvankin, 2001). In the special case of elliptical anisotropy, with vertical symmetry axis, e ¼ d, the expressions for NMO velocity become (Tsvankin, 2001): pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;P ð0Þ ¼ VP0 1 þ 2" VNMO;SV ð0Þ ¼ VS0 VNMO;SH ð0Þ ¼ VS0

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 2

Vertical symmetry axis with dipping reflector A dipping reflector creates an azimuthal dependence for the NMO velocity, even with a VTI medium. For general VTI symmetry (vertical symmetry axis) and assuming weak anisotropy, the NMO velocities overlying a dipping reflector are given for P and SV modes by Tsvankin (2001). In the dip direction,

VP0 ð1 þ Þ 1 þ sin2 þ 3ð" Þ sin2 2 sin2 cos 2

VS0 ð1 þ Þ VP0 VNMO;SV ð0; Þ ¼ ð" Þ 1 5 sin2 þ 3 sin4 ; ¼ VS0 cos

VNMO;P ð0; Þ ¼

where f is the dip, VP0 and VS0 are the vertical P- and S-wave velocities, and e, d, and g are the Thomsen parameters for VTI media. In the strike direction with VTI symmetry:

VNMO;P ; ¼ VP0 ð1 þ Þ 1 þ ð" Þ sin2 2 sin2 2 2

VP0 VNMO;SV ; ¼ VS0 ð1 þ Þ 1 sin2 2 sin2 ; ¼ ð" Þ VS0 2 The complete azimuthal dependence can be easily written as an ellipse, where the strike and dip NMO velocities are the semiaxes. For the special case of elliptical symmetry (e ¼ d) of arbitrary strength anisotropy, the dip-direction NMO velocities are (Tsvankin, 2001): pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VP0 1 þ 2" VNMO;P ð0; Þ ¼ 1 þ 2" sin2 cos VS0 cos pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VS0 1 þ 2 VNMO;SH ð0; Þ ¼ 1 þ 2 sin2 cos

VNMO;SV ð0; Þ ¼

91

3.3 NMO in isotropic and anisotropic media

where f is the layer dip angle. In the strike direction, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;P ; ¼ VP0 1 þ 2" 2 VNMO;SV ; ¼ VS0 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;SH ; ¼ VS0 1 þ 2 2

Tilted TI symmetry with dipping reflector For TI symmetry with symmetry axis tilting an angle n from the vertical and assuming weak anisotropy, the NMO velocities are given by Tsvankin (2001). In the dip direction

VP0 1 þ þ sin2 ð nÞ þ 3ð" Þ sin2 ð nÞ 2 sin2 ð nÞ VNMO;P ð0; Þ ¼ cos

2 sin n sinð nÞ 2 þ 2ð" Þ sin ð nÞ þ cos

VS0 1 þ þ sin2 ð nÞ 3 sin2 ð nÞ 2 sin2 ð nÞ VNMO;SV ð0; Þ ¼ cos sin n sinð nÞ cos 2ð nÞ þ 2 cos

where f is the reflector dip, VP0 and VS0 are the symmetry axis P- and S-wave velocities, and e, d, and g are the Thomsen parameters for TI media. It is assumed that the azimuth of the symmetry axis tilt is the same as the azimuth of the reflector dip. In the strike direction, for weak tilted TI symmetry

VNMO;P ; ¼ VP0 ð1 þ Þ 1 þ ð" Þ sin2 ð nÞ 2 sin2 ð nÞ 2

VNMO;SV ; ¼ VS0 ð1 þ Þ 1 sin2 ð nÞ 2 sin2 ð nÞ 2 2 VP0 ð" Þ ¼ VS0 In the special case of tilted elliptical symmetry with a tilted symmetry axis, the NMO velocity in the dip direction is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VP0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin n sinð nÞ 1 2 1 þ 2 1 þ 2 sin ð nÞ 1 2 VNMO;P ð0; Þ ¼ cos cos VS0 VNMO;SV ðÞ ¼ cos qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VS0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin n sinð nÞ 1 VNMO;SH ð0; Þ ¼ 1 þ 2 1 þ 2 sin2 ð nÞ 1 2 cos cos

92

Seismic wave propagation

In the strike direction for tilted elliptical symmetry VNMO;P ; ¼ VP0 ð1 þ Þ 2 VNMO;SV ; ¼ VS0 2

Orthorhombic symmetry with horizontal reflector We now consider an orthorhombic layer of arbitrary strength anisotropy over a horizontal reflector. A symmetry plane of the orthorhombic medium is also horizontal. The vertically propagating (along the x3-axis) P-wave velocity (VP0) and vertically propagating S-wave velocities, polarized in the x1 (VS0) and x2 (VS1) directions are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VP0 ¼ c33 =; VS0 ¼ c55 =; VS1 ¼ c44 = where r is the density. Additional constants necessary for this discussion are "ð2Þ ¼

c11 c33 2c33

"ð1Þ ¼

c22 c33 2c33

ð2Þ ¼

ðc13 þ c55 Þ2 ðc33 c55 Þ2 2c33 ðc33 c55 Þ

ð1Þ ¼

ðc23 þ c44 Þ2 ðc33 c44 Þ2 2c33 ðc33 c44 Þ

ð2Þ ¼

c66 c44 2c44

ð1Þ ¼

c66 c55 2c55

ð2Þ

2 h i VP0 ¼ "ð2Þ ð2Þ VS0

ð1Þ

VP0 ¼ VS1

2 h

"ð1Þ ð1Þ

i

For a CMP line in the x1 direction, the NMO velocities are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;P ð ¼ 0; ¼ 0Þ ¼ VP0 1 þ 2 ð2Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;SV ð ¼ 0; ¼ 0Þ ¼ VS0 1 þ 2ð2Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;SH ð ¼ 0; ¼ 0Þ ¼ VS1 1 þ 2 ð2Þ where f is the dip and z is the azimuth measured from the x1-axis. For a line in the x2 direction, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;P ¼ ; ¼ 0 ¼ VP0 1 þ 2 ð1Þ 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;SV ¼ ; ¼ 0 ¼ VS1 1 þ 2ð1Þ 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ VNMO;SH ¼ ; ¼ 0 ¼ VS0 1 þ 2 ð1Þ 2

93

3.4 Impedance, reflectivity, and transmissivity

NMO in a horizontally layered anisotropic Earth The effective NMO for a stack of horizontal homogeneous layers above a dipping reflector can be written as (Tsvankin, 2001) 2 VNMO ¼

N h i2 1X ðiÞ VNMO ð pÞ ti0 ð pÞ t0 i¼1

where ti0 ð pÞ is the interval travel time in layer i computed along the zero-offset ray, P ðiÞ t0 ¼ Ni¼1 ti0 is the total zero-offset time, and VNMO ð pÞ is the interval NMO velocity for the ray parameter p of the zero-offset ray. The ray parameter for the zero-offset ray is p¼

sin VN ðÞ

where f is the reflector dip, and VN ðÞ is the velocity at angle f from the vertical in the Nth layer at the reflector.

Synopsis The expressions for NMO in a layered and/or anisotropic elastic medium generally work best for small offsets.

3.4

Impedance, reflectivity, and transmissivity Synopsis The impedance, I, of an elastic medium is the ratio of the stress to the particle velocity (Aki and Richards, 1980) and is given by rV, where r is the density and V is the wave propagation velocity. At a plane interface between two thick, homogeneous, isotropic, elastic layers, the normal incidence reflectivity for waves traveling from medium 1 to medium 2 is the ratio of the displacement amplitude, Ar , of the reflected wave to that of the incident wave, Ai, and is given by R12 ¼

Ar I2 I1 2 V2 1 V1 ¼ ¼ Ai I2 þ I1 2 V2 þ 1 V1 1 lnðI2 =I1 Þ 2

The logarithmic approximation is reasonable for |R| < 0.5 (Castagna, 1993). A normally incident P-wave generates only reflected and transmitted P-waves. A normally incident S-wave generates only reflected and transmitted S-waves. There is no mode conversion.

94

Seismic wave propagation

The above expression for the reflection coefficient is obtained when the particle displacements are measured with respect to the direction of the wave vector (equivalent to the slowness vector or the direction of propagation). A displacement is taken to be positive when its component along the interface has the same phase (or the same direction) as the component of the wave vector along the interface. For P-waves, this means that positive displacement is along the direction of propagation. Thus, a positive reflection coefficient implies that a compression is reflected as a compression, whereas a negative reflection coefficient implies a phase inversion (Sheriff, 1991). When the displacements are measured with respect to a space-fixed coordinate system, and not with respect to the wave vector, the reflection coefficient is given by R12 ¼

Ar I1 I2 1 V1 2 V2 ¼ ¼ Ai I2 þ I1 2 V2 þ 1 V1

The normal incidence transmissivity in both coordinate systems is T12 ¼

At 2I1 21 V1 ¼ ¼ Ai I2 þ I1 2 V2 þ 1 V1

where At is the displacement amplitude of the transmitted wave. Continuity at the interface requires Ai þ Ar ¼ At 1þR¼T This choice of signs for Ai and Ar is for a space-fixed coordinate system. Note that the transmission coefficient for wave amplitudes can be greater than 1. Sometimes the reflection and transmission coefficients are defined in terms of scaled displacements A0 , which are proportional to the square root of energy flux (Aki and Richards, 1980; Kennett, 1983). The scaled displacements are given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A0 ¼ A V cos where y is the angle between the wave vector and the normal to the interface. The normal incidence reflection and transmission coefficients in terms of these scaled displacements are R012 ¼ 0 T12 ¼

A0r ¼ R12 A0i

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 V2 2 1 V1 2 V2 A0t p ﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ¼ ¼ T 12 A0i 1 V1 2 V2 þ 1 V1

Reflectivity and transmissivity for energy fluxes, Re and T e, respectively, are given by the squares of the reflection and transmission coefficients for scaled displacements. For normal incidence they are

95

3.4 Impedance, reflectivity, and transmissivity

Re12 ¼

Er 0 2 ð1 V1 2 V2 Þ2 ¼ R12 ¼ Ei ð2 V2 þ 1 V1 Þ2

e T12 ¼

Et 0 2 41 V1 2 V2 ¼ T12 ¼ Ei ð2 V2 þ 1 V1 Þ2

where Ei, Er , and Et are the incident, reflected, and transmitted energy fluxes, respectively. Energy reflection coefficients were first given by Knott (1899). Conservation of energy at an interface where no trapping of energy occurs requires that Ei ¼ Er þ Et 1 ¼ Re þ T e The reflection and transmission coefficients for energy fluxes can never be greater than 1.

Simple band-limited inverse of reflectivity time series Consider a flat-layered Earth, with an impedance time series In ¼ ðntÞV ðntÞ, where r is density, V is velocity (either P or S), and t ¼ nDt, where n ¼ 0,1,2, . . . ,N is the normal incidence two-way travel time, equally sampled in intervals t. In these expressions, nDt is the argument of r (·) and V(·), not a factor multiplying r and V. The reflectivity time series can be approximated as Rnþ1 ¼

Inþ1 In Inþ1 þ In

Solving for Inþ1 yields a simple recursive algorithm Inþ1 ¼ In

ð1 þ Rnþ1 Þ ; ð1 Rnþ1 Þ

n ¼ 0: N 1

where I0 , the impedance at time t ¼ 0, must be supplied. For small t and small Rn , this equation can be written as " # n

X 2 3 2 5 In I0 exp 2Ri þ 3 Ri þ 5 Ri þ i¼1

In contrast, a simple running summation of Rn leads to " # n X ð2Ri Þ In I0 exp i¼1

which is a low-order approximation of the first algorithm.

96

Seismic wave propagation

Rough surfaces Random interface roughness at scales smaller than the wavelength causes incoherent scattering and a decrease in amplitude of the coherent reflected and transmitted waves. This could be one of the explanations for the observation that amplitudes of multiples in synthetic seismograms are often larger than the amplitudes of corresponding multiples in the data (Frazer, 1994). Kuperman (1975) gives results that modify the reflectivity and transmissivity to include scattering losses at the interface. With the mean-squared departure from planarity of the rough interface denoted by s2, the modified coefficients are R~12 ¼ R12 ð1 2k12 2 Þ ¼ R12 ½1 82 ð=l1 Þ2 " 2 # 1 T~12 ¼ T12 1 ðk1 k2 Þ2 2 ¼ T12 1 22 ðl2 l1 Þ2 2 l 1 l2 where k1 ¼ o/V1, k2 ¼ o/V2 are the wavenumbers, and l1 and l2 are the wavelengths in media 1 and 2, respectively.

Uses The equations presented in this section can be used for the following purposes: to calculate amplitudes and energy fluxes of reflected and transmitted waves at interfaces in elastic media; to estimate the decrease in wave amplitude caused by scattering losses during reflection and transmission at rough interfaces.

Assumptions and limitations The equations presented in this section apply only under the following conditions: normal incidence, plane-wave, time-harmonic propagation in isotropic, linear, elastic media with a single interface, or multiple interfaces well separated by thick layers with thickness much greater than the wavelength; no energy losses or trapping at the interface; and rough surface results are valid for small deviations from planarity (small s).

3.5

Reflectivity and amplitude variations with offset (AVO) in isotropic media Synopsis The seismic impedance is the product of velocity and density (see Section 3.4), as expressed by

97

3.5 Reflectivity and AVO in isotropic media

r1⬘ V1 r2⬘ V2

Figure 3.5.1 Reflection of a normal-incidence wave at an interface between two thick homogeneous, isotropic, elastic layers.

IP ¼ VP

IS ¼ VS

where IP , IS are P- and S-wave impedances, VP , VS are P- and S-wave velocities, and r is density. At an interface between two thick homogeneous, isotropic, elastic layers, the normal incidence reflectivity, defined as the ratio of the reflected wave amplitude to the incident wave amplitude, is RPP ¼ RSS ¼

2 VP2 1 VP1 IP2 IP1 ¼ 2 VP2 þ 1 VP1 IP2 þ IP1 1 lnðIP2 =IP1 Þ 2 2 VS2 1 VS1 IS2 IS1 ¼ 2 VS2 þ 1 VS1 IS2 þ IS1 1 lnðIS2 =IS1 Þ 2

where RPP is the normal incidence P-to-P reflectivity, RSS is the S-to-S reflectivity, and the subscripts 1 and 2 refer to the first and second media, respectively (Figure 3.5.1). The logarithmic approximation is reasonable for |R| < 0.5 (Castagna, 1993). A normally incident P-wave generates only reflected and transmitted P-waves. A normally incident S-wave generates only reflected and transmitted S-waves. There is no mode conversion.

AVO: amplitude variations with offset For non-normal incidence, the situation is more complicated. An incident P-wave generates reflected P- and S-waves and transmitted P- and S-waves. The reflection

98

Seismic wave propagation

VP1, VS1, r1

qS1

Reflected S-wave

q1

Reflected P-wave

Incident P-wave

Transmitted P-wave q2

qS2

Transmitted S-wave

VP2, VS2, r2

Figure 3.5.2 The angles of the incident, reflected, and transmitted rays of a P-wave with non-normal incidence.

and transmission coefficients depend on the angle of incidence as well as on the material properties of the two layers. An excellent review is given by Castagna (1993). The angles of the incident, reflected, and transmitted rays (Figure 3.5.2) are related by Snell’s law as follows: p¼

sin 1 sin 2 sin S1 sin S2 ¼ ¼ ¼ VP1 VP2 VS1 VS2

where p is the ray parameter. y and yS are the angles of P- and S-wave propagation, respectively, relative to the reflector normal. Subscripts 1 and 2 indicate angles or material properties of layers 1 and 2, respectively. The complete solution for the amplitudes of transmitted and reflected P- and S-waves for both incident P- and S-waves is given by the Knott–Zoeppritz equations (Knott, 1899; Zoeppritz, 1919; Aki and Richards, 1980; Castagna, 1993). Aki and Richards (1980) give the results in the following convenient matrix form: 0 1 #" "" #" "" B PP SP PP SP C B #" #" "" "" C B PS SS PS SS C B C B ## ## "# "# C ¼ M1 N B C B PP SP PP SP C @ A "#

"#

PS SS PS

##

##

SS

where each matrix element is a reflection or transmission coefficient for displacement amplitudes. The first letter designates the type of incident wave, and the second letter designates the type of reflected or transmitted wave. The arrows indicate downward # and upward ↑ propagation, so that a combination #" indicates a reflection coefficient, while a combination ## indicates a transmission coefficient. The matrices M and N are given by

99

3.5 Reflectivity and AVO in isotropic media 0

cos S1

sin 2

sin S1

cos 2

1 VS1 1 2 sin2 S1

22 VS2 sin S2 cos 2

1 VS1 sin 2 S1

2 VP2 1 2 sin2 S2

sin 1

B B B cos 1 B M ¼B B B 21 VS1 sin S1 cos 1 B @

1 VP1 1 2 sin2 S1 0

sin 1 B B B cos 1 B N ¼B B B 21 VS1 sin S1 cos 1 B @

1 VP1 1 2 sin2 S1

cos S2

1

C C C C C

C 2 VS2 1 2 sin2 S2 C C A 2 VS2 sin 2 S2 sin S2

cos S1

sin 2

cos S2

sin S1

cos 2

1 VS1 1 2 sin2 S1

22 VS2 sin S2 cos 2

1 VS1 sin 2 S1

2 VP2 1 2 sin2 S2

1

C C C C C

C 2 VS2 1 2 sin2 S2 C C A 2 VS2 sin 2 S2 sin S2

Results for incident P and incident S, given explicitly by Aki and Richards (1980), #"

#"

##

##

#"

#"

are as follows, where RPP ¼ PP; RPS ¼ PS; TPP ¼ PP; TPS ¼ PS; RSS ¼ SS; RSP ¼ SP; ##

##

TSP ¼ SP; TSS ¼ SS:

RPP

. cos 1 cos 2 cos 1 cos S2 2 ¼ b c F aþd Hp D VP1 VP2 VP1 VS2

. cos 1 cos 2 cos S2 ab þ cd pVP1 ðVS1 DÞ RPS ¼ 2 VP1 VP2 VS2 TPP ¼ 21

cos 1 FVP1 =ðVP2 DÞ VP1

TPS ¼ 21

cos 1 HpVP1 =ðVS2 DÞ VP1

RSS ¼

RSP

. cos S1 cos S2 cos 2 cos S1 E aþd Gp2 D b c VS1 VS2 VP2 VS1

cos S1 cos 2 cos S2 pVS1 =ðVP1 DÞ ¼2 ab þ cd VS1 VP2 VS2

TSP ¼ 21 TSS ¼ 21

cos S1 GpVS1 =ðVP2 DÞ VS1

cos S1 EVS1 =ðVS2 DÞ VS1

100

Seismic wave propagation

where

a ¼ 2 1 2 sin2 S2 1 1 2 sin2 S1

b ¼ 2 1 2 sin2 S2 þ 21 sin2 S1

c ¼ 1 1 2 sin2 S1 þ 22 sin2 S2

2 2 d ¼ 2 2 VS2 1 VS1 D ¼ EF þ GHp2 ¼ ðdet MÞ=ðVP1 VP2 VS1 VS2 Þ cos 1 cos 2 þc VP1 VP2 cos S1 cos S2 F¼b þc VS1 VS2 cos 1 cos S2 G¼ad VP1 VS2 cos 2 cos S1 H ¼ad VP2 VS1 E¼b

and p is the ray parameter.

Approximate forms Although the complete Zoeppritz equations can be evaluated numerically, it is often useful and more insightful to use one of the simpler approximations. Bortfeld (1961) linearized the Zoeppritz equations by assuming small contrasts between layer properties as follows: 1 VP2 2 cos 1 sin 1 2 2 lnð2 =1 Þ 2 RPP ð 1 Þ ln þ ðVS1 VS2 Þ 2þ VP1 1 cos 2 VP1 2 lnðVS2 =VS1 Þ Aki and Richards (1980) also derived a simplified form by assuming small layer contrasts. The results are conveniently expressed in terms of contrasts in VP, VS, and r as follows: RPP ð Þ

1

1 VP VS þ 4p2 VS2 1 4p2 VS2 2 2 2 cos VP VS

pVP RPS ð Þ 2 cos S

2 2 2 cos cos S 1 2VS p þ 2VS VP VS

2 2 2 cos cos S VS 4p VS 4VS VP VS VS 1 1 VP þ 1 TPP ð Þ 1 2 2 2 cos VP

101

3.5 Reflectivity and AVO in isotropic media

pVP TPS ð Þ 2 cos S

2 2 2 cos cos S 1 2VS p 2VS VP VS

2 2 2 cos cos S VS 4p VS þ 4VS VP VS VS RSP ð Þ

cos S VS RPS ð Þ VP cos

1

1 2 2 2 2 VS 4p VS RSS ð Þ 1 4p VS 2 2 cos2 S VS TSP ð Þ

cos S VS TPS VP cos

1 1 VS þ TSS ð Þ 1 1 2 2 2 cos S VS where p¼

sin 1 sin S1 ¼ VP1 VS1

¼ ð 2 þ 1 Þ=2

S ¼ ð S2 þ S1 Þ=2

¼ 2 1

¼ ð2 þ 1 Þ=2

VP ¼ VP2 VP1

VP ¼ ðVP2 þ VP1 Þ=2

VS ¼ VS2 VS1

VS ¼ ðVS2 þ VS1 Þ=2

Often, the mean P-wave angle y is approximated as y1, the P-wave angle of incidence. The result for P-wave reflectivity can be rewritten in the familiar form: RPP ð Þ RP0 þ B sin2 þ Cðtan2 sin2 Þ or 1 VP 1 VP VS VS2 þ 2 2 2 þ þ sin2 RPP ð Þ 2 VP 2 VP VP VS þ

1 VP 2 2 tan

sin

2 VP

This form can be interpreted in terms of different angular ranges (Castagna, 1993). In the above equations RP0 is the normal incidence reflection coefficient as expressed by

102

Seismic wave propagation

RP0 ¼

IP2 IP1 IP 1 VP þ IP2 þ IP1 2IP 2 VP

The parameter B describes the variation at intermediate offsets and is often called the AVO gradient, and C dominates at far offsets near the critical angle. Shuey (1985) presented a similar approximation where the AVO gradient is expressed in terms of the Poisson ratio n as follows: " # n 1 VP 2 2 RPP ð 1 Þ RP0 þ ERP0 þ tan

sin

sin2 1 þ 1 1 2 2 VP ð1 nÞ where 1 VP þ RP0 2 VP 1 2n E ¼ F 2ð1 þ FÞ 1 n F¼

VP =VP VP =VP þ =

and n ¼ n2 n1 n ¼ ðn2 þ n1 Þ=2 The coefficients E and F used here in Shuey’s equation are not the same as those defined earlier in the solutions to the Zoeppritz equations. Smith and Gidlow (1987) offered a further simplification to the Aki–Richards equation by removing the dependence on density using Gardner’s equation (see Section 7.10) as follows: / V 1=4 giving VP VS RPP ð Þ c þ d VP VS where 5 1 V2 1 c ¼ S2 sin2 þ tan2 8 2 VP 2 V2 d ¼ 4 S2 sin2 VP

103

3.5 Reflectivity and AVO in isotropic media

Wiggins et al. (1983) showed that when VP 2VS, the AVO gradient is approximately (Spratt et al., 1993) B RP0 2RS0 given that the P and S normal incident reflection coefficients are 1 VP þ RP0 2 VP 1 VS RS0 þ 2 VS Hilterman (1989) suggested the following slightly modified form: RPP ð Þ RP0 cos2 þ PR sin2 where RP0 is the normal incidence reflection coefficient and PR ¼

n2 n1 ð1 nÞ2

This modified form has the interpretation that the near-offset traces reveal the P-wave impedance, and the intermediate-offset traces image contrasts in Poisson ratio (Castagna, 1993). Gray et al. (1999) derived linearized expressions for P–P reflectivity in terms of the angle of incidence, y, and the contrast in bulk modulus, K, shear modulus, m, and bulk density, r: 2 VS 1 1 VS2 2 K 1 2 2 Rð Þ ¼ 2 sec þ 2 sec 2 sin 4 3 VP K 3 VP

1 1 2 þ sec 2 4 Similarly, their expression in terms of Lame´’s coefficient, l, shear modulus, and bulk density is 2 VS 1 1 VS2 2 l 1 2 2 þ 2 sec 2 sin Rð Þ ¼ 2 sec 4 2 VP l 2 VP þ

1 1 2 sec 2 4

In the same assumptions of small layer contrast and limited angle of incidence, we can write the linearized SV-to-SV reflection (Ru¨ger, 2001): 1 IS 7 VS 1 VS 2 RSV-iso ð S Þ þ þ2 sin S tan2 S sin2 S 2 IS 2 VS 2 VS

104

Seismic wave propagation

where yS is the SV-wave phase angle of incidence and IS ¼ VS is the shear impedance. Similarly, for SH-to-SH reflection (Ru¨ger, 2001): RSH ð S Þ ¼

1 IS 1 VS tan2 S þ 2 IS 2 VS

where yS is the SH-wave phase angle of incidence. For P-to-SV converted shear-wave reflection (Aki and Richards, 1980): 0 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 tan S @ VS 2 sin2 S A RPS ð S Þ 1 2 sin S þ 2 cos S 2VS =VP VP 0

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 VS tan S VS þ @4 sin2 S 4 cos S sin2 S A VP VS 2VS =VP where yS is the S-wave phase angle of reflection. This can be rewritten as (Gonza´lez, 2006) RPS ð Þ

sin qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðVP =VS Þ ðVP =VS Þ sin2 82 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2 < 1 V 2 VP P 2

4 sin

þ cos

sin2 5 : 2 VS VS 9 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2 = VP 2 5 VS 42 sin2 2 cos sin

VS VS ; 2

where y is the P-wave phase angle of incidence. For small angles, Duffaut et al. (2000) give the following expression for RPS(y): VS VS VS 1 2 sin RPS ð Þ 1þ2 2 VP VP VS VS þ VP

1 VS VS 1 2VS þ sin3 þ 4 VP VP 2 VS

which can be simplified further (Jı´lek, 2002b) to

1 VS VS VS RPS ð Þ 1þ2 2 sin 2 VP VP VS

105

3.6 Plane-wave reflectivity in anisotropic media

Assumptions and limitations The equations presented in this section apply in the following cases: the rock is linear, isotropic, and elastic; plane-wave propagation is assumed; and most of the simplified forms assume small contrasts in material properties across the boundary and angles of incidence of less than about 30 . The simplified form for P-to-S reflection given by Gonza´lez (2006) is valid for large angles of incidence.

3.6

Plane-wave reflectivity in anisotropic media Synopsis An incident wave at a boundary between two anisotropic media (Figure 3.6.1) can generate reflected quasi-P-waves and quasi-S-waves as well as transmitted quasiP-waves and quasi-S-waves (Auld, 1990). In general, the reflection and transmission coefficients vary with offset and azimuth. The AVOA (amplitude variation with offset and azimuth) can be detected by three-dimensional seismic surveys and is a useful seismic attribute for reservoir characterization. Brute-force modeling of AVOA by solving the Zoeppritz (1919) equations can be complicated and unintuitive for several reasons: for anisotropic media in general, the two shear waves are separate (shear-wave birefringence); the slowness surfaces are nonspherical and are not necessarily convex; and the polarization vectors are neither parallel nor perpendicular to the propagation vectors. Schoenberg and Prota´zio (1992) give explicit solutions for the plane-wave reflection and transmission problem in terms of submatrices of the coefficient matrix of the Zoeppritz equations. The most general case of the explicit solutions is applicable to Reflected qS-wave

Incident qP-wave

Reflected qP-wave

q1

r1, a1, b1, e1, d1, g1 r2, a2, b2, e2, d2, g2 q2

Transmitted qP-wave

Transmitted qS-wave

Figure 3.6.1 Reflected and transmitted rays caused by a P-wave incident at a boundary between two anisotropic media.

106

Seismic wave propagation

monoclinic media with a mirror plane of symmetry parallel to the reflecting plane. Let R and T represent the reflection and transmission matrices, respectively, 2

RPP

6 R ¼4 RPS RPT 2

TPP T ¼ 4 TPS TPT

RSP

RTP

3

RSS

7 RTS 5

RST

RTT

TSP TSS TST

3 TTP TTS 5 TTT

where the first subscript denotes the type of incident wave and the second subscript denotes the type of reflected or transmitted wave. For “weakly” anisotropic media, the subscript P denotes the P-wave, S denotes one quasi-S-wave, and T denotes the other quasi-S-wave (i.e., the tertiary or third wave). As a convention for real s23P , s23S , and s23T , s23P < s23S < s23T where s3i is the vertical component of the phase slowness of the ith wave type when the reflecting plane is horizontal. An imaginary value for any of the vertical slownesses implies that the corresponding wave is inhomogeneous or evanescent. The impedance matrices are defined as 2 3 eP1 eS1 eT1 6 7 eP2 eS2 eT2 6 7 6 7 X ¼6 fðC13 eP1 þ C36 eP2 Þs1 fðC13 eS1 þ C36 eS2 Þs1 fðC13 eT1 þ C36 eT2 Þs1 7 6 7 4 ðC23 eP2 þ C36 eP1 Þs2 ðC23 eS2 þ C36 eS1 Þs2 ðC23 eT2 þ C36 eT1 Þs2 5 C33 eP3 s3P g 2

fðC55 s1 þ C45 s2 ÞeP3 6 ðC55 eP1 þ C45 eP2 Þs3P g 6 6 6 Y ¼ 6 fðC s þ C s Þe 45 1 44 2 P3 6 6 ðC e þ C e Þs g 45 P1 44 P2 3P 4 eP3

C33 eS3 s3S g fðC55 s1 þ C45 s2 ÞeS3 ðC55 eS1 þ C45 eS2 Þs3S g fðC45 s1 þ C44 s2 ÞeS3 ðC45 eS1 þ C44 eS2 Þs3S g eS3

C33 eT3 s3T g 3 fðC55 s1 þ C45 s2 ÞeT3 ðC55 eT1 þ C45 eT2 Þs3T g 7 7 7 7 fðC45 s1 þ C44 s2 ÞeT3 7 7 ðC45 eT1 þ C44 eT2 Þs3T g 7 5 eT3

where s1 and s2 are the horizontal components of the phase slowness vector; eP, eS, and eT are the associated eigenvectors evaluated from the Christoffel equations (see Section 3.2), and CIJ denotes elements of the stiffness matrix of the incident medium. X0 and Y0 are the same as above except that primed parameters (transmission medium) replace unprimed parameters (incidence medium). When neither X nor Y is singular and (X–1X0 þ Y–1Y0 ) is invertible, the reflection and transmission coefficients can be written as

107

3.6 Plane-wave reflectivity in anisotropic media

T ¼ 2ðX1 X0 þ Y1 Y0 Þ1 R ¼ ðX1 X0 Y1 Y0 ÞðX1 X0 þ Y1 Y0 Þ1 Schoenberg and Prota´zio (1992) point out that a singularity occurs at a horizontal slowness for which an interface wave (e.g., a Stoneley wave) exists. When Y is singular, straightforward matrix manipulations yield T ¼ 2Y01 YðX1 X0 Y01 Y þ IÞ1 R ¼ ðX1 X0 Y01 Y þ IÞ1 ðX1 X0 Y01 Y IÞ Similarly, T and R can also be written without X–1 when X is singular as T ¼ 2X01 XðI þ Y1 Y0 X01 XÞ1 R ¼ ðI þ Y1 Y0 X01 XÞ1 ðI Y1 Y0 X01 XÞ Alternative solutions can be found by assuming that X0 and Y0 are invertible R ¼ ðY01 Y þ X01 XÞ1 ðY01 Y X01 XÞ T ¼ 2X01 XðY01 Y þ X01 XÞ1 Y01 Y ¼ 2Y01 YðY01 Y þ X01 XÞ1 X01 X These formulas allow more straightforward calculations when the media have at least monoclinic symmetry with a horizontal symmetry plane. For a wave traveling in anisotropic media, there will generally be out-of-plane motion unless the wave path is in a symmetry plane. These symmetry planes include all vertical planes in VTI (transversely isotropic with vertical symmetry axis) media and the symmetry planes in HTI (transversely isotropic with horizontal symmetry axis) and orthorhombic media. In this case, the quasi-P- and the quasi-S-waves in the symmetry plane uncouple from the quasi-S-wave polarized transversely to the symmetry plane. For weakly anisotropic media, we can use simple analytical formulas (Banik, 1987; Thomsen, 1993; Chen, 1995; Ru¨ger, 1995, 1996) to compute AVOA (amplitude variation with offset and azimuth) responses at the interface of anisotropic media that can be either VTI, HTI, or orthorhombic. The analytical formulas give more insight into the dependence of AVOA on anisotropy. Vavrycˇuk and Psˇencˇ´ık (1998) and Psˇencˇ´ık and Martins (2001) provide formulas for arbitrary weak anisotropy.

108

Seismic wave propagation

Transversely isotropic media: VTI Thomsen (1986) introduced the following notation for weakly anisotropic VTI media with density r: sﬃﬃﬃﬃﬃﬃﬃ C33 C11 C33 ¼ ; "¼ 2C33 sﬃﬃﬃﬃﬃﬃﬃ C44 C66 C44 ; ¼ ¼ 2C44 ðC13 þ C44 Þ2 ðC33 C44 Þ2 2C33 ðC33 C44 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ In this chapter we use ¼ C33 = to refer to the P-wave velocity along the x3(vertical) axis in anisotropic media. We use b to refer to the S-wave velocity along the x3-axis. The P-wave reflection coefficient for weakly anisotropic VTI media in the limit of small impedance contrast was derived by Thomsen (1993) and corrected by Ru¨ger (1997): ¼

RPP ð Þ ¼ RPP-iso ð Þ þ RPP-aniso ð Þ " 2 # 1 Z 1 2 þ sin2 RPP-iso ð Þ 2 Z 2 þ RPP-aniso ð Þ

1 sin2 tan2 2

2 " 2 sin þ sin tan2 2 2

where

¼ ð 1 þ 2 Þ=2

" ¼ "2 "1

¼ ð1 þ 2 Þ=2

¼ 2 1

¼ 2 1

¼ ð1 þ 2 Þ=2

¼ 2 1

¼ 2 1

¼ ð1 þ 2 Þ=2

¼ 2 1

¼ 2

Z ¼ ðZ1 þ Z2 Þ=2

Z ¼ Z2 Z1

Z ¼

ZS ¼ ðZS1 þ ZS2 Þ=2

ZS ¼ ZS2 ZS1

ZS ¼

¼ ð1 þ 2 Þ=2

¼ 2 1

In the preceding and following equations, D indicates a difference in properties between layer 1 (above) and layer 2 (below) and an overbar indicates an average of

109

3.6 Plane-wave reflectivity in anisotropic media

the corresponding quantities in the two layers. y is the P-wave phase angle of incidence, Z ¼ is the vertical P-wave impedance, ZS ¼ is the vertical S-wave impedance, and ¼ 2 is the vertical shear modulus. Note that the VTI anisotropy causes perturbations of both the AVO gradient and far-offset terms. Ru¨ger (2001) derived the corresponding SV-to-SV reflectivity in VTI media, again under the assumption of weak anisotropy and small-layer contrast: RSV ð S Þ ¼ RSV-iso ð S Þ þ RSV-aniso ð S Þ 1 ZS 7 1 sin2 S tan2 S sin2 S RSV-iso ð S Þ þ þ2 2 ZS 2 2 RSV-aniso ð S Þ ¼

2 1 ð" Þ sin2 S 2 2

In the S-wave reflectivity equations, note that yS is the S-wave angle of incidence and ZS ¼ is the shear impedance. The weak anisotropy, weak contrast SH-to-SH reflectivity is (Ru¨ger, 2001) 1 ZS 1 þ þ tan2 S RSH ð S Þ ¼ 2 ZS 2 The P–S converted wave reflectivity in VTI media is given by (Ru¨ger, 2001) RPS ð S Þ ¼ RPS-iso ð S Þ þ RPS-aniso ð S Þ 1 sin P sin P cos P 2 þ RPS-iso ð S Þ 2 sin S 2 sin3 P þ 2 þ cos S " # ! 2 cos P

RPS-aniso ð S Þ ¼ 2 2 ð 2 1 Þ sin P 2 2 cos S 2 2 "

# cos P þ 2 2 ð 2 1 þ "1 "2 Þ sin3 P " # 2 2 2 ð 2 1 þ "1 "2 Þ sin3 P cos S " # 2 þ 2 2 ð 1 2 Þ sin3 P cos S 2 " # 2 þ 2 2 ð 2 1 þ "1 "2 Þ sin5 P cos S

110

Seismic wave propagation

Transversely isotropic media: HTI In HTI (transversely isotropic with horizontal symmetry axis) media, reflectivity will vary with azimuth, z, as well as offset or incident angle y. Ru¨ger (1995, 1996) and Chen (1995) derived the P-wave reflection coefficient in the symmetry planes for reflections at the boundary of two HTI media sharing the same symmetry axis. At a horizontal interface between two HTI media with horizontal symmetry axis x1 and vertical axis x3, the P-wave reflectivity for propagation in the vertical symmetry axis plane (x1–x3 plane) parallel to the x1 symmetry axis can be written as " # ? 2 ? 1 Z 1 2 ðVÞ þ RPP ð ¼ 0; Þ þ ? 2 Z 2 1 2 ðVÞ þ " sin2 tan2

sin þ 2 " # 2 1 Z 1 2 ðVÞ þ 2 þ 2 Z 2 1 2 ðVÞ sin2 tan2

sin þ þ " 2 where the azimuth angle z is measured from the x1-axis and the incident angle y is defined with respect to x3. The isotropic part RPP-iso(y) is the same as before. In the preceding expression sﬃﬃﬃﬃﬃﬃﬃ C33 C11 C33 ¼ ; "ðVÞ ¼ 2C33 sﬃﬃﬃﬃﬃﬃﬃ C44 ðC13 þ C55 Þ2 ðC33 C55 Þ2 ¼ ; ðVÞ ¼ 2C33 ðC33 C55 Þ sﬃﬃﬃﬃﬃﬃﬃ C55 C66 C44 ? ¼ ; ðVÞ ¼ ¼ 2C44 1 þ 2 ¼

C44 C66 2C66

where b is the velocity of a vertically propagating shear wave polarized in the x2 direction and ? is the velocity of a vertically propagating shear wave polarized in the x1 direction. ¼ c44 and ? ¼ c55 are the shear moduli corresponding to the shear velocities b and ? . In the vertical symmetry plane (x2–x3 plane), perpendicular to the symmetry axis, the P-wave reflectivity resembles the isotropic solution: " 2 # 1 Z 1 2 1 2 þ sin þ sin2 tan2 RPP ¼ ; 2 2 Z 2 2

111

3.6 Plane-wave reflectivity in anisotropic media

In nonsymmetry planes, Ru¨ger (1996) derived the P-wave reflectivity RPP ð ; Þ using a perturbation technique as follows: " # 2 2 ! 1 Z 1 2 2 þ RPP ð ; Þ cos2 sin2 þ ðVÞ þ 2 2 Z 2 1 ðVÞ 4 ðVÞ 2 2 þ " cos þ sin cos sin2 tan2 þ 2 where z is the azimuth relative to the x1-axis. The reflectivity for SV- and SH-waves in the symmetry axis plane (x1–x3 plane) of the HTI medium is given by (Ru¨ger, 2001) 2 1 ðVÞ ðVÞ RSV ð S Þ RSV-iso ð S Þ þ " sin2 S 2 ? 1 RSH ð S Þ RSH-iso ð S Þ þ ðVÞ tan2 S 2 where

1 ZS? 7 ? 1 ? 2 2 sin þ þ 2

tan S sin2 S S ? 2 ZS 2 ? 2 ? 1 ZS 1 RSH-iso ð S Þ ¼ þ tan2 S 2 ZS 2 RSV-iso ð S Þ

For waves propagating in the (x2–x3) plane 1 ZS 7 1 sin2 S tan2 S sin2 S RSV ð S Þ þ þ2 2 ZS 2 2 1 Z? 1 ? tan2 S RSH ð S Þ ¼ ?S þ 2 ZS 2 ?

Orthorhombic media The anisotropic parameters in orthorhombic media are given by Chen (1995) and Tsvankin (1997) as follows: sﬃﬃﬃﬃﬃﬃﬃ C33 ¼ ;

sﬃﬃﬃﬃﬃﬃﬃ C44 ¼ ;

sﬃﬃﬃﬃﬃﬃﬃ C55 ? ¼

"ð1Þ ¼

C22 C33 ; 2C33

ð1Þ ¼

ðC23 þ C44 Þ2 ðC33 C44 Þ2 2C33 ðC33 C44 Þ

"ð2Þ ¼

C11 C33 2C33

112

Seismic wave propagation

ð2 Þ ¼ ¼

ðC13 þ C55 Þ2 ðC33 C55 Þ2 2C33 ðC33 C55 Þ

C44 C55 ; 2C55

ð1Þ ¼

C66 C55 ; 2C55

ð2Þ ¼

C66 C44 2C44

The parameters e(1) and d(1) are Thomsen’s parameters for the equivalent TIV media in the x2–x3 plane. Similarly, e(2) and d(2) are Thomsen’s parameters for the equivalent TIV media in the x1–x3 plane; g2 represents the velocity anisotropy between two shearwave modes traveling along the z-axis. The difference in the approximate P-wave reflection coefficient in the two vertical symmetry planes (with x3 as the vertical axis) of orthorhombic media is given by Ru¨ger (1995, 1996) in the following form: "

ðx ;x Þ RPP1 3 ð Þ

ðx ;x Þ RPP2 3 ð Þ

ð2Þ 2 # ð2Þ ð1Þ 2 " "ð1Þ sin2 tan2 sin2 þ þ 2 2

The equations are good approximations for angles of incidence of up to 30 –40 . Ru¨ger (2001) gives the following expressions for P-wave reflectivity in the orthorhombic symmetry planes: ðx ;x Þ RPP1 3 ð Þ

ðx ;x Þ RPP2 3 ð Þ

" # ? 2 ? 1 Z 1 2 1 ð2Þ 2 ð2Þ þ sin2 tan2 sin þ þ þ " ? 2 Z 2 2 " # 2 1 Z 1 2 1 ð1Þ 2 ð1Þ þ sin2 tan2 sin

þ þ þ " 2 Z 2 2

Jı´lek (2002a, b) derived expressions for P-to-SV reflectivity for arbitrary weak anisotropy and weak layer contrasts. In the case of orthorhombic and higher symmetries with aligned vertical symmetry planes, the expressions for P–SV in the vertical symmetry planes simplify to RPSV ¼ B1

sin P sin3 P sin5 P þ B2 cos P sin P þ B3 þ B4 cos P sin3 P þ B5 cos S cos S cos S

where P is the P-wave incident phase angle and S is the converted S-wave reflection phase angle. In the x–z symmetry plane (the z-axis is vertical), the coefficients Bi are B1 ¼

2 1 ð2Þ ð2 Þ þ 2 2 ~2 ~1 2 2

B2 ¼

~2ð2Þ ~1ð2Þ 2 2 2 2

113

3.6 Plane-wave reflectivity in anisotropic media

B3 ¼

2 2 2 ~ð2Þ ~ð2Þ

þ 2 þ 1 2 2 2 2 2 2 h i 2 ð2Þ ð2 Þ ð2Þ ð2Þ þ 2 2 "2 "1 þ ~1 ~2

B4 ¼

h i "ð22Þ "ð12Þ þ ~1ð2Þ ~2ð2Þ 2 2

B5 ¼

h i 2 "ð22Þ "ð12Þ þ ~1ð2Þ ~2ð2Þ 2 2

and in the y–z symmetry plane, the coefficients are B1 ¼

2 1 ð1Þ ð1Þ þ 2 2 ~2 ~1 2 2

B2 ¼

ðSÞ ðSÞ

~2ð1Þ ~1ð1Þ 2 2 1 2 2 2 2

B3 ¼

2 2 2 ~ð1Þ ~ð1Þ

þ 2 þ 1 2 2 2 2 2 2 þ

2 2 "ð21Þ "ð11Þ þ ~1ð1Þ ~2ð1Þ þ 2 2 2ðSÞ 1ðSÞ 2 2

ð1 Þ ð1Þ ð1Þ ð1Þ ~ ~ B4 ¼ 2 2 "2 "1 þ 1 2 B5 ¼

2 "ð21Þ "ð11Þ þ ~1ð1Þ ~2ð1Þ 2 2

ðmÞ ðmÞ ðSÞ In these expressions, a subscript 1 on "1 , ~1 , and 1 refers to properties of the ðmÞ ~ðmÞ ðSÞ incident layer and a subscript 2 on "2 , 2 , and 2 refers to properties of the reflecting layer. “Background” isotropic P- and S-wave velocities are defined within each layer (I = 1, 2 refer to the incident and reflecting layers, respectively) and can be taken to be equal to the actual vertical velocities within those layers:

I ¼

qﬃﬃﬃﬃﬃﬃﬃﬃ ðIÞ A33 ;

I ¼

qﬃﬃﬃﬃﬃﬃﬃﬃ ðIÞ A55

In addition, the overall isotropic “background” P-wave velocity, S-wave velocity, and density are defined as averages over the two layers: 1 ¼ ð1 þ 2 Þ; 2

1 ¼ ð1 þ 2 Þ; 2

1 ¼ ð1 þ 2 Þ 2

114

Seismic wave propagation

and contrasts across the boundary are defined as ¼ 2 1 ;

¼ 2 1 ;

¼ 2 1

The anisotropic parameters of each layer are defined as ðIÞ

ð1Þ

"I ¼

ðSÞ

ðIÞ

2A33 ðIÞ

ð2Þ

"I ¼ I

ðIÞ

A22 A33

ðIÞ

2A33 ðIÞ

¼

ðIÞ

2A55

ðIÞ

ðIÞ

ðIÞ

ðIÞ

ðIÞ

ðIÞ

ðIÞ

;

A þ 2A55 A33 ð2Þ ~I ¼ 13 ðIÞ A33

;

A þ 2A66 A11 ð3Þ ~I ¼ 12 ðIÞ A11

ðIÞ

A44 A55

ðIÞ

A þ 2A44 A33 ð1Þ ~I ¼ 23 ðIÞ A33

ðIÞ

A11 A33

ðIÞ

;

ðIÞ

where Ajk are the density-normalized Voigt-notation elastic constants of each layer. In the approximation of weak anisotropy and weak layer contrasts, S can be related approximately to P using the linearized isotropic Snell’s law: 1 1 1 2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ sin P 2 cos S 2 1 ð 2 =2 Þ sin2 P

Arbitrary anisotropy Vavrycˇuk and Psˇencˇik (1998) derived P-wave reflectivity for arbitrary weak anisotropy and weak layer contrast: " 2 # 1 Z 1 1 2 4 sin2 P þ tan P sin2 P RPP ð ; P Þ ¼ þ 2 Z 2 2 1 A13 þ 2A55 A33 cos2

þ 2 A33 A23 þ 2A44 A33 A44 A55 þ 8 sin2

A33 2A33 A36 þ 2A45 A45 þ2 4 cos sin sin2 P A33 A33 1 A11 A33 A22 A33 4 cos þ sin4

þ 2A33 2A33 2 A12 þ 2A66 A33 A16 þ cos2 sin2 þ 2 cos3 sin

A33 A33 A26 3 þ 2 cos sin sin2 P tan2 P A33

115

3.7 Elastic impedance

where Aij ¼ cij = is the Voigt-notation elastic constant divided by density. P is the P-wave angle of incidence and z is the azimuth relative to the x1-axis. D signifies and r are , the contrast of any parameter across the reflector, e.g., ¼ 2 1 . , the background average isotropic P-wave velocity, S-wave velocity, and density, respectively. The accuracy of the result varies slightly with the choice of the background values, but a reasonable choice is the average of the vertical P-wave the average of the vertical S-wave velocities velocities across the two layers for , across the reflector for , and the average of the densities across the two layers for r. The corresponding expression for the P-wave transmission coefficient for arbitrary anisotropy is given by Psˇencˇik and Vavrycˇuk (1998). Jı´lek (2002a, b) derived expressions for P-to-SV reflectivities for arbitrary weak anisotropy and weak layer contrasts.

Assumptions and limitations The equations presented in this section apply under the following conditions: the rock is linear elastic; approximate forms apply to the P–P reflection at near offset for slightly contrasting, weakly anisotropic media.

3.7

Elastic impedance The elastic impedance is a pseudo-impedance attribute (Connolly, 1998; Mukerji et al., 1998) and is a far-offset equivalent of the more conventional zero-offset acoustic impedance. This far-offset impedance has been called the “elastic impedance” (EI), as it contains information about the VP =VS ratio. One can obtain the elastic impedance cube from a far-offset stack using the same trace-based one-dimensional inversion algorithm used to invert the near-offset stack. Though only approximate, the inversion for elastic impedance is economical and simple compared to full prestack inversion. As with any impedance inversion, the key to effectively using this extracted attribute for quantitative reservoir characterization is calibration with log data. The acoustic impedance, Ia ¼ VP , can be expressed as Z Ia ¼ exp 2 RPP ð0Þ dt where RPP ð0Þ is the time sequence of normal-incidence P-to-P reflection coefficients. Similarly, the elastic impedance may be defined in terms of the sequence of nonnormal P–P reflection coefficients, RPP ð Þ, at incidence angle y as Z Ie ð Þ ¼ exp 2 RPP ðyÞ dt

116

Seismic wave propagation

Substituting into this equation one of the well-known approximations for RPP ð Þ (e.g., Aki and Richards, 1980), RPP ð Þ RPP ð0Þ þ M sin2 þ N tan2 RPP ð0Þ ¼

1 VP þ 2 VP

2 VS 2VS M ¼ 2 þ VP VS N¼

1 VP 2 VP

leads to the following expression for Ie:

Z

Ie ð Þ ¼VP exp tan 2

2 dðln VP Þ exp 4 sin ðVS =VP Þ 2

Z 2dðln VS Þ

Z 2 2

exp 4 sin ðVS =VP Þ dðln Þ or ð1þtan2 Þ ð14K sin2 Þ

Ie ð Þ ¼ VP

ð8K sin2 Þ

VS

2 where K ¼ ðVS =VP Þ is taken to be a constant. The elastic impedance reduces to the usual acoustic impedance, Ia ¼ VP , when y ¼ 0. Unlike the acoustic impedance, the elastic impedance is not a function of the rock properties alone but also depends on the angle of incidence. Using only the first two terms in the approximation for Rð Þ gives a similar expression for Ie with the tan2 terms being replaced by sin2 ; ð1þsin2 Þ ð14K sin2 Þ

Ie ð Þ ¼ VP

ð8K sin2 Þ

VS

This last expression has been referred to as the first-order elastic impedance (Connolly, 1999), and it goes to ðVP =VS Þ2 at ¼ 90 , assuming K ¼ 14. Expressions for P-to-S converted-wave elastic impedance have been given by Duffaut et al. (2000) and Gonza´lez (2006). The elastic impedance has strange units and dimensions

117

3.7 Elastic impedance

that change with angle. Whitcombe (2002) defines a useful normalization for the elastic impedance:

VP Ie ð Þ ¼ ½VP0 0 VP0

ð1þtan2 Þ ð14K sin2 Þ ð8K sin2 Þ VS VS0 0

where the normalizing constants VP0, VS0, and r0 may be taken to be either the average values of velocities and densities over the zone of interest, or the values at the top of the target zone. With this normalization, the elastic impedance has the same dimensionality as the acoustic impedance. The extended elastic impedance (EEI) (Whitcombe et al., 2002) is defined over an angle w, ranging from –90 to þ90 . The angle w should not be interpreted as the actual reflection angle, but rather as the independent input variable in the definition of EEI. The EEI is expressed as ðcos þsin Þ ðcos 4K sin Þ ð8K sin Þ VP VS Ie ðÞ ¼ ½VP0 0 VP0 VS0 0 Under certain approximations, the EEI for specific values of w becomes proportional to rock elastic parameters such as the bulk modulus and the shear modulus. In applications to reservoir characterization, care must be taken to filter the logs to match the seismic frequencies, as well as to account for the differences in frequency content in near- and far-offset data. Separate wavelets should be extracted for the near- and far-offset angle stacks. The inverted acoustic and elastic impedances co-located at the wells must be calibrated with the known facies and fluid types in the well, before classifying the seismic cube in the interwell region. Major limitations in using partial-stack elastic-impedance inversion arise from the assumptions of the one-dimensional convolutional model for far offsets, the assumption of a constant value for K, and errors in estimates of the incidence angle. The convolutional model does not properly handle all the reflections at far offsets, because the primary reflections get mixed with other events. The approximations used to derive the expressions for elastic impedance become less accurate at larger angles. The first-order, two-term elastic impedance has been found to give stabler results than the three-term elastic impedance (Mallick, 2001). Mallick compares prestack inversion versus partial-stack elastic impedance inversions and recommends a hybrid approach. In this approach, full prestack inversion is done at a few control points to get reliable estimates of P and S impedance. These prestack inversions are used as anchors for cheaper, one-dimensional, trace-based inversions over large data volumes. Based on these results, small zones may be selected for detailed analysis by prestack inversions.

118

Seismic wave propagation

Elastic impedance expressions: isotropic Table 3.7.1 Exponents in expressions for elastic impedance of an isotropic, flat-layered Earth, EI VPA VSB C . In all expressions, y is the P-wave angle of incidence and yS is the shear-wave angle of incidence. Expressions are given for P–P reflection, converted wave reflections P–S and S–P, and shear-wave reflections SV–SV and SH–SH. 2 2 VS =VP and hVS =VP i are assumed to be constant, often taken as the average of the corresponding quantities. P–P

P–SV

SV–P

A

1 þ tan2

B

2 V 8 S2 sin2 VP

C

2 V 1 4 S2 sin2 VP

A

0

B

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! 2 2 sin V V VS S 2 2 S qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 cos

1 sin

4 sin VP2 VP2 VP 1 VS2 =VP2 sin2

C

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! 2 2 VS VS VS sin 2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 2 sin þ 2 cos 1 2 sin VP VP VP 1 VS2 =VP2 sin2

A

0

B

0

1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 V V S sin S @4 sin2 S 4 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos S 1 S2 sin2 S A 1 VP VP 1 V2 =V2 sin2 P

S

C

0 1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 S V V sin

2 2 S S @1 2 sin S þ 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos S 1 2 sin S A 1 VP VP 1 V2 =VP2 sin2 S S

SV–SV

SH–SH

S

A

0

B

C

1 þ 4 sin2 S

A

0

B

1 þ tan2 S

C

1

1 þ 8 sin2 S cos2 S

119

3.7 Elastic impedance

Elastic impedance expressions: VTI anisotropic Exponents in expressions for elastic

impedance of a VTI, flat-layered Earth, EI VPA VSB C exp D" þ E þ F ðVÞ . y is the P-wave angle of incidence, and yS is the shear-wave angle of incidence. Expressions are given for P–P reflection, and shear-wave reflections SV–SV and SH–SH. VS2 =VP2 is assumed to be constant, often taken as the average of the squared velocity ratio. Table 3.7.2

P–P

A B

C

SV–SV

D

sin2 tan2

E

sin2

F

0

A

0

B

C

1 þ 4 sin2 S 2 V sin2 S = S2 VP 2 V sin2 S = S2 VP

D

E

SH–SH

1 þ tan2 2 V 8 S2 sin2 VP 2 V 1 4 S2 sin2 VP

1 þ 8 sin2 S cos2 S

F

0

A B

0

C

1

D

0

E

0

F

tan2 S

1 þ tan2 S

120

Seismic wave propagation

VP ¼ ¼ "¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c33 =;

c11 c33 ; 2c33

VS ¼ ¼

ðVÞ ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c44 =

c66 c44 ; 2c44

¼

ðc13 þ c44 Þ2 ðc33 c44 Þ2 2c33 ðc33 c44 Þ

Elastic impedance expressions: orthorhombic x1–x3 symmetry plane Table 3.7.3 Exponents in expressions for P–P elastic impedance of an

ð2Þ A ?B C ð2Þ . y is the V exp D" þ E orthorhombic, flat-layered Earth, EI V P S 2 2 toﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ be constant, often taken as P-wave angle of incidence. VS =VP is assumedp pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the average of the squared velocity ratio. VP ¼ c33 =; VS? ¼ ? ¼ c55 = P–P

A B C

1 þ tan2 ?2 V 8 S2 sin2 VP ?2 V 1 4 S2 sin2 VP

D

sin2 tan2

E

sin2

Elastic impedance expressions: orthorhombic x2–x3 symmetry plane Table 3.7.4 Exponents in expressions for P–P elastic impedance of an

ð1Þ A B C ð1Þ orthorhombic, flat-layered Earth, ?2 EI2 VP VS exp D" þ E . y is the be constant, often taken as P-wave angle of incidence. VS =VP is assumed pto ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the average of the squared velocity ratio. VP ¼ c33 =; VS ¼ ¼ c44 = P–P

A B C

1 þ tan2 2 V 8 S2 sin2 VP 2 V 1 4 S2 sin2 VP

D

sin2 tan2

E

sin2

121

3.8 Viscoelasticity and Q

"ð2Þ ¼

c11 c33 ; 2c33

"ð1Þ ¼

c22 c33 2c33

ð2Þ ¼

ðc13 þ c55 Þ2 ðc33 c55 Þ2 ; 2c33 ðc33 c55 Þ

ð1Þ ¼

ðc23 þ c44 Þ2 ðc33 c44 Þ2 2c33 ðc33 c44 Þ

ð2Þ ¼

c66 c44 ; 2c44

ð1Þ ¼

c66 c55 2c55

ð3Þ

ðc12 þ c66 Þ2 ðc11 c66 Þ2 ¼ 2c11 ðc11 c66 Þ

Uses The elastic impedance provides an alternative for interpreting far-offset data. As with any inversion, the elastic impedance removes some wavelet effects and attempts to determine interval properties from the band-limited reflectivity (seismic traces).

Assumptions and limitations Limitations of using partial-stack elastic impedance inversion arise from the assumptions of the one-dimensional convolutional model for far offsets, the assumption of a constant value for K ¼ ðVS =VP Þ2 , and errors in estimates of the incidence angle. The convolutional model does not properly handle all the reflections at far offsets because the primary reflections get mixed with other events. Furthermore, approximations used to derive the expressions for elastic impedance get worse at larger angles. The first-order two-term elastic impedance has been found to give more stable results than the three-term elastic impedance (Mallick, 2001).

3.8

Viscoelasticity and Q Synopsis Materials are linear elastic when the stress is proportional to the strain: 11 þ 22 þ 33 ¼ Kð"11 þ "22 þ "33 Þ volumetric 3 ij ¼ 2"ij ;

i 6¼ j

ij ¼ l ij "kk þ 2"ij

shear general isotropic

where sij and eij are the stress and strain, K is the bulk modulus, m is the shear modulus, and l is Lame´’s coefficient.

122

Seismic wave propagation

E1 E2

h

Figure 3.8.1 Schematic of a spring and dashpot system for which the force – displacement relation is described by the same equation as the standard linear solid.

In contrast, linear, viscoelastic materials also depend on rate or history, which can be incorporated by using time derivatives. For example, shear stress and shear strain may be related by using one of the following simple models: "_ ij ¼

_ ij ij þ 2 2

Maxwell solid

ij ¼ 2"_ ij þ 2"ij

Voigt solid

_ ij þ ðE1 þ E2 Þij ¼ E2 ð"_ ij þ E1 "ij Þ standardlinearsolid

ð3:8:1Þ ð3:8:2Þ ð3:8:3Þ

where E1 and E2 are additional elastic moduli and is a material constant resembling viscosity. More generally, one can incorporate higher-order derivatives, as follows: N X i¼0

ai

M @i" X @j ¼ b j @ti @tj j¼0

Similar equations would be necessary to describe the generalizations of other elastic constants such as K. It is customary to represent these equations with mechanical spring and dashpot models such as that for the standard linear solid shown in Figure 3.8.1. Consider a wave propagating in a viscoelastic solid so that the displacement, for example, is given by uðx; tÞ ¼ u0 exp½ðoÞx exp½iðot kxÞ

ð3:8:4Þ

Then at any point in the solid the stress and strain are out of phase ¼ 0 exp½iðot kxÞ

ð3:8:5Þ

" ¼ "0 exp½iðot kx ’Þ

ð3:8:6Þ

The ratio of stress to strain at the point is the complex modulus, M(o).

123

3.8 Viscoelasticity and Q

The quality factor, Q, is a measure of how dissipative the material is. The lower Q, the larger the dissipation is. There are several ways to express Q. One precise way is as the ratio of the imaginary and real parts of the complex modulus: 1 MI ¼ Q MR In terms of energies, Q can be expressed as 1 W ¼ Q 2W where DW is the energy dissipated per cycle of oscillation and W is the peak strain energy during the cycle. In terms of the spatial attenuation factor, a, in equation (3.8.4), 1 V Q f where V is the velocity and f is the frequency. In terms of the wave amplitudes of an oscillatory signal with period t, 1 1 uðtÞ ln Q uðt þ tÞ which measures the amplitude loss per cycle. This is sometimes called the logarithmic decrement. Finally, in terms of the phase delay ’ between the stress and strain, as in equations (3.8.5) and (3.8.6) 1 tanð’Þ Q Winkler and Nur (1979) showed that if we define QE ¼ ER/EI, QK ¼ KR/KI, and Qm ¼ mR/mI, where the subscripts R and I denote real and imaginary parts of the Young, bulk, and shear moduli, respectively, and if the attenuation is small, then the various Q factors can be related through the following equations: ð1 vÞð1 2vÞ ð1 þ vÞ 2vð2 vÞ ¼ ; QP QE QS ð1 þ vÞ 3ð1 vÞ 2ð1 2vÞ ¼ ; QK QP QS

QP I VP2 ¼ QS KI þ 43I VS2 3 ð1 2vÞ 2ð1 þ vÞ ¼ þ QE QK QS

One of the following relations always occurs (Bourbie´ et al., 1987): QK > QP > QE > QS

for high VP =VS ratios

QK < QP < QE < QS

for low VP =VS ratios

QK ¼ QP ¼ QE ¼ QS

Seismic wave propagation

Spectra (dB)

124

S(x1) S(x2) ln S(x1)/S(x2) Frequency

Figure 3.8.2 The slope of the log of the spectral ratio (difference of the spectra in dB) can be interpreted in terms of Q.

The spectral ratio method is a popular way to estimate Q in both the laboratory and the field. Because 1/Q is a measure of the fractional loss of energy per cycle of oscillation, after a fixed distance of propagation there is a tendency for shorter wavelengths to be attenuated more than longer wavelengths. If the amplitude of the propagating wave is f uðx; tÞ ¼ u0 exp½ðoÞx ¼ u0 exp x VQ we can compare the spectral amplitudes at two different distances and determine Q from the slope of the logarithmic decrement (Figure 3.8.2): Sð f ; x2 Þ f ¼ ln ðx2 x1 Þ Sð f ; x1 Þ QV A useful illustrative example is the standard linear solid (Zener, 1948) in equation (3.8.3). If we assume sinusoidal motion " ¼ "0 eiot ¼ 0 eiot and substitute into equation (3.8.3), we can write 0 ¼ MðoÞ"0 with the complex frequency-dependent modulus MðoÞ ¼

E2 ðE1 þ ioÞ E1 þ E2 þ io

In the limits of low frequency and high frequency, we obtain the limiting moduli M0 ¼

E2 E1 ; E1 þ E2

M1 ¼ E 2 ;

o!0 o!1

3.8 Viscoelasticity and Q 2

0.6

0.4

1/Q M

1.6 ∆M 1.2

0.3 0.8

0.2

0.4

0.1 0 −5

Modulus

0.5 1/Q (∆M/M )

125

0 log(w/w r)

5

0

Figure 3.8.3 Schematic of the standard linear solid in the frequency domain.

Note that at very low frequencies and very high frequencies the moduli are real and independent of frequency, and thus in these limits the material behaves elastically. It is useful to rewrite the frequency-dependent complex modulus in terms of these limits: h i M1 M0 þ ioor ðM1 M0 Þ1=2 MðoÞ ¼ M1 þ ioor ðM1 M0 Þ1=2 and Re½MðoÞ ¼

Im½MðoÞ ¼

h i M0 M1 1 þ ðo=or Þ2 M1 þ ðo=or Þ2 M0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðo=or Þ M0 M1 ðM1 M0 Þ M1 þ ðo=or Þ2 M0

where pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E1 ðE1 þ E2 Þ or ¼ Similarly we can write Q as a function of frequency (Figure 3.8.3): 1 MI ðoÞ ¼ Q MR ðoÞ 1 E2 ðo=or Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Q E1 ðE1 þ E2 Þ 1 þ ðo=or Þ2 The maximum attenuation 1 1 E2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Q max 2 E1 ðE1 þ E2 Þ

Seismic wave propagation

0.6

2 1.6 M 1.2

0.3 0.8

Modulus

0.4

nearly constant

1/Q

0.5 1/Q (∆M/M)

126

0.2 0.4

0.1 0 −6

−4

−2

0 2 log(w)

4

6

8

0

Figure 3.8.4 Schematic of the nearly constant Q model in the frequency domain.

1 1 M1 M0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Q max 2 M1 M0 occurs at o = or . This is sometimes written as 1 1 M ¼ Q max 2 M

ð3:8:7Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ðM1 M0 Þ=M is the modulus defect and M ¼ M1 M0 : where M=M Liu et al. (1976) considered the nearly constant Q model in which simple attenuation mechanisms are combined so that the attenuation is nearly a constant over a finite range of frequencies (Figure 3.8.4). One can then write VðoÞ 1 o ¼1þ ln Vðo0 Þ Q o0 which relates the velocity dispersion within the band of constant Q to the value of Q and the frequency. For large Q, this can be approximated as 1 1 M M0 Q max logðo=o0 Þ 2 M0

ð3:8:8Þ

where M and M0 are the moduli at two different frequencies o and o0 within the band where Q is nearly constant. Note the resemblance of this expression to equation (3.8.7) for the standard linear solid. Kjartansson (1979) considered the constant Q model in which Q is strictly constant (Figure 3.8.5). In this case the complex modulus and Q are related by MðoÞ ¼ M0

io o0

2

127

3.9 Kramers–Kronig relations between velocity and Q

M(w)

Q −1(w)

1/Q

M

log(w)

Figure 3.8.5 Schematic of the constant Q model in the frequency domain.

where 1 1 1 ¼ tan Q For large Q, this can be approximated as 1 1 M M0 Q max logðo=o0 Þ 2 M0 where M and M0 are the moduli at two different frequencies o and o0. Note the resemblance of this expression to equation (3.8.7) for the standard linear solid and equation (3.8.8) for the nearly constant Q model.

Uses The equations presented in this section are used for phenomenological modeling of attenuation and velocity dispersion of seismic waves.

Assumptions and limitations The equations presented in this section assume that the material is linear, dissipative, and causal.

3.9

Kramers–Kronig relations between velocity dispersion and Q Synopsis For linear viscoelastic systems, causality requires that there be a very specific relation between velocity or modulus dispersion and Q; that is, if the dispersion is

128

Seismic wave propagation

completely characterized for all frequencies then Q is known for all frequencies and vice versa. We can write a viscoelastic constitutive law between the stress and strain components as ðtÞ ¼

dr "ðtÞ dt

where r(t) is the relaxation function and * denotes convolution. Then in the Fourier domain we can write ~ðoÞ ¼ MðoÞ "~ðoÞ where M(o) is the complex modulus. For r(t) to be causal, in the frequency domain the real and imaginary parts of M(o)/(io) must be Hilbert transform pairs (Bourbie´ et al., 1987): Z o þ1 MR ðÞ MR ð0Þ d MI ðoÞ ¼ 1 o o MR ðoÞ MR ð0Þ ¼

Z

þ1 1

M1 ðÞ d o

where MR(0) is the real part of the modulus at zero frequency, which results because there is an instantaneous elastic response from a viscoelastic material. If we express this in terms of Q1 ¼

MI ðoÞ sgnðoÞ MR ðoÞ

then we obtain joj Q ðoÞ ¼ MR ðoÞ 1

Z

þ1 1

and its inverse o MR ðoÞ MR ð0Þ ¼

Z

MR ðÞ MR ð0Þ d o þ1

1

Q1 ðÞMR ðÞ d jj ð oÞ

ð3:9:1Þ

ð3:9:2Þ

From these we see the expected result that a larger attenuation is generally associated with larger dispersion. Zero attenuation requires zero velocity dispersion. One never has more than partial information about the frequency dependence of velocity and Q, but the Kramers–Kronig relation allows us to put some constraints on the material behavior. For example, Lucet (1989) measured velocity and attenuation at two frequencies ( 1 kHz and 1 MHz) and used the Kramers–Kronig relations to compare the differences with various viscoelastic models, as shown schematically in Figure 3.9.1. Using equation (3.9.1), we can determine the expected ratio of

3.10 Full-waveform synthetic seismograms

Dispersion Vlf /Vhf

129

Constant Q

Other

Nearly constant Q 1/Q

Figure 3.9.1 Lucet’s (1989) use of the Kramers–Kronig relations to compare high- and low-frequency measured velocities and Q with various viscoelastic models.

low-frequency modulus or velocity, Vlf, and high-frequency modulus or velocity, Vhf, for various functional forms of Q (for example, constant Q or nearly constant Q). In all cases, linear viscoelastic behavior should lead to an intercept of Vlf/Vhf ¼ 1 at 1/Q ¼ 0. Mechanisms with peaked attenuation curves between the measurement points will generally cause a larger dispersion, which appears as a steeper negative slope.

Uses The Kramers–Kronig equations can be used to relate velocity dispersion and Q in linear viscoelastic materials.

Assumptions and limitations The Kramers–Kronig equations apply when the material is linear and causal.

3.10

Waves in layered media: full-waveform synthetic seismograms Synopsis One of the approaches for computing wave propagation in layered media is the use of propagator matrices (Aki and Richards, 1980; Claerbout, 1985). The wave variables of interest (usually stresses and particle velocity or displacements) at the top and bottom of the stack of layers are related by a product of propagator matrices, one for each layer. The calculations are done in the frequency domain and include the effects of all multiples. For waves traveling perpendicularly to n layers with layer velocities, densities, and thicknesses Vk , rk, and dk, respectively Y n S S ¼ A W n k¼1 k W 1 where S and W are the Fourier transforms of the wave variables s and w, respectively. For normal-incidence P-waves, s is interpreted as the normal stress across each interface, and w is the normal component of the particle velocity. For normal-incidence

130

Seismic wave propagation

S-waves, s is the shear traction across each interface, and w is the tangential component of the particle velocity. Each layer matrix Ak has the form 2 3 odk k V sin cos od i k k Vk Vk 7 6 5 Ak ¼ 4 odk i k cos od k Vk sin Vk Vk where o is the angular frequency. Kennett (1974, 1983) used the invariant imbedding method to generate the response of a layered medium recursively by adding one layer at a time (Figure ^ D and T ^ D , respectively, 3.10.1). The overall reflection and transmission matrices, R for downgoing waves through a stack of layers are given by the following recursion relations: h i1 ^ ðkÞ ¼ RðkÞ þ TðkÞ EðkÞ R ^ ðkþ1Þ EðkÞ I RðkÞ EðkÞ R ^ ðkþ1Þ EðkÞ TðkÞ R D D U D D D U D D D D 1 ðkÞ ðkþ1Þ ðkÞ ðkÞ ðkÞ ^ ðkþ1Þ ðkÞ ðkÞ ^ ^ TD TD ¼ TD ED I RU ED RD ED ðkÞ

ðkÞ

ðkÞ

ðkÞ

where RD ; TD ; RU ; and TU are just the single-interface downward and upward reflection and transmission matrices for the kth interface: 2 3 #" VPðk1Þ cos k1 1=2 #" SP PP 6 7 VSðk1Þ cos k1 6 7 ðkÞ 7 RD ¼6 6 7 1=2 #" 4 #" VSðk1Þ cos k1 5 PS SS VPðk1Þ cos k1 2 1=2 ## 1=2 3 ## k VPðkÞ cos k k VPðkÞ cos k SP 6 PP 7 6 7 k1 VPðk1Þ cos k1 k1 VSðk1Þ cos k1 6 7 ðkÞ 7 TD ¼6 ## 6 1=2 1=2 7 6 ## 7 ## k VSðkÞ cos k k VSðkÞ cos k 4 5 PS SS k1 VPðk1Þ cos k1 k1 VSðk1Þ cos k1 2 3 "# VPðkÞ cos k 1=2 "# SP PP 6 7 VSðkÞ cos k 6 7 ðkÞ 7 RU ¼6 6 7 1=2 "# 4 "# VSðkÞ cos k 5 PS SS VPðkÞ cos k 2 3 "" k1 VPðk1Þ cos k1 1=2 "" k1 VPðk1Þ cos k1 1=2 SP 6 PP 7 6 7 k VPðkÞ cos k k VSðkÞ cos k 6 7 ðkÞ 6 7 TU ¼6 ## 1=2 1=2 7 6 "" 7 "" k1 VSðk1Þ cos k1 k1 VSðk1Þ cos k1 4 5 PS SS k VPðkÞ cos k k VSðkÞ cos k

131

3.10 Full-waveform synthetic seismograms

Homogeneous half-space Reflected

Incident Interface 1 Interface 2

Layer 1 Layer 2

(1)

RD

(k)

Interface k

Layer k

Interface n − 1 Interface n Interface n + 1

Layer n − 1 Layer n Transmitted

RD

(n)

RD

(n + 1) RD

Homogeneous half-space

Figure 3.10.1 Recursively determined transfer functions in a layered medium.

with 0#" #" "" ""1 B PP SP PP SP C B#" #" "" ""C B PS SS PS SS C 1 B C B## ## "# "#C¼M N B PP SP PP SP C @ A ##

##

"#

"#

PS SS PS SS 2

sin k1

6 6 cos k1 6 M ¼6 6 2ISðk1Þ sin k1 cos k1 4 IPðk1Þ ð1 2 sin2 k1 Þ 2

sin k1

6 cos k1 6 N¼6 6 2I 4 Sðk1Þ sin k1 cos k1 IPðk1Þ ð1 2 sin2 k1 Þ

cos k1

sin k

sin k1

cos k

ISðk1Þ ð1 2 sin2 k1 Þ

2ISðkÞ sin k cos k

ISðk1Þ sin 2k1

IPðkÞ ð1 2 sin2 k Þ

cos k

3

7 7 7 7 2 ISðkÞ ð1 2 sin k Þ 7 5 sin k

ISðkÞ sin 2k

cos k1

sin k

cos k

sin k1

cos k

sin k

3

7 7 7 ISðk1Þ ð1 2 sin2 k1 Þ 2ISðkÞ sin k cos k ISðkÞ ð1 2 sin2 k Þ 7 5 2 ISðk1Þ sin 2k1 IPðkÞ ð1 2 sin k Þ ISðkÞ sin 2k

where IP(k) = rkVP(k) and IS(k) = rkVS(k) are the P and S impedances, respectively, of the kth layer, and yk and fk are the angles made by the P- and S-wave vectors with the normal to the kth interface. The elements of the reflection and transmisðkÞ ðkÞ ðkÞ ðkÞ sion matrices RD ; TD ; RU ; and TU are the reflection and transmission coefficients for scaled displacements, which are proportional to the square root of the energy flux. The scaled displacement u0 is related to the displacement u by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u0 ¼ u V cos :

132

Seismic wave propagation

For normal-incidence wave propagation with no mode conversions, the reflection and transmission matrices reduce to the scalar coefficients: ðkÞ

RD ¼ ðkÞ TU

k1 Vk1 k Vk ; k1 Vk1 þ k Vk

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 k1 Vk1 k Vk ¼ ; k1 Vk1 þ k Vk

ðkÞ

ðkÞ

RU ¼ RD ðkÞ

ðkÞ

TU ¼ TD

ðkÞ

The phase shift operator for propagation across each new layer is given by ED ðkÞ ED

expðiodk cos k =VPðkÞ Þ ¼ 0

0 expðiodk cos k =VSðkÞ Þ

where yk and fk are the angles between the normal to the layers and the directions of ^ D and T ^ D are functions of o propagation of P- and S-waves, respectively. The terms R and represent the overall transfer functions of the layered medium in the frequency domain. Time-domain seismograms are obtained by multiplying the overall transfer function by the Fourier transform of the source wavelet and then performing an inverse transform. The recursion starts at the base of the layering at interface n þ 1 ðnþ1Þ ^ ðnþ1Þ ¼ 0 and Tðnþ1Þ ¼ T ^ ðnþ1Þ ¼ I simulates a (Figure 3.10.1). Setting RD ¼R D D D stack of layers overlying a semi-infinite homogeneous half-space with properties equal to those of the last layer, layer n. The recursion relations are stepped up through ^ ð1Þ and T ^ ð1Þ , the overall reflection and the stack of layers one at a time to finally give R D D transmission response for the whole stack. Calculate the P-wave normal-incidence overall reflection and transmission functions ð1Þ ð1Þ R^D and T^D recursively for a three-layered medium with layer properties as follows: VPð1Þ ¼ 4000 m=s;

1 ¼ 2300 kg=m3 ;

d1 ¼ 100 m

VPð2Þ ¼ 3000 m=s;

2 ¼ 2100 kg=m ;

d2 ¼ 50 m

VPð3Þ ¼ 5000 m=s;

3 ¼ 2500 kg=m ;

d3 ¼ 200 m

3 3

ð4Þ ð4Þ The recursion starts with R^D = 0 and T^D = 1. The normal-incidence reflection and transmission coefficients at interface 3 are

ð3Þ

2 VPð2Þ 3 VPð3Þ ð3Þ ð3Þ ¼ 0:33; RU ¼ RD 2 VPð2Þ þ 3 VPð3Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 VPð2Þ 3 VPð3Þ ð3Þ ð3Þ ¼ ¼ 0:94; TU ¼ TD 2 VPð2Þ þ 3 VPð3Þ

RD ¼ ð3Þ

TD

and the phase factor for propagation across layer 3 is

133

3.10 Full-waveform synthetic seismograms

ð3Þ

ED ¼ expði2fd3 =VPð3Þ Þ ¼ expði2f 200=5000Þ where f is the frequency. The recursion relations give ð3Þ ð3Þ R^D ¼ RD þ

ð3Þ T^D ¼

ð3Þ ð3Þ ð4Þ ð3Þ TU ED R^D ED ð3Þ T ð3Þ ð3Þ ^ð4Þ ð3Þ D 1 RU ED RD ED

ð4Þ ð3Þ T^D ED ð3Þ T ð3Þ ð3Þ ^ð4Þ ð3Þ D 1 RU ED RD ED

ð1Þ The recursion is continued in a similar manner until finally we obtain R^D and

ð1Þ T^D .

The matrix inverse ðkÞ ðkÞ ^ ðkþ1Þ ðkÞ 1 ED ½I RU ED R D

is referred to as the reverberation operator and includes the response caused by all internal reverberations. In the series expansion of the matrix inverse ðkÞ ðkÞ ^ ðkþ1Þ ðkÞ 1 ðkÞ ðkÞ ^ ðkþ1Þ ðkÞ ED ¼ I þ RU ED R ED ½I RU ED R D D ðkÞ ðkÞ ^ ðkþ1Þ ðkÞ ðkÞ ðkÞ ^ ðkþ1Þ ðkÞ þ RU ED R ED RU ED RD ED þ D

the first term represents the primaries and each successive term corresponds to higher-order multiples. Truncating the expansion to m þ 1 terms includes m internal multiples in the approximation. The full multiple sequence is included with the exact matrix inverse.

Uses The methods described in this section can be used to compute full-wave seismograms, which include the effects of multiples for wave propagation in layered media.

Assumptions and limitations The algorithms described in this section assume the following: layered medium with no lateral heterogeneities; layers are isotropic, linear, elastic; and plane-wave, time-harmonic propagation.

134

Seismic wave propagation

3.11

Waves in layered media: stratigraphic filtering and velocity dispersion Synopsis Waves in layered media undergo attenuation and velocity dispersion caused by multiple scattering at the layer interfaces. The effective phase slowness of normally incident waves through layered media depends on the relative scales of the wavelength and layer thicknesses and may be written as Seff ¼ Srt þ Sst. The term Srt is the ray theory slowness of the direct ray that does not undergo any reflections and is just the thickness-weighted average of the individual layer slownesses. The individual slownesses may be complex to account for intrinsic attenuation. The excess slowness Sst (sometimes called the stratigraphic slowness) arises because of multiple scattering within the layers. A flexible approach to calculating the effective slowness and travel time follows from Kennett’s (1974) invariant imbedding formulation for the transfer function of a layered medium. The layered medium, of total thickness L, consists of layers with velocities (inverse slownesses), densities, and thicknesses, Vj, rj, and lj, respectively. The complex stratigraphic slowness is frequency dependent and can be calculated recursively (Frazer, 1994) by ! n tj 1 X Sst ¼ ln ioL j¼1 1 Rj 2j rj As each new layer j þ 1 is added to the stack of j layers, R is updated according to Rjþ1 ¼ rjþ1 þ

Rj 2jþ1 t2jþ1 1 Rj 2jþ1 rjþ1

(with R0 ¼ 0) and the term ln½tjþ1 ð1 Rjþ1 2jþ1 rjþ1 Þ1 is accumulated in the sum. In the above expressions, tj and rj are the transmission and reflection coefficients defined as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 j Vj jþ1 Vjþ1 tj ¼ j Vj þ jþ1 Vjþ1 rj ¼

jþ1 Vjþ1 j Vj j Vj þ jþ1 Vjþ1

whereas yj ¼ exp(iolj/Vj) is the phase shift for propagation across layer j and o is the angular frequency. The total travel time is T ¼ Trt þ Tst, where Trt is the ray theory travel time given by

135

3.11 Stratigraphic filtering and velocity dispersion

Trt ¼

n X lj V j¼1 j

and Tst is given by "

n tj 1 X Tst ¼ Re ln io j¼1 1 Rj 2j rj

!#

The deterministic results given above are not restricted to small perturbations in the material properties or statistically stationary geology.

Calculate the excess stratigraphic travel time caused by multiple scattering for a normally incident P-wave traveling through a three-layered medium with layer properties as follows: VPð1Þ ¼ 4000 m=s;

1 ¼ 2300 kg=m3 ;

l1 ¼ 100 m

VPð2Þ ¼ 3000 m=s;

2 ¼ 2100 kg=m3 ;

l2 ¼ 50 m

VPð3Þ ¼ 5000 m=s;

3 ¼ 2500 kg=m3 ;

l3 ¼ 200 m

The excess travel time is given by " !# n tj 1 X Tst ¼ Re ln io j¼1 1 Rj 2j rj The recursion begins with R0 ¼ 0, R1 ¼ r1 þ

R0 21 t21 1 R0 21 r1

where t1 ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 V1 2 V2 ¼ 0:98 1 V1 þ 2 V2

r1 ¼

2 V2 1 V1 ¼ 0:19 1 V1 þ 2 V2

y1 ¼ exp(i2pfl1/V1) ¼ exp(i2pf100/4000) with f as the frequency. The recursion is continued to obtain R2 and R3. Setting t3 ¼ 1 and r3 ¼ 0 simulates an impedance-matching homogeneous infinite half-space beneath layer 3.

136

Seismic wave propagation

Finally, the excess travel time, which is a function of the frequency, is obtained by taking the real part of the sum as follows: 1 t1 t2 t3 Tst ¼ Re þ ln þ ln ln i2f 1 R1 21 r1 1 R2 22 r2 1 R3 23 r3 The effect of the layering can be thought of as a filter that attenuates the input wavelet and introduces a delay. The function AðoÞ ¼ expðioxSst Þ ¼ expðioTrt Sst =Srt Þ (where Srt is assumed to be real in the absence of any intrinsic attenuation) is sometimes called the stratigraphic filter. The O’Doherty–Anstey formula (O’Doherty and Anstey, 1971; Banik et al., 1985) ^ jAðoÞj expðRðoÞT rt Þ approximately relates the amplitude of the stratigraphic filter to the power spectrum ^ RðoÞ of the reflection coefficient time series r() where Z x ðxÞ ¼ dx0 =Vðx0 Þ 0

is the one-way travel time. Initially the O’Doherty–Anstey formula was obtained by a heuristic approach (O’Doherty and Anstey, 1971). Later, various authors substantiated the result using statistical ensemble averages of wavefields (Banik et al., 1985), deterministic formulations (Resnick et al., 1986), and the concepts of self-averaged values and wave localization (Shapiro and Zien, 1993). Resnick et al. (1986) showed that the O’Doherty–Anstey formula is obtained as an approximation from the exact frequency-domain theory of Resnick et al. by neglecting quadratic terms in the Riccatti equation of Resnick et al. Another equivalent way of expressing the O’Doherty–Anstey relation is ^ RðoÞ ImðSst Þ 1 1 ^ ¼ oMð2oÞ o Srt 2Q 2 ^ Here 1/Q is the scattering attenuation caused by the multiples, and MðoÞ is the power spectrum of the logarithmic impedance fluctuations of the medium, ln½ðÞVðÞ hln½ðÞVðÞi, where hi denotes a stochastic ensemble average. Because the filter is minimum phase, o Re(Sst) and o Im(Sst) are a Hilbert transform pair, ^ ReðSst Þ t HfRðoÞg Srt Trt o where H{·} denotes the Hilbert transform and dt is the excess travel caused by multiple reverberations.

137

3.11 Stratigraphic filtering and velocity dispersion

Shapiro and Zien (1993) generalized the O’Doherty–Anstey formula for nonnormal incidence. The derivation is based on a small perturbation analysis and requires the fluctuations of material parameters to be small (<30%). The generalized formula for plane pressure (scalar) waves in an acoustic medium incident at an angle y with respect to the layer normal is ^ cos Þ Rðo jAðoÞj exp Trt cos4 whereas "

^ cos Þ ð2 cos2 1Þ2 Rðo Trt jAðoÞj exp cos4

#

for SH-waves in an elastic medium (Shapiro et al., 1994). For a perfectly periodic stratified medium made up of two constituents with phase velocities V1, V2; densities r1, r2; and thicknesses l1, l2, the velocity dispersion relation may be obtained from the Floquet solution (Christensen, 1991) for periodic media: cos

oðl1 þ l2 Þ ol1 ol2 ol1 ol2 cos sin sin ¼ cos V V1 V2 V1 V2 ¼

ð1 V1 Þ2 þ ð2 V2 Þ2 21 2 V1 V2

The Floquet solution is valid for arbitrary contrasts in the layer properties. If the spatial period (l1 þ l2) is an integer multiple of one-half wavelength, multiple reflections are in phase and add constructively, resulting in a large total accumulated reflection. The frequency at which this Bragg scattering condition is satisfied is called the Bragg frequency. Waves cannot propagate within a stop-band around the Bragg frequency.

Uses The results described in this section can be used to estimate velocity dispersion and attenuation caused by scattering for normal-incidence wave propagation in layered media.

Assumptions and limitations The methods described in this section apply under the following conditions: layers are isotropic, linear elastic with no lateral variation; propagation is normal to the layers except for the generalized O’Doherty–Anstey formula; and plane-wave, time-harmonic propagation is assumed.

138

Seismic wave propagation

3.12

Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation Synopsis Waves in layered media undergo attenuation and velocity dispersion caused by multiple scattering at the layer interfaces. Thinly layered media also give rise to velocity anisotropy. At low frequencies this phenomenon is usually described by the Backus average. Velocity anisotropy and dispersion in a multilayered medium are two aspects of the same phenomenon and are related to the frequency- and angledependent transmissivity resulting from multiple scattering in the medium. Shapiro et al. (1994) and Shapiro and Hubral (1995, 1996, 1999) have presented a wholefrequency-range statistical theory for the angle-dependent transmissivity of layered media for scalar waves (pressure waves in fluids) and elastic waves. The theory encompasses the Backus average in the low-frequency limit and ray theory in the high-frequency limit. The formulation avoids the problem of ensemble averaging versus measurements for a single realization by working with parameters that are averaged by the wave-propagation process itself for sufficiently long propagation paths. The results are obtained in the limit when the path length tends to infinity. Practically, this means the results are applicable when path lengths are very much longer than the characteristic correlation lengths of the medium. The slowness (s) and density (r) distributions of the stack of layers (or a continuous inhomogeneous one-dimensional medium) are assumed to be realizations of random stationary processes. The fluctuations of the physical parameters are small (<30%) compared with their constant mean values (denoted by subscripts 0): 1 s2 ðzÞ ¼ 2 ½1 þ "s ðzÞ c0 ðzÞ ¼ 0 ½1 þ " ðzÞ where the fluctuating parts es(z) (the squared slowness fluctuation) and er (z) (the density fluctuation) have zero means by definition. The depth coordinate is denoted by z, and the x- and y-axes lie in the plane of the layers. The velocity 1=2 c0 ¼ s2 corresponds to the average squared slowness of the medium. Instead of the squared slowness fluctuations, the random medium may also be characterized by the P- and S-velocity fluctuations, a and b, respectively, as follows: ðzÞ ¼ hi½1 þ " ðzÞ ¼ 0 ½1 þ " ðzÞ ðzÞ ¼ hi½1 þ " ðzÞ ¼ 0 ½1 þ " ðzÞ In the case of small fluctuations " "s =2 and hi c0 ð1 þ 32 =2Þ, where the normalized variance (the variance divided by the square of the mean)

2 is

139

3.12 Anisotropy, dispersion, and attenuation

of the velocity fluctuations. The horizontal wavenumber kx is related to the incidence angle y by kx ¼ k0 sin ¼ op where k0 ¼ o/c0, p ¼ sin y/c0 is the horizontal component of the slowness (also called the ray parameter), and o is the angular frequency. For elastic media, depending on the type of the incident wave, p ¼ sin y/a0 or p ¼ sin y/b0. The various autocorrelation and cross-correlation functions of the density and velocity fluctuations are denoted by B ðÞ ¼ " ðzÞ" ðz þ Þ B ðÞ ¼ h" ðzÞ" ðz þ Þi B ðÞ ¼ " ðzÞ" ðz þ Þ B ðÞ ¼ " ðzÞ" ðz þ Þ B ðÞ ¼ " ðzÞ" ðz þ Þ B ðÞ ¼ " ðzÞ" ðz þ Þ These correlation functions can often be obtained from sonic and density logs. The corresponding normalized variances and cross-variances are given by 2 ¼ B ð0Þ;

2 ¼ B ð0Þ;

2 ¼ B ð0Þ;

2 ¼ B ð0Þ;

2 ¼ B ð0Þ;

2 ¼ B ð0Þ

The real part of the effective vertical wavenumber for pressure waves in acoustic media is (neglecting higher than second-order powers of the fluctuations) Z 1 stat 2 2 kz ¼ kz k0 cos d BðÞ sinð2k0 cos Þ 0

kzstat ¼ k0 cos ð1 þ 2 =2 þ 2 = cos2 Þ BðÞ ¼ B ðÞ þ 2B ðÞ= cos2 þ B ðÞ= cos4 For waves in an elastic layered medium, the real part of the vertical wavenumber for P-, SV-, and SH-waves is given by Z 1 P 2 ^ kz ¼ la þ oAP o d ½BP ðÞ sinð2 la Þ þ BBB ðÞ sinðl Þ þ BDD ðÞ sinðlþ Þ 0

kzSV

¼ lb þ oA^SV o2

kzSH ¼ lb þ oA^SH o2

Z Z

1

d ½BSV ðÞ sinð2lb Þ BBB ðÞ sinðl Þ þ BDD ðÞ sinðlþ Þ

0 1

d ½BSH ðÞ sinð2lb Þ

0

and the imaginary part of the vertical wavenumber (which is related to the attenuation coefficient due to scattering) is

140

Seismic wave propagation

Z P ¼ o2

1

d ½BP ðÞ cosð2la Þ þ BBB ðÞ cosðl Þ þ BDD ðÞ cosðlþ Þ

0

Z

1

SV ¼ o

2

Z

d ½BSV ðÞ cosð2lb Þ þ BBB ðÞ cosðl Þ þ BDD ðÞ cosðlþ Þ

0 1

SH ¼ o2

d ½BSH ðÞ cosð2lb Þ

0

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where la ¼ o 1=20 p2 ; lb ¼ o 1=02 p2 ; lþ ¼ la þ lb , and l ¼ lb la (only real-valued la,b are considered). The other quantities in the preceding expressions are BP ðÞ ¼ ðX2 20 Þ1 ½B ðÞC21 þ 2B ðÞC1 þ 2B ðÞC1 C2 þ 2B ðÞC2 þ B ðÞ þ B ðÞC22 BSV ðÞ ¼ ðY 2 02 Þ1 ½B ðÞC27 þ 2B ðÞC7 C8 þ B ðÞC28 BSH ðÞ ¼ ðY 2 02 Þ1 ½B ðÞY 4 þ 2B ðÞC10 Y 2 þ 2B ðÞC210 BBB ðÞ ¼ p2 0 ð4XY0 Þ1 ½B ðÞC23 þ 2B ðÞC3 C4 þ B ðÞC24 BDD ðÞ ¼ p2 0 ð4XY0 Þ1 ½B ðÞC25 þ 2B ðÞC5 C6 þ B ðÞC26 A^P ¼ ð2X0 Þ1 AP ;

A^SV ¼ ð2Y0 Þ1 ASV ;

A^SH ¼ ð2Y0 Þ1 ASH

AP ¼ 2 ð1 4p2 02 ZÞ þ 22 ð1 4p2 02 Þ þ 82 p2 02 ð1 2ZÞ þ 32 16p2 02 2 þ 16p2 02 2 ð1 ZÞ ASV ¼ 2 ð1 C9 Þ þ 2 ð2 4C9 Þ þ 2 ð3 4C9 Þ ASH ¼ 2 Y 2 þ 22 ð2Y 2 1Þ þ 2 ð4Y 2 1Þ where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X ¼ ð1 p2 20 Þ;

Y¼

C1 ¼ X2 ð1 4p2 02 Þ;

C2 ¼ 8p2 02 X2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 p2 02 Þ;

Z ¼ p2 ð20 02 Þ

C3 ¼ Xð3 402 p2 Þ Y½ð0 =0 Þ þ ð20 =0 Þ 40 0 p2 C4 ¼ 4Xð1 202 p2 Þ 4Y½ð0 =0 Þ 20 0 p2 C5 ¼ Xð402 p2 3Þ Y½ð0 =0 Þ þ ð20 =0 Þ 40 0 p2 C6 ¼ 4Xð202 p2 1Þ 4Y½ð0 =0 Þ 20 0 p2 ; C8 ¼ 1 8p2 02 Y 2 ;

C9 ¼ 4Y 2 Z02 =20 ;

C7 ¼ Y 2 ð1 4p2 02 Þ

C10 ¼ 1 2p2 02

141

3.12 Anisotropy, dispersion, and attenuation

For multimode propagation (P–SV), neglecting higher-order terms restricts the range of applicable pathlengths L. This range is approximately given as maxfl; ag < L < Y 2 maxfl; ag=2 where l is the wavelength, a is the correlation length of the medium, and s2 is the variance of the fluctuations. The equations are valid for the whole frequency range, and there is no restriction on the wavelength to correlation length ratio. The angle- and frequency-dependent phase and group velocities are given by o 1 cphase ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 kx þ kz p þ kz2 =o2

c

group

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s 2ﬃ @o 2 @o ¼ þ @kx @kz

In the low- and high-frequency limits the phase and group velocities are the same. The low-frequency limit for pressure waves in acoustic media is h i freq 2 2 2 c 1 =2 cos

clow 0 fluid and for elastic waves freq ¼ 0 ð1 AP =2Þ clow P freq clow ¼ 0 ð1 ASV =2Þ SV freq ¼ 0 ð1 ASH =2Þ clow SH

These limits are the same as the result obtained from Backus averaging with higher-order terms in the medium fluctuations neglected. The high-frequency limit of phase and group velocities is

freq chigh ¼ c0 1 þ 2 =2 cos2 fluid for fluids, and chigh P

freq

¼ 0 1 2 1 3p2 20 =2 = 1 p2 20

freq chigh ¼ 0 1 2 1 3p2 02 =2 = 1 p2 02 SV;SH for elastic media, which are in agreement with ray theory predictions, again neglecting higher-order terms.

142

Seismic wave propagation

Shear-wave splitting or birefringence and its frequency dependence can be characterized by Sðo; pÞ ¼

cSV ðo; pÞ cSH ðo; pÞ cSV ðo; pÞ cSH ðo; pÞ cSV ðo; pÞ 0

In the low-frequency limit Slow freq(o, p) (ASH – ASV)/2, which is the shear-wave splitting in the transversely isotropic medium obtained from a Backus average of the elastic moduli. In the high-frequency limit S high freq(o, p) ¼ 0. For a medium with exponential correlation functions s2 exp (–x/a) (where a is the correlation length) for the velocity and density fluctuations with different variances s2 but the same correlation length a, the complete frequency dependence of the phase and group velocities is expressed as cphase fluid

ð2k02 a2 2 2 12 cos2 2

¼ c0 1 þ

1 þ 4k02 a2 cos2

" cgroup fluid

N¼

¼ c0 1 þ

!

# N ð1 þ 4k02 a2 cos2 Þ2

2k02 a2 2 ð3

þ

4k02 a2

cos Þ þ 2

! 2 cos2 2 þ ð4k02 a2 cos2 1Þ 2

for pressure waves in acoustic media. For elastic P-, SV-, and SH-waves in a randomly layered medium with exponential spatial autocorrelation with correlation length a, the real part of the vertical wavenumber (obtained by Fourier sine transforms) is given as (Shapiro and Hubral, 1995, 1996, 1999) " # 2la l lþ P 2 2 þ BBB ð0Þ þ BDD ð0Þ kz ¼ la þ oAP o a BP ð0Þ 1 þ 4a2 l2a 1 þ a2 l2 1 þ a2 l2þ "

kzSV

2lb l lþ ¼ lb þ oASV o a BSV ð0Þ BBB ð0Þ þ BDD ð0Þ 2 2 2 2 1 þ 4a lb 1 þ a l 1 þ a2 l2þ

#

2 2

kzSH ¼ lb þ oASH o2 a2 BSH ð0Þ

2lb 1 þ 4a2 l2b

and the imaginary part of the vertical wavenumber (which is related to the attenuation coefficient due to scattering) is " # 1 1 1 P ¼ o2 a BP ð0Þ þ BBB ð0Þ þ BDD ð0Þ 1 þ 4a2 l2a 1 þ a2 l2 1 þ a2 l2þ

143

3.12 Anisotropy, dispersion, and attenuation

" SV

#

1 1 1 ¼ o a BSV ð0Þ þ BBB ð0Þ þ BDD ð0Þ 2 2 1 þ 4a2 lb 1 þ a2 l 1 þ a2 l2þ 2

SH ¼ o2 aBSH ð0Þ

1 1 þ 4a2 l2b

The shear-wave splitting for exponentially correlated randomly layered media is ( 2lb low freq 2 þ oa 0 Y ½BSV ð0Þ BSH ð0Þ Sðo; pÞ S 1 þ 4a2 l2b )

lþ l BDD ð0Þ þ BBB ð0Þ 1 þ a2 l2þ 1 þ a2 l2

These equations reveal the general feature that the anisotropy (change in velocity with angle) depends on the frequency, and the dispersion (change in velocity with frequency) depends on the angle. Stratigraphic filtering causes the transmitted amplitudes to decay as expð LÞ, where L is the path length and g is the attenuation coefficient for the different wave modes as described in the equations above.

One-dimensional layered poroelastic medium The small-perturbation statistical theory has been extended to one-dimensional layered poroelastic media (Gurevich and Lopatnikov, 1995; Gelinsky and Shapiro, 1997b; Gelinsky et al., 1998). In addition to the attenuation due to multiple scattering in random elastic media, waves in a random porous saturated media cause inter-layer flow of pore fluids, leading to additional attenuation and velocity dispersion. The constituent poroelastic layers are governed by the Biot equations (see Section 6.1), and can support two P-waves, the fast and the slow P-wave. The poroelastic parameters of the random one-dimensional medium consist of a homogeneous background (denoted by subscript 0) upon which is superposed a zero-mean fluctuation. The fluctuations are characterized by their variance and a normalized spatial correlation function B(r/a), such that B(0) ¼ 1, where a is the correlation length. All parameters of the medium are assumed to have the same normalized correlation function and the same correlation length, but can have different variances. The poroelastic material parameters include: f, porosity; r, saturated bulk rock density; , permeability; f ; fluid density; , fluid viscosity, and Kf, fluid bulk modulus. Pd ¼ Kd þ 43 d is the dry (drained) Pwave modulus, with Kd and md being the dry bulk and shear moduli, respectively. ¼ 1 Kd =K0 is the Biot coefficient (note that in this section, a denotes the Biot coefficient, not the P-wave velocity). K0 is the mineral bulk modulus; M ¼ ½=Kf þ ð Þ=K0 1 ; H ¼ Pd þ 2 M is the saturated P-wave modulus

144

Seismic wave propagation

(equivalent to Gassmann’s equation). N ¼ MPd =H; o0 ¼ N=a2 is the characteristic frequency separating inter-layer-flow and no-flow regimes. oc ¼ =f is the Biot critical frequency. Plane P-waves are assumed to be propagating vertically (along the z-direction) normal to the stack of horizontal layers. The fast P-wavenumber and attenuation coefficient g are given by (Gelinsky et al., 1998): ¼ k1R þ A Z 1 h pﬃﬃﬃ I R Bðz=aÞ 2D expðzk Þ cosðzk =4Þ 0

i pﬃﬃﬃ I R þ 2D expðzkþ Þ cosðzkþ =4Þ þ C expð2zk1I Þ sinð2zk1R Þ dz ¼ k1I

Z

1

þ 0

h pﬃﬃﬃ I R Bðz=aÞ 2D expðzk Þ cosðzk þ =4Þ

i pﬃﬃﬃ I R Þ cosðzkþ þ =4Þ þ C expð2zk1I Þ cosð2zk1R Þ dz 2D expðzkþ where superscripts R and I denote real and imaginary parts; k~1 ¼ k1R þ ik1I and k~2 ¼ k2R þ ik2I are the complex wavenumbers for the Biot fast and slow PR I waves in the homogeneous background medium; k~þ ¼ kþ þ ikþ ¼ k~2 þ k~1 ; R I ~ ~ ~ k ¼ k þ ik ¼ k2 k1 . The quantities A, C, and D involve complicated functions of frequency and linear combinations of the variances and covariances of the medium fluctuations. Approximations for these quantities are discussed below. The above expressions assume small fluctuations in the poroelastic parameters but are not limited by any restriction on the relation between wavelength and the correlation length of the medium fluctuations. The phase velocity for the fast P-wave is given as VP ¼ o= . For a medium with an exponential correlation function with correlation length a, the expressions after carrying out the integrations are (Gelinsky et al., 1998):

R I Da 1 þ aðk þ k Þ o R ¼ k1 þ A I þ a2 ½ðkR Þ2 þ ðkI Þ2 VP 1 þ 2ak

R I Da 1 þ aðkþ þ kþ Þ 2

2Ca2 k1R 2

I þ a2 ðk I 1 þ 2akþ 1 þ 4ak1I þ 4a2 ðk1R þ k1I Þ þ þ kþ Þ

R I Da 1 aðk k Þ I ¼ k1 þ I þ a2 ðkR2 þ kI2 Þ 1 þ 2ak

R I Da 1 aðkþ kþ Þ Cað1 þ 2ak1I Þ þ þ 2 2 2 I þ a2 ðk R2 þ k I Þ 1 þ 4ak1I þ 4a2 ðk1R þ k1I Þ 1 þ 2akþ þ þ R2

2

145

3.12 Anisotropy, dispersion, and attenuation

Gelinsky et al. (1998) introduce approximate expressions for A, C, and D, valid in the frequency range below Biot’s critical frequency. Other approximations pﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ~þ k~ k~2 ; k~2 ð1 þ iÞk2 ; k~1 kR ¼ k1 ; k1 ¼ o =H ; and k2 ¼ used are: k 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ooc f =ð2NÞ: o A¼ 2

rﬃﬃﬃﬃﬃﬃ 40 M02 2 Pd0 2 20 M0 2 2 þ MM þ 2P þ 2 Pd0 H0 H0 PP Pd0

2D o ¼ k2 2

" rﬃﬃﬃﬃﬃﬃ Pd0 20 M0 2 Pd0 20 M0 2 2 2 2 þ 2 P 2M PP PM MM 2 Pd0 H0 H0

#

2 Pd0 20 M0 2 þ P2d0 C o ¼ 2k1 8

rﬃﬃﬃﬃﬃﬃ 2 40 M02 2 Pd0 2 20 M0 2 2 2 2 þ2 PM þ 2P þ 2 MM þ 4 þ 4M Pd0 H0 H02 PP Pd0

The phase velocity has three limiting values: a quasi-static value for o o0 and o ! 0, Vqs ¼ o½k1 þ A1 ; an intermediate no-flow velocity for o o0 , Vnf ¼ o½k1 þ A 2D=k2 1 , and a ray-theoretical limit Vray ¼ o½k1 þ A 2D=k2 C= ð2k1 Þ1 . The quasi-static limit is equivalent to the poroelastic Backus average (see Section 4.15). The inter-layer flow effect contributes significantly to the total attenuation in the seismic frequency range for highly permeable thin layers with correlation lengths of a few centimeters.

Uses The equations described in this section can be used to estimate velocity dispersion and frequency-dependent anisotropy for plane-wave propagation at any angle in randomly layered, one-dimensional media. They can also be used to apply angledependent amplitude corrections to correct for the effect of stratigraphic filtering in amplitude-versus-offset (AVO) modeling of a target horizon below a thinly layered overburden. The corrected amplitudes are obtained by multiplying the transmissivity by expð LÞ for the down-going and up-going ray paths (Widmaier et al., 1996). The equations for poroelastic media can be used to compute velocity dispersion and attenuation in heterogeneous, fluid-saturated one-dimensional porous media.

Assumptions and limitations The results described in this section are based on the following assumptions: layers are isotropic, linear elastic or poroelastic with no lateral variation; the layered medium is statistically stationary with small fluctuations (<30%) in the material properties;

146

Seismic wave propagation

3.13

the propagation path is very much longer than any characteristic correlation length of the medium; and incident plane-wave propagation is assumed. The expressions for poroelastic media assume plane waves along the normal to the layers (0 angle of incidence). Numerical calculations (Gelinsky et al., 1998) indicate that the expressions give reasonable results for angles of incidence up to 20 .

Scale-dependent seismic velocities in heterogeneous media Synopsis Measurable travel times of seismic events propagating in heterogeneous media depend on the scale of the seismic wavelength relative to the scale of the geological heterogeneities. In general, the velocity inferred from arrival times is slower when the wavelength, l, is longer than the scale of the heterogeneity, a, and faster when the wavelength is shorter (Mukerji et al., 1995b).

Layered (one-dimensional) media For normal-incidence propagation in stratified media, in the long-wavelength limit ðl=a 1Þ, where a is the scale of the layering, the stratified medium behaves as a homogeneous effective medium with a velocity given by effective medium theory as VEMT

MEMT 1=2 ¼ av

The effective modulus MEMT is obtained from the Backus average. For normalincidence plane-wave propagation, the effective modulus is given by the harmonic average !1 X fk MEMT ¼ Mk k X fk 1 ¼ 2 av VEMT k Vk2 k X av ¼ fk k k

where fk, rk, Mk, and Vk are the volume fractions, densities, moduli, and velocities of each constituent layer, respectively. The modulus M can be interpreted as C3333 or K þ 4m/3 for P-waves and as C2323 or m for S-waves (where K and m are the bulk and shear moduli, respectively).

147

3.13 Scale-dependent seismic velocities

In the short-wavelength limit l=a 1, the travel time for plane waves traveling perpendicularly to the layers is given by ray theory as the sum of the travel times through each layer. The ray theory or short-wavelength velocity through the medium is, therefore, X fk 1 ¼ Vk VRT k The ray theory velocity involves averaging slownesses, whereas the effective medium velocity involves averaging compliances (slownesses squared). The result is that VRT is always faster than VEMT. In two- and three-dimensional heterogeneous media, there is also the path effect as a result of Fermat’s principle. Shorter wavelengths tend to find fast paths and diffract around slower inhomogeneities, thus biasing the travel times to lower values (Nolet, 1987; Mu¨ller et al. 1992). This is sometimes referred to as the “Wielandt effect,” “fast path effect,” or “velocity shift.” The velocity shift was quantified by Boyse (1986) and by Roth et al. (1993) using an asymptotic ray-theoretical approach. The heterogeneous random medium is characterized by a spatially varying, statistically stationary slowness field nðrÞ ¼ n0 þ "n1 ðrÞ where r is the position vector, n1(r) is a zero-mean small fluctuation superposed on the constant background slowness n0, and " 1 is a small perturbation parameter. The spatial structure of the heterogeneities is described by the isotropic spatial autocorrelation function: h"n1 ðr1 Þ "n1 ðr2 Þi ¼ "2 n1 2 Nðjr1 r2 jÞ and the coefficient of variation (normalized standard deviation) is given by

n ¼

"

qﬃﬃﬃﬃﬃﬃﬃﬃﬃ n21 n0

where hi denotes the expectation operator. For short-wavelength, l=a 1, initially plane waves traveling along the x-direction, the expected travel time (spatial average of the travel time over a plane normal to x) at distance X is given as (Boyse, 1986) Z x N 0 ðÞ ðX Þ2 d þ Oð"3 Þ hT i ¼ n0 X þ 2n 0 where ¼ 1; 12, and 0 for three, two, and one dimension(s), respectively, and the prime denotes differentiation. When the wave has traveled a large distance compared

148

Seismic wave propagation

with the correlation length a of the medium X a, and when the autocorrelation function is such that N(x) N(0) for x > a, then hT i n0 ðX X2 2n DÞ Z D¼

1 0

N 0 ðÞ d > 0

The ray-theory slowness calculated from the average ray-theory travel time is given by nRT = (T)/X. Scaling the distance by the correlation length, a, results in the following equation: nRT X ^ ¼ 1 2n D n0 a where D^ is defined similarly to D but with the autocorrelation function being of unit pﬃﬃﬃ correlation length (a ¼ 1). For a Gaussian autocorrelation function D^ ¼ . In one dimension a ¼ 0, and the ray-theory slowness is just the average slowness. In two and ^ In this case, the three dimensions the path effect is described by the term 2n ðX=aÞD. wave arrivals are on average faster in the random medium than in a uniform medium having the same mean slowness. The expected travel time in three dimensions is less than that in two dimensions, which in turn is less than that in one dimension. This is because in higher dimensions more admissible paths are available to minimize the travel time. Mu¨ller et al. (1992) have established ray-theoretical results relating N(|r|) to the spatial autocorrelation function f(|r|) of the travel time fluctuations around the mean travel time. Using first-order straight-ray theory, they show Z 1 ð Þ ¼ 2X NðÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d 2 2

For a medium with a Gaussian slowness autocorrelation function, N(r) ¼ exp(–r2/ a2), the variance of travel time fluctuations f(0) is related to the variance of the slowness fluctuations by pﬃﬃﬃ ð0Þ ¼ X a n20 2n The expected travel time hTi of the wavefield U(n) in the heterogeneous medium with random slowness n is distinct from the travel time T of the expected wavefield hU(n)i in the random medium. The expected wave, which is an ensemble average of the wavefield over all possible realizations of the random medium, travels slower than the wavefield in the average medium hni (Keller, 1964). Thus, hTðUðnÞÞi TðUðhniÞÞ TðhUðnÞiÞ

149

3.13 Scale-dependent seismic velocities

Gold et al. (2000) analyzed wave propagation in random heterogeneous elastic media in the limit of small fluctuations and low frequencies (low-frequency limit of the elastic Bourret approximation) to obtain the effective P- and S-wave slownesses for the coherent wavefield. The random medium is characterized by zero-mean fluctuations of Lame´ parameters and density superposed on a homogeneous background with constant mean values of Lame´ parameters ðl0 ; 0 Þ and density ð0 Þ: lðrÞ ¼ l0 ð1 þ "l ðrÞÞ;

h"l i ¼ 0

ðrÞ ¼ 0 ð1 þ " ðrÞÞ;

" ¼ 0

ðrÞ ¼ 0 ð1 þ " ðrÞÞ;

" ¼ 0

The effective slownesses for P- and S-waves for three-dimensional random media in the low-frequency, small-fluctuation limit are expressed as (Gold et al., 2000): " SPe

¼

1þ

SP0

1 l20 2 l0 0 2 20 2 2 þ þ 2 2 ðl0 þ 20 Þ2 ll 3 ðl0 þ 20 Þ2 l 5 ðl0 þ 20 Þ2

4 0 2 þ 15 ðl0 þ 20 Þ

#

1 2 0 2 SSe ¼ SS0 1 þ 2 þ 5 15 ðl0 þ 20 Þ For two-dimensional random media Gold et al. give the following expressions: SPe

¼

1þ

SP0

1 l20 l0 0 3 20 2 2 þ þ 2 ll l 2 ðl0 þ 20 Þ2 4 ðl0 þ 20 Þ2 ðl0 þ 20 Þ2

1 0 þ 2 4 ðl0 þ 20 Þ

1 2 1 0 2 ¼ 1 þ þ 4 4 ðl0 þ 20 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ In the above equations, SP0 ¼ 0 =ðl0 þ 20 Þ and SS0 ¼ 0 =0 are the P- and S-wave slownesses in the homogeneous background medium, respectively. The normalized variances and covariances of the fluctuations of parameters x and y are denoted by xy . In this small-contrast, low-frequency limit, density heterogeneities do not affect the properties of the effective medium. The derivation assumes isotropic spatial correlation of the heterogeneities. SSe

SS0

150

Seismic wave propagation

Uses The results described in this section can be used for the following purposes: to estimate the velocity shift caused by fast path effects in heterogeneous media; to relate statistics of observed travel times to the statistics of the heterogeneities; and to smooth and upscale heterogeneous media in the low-frequency limit.

Assumptions and limitations The equations described in this section apply under the following conditions: small fluctuations in the material properties of the heterogeneous medium; isotropic spatial autocorrelation function of the fluctuations; ray-theory results are valid only for wavelengths much smaller than the spatial correlation length of the media; and low-frequency, effective-smoothing results are valid only for wavelengths much larger than the spatial correlation length of the medium, and for a smoothing window that is much smaller than the wavelength, but much larger than the spatial correlation length.

3.14

Scattering attenuation Synopsis The attenuation coefficient, gs ¼ p f/QV (where Q is the quality factor, V is the seismic velocity, and f is frequency) that results from elastic scattering depends on the ratio of seismic wavelength, l, to the diameter, ds, of the scattering heterogeneity. Roughly speaking there are three domains: Rayleigh scattering, where l > ds and s / d 3 f 4 ; s stochastic/Mie scattering, where l ds and gs / ds f 2; diffusion scattering, where l < ds and gs / 1/ds. When l ds , the heterogeneous medium behaves like an effective homogeneous medium, and scattering effects may be negligible. At the other limit, when l ds , the heterogeneous medium may be treated as a piecewise homogeneous medium. Figure 3.14.1 shows schematically the general scale dependence (or, equivalently, frequency dependence) of wave velocity that is expected owing to scattering in heterogeneous media. At very long wavelengths ðl ds Þ the phase velocity is nondispersive and is close to the static effective medium result. As the wavelength decreases (frequency increases), scattering causes velocity dispersion. In the Rayleigh scattering domain (l/ds 2p), the velocity shows a slight decrease with increasing frequency. This is usually followed by a rapid and much larger increase in

3.14 Scattering attenuation

1.05 Mie scattering 1.0 Velocity (normalized)

151

Ray theory 0.95 0.90 0.85 Effective medium

0.80 0.75 0.70 0.1

Rayleigh scattering 1

10

100

1000

l/d S

Figure 3.14.1 Scale dependence (or, equivalently, frequency dependence) of the wave velocity due to scattering in heterogeneous media.

phase velocity owing to resonant (or Mie) scattering ðl ds Þ. When l ds (specular scattering or ray theory), the velocity is again nondispersive or weakly dispersive and is usually significantly higher than its long-wavelength limit. It is usually assumed that the long-wavelength Rayleigh limit is most appropriate for analyzing laboratory rock physics results because the seismic wavelength is often much larger than the grain size. However, Lucet and Zinszner (1992) and Blair (1990), among others, have shown that the scattering heterogeneities can be clusters of grains that are comparable to, or larger than, the wavelength. Certainly any of the domains are possible in situ. Blair (1990) suggests a simple (ad hoc) expression that is consistent with both the Rayleigh and diffusion scattering limits: s ð f Þ ¼

Cs ð f =fd Þ4 ; ds ð1 þ f =fd Þ4

fd ¼

ks V ks f l ¼ ds ds

where Cs and ks are constants. Many theoretical estimates of scattering effects on velocity and attenuation have appeared (see Mehta, 1983, and Berryman, 1992b, for reviews). Most are in the longwavelength limit, and most assume that the concentration of scatterers is small, and thus only single scattering is considered. The attenuation of P-waves caused by a low concentration of small spherical inclusions is given by (Yamakawa, 1962; Kuster and Tokso¨z, 1974)

sph

3o o 3 2 ð2 þ 3 5 Þ 2 2 3 2 ¼c a 2B0 þ ð1 þ 2 ÞB1 þ B2 4VP VP 3 5

152

Seismic wave propagation

where B0 ¼

K K0 3K 0 þ 4

B1 ¼

0 3

B2 ¼

20 ð0 Þ 3 60 ðK þ 2Þ þ ð9K þ 8Þ

¼ VP =VS The terms VP and VS are the P- and S-velocities of the host medium, respectively, c is the volume concentration of the spheres, a is their radius, 2f ¼ o is the frequency, r is the density, K is the bulk modulus, and m is the shear modulus. The unprimed moduli refer to the background host medium and the primed moduli refer to the inclusions. In the case of elastic spheres in a linear viscous fluid, with viscosity , the attenuation is given by (Epstein, 1941; Epstein and Carhart, 1953; Kuster and Tokso¨z, 1974) o i þ b0 ib02 =3 sph ¼ c ð 0 Þ Real 2VP ð1 ib0 Þ ð þ 20 Þb02 =9 where

sﬃﬃﬃﬃﬃﬃﬃﬃ f b0 ¼ ð1 þ iÞa Hudson (1981) gives the attenuation coefficient for elastic waves in cracked media (see Section 4.10 on Hudson). For aligned penny-shaped ellipsoidal cracks with normals along the 3-axis, the attenuation coefficients for P-, SV-, and SH-waves are " 3 2 2 # o oa 1 2 2 2 VP 2 P ¼ " AU1 sin 2 þ BU3 2 2 sin VS VP 30 VS SV SH

3 o oa 1 2 ¼ " AU1 cos2 2 þ BU32 sin2 2 VS VS 30 3 o oa 1

¼ " AU1 2 cos2 VS VS 30

3 V5 A ¼ þ S5 2 VP B¼2þ

15 VS VS 3 VS 5 10 3 þ 8 5 4 VP VP VP

153

3.14 Scattering attenuation

In these expressions, y is the angle between the direction of propagation and the 3-axis (axis of symmetry), and e is the crack density parameter: "¼

N 3 3 a ¼ V 4

where N/V is the number of penny-shaped cracks of radius a per unit volume, f is the crack porosity, and a is the crack aspect ratio. U1 and U3 depend on the crack conditions. For dry cracks U1 ¼

16ðl þ 2Þ ; 3ð3l þ 4Þ

U3 ¼

4ðl þ 2Þ 3ðl þ Þ

For “weak” inclusions (i.e., when =ðK 0 þ 43 0 Þ is of the order 1 and is not small enough to be neglected) U1 ¼

16ðl þ 2Þ 1 ; 3ð3l þ 4Þ ð1 þ MÞ

U3 ¼

4ðl þ 2Þ 1 3ðl þ Þ ð1 þ Þ

where M¼

40 ðl þ 2Þ ; ð3l þ 4Þ

¼

ðK 0 þ 43 0 Þðl þ 2Þ ðl þ Þ

with K0 and m0 equal to the bulk and shear moduli of the inclusion material. The criterion for an inclusion to be “weak” depends on its shape, or aspect ratio a, as well as on the relative moduli of the inclusion and matrix material. Dry cavities can be modeled by setting the inclusion moduli to zero. Fluid-saturated cavities are simulated by setting the inclusion shear modulus to zero. Remember that these give only the scattering losses and do not incorporate other viscous losses caused by the pore fluid. Hudson also gives expressions for infinitely thin fluid-filled cracks: U1 ¼

16ðl þ 2Þ ; 3ð3l þ 4Þ

U3 ¼ 0

These assume no discontinuity in the normal component of crack displacements and therefore predict no change in the compressional modulus with saturation. There is, however, a shear displacement discontinuity and a resulting effect on shear stiffness. This case should be used with care. For randomly oriented cracks (isotropic distribution) the P- and S-attenuation coefficients are given as 3 o oa 4 1 VP5 2 2 P ¼ " þ BðB 2ÞU AU 1 3 VS VP 152 2 VS5 3 o oa 1 2 1 S ¼ " AU1 þ 3 BU32 VS VS 75 The fourth-power dependence on o is characteristic of Rayleigh scattering.

Seismic wave propagation

104 a=L 1000

Homogeneous

D<1 Ray theory

D=1

100 ka

154

Strong scattering

10 1

Weak scattering

∆E/E = 0.1 0.1 0.01

∆E/E < 0.1 1

10

Effective medium 100

1000

104

105

kL

Figure 3.14.2 Domains of applicability for various scattering theories.

Random heterogeneous media with spatially varying velocity c ¼ c0 þ c0 may be characterized by the autocorrelation function NðrÞ ¼

hðr 0 Þðr 0 þ rÞi h2 i

where x ¼ –c0 /c0 and c0 denotes the mean background velocity. For small fluctuations, the fractional energy loss caused by scattering is given by (Aki and Richards, 1980) 8 2 k4 a3 L E ; for NðrÞ ¼ er=a ¼ 1 þ 4k2 a2 E E pﬃﬃﬃ 2 2 2 2 ¼ k aLð1 ek a Þ; E

for NðrÞ ¼ er

2

=a2

These expressions are valid for small DE/E values as they are derived under the Born approximation, which assumes that the primary incident waves are unchanged as they propagate through the heterogeneous medium. Aki and Richards (1980) classify scattering phenomena in terms of two dimensionless numbers ka and kL, where k ¼ 2p/l is the wavenumber, a is the characteristic scale of the heterogeneity, and L is the path length of the primary incident wave in the heterogeneous medium. Scattering effects are not very important for very small or very large ka, and they become increasingly important with increasing kL. Scattering problems may be classified on the basis of the fractional energy loss caused by scattering, DE/E, and the wave parameter D defined by D ¼ 4L/ka2. The wave parameter is the ratio of the first Fresnel zone to the scale length of the heterogeneity. Ray theory is applicable when D < 1. In this case the inhomogeneities are smooth enough to be treated as piecewise homogeneous. Effective medium theories are appropriate when ka and DE/E are small. These domains are summarized in Figure 3.14.2. Scattering becomes complex when heterogeneity scales are comparable with the wavelength and when the path lengths are long. Energy diffusion models are used for long path lengths and strong scattering.

155

3.15 Waves in cylindrical rods: the resonant bar

Uses The results described in this section can be used to estimate the seismic attenuation caused by scattering.

Assumptions and limitations The results described in this section have the following limitations: formulas for spherical and ellipsoidal inclusions are limited to low pore concentrations and wavelengths much larger than the scatterer diameter; formulas for fractional energy loss in random heterogeneous media are limited to weak scattering.

3.15

Waves in cylindrical rods: the resonant bar Synopsis Time-harmonic waves propagating in the axial direction along a circular cylindrical rod involve radial, circumferential, and axial components of displacement, ur , uy, and uz, respectively. Motions that depend on z but are independent of y may be separated into torsional waves involving uy only and longitudinal waves involving ur and uz. Flexural waves consist of motions that depend on both z and y.

Torsional waves Torsional waves involve purely circumferential displacements that are independent of y. The dispersion relation (for free-surface boundary conditions) is of the form (Achenbach, 1984) saJ0 ðsaÞ 2J1 ðsaÞ ¼ 0 s2 ¼

o2 k2 VS2

where Jn ðÞ are Bessel functions of the first kind of order n, a is the radius of the cylindrical rod, VS is the S-wave velocity, and k is the wavenumber for torsional waves. For practical purposes, the lowest mode of each kind of motion is important. The lowest torsional mode consists of displacement proportional to the radius, and the motion is a rotation of each cross-section of the cylinder about its center. The phase velocity of the lowest torsional mode is nondispersive and is given by rﬃﬃﬃ Vtorsion ¼ VS ¼ where m and r are the shear modulus and density of the rod, respectively.

156

Seismic wave propagation

Longitudinal waves Longitudinal waves are axially symmetric and have displacement components in the axial and radial directions. The dispersion relation (for free-surface boundary conditions), known as the Pochhammer equation, is (Achenbach, 1984) 2pa½ðsaÞ2 þ ðkaÞ2 J1 ð paÞJ1 ðsaÞ ½ðsaÞ2 ðkaÞ2 2 J0 ð paÞJ1 ðsaÞ 4ðkaÞ2 ð paÞðsaÞJ1 ð paÞJ0 ðsaÞ ¼ 0 p2 ¼

o2 k2 VP2

where VP is the P-wave velocity. The phase velocity of the lowest longitudinal mode for small ka (ka 1) can be expressed as sﬃﬃﬃ i Eh 1 14 v2 ðkaÞ2 þ O½ðKaÞ4 Vlong ¼ where E is the Young modulus of the cylindrical rod and n is the Poisson ratio of the cylindrical rod. In the limit as p(ka) ﬃﬃﬃﬃﬃﬃﬃﬃ ! 0, the phase velocity tends to the bar velocity or extensional velocity VE ¼ E=: For very large ka (ka 1), Vlong approaches the Rayleigh wave velocity.

Flexural waves Flexural modes have all three displacement components – axial, radial, and circumferential and involve motion that depends on both z and y. The phase velocity of the lowest flexural mode for small values of ka (ka 1) may be written as sﬃﬃﬃ 1 E Vflex ¼ ðkaÞ þ O½ðkaÞ3 2 The phase velocity of the lowest flexural mode goes to zero as (ka) ! 0 and approaches the Rayleigh wave velocity for large ka values.

Bar resonance Resonant modes (or standing waves) occur when the bar length is an integer number of half-wavelengths: V ¼ lf ¼

2Lf n

where V is velocity, l is wavelength, f is the resonant frequency, L is the bar length, and n is a positive integer.

157

3.15 Waves in cylindrical rods: the resonant bar

In practice, the shear or extensional velocity is calculated from the observed resonant frequency, most often at the fundamental mode, where n ¼ 1.

Porous, fluid-saturated rods Biot’s theory has been used to extend Pochhammer’s method of analysis for fluidsaturated porous rods (Gardner, 1962; Berryman, 1983). The dependence of the velocity and attenuation of longitudinal waves on the skeleton and fluid properties is rather complicated. The motions of the solid and the fluid are partly parallel to the axis of the cylinder and partly along the radius. The dispersion relations are obtained from plane-wave solutions of Biot’s equations in cylindrical (r, y, z) coordinates. For an open (unjacketed) surface boundary condition the o–kz dispersion relation is given by (in the notation of Berryman, 1983) a11 a12 a13 Dopen ¼ a21 a22 0 ¼ 0 a31 a32 a33 a11 ¼

½ðC HÞkþ2 þ 2fr kz2 J0 ðkþ aÞ þ 2fr krþ J1 ðkþ aÞ=a ðþ Þ

a12 ¼

½ðH Cþ Þk2 2fr kz2 J0 ðk aÞ 2fr kr J1 ðk aÞ=a ðþ Þ

a13 ¼ 2fr ksr ½ksr J0 ðksr aÞ J1 ðksr aÞ=a a21 ¼

2 ðM CÞkþ J0 ðkþ aÞ ðþ Þ

a22 ¼

ðC Mþ Þk2 J0 ðk aÞ ðþ Þ

a23 ¼ 0 a31 ¼

2ifr kz krþ J1 ðkþ aÞ ðþ Þ

a32 ¼

2ifr kz kr J1 ðk aÞ ðþ Þ

a33 ¼ fr ðks2 2kz2 Þksr J1 ðksr aÞ=ðikz Þ 2 2 ¼ k kz2 ; kr

ksr2 ¼ ks2 kz2

ks2 ¼ o2 ð fl 2 =qÞfr qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 kþ ¼ 2 b þ f ðb f Þ2 þ 4cd

158

Seismic wave propagation

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 k ¼ 12 b þ f þ ðb f Þ2 þ 4cd b ¼ o2 ðM fl CÞ= c ¼ o2 ðfl M qCÞ= d ¼ o2 ðfl H CÞ= f ¼ o2 ðqH fl CÞ= ¼ MH C2 ¼ d=ðk2 bÞ ¼ ðk2 f Þ=c 4 ðK0 Kfr Þ2 H ¼ Kfr þ fr þ ðD Kfr Þ 3 C¼

ðK0 Kfr ÞK0 ðD Kfr Þ

M¼

K02 ðD Kfr Þ

D ¼ K0 ½1 þ ðK0 =Kfl 1Þ ¼ ð1 Þ0 þ fl q¼

fl iFð Þ o

where Kfr , mfr ¼ effective bulk and shear moduli of rock frame: either the dry frame or the highfrequency unrelaxed “wet frame” moduli predicted by the Mavko–Jizba squirt theory K0 ¼ bulk modulus of mineral material making up rock Kfl ¼ effective bulk modulus of pore fluid f ¼ porosity r0 ¼ mineral density rfl ¼ fluid density a ¼ tortuosity parameter (always greater than 1) ¼ viscosity of the pore fluid k ¼ absolute permeability of the rock o ¼ angular frequency of the plane wave. The viscodynamic operator F(z) incorporates the frequency dependence of viscous drag and is defined by Fð Þ ¼

1

Tð Þ 4 1 þ 2iTð Þ=

159

3.15 Waves in cylindrical rods: the resonant bar

Tð Þ ¼

ber0 ð Þ þ ibei0 ð Þ ei3=4 J1 ð ei=4 Þ ¼ berð Þ þ ibeið Þ J0 ð ei=4 Þ

¼ ðo=or Þ1=2 ¼

2 1=2 oh fl

where ber( ) and bei( ) are the real and imaginary parts of the Kelvin function, respectively, Jn( ) is a Bessel function of order n, and h is the pore-size parameter. The pore-size parameter h depends on both the dimensions and shape of the pore space. Stoll (1974) found that values of between 16 and 17 for the mean grain diameter gave good agreement with experimental data from several investigators. For spherical grains, Hovem and Ingram (1979) obtained h ¼ fd/[3(1 – f)], where d is the grain diameter. This dispersion relation gives the same results as Gardner (1962). When the surface pores are closed (jacketed) the resulting dispersion relation is a11 a12 a13 Dclosed ¼ a31 a32 a33 ¼ 0 a41 a42 a43 a41 ¼

krþ J1 ðkþ aÞ ; þ

a42 ¼

kr þ J1 ðk aÞ ; þ

a43 ¼ ksr J1 ðksr aÞfl =q

For open-pore surface conditions the vanishing of the fluid pressure at the surface of the cylinder causes strong radial motion of the fluid relative to the solid. This relative motion absorbs energy, causing greater attenuation than would be present in a plane longitudinal wave in an extended porous saturated medium (White, 1986). Narrow stop bands and sharp peaks in the attenuation can occur if the slow P-wave has wavelength l < 2.6a. Such stop bands do not exist in the case of the jacketed, closed-pore surface. A slow extensional wave propagates under jacketed boundary conditions but not under the open-surface condition.

Uses The results described in this section can be used to model wave propagation and geometric dispersion in resonant bar experiments.

Assumptions and limitations The results described in this section assume the following: an isotropic, linear, homogeneous, and elastic–poroelastic rod of solid circular cross-section; for elastic rods the cylindrical surface is taken to be free of tractions; and for porous rods, unjacketed and jacketed surface boundary conditions are assumed.

160

Seismic wave propagation

3.16

Waves in boreholes Synopsis Elastic wave propagation in the presence of a cylindrical fluid-filled borehole involves different modes caused by internal refraction, constructive interference and trapping of wave energy in the borehole. The theory of borehole wave propagation has been described in the books by White (1983), Paillet and Cheng (1991), and Tang and Cheng (2004), where references to the original literature may be found. The dispersion characteristics of borehole wave modes depend strongly on the shear wave velocity of the elastic medium surrounding the borehole. Two scenarios are usually considered: “fast” formation when the S-wave velocity in the formation is greater than the borehole fluid velocity and “slow” formation when the S-wave velocity in the formation is slower than the borehole fluid velocity. Wave modes are guided by the borehole only when the formation S-wave velocity is greater than the phase velocity of the modes; otherwise the modes become leaky modes, radiating energy into the formation. In fast formations, pseudo-Rayleigh modes or shear normal modes exist above characteristic cut-off frequencies. The pseudo-Rayleigh mode is strongly dispersive and is a combined effect of reflected waves in the fluid and critical refraction along the borehole walls. The phase velocity of the pseudo-Rayleigh wave at the cut-off frequency drops from the shear wave velocity of the formation and approaches the fluid velocity at high frequencies. In slow formations pseudo-Rayleigh modes do not exist. Stoneley waves in boreholes refer to waves along the borehole interface. At low frequencies the Stoneley waves are referred to as tube waves. Stoneley waves exist at all frequencies and in both fast and slow formations. “Leaky P” modes exist in slow formations and are dominated by critical refraction of P-waves at the borehole wall. They lose energy by conversion to shear waves. Higher-order modes include the dipole or flexural mode and the quadrupole or screw mode (Tang and Cheng, 2004).

Isotropic elastic formation A low-frequency (static) analysis for a thick-walled elastic tube of inner radius b and outer radius a, with Young’s modulus E, and Poisson ratio n, containing fluid with bulk modulus B, and density r, gives the speed of tube waves as (White, 1983) 1 1 1=2 ct ¼ þ B M M¼

Eða2 b2 Þ 2½ð1 þ nÞða2 þ b2 Þ 2nb2

For a thin-walled tube with thickness h ¼ ða bÞ; a b; and M Eh=2b: For a borehole in an infinite solid a b and the speed of the tube wave in the lowfrequency limit is

161

3.16 Waves in boreholes

1 1 1=2 ct ¼ þ B where m is the shear modulus of the formation. For a borehole with a casing of thickness h, inner radius b, and Young’s modulus E, the tube wave velocity (in the low-frequency limit) is 1=2 1 1 ct ¼ þ B þ Eh=2b The o–k dispersion relation gives a more complete description of the modes. The dispersion relation (or period equation) yields characteristic cut-off frequencies and phase velocities as a function of frequency for the different modes (Paillet and Cheng, 1991; Tang and Cheng, 2004). For a cylindrical borehole of radius R, in an infinite, isotropic, elastic formation with Lame´ constants l and m, and density r, and openhole boundary conditions at the interface, the dispersion relation is given by (in the notation of Tang and Cheng, 2004) M11 M Dðo; kÞ ¼ 21 M31 M41

M11 ¼

M12 M22 M32 M42

M13 M23 M33 M43

M14 M24 ¼0 M34 M44

n In ð fRÞ fInþ1 ð fRÞ R

M12 ¼ pY1 ð pRÞ n M13 ¼ Kn ðsRÞ R M14 ¼ iksY1 ðsRÞ M21 ¼ f o2 In ð fRÞ

2p 2 Y2 ð pRÞ M22 ¼ 2k2 2 o2 Kn ðpRÞ þ R M23 ¼

2ns2 Y3 ðsRÞ R

M24 ¼ 2iks2 2 Kn ðsRÞ þ M31 ¼ 0 M32 ¼

2np 2 Y3 ðpRÞ R

M33 ¼ s2 2 Y4 ðsRÞ M34 ¼

2ikns 2 Y3 ðsRÞ R

2iks 2 Y2 ðsRÞ R

162

Seismic wave propagation

M41 ¼ 0 M42 ¼ 2ikp2 Y1 ðpRÞ ikn2 Kn ðsRÞ R

2 ¼ k þ s2 s2 Y1 ðsRÞ

M43 ¼ M44

where In and Kn are the modified Bessel functions of the first and second kind, respectively, of order n. The azimuthal order number, n, controls the azimuthal variations, with n ¼ 0, 1, and 2 corresponding to monopole, dipole, and quadrupole modes, respectively. In the above expressions Yi denotes the following combinations of modified Bessel functions: n Y1 ð xÞ ¼ Kn ðxÞ þ Knþ1 ð xÞ x nðn 1Þ Y2 ð x Þ ¼ Kn ð xÞ þ Knþ1 ð xÞ x 1n Kn ðxÞ þ Knþ1 ð xÞ Y3 ð x Þ ¼ x 2nðn 1Þ 2 Kn ðxÞ þ Knþ1 ð xÞ Y4 ð x Þ ¼ 1 þ x2 x Other terms in the expressions are: o ¼ angular frequency k ¼ axial wavenumber

1=2 p ¼ k2 k2 ¼ compressional radial wavenumber

2 1=2 ¼ shear radial wavenumber s ¼ k k2 1=2

¼ radial wavenumber in the borehole fluid f ¼ k2 kf2 k ¼ o= ¼ compressional wavenumber k ¼ o= ¼ shear wavenumber kf ¼ o=f ¼ acoustic wavenumber in borehole fluid where a and b are the P- and S-wave velocities in the formation, and f is the compressional wave velocity in the borehole fluid. From the roots of the dispersion relation Dðo; kÞ ¼ 0, the phase and group velocities of the wave modes are obtained as Vphase ¼ Vgroup ¼

o kðoÞ

do @D @D ¼ dk @k @o

The roots and the partial derivatives have to be calculated numerically.

Transversely isotropic (TI) elastic formation The elastic properties of the transversely isotropic formation are given by the five independent components of the elastic stiffness tensor (c11, c13, c33, c44, c66), and

163

3.16 Waves in boreholes

r denotes the density. The TI symmetry axis is along the z-axis and the borehole axis coincides with the symmetry axis. The borehole radius is R, with borehole fluid density f ; and compressional wave velocity f : For open-hole boundary conditions the dispersion relation is given by (Tang and Cheng, 2004) Q11 Q12 Q13 Q14 Q21 Q22 Q23 Q24 Dðo; kÞ ¼ ¼0 Q31 Q32 Q33 Q34 Q Q Q Q 41

42

43

44

n Q11 ¼ In ð fRÞ fInþ1 ð fRÞ R

Q12 ¼ ð1 þ ika0 Þqp Y1 qp R Q13 ¼

n Kn ðqsh RÞ R

Q14 ¼ ðik þ b0 Þqsv Y1 ðqsv RÞ Q21 ¼ f o2 In ð fRÞ h i

Q22 ¼ c11 q2p c13 k2 þ ðc11 c13 Þika0 q2p Kn qp R þ

2c66 qp ð1 þ ika0 ÞY2 qp R R

2c66 nqsh Y3 ðqsh RÞ R

¼ c11 q2sv c13 k2 b0 þ ðc11 c13 Þikq2sv Kn ðqsv RÞ

Q23 ¼ Q24

þ

2c66 qsv ðik þ b0 ÞY2 ðqsv RÞ R

Q31 ¼ 0 Q32 ¼

2c66 nqp ð1 þ ika0 ÞY3 qp R R

Q33 ¼ c66 q2sh Y4 ðqsh RÞ 2c66 nqsv ðik þ b0 ÞY3 ðqsv RÞ R h i

¼ c44 qp 2ik a0 k2 þ q2p Y1 qp R

Q34 ¼ Q42

164

Seismic wave propagation

iknc44 Kn ðqsh RÞ R

¼ c44 qsv k2 þ q2sv 2ikb0 Y1 ðqsv RÞ

Q43 ¼ Q44

a0 ¼

1 ðc13 þ 2c44 Þk2 c11 q2p o2 ik c44 k2 ðc11 c13 c44 Þq2p o2

c44 k2 ðc11 c13 c44 Þq2sv o2 ðc13 þ 2c44 Þk2 c11 q2sv o2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c44 k2 o2 qsh ¼ c66 b0 ¼ ik

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V þ V 2 4UW qp ¼ o 2U sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V V 2 4UW qsv ¼ o 2U

1=2 f ¼ k2 kf2 ¼ radial wavenumber in the borehole fluid kf ¼ o=f ¼ acoustic wavenumber in the borehole fluid U ¼ c11 c44

V ¼ ðc11 þ c44 Þ c11 c33 c213 2c13 c44 k2 =o2

W ¼ c33 c44 =c44 k2 =o2 =c33 k2 =o2 where In and Kn are the modified Bessel functions of the first and second kind, respectively, of order n. The azimuthal order number, n, controls the azimuthal variations, with n ¼ 0, 1, and 2 corresponding to monopole, dipole, and quadrupole modes, respectively. In the above expressions Yi denotes the same combinations of modified Bessel functions as described for the isotropic case.

Isotropic, poroelastic, permeable formation Borehole wave in permeable formations interacts with the formation permeability through the propagation of Biot slow waves in the pore fluid. The Biot–Rosenbaum theory (Rosenbaum, 1974) was the first to model monopole excitation in an open borehole within an isotropic poroelastic formation. The theory was later extended to transversely isotropic poroelastic formations with multipole excitations (Schmitt, 1989). The Biot–Rosenbaum model solves the Biot poroelastic equations for a cylindrical borehole geometry and boundary conditions. For the monopole case in an open borehole of radius R, within a permeable formation, the dispersion relation is given by (Tang and Cheng, 2004)

165

3.16 Waves in boreholes

N11 N21 Dðo; kÞ ¼ N 31 N41

N12

N13

N22

N23

N32

N33

N42

N43

N14 N24 ¼0 N34 N44

N11 ¼ fI1 ð fRÞ N12 ¼ qfast ð1 þ bfast ÞK1 ðqfast RÞ N13 ¼ qslow ð1 þ bslow ÞK1 ðqslow RÞ N14 ¼ ikð1 þ bs ÞK1 ðqs RÞ N21 ¼ f o2 I0 ð fRÞ

N22 ¼ 2 q2fast K0 ðqfast RÞ þ qfast K1 ðqfast RÞ=R 2 þ kfast l þ ð þ bfast Þ K0 ðqfast RÞ 2

N23 ¼ 2 qslow K0 ðqslow RÞ þ qslow K1 ðqslow RÞ=R 2 þ kslow l þ ð þ bslow Þ K0 ðqslow RÞ 2

N24 ¼ 2ik qs K0 ðqs RÞ þ qs K1 ðqs RÞ=R N31 ¼ 0 N32 ¼ 2ikqfast K1 ðqfast RÞ N33 ¼ 2ikqslow K1 ðqslow RÞ

2 qs K1 ðqs RÞ N34 ¼ 2k2 kshear N41 ¼ f o2 I0 ð fRÞ 2 ½ð þ bfast Þ=K0 ðqfast RÞ N42 ¼ kfast 2 N43 ¼ kslow ½ð þ bslow Þ=K0 ðqslow RÞ

N44 ¼ 0 where In and Kn are the modified Bessel functions of the first and second kind, respectively, of order n, k is the axial wavenumber along the borehole axis, and o is the angular frequency. The radial wavenumbers are given by

166

Seismic wave propagation

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 k2 kfast qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 qslow ¼ k2 kslow qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 qs ¼ k2 kshear qfast ¼

1=2 ¼ the radial wavenumber in the borehole fluid f ¼ k2 kf2 kf ¼ o=f ¼ the acoustic wavenumber in the borehole fluid sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ bfast pf = kfast ¼ kp0 1 bfast =b0 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ bslow pf = kslow ¼ kp0 1 bslow =b0 o kp0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðl þ 2 þ 2 = Þ= pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi 1h b fast ¼ c c2 4ð1 cÞ=b0 slow 2 ðl þ 2 þ 2 = Þ

bs = pf b0 c¼ þ bs b0 ¼

bs ¼ pf o2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kshear ¼ o= = þ 2pf o2

¼ 1 Kdry =Km ¼ ð Þ=Km þ =Kpf ¼ ð1 Þ0 þ pf Kdry ¼ dry (drained) bulk modulus l; ¼ Lame´ constants of the dry (drained) porous formation Km ¼ mineral bulk modulus r0 ¼ mineral density Kpf ¼ pore fluid bulk modulus pf ¼ pore fluid density f ¼ porosity

¼ iðoÞ=o ¼ fluid viscosity ðoÞ ¼ dynamic permeability

167

3.16 Waves in boreholes

The dynamic permeability measures fluid transport in a porous medium under dynamic wave excitation. It is given by (Johnson et al., 1987) ðoÞ ¼

1

0

1=2 2 2 2 4i 0 pf o=ð 2 Þ it0 pf o=ðÞ

where 0 is the static Darcy permeability, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃtﬃ is tortuosity, L is a measure of the pore size given approximately by 80 =. In the low-frequency limit the dynamic permeability goes to the static Darcy permeability, while in the high-frequency limit

ðoÞ ! i= tpf o After finding the roots of the dispersion equation numerically, the Stoneley wave phase velocity and attenuation are given by Vstoneley ¼ o=ReðkÞ Q1 stoneley ¼ 2 ImðkÞ=ReðkÞ Tang et al. (1991) present a simplified Biot–Rosenbaum model by decoupling the elastic and flow effects in the full Biot–Rosenbaum model. This reduces the complexity of root finding to an approximate analytical expression for the Stoneley phase wavenumber k: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ u qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K Rpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u io=D þ ke2 2ipf oðoÞR 1 t 2 2 k¼ ke þ io=D þ ke

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðR2 a2 Þ K0 R io=D þ ke2 D¼

1þ

ðoÞKpf =ðÞ h i 4=3Kdry ðlþ2Þ 1þ Km

Kpf ðlþ2Þ

a ¼ tool radius where D is the diffusivity modified to correct for the elasticity of the medium, and K0 and K1 are modified Bessel functions of the second kind. The parameter ke in the above expression is the elastic Stoneley wavenumber, which can be estimated from the density and effective elastic moduli (or P- and S-wave velocities) of the equivalent formation (Tang and Cheng, 2004). The tool radius a can be set to 0 to model a fluid-filled borehole without a tool. A low-frequency approximation to the Stoneley wave slowness (assuming a ¼ 0) is given by (Tang and Cheng, 2004) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2 K p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1 R io=D pf pf 2ipf 0 io=D pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ5 þ S2 ¼ þ4 Kpf oR K0 R io=D

168

Seismic wave propagation

The first term is the elastic component while the second term represents the effect of formation permeability and fluid flow. The second term is usually small compared with the first term. Therefore it is difficult to estimate formation permeability using this expression for Stoneley wave slowness. The Stoneley wavenumber k obtained from the simplified theory matches very well the exact wavenumber for fast formations. For slow formations, a better match is obtained by applying an empirical correction to account for borehole compliance. With the empirical correction, the expression for the Stoneley wavenumber is (Tang and Cheng, 2004) vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 u þ k2 t 2 2ipf oðoÞR io=D þ ke K1 Rpio=D ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃeﬃ

k ¼ ke þ 2 2 1þB ðR a Þ K0 R io=D þ k2 e

I1 ðfe RÞ I0 ðfe RÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fe ¼ ke2 kf2 ¼ the radial wavenumber for an equivalent elastic formation kf ¼ o=f ¼ the acoustic wavenumber in the borehole fluid ¼ VS =f ¼ the ratio of the formation S-wave velocity to the acoustic velocity in the borehole fluid. B ¼ fe R

Uses The results described in this section can be used to model wave propagation and geometric dispersion in boreholes.

Assumptions and limitations The results described in this section assume the following: isotropic or TI, linear, homogeneous, elastic or poroelastic formation; a borehole of circular cross-section; for the TI dispersion relation, the borehole axis is along the symmetry axis; and open-hole boundary conditions at the borehole interface.

Extensions The Biot–Rosenbaum theory has been extended to anisotropic poroelastic formations with multipole excitations (Schmitt, 1989).

4

Effective elastic media: bounds and mixing laws

4.1

Hashin–Shtrikman–Walpole bounds Synopsis If we wish to predict the effective elastic moduli of a mixture of grains and pores theoretically, we generally need to specify: (1) the volume fractions of the various phases, (2) the elastic moduli of the various phases, and (3) the geometric details of how the phases are arranged relative to each other. If we specify only the volume fractions and the constituent moduli, the best we can do is to predict the upper and lower bounds (shown schematically in Figure 4.1.1). At any given volume fraction of constituents the effective modulus will fall between the bounds (somewhere along the vertical dashed line in the plot of bulk modulus, Figure 4.1.1), but its precise value depends on the geometric details. We use, for example, terms like “stiff pore shapes” and “soft pore shapes.” Stiffer shapes cause the value to be higher within the allowable range; softer shapes cause the value to be lower. The best bounds for an isotropic linear elastic composite, defined as giving the narrowest possible range without specifying anything about the geometries of the constituents, are the Hashin–Shtrikman bounds (Hashin and Shtrikman, 1963). When there are only two constituents, the bounds are written as K HS ¼ K1 þ mHS ¼ m1 þ

f2 1

ðK2 K1 Þ

þ f1 ðK1 þ 43m1 Þ1 f2

ðm2 m1 Þ

1

þ 2f1 ðK1 þ 2m1 Þ=½5m1 ðK1 þ 43 m1 Þ

ð4:1:1Þ ð4:1:2Þ

where K1 and K2 are the bulk moduli of individual phases; m1 and m2 are the shear moduli of individual phases; and f1 and f2 are the volume fractions of individual phases. With equations (4.1.1) and (4.1.2), upper and lower bounds are computed by interchanging which material is termed 1 and which is termed 2. The expressions yield the upper bound when the stiffest material is termed 1 and the lower bound when the softest material is termed 1. Use of the above equations assumes that the constituent with the larger bulk modulus also has the larger shear modulus and the 169

K1 Upper bound Softer Stiffer K2

Lower bound

Effective shear modulus

Effective elastic media: bounds and mixing laws

Effective bulk modulus

170

Volume fraction of material 2

m1

Upper bound

Lower bound m2 Volume fraction of material 2

Figure 4.1.1 Schematic representation of the upper and lower bounds on the elastic bulk and shear moduli.

constituent with the smaller bulk modulus also has the smaller shear modulus. Slightly more general forms, sometimes called the Walpole (1966) bounds or the Hashin–Shtrikman–Walpole bounds, can be written as K HS ¼ K1 þ

f2 ðK2 K1 Þ

1

mHS ¼ m1 þ 1

ðm2 m1 Þ

þ f1 ðK1 þ 43mm Þ1

f2

mm 9Km þ 8mm 1 þ f1 m 1 þ 6 Km þ 2mm

ð4:1:3Þ ð4:1:4Þ

where the subscripts 1 and 2 again refer to the properties of the two components. Equations (4.1.3) and (4.1.4) yield the upper bound when Km and mm are the maximum bulk and shear moduli of the individual constituents, and the lower bound when Km and mm are the minimum bulk and shear moduli of the constituents. The maximum (minimum) shear modulus might come from a different constituent from the maximum (minimum) bulk modulus. This would be the case, for example, for a mixture of calcite ðK ¼ 71; m ¼ 30 GPaÞ and quartz ðK ¼ 37; m ¼ 45 GPaÞ: Equation (4.1.4) reduces to equation (4.1.2) when one constituent has both the maximum bulk and shear moduli, while the other constituent has the minimum bulk and shear moduli. The physical interpretation of the Hashin–Shtrikman bounds for bulk modulus is shown schematically in Figure 4.1.2. The space is filled by an assembly of spheres of material 2, each surrounded by a shell of material 1. Each sphere and its shell have precisely the volume fractions f1 and f2. The upper bound is realized when the stiffer material forms the shell; the lower bound is realized when it is in the core. The physical interpretation implies a very wide distribution of sizes of the coated spheres such that they fill all the space. A more general form of the Hashin–Shtrikman–Walpole bounds, which can be applied to mixtures of more than two phases (Berryman, 1995), can be written as K HSþ ¼ ðmmax Þ;

K HS ¼ ðmmin Þ

mHSþ ¼ ðzðKmax ; mmax ÞÞ;

mHS ¼ ðzðKmin ; mmin ÞÞ

171

4.1 Hashin–Shtrikman–Walpole bounds

Figure 4.1.2 Physical interpretation of the Hashin–Shtrikman bounds for the bulk modulus of a two-phase material.

where

*

+1 1 4 z ðzÞ ¼ 4 3 KðrÞ þ 3z 1 1 ðzÞ ¼ z mðrÞ þ z m 9K þ 8m zðK; mÞ ¼ 6 K þ 2m The brackets hi indicate an average over the medium, which is the same as an average over the constituents weighted by their volume fractions. Compute the Hashin–Shtrikman upper and lower bounds on the bulk and shear moduli for a mixture of quartz, calcite, and water. The porosity (water fraction) is 27%, quartz is 80% by volume of the solid fraction, and calcite is 20% by volume of the solid fraction. The moduli of the individual constituents are: Kquartz ¼ 36 GPa

Kcalcite ¼ 75 GPa Kwater ¼ 2:2 GPa;

mquartz ¼ 45 GPa

mcalcite ¼ 31 GPa mwater ¼ 0 GPa

Hence mmin ¼ 0 GPa; mmax ¼ 45 GPa; K

HS

Kmin ¼ 2:2 GPa; and Kmax ¼ 75 GPa

¼ ðmmin Þ ð1 Þð0:8Þ ð1 Þð0:2Þ 1 ¼ þ þ 2:2 36:0 75:0 ¼ 7:10 GPa

172

Effective elastic media: bounds and mixing laws

K HSþ ¼ ðmmax Þ "

#1 ð1 Þð0:8Þ ð1 Þð0:2Þ 4

þ

þ

¼ 45 3 2:2 þ 43 45 36:0 þ 43 45 75:0 þ 43 45:0 ¼ 26:9 GPa 45 9 75 þ 8 45 ¼ 47:0 GPa zðKmax ; mmax Þ ¼ 6 75 þ 2 45 zðKmin ; mmin Þ ¼ 0 GPa mHSþ ¼ ðzðKmax ; mmax ÞÞ

ð1 Þð0:8Þ ð1 Þð0:2Þ 1 ¼ 47:0 þ þ 47:0 45:0 þ 47:0 31:0 þ 47:0 ¼ 24:6 GPa mHS ¼ ðzðKmin ; mmin ÞÞ ¼0

The separation between the upper and lower bounds depends on how different the constituents are. As shown in Figure 4.1.3, the bounds are often quite similar when mixing solids, for the moduli of common minerals are usually within a factor of 2 of each other. Because many effective-medium models (e.g., Biot, Gassmann, Kuster– Tokso¨z, etc.) assume a homogeneous mineral modulus, it is often useful (and adequate) to represent a mixed mineralogy with an “average mineral” equal to either one of the bounds or to their average (MHSþ þ MHS–)/2. On the other hand, when the constituents are quite different (such as minerals and pore fluids), the bounds become quite separated, and we lose some of the predictive value. Note that when mmin ¼ 0, KHS– is the same as the Reuss bound. In this case, the Reuss or Hashin–Shtrikman lower bound exactly describes the moduli of a suspension of grains in a pore fluid (see Section 4.2 on Voigt–Reuss bounds and also Section 4.3 on Wood’s relation). These also describe the moduli of a mixture of fluids or gases, or both. When all phases have the same shear modulus, m ¼ mmin ¼ mmax, the upper and lower bounds become identical, and we obtain the expression from Hill (1963) for the effective bulk modulus of a composite with uniform shear modulus (see Section 4.5 on composites with uniform shear modulus). The Hashin–Shtrikman bounds are computed for the bulk and shear moduli. When bounds on other isotropic elastic constants are needed, the correct procedure is to

4.1 Hashin–Shtrikman–Walpole bounds

80

80 Calcite + Dolomite

78 76

Upper bound

74

Lower bound

72 70 68

Calcite + Water

70 Bulk modulus (GPa)

Bulk modulus (GPa)

173

60 50 40

Upper bound

30 Lower bound

20 10

0

0.2

0.4 0.6 0.8 Fraction of dolomite

0

1

0

0.2

0.4 0.6 Porosity

0.8

1

Figure 4.1.3 Hashin–Shtrikman bounds for similar constituents (left) and dissimilar constituents (right).

K

+

K

–

n +, l+

ν

E+

E

n = 12 – 6K +

E

n = 12 – 6K –

0.5 0 E–

E

n ,l –

–

n = –1+ m–

m+ –1

l = K +–2m /3

λ

n = –1 +

E – 2m

E + 2m

l = K ––2m /3 m–

m+

Figure 4.1.4 Schematic representation of the Hashin–Shtrikman bounds in the (K, m), (n, E) and (l, m) planes.

compute the bounds for bulk and shear moduli and then compute the other elastic constants from these. Figure 4.1.4 shows schematically the bounds in the planes (K, m), (n, E), and (l, m). In each case, the allowed pairs of elastic constants lie within the polygons. Notice how the upper and lower bounds for (K, m) are uncoupled, i.e., any value K K K þ can exist with any value m m mþ . In contrast, the bounds on ðn; EÞ both depend on (K, m) – the upper and lower bounds on n depend on the value of E, and vice versa. Similarly, the bounds on l are coupled to the value of m. K and m are more “orthogonal” than l and m.

Uses The bounds described in this section can be used for the following: to compute the estimated range of average mineral modulus for a mixture of mineral grains; to compute the upper and lower bounds for a mixture of mineral and pore fluid.

174

Effective elastic media: bounds and mixing laws

Assumptions and limitations The bounds described in this section apply under the following conditions: each constituent is isotropic, linear, and elastic; the rock is isotropic, linear, and elastic.

4.2

Voigt and Reuss bounds Synopsis If we wish to predict theoretically the effective elastic moduli of a mixture of grains and pores, we generally need to specify: (1) the volume fractions of the various phases, (2) the elastic moduli of the various phases, and (3) the geometric details of how the phases are arranged relative to each other. If we specify only the volume fractions and the constituent moduli, the best we can do is to predict the upper and lower bounds (shown schematically in Figure 4.2.1). At any given volume fraction of constituents, the effective modulus will fall between the bounds (somewhere along the vertical dashed line in Figure 4.2.1), but its precise value depends on the geometric details. We use, for example, terms like “stiff pore shapes” and “soft pore shapes.” Stiffer shapes cause the value to be higher within the allowable range; softer shapes cause the value to be lower. The simplest, but not necessarily the best, bounds are the Voigt and Reuss bounds. (See also Section 4.1 on Hashin–Shtrikman bounds, which are narrower.) The Voigt upper bound of the effective elastic modulus, MV, of N phases is MV ¼

N X

fi Mi

i¼1

where fi is the volume fraction of the ith phase and Mi is the elastic modulus of the ith phase. The Voigt bound is sometimes called the isostrain average because it gives the ratio of the average stress to the average strain when all constituents are assumed to have the same strain. The Reuss lower bound of the effective elastic modulus, MR, is (Reuss, 1929) N X 1 fi ¼ Mi MR i¼1

The Reuss bound is sometimes called the isostress average because it gives the ratio of the average stress to the average strain when all constituents are assumed to have the same stress. When one of the constituents is a liquid or a gas with zero shear modulus, the Reuss average bulk and shear moduli for the composite are exactly the same as given by the Hashin–Shtrikman lower bound.

4.3 Wood’s formula

Effective bulk modulus

175

K1 Upper bound Softer Stiffer Lower bound

K2

Volume fraction of material 2

Figure 4.2.1 Effect of changing the volume fraction of constituent materials. The bulk modulus will move along the vertical dotted line between the two bounds.

The Reuss average exactly describes the effective moduli of a suspension of solid grains in a fluid. It also describes the moduli of “shattered” materials in which solid fragments are completely surrounded by the pore fluid. When all constituents are gases or liquids, or both, with zero shear moduli, the Reuss average gives the effective moduli of the mixture exactly. In contrast to the Reuss average, which describes a number of real physical systems, real isotropic mixtures can never be as stiff as the Voigt bound (except for the single phase end-members). Mathematically M in the Reuss average formula can represent any modulus: K, m, E, and so forth. However, it makes most sense to compute only the Reuss averages of the shear modulus, M ¼ m, and the bulk modulus, M ¼ K, and then compute the other moduli from these.

Uses The methods described in this section can be used for the following purposes: to compute the estimated range of the average mineral modulus for a mixture of mineral grains; to compute the upper and lower bounds for a mixture of mineral and pore fluid.

Assumptions and limitations The methods described in this section presuppose that each constituent is isotropic, linear, and elastic.

4.3

Wood’s formula Synopsis In a fluid suspension or fluid mixture, where the heterogeneities are small compared with a wavelength, the sound velocity is given exactly by Wood’s (1955) relation

176

Effective elastic media: bounds and mixing laws

sﬃﬃﬃﬃﬃﬃ KR V¼ where KR is the Reuss (isostress) average of the composite N X 1 fi ¼ K KR i¼1 i

and r is the average density defined by ¼

N X

fi i

i¼1

fi, Ki, and ri are the volume fraction, bulk moduli, and densities of the phases, respectively. Use Wood’s relation to estimate the speed of sound in a water-saturated suspension of quartz particles at atmospheric conditions. The quartz properties are Kquartz ¼ 36 GPa and rquartz ¼ 2.65 g/cm3. The water properties are Kwater ¼ 2.2 GPa and rwater ¼ 1.0 g/cm3. The porosity is f¼ 0.40. The Reuss average bulk modulus of the suspension is given by 1 1 0:4 0:6 1 þ ¼ ¼ 5:04 GPa KReuss ¼ þ Kwater Kquartz 2:2 36 The density of the suspension is ¼ water þ ð1 Þquartz ¼ 0:4 1:0 þ 0:6 2:65 ¼ 1:99 g=cm3 This gives a sound speed of pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V ¼ K= ¼ 5:04=1:99 ¼ 1:59 km=s

Use Wood’s relation to estimate the speed of sound in a suspension of quartz particles in water with 50% saturation of air at atmospheric conditions. The quartz properties are Kquartz ¼ 36 GPa and rquartz ¼ 2.65 g/cm3. The water properties are Kwater ¼ 2.2 GPa and rwater ¼ 1.0 g/cm3. The air properties are Kair ¼ 0.000131 GPa and rair ¼ 0.00119 g/cm3. The porosity is f ¼ 0.40. The Reuss average bulk modulus of the suspension is given by 0:5 0:5 1 1 KReuss ¼ þ þ Kwater Kair Kquartz 0:5 0:4 0:5 0:4 0:6 1 ¼ ¼ 0:000 65 GPa þ þ 2:2 0:000 131 36

177

4.4 Voigt–Reuss–Hill average moduli estimate

The density of the suspension is ¼ 0:5water þ 0:5air þ ð1 Þquartz ¼ 0:5 0:4 1:0 þ 0:5 0:4 0:00119 þ 0:6 2:65 ¼ 1:79 g=cm3 This gives a sound speed of pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V ¼ K= ¼ 0:00065=1:79 ¼ 0:019 km=s

Uses Wood’s formula may be used to estimate the velocity in suspensions.

Assumptions and limitations Wood’s formula presupposes that composite rock and each of its components are isotropic, linear, and elastic.

4.4

Voigt–Reuss–Hill average moduli estimate Synopsis The Voigt–Reuss–Hill average is simply the arithmetic average of the Voigt upper bound and the Reuss lower bound. (See the discussion of the Voigt–Reuss bounds in Section 4.2.) This average is expressed as MVRH ¼

MV þ MR 2

where MV ¼

N X i¼1

fi Mi

N X 1 fi ¼ Mi MR i¼1

The terms fi and Mi are the volume fraction and the modulus of the ith component, respectively. Although M can be any modulus, it makes most sense for it to be the shear modulus or the bulk modulus. The Voigt–Reuss–Hill average is useful when an estimate of the moduli is needed, not just the allowable range of values. An obvious extension would be to average, instead, the Hashin–Shtrikman upper and lower bounds. This resembles, but is not exactly the same as the average of the algebraic and harmonic means of velocity used by Greenberg and Castagna (1992) in their empirical VP–VS relation (see Section 7.9).

178

Effective elastic media: bounds and mixing laws

Uses The Voigt–Reuss–Hill average is used to estimate the effective elastic moduli of a rock in terms of its constituents and pore space.

Assumptions and limitations The following limitation and assumption apply to the Voigt–Reuss–Hill average: the result is strictly heuristic. Hill (1952) showed that the Voigt and Reuss averages are upper and lower bounds, respectively. Several authors have shown that the average of these bounds can be a useful and sometimes accurate estimate of rock properties; the rock is isotropic.

4.5

Composite with uniform shear modulus Synopsis Hill (1963) showed that when all of the phases or constituents in a composite have the

same shear modulus, m, the effective P-wave modulus, Meff ¼ Keff þ 43 meff , is given exactly by * + N X 1 xi 1 ¼ ¼ ðKeff þ 43 meff Þ K þ 43 m ðKi þ 43 mÞ i¼1 where xi is the volume fraction of the ith component, Ki is its bulk modulus, and hi refers to the volume average. Because meff ¼ mi ¼ m, any of the effective moduli can then be easily obtained from Keff and meff. This is obviously the same as 1 1 ¼ ðVP2 Þeff VP2 This striking result states that the effective moduli of a composite with uniform shear modulus can be found exactly if one knows only the volume fractions of the constituents independent of the constituent geometries. There is no dependence, for example, on ellipsoids, spheres, or other idealized shapes. Hill’s equation follows simply from the expressions for Hashin–Shtrikman bounds (see Section 4.1 on Hashin–Shtrikman bounds) on the effective bulk modulus: * + 1 1 ¼ K HS þ 43 mfmax g K þ 43 mfmax g min

min

where mmin and mmax are the minimum and maximum shear moduli of the various constituents, yielding, respectively, the lower and upper bounds on the bulk

179

4.6 Rock and pore compressibilities and some pitfalls

modulus, KHS. Any composite must have an effective bulk modulus that falls between the bounds. Because here m ¼ mmin ¼ mmax, the two bounds on the bulk modulus are equal and reduce to the Hill expression above. In the case of a mixture of liquids or gases, or both, where m ¼ 0 for all the constituents, the Hill’s equation becomes the well-known isostress equation or Reuss average: N X 1 xi ¼ ¼ K Keff i¼1 i

1 K

A somewhat surprising result is that a finely layered medium, where each layer is isotropic and has the same shear modulus but a different bulk modulus, is isotropic with a bulk modulus given by Hill’s equation. (See Section 4.13 on the Backus average.)

Uses Hill’s equation can be used to calculate the effective low-frequency moduli for rocks with spatially nonuniform or patchy saturation. At low frequencies, Gassmann’s relations predict no change in the shear modulus between dry and saturated patches, allowing this relation to be used to estimate K.

Assumptions and limitations Hill’s equation applies when the composite rock and each of its components are isotropic and have the same shear modulus.

4.6

Rock and pore compressibilities and some pitfalls Synopsis This section summarizes useful relations among the compressibilities of porous materials and addresses some commonly made mistakes. A nonporous elastic solid has a single compressibility b¼

1 @V V @s

where s is the hydrostatic stress (defined as being positive in tension) applied on the outer surface and V is the sample bulk volume. In contrast, compressibilities for porous media are more complicated. We have to account for at least two pressures (the

180

Effective elastic media: bounds and mixing laws

external confining pressure and the internal pore pressure) and two volumes (bulk volume and pore volume). Therefore, we can define at least four compressibilities. Following Zimmerman’s (1991a) notation, in which the first subscript indicates the volume change (b for bulk and p for pore) and the second subscript denotes the pressure that is varied (c for confining and p for pore), these compressibilities are 1 @Vb bbc ¼ Vb @c p bbp ¼

1 @Vb Vb @p c

1 @p bpc ¼ p @c p bpp

1 @p ¼ p @p c

Note that the signs are chosen to ensure that the compressibilities are positive when tensional stress is taken to be positive. Thus, for instance, bbp is to be interpreted as the fractional change in the bulk volume with respect to change in the pore pressure while the confining pressure is held constant. These are the dry or drained bulk and pore compressibilities. The effective dry bulk modulus is Kdry ¼ 1/bbc, and the drypore-space stiffness is Kf ¼ 1/bpc. In addition, there is the saturated or undrained bulk compressibility when the mass of the pore fluid is kept constant as the confining pressure changes: bu ¼

1 Ksat low f

¼

1 @Vb Vb @c m-fluid

This equation assumes that the pore pressure is equilibrated throughout the pore space, and the expression is therefore appropriate for very low frequencies. At high frequencies, with unequilibrated pore pressures, the appropriate bulk modulus is Ksat hi f calculated from some high-frequency theory such as the squirt, Biot, or inclusion models, or some other viscoelastic model. The subscript m-fluid indicates constant mass of the pore fluid. The moduli Kdry, Ksat low f, Ksat hi f, and Kf are the ones most useful in wavepropagation rock physics. The other compressibilities are used in calculations of subsidence caused by fluid withdrawal and reservoir-compressibility analyses. Some of the compressibilities can be related to each other by linear superposition and reciprocity. The well-known Gassmann’s equation relates Kdry to Ksat through the mineral and fluid bulk moduli K0 and Kfl. A few other relations are (for simple derivations, see Zimmerman, 1991a)

181

4.6 Rock and pore compressibilities and some pitfalls

bbp ¼ bbc

1 K0

bbp 1 . ¼ bbc ð1 þ Þ K0

bpc ¼ bpp

More on dry-rock compressibility The effective dry-rock compressibility of a homogeneous, linear, porous, elastic solid with any arbitrarily shaped pore space (sometimes called the “drained” or “frame” compressibility) can be written as 1 1 1 @p ¼ þ Kdry K0 Vb @c p or 1 1 ¼ þ Kdry K0 K

ð4:6:1Þ

where 1 1 @p ¼ K p @c p is defined as the dry pore-space compressibility (Kf is the dry pore-space stiffness), Kdry ¼ 1/bbc ¼ effective bulk modulus of dry porous solid K0 ¼ bulk modulus of intrinsic mineral material Vb ¼ total bulk volume p ¼ pore volume f ¼ p/Vb ¼ porosity sc, sp ¼ hydrostatic confining stress and pore stress (pore pressure) We assume that no inelastic effects such as friction or viscosity are present. These equations are strictly true, regardless of pore geometry and pore concentration.

Caution: “Dry rock” is not the same as gas-saturated rock

The dry-frame modulus refers to the incremental bulk deformation resulting from an increment of applied confining pressure while the pore pressure is held constant. This corresponds to a “drained” experiment in which pore fluids can flow freely in or out of the sample to ensure constant pore pressure. Alternatively, it can correspond to an undrained experiment in which the pore fluid has zero bulk

182

Effective elastic media: bounds and mixing laws

modulus and thus the pore compressions do not induce changes in pore pressure, which is approximately the case for an air-filled sample at standard temperature and pressure. However, at reservoir conditions (high pore pressure), the gas takes on a non-negligible bulk modulus and should be treated as a saturating fluid.

Caution

The harmonic average of the mineral and dry-pore moduli, which resembles equation (4.6.1) above, is incorrect: 1 ? 1 ¼ þ Kdry K0 K

ðincorrectÞ

This equation is sometimes “guessed” because it resembles the Reuss average, but it has no justification from elasticity analysis. It is also incorrect to write 1 ? 1 @ ¼ þ Kdry K0 @c

ðincorrectÞ

ð4:6:2Þ

The correct expression is 1 1 1 @ ¼ þ Kdry ð1 Þ K0 @c The incorrect equation (4.6.2) appears as an intermediate result in some of the classic literature of rock physics. The notable final results are still correct, for the actual derivations are done in terms of the pore volume change, @p =@c , and not @=@c . Not distinguishing between changes in differential pressure, sd ¼ sc – sp, and confining pressure, sc, can lead to confusion. Changing sc while sp is kept constant (dsp ¼ 0) is not the same as changing sc with dsp ¼ dsc (i.e., the differential stress is kept constant). In the first situation the porous medium deforms with the effective dry modulus Kdry. The second situation is one of uniform hydrostatic pressure outside and inside the porous rock. For this stress state, the rock deforms with the intrinsic mineral modulus K0. Not understanding this can lead to the following erroneous results: 1 ? 1 1 @ ¼ K0 Kdry ð1 Þ @c

ðincorrectÞ

or @ ? 1 1 ðincorrectÞ ¼ð1 Þ @c Kdry K0

183

4.7 Kuster and Tokso¨z formulation for effective moduli

Table 4.6.1 summarizes some correct and incorrect relations. Table 4.6.1

Correct and incorrect versions of the fundamental equations.

Incorrect

Correct

1 ? 1 @ ¼ þ Kdry K0 @c

1 1 1 @p ¼ þ Kdry K0 Vb @c

1 ? 1 ¼ þ Kdry K0 K

1 1 ¼ þ Kdry K0 K 1 1 1 @ ¼ þ Kdry ð1 Þ K0 @c

1 ? 1 1 @ ¼ þ Kdry K0 ð1 Þ @c @ ? 1 1 ð1 Þ ¼ @c Kdry K0

@ 1 1 ¼ @c Kdry K0

Assumptions and limitations The following presuppositions apply to the equations presented in this section: they assume isotropic, linear, porous, and elastic media; all derivations here are in the context of linear elasticity with infinitesimal, incremental strains and stresses. Hence Eulerian and Lagrangian formulations are equivalent; it is assumed that the temperature is always held constant as the pressure varies; and inelastic effects such as friction and viscosity are neglected.

4.7

Kuster and Tokso¨z formulation for effective moduli Synopsis Kuster and Tokso¨z (1974) derived expressions for P- and S-wave velocities by using a long-wavelength first-order scattering theory. A generalization of their expressions for the effective moduli KKT and mKT for a variety of inclusion shapes can be written as (Kuster and Tokso¨z, 1974; Berryman, 1980b) ðKKT Km Þ

ðmKT mm Þ

N X ðKm þ 43 mm Þ xi ðKi Km ÞPmi ¼ þ 4m Þ ðKKT 3 m i¼1

N X ðmm þ zm Þ ¼ xi ðmi mm ÞQmi ðmKT þ zm Þ i¼1

where the summation is over the different inclusion types with volume concentration xi, and

184

Effective elastic media: bounds and mixing laws

Coefficients P and Q for some specific shapes. The subscripts m and i refer to the background and inclusion materials (from Berryman, 1995).

Table 4.7.1

Inclusion shape

Pmi

Qmi

Spheres

Km þ 43mm Ki þ 43mm

mm þ zm mi þ zm

Needles

Km þ mm þ 13mi Ki þ mm þ 13mi

Ki þ 43mm 1 4mm m þ gm þ2 m þ mi þ gm Ki þ mm þ 13mi 5 mm þ mi

Disks

Km þ 43mi Ki þ 43mi

mm þ zi mi þ zi

Penny cracks

Km þ 43mi Ki þ 43mi þ pabm

Ki þ 23ðmi þ mm Þ 1 8mm þ2 1þ 4mi þ paðmm þ 2bm Þ 5 Ki þ 43mi þ pabm

!

!

Notes: ð3KþmÞ ð3KþmÞ b ¼ m ð3Kþ4mÞ ; g ¼ m ð3Kþ7mÞ ; z ¼ m6 ð9Kþ8mÞ ðKþ2mÞ ; a ¼ crack aspect ratio, a disk is a crack of zero thickness.

z¼

m ð9K þ 8mÞ 6 ðK þ 2mÞ

The coefficients Pmi and Qmi describe the effect of an inclusion of material i in a background medium m. For example, a two-phase material with a single type of inclusion embedded within a background medium has a single term on the right-hand side. Inclusions with different material properties or different shapes require separate terms in the summation. Each set of inclusions must be distributed randomly, and thus its effect is isotropic. These formulas are uncoupled and can be made explicit for easy evaluation. Table 4.7.1 gives expressions for P and Q for some simple inclusion shapes. Dry cavities can be modeled by setting the inclusion moduli to zero. Fluidsaturated cavities are simulated by setting the inclusion shear modulus to zero.

Caution

Because the cavities are isolated with respect to flow, this approach simulates very high-frequency saturated rock behavior appropriate to ultrasonic laboratory conditions. At low frequencies, when there is time for wave-induced pore-pressure increments to flow and equilibrate, it is better to find the effective moduli for dry cavities and then saturate them with the Gassmann low-frequency relations (see Section 6.3). This should not be confused with the tendency to term this approach a low-frequency theory, for crack dimensions are assumed to be much smaller than a wavelength.

185

4.8 Self-consistent approximations of effective moduli

Calculate the effective bulk and shear moduli, KKT and mKT , for a quartz matrix with spherical, water-filled inclusions of porosity 0.1:

Km ¼ 37 GPa mm ¼ 44 GPa

Ki ¼ 2:25 GPa

mi ¼ 0 GPa:

Volume fraction of spherical inclusions x1 ¼ 0.1 and N ¼ 1. The P and Q values for spheres are obtained from the table as follows: ð37 þ 43 44Þ ¼ 1:57 ð2:25 þ 43 44Þ 44 ð9 37 þ 8 44Þ ¼ 40:2 zm ¼ 6 ð37 þ 2 44Þ ð44 þ 40:2Þ Qm1 ¼ ¼ 2:095 ð0 þ 40:2Þ Pm1 ¼

Substituting these in the Kuster–Tokso¨z equations gives ¼ 31:84 GPa mKT ¼ 35:7 GPa KKT

The P and Q values for ellipsoidal inclusions with arbitrary aspect ratio are the same as given in Section 4.8 on self-consistent methods. Note that for spherical inclusions, the Kuster–Tokso¨z expressions for bulk modulus are identical to the Hashin–Shtrikman upper bound, even though the Kuster–Tokso¨z expressions are formally limited to low porosity.

Assumptions and limitations The following presuppositions and limitations apply to the Kuster–Tokso¨z formulations: they assume isotropic, linear, and elastic media; they are limited to dilute concentrations of the inclusions; and they assume idealized ellipsoidal inclusion shapes.

4.8

Self-consistent approximations of effective moduli Synopsis Theoretical estimates of the effective moduli of composite or porous elastic materials generally depend on: (1) the properties of the individual components of the composite, (2) the volume fractions of the components, and (3) the geometric details of the shapes and spatial distributions of the components. The bounding methods (see discussions of the Hashin–Shtrikman and Voigt–Reuss bounds, Sections 4.1 and 4.2) establish upper and lower bounds when only (1) and (2) are known, with no geometric details. A second approach improves these estimates by adding statistical information about the phases (e.g., Beran and Molyneux, 1966; McCoy, 1970; Corson,

186

Effective elastic media: bounds and mixing laws

1974; Watt et al., 1976). A third approach is to assume very specific inclusion shapes. Most methods use the solution for the elastic deformation of a single inclusion of one material in an infinite background medium of the second material and then use one scheme or another to estimate the effective moduli when there is a distribution of these inclusions. These estimates are generally limited to dilute distributions of inclusions, owing to the difficulty of modeling or estimating the elastic interaction of inclusions in close proximity to each other. A relatively successful, and certainly popular, method to extend these specific geometry methods to slightly higher concentrations of inclusions is the self-consistent approximation (Budiansky, 1965; Hill, 1965; Wu, 1966). In this approach one still uses the mathematical solution for the deformation of isolated inclusions, but the interaction of inclusions is approximated by replacing the background medium with the as-yet-unknown effective medium. These methods were made popular following a series of papers by O’Connell and Budiansky (see, for example, O’Connell and Budiansky, 1974). Their equations for effective bulk and shear moduli, KSC and mSC , respectively, of a cracked medium with randomly oriented dry penny-shaped cracks (in the limiting case when the aspect ratio a goes to 0) are KSC 16 1 v2 SC ¼ 1 e K 9 1 2vSC mSC 32 ð1 vSC Þð5 vSC Þ ¼ 1 e 45 ð2 vSC Þ m where K and m are the bulk and shear moduli, respectively, of the uncracked medium, and e is the crack density parameter, which is defined as the number of cracks per unit volume times the crack radius cubed. The effective Poisson ratio vSC is related to e and the Poisson ratio v of the uncracked solid by e¼

45 ðv vSC Þð2 vSC Þ 16 ð1 v2 SC Þð10v 3vvSC vSC Þ

This equation must first be solved for vSC for a given e, after which KSC and mSC can be evaluated. The nearly linear dependence of vSC on e is well approximated by 16 nSC n 1 e 9

and this simplifies the calculation of the effective moduli. For fluid-saturated, infinitely thin penny-shaped cracks KSC ¼1 K mSC 32 1 vSC e ¼1 m 15 2 vSC

e¼

45 ðvSC vÞð2 vSC Þ 32 ð1 v2 SC Þð1 2vÞ

187

4.8 Self-consistent approximations of effective moduli

Coefficients P and Q for some specific shapes. The subscripts m and i refer to the background and inclusion materials (from Berryman 1995).

Table 4.8.1

Inclusion shape

Pmi

Qmi

Spheres

Km þ 43mm Ki þ 43mm

mm þ zm mi þ zm

Needles

Km þ mm þ 13mi Ki þ mm þ 13mi

Ki þ 43mm 1 4mm m þ gm þ2 m þ mi þ gm Ki þ mm þ 13mi 5 mm þ mi

Disks

Km þ 43mi Ki þ 43mi

Penny cracks

Km þ 43mi Ki þ 43mi þ pabm

mm þ zi mi þ zi " # Ki þ 23ðmi þ mm Þ 1 8mm þ2 1þ 4mi þ paðmm þ 2bm Þ 5 Ki þ 43mi þ pabm

!

Notes: ð3KþmÞ ð3KþmÞ b ¼ m ð3Kþ4mÞ ; g ¼ m ð3Kþ7mÞ ; z ¼ m6 ð9Kþ8mÞ ðKþ2mÞ , a ¼ crack aspect ratio, a disk is a crack of zero thickness.

However, this result is inadequate for small aspect ratio cracks with soft-fluid saturation, such as when the parameter o ¼ Kfluid/aK is of the order of 1. Then the appropriate equations given by O’Connell and Budiansky are KSC 16 ð1 v2 SC Þ ¼1 De K 9 ð1 2vSC Þ mSC 32 3 ¼ 1 ð1 vSC Þ D þ e m 45 ð2 vSC Þ

45 ðv vSC Þ ð2 vSC Þ

16 ð1 vSC 2 Þ Dð1 þ 3vÞð2 vSC Þ 2ð1 2vÞ 1 4 ð1 v2 SC Þ K D¼ 1þ o 3p ð1 2vSC Þ KSC

e¼

Wu’s self-consistent modulus estimates for two-phase composites may be expressed as (m ¼ matrix, i ¼ inclusion) ¼ Km þ xi ðKi Km ÞPi KSC

mSC ¼ mm þ xi ðmi mm ÞQi Berryman (1980b, 1995) gives a more general form of the self-consistent approximations for N-phase composites: N X xi ðKi KSC ÞPi ¼ 0 i¼1 N X

xi ðmi mSC ÞQi ¼ 0

i¼1

where i refers to the ith material, xi is its volume fraction, P and Q are geometric factors given in Table 4.8.1, and the superscript *i on P and Q indicates that the

188

Effective elastic media: bounds and mixing laws

factors are for an inclusion of material i in a background medium with self-consistent effective moduli KSC and mSC . The summation is over all phases, including minerals and pores. These equations are coupled and must be solved by simultaneous iteration. Although Berryman’s self-consistent method does not converge for fluid disks (m2 ¼ 0), the formulas for penny-shaped fluid-filled cracks are generally not singular and converge rapidly. However, his estimates for needles, disks, and penny cracks should be used cautiously for fluid-saturated composite materials. Dry cavities can be modeled by setting the inclusion moduli to zero. Fluidsaturated cavities are simulated by setting the inclusion shear modulus to zero. Caution

Because the cavities are isolated with respect to flow, this approach simulates very-high-frequency saturated rock behavior appropriate to ultrasonic laboratory conditions. At low frequencies, when there is time for wave-induced pore-pressure increments to flow and equilibrate, it is better to find the effective moduli for dry cavities and then saturate them with the Gassmann low-frequency relations. This should not be confused with the tendency to term this approach a low-frequency theory, for crack dimensions are assumed to be much smaller than a wavelength.

Calculate the self-consistent effective bulk and shear moduli, KSC , and mSC , for a water-saturated rock consisting of spherical quartz grains (aspect ratio a ¼ 1) and total porosity 0.3. The pore space consists of spherical pores (a ¼ 1) and thin, penny-shaped cracks (a ¼ 10–2). The thin cracks have a porosity of 0.01, whereas the remaining porosity (0.29) is made up of the spherical pores. The total number of phases, N, is 3. K1(quartz) ¼ 37 GPa, m1(quartz) ¼ 44 GPa, a1 ¼ 1, x1 (volume fraction) ¼ 0.7 K2(water, spherical pores) ¼ 2.25 GPa, m2(water, spherical pores) ¼ 0 GPa, a2(spherical pores) ¼ 1, x2 (volume fraction) ¼ 0.29 K3(water, thin cracks) ¼ 2.25 GPa, m3(water, thin cracks) ¼ 0 GPa, a3(thin cracks) ¼102, x3 (volume fraction) ¼ 0.01 The coupled equations for KSC and mSC are ÞP1 þ x2 ðK2 KSC ÞP2 þ x3 ðK3 KSC ÞP3 ¼ 0 x1 ðK1 KSC

x1 ðm1 mSC ÞQ1 þ x2 ðm2 mSC ÞQ2 þ x3 ðm3 mSC ÞQ3 ¼ 0 The P’s and Q’s are obtained from Table 4.8.1 or from the more general equation for ellipsoids of arbitrary aspect ratio. In the equations for P and Q, Km and mm are

189

4.8 Self-consistent approximations of effective moduli

replaced everywhere by KSC and mSC , respectively. The coupled equations are solved iteratively, starting from some initial guess for KSC and mSC . The Voigt average may be taken as the starting point. The converged solutions (known as the fixed points of the coupled equations) are KSC ¼ 16.8 GPa and mSC ¼ 11.6 GPa.

The coefficients P and Q for ellipsoidal inclusions of arbitrary aspect ratio are given by P ¼ 13 Tiijj

Q ¼ 15 Tijij 13 Tiijj where the tensor Tijkl relates the uniform far-field strain field to the strain within the ellipsoidal inclusion (Wu, 1966). Berryman (1980b) gives the pertinent scalars required for computing P and Q as Tiijj ¼

3F1 F2

1 2 1 F4 F5 þ F6 F7 F8 F9 Tijij Tiijj ¼ þ þ F2 F4 3 F3 F4 where

F1 ¼ 1 þ A 32ð f þ yÞ R 32 f þ 52y 43

F2 ¼ 1 þ A 1 þ 32ð f þ yÞ 12 Rð3f þ 5yÞ þ Bð3 4RÞ 1 þ AðA þ 3BÞð3 4RÞ½ f þ y Rð f y þ 2y2 Þ 2

F3 ¼ 1 þ A 1 f þ 32y þ Rð f þ yÞ 1 F4 ¼ 1 þ A½ f þ 3y Rð f yÞ 4

F5 ¼ A f þ R f þ y 43 þ Byð3 4RÞ F6 ¼ 1 þ A½1 þ f Rð f þ yÞ þ Bð1 yÞð3 4RÞ 1 F7 ¼ 2 þ A½3f þ 9y Rð3f þ 5yÞ þ Byð3 4RÞ 4 F8 ¼ A½1 2R þ 12 f ðR 1Þ þ 12 yð5R 3Þ þ Bð1 yÞð3 4RÞ F9 ¼ A½ðR 1Þf Ry þ Byð3 4RÞ

190

Effective elastic media: bounds and mixing laws

with A, B, and R given by A ¼ mi =mm 1 1 Ki mi B¼ 3 Km m m and R¼

ð1 2vm Þ 2ð1 vm Þ

The functions y and f are given by 8 h i a 1=2 1 2 > aða 1Þ cosh a > > < ða2 1Þ3=2 y¼ h i a > 1 2 1=2 > > cos a að1 a Þ : ð1 a2 Þ3=2 for prolate and oblate spheroids, respectively, and f ¼

a2 ð3y 2Þ 1 a2

Note that a < 1 for oblate spheroids and a > 1 for prolate spheroids.

Assumptions and limitations The approach described in this section has the following presuppositions: idealized ellipsoidal inclusion shapes; isotropic, linear, elastic media; and cracks are isolated with respect to fluid flow. Pore pressures are unequilibrated and adiabatic, which is appropriate for high-frequency laboratory conditions. For lowfrequency field situations, use dry inclusions and then saturate by using Gassmann relations. This should not be confused with the tendency to term this approach a low-frequency theory, for crack dimensions are assumed to be much smaller than a wavelength.

4.9

Differential effective medium model Synopsis The differential effective medium (DEM) theory models two-phase composites by incrementally adding inclusions of one phase (phase 2) to the matrix phase (Cleary et al., 1980; Norris, 1985; Zimmerman, 1991a). The matrix begins as phase 1 (when the concentration of phase 2 is zero) and is changed at each step as a new increment of phase 2 material is added. The process is continued until the desired proportion of

191

4.9 Differential effective medium model

Coefficients P and Q for some specific shapes. The subscripts m and i refer to the background and inclusion materials (from Berryman 1995).

Table 4.9.1

Inclusion shape

Pmi

Qmi

Spheres

Km þ 43mm Ki þ 43mm

mm þ zm mi þ zm

Needles

Km þ mm þ 13mi Ki þ mm þ 13mi

Ki þ 43mm 1 4mm m þ gm þ2 m þ mi þ gm Ki þ mm þ 13mi 5 mm þ mi

Disks

Km þ 43mi Ki þ 43mi

Penny cracks

Km þ 43mi Ki þ 43mi þ pabm

mm þ zi mi þ zi " # Ki þ 23ðmi þ mm Þ 1 8mm þ2 1þ 4mi þ paðmm þ 2bm Þ 5 Ki þ 43mi þ pabm

!

Notes: ð3KþmÞ ð3KþmÞ ; g ¼ m ð3Kþ7mÞ ; z ¼ m6 ð9Kþ8mÞ b ¼ m ð3Kþ4mÞ ðKþ2mÞ , a ¼ crack aspect ratio, a disk is a crack of zero thickness.

the constituents is reached. The DEM formulation does not treat each constituent symmetrically. There is a preferred matrix or host material, and the effective moduli depend on the construction path taken to reach the final composite. Starting with material 1 as the host and incrementally adding inclusions of material 2 will not, in general, lead to the same effective properties as starting with phase 2 as the host. For multiple inclusion shapes or multiple constituents, the effective moduli depend not only on the final volume fractions of the constituents but also on the order in which the incremental additions are made. The process of incrementally adding inclusions to the matrix is really a thought experiment and should not be taken to provide an accurate description of the true evolution of rock porosity in nature. The coupled system of ordinary differential equations for the effective bulk and shear moduli, K* and m*, respectively, are (Berryman, 1992b) ð1 yÞ

d ½K ðyÞ ¼ ðK2 K ÞPð2Þ ðyÞ dy

ð1 yÞ

d ½m ðyÞ ¼ ðm2 m ÞQð2Þ ðyÞ dy

with initial conditions K*(0) ¼ K1 and m*(0) ¼ m1, where K1 and m1 are the bulk and shear moduli of the intial host material (phase 1), K2 and m2 are the bulk and shear moduli of the incrementally added inclusions (phase 2), and y is the concentration of phase 2. For fluid inclusions and voids, y equals the porosity, f. The terms P and Q are geometric factors given in Table 4.9.1, and the superscript *2 on P and Q indicates that the factors are for an inclusion of material 2 in a background medium with effective moduli K* and m*. Dry cavities can be modeled by setting the inclusion moduli to zero. Fluid-saturated cavities are simulated by setting the inclusion shear modulus to zero.

192

Effective elastic media: bounds and mixing laws

Caution

Because the cavities are isolated with respect to flow, this approach simulates veryhigh-frequency saturated rock behavior appropriate to ultrasonic laboratory conditions. At low frequencies, when there is time for wave-induced pore-pressure increments to flow and equilibrate, it is better to find the effective moduli for dry cavities and then saturate them with the Gassmann low-frequency relations. This should not be confused with the tendency to term this approach a low-frequency theory, for inclusion dimensions are assumed to be much smaller than a wavelength. The P and Q for ellipsoidal inclusions with arbitrary aspect ratio are the same as given in Section 4.8 for the self-consistent methods, and are repeated in Table 4.9.1. Norris et al. (1985) have shown that the DEM is realizable and therefore is always consistent with the Hashin–Shtrikman upper and lower bounds. The DEM equations as given above (Norris, 1985; Zimmerman, 1991b; Berryman et al., 1992) assume that, as each new inclusion (or pore) is introduced, it displaces on average either the host matrix material or the inclusion material, with probabilities (1 – y) and y, respectively. A slightly different derivation by Zimmerman (1984) (superseded by Zimmerman (1991a)) assumed that when a new inclusion is introduced, it always displaces the host material alone. This leads to similar differential equations with dy/(1 – y) replaced by dy. The effective moduli predicted by this version of DEM are always slightly stiffer (for the same inclusion geometry and concentration) than the DEM equations given above. They both predict the same firstorder terms in y but begin to diverge at concentrations above 10%. The dependence of effective moduli on concentration goes as e2y ¼ ð1 2y þ 2y2 Þ for the version without (1 y), whereas it behaves as (1 y)2 ¼ (1 2y þ y2) for the version with (1 y). Including the (1 y) term makes the results of Zimmerman (1991a) consistent with the Hashin–Shtrikman bounds. In general, for a fixed inclusion geometry and porosity, the Kuster–Tokso¨z effective moduli are stiffer than the DEM predictions, which in turn are stiffer than the Berryman self-consistent effective moduli. An important conceptual difference between the DEM and self-consistent schemes for calculating effective moduli of composites is that the DEM scheme identifies one of the constituents as a host or matrix material in which inclusions of the other constituent(s) are embedded, whereas the self-consistent scheme does not identify any specific host material but treats the composite as an aggregate of all the constituents.

Modified DEM with critical porosity constraints In the usual DEM model, starting from a solid initial host, a porous material stays intact at all porosities and falls apart only at the very end when y ¼ 1 (100% porosity). This is because the solid host remains connected, and therefore load bearing. Although DEM is a good model for materials such as glass foam (Berge et al., 1993) and oceanic basalts (Berge et al., 1992), most reservoir rocks fall apart at

4.9 Differential effective medium model

1

Bulk moduli (normalized)

193

0.8 0.6

0.4

DEM

fc = 0.2 0.2

Reuss

0.4 0.6

0.8

1.0

0 0

0.2

0.4

0.6

0.8

1

Porosity

Figure 4.9.1 Normalized bulk-modulus curves for the conventional DEM theory (percolation at f ¼ 1) and for the modified DEM (percolation at fc < 1) for a range of fc values.

a critical porosity, fc, significantly less than 1.0 and are not represented very well by the conventional DEM theory. The modified DEM model (Mukerji et al., 1995a) incorporates percolation behavior at any desired fc by redefining the phase 2 endmember. The inclusions are now no longer made up of pure fluid (the original phase 2 material) but are composite inclusions of the critical phase at fc with elastic moduli (Kc, mc). With this definition, y denotes the concentration of the critical phase in the matrix. The total porosity is given by f ¼ yfc. The computations are implemented by replacing (K2, m2) with (Kc, mc) everywhere in the equations. Integrating along the reverse path, from f ¼ fc to f ¼ 0 gives lower moduli, for now the softer critical phase is the matrix. The moduli of the critical phase may be taken as the Reuss average value at fc of the pure end-member moduli. Because the critical phase consists of grains just barely touching each other, better estimates of Kc and mc may be obtained from measurements on loose sands, or from models of granular material. For porosities greater than fc, the material is a suspension and is best characterized by the Reuss average (or Wood’s equation). Figure 4.9.1 shows normalized bulk-modulus curves for the conventional DEM theory (percolation at f ¼ 1) and for the modified DEM (percolation at fc < 1) for a range of fc values. When fc ¼ 1, the modified DEM coincides with the conventional DEM curve. The shapes of the inclusions were taken to be spheres. The path was from 0 to fc, and (Kc, mc) were taken as the Reuss-average values at fc. For this choice of (Kc, mc), estimates along the reversed path coincide with the Reuss curve.

Uses The purpose of the differential effective-medium model is to estimate the effective elastic moduli of a rock in terms of its constituents and pore space.

194

Effective elastic media: bounds and mixing laws

Assumptions and limitations The following assumptions and limitations apply to the differential effective-medium model: the rock is isotropic, linear, and elastic; the process of incrementally adding inclusions to the matrix is a thought experiment and should not be taken to provide an accurate description of the true evolution of rock porosity in nature; idealized ellipsoidal inclusion shapes are assumed; and cracks are isolated with respect to fluid flow. Pore pressures are unequilibrated and adiabatic. The model is appropriate for high-frequency laboratory conditions. For lowfrequency field situations, use dry inclusions and then saturate using Gassmann relations. This should not be confused with the tendency to term this approach a low-frequency theory, for crack dimensions are assumed to be much smaller than a wavelength.

4.10

Hudson’s model for cracked media Synopsis Hudson’s model is based on a scattering-theory analysis of the mean wavefield in an elastic solid with thin, penny-shaped ellipsoidal cracks or inclusions (Hudson, 1980, 1981). The effective moduli ceff ij are given as 0 1 2 ceff ij ¼ cij þ cij þ cij

where c0ij are the isotropic background moduli, and c1ij , c2ij are the first- and secondorder corrections, respectively. (See Section 2.2 on anisotropy for the two-index notation of elastic moduli. Note also that Hudson uses a slightly different definition and that there is an extra factor of 2 in his c44, c55, and c66. This makes the equations given in his paper for c144 and c244 slightly different from those given below, which are consistent with the more standard notation described in Section 2.2 on anisotropy.) For a single crack set with crack normals aligned along the 3-axis, the cracked media show transverse isotropic symmetry, and the corrections are l2 eU3 m lðl þ 2mÞ ¼ eU3 m

c111 ¼ c113

ðl þ 2mÞ2 eU3 m ¼ meU1

c133 ¼ c144

c166 ¼ 0

195

4.10 Hudson’s model for cracked media

and (superscripts on cij denote second order, not quantities squared) c211 ¼

q l2 ðeU3 Þ2 15 ðl þ 2mÞ

c213 ¼

q lðeU3 Þ2 15

c233 ¼

q ðl þ 2mÞðeU3 Þ2 15

c244 ¼

2 mð3l þ 8mÞ ðeU1 Þ2 15 l þ 2m

c266 ¼ 0 where q ¼ 15 e¼

l2 l þ 28 þ 28 2 m m

N 3 3 a ¼ ¼ crack density V 4pa

The isotropic background elastic moduli are l and m, and a and a are the crack radius and aspect ratio, respectively. The corrections c1ij and c2ij obey the usual symmetry properties for transverse isotropy or hexagonal symmetry (see Section 2.2 on anisotropy).

Caution

The second-order expansion is not a uniformly converging series and predicts increasing moduli with crack density beyond the formal limit (Cheng, 1993). Better results will be obtained by using just the first-order correction rather than inappropriately using the second-order correction. Cheng gives a new expansion based on the Pade´ approximation, which avoids this problem. The terms U1 and U3 depend on the crack conditions. For dry cracks U1 ¼

16ðl þ 2mÞ 3ð3l þ 4mÞ

U3 ¼

4ðl þ 2mÞ 3ðl þ mÞ

For “weak” inclusions (i.e., when ma=ðK 0 þ 43 m0 Þ is of the order of 1 and is not small enough to be neglected) U1 ¼

16ðl þ 2mÞ 1 3ð3l þ 4mÞ ð1 þ MÞ

U3 ¼

4ðl þ 2mÞ 1 3ðl þ mÞ ð1 þ kÞ

196

Effective elastic media: bounds and mixing laws

where M¼

4m0 ðl þ 2mÞ pam ð3l þ 4mÞ

k¼

ðK 0 þ 43 m0 Þðl þ 2mÞ pamðl þ mÞ

where K0 and m0 are the bulk and shear modulus of the inclusion material. The criteria for an inclusion to be “weak” depend on its shape or aspect ratio a as well as on the relative moduli of the inclusion and matrix material. Dry cavities can be modeled by setting the inclusion moduli to zero. Fluid-saturated cavities are simulated by setting the inclusion shear modulus to zero.

Caution

Because the cavities are isolated with respect to flow, this approach simulates veryhigh-frequency behavior appropriate to ultrasonic laboratory conditions. At low frequencies, when there is time for wave-induced pore-pressure increments to flow and equilibrate, it is better to find the effective moduli for dry cavities and then saturate them with the Brown and Korringa low-frequency relations (Section 6.5). This should not be confused with the tendency to term this approach a low-frequency theory, for crack dimensions are assumed to be much smaller than a wavelength. Hudson also gives expressions for infinitely thin, fluid-filled cracks: U1 ¼

16ðl þ 2mÞ 3ð3l þ 4mÞ

U3 ¼ 0

These assume no discontinuity in the normal component of crack displacements and therefore predict no change in the compressional modulus with saturation. There is, however, a shear displacement discontinuity and a resulting effect on shear stiffness. This case should be used with care. The first-order changes l1 and m1 in the isotropic elastic moduli l and m of a material containing randomly oriented inclusions are given by m1 ¼

2m eð3U1 þ 2U3 Þ 15

3l1 þ 2m1 ¼

ð3l þ 2mÞ2 eU3 3m

These results agree with the self-consistent results of Budiansky and O’Connell (1976). For two or more crack sets aligned in different directions, corrections for each crack set are calculated separately in a crack-local coordinate system with the 3-axis normal to the crack plane and then rotated or transformed back (see Section 1.4 on coordinate transformations) into the coordinates of ceff ij ; finally the results are added to

197

4.10 Hudson’s model for cracked media

obtain the overall correction. Thus, for three crack sets with crack densities e1, e2, and e3 with crack normals aligned along the 1-, 2-, and 3-axes, respectively, the overall 1ð3setsÞ first-order corrections to c0ij , cij may be given in terms of linear combinations of the corrections for a single set, with the appropriate crack densities as follows (where we have taken into account the symmetry properties of c1ij ): 1ð3setsÞ

¼ c133 ðe1 Þ þ c111 ðe2 Þ þ c111 ðe3 Þ

1ð3setsÞ

¼ c113 ðe1 Þ þ c113 ðe2 Þ þ c112 ðe3 Þ

1ð3setsÞ

¼ c113 ðe1 Þ þ c112 ðe2 Þ þ c113 ðe3 Þ

1ð3setsÞ

¼ c111 ðe1 Þ þ c133 ðe2 Þ þ c111 ðe3 Þ

1ð3setsÞ

¼ c112 ðe1 Þ þ c113 ðe2 Þ þ c113 ðe3 Þ

1ð3setsÞ

¼ c111 ðe1 Þ þ c111 ðe2 Þ þ c133 ðe3 Þ

1ð3setsÞ

¼ c144 ðe2 Þ þ c144 ðe3 Þ

1ð3setsÞ

¼ c144 ðe1 Þ þ c144 ðe3 Þ

1ð3setsÞ

¼ c144 ðe1 Þ þ c144 ðe2 Þ

c11 c12 c13 c22 c23 c33 c44 c55 c66

Note that c166 ¼ 0 and c112 ¼ c111 2c166 ¼ c111 . Hudson (1981) also gives the attenuation coefficient (g ¼ oQ–1/2V) for elastic waves in cracked media. For aligned cracks with normals along the 3-axis, the attenuation coefficients for P-, SV-, and SH-waves are " 2 2 # o oa 3 1 2 2 2 VP 2 gP ¼ e AU1 sin 2y þ BU3 2 2 sin y VS VP 30p VS o oa 3 1 e ðAU12 cos2 2y þ BU32 sin2 2yÞ VS VS 30p o oa 3 1 ¼ e ðAU12 cos2 yÞ VS VS 30p

gSV ¼ gSH

A¼

3 VS5 þ 2 VP5

B¼ 2þ

15 VS V3 V5 10 S3 þ 8 S5 4 VP VP VP

In the preceding expressions, VP and VS are the P and S velocities in the uncracked isotropic background matrix, o is the angular frequency, and y is the angle between the direction of propagation and the 3-axis (axis of symmetry).

198

Effective elastic media: bounds and mixing laws

For randomly oriented cracks (isotropic distribution), the P and S attenuation coefficients are given as o oa 3 4 1 VP5 2 2 gP ¼ e þ BðB 2ÞU AU 1 3 VS VP 152 p 2 VS5 o oa 3 1 1 2 2 e gS ¼ AU1 þ BU3 VS VS 75p 3 The fourth-power dependence on o is characteristic of Rayleigh scattering. Hudson (1990) gives results for overall elastic moduli of material with various distributions of penny-shaped cracks. If conditions at the cracks are taken to be uniform, so that U1 and U3 do not depend on the polar and azimuthal angles y and f, the first-order correction is given as A U3 ½l2 dij dpq þ 2lmðdij~epq þ dpq~eij Þ þ 4m2~eijpq m

c1ijpq ¼

AmU1 ðdjq~eip þ djp~eiq þ diq~ejp þ dip~ejq 4~eijpq Þ where Z A¼

2p Z p=2

0

~eij ¼

1 A

~eijpq ¼

eðy; Þ sin y dy d

0 2p Z p=2

Z 0

1 A

eðy; Þ ni nj sin y dy d

0

Z

2p Z p=2

0

eðy; Þ ni nj np nq sin y dy d

0

and ni are the components of the unit vector along the crack normal, n ¼ (sin y cos f, sin y sin f, cos y), whereas e(y, f) is the crack density distribution function, so that e(y, f) sin y dy df is the density of cracks with normals lying in the solid angle between (y, y þ dy) and (f, f þ df).

Special cases of crack distributions (a) Cracks with total crack density et, which have all their normals aligned along y ¼ y0, f ¼ f0: eðy; Þ ¼ et

dðy y0 Þ dð 0 Þ sin y

A ¼ et ~eij ¼ n0i n0j ~eijpq ¼ n0i n0j n0p n0q where n01 ¼ sin y0 cos 0 ; n02 ¼ sin y0 sin 0 ; and n03 ¼ cos y0 :

199

4.10 Hudson’s model for cracked media

(b) Rotationally symmetric crack distributions with normals symmetrically distributed about y ¼ 0, that is, where e is a function of y only: Z

p=2

A ¼ 2p

eðyÞ sin y dy 0

~e12 ¼ ~e23 ¼ ~e31 ¼ 0 Z p p=2 ~e11 ¼ ~e22 ¼ eðyÞ sin3 y dy ¼ 12 ð1 ~e33 Þ A 0 Z 3p p=2 ~e1111 ¼ ~e2222 ¼ eðyÞ sin5 y dy ¼ 3~e1122 ¼ 3~e1212 ; etc: 4A 0 ~e3333 ¼ 83 ~e1111 4~e11 þ 1 ~e1133 ¼ ~e11 43 ~e1111 ¼ ~e2233 ¼ ~e1313 ¼ ~e2323 ; etc: Elements other than those related to the preceding elements by symmetry are zero. A particular rotationally symmetric distribution is the Fisher distribution, for which e(y) is et eðcos yÞ= 2p 2 ðe1=2 1Þ 2

eðyÞ ¼

For small s2, this is approximately a model for a Gaussian distribution on the sphere ey =2 eðyÞ et 2p2 2

2

The proportion of crack normals outside the range 0 y 2s is approximately 1/e2. For this distribution, A ¼ et 2

2

1 þ 22 e1= 24 ðe1= 1Þ 2 2ðe1=2 1Þ " # 2 2 2 3 1 þ 44 ð2e1= þ 1Þ 246 e1= þ 248 ðe1= 1Þ ¼ 34 8 ðe1=2 1Þ

~e11 ¼ ~e1111

This distribution is suitable when crack normals are oriented randomly with a small variance about a mean direction along the 3-axis. (c) Cracks with normals randomly distributed at a fixed angle from the 3-axis forming a cone. In this case e is independent of f and is zero unless y ¼ y0, 0 f 2p. eðyÞ ¼ et

dðy y0 Þ 2p sin y

200

Effective elastic media: bounds and mixing laws

which gives A ¼ et ~e11 ¼ 12 sin2 y0 ~e1111 ¼ 38 sin4 y0 and the first-order corrections are c11111 ¼

et ½U3 ð2l2 þ 4lm sin2 y0 þ 3m2 sin4 y0 Þ þ U1 m2 sin2 y0 ð4 3 sin2 y0 Þ 2m

¼ c12222 c13333 ¼

et ½U3 ðl þ 2m cos2 y0 Þ2 þ U1 m2 4 cos2 y0 sin2 y0 m

c11122 ¼

et ½U3 ð2l2 þ 4lm sin2 y0 þ m2 sin4 y0 Þ U1 m2 sin4 y0 2m

c11133 ¼

et ½U3 ðl þ m sin2 y0 Þðl þ 2m cos2 y0 Þ U1 m2 2 sin2 y0 cos2 y0 m

¼ c12233 c12323 ¼

et m½U3 4 sin2 y0 cos2 y0 þ U1 ðsin2 y0 þ 2 cos2 y0 4 sin2 y0 cos2 y0 Þ 2

¼ c11313 c11212 ¼

et m½U3 sin4 y0 þ U1 sin2 y0 ð2 sin2 y0 Þ 2

Heavily faulted structures Hudson and Liu (1999) provide a model for the effective elastic properties of rocks with an array of parallel faults based on the averaging process of Schoenberg and Douma (1988). The individual faults themselves are modeled using two different approaches: model 1 as a planar distribution of small circular cracks; model 2 as a planar distribution of small circular welded contacts. Hudson and Liu also give expressions where both models 1 and 2 are replaced by an equivalent layer of constant thickness and appropriate infill material, which can then be averaged using the Backus average. Model 2 (circular contacts) provides expressions for effective elastic properties of heavily cracked media with cracks aligned and confined within the fault planes. Assuming the fault normals to be aligned along the 3-axis, the elements of the effective stiffness tensor are (Hudson and Liu, 1999)

201

4.10 Hudson’s model for cracked media

c1111 ¼ c2222 ¼ l þ 2m l þ 2m c3333 ¼ 1 þ EN c1122 ¼ l l c1133 ¼ c2233 ¼ 1 þ EN m c2323 ¼ c1313 ¼ 1 þ ET c1212 ¼ m The quantities EN and ET depend on the fault model. For model 1, where the fault plane consists of an array of circular cracks, s 3

3=2 na l þ 2m m 1 U3 1 þ pU3 ns a2 EN ¼ H m l þ 2m s 3 3=2 na p

m 32 ET ¼ U1 1 þ U1 ns a2 H 4 l þ 2m where a is the crack radius, ns is the number density of cracks on the fault surface, H is the spacing between parallel faults, and l and m are the Lame´ parameters of the uncracked background material. The overall number density of cracks, n ¼ ns =H and the overall crack density e ¼ na3 ¼ ns a3 =H. The relative area of cracking on the fault is given by r ¼ ns pa2 . Alternatively, the faults may be replaced by an elastically equivalent layer with bulk and shear moduli given by 1=2 # " pam m 4 r3 0 K ¼K 1 4 l þ 2m 3 p 3 1=2 # " 3pam 3l þ 4m 4 r m ¼ m 0 þ 1 16 l þ 2m 3 p In the above expressions K0 and m0 are, respectively, the bulk and shear moduli of the material filling the cracks, of aspect ratio a, within the faults. Viscous fluid may be modeled by setting m0 to be imaginary. Setting both moduli to zero represents dry cracks. The equivalent layers can be used in Backus averaging to estimate the overall stiffness. These give results very close (but not completely identical, because of the approximations involved) to the results obtained from the equations for cijkl given above. For model 2, which assumes a planar distribution of small, circular, welded contacts: EN ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 ðl þ 2mÞ2 1 1 þ 2 nw b2 4mðl þ mÞ ðnw HbÞ

202

Effective elastic media: bounds and mixing laws

ET ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 ð3l þ 4mÞ 1 1 þ 2 nw b2 w 8ðl þ mÞ ðn HbÞ

where b is the radius of the welded contact region and nw is the number density of welded contacts on the fault surface. In this case, the relative area of cracking (noncontact) on the fault is given by r ¼ 1 nw pb2 . Again, as an alternate representation, the faults in model 2 may be replaced by an elastically equivalent layer with bulk and shear moduli given by " 1=2 # 4m ð d=b Þr ð 1 r Þ ð l þ m Þ ð 4m l Þ 1 r K ¼ K0 1þ2 3p ð3l þ 4mÞðl þ 2mÞ p " # 8mðd=bÞr ð1 r Þ l þ m 1 r 1=2 m ¼m þ 1þ2 p 3l þ 4m p

0

where d is the aperture of the cracked (noncontact) area. Note that d/b is not the aspect ratio of cracks since b is the radius of the welded regions. The above equations are valid for small d/b. The equivalent layers can be used in Backus averaging to estimate the overall stiffness. The equations for model 1 are valid for r 1 (dilute cracks, predominantly welded) while the equations for model 2 are valid for (1 r) 1 (dilute contacts, heavily cracked). Hudson and Liu (1999) show that, to first order, the elastic response of heavily cracked faults (model 2) is the same as that of a cubic packing of identical spheres with the same number density and size of contact areas. To the first order, the actual distribution of contact regions and the shape of cavities between the contact regions do not affect the overall elastic moduli.

Uses Hudson’s model is used to estimate the effective elastic moduli and attenuation of a rock in terms of its constituents and pore space.

Assumptions and limitations The use of Hudson’s model requires the following considerations: an idealized crack shape (penny-shaped) with small aspect ratios and either small crack density or small contact density are assumed. The crack radius and the distance between cracks are much smaller than a wavelength. The formal limit quoted by Hudson for both first- and second-order terms is e less than 0.1; the second-order expansion is not a uniformly converging series and predicts increasing moduli with crack density beyond the formal limit (Cheng, 1993). Better results will be obtained by using just the first-order correction rather than inappropriately

203

4.11 Eshelby–Cheng model for cracked anisotropic media

4.11

using the second-order correction. Cheng gives a new expansion based on the Pade´ approximation, which avoids this problem; cracks are isolated with respect to fluid flow. Pore pressures are unequilibrated and adiabatic. The model is appropriate for high-frequency laboratory conditions. For low-frequency field situations, use Hudson’s dry equations and then saturate by using the Brown and Korringa relations (Section 6.5). This should not be confused with the tendency to think of this approach as a low-frequency theory, because crack dimensions are assumed to be much smaller than a wavelength; and sometimes a single crack set may not be an adequate representation of crackinduced anisotropy. In this case we need to superpose several crack sets with angular distributions.

Eshelby–Cheng model for cracked anisotropic media Synopsis Cheng (1978, 1993) has given a model for the effective moduli of cracked, transversely isotropic rocks based on Eshelby’s (1957) static solution for the strain inside an ellipsoidal inclusion in an isotropic matrix. The effective moduli ceff ij for a rock containing fluid-filled ellipsoidal cracks with their normals aligned along the 3-axis are given as 0 1 ceff ij ¼ cij cij

where f is the porosity and c0ij are the moduli of the uncracked isotropic rock. The corrections c1ij are c111 ¼ lðS31 S33 þ 1Þ þ c133 ¼

2mE DðS12 S11 þ 1Þ

ðl þ 2mÞðS12 S11 þ 1Þ þ 2lS13 þ 4mC D

ðl þ 2mÞðS13 þ S31 Þ 4mC þ lðS13 S12 S11 S33 þ 2Þ 2D m m ¼ c166 ¼ 1 2S1313 1 2S1212

c113 ¼ c144

with C¼

Kfl 3ðK Kfl Þ

D ¼ S33 S11 þ S33 S12 2S31 S13 ðS11 þ S12 þ S33 1 3CÞ C½S11 þ S12 þ 2ðS33 S13 S31 Þ

204

Effective elastic media: bounds and mixing laws

E ¼ S33 S11 S31 S13 ðS33 þ S11 2C 1Þ þ CðS31 þ S13 S11 S33 Þ 4p S11 ¼ QIaa þ RIa S33 ¼ Q 2Iac a2 þ Ic R 3 S12 ¼ QIab RIa

S13 ¼ QIac a2 RIa

S31 ¼ QIac RIc

S1212 ¼ QIab þ RIa

S1313 ¼ Ia ¼

Qð1 þ a2 ÞIac RðIa þ Ic Þ þ 2 2

2paðcos1 a aSa Þ S3a

Iac ¼

Ic Ia 3S2a

¼

3K 2m 6K þ 2m

R¼

1 2 8pð1 Þ

Iaa ¼ p Sa ¼

Ic ¼ 4p 2Ia 3Iac 4

Iab ¼

Iaa 3

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 a2

Q¼

3R 1 2

In the preceding equations, K and m are the bulk and shear moduli of the isotropic matrix, respectively; Kfl is the bulk modulus of the fluid; and a is the crack aspect ratio. Dry cavities can be modeled by setting the inclusion moduli to zero. Do not confuse S with the anisotropic compliance tensor. This model is valid for arbitrary aspect ratios, unlike the Hudson model, which assumes very small aspect ratio cracks (see Section 4.10 on Hudson’s model). The results of the two models are essentially the same for small aspect ratios and low crack densities (< 0.1), as long as the “weak inclusion” form of Hudson’s theory is used (Cheng, 1993).

Uses The Eshelby–Cheng model is used to obtain the effective anisotropic stiffness tensor for transversely isotropic, cracked rocks.

Assumptions and limitations The following presuppositions and limitations apply to the Eshelby–Cheng model: the model assumes an isotropic, homogeneous, elastic background matrix and an idealized ellipsoidal crack shape; the model assumes low crack concentrations but can handle all aspect ratios; and because the cavities are isolated with respect to flow, this approach simulates very-high-frequency behavior appropriate to ultrasonic laboratory conditions.

205

4.12 T-matrix inclusion models for effective moduli

At low frequencies, when there is time for wave-induced pore-pressure increments to flow and equilibrate, it is better to find the effective moduli for dry cavities and then saturate them with the Brown and Korringa low-frequency relations. This should not be confused with the tendency to term this approach a low-frequency theory, for crack dimensions are assumed to be much smaller than a wavelength.

Extensions The model has been extended to a transversely isotropic background by Nishizawa (1982).

4.12

T-matrix inclusion models for effective moduli Synopsis One approach for estimating effective elastic constants for composites is based on the integral equation or T-matrix approach of quantum scattering theory. This approach takes into account interactions between inclusions based on multiple-point correlation functions. The integral equation for effective elastic constants of macroscopically homogeneous materials with statistical fluctuation of properties at the microscopic level is very similar to the Lippmann–Schwinger–Dyson equation of multiple scattering in quantum mechanics. The theory in the context of elastic composites was developed by Eimer (1967, 1968), Kro¨ner (1967, 1977, 1986), Zeller and Dederichs (1973), Korringa (1973), and Gubernatis and Krumhansl (1975). Willis (1977) used a T-matrix approach to obtain bounds and estimates for anisotropic composites. Middya and Basu (1986) applied the T-matrix formalism for polycrystalline aggregates. Jakobsen et al. (2003a) synthesized many of the existing effective medium approximations and placed them on a common footing using the T-matrix language. They also applied the formalism to model elastic properties of anisotropic shales. A homogeneous anisotropic matrix with elastic stiffness tensor Cð0Þ has embedded inclusions, divided into families r ¼ 1,2, . . . ,N having concentrations nðrÞ and shapes aðrÞ . For ellipsoidal inclusions aðrÞ denotes the aspect ratio of the rth family of inclusions. While the general theory is not limited to ellipsoidal inclusions, most practical calculations are carried out assuming idealized ellipsoidal shapes. The effective elastic stiffness is given by Jakobsen et al. (2003a) as 1 CT ¼ Cð0Þ þ hT1 i I hT1 i1 X ð4:12:1Þ where hT1 i ¼

X r

nðrÞ tðrÞ

206

Effective elastic media: bounds and mixing laws

X¼

XX r

ðrsÞ

nðrÞ tðrÞ Gd tðsÞ nðsÞ

s

and I ¼ Ijklm ¼ 12 ðdjl dkm þ djm dkl Þ is the fourth-rank identity tensor. The T-matrix t(r) for a single inclusion of elastic stiffness CðrÞ is given by 1 tðrÞ ¼ dCðrÞ I GðrÞ dCðrÞ where dCðrÞ ¼ CðrÞ Cð0Þ The fourth-rank tensor G(r) (not the same as the tensor Gd(rs)) is a function of only C(0) and the inclusion shape. It is computed using the following equations: i 1 h ðrÞ ðrÞ ðrÞ ðrÞ GðrÞ ¼ þ E þ E þ E E pqrs pqsr qprs qpsr 4 pqrs Z p Z 2p ðrÞ EðrÞ ¼ dy sin y d D1 pqrs qs ðkÞkp kr A ðy; Þ 0

0

Dqs ¼ Cð0Þ qmsn km kn where k, y, and f are spherical coordinates in k-space (Fourier space), and the components of the unit k-vector are given as usual by ½sin y cos ; sin y sin ; cos y. The term A(r)(y,f) is a shape factor for the rth inclusion and is independent of elastic constants. For an oblate spheroidal inclusion with short axis b3 aligned along the x3-axis, long axes b1 ¼ b2, and aspect ratio a(r) ¼ b3/b1, A(r) is independent of f and is given by AðrÞ ðyÞ ¼

1 h 4p

aðrÞ

i3=2 sin2 y þ ðaðrÞ Þ2 cos2 y

For a spherical inclusion A(r) ¼ 1/(4p). For the case when C(0) has hexagonal symmetry and the principal axes of the oblate spheroidal inclusions are aligned along the symmetry axes of C(0), the nonzero elements of Epqrs are given by definite integrals of polynomial functions (Mura, 1982): Z p 1 E1111 ¼ E2222 ¼ ð1 x2 Þf½ f ð1 x2 Þ þ hg2 x2 2 0

½ð3e þ dÞð1 x2 Þ þ 4f g2 x2 g2 g2 x2 ð1 x2 Þg dx Z

1

E3333 ¼ 4p

g2 x2 ½dð1 x2 Þ þ f g2 x2 ½eð1 x2 Þ þ f g2 x2 dx

0

E1122 ¼ E2211

p ¼ 2

Z

1

ð1 x2 Þf½ f ð1 x2 Þ þ hg2 x2

0

½ðe þ 3dÞð1 x2 Þ þ 4f g2 x2 3g2 g2 x2 ð1 x2 Þg dx

207

4.12 T-matrix inclusion models for effective moduli

Z

1

E1133 ¼ E2233 ¼ 2p

g2 x2 f½ðd þ eÞð1 x2 Þ þ 2f g2 x2

0

½ f ð1 x2 Þ þ hg2 x2 g2 g2 x2 ð1 x2 Þg dx Z

1

E3311 ¼ E3322 ¼ 2p

ð1 x2 Þ½dð1 x2 Þ þ f g2 x2

0

½eð1 x2 Þ þ f g2 x2 dx E1212

p ¼ 2

Z

1

ð1 x2 Þ2 fg2 g2 x2 ðd eÞ½ f ð1 x2 Þ þ hg2 x2 g dx

0

Z E1313 ¼ E2323 ¼ 2p

1

gg2 x2 ð1 x2 Þ½eð1 x2 Þ þ f g2 x2 dx

0

where 1 ¼ ½eð1 x2 Þ þ f g2 x2 f½dð1 x2 Þ þ f g2 x2 ½ f ð1 x2 Þ þ hg2 x2 g2 g2 x2 ð1 x2 Þg ð0Þ

e ¼ ½C11 C12 =2

ð0Þ

g ¼ C13 þ C44

d ¼ C11 f ¼ C44

ð0Þ

ð0Þ

ð0Þ

ð0Þ

ð0Þ

h ¼ C33

and x is the integration variable while g ¼ a is the inclusion aspect ratio. Using these equations, the tensor Epqrs (and hence Gpqrs) can be obtained by numerical or symbolic integration. For numerical computations, instead of the more common Voigt notation, the Kelvin notation of fourth-rank tensors in terms of 6 6 matrices is more efficient, since matrix operations can be performed on the Kelvin 6 6 matrices according to the usual matrix rules (see also Section 2.2). In the equation for the effective elastic stiffness, the term X involving the tensor (rs) Gd represents the two-point interaction between the rth set and the sth set of (r) inclusions. The tensor G(rs) is computed in the same way as the tensor Gijkl except d that the aspect ratio of the associated inclusion is set to be the aspect ratio, ad, of the ellipsoidal two-point spatial correlation function, representing the symmetry of p(s|r)(xx0 ), the conditional probability of finding an inclusion of type s at x0 , given that there is an inclusion of type r at x. The aspect ratio of the spatial correlation function can be different from the aspect ratio of the inclusions. The effect of spatial distribution is of higher order in the volume concentrations than the effect of inclusion shapes. The effect of spatial distribution becomes important when the inclusion concentration increases beyond the dilute limit. However, even the T-matrix approximation given by equation (4.12.1) may be invalid at very high concentration, as it neglects the higher-order interactions beyond two-point interactions (Jakobsen et al., 2003a). According to Jakobsen et al. (2003a) it appears to be difficult to determine the range of validity for the various T-matrix approximations. For the case

208

Effective elastic media: bounds and mixing laws

of a matrix phase with just one type of inclusion, Ponte-Castaneda and Willis (1995) found that the T-matrix approximation is consistent only for certain combinations of ad, (aspect ratio of the correlation function), a (inclusion aspect ratio), and n (inclusion concentration). When ad > a, the minimum possible value for inclusion aspect ratio is amin ¼ adn. Using a series expansion of the reciprocal, the equation for the effective stiffness may be written as X XX 3 ð0Þ ðrÞ ðrÞ ðrÞ ðrÞ ðrsÞ ðsÞ ðsÞ CT ¼ C þ v t n t Gd t n þ O vðrÞ r

r

s

which agrees with Eshelby’s (1957) dilute estimate to first order in concentration. When the spatial distribution is assumed to be the same for all pairs of interðrsÞ acting inclusions, Gd ¼ Gd for all r and s. Under this assumption the equation for CT becomes " CT

¼C

ð0Þ

þ

X

#( ðrÞ ðrÞ

I þ Gd

v t

" X

r

#)1 ðsÞ ðsÞ

t n

s

or CT

¼C

ð0Þ

þ

X

! ðrÞ ðrÞ

v t

r

X r

! ðrÞ ðrÞ

v t

Gd

X

! ðsÞ ðsÞ

v t

3 þ O nðrÞ

s

These non-self-consistent, second-order T-matrix approximations with a well-defined C(0) are sometimes termed the generalized optical potential approximation (OPA) (Jakobsen et al., 2003a). The generalized OPA equations reduce to Nishizawa’s (1982) differential effective medium formulation when the second-order correction terms are dropped, and the calculations are done incrementally, increasing the inclusion concentration in small steps, and taking C(0) to be the effective stiffness from the previous incremental step. This formulation can be written as a system of ordinary differential equations for the effective stiffness CDEM (Jakobsen et al., 2000): h

1 nðrÞ

i dC

DEM dnðrÞ

h ih i1 ¼ CðrÞ CDEM I GðDEMÞ ðCðrÞ CDEM Þ

with initial conditions CDEM ðnðrÞ ¼ 0Þ ¼ Cð0Þ The superscript (DEM) on the tensor G indicates that it is computed for a medium having elastic stiffness CDEM . When the multiphase aggregate does not have a clearly defined matrix material and each phase is in the form of inclusions embedded in a sea of inclusions, one can set

209

4.12 T-matrix inclusion models for effective moduli

Cð0Þ ¼ CT to give the self-consistent T-matrix approximation, also called the generalized coherent potential approximation (CPA): 91 " #8 " #1 = < X XX X ðpqÞ vðrÞ tðrÞ vðsÞ tðsÞ nðpÞ tðpÞ Gd nðqÞ tðqÞ ¼0 Iþ ; : r s p q or " X

# vðrÞ tðrÞ

r

XX p

ðpqÞ

nðpÞ tðpÞ Gd nðqÞ tðqÞ 0

q

where the subscript * on the tensor quantities t and G indicates that they are computed for a reference medium having the (as yet unknown) elastic stiffness CT . The generalized CPA equations are an implicit set of equations for the effective stiffness, and must be solved iteratively. The generalized CPA reduces to the firstorder CPA, ignoring higher-order terms: X vðrÞ tðrÞ 0 r

which is equivalent to Berryman’s symmetric self-consistent approximation (see Section 4.8). When the rth set of inclusions have a spatial distribution described by an orientation distribution function OðrÞ ðy; ; Þ, the T-matrix for the individual inclusion, tðrÞ , in the above equations for the first- and second-order corrections have to be replaced by the orientation averaged tensor tðrÞ given by Z p Z 2p Z 2p tðrÞ ¼ dy sin y d d OðrÞ ðy; ; Þ tðrÞ ðy; ; Þ 0

0

0

The three Euler angles ðy; ; Þ define the orientation of the ellipsoid with principal axes X1X2X3 with respect to the fixed global coordinates x1x2x3, where y is the angle between the short axis of the ellipsoid and the x3-axis. Computing the orientation average T-matrix involves coordinate transformation of the inclusion stiffness tensor CðrÞ from the local coordinate system of the inclusion to the global coordinates using the usual transformation laws for the stiffness tensor. Jakobsen et al. (2003b) have extended the T-matrix formulation to take into account fluid effects in porous rocks. For a single fully saturated ellipsoidal inclusion of the rth set that is allowed to communicate and exchange fluid mass with other ðrÞ ðrÞ cavities, the T-matrix, tsat is given as the dry T-matrix tdry plus an extra term accounting for fluid effects. The T-matrix for dry cavities is obtained by setting CðrÞ to zero in the above equations. The saturated T-matrix is expressed as ðrÞ

ðrÞ

tsat ¼ tdry þ

ZðrÞ þ iokf XðrÞ 1 þ iogðrÞ

210

Effective elastic media: bounds and mixing laws ðrÞ

ðrÞ

XðrÞ ¼ tdry Sð0Þ ðI2 I2 ÞSð0Þ tdry ðrÞ

Z

¼

ðrÞ tdry Sð0Þ ðI2

¼ kf

8 < :

ð0Þ

I2 ÞS

1 kf Sð0Þ uuvv

!

ðrÞ

X

nðrÞ tdry

r

1 þ iogðrÞ

X r

ðrÞ

n 1 þ iogðrÞ

!

1 91 0 ðrÞ = X nðrÞ Kd ik k k u v uv f uuvv A þ kf @ of ; 1 þ iogðrÞ r

1 ðrÞ Kd ¼ I4 þ GðrÞ Cð0Þ Sð0Þ ðrÞ gðrÞ ¼ 1 þ kf Kd Sð0Þ

uuvv

1 In the above equations, Sð0Þ ¼ Cð0Þ is the elastic compliance of the background medium, kf and f are the bulk modulus and viscosity of the fluid, o is the frequency, t is a flow-related relaxation-time constant, ku, kn are components of the wave vector, and Guv is the (possibly anisotropic) permeability tensor of the rock. Repeated subscripts u and v imply summation over u, v ¼ 1, 2, 3. The tensors I2 and I4 are second- and fourth-rank identity tensors, respectively. The dyadic product is denoted by and ðI2 I2 Þijkl ¼ dij dkl . Usually the relaxation-time constant t is poorly defined and has to be calibrated from experimental data. Jakobsen et al. (2003b) show that the first-order approximations of these expressions reduce to the Brown and Korringa results (see Section 6.5) in the low-frequency limit.

Uses The results described in this section can be used to model effective-medium elastic properties of composites made up of anisotropic ellipsoidal inclusions in an anisotropic background. The formulation for saturated, communicating cavities can be used to model fluid-related velocity dispersion and attenuation in porous rocks.

Assumptions and limitations The results described in this section assume the following: idealized ellipsoidal inclusion shape; linear elasticity and a Newtonian fluid.

4.13

Elastic constants in finely layered media: Backus average Synopsis A transversely isotropic medium with the symmetry axis in the x3-direction has an elastic stiffness tensor that can be written in the condensed Voigt matrix form (see Section 2.2 on anisotropy):

211

4.13 Backus average

2

a 6b 6 6f 6 60 6 40 0

b a f 0 0 0

f 0 f 0 c 0 0 d 0 0 0 0

3 0 07 7 07 7; 07 7 05 m

0 0 0 0 d 0

1 m ¼ ða bÞ 2

where a, b, c, d, and f are five independent elastic constants. Backus (1962) showed that in the long-wavelength limit a stratified medium composed of layers of transversely isotropic materials (each with its symmetry axis normal to the strata) is also effectively anisotropic, with effective stiffness as follows: 2 3 A B F 0 0 0 6B A F 0 0 0 7 6 7 6F F C 0 0 0 7 1 6 7 6 0 0 0 D 0 0 7; M ¼ 2 ðA BÞ 6 7 40 0 0 0 D 0 5 0 0 0 0 0 M where 1 1 2 fc A ¼ a f 2 c1 þ c1 1 1 2 B ¼ b f 2 c1 þ c1 fc 1 C ¼ c1 1 1 F ¼ c1 fc 1 D ¼ d 1 M ¼ hmi The brackets hi indicate averages of the enclosed properties weighted by their volumetric proportions. This is often called the Backus average. If the individual layers are isotropic, the effective medium is still transversely isotropic, but the number of independent constants needed to describe each individual layer is reduced to 2: a ¼ c ¼ l þ 2m;

b ¼ f ¼ l;

d¼m¼m

giving for the effective medium 1 2 4mðl þ mÞ 1 l A¼ þ l þ 2m l þ 2m l þ 2m 1 2 2ml 1 l B¼ þ l þ 2m l þ 2m l þ 2m

212

Effective elastic media: bounds and mixing laws

C¼

1 l þ 2m

1 F¼ l þ 2m 1 1 D¼ m

1 1

l l þ 2m

M ¼ hmi In terms of the P- and S-wave velocities and densities in the isotropic layers (Levin, 1979), a ¼ VP2 d ¼ VS2 f ¼ ðVP2 2VS2 Þ the effective parameters can be rewritten as 2 VS2 VS2 D 2 1 E1 2 þ 12 2 VP A ¼ 4VS 1 2 VP VP B¼

2 2V 2 V 2 D 2 1 E1 þ 1 2 S2 2VS2 1 2S VP VP VP

C¼

D

1 E1 VP2

F¼

E1 V2 D 1 2 S2 ðVP2 Þ1 VP

D¼

D

VS2

1 E1

M ¼ VS2 The P- and S-wave velocities in the effective anisotropic medium can be written as pﬃﬃﬃﬃﬃﬃﬃﬃﬃ VSH;h ¼ M= pﬃﬃﬃﬃﬃﬃﬃﬃﬃ VSH;v ¼ VSV;h ¼ VSV;v ¼ D= pﬃﬃﬃﬃﬃﬃﬃﬃ VP;h ¼ A= pﬃﬃﬃﬃﬃﬃﬃﬃﬃ VP;v ¼ C= where r is the average density; VP,v is for the vertically propagating P-wave; VP,h is for the horizontally propagating P-wave; VSH,h is for the horizontally propagating, horizontally polarized S-wave; VSV,h is for the horizontally propagating, vertically polarized S-wave; and VSV,v and VSH,v are for the vertically propagating S-waves of any polarization (vertical is defined as being normal to the layering).

213

4.13 Backus average

Calculate the effective anisotropic elastic constants and the velocity anisotropy for a thinly layered sequence of dolomite and shale with the following layer properties: VPð1Þ ¼ 5200 m=s; VPð2Þ ¼ 2900 m=s;

VSð1Þ ¼ 2700 m=s; 1 ¼ 2450 kg=m3 ; d1 ¼ 0:75 m VSð2Þ ¼ 1400 m=s; 2 ¼ 2340 kg=m3 ; d2 ¼ 0:5 m

The volumetric fractions are f1 ¼ d1 =ðd1 þ d2 Þ ¼ 0:6;

f2 ¼ d2 =ðd1 þ d2 Þ ¼ 0:4

If one takes the volumetric weighted averages of the appropriate properties, ! ! 2 2 VSð1Þ VSð2Þ 2 2 A ¼ f1 41 VSð1Þ 1 2 þ f2 42 VSð2Þ 1 2 VPð1Þ VPð2Þ " þ f1 1 2

2 VSð1Þ 2 VPð1Þ

! þ f2 1 2

2 VSð2Þ 2 VPð2Þ

!#2 2 Þ f1 =ð1 VPð1Þ

1 2 Þ þ f2 =ð2 VPð2Þ

A ¼ 45:1 GPa Similarly, computing the other averages we obtain C ¼ 34.03 GPa, D ¼ 8.28 GP