Fractured Rock Hydraulics
© 2010 Taylor & Francis Group, London, UK
Fractured Rock Hydraulics
F.O. Franciss Progeo – Consultoria de Engenharia Ltda, Rio de Janeiro, Brazil
© 2010 Taylor & Francis Group, London, UK
Taylor & Francis is an imprint of the Taylor & Francis Group, an informa business © 2010 Taylor & Francis Group, London, UK Typeset by Macmillan Publishing Solutions, Chennai, India Printed and bound in Great Britain by Antony Rowe (A CPI Group Company), Chippenham, Wiltshire All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Library of Congress Cataloging-in-Publication Data Franciss, F. O. Fractured rock hydraulics / Fernando Olavo Franciss. p. cm. Includes bibliographical references. ISBN 978-0-415-87418-2 (hardcover : alk. paper) – ISBN 978-0-203-85941-4 (eBook) 1. Seepage–Mathematical models. 2. Groundwater flow–Mathematical models. 3. Rocks. I. Title. TC176.F727 2010 627—dc22 2009036651 Published by:
CRC Press/Balkema P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail:
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ISBN: 978-0-415-87418-2 (Hbk) ISBN: 978-0-203-85941-4 (eBook)
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Table of Contents
About the author Preface
IX XI
Introduction
1
Fractured rock hydraulics Scope
1 2
Chapter 1
3
Fundamentals
1.1
Basic concepts 1.1.1 Pseudo-continuity 1.1.2 Observation scale 1.1.3 Description at different scales 1.1.4 Representative elementary volume 1.1.5 Hydraulic variables 1.1.5.1 Introduction 1.1.5.2 Specific discharge 1.1.5.3 Hydraulic gradient 1.1.6 Hydraulic conductivity 1.1.6.1 Introduction 1.1.6.2 Fractures and conduits 1.2 Governing equations 1.2.1 Preliminaries 1.2.2 Energy conservation principle: Darcy’s law 1.2.3 Mass conservation principle: continuity equation 1.2.3.1 General equation 1.2.3.2 Dupuit’s approximation 1.2.4 Boundary and initial conditions 1.2.4.1 Main boundary types 1.2.4.2 Submerged boundaries 1.2.4.3 Impervious boundaries 1.2.4.4 Seepage boundaries 1.2.4.5 Unconfined groundwater-air interface
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3 3 4 8 8 8 8 9 10 17 17 19 28 28 28 30 30 31 35 35 36 36 37 37
VI
1.3
Table of Contents
Addenda to Chapter 1 1.3.1 Addendum 1.1: Effective velocity and specific discharge 1.3.2 Addendum 1.2: Hydrodynamic gradient 1.3.3 Addendum 1.3: Hydraulic conductivity for randomly fractured subsystems 1.3.4 Addendum 1.4: Energy conservation principle 1.3.5 Addendum 1.5: Mass conservation principle
Chapter 2
Approximate solutions
2.1 2.2 2.3 2.4 2.5
Overview Differential operators Uniqueness of solutions Approximate solution errors Approximation methods 2.5.1 Preliminaries 2.5.2 Collocation method 2.5.3 Least squares method 2.5.4 Galerkin’s method 2.5.4.1 Orthogonality 2.5.4.2 Galerkin’s approach 2.5.4.3 “Weak solutions’’ 2.5.4.4 Variational notation 2.5.5 Time-dependent solutions 2.6 Addenda to Chapter 2 2.6.1 Addendum 2.1: Classification of second order linear partial differential equations 2.6.2 Addendum 2.2: Minimisation of the sum of the squared residuals 2.6.3 Addendum 2.3: Minimisation of the sum of the squared residuals transformed by the differential operators DV and DN 2.6.4 Addendum 2.4: The concept of “orthogonality’’ Chapter 3
Data analysis
3.1 3.2 3.3
Preliminaries Analysing geological features Handling of hydraulic head data 3.3.1 Variation in time 3.3.2 Variation in space 3.4 Handling of flow rate data 3.5 Handling of hydraulic conductivity data 3.5.1 Preliminaries 3.6 Hydraulic transmissivity and connectivity 3.6.1 Preliminaries 3.6.2 Hydraulic conductivity appraisal 3.6.2.1 Hydraulic tests at “core sample’’ scale 3.6.2.2 Hydraulic tests at “borehole integral core’’ scale 3.6.2.3 Hydraulic tests at “cluster of boreholes’’ scale 3.6.2.4 Hydraulic tests at “aquifer’’ scale
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39 39 40 41 42 43 47 47 48 53 53 59 59 60 65 71 71 71 76 77 80 87 87 88 89 89 91 91 91 91 93 96 105 107 107 108 108 108 109 109 112 115
T a b l e o f C o n t e n t s VII
3.6.3
3.7
Hydraulic connectivity appraisal 3.6.3.1 Dynamic correlations of WT time series 3.6.3.2 Filtering WT contour maps Modelling hydrogeological systems 3.7.1 Concepts 3.7.2 Guidelines to conceptual models
Chapter 4
Finite differences
4.1 4.2
Preliminaries Finite difference basics 4.2.1 Difference equations 4.2.2 Finite differences 4.2.3 Difference equations for steady-state systems 4.2.4 Difference equations for unsteady-state systems 4.2.5 Difference equations for boundary conditions 4.2.6 Simultaneous difference equations 4.2.6.1 Preliminaries 4.2.6.2 Gauss-Seidel iterative routine 4.2.6.3 Crank-Nicholson iterative routine 4.3 Finite differences algorithms for fractured rock masses 4.3.1 Preliminaries 4.3.2 Steady-state solutions 4.3.2.1 Dupuit’s approximation 4.3.2.2 3D algorithms 4.3.3 Transient solutions
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About the author
Born in 1935, Fernando Olavo Franciss grew up in Rio de Janeiro and was educated as a civil engineer in the Pontifical Catholic University of Rio de Janeiro, Brazil. He started his professional career in 1959 by his education in applied geology by the late and distinguished Prof. Reynold Barbier at the Institut Dolomieu of the University of Grenoble, France. Ten years later, in 1970, he obtained his doctoral degree from the same university. A leading rock engineer, he has gathered a lifetime of international experience in civil engineering practice, often while crossing with other fields such as engineering geology, underground mining and oil reservoir engineering. From 1964 to 1980, he worked as a part-time professor at the Pontifical Catholic University of Rio de Janeiro. Until 1991 he worked at Sondotecnica, a reputed Brazilian consulting bureau, and since then as an independent consultant. Many now well-known Brazilian experts in civil, earth and water engineering started their professional careers closely working with Prof. Franciss, a fact that pleases him very much. During his career, Dr. Franciss has had the chance to devote part of his time to an investigation of the hydraulics of fractured rocks related to civil works, mining, oil and gas storage caverns and the interactions of hydrothermal resources with dam reservoirs. He has accordingly developed a tensor approach to describe the hydraulic properties of fractured rocks and unique finite difference matrix-algorithms to model the hydraulic and hydrothermal behaviour of randomly fractured rock masses. He is member of the Brazilian Society for Soil Mechanics and Geotechnical Engineering, the Brazilian Society for Engineering and Environmental Geology, the National Academy of Engineering and the International Society for Rock Mechanics. He has won a number of prestigious awards in Brazil, and has written several papers and a number of books: Soil and Rock Hydraulics (Balkema, Rotterdam, 1985), Weak Rock Tunnelling (Balkema, Rotterdam, 1994) and a co-authored contribution with Manoel Rocha on rock mass permeability in Structural and Geotechnical Mechanics by W.J. Hall, ed. (Prentice Hall, New Jersey, 1976).
© 2010 Taylor & Francis Group, London, UK
Preface
As for any other book, the history of this one helps to understand why it was written, despite the excellent works that were published on this subject already. In fact, my interest on fracture hydraulics arose when, challenged by practical problems, I developed in 1962 an electrical analogue by simply drawing fracture traces with special electro-resistive ink and plate condensers with electro-conductive ink on very thin polyester paper. Later, in 1969, while working on my thesis on the same subject at the National Civil Engineering Laboratory of Lisbon (LNEC), I was captivated by the possibilities of the tensorial language to correctly describe anisotropic physical properties. Inspired by Ferrandon’s model1 , I tried to fit second order tensors to the hydraulic transmissivity of isolated fractures aiming at the description of the equivalent permeability of a group of fractures. In 1972, the Mines and Applied Geology Division (DMGA) of the Technological Research Institute of São Paulo (IPT) invited me, encouraged by the favourable opinion of the late but ever prized Pierre Londe, to give a course on hard rock hydraulics at the Mines Department of the Polytechnic School of the University of São Paulo (EPUSP). At the same time, I had the honour to closely work with the late and distinguished Manoel Rocha. In that fertile period, I introduced in Brazil the LNEC integral sampling method and large-scale testing techniques for rock masses. After gaining a bit more of experience as a part-time professor at the Catholic University of Rio de Janeiro, I revised the original EPUSP course and offered it in 1978 at the Institut Dolomieu, University of Grenoble, on the kind invitation of Prof. Jean Sarrot. Except for the simplified electro-analogue, the Grenoble course, published by Balkema in 1984, was incomplete in several aspects, notably because an appropriate link between fundamental concepts and tailor-made solving techniques for randomly fractured hard rock was missing. However, due to private commitments, I could not spend more time thinking about the problem. To support my colleagues at Sondotecnica, a very reputed and traditional consulting bureau in Brazil, I left the Catholic University of Rio de Janeiro to devote myself to professional activities only. Since 1994 until presently now as an independent consultant, I have had the opportunity to be deeply involved in the complex dewatering of an underground mine in fractured, faulted and karstified dolomites. Moreover, thanks to Votorantim Metais Zinco from Brazil and to Intraconsult Associates from South Africa, I had the chance to
1 Ferrandon, J. (1948), Les lois de l’écoulement de filtration, Le Génie Civil, no 2, 125, 24–28.
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XII P r e f a c e
compare the characteristics of this particular problem to some South African groundwater cases in dolomite land. During the same period, I faced many seepage problems related to dam foundation, interaction of dam reservoirs and hydrothermal resources and underground LPG storage. Thus, I was again compelled to think about fractured rock hydraulics. By fate, my clients offered me a new chance to complete my interrupted work, now finally presented in this current publication. Hydraulic behaviour of inhomogeneous and anisotropic randomly fractured rock masses is usually governed by major contacts, faults and discontinuities. On the other hand, fine discretisation of any hydrogeological system into small cuboids, a sort of 3-D bit-mapping, reasonably describes their complex discontinuous architecture and, as a natural consequence, encourages the use of finite differences to simulate their hydraulic behaviour, if their heterogeneity and the tensor character of the transmissivity of all their singularities are duly taken into account. Based on that approach, I developed and thoroughly tested specific finite difference algorithms to deal with common problems concerning the hydraulics of hard rock without restrictions to the 3D geometry and to the properties of their discontinuities. Their inclusion in the present book completes my previous but unfinished work with no other pretension than to share my personal experience with other professionals and students. To condense this information in a 12 lectures course, a concise explanation was adopted. Besides all the friends and organisations referred above, I wish to express my gratitude to many students, colleagues and clients who helped me with due criticism and sensible proposals. Finally, my thanks also go to Germaine Seijger, Senior Editor of Engineering, Water & Earth Sciences, CRC Press/Balkema and also to the Balkema team, for their patient support and incentive during the gestation period of this book.
F.O. Franciss Rio de Janeiro, August 2009
© 2010 Taylor & Francis Group, London, UK
Introduction
Fractured rock hydraulics Acid igneous rocks, like granites, basalts and gabbros, as well as metamorphic rocks, like gneisses, schists and slates, form more than 95% of the earth’s crust and make up the rigid basements of the continents, attaining more than 70 km depth at the root of high mountain chains. Basic igneous rocks, like petrified seafloor basalt extrusions, pile up to 7 km thickness under the oceans. However, igneous and metamorphic rocks emerge as outcrops only over 15% of the earth’s surface and are frequently covered by layers of sedimentary rocks, like sandstones, siltstones and limestone, up to hundreds metres thick, and also by less resistant thinner veneers, such as weathered rocks and residual soils or unconsolidated sediments like gravels, sand and clays. Yet, rock masses near the earth’s surface, up to tens or hundreds metres depth, are not massive. Indeed, distinct groups of almost planar discontinuities split all igneous and metamorphic rock masses into contiguous and practically impervious blocks of quite resistant intact rock. From a strict mechanical point of view, even if genetically dissimilar, these rock masses, as well as some types of crystalline limestone and dolomites, can be collectively classified as “hard rocks’’ or, simply, as “fractured rocks’’. By custom, the generic term fracture stands for all kinds of restricted partitions or discontinuities within rock masses. Irrelevant and small amounts of groundwater, almost clogged, accumulate inside the minute voids and micro cracks of intact rock blocks and hardly drop under the action of gravity. This kind of water remains practically unavailable for exploitation. On the other hand, significant quantities of groundwater may saturate the fractures of rock masses and other sorts of open discontinuities. This kind of water, called free water, may drain and percolate under its self-weight. Only this type of “gravity-driven’’ groundwater is the focus of the present book. In the last 50 years, many technical publications have dealt with the theoretical principles and practical field and laboratory tests related to the appraisal of the groundwater flow through sedimentary rocks and unconsolidated sediments. In the same period, as “hard rocks’’ are not good aquifers by themselves, few publications have considered fractured rock hydraulics. In fact, groundwater storage volume is proportional to the porosity of the rock reservoir, including interconnected interstitial voids, fractures and dissolution discontinuities. For hard rocks, porosities seldom attain 2% of their total volume while may range from 5 to 15% for consolidated clastic rocks and up to 25%
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Fractured rock hydraulics
for unconsolidated sands and gravels. Yet, during the last decades, the growing need of groundwater supply in “hard rock’’ land, mainly in poor countries, coupled with a better valuation of the seasonal groundwater recharge and corresponding transient storage at the top of deep weathered “hard rock’’ land in wet inter-tropical climate, has altered attitudes. Moreover, in the same period, new investigation methods and new technologies based on sounder theoretical concepts have been devised to improve “hard-rock’’ hydraulics models, commonly motivated by attempts to enhance the structural and environmental safety factors of increasingly larger engineering works founded and excavated in these rocks. Civil or mining facilities under the sea level or under the groundwater table, such as access drifts, stopes, panels, galleries, tunnels, shafts, underground hydropower plants, oil and gas storage caverns, and classed nuclear waste deep disposal, are examples of excavation works where their structural behaviour highly interacts with their hydraulics.
Scope This book is written for readers with some knowledge of hydraulics, geology, hydrogeology and soil and rock mechanics. It has an introductory level. Chapter 1 covers the fundamentals of fractured rock hydraulics under a tensor approach. Chapter 2 presents some key concepts about approximate solutions. Chapter 3 discuss some data analysis techniques applied to groundwater modelling. Chapter 4 presents 3D finite difference matrix algorithms to simulate practical problems concerning heterogeneous and anisotropic fractured rock masses.
© 2010 Taylor & Francis Group, London, UK
Chapter 1
Fundamentals
1.1 Basic concepts 1.1.1
Ps eu d o - c o n t i n u i t y
To construct uncomplicated models, the governing equations of groundwater movement must be based on the simplified assumptions of classical continuum mechanics. According to its fundamental postulate – the continuum concept of matter – the observation and the prediction of the groundwater behaviour in space and time must be described from a “macroscopic’’ point of view but taking implicitly into account all interactions, heterogeneities, anisotropies and discontinuities influencing the groundwater movement from a “microscopic’’ point of view. Boundaries separating the “macro-scale’’ from the “micro-scale’’ depend simultaneously on the purposes of the analysis and on the characteristics of the pervious media. In addition, the application of the continuum concept to any material system assumes an unbroken distribution of its matter in space and its uninterrupted existence in time. Consequently, to model a groundwater system according to these principles, two key conditions must be fulfilled: • •
First, the size of its subsystems must be set above an almost empirical minimum size demarcating the interface between the macro and the micro observation scales. Second, a fictitious continuous system, i.e. a pseudo-continuum, having an almost equivalent hydraulic behaviour, must substitute for the real discontinuous matter.
If these two conditions are satisfied, it is possible to construct an acceptable numerical model by observing the following preliminary steps: • • •
First, select the most adequate observation scale for the pseudo-continuous system, taking into account the compatibility between the model size and the desired level of detail. Second, split the pseudo-continuous system into several smaller pseudo-continuous subsystems. Third, describe the hydraulic properties of these subsystems and formulate their corresponding governing flow equations so as to approximately replicate the factual hydraulic behaviour of the system at the elected observation scale.
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Fractured rock hydraulics
Crushed zones and major fracture zones Fracture zones or shear zones 500 m
N
Figure 1.1 Example of a typical mosaic pattern delineated by shear zones and fractures affecting the underlying hard rock basement of Norway, Sweden, Finland, and Northern Denmark, over 3.1 billion years old, dubbed Fennoscandia (figure adapted from S, Johansson, Large Rock Caverns at Neste Oy’s Porvoo Works, Large Rock Caverns Proceedings of the International Symposium, Helsinki, 1986, structural features adapted from fig. 2 of Volume 1).
For most numerical models, the modal (most observed or most frequent class of values) linear dimension of the pseudo-continuous subsystems generally do not exceed 1/30 to 1/300 of the greatest linear dimension of the model. 1.1.2
Ob s er va t i o n s c a l e
Natural discontinuities occurring in hard rock basements delineate several interlaced and hierarchical nets. Based on their relevance, it is possible to distinguish a number of well-defined families of discontinuities, from major lineaments, as tension fractures and shear zones, to minor joint systems. The major discontinuities outline contiguous hard rock mosaics, from high order size, between 10 to 100 km, to low order size, between 100 to 1000 m. Several minor discontinuities, classified by frequency and magnitude, still crack these mosaics into smaller and less relevant units, imparting structural complexity to hard rock domains (see see fig. 1.1). As a result, it is difficult to construct adequate pseudo-continuous models for these intermingled lithologic and structural features. However, there are rough guidelines for setting the minimum size for their pseudo-continuous subsystems. This minimum is related to the extent of the model, the length and frequency of the discontinuities and the desired simulation details. In plain words: it depends on the observation scale. Common observation scales may be classified as follows: •
Class I – “infilling scale’’: Consider, for example, a major discontinuity plugged by a pervious granular soil (see fig. 1.2). To simulate the hydraulic behavior of that confined layer, the size of their pseudo-continuous subsystems must be greater than 10 to 30 times the soil modal pore diameter. Their hydraulic properties must
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Fundamentals
Intact rock
5
Fault gauge ≈2m
Pseudo-continuous subsystem
Figure 1.2 Some simulations require a detailed appraisal of the hydraulic influence of the infillings of intersecting major discontinuities under high differential pressures. An assemblage of pseudo-continuous subsystems at a very small observation scale may model these pervious granular layers adequately (adapted illustration from NTNU-Project Report 1D-98).
•
•
•
reflect the integrated effect of all individual grains and pores they contain. Civil engineers or hydrogeologists hardly ever perform this type of analysis. However, this very small-scale approach may be useful to evaluate and predict effects of burial pressure changes on the permeability and porosity of major discontinuities in oil reservoirs. Class II – “intact rock scale’’: Consider, for example, the tension gashes associated with stylolites in carbonate rock blocks. These small structural features play an important role in some oil reservoirs (see fig. 1.3). Their equivalent hydraulic properties are sensibly anisotropic. The smallest size of their pseudo-continuous subsystems must be about 10 to 30 times the modal discontinuity spacing. Class III – “discontinuity scale’’: Consider, for example, the observation of the hydraulic behavior of partially filled and partially sealed discontinuities taken one by one. In this case, the major dimension of their flat rectangular pseudocontinuous subsystems must be greater than 10 to 30 times the modal discontinuity thickness (see fig. 1.4). Their hydraulic properties reflect the integrated effect of their incomplete infilling, imperfect sealing and erratic voids. This type of analysis may be employed to predict the hydraulic behaviour of discontinuities in close proximity to underground civil engineering works or groups of wells. Class IV – “rock mass scale’’: Consider, for example, a pseudo-continuous cuboid subsystem enclosing a volume of rock mass partitioned by several discontinuities at random, separating intact rock blocks (see fig. 1.5). For one or more erratic discontinuities, the equivalent hydraulic property of that pseudo-continuous subsystem results from specific construction rules depending on the hydraulic properties of every discontinuity, as discussed in section 2.5.4. In this case, the minimum subsystem size must be greater than 3 to 10 times the thickness of the most relevant discontinuity. For quasi-systematic fracture sets, the equivalent hydraulic property
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Fractured rock hydraulics
Figure 1.3 Stylolites associated with secondary fissures. These features are jagged discontinuities but are not structural fractures. They occur in limestone and other sedimentary rocks and typically delineate irregular and interlocked small columns, pits and teeth-like projections, probably formed diagenetically by differential movement under pressure, accompanied by solution.They may be widened by subsequent groundwater flow. A valuable discussion about the 3D effect of intersecting stylolites on the permeability of whole cores can be found in Nelson, 2001.
Pseudocontinuous subsystem size
Vast discontinuity with variable thickness and infillings with variable hydraulic properties
N
Figure 1.4 Some simulations require the detailed appraisal of the hydraulic influence of major discontinuities that crosses the flow domain. In these very special cases, an assemblage of pseudo-continuous subsystems with variable hydraulic properties may model the pervious discontinuity adequately (Canadian Shield, figure adapted from Les Eaux Souterraines des Roched Dures du Socle, UNESCO, 1987, approximate scale 1:8000).
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Fundamentals
7
Erratically discontinuous rock mass subsystem
Random fractures Pseudo-continuous subsystem hydraulically equivalent
Figure 1.5 An erratically discontinuous rock mass subsystem may be substituted by a pseudocontinuous subsystem having an almost equivalent hydraulic behaviour.
Bearing II
Bearing I
Bearing IV
Bearing III
Kmin E-8 m/s
Kint 1.2E-8 m/s
Kmax 1.7E-8 m/s
Bearing V 7.367.000 N 300 m 400 m 200 m
20 m
7.366.00 N 100 m 454.000 E
455.000 E 0m
Atlantic Ocean
Figure 1.6 A mixed model may adequately simulate the hydraulic behaviour of the small rectangular area enclosing well-defined discontinuities but correctly coupled to the hard rock mosaics (South Atlantic Brazilian coast).
•
may also be statistically inferred. In this case, the subsystem must be greater than 10 to 30 times their modal spacing. Class V – “hard rock basement scale’’: Consider, for example, a hard rock basement depicting mosaics separated by several lineaments. In this case, more than one observation scale may be combined into a single model: smaller and restricted models for a group of lineaments and a greater integrated model for all mosaics (see fig. 1.6).
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Fractured rock hydraulics
In practice, the minimum size of the pseudo-continuous subsystems for numerical simulations normally exceeds the minimum imposed by the rules set out on pages 4–7. There are no simple criteria to define the best size. Usually, a satisfactory limit is elected after judging the most convenient size for the model, the costs of getting the geological data, the desired output details, the trade-off between truncation errors and round-off errors, costs of processing time and storage capacity, etc. To reduce costs, it is always possible to refine a previous simulation only within certain subdomains where this is essential. It is important to note that the size of a pseudo-continuous subsystem may range from few centimetres (for a granular seam) to tens or hundreds of metres (for a huge aquifer). In transient processes, the system variables change from an unbalanced condition to a final state of equilibrium. In this case, the observation time interval may vary from a few seconds or minutes (for example, during the equalisation of neutral pressure transients in saturated cores tested in a triaxial chamber) to several hours or days (for example, during the progressive dewatering of an underground mine). 1.1.3
Des c r i p t i o n a t d i f f e r e n t s c a l e s
All variables describing the hydraulic properties and the physical state of a pseudocontinuous subsystem, such as porosities, pressure heads, flow rates and hydraulic conductivities, not only must be referred to its geometrical centre and to the midpoint of the time interval but also must be, in some way, properly averaged in space and time. Therefore, if the output of a measuring device can only be considered as a fair average for a volume sensibly smaller than the volume of the modelled subsystem, this single output cannot be extended to the whole subsystem. Similar considerations apply to an observation time interval. To be consistent, the size of the rock mass affected by the measurement and the size of the subsystem must be quite comparable. If not, a coherent handling of the measurement data is imperative. These remarks must be kept in mind when calibrating numerical models against a few field measurements or test results located in strategic points of the flow domain. 1.1.4
Rep r e s e n t a t i ve e l e m e n t a r y vo l um e
Bear introduced the elegant concept of the representative elementary volume, abbreviated as REV (Bear, 1972). For quasi-homogeneous rock masses, this concept can be recognized in a physical or a statistical sense, but always associated to an elected class of lithologic or structural features. In a physical sense, REV corresponds to the minimum size of a test volume that will probably give significant average outcomes that can be extended to the whole volume of a quasi-homogeneous fractured rock mass. In a statistical sense, REV corresponds to the smallest sampling volume allowing a confident statistical description of the character and physical properties of the whole volume. Depending on the chosen observation scale, the REV size and the pseudo-continuous subsystem size may be of the same order. However, for an erratically inhomogeneous fractured rock mass the REV concept can hardly be applied. 1.1.5
H yd r au l i c va r i a b l e s
1.1.5.1 Int ro d u c ti on In brief, a pseudo-continuous groundwater model simulates the space and time variation of two dependent measurable physical quantities within the flow domain. The first
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Fundamentals
9
x3
x2 x1
Pseudo-continuous subsystem defined by the extremity of a position vector r
Figure 1.7 Coordinates (x1 , x2 , x3 ) of the extremity of a position vector r defines the geometric centre of a pseudo-continuous subsystem.
one, dubbed hydraulic gradient, measures the spatial decay rate of the energy carried by a percolating water particle. The second one, called the specific discharge, measures the intensity of the resulting flow. Both variables are interconnected by a third parameter, called the hydraulic conductivity. Jointly, these three interrelated variables adequately describe the hydraulics of a groundwater system. They must be referred to the geometrical centre of each pseudo-continuous subsystem (see fig. 1.7). The hydraulic head is described by a scalar quantity, i.e. by a single number, but the hydraulic gradient and the associated specific discharge are described by vector quantities. A vector quantity may be split into three components referred to a rectangular Cartesian coordinate system. These three components can be converted to magnitude and direction measurements. The hydraulic conductivity that typifies an isotropic pervious media is condensed in a simple scalar quantity. However, for an anisotropic media this quantity can only be described by a second order tensor. Generally, the description of a second order tensor requires nine components properly referred to an arbitrary rectangular Cartesian coordinate system. These nine components can be reduced to three if suitably referred to the particular directions of three privileged axes. 1.1.5.2 S pe c i f ic d i s c h a r g e As for soil mechanics, the void volume Vv contained in the interior of a volume V of a pseudo-continuous subsystem corresponds to the total volume that is occupied by the non-solid phase, i.e. gas and water. It may be associated with conduits, fractures and other types of discontinuities, including minute openings. The effective void volume Ve excludes from the void volume Vv not only all hydraulically isolated pores or closed discontinuities but also all immobilised and very thin water layers bonded to the solid phase surface. Consequently, Ve is somewhat smaller than Vv . The effective porosity ne equals the ratio between the effective volume Ve and the total volume V, i.e. ne = Ve /V. It has no units and for hard rocks effective porosities seldom attain 1 to 2% of total volume, while they may range from 5 to 15% for consolidated clastic rocks and up to 25% for unconsolidated sands and gravels (see fig. 1.8). As formulated by Bear, the groundwater average effective velocity ve ascribed to the centre of the pseudo-continuous subsystem corresponds to the vector average of the
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Fractured rock hydraulics
Free liquid phase: flowing water Solid phase: intact rock
Adherent liquid phase: pellicle water
Figure 1.8 Schematic association of the intact rock, the percolating water and the immobilised pellicle water. Free water particles seep with varying speeds. The average effective velocity ve of the moving water ascribed to the centre of the pseudo-continuous subsystem corresponds to the vector-average of the velocities of all water particles vP percolating throughout the effective voids during an observation time interval δt.
velocities of all water particles vP percolating throughout the effective voids during an observation time interval δt (Bear et al., 1968). The specific discharge q, also called Darcy’s velocity (Darcy, 1856), may be determined by the product of the effective velocity ve and the effective porosity ne , i.e. q = ne · ve (see addendum 1.1 on page 39). It is important to note that the specific discharge q is always smaller than the effective velocity ve . The effective velocity ve and the specific discharge q are collinear vector quantities, carrying the units of a velocity LT−1 . Both are tangent vectors to the pseudostreamlines. They describe vector fields q(r, t) and ve (r, t) referred to an arbitrary Cartesian frame that are functionally related to the space-time coordinates (x1 , x2 , x3 , t) or, in compact symbolic lettering, to (r, t). Percolating water, entering or leaving the boundaries of a pseudo-continuous system, can only seep across the voids of the boundary elements δS (see fig. 1.9). Then, the elementary discharge δQ traversing an area element δS result from the scalar product of the effective voids area ne δS by the effective velocity ve : δQ = ve ne δ S = |q| |δ S| cos (ve , us )
(1.1)
In the above expression (ve , uS ) denotes the angle between the vector ve and the exterior normal uS . By the convention adopted in this book, δQ is positive for inflows and negative for outflows.
1.1.5.3 H y d ra u l i c g r a d i en t A percolating water particle consumes part of its own energy as it moves and as time elapses. The total stored energy associated to a percolating water particle, dubbed hydraulic head, customarily symbolised by H, is a scalar quantity that results from
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Fundamentals
δS ⫽ us·δS
11
Ve
δS
δS
Figure 1.9 Any area element δS of a pseudo-continuous subsystem boundary S is a vector quantity defined by the product of its area δS and its exterior normal uS . Table 1.1 Common normalised expressions of the total hydraulic head H. Normalised hydraulic head H
Type of energy
Referred to the density of water ρ (L2 T−2 )
Referred to the unit weight of water ρg (L)
Referred to the unit volume L3 (ML−1 T−2 )
Gravitational Piezometric Kinetic
gZ p/ρ v2e /2
Z p/ρg v2e /2g
ρgZ p ρ v2e /2
the sum of three kinds of energy: gravitational, piezometric and kinetic. The gravitational head corresponds to the potential energy acquired by a water particle when it is raised from an arbitrary reference level (such as the mean sea level) to an elevation Z. The piezometric head measures the elastic energy stored in a water particle when compressed from the atmospheric pressure Patm to a confining pressure P. To make things easier, a relative scale p is adopted for the piezometric head. In that scale, the barometric pressure rises and falls are bypassed by reckoning the absolute pressure P from the atmospheric pressure Patm , i.e. the absolute atmospheric pressure Patm is taken as the zero origin of the relative pressure scale of p (gauge pressure). Therefore, relative patm = 0 always and the absolute pressure P = p + Patm . The kinetic head measures the additional energy due to its motion. As typical effective velocities ve are very small, usually from 10−10 to 1 m/s, kinetic energies carried by percolating water particles in soils and rocks are commonly neglected. In this case, the total head H is only a potential head. To be homogeneously written, as in table 1.1, these energies should be normalised by the density or unit weight or unit volume of water. A transient or unsteady-state head field H(r, t) characterises a flow domain where the potential head H varies both in space and time. For any point in this field, its
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Fractured rock hydraulics
potential head H may be described by a functional relationship involving its spacecoordinate r and time-coordinate t. A permanent or a steady-state head field H(r) characterises a flow domain where H solely varies in space and may be described by the space-coordinates r only. Points on both types of fields storing the same potential head H configure an equipotential surface: a moving one for a transient condition or a stable one for a permanent condition. In a zenithal orthogonal coordinate system, the vertical Z-axis points to the zenith. When written in L units, the head H at a generic point (E, N, Z) results from the sum of the gravitational head Z plus the piezometric head p/ρg: H=Z+
p ρg
(1.2)
In another reference frame xi (i = 1, 2, 3) the head H at a generic point (x1 , x2 , x3 ) is expressed as: H = Z0 + x1 cos (Z, x1 ) + x2 cos (Z, x2 ) + x3 cos (Z, x3 )
(1.3)
In the above expression, Z0 denotes the origin elevation and cos(Z, xi ) corresponds to direction cosines of the axes xi referred to the zenithal axis Z. They are measured in the trigonometric sense. The head decay per unit length of a moving water particle, its spatial head decay rate, may be measured by the space gradient of the hydraulic head. Consider two points A and B of a head field H(r, t) joined by a straight line AB to which are respectively associated two position vectors rA and rB and two heads HA and HB . Their separation δr and their differential head δH can be respectively calculated by the vector distance (rB − rA ) and the scalar difference (HB − HA ). Now, assume that A and B are selected in such a manner that the straight line AB is perpendicular to the equipotential surface HA through the point A. In this particular configuration, the scalar-to-vector ratio δH/δr measures the average spatial head decay rate between these two points. Although it is mathematically inappropriate to derive a scalar quantity H by a vector quantity r, or, more generally, to derive a tensor P by another tensor Q, it is convenient to define the symbolic partial derivative ∂P/∂Q as a tensor R whose rank equals the sum of the ranks of P and Q (Koerber, 1962). Then, the scalar-to-vector ratio δH/δr is a vector quantity and has the same direction as δr but pointing towards B if HB > HA and vice versa. As the point B gradually recedes toward the point A all along the normal AB, this vector rate-of-change continuously tends to a limit and when B finally converges to A it defines the local spatial head decay rate. To be in agreement with the general sense of the groundwater flow, practical formulas for flow calculation substitute −∂H/∂r (note the negative sign) for the hydraulic gradient J (see fig. 1.10). For any other point B of any other straight line AB but not perpendicular to the equipotential surface at the point A it is also possible to apply a similar reasoning and define another directional gradient J A . However, the absolute value of J A will always be smaller than the absolute value of JA . Indeed, J A = JA cos (α) where α is the angle BAB . So JA > J A . Being a vector, J results from the vector sum [(∂H/∂x1 )u1 + (∂H/∂x2 )u2 + (∂H/∂x3 )u3 ] where the unit vectors (u1 , u2 , u3 ) are referred to an arbitrary reference frame
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Fundamentals
13
Equipotential surface by point A Head: HA Coordinate: rA
Equipotential surface by point B Head: HB Coordinate: rB Pseudo-streamline
Figure 1.10 For homogeneous and isotropic pseudo-systems, all pseudo-streamlines traverse all equipotential surfaces perpendicularly. In this case, the limit of the symbolic ratio −δH/δr when B recedes toA defines the hydraulic gradient JA atA,tangent to the pseudo-streamline.
(x1 , x2 , x3 ). Writing H in L units, these components if referred to a zenithal frame ENZ are: ∂H
1 ∂p − − cos (x, Z) + ∂x ρ g ∂x Jx ∂H
= Jy = − 1 ∂p ∂y − cos (y, Z) + Jz ρ g ∂y ∂H − cos (z, Z) − ∂z
(1.4)
In the above expressions, the terms −[(∂p/∂x)/ρg] and −[(∂p/∂x)/ρg] correspond to the components of the piezometric gradient and the term −cos(z, Z) = −1 in the last row of the column vector of the right member corresponds to the vertical Z-component of the gravitational pull. However, in an arbitrary non-zenithal frame (x1 , x2 , x3 ), these three components of the gravitational gradient are [−cos(z, x1 ), −cos(z, x2 ), −cos(z, x3 )]T . It must be kept in mind that the hydraulic gradient has no units only for heads H written in L units. The head decay per unit time of a moving water particle, its temporal head decay rate, may be measured by the hydrodynamic gradient DH/dt resulting from the summation of the time gradient ∂H/∂t and the product of spatial gradient ∂H/∂r by the specific discharge q, i.e. DH/dt = ∂H/∂t + (∂H/∂r)·q (see fig. 1.11). The scalar product (∂H/∂r)·q measures the head spatial decay as the water particle moves during a unit time. In addition, for q = 1·LT−1 the negative value of this product
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Fractured rock hydraulics
H t
t
t
Water particle
t t
t
(∂H/∂ t )δ t
Pseudo-streamline
(∂H/∂r)qδ t
r
δt
t
Figure 1.11 This schematic figure shows a transient 4D head field H(r, t) and helps to understand the meaning of DH/dt = ∂H/∂t + (∂H/∂r) · q. Since head decays in both time and space, the observer must keep track of two kinds of motion: he must stand still in a fixed point to watch the head time decay or he must go after the moving water particle to watch the head spatial decay.
Cell centre: i, j, k z
z y x x
Figure 1.12 3D cubic cell (left) and 2D slab lattices (right) referred to an arbitrary coordinate system xyz. Hydraulic gradients J at the centre of these cell types can be numerically approximated as linear functions of the total head values at its neighbourhood.
is the hydraulic gradient J. Then, J may formally be defined as a measure of the consumption of the potential energy stored in a percolating water particle when it travels a unit length during a unit time with a unit apparent velocity (see addendum 1.2 on page 40). Numerical approximations of hydraulic gradients are relatively simple but require attention. For 3D cubic lattices referred to an arbitrary frame xyz (see fig. 1.12 left), devised to simulate the hydraulic behaviour of pseudo-continuous pervious media, the hydraulic gradient J components at the lattice centre, defined by the subscripts (i, j, k),
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Fundamentals
15
can be estimated by the head values H at its neighbourhood: (Jx )i,j,k = −
Hi+1,j,k − Hi−1,j,k xi+1,j,k − xi−1,j,k
(Jy )i,j,k = −
Hi,j+1,k − Hi,j−1,k yi,j+1,k − yi,j−1,k
(Jz )i,j,k = −
Hi,j,k+1 − Hi,j,k−1 zi,j,k+1 − zi,j,k−1
(1.5)
As the head H corresponds to the sum of the piezometric head p and the gravitational head Z (elevation measured in the zenithal frame):
Zi+1,j,k − Zi−1,j,k 1 pi+1,j,k − pi−1,j,k + (Jx )i,j,k = − ρ g xi+1,j,k − xi−1,j,k xi+1,j,k − xi−1,j,k
Zi,j+1,k − Zi,j−1,k 1 pi,j+1,k − pi,j−1,k (1.6) (Jy )i,j,k = − + ρ g yi,j+1,k − yi,j−1,k yi,j+1,k − yi,j−1,k
Zi,j,k+1 − Zi,j,k−1 1 pi,j,k+1 − pi,j,k−1 (Jz )i,j,k = − + ρ g zi,j,k+1 − zi,j,k,1 zi,j,k+1 − zi,j,k,1 Expressing the components of the gravitational gradient by the direction cosines of the axes xi referred to the zenithal axis Z:
1 pi+1,j,k − pi−1,j,k + cos (x, Z) (Jx )i,j,k = − ρ g xi+1,j,k − xi−1,j,k
1 pi,j+1,k − pi,j−1,k (Jy )i,j,k = − + cos (y, Z) (1.7) ρ g yi,j+1,k − yi,j−1,k
1 pi,j,k+1 − pi,j,k−1 (Jz )i,j,k = − + cos (z, Z) ρ g zi,j,k+1 − zi,j,k,1 The components of the gravitational gradient, i.e. cos(x, Z), cos(y, Z) and cos(z, Z), only depend on the direction of the z-axis with respect to the Z-axis of the zenithal frame. If, instead of an arbitrary xyz coordinate system, the spatial lattice is consistent with the ENZ zenithal frame, these components are: (JE )i,j,k = −
1 pi+1,j,k − pi−1,j,k ρ g Ei+1,j,k − Ei−1,j,k
1 pi,j+1,k − pi,j−1,k ρ g Ni,j+1,k − Ni,j−1,k
1 pi,j,k+1 − pi,j,k−1 =− +1 ρ g Zi,j,k+1 − Zi,j,k,1
(JN )i,j,k = − (JZ )i,j,k
(1.8)
For 2D slab lattices referred to an arbitrary xyz frame, as very thin flattened cells attached to the plane xy, as shown in fig. 1.12 right, it is assumed that the confined
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water pressure p in between the fracture walls may be taken as unvarying in the z-direction. Indeed, the fracture aperture δz is relatively very small if compared to its x and y dimensions. In this case, conceived to simulate the hydraulic behaviour of a planar fracture attached to the plane xy, the hydraulic gradient J components at the cell centre (i, j, k) are:
1 pi+1,j,k − pi−1,j,k + cos (x, Z) ρ g xi+1,j,k − xi−1,j,k
1 pi,j+1,k − pi,j−1,k =− + cos (y, Z) ρ g yi,j+1,k − yi,j−1,k
δz − cos (z, Z) =− δz
(Jx )i,j,k = − (Jy )i,j,k (Jz )i,j,k
(1.9)
For a horizontal fracture, referred to the zenithal axis Z: (JE )i,j,k = −
1 pi+1,j,k − pi−1,j,k ρ g Ei+1,j,k − Ei−1,j,k
(JN )i,j,k = −
1 pi,j+1,k − pi,j−1,k ρ g Ni,j+1,k − Ni,j−1,k
(1.10)
(JZ )i,j,k = −1 For linear tubes conceived to simulate the hydraulic behaviour of straight elements of solution conduits, shafts and galleries and even rivers, all previous procedures can be simply adapted. A hydraulic gradient J referred to an arbitrary frame xyz may be described in the zenithal frame ENZ by the following matrix operation:
JE Jx JN = R Jy JZ Jz
(1.11)
The rotation matrix [R] is defined by:
cos (x, E) cos (y, E) cos(z, E) R = cos (x, N) cos (y, N) cos(z, N) cos (x, Z) cos (y, Z) cos(z, Z)
(1.12)
The angular measures (x, E), (x, N) . . . (z, Z) comply with the trigonometric rules. The magnitude and direction of J remains invariant under this rotation.
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Pseudo-streamline
Water particle Hydraulic gradient J
Specific discharge q
Figure 1.13 Specific discharge q and hydraulic gradient J at an arbitrary point. Except for homogeneous and isotropic pervious soils and rock masses, both vectors q and J are normally noncollinear.
1.1.6
H yd r a ul i c c o n d u c t i vi t y
1.1.6.1 In t ro d u c ti on To any pervious pseudo-continuous subsystem can be attributed a physical property, called its hydraulic conductivity, that measures its ability to transfer groundwater from one point to another through its interconnected pores, fractures, conduits and other open discontinuities. Assuming a laminar flow, the specific discharge q, the hydraulic gradient J and the hydraulic conductivity [k] are linearly interrelated: [hydraulic conductivity] α [specific discharge]/[hydraulic gradient]
(1.13)
This linear relationship was empirically discovered in 1856 by the French engineer Henry Philibert Gaspard Darcy and bears his name as Darcy’s law. It gives a scalar quantity, i.e. a zero order tensor, for homogeneous and isotropic pervious media. However, as the specific discharge q and the hydraulic gradient J are vectors, i.e. first order tensors, their vector-to-vector ratio implies a second order tensor, generally written as a matrix [k] (see fig. 1.13). Referred to an arbitrary orthogonal frame xyz, the relationship between the specific discharge q, the hydraulic gradient J and the general hydraulic conductivity [k] is written in full matrix lettering as:
qx kxx qy = kyx qz kzx
kxy kyy kzy
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Jx kxz kyz Jy kzz Jz
(1.14)
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Fractured rock hydraulics
Expanding the right-hand product: kxx Jx + kxy Jy + kxz Jz qx qy = kyx Jx + kyy Jy + kyx Jz qz kzx Jx + kzy Jy + kzz Jz
(1.15)
Note that each component qi of the specific discharge q comes from the sum of three summands qij = kij Jj each one parallel to an i-direction. However, each group of the three summands parallel to the same direction is driven by all components of the hydraulic gradient J: qxx + qxy + qxz qx qy = qyx + qyy + qyz qz qzx + qzy + qzz
(1.16)
The matrix [k] is symmetric, i.e. kij = kji (i or j = x, y, z). The diagonal elements kii are always positive but the off-diagonal kij elements can be positive or negative. The magnitude of each component kij is proportional to the size of the interconnected discontinuities (pores and/or fractures and/or conduits) of the pervious soil or rock mass. The quantity, the value, the algebraic signal and the location of each off-diagonal element kij depend on the geometric symmetry of the 3D pattern and the size of these discontinuities. As the main diagonal components kii are always positive quantities, all diagonal ii-summands qii point to the same direction of Ji . However, to match the correct flow direction, all off-diagonal ij-summands qij make ±90◦ with respect to Jj depending on the algebraic signal of the off-diagonal components kij (+90◦ for positive kij and −90◦ for negative kij ). A positive signal indicates that the strike and dip of the fracture constrain the direction of the groundwater flow to point to the same direction of the hydraulic gradient. A negative signal indicates the contrary. In a particular frame, called principal axes, only the elements kii of the main diagonals are not equal to zero. These non-zero elements are called the eigenvalues of [k] (normally lettered as ki , i.e. as k1 , k2 and k3 ) and they measure the maximum, in between and minimum hydraulic conductivities along the principal axes of [k]. Each eigenvalue ki is defined by a set of three direction cosines, i.e. ai1 , ai2 and ai3 , called its eigenvectors (see matrix diagonalisation techniques for finding eingenvalues and corresponding eigenvectors). Using matrix notation:
k1 eigenvalues of [k] = 0 0
0 k2 0
0 0 k3
ai1 eigenvectors of [ki ] [i = 1, 2, 3] = ai2 ai3
(1.17)
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(1.18)
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19
1.1.6.2 F ra c t u r es a n d c on d u i ts The concept of transmissivity applied to a horizontal and isotropic fracture is quite similar to the one applied, for example, to a flat bed of pervious sedimentary rock. It is applied to a unit wide strip and its dimension is L2 T−1 . By simple premises, it is possible to presume that the magnitude of the transmissivity of a fracture depends on its “equivalent’’ hydraulic aperture e, on the interstitial permeability k of its infilling and on the extent κ of the distribution of its infilling over the fracture area. The parameter κ is the ratio between the partially filled area Aκ and the total area A, i.e. κ = Aκ /A. As the partially filled area Aκ varies between 0 and A, the parameter κ varies between 0 and 1, i.e. 0 ≤ κ ≤ 1. For completely filled fractures, i.e. defined by κ = 1, their average transmissivity T1 may be estimated by: T1 = k e
(1.19)
For completely clean fractures, i.e. for κ = zero, their average transmissivity T0 may be estimated by: T0 = c e3
(1.20)
In formula 1.13, c is an empirical coefficient that must be dimensionally consistent with the employed units (see Louis, 1969 and Quadros, 1982). As frequently observed at leaking rock outcrops or at underground excavations, groundwater usually percolates along a network of irregular and braided flow paths within the fracture itself. Moreover, the thickness of the open fractures and fracture zones varies from place to place, normally from 0.01 mm to 1 mm for single fractures and from 1 mm to 100 mm for narrow fracture zones. Consequently, the so-called “equivalent’’ hydraulic aperture e of a fracture or fracture zone is an ideal concept and corresponds to a value that may be cautiously employed to estimate the average fracture transmissivity T, as defined by the above functional relationships. For partially filled fractures, i.e. for 0 < κ < 1, the average transmissivity Tκ may be approximated by the Maxwell mixture rule: Tk = (T0 )1−κ (T1 )κ
(1.21)
In fact, more or less reliable estimates of fracture transmissivities for laminar flows require ingenious field tests accurately performed, as discussed later. Actually, the hydraulic regime of the groundwater flow throughout a pervious rough and uniform fracture depends on the magnitude of the hydraulic gradient J, on the fracture average aperture e and on its relative roughness δe/e, where δe is the modal height of the fracture asperities (see fig. 1.14). For clean fractured rocks, normally defined by “equivalent’’ hydraulic apertures of e < 1 mm and δe/e around 1/3, the groundwater flow far from wells or springs may normally be taken as laminar because the natural hydraulic gradients J seldom exceed 1 (Franciss, 1970; for a detailed account about these influences, see Louis, 1969).
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Fractured rock hydraulics
Average fracture aperture (mm)
6 5 TURBULENT
4 3 2 LAMINAR
1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hydraulic gradient (-)
Figure 1.14 Approximate empirical boundary, as a function of the average fracture aperture e and the hydraulic gradients J, separating the laminar from the turbulent flow domain for relative roughness δe/e around 1/3 (numerical data from Louis, 1969).
The hydraulic transmissivity of a non-horizontal and non-isotropic fracture must be treated as a second order tensor [T]. For instance, the anisotropic hydraulic transmissivity [T] of an almost planar fracture (referred to an orthogonal coordinate system xyz, whose axis z is normal to the fracture plane and the axes x and y coincide with the maximum and minimum transmissivity directions) is (see fig. 1.15):
Tx [T] = 0 0
0 Ty 0
0 0 0
(1.22)
In this case, referred to the fracture proper axes xyz, the relationship between the fracture discharge θ per unit width, the hydraulic gradient J and the hydraulic transmissivity [T] is: θx Tx θy = 0 θz 0
0 Ty 0
0 Jx 0 Jy Jz 0
(1.23)
As intuitively expected, the z-component Tz of the tensor [T] referred to the fracture proper frame xyz must be zero. As a result, the z-component θz of the fracture discharge vector referred to this frame is also zero, as results from the right-hand product, despite the fact that the z-component of the gravity driving pull Jz is not zero: θx Jx Tx θy = Jy Ty θz 0
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(1.24)
Fundamentals
21
z Z-zenithal axis
x Fracture discharge per unit width (in the fracture plane) Hydraulic gradient (not in the fracture plane)
Component of the hydraulic gradient (in the fracture plane)
y
Almost planar fracture Gravitational Z-component of the hydraulic gradient
Figure 1.15 Almost planar and hydraulically anisotropic fracture.The orthogonal coordinate system xyz has its z-axis normal to the fracture plane and axes x and y defined by the maximum and minimum transmissivity directions.The fracture discharge θ per unit width stays confined to the fracture plane and is only driven by the component of the hydraulic gradient J projected on the fracture plane. For a hydraulically isotropic fracture, the specific discharge and the projected gradient are aligned. Note the out-of-plane gravitational components of the hydraulic gradient referred to the proper axes xyz and to the zenithal axis Z.
The components of the transmissivity tensor [T] with respect to a zenithal frame ENZ come from the following matrix rotation: TEE TEN TEZ Tx 0 0 TNE TNN TNZ = R 0 Ty 0 RT (1.25) 0 0 0 TZE TZN TZZ The rotation matrix [R] is the same previously defined (see equation 1.17). The resulting matrix is symmetric, i.e., TEN = TNE , TEZ = TZE and TNZ = TZN . This matrix is an orthogonal square matrix and the elements of its three columns correspond respectively to the direction cosines of the unit vectors of the x-axis, y-axis and z-axis with respect to the zenithal reference frame ENZ. Then, its transpose equals its inverse, i.e. RT = R−1 . Consequently, the matrix transformation R{q} = R[T]RT· R{J} holds. As a result, referred to the zenithal frame ENZ, the relationship between the discharge θ per unit width, the hydraulic gradient J and the hydraulic transmissivity [T] can be written as (Rocha and Franciss, 1977): θE TEE TEN TEZ JE θN = TNE TNN TNZ JN (1.26) θZ TZE TZN TZZ JZ It is important to keep in mind that all matrix equations of the type θ = [T] · J, either referred to the fracture “proper’’ coordinate system xyz or to the zenithal frame ENZ, relate the same vectors θ and J via the same tensor [T].
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Modelling groundwater flow by taking into account the interplay of many arbitrary fractures and applying for each one a particular relationship θ = [T]·J is difficult. However, if less accuracy and less detail are accepted and if Darcy’s law is adapted to discontinuous systems, it is relatively simple to construct a pseudo-continuous model for a rock mass implicitly incorporating several arbitrary discontinuities (Franciss, 1970). To do this, the transmissivity [T] of an arbitrary discontinuity traversing a subsystem of volume δV = δE·δN·δZ may be substituted by a hydraulically equivalent conductivity tensor [k] with the following expression: TEE TEN TEZ √ √ √ δZδE δEδN δNδZ k11 k12 k13 T TNN TNZ NE √ √ |k| = k21 k22 k23 = √ (1.27) δEδZ δZδE δEδN k31 k31 k33 TZE TZN TZZ √ √ √ δNδE δNδE δEδN Then, a more tractable relationship q = [k]·J holds. Example 1.1 helps to understand the above concepts. Example 1.1 If referred to the fracture “proper’’ coordinate system, the anisotropic transmissivity tensor [T] of an almost planar and anisotropic fracture attached to the plane xy is described by (see fig. 1.16): Txx 0 0 0 0 1 × 10−4 2 m T = 0 Tyy 0 = (1.28) 0 5 × 10−5 0 s 0 0 0 0 0 0 z
y Tyy
Txx
x
Figure 1.16 Planar fracture referred to its “proper’’ reference frame xyz. The Txx and Tyy transmissivity directions match the x-axis and y-axis. The z-axis is normal to the fracture plane. The hydraulic conductivity anisotropy may be measured by the ratio |Txx |/|Tyy |.
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F u n d a m e n t a l s 23
As shown in fig. 1.17 the attitude of this fracture, referred to the zenithal ENZ frame, may be defined by its dip φ and dip direction χ. Then, the transmissivity tensor T referred to this frame can be determined by a simple matrix rotation operation. For dip φ = 30◦ and dip direction χ = 130◦ , the rotation operator is: Z
Fracture dip
N
Rotation angle: + from X to N Dip direction: + from N to X E
Figure 1.17 Planar fracture referred to the zenithal ENZ frame. The dip and dip direction are indicated.
cos (χ) R = −sin (χ) 0
cos (φ)sin (χ) cos (χ)cos (φ) −sin (φ)
sin (χ)sin (φ) −0.643 0.663 cos (χ)sin (φ) = −0.766 −0.557 cos (φ) 0 −0.5
0.383 −0.321 0.866
(1.29) Then, the transmissivity tensor T referred to the ENZ frame is: TEE TEN TEZ T TNE TNN TNZ = R T R TZE TZN TZZ 6.332 × 10−5 3.078 × 10−5 −1.659 × 10−5 m = 3.078 × 10−5 7.418 × 10−5 1.392 × 10−5 s −1.659 × 10−5 1.392 × 10−5 1.25 × 10−5 (1.30) Considering a pseudo-continuous cuboid subsystem of volume δV = δE · δN · δZ defined by edge lengths δE = 100 m, δN = 125 m and δZ = 150 m, the above hydraulic transmissivity tensor can be substituted by an equivalent hydraulic conductivity [k]: TEE TEN TEZ √ √ √ δNδZ δZδE δEδN 4.72 × 10−7 2.513 × 10−7 −1.514 × 10−7 T TNN TNZ NE √ √ = 2.513 × 10−7 6.057 × 10−7 1.271 × 10−7 √ δEδZ δZδE δEδN −7 −7 −7 −1.514 × 10 1.271 × 10 1.141 × 10 TZE TZN TZZ √ √ √ δNδE δNδE δEδN (1.31)
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The physical meaning of the positive or negative signs of the off-diagonal parameters of the transmissivity tensor k is as follows: • • •
kEN = 2.513 · 10−7 : positive JN induce positive flow qEN = kEN · JN along the E-axis and vice versa. kEZ = −1.514 · 10−7 : positive JZ induce negative flow qEZ = kEZ · JZ along the E-axis and vice versa. kNZ = 1.271 · 10−7 : positive JZ induce positive flow qNZ = kNZ · JZ along the N-axis and vice versa.
Observation of the geometric relationships depicted in fig. 1.17 helps how to be sure about these conclusions.
The same arguments previously applied to planar fractures hold for tubular conduits filled or partially filled with a pervious matrix depicting a linear hydraulic behaviour. To keep the same rotation formula, its axis must be attached to the y-axis of its “proper’’ reference frame and, in this case, its hydraulic transmissivity is (see fig. 1.18):
0 0 |T| = 0 Tyy 0 0
0 0 0
(1.32)
Using appropriate and specific coefficients, these tubular tensors may also model rivers segments or inclined shafts and galleries, after avoiding non-linearity by using approximate solutions and adequate inner boundary conditions. As shown example 1.2, the same theoretical principles applied to fractures may be used to describe the transmissivity tensors of tubular conduits.
y
Figure 1.18 A tubular conduit. To keep the same rotation formula, its axis must be attached to the y-axis.
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F u n d a m e n t a l s 25
Example 1.2 The transmissivity tensor [T] of a tubular conduit attached to the y-axis of the plane xy is described by (see fig. 1.19): 0 0 0 0 0 0 2 m T = 0 Tyy 0 = 0 1 × 10−3 0 (1.33) s 0 0 0 0 0 0
z
Tyy y x
Figure 1.19 Tubular conduit referred to its “proper’’ xyz reference frame. The Tyy transmissivity direction matches the y-axis.
The conduit geometry referred to the zenithal ENZ frame may also be defined by its dip φ and dip direction χ, as shown in fig.1.20. The transmissivity tensor T referred to this frame can be determined by a matrix rotation operation. In this case, dip φ = 20◦ and dip direction χ = 160◦ . Then, the rotation operator is:
cos (χ) cos (φ) · sin (χ) sin (χ) · sin (φ) −0.94 0.321 0.117 R = −sin (χ) cos (χ) · cos (φ) cos (χ) · sin (φ) = −0.342 −0.883 −0.321 0 −sin (φ) cos (φ) 0 −0.342 0.94 (1.34) Z
Conduit dip
N Rotation angle: ⫹from X to N
E
Dip direction: ⫹from N to X
Figure 1.20 Tubular conduit referred to the zenithal ENZ frame. The dip and dip direction are indicated.
© 2010 Taylor & Francis Group, London, UK
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Fractured rock hydraulics
The transmissivity tensor T referred to the ENZ frame is: TEE TEN TEZ T TNE TNN TNZ = R · T · R TZE TZN TZZ 1.033 × 10−4 −2.838 × 10−4 = −2.838 × 10−4 7.797 × 10−4 −1.099 × 10−4 3.02 × 10−4
−1.099 × 10−4 2 m 3.02 × 10−4 s 1.17 × 10−4 (1.35)
Considering a pseudo-continuous subsystem of volume δV = δE · δN · δZ delimited by δE = 100 m, δN = 125 m and δZ = 150 m, the hydraulic transmissivity can be substituted by an equivalent hydraulic conductivity [k]:
√
TEE
TEN √ δZ · δE TNN √ δZ · δE TZN √ δN · δE
δN · δZ T NE √ δE · δZ TZE √ δN · δE 7.7 × 10−7 = −2.317 × 10−6 −1.003 × 10−6
√
TEZ
δE · δN TNZ √ δE · δN TZZ √ δE · δN −2.317 × 10−6 6.366 × 10−6 2.757 × 10−6
−1.003 × 10−6 m 2.757 × 10−6 s 1.068 × 10−6
(1.36)
For erratic discontinuities labelled a, b, c . . . defined by their equivalent hydraulic conductivities |ka |, |kb |, |kc | . . ., the resulting equivalent hydraulic conductivity for a pseudo-subsystem can be approached by a “parallel coupling’’ if the hydraulic gradients within the subsystem are substituted by an average value (see addendum 1.3 on page 41): |k| = |ka | + |kb | + |kc | + . . .
(1.37)
Written in full matrix lettering:
[(k11 )a + (k11 )b + (k11 )c + . . . ] |k| = [(k21 )a + (k21 )b + (k21 )c + . . . ] [(k31 )a + (k31 )b + (k31 )c + . . . ]
[(k12 )a + . . . ] [(k22 )a + . . . ] [(k32 )a + . . . ]
[(k13 )a + . . . ] [(k23 )a + . . . ] [(k33 )a + . . . ] (1.38)
It is implied in the above summation rule that minor hydraulic head losses at the discontinuity intersections can be neglected and the “modeller’’ accepts that the accuracy and detail of the simulation results suit his needs at the selected scale of observation.
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Fundamentals
27
Discontinuity to exclude
Discontinuity to include
Figure 1.21 All discontinuities that do not traverse the subsystem from side to side must be excluded from the summation matrix. The influence of these discontinuities may be approximately incorporated in the intact rock permeability.
The hydraulic conductivity tensor |kr | of the rock mass must be added to the resulting tensor |k|. Then, the final hydraulic conductivity tensor |K| for the fractured rock mass results from: |K| = |kr | + |k|
(1.39)
To work with a stable algorithm, do not neglect the influence of |kr |. To give consistent results, the summation rule given above is restricted to uninterrupted discontinuities that cut the subsystem from side to side (see fig. 1.21). Occasionally an almost impervious discontinuity, such as a thin diabase dike, a fracture completely sealed with calcite or clay material, can restrain the percolation. In this case, the eigenvalues of the permeability tensor of the rock block may be approximately constructed as follows: •
•
First hypothesis: all pervious discontinuities that traverse the subsystem maintain their connectivity as they cross the impervious discontinuity. In this case, only the eigenvalues of the permeability tensor [kr ] of the intact rock matrix are reduced before applying the summation rule. This may be calculated approximately by multiplying [kr ] by (kf /kr )p /kr , where kr is the average rock matrix original permeability, kf << kr is the permeability of the quasi-impervious structure and p << 1 is its volumetric proportion (roughly estimated by its thickness divided by the subsystem side). Second hypothesis: some of the pervious discontinuities that traverse the subsystem do not retain their connectivity as they are crossed by the impervious discontinuity. In this case, all blocked discontinuities are excluded from the summation matrix (see. fig. 1.22). Then apply the summation rule to the remaining pervious discontinuities [ka ], [kb ], [kc ] . . . but reduce the rock matrix permeability tensor [kr ] as explained above.
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Fractured rock hydraulics
Discontinuity to exclude
Discontinuity to include
Impervious discontinuity
Figure 1.22 An impervious thick seam intercepts and restrains seepage of some pervious discontinuities. Some pervious discontinuities lose their connectivity as they are crossed by this impervious discontinuity.
Based on the geometry and hydraulic properties of the lithologic and structural features of a flow domain, it is possible to estimate the hydraulic conductivities [k] over that domain. However, by solving “inverse problems’’ (based on field tests results or monitored piezometric and discharge data) the resulting parameters are much more dependable (Hshieh et al. 1985). This procedure, presented in Chapter 3, not only takes into account the effective existing hydraulic boundary conditions but also the hidden contributions of all unseen discontinuities, without exception.
1.2 Governing equations 1.2.1
P r elim i n a r i e s
The scalar field H and the vector field q within a space domain and during a time span can be described by a closed or a numerical solution of a system of partial differential equations deduced from energy and mass conservation principles. To solve these equations it is necessary to identify some additional information about the problem in hand. First, what happened in the past? This information determines the initial conditions, i.e. values of H or q for a particular t. Secondly, what are the influences from the outside? This information determines the boundary conditions, i.e. values of H and q on the borders (or, in some cases, very far from) the flow domain. The solution of a well-posed problem, besides satisfying the governing equations, must be unique and continuously dependent on these subsidiary conditions. Unfortunately, a closed solution can only be found for simple cases. Conversely, numerical approximations are normally used for all practical applications. 1.2.2
En er g y c o n s e r va t i o n p r i n c i p l e : D ar c y ’s l aw
The empirical relationship between the specific discharge q, the hydraulic gradient J and the hydraulic conductivity [k] derives from the energy conservation principle.
© 2010 Taylor & Francis Group, London, UK
Fundamentals
External driving forces resulting from the water pressure differentials on opposite faces of the subsystem
29
Internal resisting forces drag forces – resulting from water viscosity Very unimportant inertial forces compared to the drag forces Internal driving forces resulting from gravitational pull
Figure 1.23 Resultant of internal mass forces and external boundary forces acting on a percolating pseudo-fluid enclosed by a pseudo-subsystem.
If the components Ji of the hydraulic gradient J are described as partial derivatives of the hydraulic head H, the corresponding specific discharge components qi define a system of three partial differential equations: qi = −
3
j=1
ρkij
∂H ∂xi
(1.40)
Now, consider Newton’s second law presented as the energy conservation principle: [mass] × [acceleration] − [internal forces + external forces] = zero
(1.41)
Applied to the volume of a percolating pseudo-fluid enclosed by a pseudo-subsystem, Newton’s second law equates the product of its mass and its acceleration to the resultant of the distributed forces within its mass, called internal forces, plus the forces applied to its boundaries, called external forces. For a pervious pseudo-continuous subsystem in dynamic equilibrium, it is possible to distinguish four kinds of external and internal forces (see fig. 1.23). First, the external driving forces resulting from the differential water pressure acting on its faces. Second, the internal driving forces resulting from gravitational pull. Third, the internal resisting forces, called drag forces, resulting from water viscosity. Finally, there exist unimportant inertial forces (compared to the drag forces) that only develop when the moving fluid accelerates or slows down (see addendum 1.4 on page 42). The resulting driving force per unit volume, suggestively called percolation force, corresponds to the product of the groundwater unit weight ρg and the hydraulic gradient J, i.e. ρgJ. In a steady state, the specific force ρgJ is just balanced by the specific drag D. For percolation under Darcy’s law, the coefficient D is related to the empirical hydraulic conductivity [k] by D = ρgq · [k]−1 . Here, the inverse matrix [k]−1 is called the hydraulic resistivity. In an unsteady state, an insignificant specific inertial force ρg(dq/dt) counteracts the unbalanced percolation and drag forces (both per unit volume).
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Fractured rock hydraulics
Inflow discharge qi
Outflow discharge qi ⫹ dq
Internal source or drain Q
Figure 1.24 Observing a pervious subsystem of unit volume during a unit time, the differential amount of its water content mass ρn is ∂(ρn)/∂t. Occasionally, an internal source (or sink) of Q strength per unit volume may add (or drain) another differential amount of water mass ρQ. At the same time, differential amounts of water enter and leave its 2x3 pairing i-faces totalizing ∂(ρqi )/∂xi . Balancing these differential amounts of water entering and leaving the subsystem and determining qi by Darcy’s law, i.e. qi = −kij ∂H/∂xj , yields the classical continuity equation: ∂(ρ − kij ∂H/∂xj )/∂xi + ∂(ρn)/∂t = ρQ.
1.2.3
Ma s s c o n s e r va t i o n p r i n c i p l e : c o nt i nui t y e quat i o n
1.2.3.1 Ge n e r a l eq u a ti on During a unit time interval, water percolates through a unit volume of saturated and slightly compressible pervious subsystem. In that interim, the mass conservation principle implies that the differential amount of water entering and leaving this subsystem matches the differential amount of water removed (or added) from this subsystem due to a contraction (or expansion) of its open discontinuities and due to the compressibility of water itself (see fig. 1.24). Additionally, an occasional interior source (or sink) may add in (or drain out) another differential amount of water ρQ, where Q denotes the source strength (or sink) expressed as a discharge per unit volume (positive for a source or negative for a sink). Q units are L3 T−1 /L3 . Then, the formal differential statement of the mass conservation principle, written in ML−3 T−1 units, is (see addendum 1.5 on page 43):
3 3 ∂ ∂H ∂H ρkij = V − ρQ ∂xi ∂xj ∂t i=1
(1.42)
j=1
In this equation, the parameter V , called the specific volumetric storage, is defined by:
ρ V = ρg nρ0 β + Kp
© 2010 Taylor & Francis Group, London, UK
(1.43)
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31
In equation 1.43, ρ0 (ML−3 ) denotes the water density, equal to 997.1 kg/m3 at 25◦ C under atmospheric pressure; β (M−1 LT2 ) denotes the water isothermal compressibility, equal to 4.8 · 10−10 Pa−1 at 25◦ C and Kp (ML−1 T−2 ) denotes the discontinuous subsystem bulk modulus, varying around 107 Pa for loose sands to 3 · 109 Pa for hard rocks. Strict units of v are ML−4 . However, considering the water density ρ invariant and normalising the general continuity equation by ρ, the parameter V takes a simpler expression symbolised by SV :
1 SV = ρg nβ + (1.44) Kp In this case, the units of SV reduce to L−1 , as commonly quoted in technical publications. Then, the continuity equation, also written in L−1 units, is:
3 3 ∂ ∂H ∂H kij · = SV · −Q ∂xi ∂xj ∂t i=1
(1.45)
j=1
The parameter SV measures the amount of water expelled from a unit volume subsystem, due to a water expansion and a porosity contraction, resulting from a pressure unit decrease (assuming effective stress variation always balanced by neutral stress variation). For short-range head variations, SV can be taken as a constant property to be empirically estimated. 1.2.3.2 D u pu i t’s a p p r ox i m a ti on Groundwater flow in quasi-horizontal and thin tabular groundwater systems, having planar dimensions much bigger than their thickness, are more easily described if referred to the zenithal frame ENZ. Depending on the height of the permeability contrast between its juxtaposed strata, groundwater may or may not flows under virtual confinement. When unconfined, characterised by pervious strata having very low permeability contrasts, groundwater flows without restraint over a practically impervious base. Its upper boundary, dubbed water table, moves up and down freely (see fig. 1.25). When confined, groundwater percolates “sandwiched’’ between almost impervious bottom and top strata (see fig 1.26). In both cases, confined and unconfined, very gentle hydraulic gradients drive the groundwater mass. Then, except when too close to natural water sources or pumping wells, the vertical Z-component JZ of the hydraulic gradient J at any point of the flow domain is normally very small and can be neglected. This means ∂H/∂z = zero and this approximate assumption is the starting point of the simplified hydraulic model suggested by Dupuit (Dupuit, 1863). In these cases, all equipotential surfaces H can be approximated by vertical cylindrical surfaces described by the argument (E, N, t) instead of (E, N, Z, t). 1.2.3.2.1
U N C O N F I N E D G R O U N DWAT E R
Consider an unconfined groundwater columnar cell, where B is the average height of the impervious base and H is the average height of the saturated water column.
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Fractured rock hydraulics
Previous WT
Water accretion ω
Free water table WT δH Saturated column q Impervious bottom
H
Datum B
δxi
Figure 1.25 In this i-vertical cross-section through a vertical prismatic cell of an unconfined aquifer, B is the average height of the impervious bottom (elevation referred to an arbitrary horizontal datum), H is the average saturated thickness (neglecting the capillary fringe) and B + H is the total hydraulic head. The water specific accretion is ω (in L3 /T/L2 = L/T units). The column area base is δS (in L2 units) and referred to a zenithal frame δS = δE · δN. Due to infiltration (or evaporation), the differential height associated to the rise (or fall) of the water table in a time interval δt is δH (in L units).
Then, the sum (B + H) approximates the average hydraulic head (see fig. 1.25). Over the EN domain, the variable height B only depends on the 3D geometry of the top of the impervious botton. The variable height H at each point (E, N) depends on the aquifer hydraulic properties and on the initial and boundary conditions of the flow domain. As equipotential surfaces are upright cylindrical surfaces, in view of Dupuit’s approximation, the specific discharge q is always a horizontal vector, whose components, described in a zenithal frame, are: ∂
(B + H) qE kEN k ∂E = − EE (1.46) qN kNE kNN ∂ (B + H) ∂N During a time interval δt, the differential amount of water δVw entering and leaving the saturated columnar cell is: ∂ ∂ δVw = − (ρqE HδN)δE + (ρqN HδE)δN δt (1.47) ∂E ∂N
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In the above expression, H denotes the variable height of the saturated columnar cell, δE and δN denote the lengths of the orthogonal sides of the base of that prismatic cell and δE · δN measures the base area δS. Then:
∂ ∂ δVw = − (ρqE H) + (ρqN H) δS δt ∂E ∂N
(1.48)
Recalling that ne denotes the effective porosity, the liquid phase mass within the saturated columnar cell measures ne ρHδS. During a time interval δt and disregarding groundwater and aquifer deformations induced by field stress variations, the amount of mass within the prismatic cell can only change if the water table rises or falls. In this case, the mass variation is: ∂ (ne ρH)δS δt ∂t
(1.49)
An infiltration or evaporation at the water table respectively results in the gradual rise or fall of the height of the columnar cell. The water accretion or removal rate is quantitatively defined by a specific strength ω = [∂Vw /∂t]/δS (in L3 /T/L2 = L/T units). This rate is positive for infiltration and negative for evaporation, adding or removing from the columnar cell an amount of water mass equal to ρωδSδt during a time interval δt. As a result, during a time interval δt, the differential amount of water mass entering and leaving the columnar cell matches the differential amount of water mass due to a raise or a decline of the water table plus the equivalent height of the infiltrated or evaporated water mass. Then, combining the previous equations yields the continuity equation: ∂ ∂ ∂ ρH kEE (B + H) + kEN (B + H) . . . ∂E ∂ ∂E ∂N = (ne ρH) − ρω ∂ ∂ ∂ ∂t ρH kNE (B + H) + kNN (B + H) + ∂N ∂E ∂N
(1.50)
This is a non-linear second order partial differential equation. Numerical algorithms can be modified to avoid the difficulties implied by that non-linearity, as discussed in Chapter 4. For common aquifers, the effective porosity ne may be as high as 5 to 15%. However, for fractured hard rock masses this value is normally lower than 2%. 1.2.3.2.2
C O N F I N E D G R O U N DWAT E R
The variable average height of a saturated confined columnar cell now solely depends on the aquifer 3D geometry. The saturated height denoted by HC must not be confused with the unconfined height H (see fig. 1.26). The meaning of the variable B remains unchanged and still denotes the height of the impervious base. Denoting by P the pressure head at the aquifer’s roof, the total head now corresponds to the sum (B + HC + P/ρg). However, if Pm is the pressure head measured at the middle of the confined thickness HC , instead of at the aquifer’s roof, then the total head corresponds
© 2010 Taylor & Francis Group, London, UK
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Fractured rock hydraulics
Total head H Top leakage
Impervious roof P/ρ q Impervious bottom Hc Confined and saturated column
q+δq B
Datum δXi
Figure 1.26 In this i-vertical cross-section through a columnar cell of a confined aquifer, B is the average impervious bottom height (elevation referred to an arbitrary horizontal datum), HC is the average saturated thickness of the confined aquifer and P is the pressure head at the aquifer’s roof. The sum (B + HC + P/ρg) is the total head. The specific leakage (or forced accretion) is w (in L3 /T/L2 = L/T units). The area of the column base is δS (in L2 units). In a zenithal frame δS = δE · δN.
to (B + HC /2 + Pm /ρg). As it is assumed by Dupuit’s simplification that ∂H/∂z = zero, than Pm = ρg(HC /2) + P. However, one must keep in mind that Dupuit’s simplified modelling cannot be retained for true 3D unconfined or confined modelling. The continuity equation for confined systems referred to a zenithal frame, derived as in the preceding section, has the following expression:
∂ P B + H + . . . c ∂ ∂E ρg ρHc . . .
∂ P ∂E + kEN B + Hc + ∂N ρg ∂ = (ne ρHc ) − ρω
∂t ∂ P B + Hc + ... k NE ∂E ∂ ρg ρH +
∂N c ∂ P + kNN B + Hc + ∂N ρg
kEE
(1.51)
During a time interval δt, the mass of the liquid phase within a confined saturated columnar cell measures [∂(ne ρHC )/∂t]δSδt. In a confined aquifer, that mass can only change by stress induced deformations. Applying the differentiation chain rule to
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∂(ne ρHC )/∂t and recalling that H = B + HC + P/ρg and also that B and HC may be taken as time-independent variables: ∂ ∂ ∂ (ne ρHc ) = ρg (ne ρHc ) ∂t ∂P ∂t
P ρg
= ρg
∂ ∂ (ne ρHc ) (H) ∂P ∂t
(1.52)
The first derivative in the right hand product, ρg∂(ne ρHC )/∂P, called storativity or columnar storage, is here denoted by C . Assuming that for usual groundwater pressure variations the height HC remains practically invariant, a similar reasoning as previously applied for the general continuity equation yields:
c = ρgHc
∂ n e ρ0 β + ρ n e ∂P
(1.53)
The first term inside the brackets concerns the compressibility of the water. Now, assuming that the total stresses, defined in the sense of soil mechanics, remain invariant, any water pressure variation induces an opposite but similar effective stress variation. For short-range pressure changes, C is almost constant. It may be estimated for deformable rock masses by sophisticated field pumping tests (Hshieh et al., 1985). The dimensional units of C are ML−3 . However, for constant water density ρ, the continuity equation can be divided by ρ and the parameter C takes a simpler expression symbolised by SC that has no units:
∂ Sc = ρgHc ne β + ne ∂P
1.2.4
(1.54)
B ou n d ar y a n d i n i t i a l c o n d i t i o ns
Darcy’s law and the continuity equation, which are often combined into a single equation, link the hydraulic head H to the specific discharge q. However, unique descriptions for H under unique flow circumstances are only feasible if, in addition to these two equations, supplementary conditions are properly stated. For each problem, solving strategies depend on the character of these supplementary equations. Some of them are described in the following pages. Particular expressions for boundary and initial conditions applied to heterogeneous and anisotropic pervious systems are given in the next chapter. 1.2.4.1 Ma i n b ou n d a r y ty p es Boundary conditions can be mathematically defined where free water infiltrates or emerges from pervious media and where groundwater hits impervious boundaries, traverses dissimilar pervious media, emerges into the open air or changes from a saturated to an unsaturated state. Often it is difficult to construct a model surrounded by well-defined boundaries. To reduce its size to practical limits and at the same time to retain its significance where this is relevant, heads and flow data at arbitrary boundary conditions must be based on dependable field data (see fig. 1.27).
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Fractured rock hydraulics
Free water body
Earth dam 3
2
1
4
Free water body 5
Pervious foundation
11
8 6
7
9 10
Impervious base
Figure 1.27 Cross-section of a schematic reservoir of water impounded by an earth dam. Water percolates through the earth dam and the pervious foundation. Boundary conditions are well defined at: 1-2-3 and 5-6-7-8, the free water-pervious media interface; 3-4, the unconfined groundwater-air interface; 4-5, the seepage boundary; 10-11, the impervious boundary; 2-6, the dissimilar pervious media interface. To restrain the model size, heads or flows at boundaries 1-11 and 9-10 must be based on dependable field data.
1.2.4.2 S u bm er g ed b ou n d a r i es Submerged boundaries typify interfaces where free water infiltrates (or emerges) into (or from) the pervious media (see fig.1.27: interfaces 1-2-3 and 5-6-7-8). At these boundaries, hydraulic heads H take constant values and if expressed in L units are numerically equivalent to the water level that is geometrically associated to each boundary but all referred to the same arbitrarily datum. Then, referred to a zenithal frame, for any point at an elevation Z but located on a submerged interface its boundary condition is: Z+
P = Hconstant ρg
(1.55)
For example, considering again fig. 1.27, the hydraulic head H123 at any point located on the interface 1-2-3 remains unchanged for an unchanging water level, i.e. H123 = constant. The same observation applies to the interface 5-6-7-8. However, normally H123 and H5678 have different values, i.e. H123 = H5678 . 1.2.4.3
Impe r v i ou s b ou n d a r i es
A geological contact between a more and a less pervious media can be qualified as relatively impervious, from a practical point of view, if the permeability of the less pervious media is 10 to 30 times smaller than the permeability of the more pervious (see fig. 1.27: interface 10-11). To express an impervious interface condition in mathematical terms it suffices to equate to zero the orthogonal projection of the specific discharge q on the outward normal n of this interface element δS. In other words, to annul the dot product q · n,
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i.e. q · n = 0. For isotropic media, q · n = 0 implies J · n = 0. However, for anisotropic media, q · n = 0 does not imply J · n = 0 but J · n = 0 in most cases. Defining n by its direction cosines (αx , αy , αz ), the boundary condition for a pervious media described by an anisotropic hydraulic conductivity [k] is: qx αx kxx qn = qy αy = kyx qz αz kzx
kxy kyy kzy
kxz Jx αx kyz Jy αy = 0 kzz Jz αz
(1.56)
1.2.4.4 S e e pag e b ou n d a r i es Seepage boundaries occur where groundwater emerges from a pervious media into the open air (see fig.1.27: interface 4-5). Neglecting capillary heads, the pressure p at these interfaces corresponds to the atmospheric pressure Patm , previously defined as the zero origin of the relative pressure scale adopted (gauge pressure). As a result, referred to a zenithal frame, the hydraulic head H at any point (E, N, Z) located on a immobile seepage boundary matches its elevation Z because the pressure term p of the hydraulic head is zero everywhere, i.e. H(E, N, Z)at seepage boundary = Z. In addition, assuming an invariant atmospheric pressure along the seepage boundary, i.e. p = Patm = invariant, the directional gradient ∂p/∂r must be parallel to the outward normal n of the immobile seepage boundary. Due to that parallelism, the dot product n · ∂p/∂r equals the gradient modulus |∂p/∂r|. Then, the boundary condition is n · ∂p/∂r|∂p/∂r| = 0. On the other hand, the pressure p may be expressed as p/ρg = H – Z. Then, defining n by its direction cosines (αx , αy , αz ), the full expression of the boundary condition is: ∂ ∂ H H ρg ∂x ∂x
α E ∂ ∂ ∂ ∂ ρg ρg H H p − p = αN n − = 0 ∂N ∂N ∂r ∂r αZ
∂ ∂ ρg ρg H−1 H − 1 ∂Z ∂Z
ρg
(1.57) Expressed in terms of the hydraulic gradient J components: ρgJE αE ρgJE ∂ ∂ p − p = αN ρgJN − ρgJN = 0 n ∂r ∂r ρg(JZ − 1) αZ ρg(JZ − 1)
(1.58)
1.2.4.5 U n c o n fi n ed g r ou n d wa ter - a i r i n ter f a c e Neglecting capillary fringe effects, a water table defines a steady or unsteady groundwater-air interface, depending on the supplementary equations of the problem under analysis (see fig.1.27: interface 3–4).
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Fractured rock hydraulics
As with seepage boundaries the pressure p on the water table corresponds to the atmospheric pressure Patm equal to zero. The pressure at any point (r, t) on a mobile or immobile water table is zero everywhere, i.e. Pwater table = 0. Referred to a zenithal frame, the hydraulic head H at any point of the water table WT has the same value (in L units) as its elevation Z: HWT = Z
(1.59)
Furthermore, assuming that the atmospheric pressure remains constant, i.e. Patm = invariant, the hydrodynamic derivative DP/dt is zero, i.e. DP/dt = ∂P/∂t + ∂P/∂r = 0. On the other hand, the equality H = Z + Patm /ρg holds for all points restricted to the water table. This implies ∂P/∂t = ρg · ∂H/∂t, ∂P/∂x = ρg · ∂H/∂x, ∂P/∂y = ρg · ∂H/∂y and ∂P/∂z = ρg · ∂H/∂z − 1. Then, the water table boundary condition is: ∂ P ∂E q dP ∂ 1 E ∂ qN = P+ P = 0 dt ∂t n.e q ∂N Z ∂ P ∂Z
(1.60)
Written with explicit matrix components:
k k ∂ ρg EE EN kNE kNN ρg H − ∂t ne k k ZE ZN
∂ ∂ H H ∂E ∂E kEZ ∂ ∂ H kNE =0 ∂N H ∂N
kZZ ∂ ∂ H − 1 H ∂Z ∂Z
(1.61)
This boundary condition can also be applied to a moving water table (see fig. 1.28) even when acquiring or losing an apparent vertical mass accretion rate ρg · ω (ω in LT−1 ). In this case, the vertical component (ve )Z of the effective velocity vector ve of the moving water table must be added to the effective mass accretion vector defined by ρg · ω/ne . The resulting downward or upward movement of the water table slows down or speeds up depending whether the effective mass accretion rate ρg · ω/ne is positive (infiltration) or negative (evaporation). Then, the complete boundary condition is: ∂ P ∂E q 0 dP ∂ 1 E ∂ qN − 0 = P+ ∂N P = 0 dt ∂t ne ωZ qZ ∂ P ∂Z
© 2010 Taylor & Francis Group, London, UK
(1.62)
Fundamentals
39
n
ρg·ω
δS ⫽ n·δS
Ve
(Ve)Z
Figure 1.28 Moving water table with vertical accretion. Its downward or upward movement slows down or speeds up depending whether the accretion rate ρg · ω/ne is positive (infiltration) or negative (evaporation).
Written with explicit matrix components: kEE ρg ∂ kNE ρg H − ∂t ne kZE
kEN kNN kZN
∂ ∂ H H ∂E ∂E kEZ 0 ∂ ∂ − 0 H kNZ =0 H ∂N ∂N kZZ ω Z
∂ ∂ H−1 H ∂Z ∂Z (1.63)
For steady state flows, the term ρg · ∂H/∂t disappears in both cases.
1.3 Addenda to Chapter I 1.3.1 A d d en d u m 1 . 1 : E f f e c t i ve ve l o c i t y and s p e c i f i c di s c har g e According to Bear, the groundwater effective velocity ve ascribed to the centre of a pseudo-continuous subsystem must be considered as the vector average of all water particles velocities vP percolating throughout its effective voids during an observation time interval δt (Bear et al., 1968): δt δVe 1 vP dV dt (1.64) ve = δVe δt 0 0 In this expression, the inner integral δVe stands for the void effective volume inside the total volume δVt of the pseudo-continuous subsystem. Their ratio defines the effective porosity ne , i.e. ne = δVe /δVt . The specific discharge q corresponds to the product of the effective velocity ve and the effective porosity ne and preserves the units of velocity LT−1 , i.e. q = ne ve . As the
© 2010 Taylor & Francis Group, London, UK
40
Fractured rock hydraulics
value of the effective porosity ne is less than 1, the specific discharge q is always smaller than ve . For that reason, the travelling time of a water particle along a streamline must be calculated using the effective velocity ve . Recall that a specific property can only be applied to a measurable property whose value is proportional to the size of the subsystem to which it is ascribed. This property divided by the value of a size measure of the subsystem expresses a specific property, such as weight and specific weight.
1.3.2 A d d en d u m 1.2: Hy d r o d y n a m i c g r adi e nt Darcy’s assumption allows a pervious soil or rock mass, made up of a mixture of solid, liquid and gaseous matter, to be substituted and modeled as a fictitious pseudocontinuous fluid having the same unit weight of the mixed liquid-gaseous material. Then, considering a transient head field H(r, t), the value of the consumption of the stored potential energy δH associated with a percolating water particle during a time interval δt is:
∂H ∂H ∂x ∂H ∂x ∂H ∂x δH = + + + δt (1.65) ∂t ∂x ∂t ∂y ∂t ∂z ∂t The sum inside the brackets defines the hydrodynamic gradient. It measures the head decay per unit time of a moving water particle, in other words its temporal head decay rate. Bearing in mind that q = ne · ve , the hydrodynamic gradient can be symbolically written as: dH ∂H ∂H ∂H ve ∂H = +q = + dt ∂t ∂r ∂t ne ∂r
(1.66)
The first partial derivative ∂H/∂t measures the head decay per unit time (δt = 1 T) and only appears in transient flows. Now, to understand the meaning of the symbolic product (∂H/∂r) · q it is necessary to recall that the symbolic differential ratio ∂P/∂Q between two tensors P and Q yields a third tensor R whose rank equals the sum of the ranks of P and Q. Therefore, ∂H/∂r is a vector and the scalar product (∂H/∂r) · q measures the head spatial decay as the water particle moves during a unit time. In addition, for q = 1 · LT−1 the negative value of this product is the hydraulic gradient J. Then, J may formally be defined as a measure of the consumption of the potential energy stored in a percolating water particle when it travels a unit length during a unit time with a unit apparent velocity. Alternatively, written with explicit matrix components: ∂H
∂H
∂x ∂x qx (ve )x ∂H dH ∂H ∂H ∂H 1 (v ) = + qy + = e y dt ∂t ∂t ne (v ) ∂y ∂y qz e z ∂H ∂H ∂z ∂z
© 2010 Taylor & Francis Group, London, UK
(1.67)
F u n d a m e n t a l s 41
The hydrodynamic gradient may also be referred to in the technical literature by the terms convective or advective derivative, substantial or material derivative, Lagrangian or Stokes derivative and also derivative following the motion. 1.3.3 A d d en d u m 1.3: Hy d r a u l i c c o n d uc t i v i t y f o r r ando m l y fr a ct u re d s u b s y s t e m s If the “modeller’’ accepts that minor hydraulic head losses at discontinuity intersections can be neglected and that the accuracy and details of the results of the pseudocontinuous model suit his needs at the selected scale of observation, a reasonable approach for the hydraulic conductivity of randomly fractured subsystems stems from the linearity of Darcy’s law. In this case, the following assumptions prevail: i. ii.
The interface between the intact rock and the fracture walls is not coated by impervious cakes. The hydraulic gradient vector field within the pseudo-subsystem is substituted by an average vector field J.
Then, for a pseudo-system hydraulically equivalent to one single fracture, the discharge component qi crossing a control section Si is: 3 (kij Ji ) Si Qi = qi Si =
(1.68)
j=1
Now, considering many random fractures a, b, c . . . the resulting discharge Qi is: 3 3 3 a a (kb )ij (jb )i Si + (k )ij (j )i Si + (kc )ij (jc )i Si + . . . Qi = j=1
j=1
j=1
(1.69) As assumed, Ja ≈ Jb ≈ Jc ≈ · · · ≈ J, then the superposition principle can be applied: 3 3 3 Qi = (ka )ij + (kb )ij + (kb )ij + . . . Ji S j=1
j=1
(1.70)
j=1
The summation in brackets can be substituted with a single tensor [k]: |k| = |ka | + |kb | + |kc | + . . .
(1.71)
In full matrix lettering:
[(k11 )a + (k11 )b + (k11 )c + . . . ] [(k12 )a + . . . ] |k| = [(k21 )a + (k21 )b + (k21 )c + . . . ] [(k22 )a + . . . ] [(k31 )a + (k31 )b + (k31 )c + . . . ] [(k32 )a + . . . ]
© 2010 Taylor & Francis Group, London, UK
[(k13 )a + . . . ] [(k23 )a + . . . ] [(k33 )a + . . . ]
(1.72)
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Fractured rock hydraulics
1.3.4 A d d en d u m 1.4: E n e r g y c o n s e r v at i o n p r i nc i p l e Darcy’s smart assumption allows a pervious and saturated soil or rock mass to be modelled as a pseudo-continuous fluid having the unit weight of the liquid-gaseous mixture. Then, one of its driving forces is the vector addition of all hydrodynamic fluid pressures acting on its faces. The value of the component of the driving force acting on face δSi is [(−∂p/∂xi )δxi ]δSi where δxi is the length of the subsystem i-edge, δSi is the i-face area normal to the i-axis. This component equals (−∂p/∂xi )δV where δV is the pseudo-continuous subsystem volume derived from the product δxi δSi . Another driving force is the vertical gravitational pull −ρgδV, where ρ is the fluid density and g is the gravitational acceleration. These two driving forces induce flow against a resisting viscous force whose i-component is Di δV, where D is the drag force per unit volume. If referred to the solid phase unit volume δVe , that specific drag must be written D/ne where ne is the effective porosity. In a steady-state flow, without acceleration, these driving and resisting forces are in a dynamic equilibrium satisfying the following equations, referred to zenithal frame ENZ:
∂ − p + DE δV = 0. ∂E
∂ − p + DN δV = 0. ∂N
(1.73)
∂ − p − ρg + DZ δV = 0. ∂Z Eliminating δV from these equations and keeping in mind that the first three terms inside the brackets −∂p/∂E, −∂p/∂N and (−∂p/∂Z − ρg) correspond to the components of the hydraulic gradient JE , JE and JZ , this system of equations may be reduced to the following vector equation: ρg J + D = 0
(1.74)
Therefore, in a steady state, the specific viscous drag D balances the impulse ρgJ. They are opposite vectors. In a transient state, the resultant force of ρgJ + D accelerates or slows down the groundwater flow. In other words, an insignificant specific inertial force ρg(dq/dt) counteracts the unbalanced percolation and drag forces (both per unit volume). The i-component of the resisting inertial force per unit volume is [ρg(∂qi /∂t)δSi δt]/δV. If referred to the solid phase volume, it must be written [ρg/ne (∂qi /∂t)δSi δt]/δVe , where ne is the effective porosity. However, the inertial force is unimportant compared to the viscous drag. Upon substitution of the hydraulic gradient J for D/ρg in the empirical Darcy’s law, the specific drag force D can be expressed as a function of the specific discharge q and the inverse of the hydraulic transmissivity [k]−1 , called hydraulic resistivity, giving D = ρgq · [k]−1 . Therefore, in addition to the effects due to the size and the dimension of the discontinuities of the soil and rock mass, the hydraulic resistivity also incorporates the influence of the kinetic fluid viscosity. To separate these effects from the control
© 2010 Taylor & Francis Group, London, UK
F u n d a m e n t a l s 43
of the structure of the previous media, the eigenvectors of the hydraulic conductivity may be expressed as ki = (ρg/ν) · Ki where K, dubbed the intrinsic permeability, and has dimensions L2 and only depends on the shape and spatial arrangement of pores, fractures and conduits. It must be pointed out that the vectors D and q are only collinear for isotropic and homogeneous media. 1.3.5 A d d en d u m 1.5: M a s s c o n s e r va t i o n p r i nc i p l e Consider a saturated and slightly compressible Darcy’s pseudo-continuous system. During an unit time interval, the mass conservation principle implies that the differential amount of water mass entering and leaving throughout the surface S of this system matches the differential amount of water mass within its volume V due to the contraction (or expansion) of its voids and to the compressibility of water. Occasionally, during the same time interval, distributed specific sources (or sinks) Q within the system volume V may add into (or take out from) its voids an extra amount of water. Then, the mathematical statement of the continuity equation is:
V
∂(ρ n) dV + ∂t
0
S
V
ρ q dS −
0
ρ Q dV = 0
(1.75)
0
In this expression, the first integral measures the sum of the differential changes of the amount of water mass filling up the voids within the system volume elements dV, due to variations in porosity n and water density ρ. The second integral measures the net amount of the differential inflows and outflows of water ρq throughout all the six surface elements dS. By the convention adopted in this book, the outgoing normal n of any surface elements dS is always positive. In contrast, also by a traditional convention adopted in practice, the scalar product ρq · dS is considered negative for outflows and vice versa. This somewhat arbitrary choice reflects the fact that the gradient ∂H/∂n has the opposite direction of the outgoing normal n for outflows and vice versa. The third integral measures the net amount of extra water added into or removed from the system volume V by all sources and sinks Q within the system Q is positive for an injection). Recalling Gauss’ theorem, the surface integral in this equation can be substituted for a volume integral:
S
ρq dS =
0
V
∇(ρq)dV
(1.76)
0
In this expression, the symbol ∇, called the del operator, is: ∇=
3 ∂ ui ∂xi
(1.77)
i=1
Then, the continuity statement takes the form: 0
V
∂(ρn) dV + ∂t
V
V
∇(pq)dV −
0
© 2010 Taylor & Francis Group, London, UK
0
ρQ dV = 0
(1.78)
44
Fractured rock hydraulics
Shrinking indefinitely the volume of integration V around any point inside this volume and cancelling all volume elements dV and integral signs ∫ yields the differential form of the continuity equation (provided all integrands are continuous): ∂(ρn) + ∇(ρq) − ρQ = 0 ∂t
(1.79)
Darcy’s law allows the specific discharge q in this equation to be written in terms of the hydraulic head H:
3
3 ∂(ρn) ∂ ∂H ρ −k.ij + − ρQ = 0 ∂t ∂xi ∂xj i=1
(1.80)
j=1
As H = z + p/ρg, it follows that ∂p/∂t = ρg · ∂H/∂t. Therefore, applying the “chain rule’’ to the first term of the equation:
∂(ρn) ∂(ρn) ∂p ∂(ρn) ∂H ∂ρ ∂n ∂H = = ρg = ρg n + ρ ∂t ∂p ∂t ∂p ∂t ∂p ∂p ∂t
(1.81)
In this equation, considering that the water density ρ at a confined pressure p can be estimated by a linear relation ρ0 [1 + βp], where ρ0 is the water density at atmospheric pressure and β is the isothermal compressibility coefficient of water, the first differential ratio ∂ρ/∂p in the brackets corresponds to ρ0 β. Assuming linear elastic behaviour and that any pressure relief (or pressure increase) is fully loaded on (or unloaded from) the discontinuous solid matter the second differential ratio ∂n/∂p in the brackets corresponds to the inverse of the subsystem bulk modulus Kp . In fact, supposing that any expansion (or contraction) of its open discontinuities primarily reflects variations of its effective hydrostatic stress ∂σhydrostatic (as defined in soil mechanics) and supposing that the total average stress σ remains invariant, any variation ∂p induces an equal but opposed hydrostatic stress variation −σhydrostatic . Moreover, ignoring the insignificant deformation of the solid matter itself, any expansion (or contraction) of the open discontinuities caused by −∂σhydrostatic is essentially due to its void volume change. Then, in this case, Kp = −∂σhydrostatic /[∂(nV)/V] = ∂p/∂n. Taking into account these considerations, the expression ρg[n∂ρ/∂p + ρ∂n/∂p] can approximately be substituted by a constant value v , called specific volumetric storage:
ρ V = ρg nρ0 β + Kp
(1.82)
V units are ML−4 . For constant water density, the general continuity equation can be normalized by ρ and v and takes as its simplified expression, denoted by SV , quoted in L−1 units:
1 SV = ρg nβ + Kp
© 2010 Taylor & Francis Group, London, UK
(1.83)
F u n d a m e n t a l s 45
Substituting v for ∂(ρn)/∂t, the continuity equation takes the form:
3 3 ∂ ∂H ∂H ρkij = V − ρQ ∂xi ∂xj ∂t i=1
(1.84)
j=1
For constant ρ:
3 3 ∂ ∂H ∂H kij = SV −Q ∂xi ∂xj ∂t i=1
(1.85)
j=1
For a more rigorous discussion about the mass conservation principle, see the full development presented by Bear (Bear et al., 1968).
References Bear, J, Zaslavsky, D and Irmay, S, 1968, Physical Principles of Water Percolation and Seepage, Paris, Unesco. Bear, J., 1972, Dynamics of Fluids in Porous Media, American Elsevier, New York, NY. Darcy, H, 1856, Les fontaines publiques de la ville de Dijon, Victor Dalmon, Paris. Dupuit, J., 1863. Etudes Théoriques et Pratiques sur le Mouvement des Eaux dans les Canaux Découverts et a travers les Terrains Perméables, 2◦ Ed., Dunod, Paris. Franciss, F. O., 1970, Contribution a l’Étude du Mouvement de l’Eau a travers les Milieux Fissurés, Thèse, Faculté de Sciences de l’Université de Grenoble. Hshieh, P. A., Neumann, S. P., Stiles, G. K. and Simpson, E. S., Nov/1985, Field Determination of the Three-Dimensional Hydraulic Conductivity Tensor of Anisotropic Media, Water Resources Research, Vol. 21. Koerber, G. G., 1962, Properties of Solids, Prentice-Hall, New Jersey. Louis, C., 1969, Flow phenomena in jointed media and their effect on the stability of structures and slopes in rock, Imperial College, Rock Mechanics Progress Report. Nelson, R. A., 2001, Geologic Analysis of Fractured Reservoirs, Second Edition, ButterworthHeinemann, Woburn, MA. Quadros, E., 1982, Determinação das Características do Fluxo de Água em Fraturas de Rochas, Universidade de São Paulo, Brazil. Rocha, M and Franciss, F. O.: “Determination of Permeability in Anisotropic Rock Masses from Integral Samples’’, Rock Mechanics, Vol. 9/2-3, 1977.
© 2010 Taylor & Francis Group, London, UK
Chapter 2
Approximate solutions
2.1 Overview The mathematical simulation of the hydraulic behaviour of a groundwater system requires the analytical or numerical integration of two simultaneous partial differential equations, the governing equations, connecting one or more partial derivatives of the dependent variable head H with respect to the independent variables (x1 , x2 , x3 , t). Occasionally, as for Dupuit’s assumption for unconfined flows, these partial differential equations involve the dependent variable H itself. The order of a partial differential equation corresponds to the order of its highest derivative. They are linear if the dependent variable H and the partial derivatives ∂H/∂xi and ∂H/∂t, including their products, are only raised to the first power, but they are considered quasi-linear, if their highest derivatives remain linear. The continuity equation, resulting from the combination of these two governing equations, associates the spatial and temporal coordinates (r, t) of the points of the flow domain to their hydraulic heads H, via the hydraulic conductivity tensor [k]. The integration of the continuity equation gives the spatial-temporal description of the hydraulic head H on the flow domain as a scalar field. The derivatives of the hydraulic head H give the hydraulic gradients vectors J as a vector field. Finally, the empirical relationships between [k] and J yields the specific discharges vectors q as another vector field. Domains typified by simple structures and properties favour analytical solutions but for most groundwater problems, particularly those involving fractured rocks, a closed analytical solution does not exist. In these cases, numerical techniques, may lead to approximate solutions. Table 2.1 lists the essential features of numerical and the analytical solutions, pointing out their merits and disadvantages (Chung-Yau, 1994). Fractured rock masses are inhomogeneous and anisotropic. Their hydraulic properties may vary enormously over the flow domain. Fortunately, they may be correlated to descriptors of the geological features of the flow domain, duly calibrated by field and laboratory tests. However, a consistent knowledge of these properties alone does not lead to a particular solution for a specific groundwater system. Information about its history and the external influences on the spatial and temporal boundaries must be furnished to the “modeller’’. In fact, every particular solution for the continuity equation depends on knowledge of some supplementary equations: the boundary conditions and initial values. The problem is said to be well posed if for every set of supplementary
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Fractured rock hydraulics
Table 2.1 Analytical and numerical methods: pros and cons. Analytical
Numerical
Continuous solutions, described at every point. Solutions are exact or very accurate. Solution parameters usually have a clear physical meaning. Solutions do not exist for most practical problems.
Piecewise solutions, only described at grid points, requiring interpolation from these points. Solutions are approximate but allow error control. Solution parameters usually do not have a clear physical meaning. More than one approximate solution may be found for practical problems, depending on the desired accuracy.
equations one can find a unique solution continuously dependent on this set. When boundary conditions are partially unknown, it may be possible to estimate the missing data by plausible hypotheses about the past and about the hydraulic behaviour on the borders of the groundwater system.
2.2 Differential operators Assuming constant density ρ and using a symbolic differential operator, i.e. operators that perform as ordinary algebraic symbols and comply with the fundamental laws of algebra, the general continuity equation can be compactly written as: DV · H(r, t) − Q(r) = 0
(2.1)
In this equation, the symbolic operator DV stands for a second order linear differential operator and Q(r) for a space-dependent, and sometimes also time-dependent, water source (or drain) per unit volume and per unit time, i.e. Q = lim[(δVw /δt)/δV]. In this case, the symbolic expression for DV is: DV = −
3
3 ∂ ∂ ∂ Kij · + SV ∂xi ∂xj ∂t i=1
(2.2)
i=1
Here, SV denotes the specific volumetric storage, as discussed in Chapter 1 (see equation 2.2). For homogeneous and anisotropic pervious media and for coordinate axes complying with the eigenvectors of [k], all cross products vanish and the differential operator DV takes its more traditional form:
3 ∂ ∂ ∂ DV = − ki · + SV · ∂xi ∂xi ∂t
(2.3)
i=1
If the time-dependent term SV · (∂/∂t) = 0 vanishes, the differential equation describes a steady flow. Otherwise, it describes an unsteady flow. If present, water sources or sinks Q(r) are only located at some points of the flow domain.
© 2010 Taylor & Francis Group, London, UK
Approximate solutions
49
Second order linear partial differential equations may be classified as elliptic, parabolic or hyperbolic (see addendum 2.1 on page 87). They are homogeneous if all terms contain the unknown variable. Otherwise, they are inhomogeneous. Boundary-value problems associated with the Laplace equation, in which Q(r) = 0, or with the Poisson, in which Q(r) = 0, give time-independent solutions. These equations do not explain the rate of the energy consumption needed to sustain the steady-state flow. However, this non-explicit power rate is implied by the definition of the hydraulic gradient J (see Chapter 1, addendum 1.2). The Laplace equation describes “natural’’ equilibrium processes. The maximum and minimum solution values occur at the flow boundaries. The Poisson equation describes equilibrium processes disturbed by steady sources and/or drains. Thus, the maximum and minimum solution values may occur within the flow domain. Parabolic equations describe transient flows tending to equilibrium as t tends to infinity. Maximum and minimum solution values gradually shift within the flow domain, decreasing or increasing with time. Their time-dependent nature requires the prescription of initial values to particularise solutions. In rare cases, the prescription of time-dependent boundary values is also required. If applied to a confined aquifer resting on a non-horizontal impervious bottom, Dupuit’s assumption also yields a linear second order differential equation (see fig. 2.1 and also Chapter 1, section 1.2.3.2):
∂ ∂ ∂ P P HC kEE B + HC + + kEN B + HC + ... ∂E ∂E ρg ∂N ρg −
∂ ∂ ∂ P P HC kNE B + HC + + kNN B + HC + + ∂N ∂E ρg ∂N ρg ∂ + (ne HC ) − ω = 0 (2.4) ∂t
In this equation, the independent variables HC and B denote the confined saturated height and the base elevation, respectively. The dependent variable P/ρg stands for the pressure head at the roof of the aquifer, given the dependent total head H = (B + HC + P/ρg). The dependent pressure may be referred to the middle of the confined thickness HC and denoted by Pm . The differential equation must change accordingly. Recalling that as ∂H/∂z = zero, Pm = ρg(HC /2) + P. Collecting the unknown variable H and known variables B and HC and introducing another differential operator DC , this equation can also be written as: ∂ ∂ ∂ HC kEE B + kEN B ... ∂E ∂ ∂E ∂N DC H − H ) − ω =0 + (n e C ∂t ∂ ∂ ∂ + HC kNE B + kNN B ∂N ∂E ∂N
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(2.5)
50
Fractured rock hydraulics
Previous position of falling water table
Piezometric head
Subsequent position of falling water table Impervious top boundary
i-component of differential groundwater flow
Impervious bottom boundary
Figure 2.1 Dupuit’s hypothesis for an unconfined (left) and a confined (right) aquifer over an impervious base.
The symbolic linear differential operator DC is: ∂ ∂ ∂ HC kEE +kEN ... ∂E ∂E ∂N DC = − ∂ ∂ ∂ + HC kNE +kNN ∂N ∂E ∂N
(2.6)
All preceding symbolic operators are linear. However, Dupuit’s approximation when applied to an unconfined aquifer yields a non-linear second order differential equation. Whenever possible, a simple change of variable may transform and approximately linearise this equation. Indeed, considering the general continuity equation for an
© 2010 Taylor & Francis Group, London, UK
A p p r o x i m a t e s o l u t i o n s 51
unconfined flow: ∂ ∂ ∂ H kEE (B + H) + kEN (B + H) ... ∂E ∂E ∂N ∂ − + (ne H) − ω = 0 ∂ ∂t ∂ ∂ + H kNE (B + H) + kNN (B + H) ∂N ∂E ∂N (2.7) By the differentiation rule of a variable raised to a power and for ne assumed constant, H · ∂H/∂ . . . = (1/2) · ∂H2 / ∂ … Then, this equation can be rewritten as:
∂ ∂ ∂ H k B + k B . . . EE EN ∂E ∂E ∂N
. . . ∂ ∂ ∂ H kNE B + kNN B + ∂N ∂E ∂N ne ∂ 2 − (H ) − ω = 0 +
2H ∂t ∂ 2 ∂ 2 ∂ ... kEE H + kEN H 1 ∂E ∂E ∂N +
2 ∂ ∂ 2 ∂ 2 + kNE H + kNN H ∂N ∂E ∂N (2.8) For most unconfined aquifers where Dupuit’s hypothesis can be adequately applied, the areal variations of H are relatively small compared to the aquifer thickness. Then, substituting a new auxiliary variable V for H2 and considering an average value for H ≈ Hav :
∂ ∂ ∂ k H B + k B ... av EE EN ∂E ∂E ∂N
... ∂ ∂ ∂ Hav kNE B + kNN B + ∂N ∂E ∂N 1 ∂ −
+ 2H ∂t (ne V) − ω = 0 av ∂ ∂ ∂ ... 1 ∂E kEE ∂E V + kEN ∂N V +
2 ∂ ∂ ∂ + kNE V + kNN V ∂N ∂E ∂N (2.9) This transformed equation can also be compactly written as a linear if one:
∂ ∂ ∂ H k B + k B ... av EE EN ∂E ∂E ∂N
DU V − ∂ −ω=0 ∂ ∂ Hav kNE B + kNN B + ∂N ∂E ∂N
© 2010 Taylor & Francis Group, London, UK
(2.10)
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Fractured rock hydraulics
Now, the linear differential operator DU is:
∂ ∂ ∂ k + k ... EN ∂E EE ∂E ∂N 1 1 ∂
DU = − ∂ (ne ) + ∂ ∂ 2Hav ∂t 2 + kNE + kNN ∂N ∂E ∂N
(2.11)
This variable change implies also changing all specified boundary conditions in terms of V instead of H2 . The square root of the final solution V(r, t) re-establishes the head H(r, t). Boundary conditions can also be written in a compact symbolic form. As explained in Chapter I, the 3D equation of continuity results from the application of the mass conservation principle to an elementary volume δV within a flow domain. By a similar reasoning, the 2D Neumann or natural boundary condition may result from the application of the same principle to an elementary surface δS on the boundary of the flow domain. In fact, if αni denote the direction cosines of the outward normal n to a boundary element δS, a specified incoming (or outgoing) normal flow qn · δS must be equal to a corresponding Darcy’s flow component at the back side of δS, defined by the inner product kij Jj · αni · δS. Then, equating both expressions and removing δS: 3 3
(kjj jj ) αni − qn = 0
(2.12)
i=1 j=1
Considering the hydraulic gradient J expressed as a function of the hydraulic head H: −
3
3 i=1 j=1
∂ kjj H ∂xj
αni − qn = 0
(2.13)
This equation translates the Neumann or natural boundary condition, and is compactly written as: DN H(r, t) − qn (r) = 0
(2.14)
The differential operator DN is: DN = −
3 3
i=1 j=1
kjj
∂ αni ∂xj
(2.15)
On the other hand, the Dirichlet or essential boundary condition prescribes the hydraulic head values, at least at one single boundary point, to guarantee a unique solution.
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Approximate solutions
53
2.3 Uniqueness of solutions Two apparently dissimilar particular solutions for the same domain and the same boundary and/or initial conditions must be identical: i.e. there must be a unique solution. Statements and proofs of uniqueness theorems are a well-developed branch of mathematics. However, the validity of uniqueness theorems can be more or less intuitively perceived. What is important to keep in mind is that properly defined supplementary equations, i.e. well-posed boundary and/or initial conditions, are crucial to assure a unique solution. Correct prescriptions for boundary and initial conditions depend on the type of the partial differential equation, hyperbolic, elliptic or parabolic, and on the character of the domain boundaries, open or closed. An open border extends indefinitely in one or more dimensions and a closed border surrounds the entire domain. Dirichlet and Neumann prescriptions for boundary conditions are usually required to guarantee unique solutions for groundwater flow in a state of equilibrium: steadystate solutions. For anisotropic rock masses, the Neumann condition must take into account the non-parallelism between the hydraulic gradient and the specific discharge. A third kind of boundary condition, called Cauchy or Robbins or mixed condition, stipulates linear combinations of Dirichlet and Neumann. If not compatible, these prescriptions may not fit to concrete situations or may imply forcing functions. This is a common source of groundwater simulation errors, for example, if gravity-induced draining shafts are substituted for pumping wells. Prescribed values of the hydraulic head and its derivatives over the flow domain at an elected time, dubbed initial conditions, are essential for unique non-equilibrium simulations: unsteady-state solutions. These solutions generally tend to a state of final equilibrium as time goes on: the initial configuration causing subsequent effects until a virtually stable state is reached. Then, a fixed boundary condition for a boundary value problem may be interpreted as describing the final state of equilibrium that results in time from the initial condition. A third type of problems, called initial boundary problems require concomitant prescriptions of initial conditions and well-matched boundary conditions. Even if a unique solution exists, at least theoretically, it may not be found or may be hard to get. This is particularly the case for complex systems that do not conform to simple classification schemes, as their erratic properties imply differential equations with varying coefficients. In this situation, more than one approximate solution, each one with its own accuracy level, may suit the “modeller’s’’ needs. Note, even here the boundary conditions must be consistently defined. Given without proof, table 2.2 summarises the uniqueness theorems that apply to the several types of boundaries, boundary conditions and differential equations (Band, 1960).
2.4 Approximate solution errors When an analytical solution is too complex or unknown, it may be advantageous to find an approximate solution (also known as an approximant). Simple approximants are cast as linear polynomials or interpolation polynomials, allowing easy and fast computations by plain algebraic rules at a known accuracy level. Attentive inspection of how well different approximants simulate the same physical system under the same
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Table 2.2 Uniqueness theorems Boundary condition
Boundary type
Hyperbolic equation
Dirichlet or Neumann
Open
Dirichlet or Neumann
Closed
Underspecified (therefore many solutions)
Cauchy
Open
Unique Solution
Cauchy
Closed
Elliptic equation
Underspecified (therefore many solutions) Unique Solution
Parabolic equation Unique Solution Overspecified (usually no solution possible)
Overspecified (usually no solution possible)
Overspecified (usually no solution possible)
conditions can reveal dominant trends that may led to the discovery of more significant facts. If a linear approximant h(r, t) is substituted for the exact but unknown solution H(r, t), the following approximant error measures must be properly considered: – – – –
For the approximate solution itself within the solution domain: ε = h(r, t) − H(r, t) For the Dirichlet condition on the boundary: εD = h(r, t) − HD (r, t) For the continuity condition within the solution domain: εV = DV · h(r, t) − Q(r) For the Neumann condition on the boundary: εN = DN · h(r, t) − qn (r)
Obviously, the error measures ε and εD have the same meaning, and are distinguished here only for their specific location. Well-known numerical methods, differing in the way they minimise these residuals, have their own advantages and inconveniences with regard to reliability, costs, performance and practicability. For simple systems, a unique approximation applied to the flow domain may be adequate. For complex systems, a finite set of simultaneous but local and specific approximants applied to its subdomains is recommended. The hydraulic behaviour of a groundwater system may be acceptably described by a linear polynomial approximation h(r, t) resulting from the finite sum of (n + 1) products of numerical coefficients ai and convenient elementary analytical functions hi (r, t): h(r, t) = a0 h0 (r, t) + a1 h1 (r, t) + a1 h3 (r, t) + · · · + an
(2.16)
Written more compactly: h(r, t) =
n
[ai hi (r, t)]
(2.17)
i=0
Adequate elementary functions hi (r, t) depend on the behaviour to be described. These functions, called base functions, share some common analytic attributes. The numerical coefficients ai need to be determined for each specific solution.
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A p p r o x i m a t e s o l u t i o n s 55
Two properties of the elementary analytical solutions of the continuity equation help to select efficient base functions. The first, the superposition principle, asserts that if elementary analytical solutions hi (r, t) solve the homogeneous equation D · hi (r, t) = 0, then a linear combination of these solutions is also a solution. Indeed, considering the linear character of the operator D, the following equality holds: n
[D[ai hi (r, t]] = D
i=0
n
[ai hi (r, t)]
(2.18)
i=0
The second property asserts that any partial derivative ∂ hi (r, t)/∂xj of an elementary solution hi (r, t) is also a solution. In fact, taking into account the linear character of the partial derivative operator, the following equality holds: n n ∂ ∂ D ai [hi (r, t)] = [D[ai hi (r, t)]] ∂xj ∂xj i=0
(2.19)
i=0
For a non-homogeneous equation DV · H(r, t) − Q(r) = 0, the forcing function Q(r) may be also approximated by a finite sum of low order terms gk (r, t): Q(r, t) =
n
[ak gk (r, t)]
(2.20)
k=j+1
If each elementary solution gk (r, t) solves DV · gk (r, t) − Q(r) = 0, then their linear combinations also do. As a result, approximants for the non-homogeneous equation can be structured by a linear combination of elementary solutions for both homogeneous and non-homogeneous equations. A natural linear set of base functions may be recursively generated by successive integrations of low order elementary analytical solutions – – –
1, x, x2 , x3 , … xn 1, x, y, x2 , y2 , xy, … xa yb 1, x, y, z, … xa yb zc
(a + b < n) (a + b + c < n)
However, these base functions do not suitably fit time-dependent solutions. In this case, they may be partially substituted with the product of a time-dependent and space-dependent elementary solution: fi (t) · ui (r). Finally, it has to be noted that trigonometric or exponential base functions, or a few terms of their infinite linear series expansions, may also give adequate approximants. Efficient base functions must be linearly independent, i.e. a base function must not result from combinations of similar base functions, to assure the construction of successively improved high order approximants without replicating the fitting ability of the low order base functions. Space-dependent power series are effective approximants. Their accuracy can be improved by adding higher order terms. Their derivatives or integrals result from the sum of the derivatives or integrals of their individual terms. The numerical evaluation of
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power series requires simple addition, subtraction and multiplication operations. However, to restrict computation, the number of their terms and corresponding numerical coefficients must be limited. Time-dependent trigonometric base functions may fit periodic behaviour of oscillatory boundary conditions. For an open time boundary, the linear combinations of the low order terms of the infinite series expansion of the exponential function may fit the long-run reaction (rise or decay) of a transient system. The choice of the most adequate set of base functions to fit the expected response of a system to an imposed field is relatively simple as sometimes it is possible to link the typical characteristics of the system’s physical reactions to base function graphs. Approximants h(r, t) can be used to describe scalar quantities or vector or high order tensor components. An s-order Cartesian tensor referred to a 3D-frame and described by a continuous function may have its 3s components approximated by 3s distinct general polynomials. An approximation polynomial h(r, t) may be alternatively written as a polynomial interpolation:
h(r, t) =
n
Hi · fi (r, t)
(2.21)
i=0
Cast in this form, the interpolating functions fi (r, t) are linear combinations of the elementary functions hi (r, t) and the numerical coefficients Hi correspond to known solution values at n arbitrarily selected points (ri , ti ). As the general polynomial and the interpolation polynomial express the same approximation, the following identity holds: n
n ai · hi (r, t) = [Hi · fi (r, t)]
i=0
(2.22)
i=0
Obviously, a Taylor series expansion defined in a closed interval can also be written as a general polynomial. Its main advantage is the immediate recognition of the upper limit of the approximation error. Example 2.1 helps to understand these concepts. It shows a Taylor series approximant and the equivalence between a general polynomial and an interpolation polynomial.
Example 2.1 A one-dimensional analytical function f(x) defined in a closed interval [a, b] may be approximated by the first three terms of a second order Taylor expansion: ∂ ∂2 [h (x )] [h0 (x0 )] 2 0 0 h(x) ≈ h(x0 ) + ∂x (x − x0 ) + ∂x (x − x0 ) 1! 2!
© 2010 Taylor & Francis Group, London, UK
(2.23)
A p p r o x i m a t e s o l u t i o n s 57
As known from differential calculus, the remainder Rn of the Taylor series gives an upper estimate of the approximation error: ∂n [h (ξ)] n 0 Rn = ∂x · (x − x0 )n n!
(2.24)
As ξ is an unknown value in between a and b, the exact error cannot be estimated but its upper value can. The first three terms of the Taylor expansion can be also written as a polynomial approximation (and vice versa) as: h(x) ≈ a0 + a1 x + a2 x
(2.25)
By simple algebraic operations, it can be shown that the coefficients a0 , a1 and a2 are: ∂ ∂ [h0 (x0 )] [h0 (x0 )] ∂x 2 a0 = h(x0 ) − ∂x · x0 + · (x0 ) 1! 2! ∂ ∂ [h0 (x0 )] [h0 · (x0 )] ∂x a1 = ∂x −2· · x0 1! 2!
(2.26)
∂2 ∂x2 [h0 (x0 )] a2 = 2! The function f(x) may also be approximated at any point x of a closed interval [x0 , x3 ] by a Lagrangian quadratic interpolation polynomial H(x). If H0 , H1 and H2 are already known values of f(x) at three non-equidistant points x0 , x1 and x2 , then: (x − x0 )(x − x2 ) (x − x1 )(x − x2 ) H(x) = H0 + H1 (x0 − x1 )(x0 − x2 ) (x1 − x2 )(x1 − x0 ) (x − x0 )(x − x1 ) + (2.27) H2 (x2 − x0 )(x2 − x1 ) The first and second derivatives of f(x) can also be estimated by the polynomial derivatives of this approximant: ∂ (x − x2 ) + (x − x0 ) (x − x2 ) + (x − x1 ) H(x) = H0 + H1 ∂x (x0 − x1 )(x0 − x2 ) (x1 − x0 )(x1 − x2 ) (x − x1 ) + (x − x0 ) + (2.28) H2 (x2 − x0 )(x2 − x1 ) 2 2 2 ∂2 H(x) = H0 + H1 + H2 ∂x2 (x0 − x1 )(x0 − x2 ) (x1 − x0 )(x1 − x2 ) (x2 − x0 )(x2 − x1 ) (2.29)
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The resulting error is: f(x)−H(x) =
(x − x0 )(x − x1 )(x − x2 ) (x − x0 )(x − x1 )(x − x2 ) ∂3 f(x) ≤ M 2 ∂x3 2
(2.30)
The quantity M never exceeds the absolute value of ∂3 f(x)/∂x3 . Residual estimates for polynomials of greater order result from generalisations of the Taylor expansion. Hermitian interpolation polynomials (and one variant known as the Padé compact formula) differ from the Lagrangian approach by using at each interpolating point not only the base functions but also their first derivatives together. Generalisations include higher order derivatives. A general polynomial can be alternatively written as an interpolation polynomial. For example, consider how to fit as a general polynomial a simple first order polynomial approximant h(x, y) to an incompletely defined higher order surface (see fig. 2.2): a h(x,y) = (1 x y) b (2.31) c H Hi Hk Hj
y
S x
Figure 2.2 Higher order surface incompletely defined by only three known values H.
Select the interpolating points (xi , yi , Hi ), (xj , yj , Hj ) and (xk , yk , Hk ). Now, express the known values Hi , Hj and Hk as functions of the coordinates (xi , yi ), (xj , yj ) and (xk , yk ) and of the coefficients of the polynomial approximant a, b and c: 1 xi yi a Hi (2.32) H j = 1 xj y j b c Hk 1 xk y k Then, the coefficients a, b and c may be calculated as: a 1 xi b = 1 xj c 1 xk
−1 yi Ai Hi 1 yj Hj = Bi 2S yk Hk Ci
© 2010 Taylor & Francis Group, London, UK
Aj Bj Cj
Ak Hi Bk H j Ck Hk
(2.33)
A p p r o x i m a t e s o l u t i o n s 59
In this expression, S denotes the area of the triangle defined by the interpolating points: 2S = [xi (yk − yj ) + xj (yi − yk ) + xk (yj − yi )].
(2.34)
The matrix of coefficients Ai … C k is:
Ai Bi Ci
Aj Bj Cj
Ak yj xk − xj yk B k = yk − y j Ck xj − x k
xi y k − y i x k yi − y k xk − x i
y i xj − x i y j yj − y i xi − x j
(2.35)
Substituting the coefficients a, b and c for their expressions in the general polynomial yields:
h(x,y) = (1
x
Ai 1 y) Bi 2S Ci
Aj Bj Cj
Ak Hi Bk Hj Ck Hk
(2.36)
2.5 Approximation methods 2.5.1
Pr elim i n a r i e s
To find an approximate description for the hydraulic behaviour of a groundwater system it is generally sufficient to know what differential operator complies with the problem in hand and what are its particular boundary and initial conditions. Then, the approximation errors or residuals may be forced to be locally zero or may be minimised in an average sense by one of the main weighted residuals methods: collocation, least squares and Galerkin (due to Boris Grigorievich Galerkin, 1871–1945, a Russian mathematician). For relatively simple flow problems, these methods can be applied to find global approximants for the entire flow domain. For complex groundwater systems, they can be applied to find local approximants for adjacent flow subdomains, using the finite difference, finite element or boundary element methods. Example 2.2 helps to understand how a cumbersome analytical solution may be replaced by a simpler one without harmful loss of accuracy.
Example 2.2 If a continuous and constant excess load is suddenly placed at the top of a fully saturated, deformable and normally consolidated horizontal soil stratum, the original groundwater hydrostatic pressure distribution grows up immediately as the soil pore water takes up the induced excess of vertical stress due to that top loading. By Terzaghi’s solution for one-dimensional consolidation, this pressure excess is progressively transferred from the pore water to the soil grains. This gradual transfer is theoretically described by the degree of consolidation as a function of the time elapsed after the top
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loading. This parameter, denoted by U and normally expressed as a percentage, is related by another normalised parameter, the time factor T, by the following expressions:
∞ U = 1 − m=0
2
2H
0
Mz −M2 T ui sin dze H 2H 100 M ui dz
(2.37)
0
T=
cv t H2
In these formulas, ui stands for the excess initial pore pressure at the centre of the soil strata, cv for the coefficient of vertical consolidation (experimentally determined by laboratory tests), H for the shortest escape path for the pressurised groundwater and t for the time elapsed since the start of the process. From U values between 0% and 60%, this percentage can be conveniently calculated by the simple approximate formula: 4T U= (2.38) π
2.5.2
C olloc a t i o n m e t h o d
The collocation method may be used when hydraulic heads and their transforms by the differential operators are respectively known at m and mV points within the flow domain and, in addition, Dirichlet and Neumann conditions are respectively known at mD and mN points at boundaries. Then, if the errors ε, εV , εN and εD for the chosen n-degree approximant are equated to zero, the resulting system of n simultaneous equations gives the numerical coefficients. If m + mV + mN + mD > n + 1, this system may be solved by a best fit model. Considering an interpolation polynomial (equation 2.21), this system of simultaneous equations is symbolically written as: ε=
n
[Hfi (r, t)] − H(r, t) = 0
i=0
εV = DV
n
for all m points within the flow domain
[Hfi (r, t)] − Q(r) = 0
for all mV points within the flow domain
(2.40)
[Hfi (r, t)] − qn (r) = 0
for all mN points at the Neuman boundaries
(2.41)
i=0
εN = DN
n i=0
εD =
n
(2.39)
[Hfi (r, t)] − HD (r, t) = 0
i=0
© 2010 Taylor & Francis Group, London, UK
for all mD points at the Dirichlet boundaries
(2.42)
Approximate solutions
61
A similar system may be written for a polynomial approximation. Example 2.3 clarifies the essence of this method.
Example 2.3 The table below assembles data from piezometers PA, PB, PC and PD installed to measure the water table elevation on a restricted area of an unconfined aquifer seeping over an almost impervious non-horizontal bottom.
Date of observation 2004-10-28
East coordinate E (m)
North coordinate N (m)
Aquifer bottom B (m)
Hydraulic head H (m)
Piezometer PA Piezometer PB Piezometer PC Piezometer PD
1.26E + 03 2.26E + 03 1.76E + 03 1.66E + 03
3.75E + 03 3.65E + 03 2.85E + 03 3.35E + 03
31.331 29.719 29.569 30.581
62.214 59.592 67.758 73.878
The hydraulic heads inside the influence area of these piezometers were previously estimated by a simple linear interpolation (see fig. 2.3): PA PB
3700 3600 3500 PD
N (m)
3400 3300 3200 3100 3000
PC
2900
1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 E (m)
Figure 2.3 Hydraulic head contour map on 2004-10-28 drawn by simple linear interpolation of the monitored head values at the piezometers PA, PB, PC and PD.
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A polynomial approximation by the collocation method may produce a best picture of the head contour map. Based on local geological evidence, the elevation B of the aquifer bottom may be approximated by a planar surface defined by the equation: B(E,N) = b0 + b1 E + b2
(2.43)
Known elevation levels B of the aquifer bottom at the four piezometers axes provide four algebraic equations to give the coefficients b0 , b1 and b2 : b0 + 1.3 103 b1 + 3.7 103 b2 − 31.331 = 0 b0 + 2.3 103 b1 + 3.6 103 b2 − 29.719 = 0 b0 + 1.8 103 b1 + 2.8 103 b2 − 29.569 = 0
(2.44)
b0 + 1.7 103 b1 + 3.3 103 b2 − 30.581 = 0 Minimising the residuals of the b coefficients in these four simultaneous equations by the least squares method gives their best estimate: 28.998 b = −1.5 × 10−3 1.125 × 10−3
(2.45)
Next, the variable hydraulic head B + H may be estimated by another polynomial approximation: B + H = a0 + a1 E + a2 N + a3 E2 + a4 N2 + a5 EN
(2.46)
The known hydraulic heads H at the four piezometers give four equations relating the six unknown coefficients a0 , a1 , a2 , a3 , a4 and a5 : a0 + 1.3 103 a1 + 3.7 103 a2 + 1.6 106 a3 + 1.4 107 a4 + 4.7 106 a5 − 62.214 = 0 a0 + 2.3 103 a1 + 3.6 103 a2 + 5.1 106 a3 + 1.3 107 a4 + 8.2 106 a5 − 59.592 = 0 a0 + 1.8 103 a1 + 2.8 103 a2 + 3.1 106 a3 + 8.1 106 a4 + 5.0 106 a5 − 67.758 = 0 a0 + 1.7 103 a1 + 3.3 103 a2 + 2.7 106 a3 + 1.1 107 a4 + 8.2 106 a5 − 73.878 = 0 (2.47) Assuming constant recharge w and invariant and isotropic hydraulic conductivity k and defining the normalised recharge as Rn = w/k, the general continuity equation of an unconfined aquifer over a variable non-horizontal impervious bottom is (see Chapter 1, section 1.2.3.2, Dupuit’s approximation): ∂ H ∂E∂ (B + H) . . . ∂E + Rn = 0 (2.48) ∂ ∂ + ∂N H ∂N (B + H)
© 2010 Taylor & Francis Group, London, UK
A p p r o x i m a t e s o l u t i o n s 63
Substituting B and B + H in the above equation for their corresponding polynomial approximations and performing the derivations yields the residual of the continuity equation: εV = (a1 + 2a3 E + a5 N − b1 )(a1 + 2a3 E + a5 N) . . . + 2(a0 + a1 E + a2 N + a3 E2 + a4 N2 + a5 E N − b0 − b1 E − b2 N)a3 . . . + (a2 + 2a4 N + a5 E − b2 )(a2 + 2a4 N + a5 E) . . . ω k (2.49)
+ 2(a0 + a1 E + a2 N + a1 E2 + a4 N2 + a5 E N − b0 − b1 E − b2 N)a4 . . . +
Equating εV to zero at the locations of the four piezometers gives four additional equations: (a1 + 2.6 103 a3 + 3.7 103 a5 − 1. b1 )(a1 + 2.6 103 a3 + 3.7 103 a5 ) . . . + (2. a0 + 2.6 103 a1 + 7.4 103 a2 + 3.4 106 a3 + 2.8 107 a4 + 9.6 106 a5 − 2. b0 − 2.6 103 b1 − 7.4 103 b2 )a3 . . . + (a2 + 7.4 103 a4 + 1.3 103 a5 − 1. b2 )(a2 + 7.4 103 a4 + 1.3 103 a5 ) . . . + (2. a0 + 2.6 103 a1 + 7.4 103 a2 + 3.4 106 a3 + 2.8 107 a4 + 9.6 106 a5 − 2. b0 − 2.6 103 b1 − 7.4 103 b2 )a4 + Rn = 0 (a1 + 4.6 103 a3 + 3.6 103 a5 − 1. b1 )(a1 + 4.6 103 a3 + 3.6 103 a5 ) . . . + (2. a0 + 4.6 103 a1 + 7.2 103 a2 + 1.1 107 a3 + 2.6 107 a4 + 1.7 106 a5 − 2. b0 − 4.6 103 b1 − 7.2 103 b2 )a3 . . . + (a2 + 7.2 103 a4 + 2.3 103 a5 − 1. b2 )(a2 + 7.2 103 a4 + 2.3 103 a5 ) . . . + (2. a0 + 4.6 103 a1 + 7.2 103 a2 + 1.1 107 a3 + 2.6 107 a4 + 1.7 107 a5 − 2. b0 − 4.6 103 b1 − 7.2 103 b2 )a4 + Rn = 0 (a1 + 3.6 103 a3 + 2.8 103 a5 − 1. b1 )(a1 + 3.6 103 a3 + 2.8 103 a5 ) . . . + (2. a0 + 3.6 103 a1 + 5.6 103 a2 + 6.4 106 a3 + 1.6 107 a4 + 1.0 107 a5 − 2. b0 − 3.6 103 b1 − 5.6 103 b2 )a3 . . . + (a2 + 5.6 103 a4 + 1.8 103 a5 − 1. b2 )(a2 + 5.6 103 a4 + 1.8 103 a5 ) . . . + (2. a0 + 3.6 103 a1 + 5.6 103 a2 + 6.4 106 a3 + 1.6 107 a4 + 1.0 107 a5 − 2. b0 − 3.6 103 b1 − 5.6 103 b2 )a4 + Rn = 0 (a1 + 3.4 103 a3 + 3.3 103 a5 − 1. b1 )(a1 + 3.4 103 a3 + 3.3 103 a5 ) . . . + (2. a0 + 3.4 103 a1 + 6.6 103 a2 + 5.8 106 a3 + 2.2 107 a4 + 1.1 107 a5 − 2. b0 − 3.4 103 b1 − 6.6 103 b2 )a3 . . . + (a2 + 6.6 103 a4 + 1.7 103 a5 − 1. b2 )(a2 + 6.6 103 a4 + 1.7 103 a5 ) . . . + (2. a0 + 3.4 103 a1 + 6.6 103 a2 + 5.8 106 a3 + 2.2 107 a4 + 1.1 107 a5 − 2. b0 − 3.4 103 b1 − 6.6 103 b2 )a4 + Rn = 0
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(2.50)
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In these equations, the normalised recharge Rn stands for the ratio w/k. Minimising the residuals of these eight simultaneous equations by the least squares method gives the best estimate of the normalised recharge Rn and of the ai coefficients:
116.944
−5.172 × 10−3 −0.02 a= −6 −3.138 × 10 9.865 × 10−7 3.5 × 10−6
(2.51)
Rn = 6.832 × 10−5 Then, the hydraulic heads H (expressed in metres) at the nodes of a 100 m × 100 m equally spaced grid covering the influence area of these piezometers can be estimated by the matrix equation:
H (E, N) = (a0
a1
a2
a3
a4
E
N a5 ) E 2 N2 EN
(2.52)
In this expression, the numerical coefficients (a0 , a1 , a2 , a3 , a4 , a5 ) as well as the variables (E, N) have compatible units. The following matrix assembles the calculated head values at the nodes of a 100 m × 100 m grid: 70.082 69.743 69.341 68.876 68.348 67.758 67.105 66.389 65.61 64.769 63.865
69.223 68.916 68.545 68.112 67.616 67.057 66.435 65.751 65.004 64.194 63.321
68.381 68.104 67.765 67.363 66.899 66.371 65.781 65.129 64.413 63.635 62.793
67.554 67.309 67.002 66.631 66.198 65.702 65.144 64.522 63.838 63.091 62.282
© 2010 Taylor & Francis Group, London, UK
66.743 66.53 66.254 65.915 65.513 65.049 64.522 63.932 63.279 62.564 61.786
65.948 65.766 65.522 65.215 64.844 64.412 63.916 63.358 62.737 62.053 61.306
65.17 65.019 64.806 64.53 64.192 63.79 63.326 62.799 62.21 61.557 60.842
64.407 64.288 64.106 63.862 63.555 63.185 62.752 62.257 61.699 61.078 60.394
63.66 63.572 63.422 63.209 62.934 62.596 62.194 61.731 61.204 60.615 59.962
62.929 62.873 62.754 62.573 62.329 62.022 61.653 61.22 60.725 60.167 59.547
62.214 62.19 62.103 61.953 61.74 61.465 61.127 60.726 60.262 59.736 59.147
Approximate solutions
65
A hydraulic head contour map can be constructed by interpolating between these grid nodes (see fig. 2.4): PA PB
3700 3600 3500 PD
N (m)
3400 3300 3200 3100 3000
PC
2900
1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 E (m)
Figure 2.4 Interpolated hydraulic head map from 100 m × 100 m equally spaced grid nodes of calculated head values.
2.5.3
L ea s t s q u a r e s m e t h o d
The collocation method is quite simple but it does not restrict the magnitude of the errors ε, εV , εN and εD outside the collocation points. An intuitive extension of this method is to equate to zero the sum of the residuals within subdomains. However, this approach may conceal quite large residuals and lead to computational difficulties (Linz, 1979). In an average sense, the least squares method overcomes this inconvenience by minimising the sum of squared errors. Consider first how to find an approximation for H(r, t) if some of its exact values are already known at few points. In this case, assume the quantity and the type of base functions hi (r, t) of the approximant h(r, t) are previously selected (according to the expected physical behaviour to be described and the desired accuracy). Then, the sum of the squared residuals ε2 only depends on arbitrary variations of the numerical coefficients aj . Therefore, the problem is reduced to the search for which set of coefficients aj minimises this sum within the flow domain and its boundaries. This set is the solution of the simultaneous equations below (see also addendum 2.2 on page 88). tf V n [ai hi (r, t)] − H(r, t) [hj (r, t)]dV dt = 0 0
0
i=0
for all j elementary solutions hj (r,t)
© 2010 Taylor & Francis Group, London, UK
(2.53)
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Fractured rock hydraulics
tf
0
V
0
n
[ai hi (r, t)] − H(r, t)
i=0
hj (r, t)
3
αk dS dt = 0
k=1
for all j elementary solutions hj (r,t)
(2.54)
In the last constraint, αk denote the direction cosines of the outward normal n to the boundary element dS. It must be stressed that as the analytic expression for H(r, t) is in fact unknown, the integrals ∫H(r, t) must be found numerically from the known values of H(r, t) at monitoring points. Example 2.4 makes this method clear for a simple case.
Example 2.4 The following graph (see fig. 2.5) shows the variation of the zinc content, from September 30, 2004 to March 31, 2005, found in groundwater samples collected each week (but not on the same day each week) at a point located 259 m downstream from a industrial process tailing dam.
Zn content (mg/l)
5 4 3 2 1 0
100
200
300
Days since 30/9/2004
Figure 2.5 Zinc content variation monitored at a control point. Faulty sampling, defective analysis or rain infiltration dilution may explain the disproportionate data spread.
To detect a general trend in time, the Zn content can be approximated by a cubic polynomial fitted to the scattered data by the least squares method, taking into account the irregular time interval between successive samples (varying from 1 to 7 days). Denoting by Pj the unknown numerical coefficients, a general polynomial expression for j varying from 0 to Np can be compactly written as (in this example Np = 4): Np −1
Zn =
(pj ,tj )
(2.55)
j=0
Minimising the sum of the squared residuals ε2 over the total time span, considering a cubic approximant, yields four simultaneous equations: T 0
N·p−1
(pj t ) t0 dt − j
j=0
© 2010 Taylor & Francis Group, London, UK
N−2
datai + datai+1 ti+1 + ti 0 =0 (ti+1 − ti ) 2 2 i=0
Approximate solutions
T
(pj t ) t1 dt − j
N·p−1
(pj t ) t2 dt − j
j=0
0
T
j=0
0
T
N·p−1
N·p−1
0
(pj t ) t3 dt − j
j=0
N−2
ti+1 + ti 1 datai + datai+1 =0 (ti+1 − ti ) 2 2 i=0
N−2
datai + datai+1 ti+1 + ti 2 (ti+1 − ti ) = 0 (2.56) 2 2 i=0
N−2
datai + datai+1 ti+1 + ti 3 (ti+1 − ti ) =0 2 2 i=0
In the above equations,T denotes the time since sampling begun. The second expression in these equations corresponds to trivial numerical integrations multiplied by the weighting factors t0 , t1 , t2 and t3 , taking into account the irregular time interval δt between consecutive Zn content values. From a technical point of view, these four simultaneous equations have a precise meaning. The first equation expresses the equivalence of the areas bounded by the approximant and by the known solution (in this particular case, defined by discrete values despite their questionable accuracy). The second equation forces the centres of gravity of these areas to be equally distant from the ordinate axis (equivalence or first moments). The third one requires equivalence of inertia moments (second moments) and the fourth, of third moments. Solving these simultaneous equations gives the unknown numerical coefficients of the cubic approximant: 0.908 0.314 (2.57) P= −0.012 1.627 × 10−4 Fig. 2.6 shows the resulting approximant.
Zn content (mg/l)
5 4 3 2 1 0
100
200
300
Days since 30/9/2004
Figure 2.6
Cubic polynomial fitted to the scattered data, taking into account the irregular variation of the time interval t between successive Zn content values.
© 2010 Taylor & Francis Group, London, UK
67
68
Fractured rock hydraulics
The preceding simultaneous equations (equation 2.53 and equation 2.54) can be compactly written using the symbolic continuous inner product notation and they may also be combined by weighting coefficients wV and wS : wV · < [ai hi (r, t) − H(r, t)], hi (r, t) >V,t − wS · < [ai hi (r, t) − H(r, t)], hi (r, t) >S,t = 0
(2.58)
For the weighted inner products, neither the minimum constraint within the flow domain nor the minimum constraints at the flow domain boundaries are exactly satisfied. How well they approximate the solution in each location depends on the relative magnitude of the weighting coefficients wV and wS . The least squares method also holds when the differential operators DV and DN are applied to the residuals [h(r, t) − H(r, t)]. In this case, the unknown coefficients aj , after minimising the sum of the squared residuals ε2V and ε2N , correspond to the solution of the following simultaneous equations (see addendum 2.3 on page 89): tf V DV 0
n
[ai hi (r, t) − Q(r, t) [hj (r, t)]dv dt = 0
i=o
0
for all j elementary solution hj (r, t)
tf S DN 0
0
n
[ai hi (r, t) − qn (r, t)
hj (r, t)
i=o
3
(2.59)
αk dS dt = 0
k=1
for all j elementary solution hj (r, t)
(2.60)
Both requirements can also be combined (occasionally using weighting coefficients). Then, using the convenient inner product notation, these conditions can be simply written as: < DV · ai hi (r, t) − Q(r, t), hj (r, t) >V,t − < DN · ai hi (r, t) − qn (r, t), hj (r, t) >S,t = 0
(2.61)
Example 2.5 clarifies this method.
Example 2.5 Parallel linear drains buried under an embankment must be designed to keep the rising water table, caused by the infiltration of 75% of a continuous and extremely exceptional 100 mm rain fall over a 4 period, below a predetermined maximum level of 0.5 m. This gives an average infiltration rate of 18.75 mm/day (see fig. 2.7).
© 2010 Taylor & Francis Group, London, UK
A p p r o x i m a t e s o l u t i o n s 69
The rising water table may be approximated by a quadratic polynomial in x (x-distance measured from the drains’ centreline) multiplied by a sine function in t (elapsed time since the start of the continuous rain):
H(x,t,T) = (a0 + a1 x + a2 x2 ) sin
πt 2T
+ HL
(2.62)
In this formula, HL is the vertical distance from the embankment’s impervious base to the drain level. Obviously, the units of the numerical coefficients must be compatible, i.e. a0 : [L], a1 : [−], a2 : [L−1 ], HL : [L] and x: [L]. Flow symmetry implies zero horizontal discharge at the drains’ centreline, i.e. qn = 0, at x = 0, corresponding to the following Neumann condition at x = 0:
−k
∂ (H + HL ) ∂x
=0
(2.63)
x=0
Substituting H for the selected approximation:
πt ∂ 2 =0 −k (a0 + a1 x + a2 x ) sin + HL ∂x 2T x=0
(2.64)
Deriving:
−k(a1 + 2a2 x) sin
1 t π 2 T
=0
(2.65)
x=0
From which a1 = 0. The draining capacity is designed to assure zero hydraulic pressure the drains’ centres. Then, the following Dirichlet condition at x = L is:
(a0 + a2 L2 ) sin
πt 2T
= HL
+ HL
(2.66)
x=L
From which a2 = −a0 /L2 . Taking into account the preceding values for a1 and a2 , the expression for the approximant is reduced to:
1 1 t H(x,t,T) = a0 1 − 2 x2 sin π + HL L 2 T
(2.67)
Now, the least squares method can be employed to find a0 . The unconfined Dupuit’s continuity equation in (x, t) and the Neumann condition at (L, t) for a 1 m wide strip are,
© 2010 Taylor & Francis Group, London, UK
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Fractured rock hydraulics
respectively: ∂ ∂ ∂ ρH −k (H + HL ) + (nρH) − ρω = 0 ∂x ∂x ∂t ∂ =0 −k [(H + HL )m] − qn ∂x x=L
(2.68)
To effectively drain, qn must balance all water accretion ωL, i.e. qn = ωL. Substituting the last expression of H(x, t,T) in these two conditions and evaluating:
(a0 )2 2 1 t 2 1 2 1 t 2 k 2 x sin k + 2ρ(a ) π 1 − sin x ... π 0 L4 2 T L2 2 T L2
1 2 1 t π 1 π − ρω = 0. + nρa0 1 − 2 x cos 2 L 2 T T
−4ρ
a0 m 1 t 2ρk 2 × sin π − ρωL = 0. L 2 T
(2.69)
(2.70)
Then, the squared residuals can be minimised within the flow domain and the drain boundaries. Making hj = 1 and combining these two requisites considering equal weighting coefficients:
(a0 )2 2 1 t 2 1 2 2 −4ρ x sin k + 2ρ(a ) π 1 − x 0 4 2 2 T T L L 2
L 1 2 1 k sin 1 π t dx dt . . . . . . + nρa0 1 − 2 x 2 2 T L 2 L
0 0 1 t π cos π − ρω 2 T T T
a0 1 t 2ρk 2 L sin π m − ρωL dt = 0 L 2 T
+
(2.71)
0
Evaluating symbolically these two integrals gives: 4 a0 m 2 ρLna0 − 2ρωLT + Tρk =0 3 π L
(2.72)
Solving for a0 : a0 = 3ωL2 T
π L2 nπ + 6T k m
(2.73)
Finally, substituting a0 in the last expression for H(x, t,T):
πt π 1 2 + HL H(x,t,T) = 3ωL T 2 1 − 2 x sin L nπ + 6T k m L 2T 2
© 2010 Taylor & Francis Group, London, UK
(2.74)
Approximate solutions
71
Results for different periods are shown in fig. 2.7. WT rise: 100 mm rain in 4 days Static WT before 4 days of 18.75 mm/day/m² infiltration WT rise after 1 day of rain infiltration WT rise after 2 days of rain infiltration WT rise after 3 days of rain infiltration WT rise after 4 days of rain infiltration Embankment top Linear parallel drain Linear parallel drain
WT rising heigth (m)
0.5 0.4 0.3 0.2 0.1 6
4
2
0
2
4
6
Distance to centre line (m)
Figure 2.7 Water table rising controlled by parallel linear buried drains.
2.5.4
Ga ler ki n’s m e t h o d
2.5.4.1 O rt ho g on a l i ty A “vector’’ of an n-dimensional vector space has n components. Similarly, a linear function h(x) can also be regarded as a “vector’’ of a ∞-dimensional function space, having an infinite number of components h(x) defined for x varying from a to b. This concept can be easily extended from the one-dimensional argument x to the fourdimensional argument (r, t). Hence, the solution H(r, t) of a linear partial differential equation and its approximant h(r, t) can be considered as “vectors’’ of “function spaces’’ in which a “plane’’ P can be defined by the “vector’’ h(r, t), for random changes of its numerical coefficients aj . If the “vector’’ H does not belong to this “plane’’ P, then there exists a unique “vector’’ h(r, t) on P, called the “projection’’ of H(r, t) on P, such that the “approximant error vector’’ ε is perpendicular to P and takes its lowest minimum, i.e. its infimum (see fig. 2.8). Then, h(r, t) is the best approximant to H(r, t) in the sense of the minimum least squares method (see addendum 2.4 on page 89). A similar reasoning can be applied for the transformed “error vectors’’ εV , εN and εD . However, the least squares method does not guarantee that the “error vectors’’ ε, εV , εN and εD are orthogonal to the same “plane’’ P nor that the “orthogonalisation’’ of one kind of error implies the “orthogonalisation’’ of the others. Additionally, the least squares method requires a continuous approximant up to the second derivative. These insufficiencies are resolved by Galerkin’s method.
2.5.4.2 Gal e rk i n’s a p p r oa c h
In Galerkin’s method, the coefficients aj of the best approximant aj hj (r, t) are given by a set of simultaneous equations expressing the orthogonality condition for ε, εV , εN and εD to a “plane’’ defined by an arbitrary set of linearly independent base
© 2010 Taylor & Francis Group, London, UK
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Fractured rock hydraulics
H h–H h
h+bjhj
bjhj
Figure 2.8 “Visualisation’’ of the orthogonality requirement to find the best approximant h(r, t) to a function H(r, t) in the sense of the minimum least squares method. Let the “plane’’ P be defined in the “function space’’ by the “vectors’’ h(r, t) or, cast as a polynomial approximant, by the “vectors’’ ah hj (r, t) as its numerical coefficients aj change arbitrarily. For a particular set of values aj the “vector’’ h(r, t) may match the “orthogonal projection’’ of H(r, t) on P, and in this moment, the “approximant error vector’’ ε = [h(r, t) − H(r, t)] takes its lowest minimum, i.e. its infimum value.
“vectors’’ φj (r, t): < ai hi (r, t) − H(r, t) , φj (r, t) >V,t − < ai hi (r, t) − H(r, t) , φj (r, t) >S,t = 0 (2.75) < DV · ai hi (r, t) − Q(r, t) , φj (r, t) >V,t − < DN · ai hi (r, t) − qn (r, t) , φj (r, t) >S,t = 0
(2.76)
The “plane’’ does not necessarily match the “plane’’ P defined by the approximant “vectors’’ aj hj (r, t) and its transforms by the operators DV and DN (see fig. 2.9). However, that “geometric congruence’’ may be forced if the set of “base vectors’’ {φj (r, t)} that define the “plane’’ is constructed by sequentially integrating the elementary solutions hj (r, t). In addition, the forcing sinks or sources Q(r, t) and the boundary inflows or outflows q(r, t)n can also be approximated by linear combinations of these solutions: DV · ai φj (r, t) − Q(r, t) ≈ DV · ai φj (r, t) − bk φk (r, t)
(2.77)
DN · ai φj (r, t) − qn (r, t) ≈ DN · ai φj (r, t) − ck φk (r, t)
(2.78)
In this case, it can be proved that if the orthogonality condition holds for the approximant error ε, then it also holds for the transformed errors εV , εN and εD . Example 2.6 clarifies Galerkin’s method.
© 2010 Taylor & Francis Group, London, UK
Approximate solutions
73
hΦ–H hP–H H
hΦ hP
Figure 2.9 “Visualisation’’ of the orthogonality requirement to find the best approximant h(r, t) to a function H(r, t) by Galerkin’s method. The “plane’’ does not necessarily coincide with the “plane’’ P defined by the approximant “vectors’’ h(r, t) and its transforms by the operators DV and DN .
Example 2.6 The variation of the hydraulic head inside a homogenous core of a rock-fill dam may be fairly described by a bilinear approximant referred to a non-zenithal frame (see fig. 2.10): h = a0 x + a 1 z + a z x
(2.79)
The narrow free surface practically matches the x-axis where the falling head can be approximated by −x · cos(i). Then, a simple orthogonality requirement for the Dirichlet condition may be applied along the free surface: < [(a0 x + a1 z + a2 xz) − (−x · cos(i))],1 >free surface = 0 Alternatively, written in full: x1 [a0 x + a1 z + a2 xz − x(−cos(i))]dx = 0
(2.80)
(2.81)
0
Integrating and taking into account that z = 0 at the free surface gives a0 = −cos(i). Applying a similar reasoning to the core upstream face where the hydraulic head is zero (equal to the head reference level): < [(a0 x + a1 z + a2 xz − 0],1 >core upstream face = 0 Alternatively, written in full: z3 (−cos(i)x + a1 z + a2 xz)dz = 0 0
© 2010 Taylor & Francis Group, London, UK
(2.82)
(2.83)
74
Fractured rock hydraulics
0 1 0
z(m )
2 0 3 0
0
4 0
1
5 0
0
6 0 0
2
0
Z zenith H0
z(
m)
x ( 10 m) 20 D Eq ata c u Eq ipo ore u t Eq ipo en Eq uipo tent tial 0 Eq uipo tenti ial 0 .2 H uip ten al .4 H m ote tia 0.6 nti l 0. H mm 8 al H Hm m
4
0
z
5 0
(0, 0)
6
0 0
(x1, 0)
nin (0, z2)
dS
(0, z3)
nout βx
i
(x2, z2)
3 0
x βz
H -H
10 x( Da m 20 Eq m co ) Eq uipo re Eq uipot tentia Eq uipo enti l 0. Eq uipo tenti al 0.42 H m uip ten al 0 H ote tia .6 m ntia l 0. H m lH 8H m m
Figure 2.10 Top left sketch: approximated head h contours inside the core. Bottom left sketch: adopted non-zenithal frame xz. The head zero reference level coincides with the water table. The total head difference at the downstream bottom of the core equals −H. The points (0, 0), (x 1 , 0), (0, z3 ) and (x 2 , z2 ) define the core geometry. The upstream core outward normal equals nin . The outward normal of the seepage face equals nout . The upstream core inclination equals i. The seepage inclination equals β. Right sketch: detail of the head h contours for this example.
Integrating and making x = 0 at the upstream core face gives a1 = 0. Then, the polynomial approximation reduces to: h = −cos(i) x + a2 x z
(2.84)
In this non-zenithal frame, the total head at any point (x, z) is (see Chapter 1, section 1.1.5.3 Hydraulic Gradient): h = ax x + αz z +
P ρg
αx = cos(Z,x) = cos(π − i) = −cos(i) π αz = cos(Z,z) = cos − i = sin(i) 2
© 2010 Taylor & Francis Group, London, UK
(2.85)
Approximate solutions
75
At the seepage face, the water seeps out into the free atmosphere. Then, the directional gradient ∂P/∂r must be parallel to the outward normal nout . Thus, the components of its direction cosines are βx = cos (nout , x) and βz = cos (nout , z): ∂ ∂ [(−cos(i)x + a·2 xz − α·x x − α·z z)ρg] [(−cos(i)x + a·2 xz − α·x x − α·z z)ρg] ∂x ∂z − =0 β ·x β ·z (2.86) Taking the derivatives and simplifying: (−cos(i) + a2 z − αx )
1 1 − (a2 x − αz ) = 0 βx βz
(2.87)
The head at the upstream seepage face is z · sin(i). Applying a mixed orthogonality requirement at that boundary: < [(a0 x + a1 z + a2 xz) − (z · sin(i))],1 >seepage face + . . . + < [(−cos(i) + a2 z + αx )/βx − (a2 x + αz )/βz ],1 >seepage face = 0
(2.88)
Alternatively, written in full:
S2 S1
[(−cos(i)x + a2 xz) − (z sin(i)))dS]
+
S2
S1
1 1 (−cos(i)x + a2 z − αx ) − (a2 x − αz ) dS = 0 βx βz
(2.89)
Integrating and solving for a2 (taking into account the geometrical relationships between units, levels, point coordinates, direction cosines of the upstream outward normal, face inclinations, etc.):
a2 = 3
sin(i)(z3 )2 + 2z3 cos(i)x1 βx + 2z2 cos(i) + 2z2 αx (βz )3 +(2x1 αz − 2x2 αz )(βx )2 2(z3 )3 (βz )4 + 3(z3 )2 x1 βx + 3(z22 ) (βz )3 + −3(x2 )2 + 3(x1 )2 ) (βx )2
(z3 )2 (βz )4 cos(i) +
(2.90) Substituting a2 in the approximate head gives a fairly analytic solution. The derivative ∂h/∂x of the analytic expression gives the exit gradient in the x direction (see fig. 2.11): sin(i)(z3 )2 + 2z3 cos(i)x1 βx + 2z2 cos(i) + 2z2 αx (βz )3 + (2x1 αz − 2x2 αz )(βx )2 h = −cos(i)x + 3 xz 2(z3 )3 (βz )4 + 3(z3 )2 x1 βx + 3(z22 ) (βz )3 + −3(x2 )2 + 3(x1 )2 ) (βx )2 (z3 )2 (βz )4 cos(i) +
(2.91)
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Fractured rock hydraulics
2.5
0
2 1.5 1 0.5
10
20
z(m)
z(m)
20
30
40
40
50 Hydraulic gradient (-) 60
Exit gradient 0
5
10 x(m)
15
20
Dam core Equipotential 0.2 H m Equipotential 0.4 H m Equipotential 0.6 H m Equipotential 0.8 H m Equipotential H m
Figure 2.11 Exit gradient in the x-direction and estimated head contours for (x 1 = 15 m, z1 = 0 m), (x 1 = 20 m, z1 = −45 m) and (x 1 = 0 m, z1 = −56.55 m).
2.5.4.3 “We a k s ol u ti on s’’ In the “finite element method’’, the flow domain is partioned in contiguous small subdomains. To each subdomain, appropriately called a “finite element’’, corresponds a local approximant of the solution of the governing equation, subjected to the finite element inherent boundary conditions. To guarantee convergence, as the size of the finite elements decrease, the inter-element continuity for their first and second order derivatives of the local approximants should be preserved. However, the full continuity constraint cannot be applied to approximants that are only continuous up to the first derivative. To avoid that difficulty, it is convenient to transform the differential form
© 2010 Taylor & Francis Group, London, UK
A p p r o x i m a t e s o l u t i o n s 77
of this constraint so as to eliminate the second order derivative from the first inner product. To do this, consider the full orthogonality constraint: < [DV · ai hi − Q], hj >V,t − < [DN · ai hi − qn ], hj >S,t = 0
(2.92)
Then, integrating by parts the terms DV · ai hi and DN · ai hi of the above requirement, the resulting transformed orthogonality condition can be applied to approximants that need to be continuous only up to the first derivative: < kk · ∂h/∂xk , grad · (hj ) >V,t − < SV · ∂h/∂t, hj >S,t + < Q, hj >V,t − < qn , hj >S,t = 0
(2.93)
These types of approximants are called “weak solutions’’ because they do not fulfil the partial differential equation itself but only its equivalent integral formulation. Moreover, the Dirichlet condition is “strongly’’ imposed by the above requirement but the Neumann condition is only approximately satisfied in the limit, in terms of an average. 2.5.4.4 Va ri a t ion a l n ota ti on If δh(r, t) denotes the total variation of an approximant h(r, t) due to arbitrary variations of its unknown coefficients ai , it follows that: δh = δa0 h0 + δa1 h1 + · · · + δan hn =
n
(δai hi )
(2.94)
i=0
On the other hand, the first orthogonality constraint, as shown below, remains unaffected if it is multiplied by an arbitrary variation δ-aj of any coefficient aj : δaj · < [ai hi − H], hj >=< [ai hi − H], δaj · hj >= 0 Adding up all these j constraints and recalling that δh =
(2.95) δaj · hj :
δaj · < [ai hi −H], hj >=< [ai hi −H], δaj ·hj >=< [ai hi −H], δh >= 0
(2.96)
This sort of integrated notation can be applied to all types of orthogonality requirements. For “weak solutions’’ it is written usually as follows: < kk · ∂h/∂xk , grad · (δh) >V,t − < SV · ∂h/∂t, δh >S,t + < Q, δh >V,t − < qn , δh >S,t = 0
(2.97)
“Variational notation’’ is employed for orthogonality requirements formulated in terms of maxima and minima of functionals (functions of functions), i.e. in terms of the principles of the calculus of variations, one of the oldest mathematical disciplines and developed almost simultaneous with differential and integral calculus. Example 2.7 clarifies the use of the variational notation.
© 2010 Taylor & Francis Group, London, UK
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Fractured rock hydraulics
Example 2.7 The hydraulic behaviour of an artesian horizontal leaking aquifer (see fig. 2.12) can be described by a one-dimensional steady flow continuity equation and two Dirichlet boundary conditions: −T
∂2 h + ch = 0 ∂x2
0<x
(2.98)
hx=0 = A hx=L = B
Head A Head h
Head B
Leakage ch
Semipervious (aquiclude)
Pervious (aquifer) Impervious (aquitard) Length L A
1
2
3
B
Figure 2.12 Unit width strip of an artesian horizontal leaking aquifer. The upward leakage at the interface aquifer-aquiclude is proportional to the head. Heads at A and B define two Dirichlet conditions.
Here, the meanings of the algebraic symbols are x: horizontal coordinate, h: hydraulic artesian head,T: aquifer transmissivity, c: leakage proportionality coefficient of the variable upward leakage ch. Pairs of coordinates xi and xj delimit subdomains δLi,j (appropriately called “finite elements’’). Their head variations can be described locally by “weak solutions’’, i.e. first order polynomial interpolations instead of twice-differentiable approximants:
x − xi x − xi H = Hi 1 − + Hj . xj − x i xj − x j
© 2010 Taylor & Francis Group, London, UK
(2.99)
A p p r o x i m a t e s o l u t i o n s 79
For each finite element, the independent coordinate x can be normalised by the subdomain length: X=
x − xi xj − x i
0≤X≤1
(2.100)
Then, the continuity equation takes a simple form: ∂2 h + Ci,j h = 0 ∂X2
(2.101)
Considering the following differentiation rule:
dX2 =
∂ X ∂X
2 =
∂ x − xi ∂x xj − xi
2 =
1 (xj − xi )2
(2.102)
The relationship between the old c and the new constant Ci,j is: Ci,j =
C (Xj − Xi )2 T
(2.103)
Similarly, the polynomial interpolation H,the variation δH and their gradients are written as: H = Hi (1 − X) + Hj X δH = δHi (1 − X) + δHj X (2.104)
∂ H = −Hi + Hj ∂X ∂ δH = −δHi + δHj ∂X Thus, applying the Galerkin method with a variational inner product notation: < ∂H/∂X,∂(δH)/∂X >L − < C,δH >V − < qk ,δH >k=i,j = 0
(2.105)
Developing in full: δHi (1 − X) (−Hi + Hj ) dX dx − Ci,j [Hi (1 − X) + Hj X] δHj X 0 0 [∂Hi (1 − X)]X=1 [∂Hi (1 − X)]X=0 − q.i =0 (2.106) − q.j (∂Hj x)X=1 (∂Hj x)X=1
1
−δHi δHj
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1
80
Fractured rock hydraulics
Integrating the above condition gives a system of two simultaneous equations for each subdomain i-j:
Ci , j Ci , j 1− − 1+ 3 6 −qi Hi − =0 (2.107)
qj Ci , j Hj Ci , j 1− − 1+ 6 3 However, as a first-degree approximant gives an average discharge q for each finite element, the terms −qi and qj cancel out. Then, applying that result to the four subdomains A-1, 1-2, 2-3 and 3-B, gives a system of three simultaneous equations for the unknowns H1 , H2 and H3 , after some matrix algebra manipulation. Instead of two Dirichlet boundary conditions, the same problem can be stated for one Dirichlet HA at A and one Neumann condition qB at B. Then, Darcy’s law could determine the missing Dirichlet condition HB : qB = −T
HB − H3 xB − x 3
(2.108)
This example illustrates the essence of the “finite element method’’.
2.5.5 Time-d e p e n d e n t s o l u t i o n s Approximate solutions also simulate unsteady groundwater flow. However, as time boundaries are usually open, with the exception of cyclic behaviour, an approximate functional relationship between time and head variation is not self evident, except in some circumstances. A common approach is to apply a first order linear approximant for very short time intervals and to establish a recursive solution. Example 2.8 illustrates this approach.
Example 2.8 The excess pore pressure dissipation within a nearly saturated and plastic clay core of a rock-fill dam can be reasonably modelled by the product of a quadratic space-dependent polynomial and a time-dependent polynomial (see fig. 2.13): (a + a x + a x2 ) (b + b t) 0 1 2 0 1 u = −L ≤ x ≤ L,t ≥ 0
(2.109)
The starting excess pore pressure u0 at any point of the centreline of the impervious core can be estimated by: u0 = ux=0, t=0 = Bγh
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(2.110)
Approximate solutions
C.L.
Excess pore pressure dissipation
h
Figure 2.13 Excess pore pressure dissipation within a nearly saturated plastic clay core of a rock-fill dam following vertical consolidation inducing horizontal drainage.
Where, B (b-bar) denotes an empirical coefficient (ranging from 0.25 to 0.75 for typical core materials), γ the fill saturated unit weight and h the soil height above the point being considered. The chosen approximant can be written as: u = A0 + A1 + x + A2 x2 + A3 t + A4 xt + A5 x2 t
(2.111)
Where the new coefficients Ai are related to the old coefficients aj and bk as: A0 = a0 b0 A 1 = a1 b 0 A 2 = a2 b 0 A 3 = a0 b 1
(2.112)
A 4 = a1 b 1 A 5 = a2 b 1 Known boundaries and initial conditions help determine four coefficients Ai : –
Initial conditions at x = 0 and t = 0: (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t)x=0, t=0 = u0
–
Dirichlet boundary condition at x = −L and t ≥ 0: (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t)(x=−L), t≥0 = 0
–
(2.113)
(2.114)
Dirichlet boundary condition at x = L and t ≥ 0: (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t)(x=−L), t≥0 = 0
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(2.115)
81
82
Fractured rock hydraulics
–
Neumann boundary condition at x = 0 (horizontal slope):
∂ (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t) ∂x
=0
(2.116)
x=0, t≥0
Solving the preceding equations, substituting the expressions for Ai (i = 0, 1, 2, 3, 4) in the new approximant and manipulating algebraically yields:
x2 t u= 1− 2 u0 − A 5 2 L L
(2.117)
Now, the Galerkin method may be employed to determine the remaining unknown coefficient A5 . If cr symbolises the experimental radial consolidation coefficient, then the continuity equation for vertical consolidation and horizontal drainage can be written as: 1 ∂2 u+ 2 ∂x cr
∂ u =0 ∂t
(2.118)
Substituting u for its approximation in the continuity equation gives: ∂2 ∂x2
1−
x2 L2
u0 − A5
t L2
+
1 cr
∂ ∂t
1−
x2 L2
u0 − A5
t L2
= 0 (2.119)
Deriving yields: 2 L2
u0 − A 5
t L2
+
1 cr
x2 L2
1−
A5 =0 L2
(2.120)
Applying a simple orthogonality constraint:
t.f 0
0
L
2 L2
u0 − A 5
t L2
+
1 cr
1−
x2 L2
A5 dx dt = 0 L2
(2.121)
Integrating, solving for A5 and substituting in the new approximant, gives the approximate closed solution:
x2 6 u= 1− 2 1− c t u0 (2.122) r L 3cr tf + 2L2 In this formula, t denotes time from 0 to tf . For a very short interval δt, the average excess pore pressure u can be estimated by making t = δt/2:
x2 1 u= 1− 2 u0 1− (2.123) 2 L 1 + 23 cLδt r
Assuming for each succeeding short time interval δt that the pore pressure at the centreline (i.e. at x = 0) can be taken as the previous average and applying the above solution
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Approximate solutions
recursively yields an expression for u after n intervals δt/2:
x2 1 u1 = 1 − 2 u0 1− 2 L 1 + 23 cLr δt 2
x2 1 u2 = 1 − 2 1− u0 2 L 1 + 23 cLδt r
(2.124)
..................................................... ..................................................... ..................................................... n
x2 1 1− u0 un = 1 − 2 2 L 1 + 23 cLδt r
Then, the excess pore pressure dissipation Un at time step n can be computed as: n un 1 u=1− = 1− (2.125) 2 u0 1 + 23 cLδt r
The graph below shows the excess pressure dissipation U at a horizontal level and at the corresponding centreline. Core width at that level is L = 10 m; the x coordinate is normalised as X = x/L; the radial coefficient of consolidation exemplified is cr = 10−3 cm/s (see fig. 2.14):
U (%)
Pore pressure dissipation 100 80 60 40 20 1
0.5
0 X ()
0.5
initial excess pore pressure
% dissipation after 180 days
% dissipation after 30 days
% dissipation after 365 days
1
% dissipation after 60 days
U (%)
Pore pressure dissipation at centreline 100 80 60 40 20 0
100
200 days
300
400
Figure 2.14 The pore pressure decay. The second-degree approximant in x gives a parabolic pore pressure graph.
© 2010 Taylor & Francis Group, London, UK
83
84
Fractured rock hydraulics
Another alternative to cope with unsteady problems is to combine a time-marching “finite difference’’ algorithm with a “finite element’’ spatial solution. Example 2.9 illustrates this common technique.
Example 2.9 The half-width of the dam core shown in example 2.8 can be partitioned into four finite elements 0-1, 1-2, 2-3 and 3-4. Using matrix notation, the excess pore pressure dissipation at each finite element can be approximated by a “weak polynomial interpolation’’ (see fig. 2.15): Un =
T
fj fj+1
Uj ,n Uj+1,n
(2.126)
C.L.
Excess pore pressure dissipation
h
j
j1
Figure 2.15 Excess pore pressure dissipation within the core of a rock-fill dam. Due to the centreline symmetry, the core half-width can be partitioned into four finite elements 0–1, 1–2, 2–3 and 3–4.
In this approximant, the symbols fj and fj+1 denote the interpolating functions at the end points j and j + 1 of each finite element:
fj = 1 − fj+1 =
x − xj xj+1 − xj
x − xj xj+1 − xj
(2.127)
The capital letters Uj,n and Uj,n denote the excess pore pressure dissipation at the points j and j + 1 at a certain moment of time defined by n · δt where δt is a short time interval and n the number of time steps elapsed since the start of the dissipation process (initial conditions).
© 2010 Taylor & Francis Group, London, UK
A p p r o x i m a t e s o l u t i o n s 85
The orthogonality requirement (equation 2.93) applied to the finite element (j, j + 1) gives: T Uj,n ∂ fj,n fj xj+1 ∂ dx xj ∂x ∂x fj+1 Uj+1,n fj+1 =
xj+1 1 ∂ xj cr ∂t
T
fj fj+1
+ qj+1,n
Uj,n fj dx . . . fj+1 Uj+1,n
fj fj+1
fj
+ qj,n
fj+1
x=xj+1
(2.128)
x=xj
As a first-degree approximant only gives an average discharge q for each finite element, the terms −qj,n and qj+1,n cancel out. Since the two nodal parameters Uj,n and Uj+1,n are independent of x, they can be taken out of the integrand. Then, considering the values of fj and fj+1 at x = xj+1 and x = xj and integrating, the above constraint yields:
M
Uj,n Uj+1,n
∂ +N ∂t
Uj,n Uj+1,n
=0
(2.129)
Where the time derivative multiplying N can be approximated by a “finite difference’’ quotient: ∂ 1 Uj,n Uj,n+1 − Uj,n = =0 (2.130) ∂t Uj+1,n δt Uj+1,n+1 − Uj+1,n The coefficients M and N are computed as:
xj+1
M = xj
T x−xj x−xj ∂ 1 − xj+1 −xj ∂ 1 − xj+1 −xj 2 dx = x−x x−x j j ∂x ∂x xj+1 − x xj+1 −xj
xj+1
N= xj
1 1 cr
x−x − xj+1 −xj j x−xj xj+1 −xj
xj+1 −xj
T
1
x−x − xj+1 −xj j x−xj xj+1 −xj
(2.131)
dx = 2 1 (xj+1 − xj ) 3 cr
Substituting the time derivative by its finite difference analogue and manipulating algebraically yields a time-marching algorithm:
M Uj,n+1 Uj,n = 1 − δt . (2.132) Uj+1,n+1 Uj+1,n N Then, applying this result to the four subdomains 0–1, 1–2, 2–3 and 3–4 gives a system of three simultaneous time-marching equations for the unknowns H0 , H1 , H2 , and H3 after some matrix algebra handling.
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Fractured rock hydraulics
If, instead of four finite elements, the time-marching algorithm is applied to only one large finite element, defined from the core centreline to the core edge, it is possible to compare the results to those obtained in example 2.8. Now, by a similar reasoning, the recurring formula is:
3 δt n U(n) : = 1 − 2 cr L 2
(2.133)
The graph below (see fig. 2.16) shows the excess pressure dissipation U at a horizontal level and at the corresponding centreline for L = 10 m; x = 0 m to 10 m; cr = 10−3 cm/s, the same parameters used in example 2.8 (see fig. 2.14):
Pore pressure dissipation
U (%)
100 80 60 40 20 10
5
0
5
10
x (m) initial excess pore pressure
% dissipation after 180 days
% dissipation after 30 days
% dissipation after 365 days
% dissipation after 60 days
Pore pressure dissipation at centreline
U (%)
100 80 60 40 20 0
100
200 Days
300
400
Figure 2.16 The pore pressure decay is almost the same in both examples. However, in this example, the first-degree finite element approximant gives a linear pore pressure graph instead of the parabolic one, as in example 2.8 (see fig. 2.14).
© 2010 Taylor & Francis Group, London, UK
A p p r o x i m a t e s o l u t i o n s 87
2.6 Addenda to Chapter 2 2.6.1 A d d en d u m 2. 1: C l a s s i f i c a t i o n o f s e c o nd o r de r lin ea r p a r t i a l d i f f e r e n t i a l e q u a t i o ns Consider that a dependent variable U is related to n independent coordinates (x1 , x2 … xn ) by a linear second order partial differential equation. By an appropriate coordinate transform (defined by the solution of an eigenvalue problem), this partial equation loses its mixed derivatives ∂2 U/∂xi ∂xj (i = j) and then can be written as: n
Ai
i=1
n
∂2 ∂ B U + U +CU+D=0 i ∂xi ∂xi2 i=1
(2.134)
Through any point (x1 , x2 … xn ) of the independent variable space pass one or two peculiar hyper-surfaces, called characteristics. For points resting on these hypersurfaces the second order derivatives of U can be mathematically defined but are indeterminate and finite (0/0 indetermination). For 3D (or 2D) systems, these hypersurfaces are associated with the form of a quadric defined by the coefficients Ai . These quadrics can have the shape of an ellipsoid (or an ellipse), a paraboloid (or a parabola) and a hyperboloid (or a hyperbola). This categorisation is generalised for an n-dimensional space. These surfaces propagate within the solution domain all information controlled by the boundary conditions. Then, at any moment of time, these characteristic surfaces separate into two distinct but continuously changing subspaces: the controlled and the controlling domains. Values of U on the controlled domain depend on the values taken by U on the controlling domain at that moment. Any change of U at one or more points on the controlling domain will affect the U values on the controlled domain. The class of a differential equation is revealed by the values and signals of the coefficients Ai of the quadric surfaces: – – –
Elliptic if all Ai are non-zero and of same signal. Hyperbolic if all Ai are non-zero and of the same signal but with one exception. Parabolic if one (or more) Ak is (are) zero but the corresponding Bk is (are) non-zero and the remaining Ai are non-zero and of the same signal.
As the coefficients Ai , Bi , C and D may be functions of the space coordinates xi and of the derivatives ∂U/∂xi but not of ∂2 U/∂xi , the classification of a linear second order partial differential equation may also change more than once within the definition domain, mainly in complex systems. For 3D groundwater flow systems, the dependent variable is the head H and the independent variables are the time-coordinate t and the space-coordinates (x, y, z). A time-independent solution can be described by an elliptic equation that has no real characteristics and no directional restrictions. In this case, its unique “controlling domain’’ merges with its unique “controlled domain’’. As a simple example, consider the steady flow induced by a penetrating well in a confined pervious layer. If this well pumps continuously but preserves a constant drawdown s at the well axis, its pumping rate Q is also constant. However, any change δs in the drawdown not only implies a change δQ in the pumping rate but also alters the heads H of all points of the confined
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88
Fractured rock hydraulics
pervious layer. On the other hand, if s never changes, any head disturbance δH at any point of the confined layer will affect all the other heads and also the pumping rate Q. For a transient flow regime, described by a parabolic equation, two identical families of real characteristics separate the “controlling domain’’ from the “controlled domain’’ by parallel planes. Consider the previous example in unsteady state. Then, at any moment tn , the “controlled domain’’ corresponds to points located above the characteristic plane (x, y, z, tn ), i.e. all future heads for t > tn . On the other hand, the “controlling domain’’ corresponds to points located below this plane, i.e. past heads for t < tn defining the history of the system, including its initial values and all precedent boundary values. Therefore, at any moment tn , past values “under’’ the characteristic plane (x, y, z, tn ) control all future values “above’’ this plane. Hyperbolic 3D equations have two real and distinct characteristics that define two crossing surfaces, separating both influencing and conditioned but confined domains. This type of equation does not occur in groundwater flow.
2.6.2 A d d en d u m 2. 2: M i n i m i s a t i o n o f t he s um o f t he squared residuals After electing the most promising approximant, the set of coefficients aj that minimises the sum of the squared residuals ε2v within the flow domain satisfies the following condition:
∂ ∂aj
tf
V
(εV )2 dV dt = 0
for all j
(2.135)
0
0
Or:
tf
0
V
εV
0
∂ εV dV dt = 0 ∂aj
for all j
(2.136)
Replacing, in the last expression, εV with [h(r, t) − H(r, t)], ∂εV / ∂aj with hj (r,t) and simplifying the resulting equation, one obtains:
tf
0
V
[h(r, t) − H(r, t)][hj (r, t)] dV dt = 0
0
for all j elementary solutions hj (r, t).
(2.137)
Writing h(r, t) as a complete polynomial approximant: 0
tf
0
V
n
[ai hi (r, t) − H(r, t) [hj (r, t)] dV dt = 0
i=0
for all j elementary solutions hj (r, t).
(2.138)
The solution of this system of (n + 1) equations defines the unknown coefficients aj .
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Approximate solutions
89
If applied to the flow domain boundaries, this method implies another set of (n + 1) simultaneous equations: 0
tf S
0
n
[ai hi (r, t) − H(r, t)
hj (r, t)
i=0
3
ak dS dt = 0
k=1
for all j elementary solutions hj (r, t)
(2.139)
In the last constraint, αk denote the direction cosines of the outward normal n to the boundary element dS. 2.6.3 A d d en d u m 2. 3: Mi n i m i s a t i o n o f t he s um o f t he s q u a r e d r e s i d u a l s t r a n s f o r m e d by t he di f f e r e nt i al oper a t o r s DV a n d DN Transforming the residuals εV within the flow domain by the differential operator DV and the residuals εS at its boundaries by differential operator DS gives: DV · [h(r, t) − H(r, t)] = DV · h(r, t) − Q DN · [h(r, t) − H(r, t)] = DN · h(r, t) − qn
(2.140)
Then the set of coefficients aj minimising the sum of the squared transformed residuals satisfies two sets of simultaneous equations: tf V n DV · [ai · hi (r, t) − Q(r, t) · [hj (r, t)]dV dt = 0 0
0
i=0
for all j elementary solutions hj (r,t) 0
tf S 0
DN ·
n
[ai · hi (r, t) − qn (r, t) · hj (r, t) ·
i=0
(2.141) 3
αk
dS dt = 0
k=1
for all j elementary solutions hj (r, t)
(2.142)
2.6.4 A d d en d u m 2. 4: T h e c o n c e p t o f “o r t ho g o nal i t y’’ Let the “plane’’ P be defined in the “function space’’ by the “vectors’’ h(r, t) or, cast as a polynomial approximant, by the “vectors’’ aj hj (r, t) as its numerical coefficients aj change arbitrarily (see fig. 2.17). Let (h + bj hj ) be any other “vector’’ on the “plane’’ P. If (h − H) is the infimum, then the “length’’ of the “error vector’’ (h − H) is smaller than the “length’’ of the vector [(h + bj hj ) − H] and the following inequality prevails: < [(h + bj hj ) − H], [(h + bj hj ) − H] > − < (h − H), (h − H) > ≥ 0
(2.143)
Taken into account that ||x||2 = <x, x > and simplifying the preceding inequality: b2j < hj , hj > −2bj < (h − −H), hj > ≥ 0
© 2010 Taylor & Francis Group, London, UK
(2.144)
90
Fractured rock hydraulics
h-H
H h
h + bjhj
bjhj
Figure 2.17 For a particular set of values aj the “vector’’ h(r, t) may match the “orthogonal projection’’ of H(r, t) on P, and in this moment, the “approximant error vector’’ ε = [h(r, t) − H(r, t)] takes its lowest minimum, i.e. its infimum value.
As that result must hold for arbitrary values of bj it necessarily implies a formal orthogonality constraint defined by a null inner product: < (h − H), hj > = 0
(2.145)
References Band, W., 1960, Introduction to Mathematical Phyisics, D. Van Nostrand, Princeton. Chung-Yau, L., 1994, Applied Numerical Methods for Partial Differential Equations, Prentice Hall. Linz, P., 1979, Theoretical Numerical Analysis, New York, Dover.
© 2010 Taylor & Francis Group, London, UK
Chapter 3
Data analysis
3.1 Preliminaries Modelling the spatial distribution of the hydraulic properties of a flow domain is not an easy task as it reflects the distribution of 3D geological features, commonly inferred from insufficient combinations of 2D field mapping and one dimensional logs. The non-conservative nature of geological features rule out simple conservation laws, as for mass and energy, which could guide the evaluation of their spatial distribution. Whenever possible, point inferences for the eigenvalues and eigenvectors of hydraulic conductivities may allow interpolations or correlation to descriptors of lithologic and structural features outside the sampling points. Given a dense spatial distribution of the hydraulic head, it may sometimes be possible to infer some characteristics of these tensor components. However, hydraulic heads are seldom monitored at many sampling points. Despite these difficulties, there are few techniques to approximate the spatial distribution of hydraulic attributes and properties within a flow domain. If the “modeller’’ bypasses this important step and substitutes the actual flow domain with a simple homogeneous and isotropic one, the resulting numerical model has little value despite its computational merits. This is particularly true regarding groundwater simulation for fractured rock hydraulics.
3.2 Analysing geological features For groundwater modelling, the traditional tools employed by geologists to picture and evaluate the chief parameters of structural features are generally adequate to correlate these attributes to the results of hydraulic tests. For example, traditional histograms that depict orientation and spacing distributions of several kinds of discontinuities may be correlated to observed head contours inferred from monitored data (see fig. 3.1 and fig. 3.2; Franciss, 1994).
3.3 Handling of hydraulic head data Hydraulic heads measurements are associated to specific times and observation points. Observation times must be chronologically recorded: year, month, day and hour of the events. Observation places must be referenced to their map coordinates, duly related to a known reference frame system. Therefore, despite the natural continuity of all implicit physical processes, head measurements H(r, t) may be considered as discrete dependent variables because they are only defined at specific locations and at some times.
© 2010 Taylor & Francis Group, London, UK
92
Fractured rock hydraulics
Statistical distribution of the orientation of the major structures 12
10
8
6
4
2
0
90
75
60
45
30
15
Megafractures
0 Faults
15
30
Dykes
45
60
75
90
Slikensides
Figure 3.1 Statistical distribution of the orientation of the major structures affecting the Brazilian Precambrian Shield resulting from field and desk studies for a planned underground LPG refrigerated cavern system (Almirante Barroso Maritime Terminal-TEBAR, São Sebastião, SP, Brazil). A vast gneissic complex is traversed by recent sub-vertical diabase dykes. Most rock masses conform to typical migmatites. The general foliation plunge is N 324◦ /27◦ (dip/dip direction). The depth of the weathered cover attains 30 to 40 m. Occasionally, deep altered zones, locally related to crossings of megafractures, faults and/or dykes, are found bellow the 100 m depth.
Relative frequence (%)
Inferred joint spacing statistical distribution 30
20
10
0
Joint spacing
0
1
2
3 Spacing (m)
4
5
6
Figure 3.2 Statistical distribution of the general joint spacing based on scan line counts, for the site in fig. 3.1. Modal value around 2 m. Two dominant joint trends, one parallel and other normal to the strike of the foliation, are N 27◦ /86◦ and N 136◦ /82◦ (dip/dip direction). Secondary joints traverse foliation at low angles.
Different interpolation and extrapolation techniques extend these data to other times and positions. Obviously, interpolated data within the observation time span and/or contained in the monitored area are more consistent than extrapolated data.
© 2010 Taylor & Francis Group, London, UK
Data analysis
0.01 1
0.10
Porosity (%) 1.00
10.00
93
100.00
Depth (m)
Porosity (%) = 8.9865[Depth (m)]-0.5559
10
100
Figure 3.3 Plot of porosity against depth. Samples extracted from a very thick soil-rock transition zone in a deep weathered quartzite 120 km2 plateau in a tropical climate. A significant amount of infiltrated rainfall is temporarily stored each year at the soil-rock interface and slowly transferred to the fractured aquifer below. Annual recharge averages, statistically inferred from four-year period, varies from 500 to 1000 litre/s.
3.3.1 V a r ia t i o n i n t i m e The rise and fall of the water table elevation in the saturated zone of a fractured rock aquifer mostly depends on the intensity and duration of the groundwater recharge and on the value of the effective specific porosity of the aquifer. Bare outcrops of fractured rocks in cold and temperate climates restrict water infiltration from direct precipitation runoff. However, groundwater recharge from long lasting snow and ice melts are important. In wet tropical climates, typified by deep chemical weathering, the “soil-rock’’ transition zone may be very thick and may acquire an open and significant secondary porosity. These horizons are able to retain enough infiltrated rainwater to give rise to perched aquifers that feed the fractured aquifer underneath. These perched aquifers may be fully depleted before the start of the next rainy season or may be recharged several times a year (see fig. 3.3). A rough estimate of the seasonal amount of the “rise and fall’’ of the water table in the saturated zone may be obtained by dividing the infiltrated rainfall height by the effective porosity (the infiltrated rainfall height, is only a small fraction of the total rainfall height, about 5 to 25%). As the effective porosity of fractured rocks is 10 to 100 times smaller than that of sedimentary rocks, water table fluctuations on fractured rocks are correspondingly greater than on sedimentary rocks (see fig. 3.4). Whenever possible, the variation of the water table elevation over time should be monitored by self-powered and fully automatic data loggers. If this is not possible,
© 2010 Taylor & Francis Group, London, UK
Fractured rock hydraulics
WT elevation (m)
94
800
750
3.45104
3.5104
3.55104 3.6104 3.65104 Day (excel serial date)
3.7104
3.75104
Observed at P6 Observed at P7 Observed at P8
WT elevation (m)
Figure 3.4 Water table rise and fall during seven years at three observation points in the saturated zone of a moderately weathered quartz-schist monocline ridge, defined by porosities lower than 1%. Yearly oscillations, only due to variations of seasonal rainfall, attained more than 10 m in two observation points. 745
740
735
3.48104 3.5104 3.52104 3.54104 3.56104 3.58104 3.6104 3.62104 3.64104 3.66104 3.68104 3.7104
Day (excel serial date) WT observed Fast Fourier transform cycles
Figure 3.5 Fast Fourier transform techniques applied to a time series of water table elevation in saturated schist not only strongly removed all irregularities and secondary cycles but also emphasised the main aspects of the seasonal cycles.
recorded levels measured at different dates may be interpolated to allow synchronous comparisons. Almost every commercial spreadsheet program incorporates friendly piecewise polynomial interpolation routines for time series analysis. Linear interpolation gives sharp corners but cubic spline interpolation at regular time intervals gives smooth curves continuous up to the second derivative. Moving average or exponential smoothing, also available in commercial spreadsheets programs, may concomitantly be used to reduce data irregularities. Fast Fourier transform algorithms, included in some mathematical software, can not only filter but also describe time series data in terms of “frequency domain’’ thus revealing all significant temporal cycles if they exist (see fig. 3.5). Future or past values, even missing gaps, of time series may be approximated by traditional interpolation or extrapolation algorithms. For almost periodic sequences, it may be possible to get plausible results using some type of forecasting technique; these
© 2010 Taylor & Francis Group, London, UK
WT elevation (m)
Data analysis
95
745
740
735 3.45104
3.5104
3.55104 3.6104 3.65104 Day (excel serial date)
3.7104
3.75104
Observed WT elevations Interpolated data
WT level (m)
Figure 3.6 Time series of continuously monitored water table levels. The missing gap was filled using Burger’s linear prediction method. In fact, predicted values only “mimic’’ past values, correctly shifted ahead in time. Nevertheless, this type of adjusted time series may allow some statistical analysis.
585 580 575 570 0
5
10
15
20
25
Observed values Mean value Mean value standard deviation Mean value standard deviation
30
35 40 Month
45
50
55
60
65
70
75
Figure 3.7 Time series of monthly water table elevations. It is possible to discern five complete repetitive cycles of 12 months each (a regular hydrologic year). Observed irregularities are due to several causes, including uneven rainfall intensity. For each month, it is possible to compute five-year statistics, as averages and standard deviations. At the end of every month, the predicted levels may be systematically re-evaluated.
techniques are included in some mathematical software (Press et al. 1994, Sophocles, 1992; see fig. 3.6). However, despite their mathematical cleverness, no prediction technique actually matches factual outcomes. If a sequence of predicted values is not required, it is sufficient to use traditional statistics to picture the probable range of future values (see fig. 3.7). Assessment of dynamic correlation between different time series may confirm their mutual interdependence or prove their cause and effect connection. Lag correlation or non-parametric correlation techniques, also available in some mathematical software, may give good results (see fig. 3.8).
© 2010 Taylor & Francis Group, London, UK
Fractured rock hydraulics
80
0.88 0.86
0.85954
0.84 0.82 0.8
0 20 40 60 80100
normalized WT elevation
Spearman correlation
96
2 Piezometer SR3 Piezometer NEB
1 0 1 2 3.4104
Time lag (days)
3.45104
3.5104
3.55104
3.6104
Excel serial date
Figure 3.8 Left graph:“Spearman’s correlation coefficient’’ x“time lag’’. Coefficients calculated for growing time lags to compare the time evolution pattern of two WT time series: piezometer NEB closed to deep pumping wells on fractured quartzite and piezometer SR3 far from the pumping wells, located at the top of a 350 m high plateau, more than 25 km apart. Right graph: Comparison of the time series of NEB and SR3, both normalised,“lagged’’ by 80 days at peak correlation. Normalisation is only useful for graphical display but unnecessary for correlation calculation and comparisons.
3.3.2 V a r ia t i o n i n s p a c e The spatial interpolation (and extrapolation) of observed quantities monitored at regular or erratic sampling locations may be achieved by different types of built-in interpolation algorithms in commercial software specifically designed to produce contour maps from rectangular arrays (“grids’’) of estimated values. Check, for example, the capabilities of Surfer – from Golden Software, Inc. – a traditional and well-proven product, or GW Contour – from Waterloo Hydrogeologic, Inc. – a less traditional product but one specifically designed for hydrogeologists. In a certain sense, almost all interpolation algorithms are weighted averages, i.e. finite interpolation polynomials where the interpolating functions fi (r) play the role of averaging weights. However, instead of being constructed as functions of the coordinates r of the estimation point, as for traditional piecewise approximants, these averaging weights are constructed as functions of the radial separation δrij (and sometimes of the azimuth αij ) between the estimation point i and a few points j around it, forming a cluster of influencing values (see fig. 3.9). The main difference between weighted averages algorithms resides in the way their weights are determined. Different weighting criteria yield comparable but unequal estimations for the same data and grid geometry. Usually, for each type of interpolation algorithm, the influence of the size, spatial distribution and inherent variability of the input data on the character of the resulting grid is properly considered in the “help section’’ of commercial software. Inverse distance is one of the simplest and intuitive weighted average methods. Each observed value j is weighted by the inverse of its separation from the estimation point i (frequently, distance squared). The closer the observation point to the estimation point, the greater its contribution to the estimated value.
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Data analysis
97
8000 Observation well j
Radial separation ij 6000 Y (m)
Estimation point i 4000
2000
2000
4000
6000
8000
10000
12000
14000
X (m)
Figure 3.9 Example of a cluster of five observations wells j around an arbitrary estimation point i.
Natural neighbour is a fine and robust weighted average suggested for sparse and irregular observation points. Optimal averaging weights are proportional to the “influence area’’ of each observation point j clustered around the estimation point i (see, for example, the fundamentals of the Thiessen method – source of the natural neighbour method – that are described on sections devoted to the subject of descriptive hydrology in common textbooks). Kriging is a more elaborated weighted average based on two basic assumptions. The first one assumes that all observations can be decomposed in two parts. One part is a deterministic value, a global average or a trend value calculated at the observation point. The other part is a “residual’’ value, defined as the difference between the observed value and the deterministic value. The second assumption presumes that it is possible to statistically infer the character of a “measure’’ of the dissimilarity of two “residuals’’ as a function of their radial separation δrij up to a certain limit (and sometimes, as a function of the radial separation δrij and the azimuth αij ). Then, based on an empirically constructed “dissimilarity measure’’ between closer observations, local averaging weights are optimised for every cluster of P points around an estimation point i (fig. 3.9). The corresponding “objective function’’ to be minimised for each cluster is a statistical measure of the local dissimilarity between predicted and observed values (error variance).
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Table 3.1 Input data set. Order
E(m)
N(m)
Z(m)
H(m)
1 2 3 4 5 6 … … …
741749.92 712850.46 708794.42 711900.74 745198.91 810899.43 … … …
7786118.54 7785151.50 7783985.81 7780999.76 7771800.78 7769598.54 … … …
670.00 568.00 568.00 640.00 660.00 502.00 … … …
615.00 530.00 530.00 604.00 648.00 502.00 … … …
159 160 161 162 163 164
706249.19 704899.72 655100.57 710000.14 703599.72 689750.07
7430951.04 7425898.83 7425700.20 7423701.23 7423150.53 7422082.14
580.00 575.00 750.00 600.00 610.00 650.00
577.00 564.00 742.00 565.00 587.00 604.00
Besides the set of observed values, additional information about the sample domain may be just ignored (simple kriging), treated as complementary random variables (cokriging) or incorporated as a paired spatial bias. Finally, it is important to note that kriging algorithms may be used to produce point estimations or to generate the averages of point estimations around cell centres, smoothing the corresponding map contours in the latter case. Excellent introductory texts on geostatistics and kriging are available, for example, Kosakowski and Bohling, 2005. Example 3.1 helps to grasp the fundamentals of almost all griding methods and illustrates the construction of a variant of an inverse distance squared algorithm.
Example 3.1 The stabilised values of the piezometric elevations Hweel at the roof of a half-confined groundwater body trapped in a sandstone formation were recorded during the perforation of 164 deep wells for oil prospecting (see table 3.1). Based on this data record it was possible to picture the most probable configuration of the piezometric surface over a vast area of nearly 70,000 km2 . An optimised inverse distance squared algorithm was chosen to interpolate (and extrapolate) the observed piezometric elevations H over a rectangular grid of 40 × 64 equidistant points, centering squares of 5000 m side. Based on that grid, a contour map of H
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Data analysis
equal values was graphed (see fig. 3.10). To get this result the following procedure was adopted:
7750000
800 m 780 m 760 m 740 m 720 m 700 m 680 m 660 m 640 m 620 m 600 m 580 m 560 m 540 m 520 m 500 m 480 m 460 m 440 m 420 m 400 m
7700000
N (m)
7650000
7600000
7550000
7500000
7450000
650000
700000
750000
800000
E (m)
Figure 3.10 Location of the observation points (X symbols) and head H contours based on the grid estimates.
Step A: Detrending The coefficients of a deterministic 3D planar trend Htrend through the cloud of the m observed heads (Ewell , Nwell , Hwell ) were found by the traditional least squares method. The resulting best-fit plane was: Htrend = −2.124223 · 10−4 · E + 2.257132 · 10−5 · N + 540.19
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(3.1)
99
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Having defined this plane, the head residuals δHwell at each well with respect to the trend plane Htrend were computed as the difference between the observed head Hwell and the calculated trend Htrend at the well location (see fig. 3.11): δHwell = (−2.124223 · 10−4 · Ewell + 2.257132 · 10−5 · Nwell + 540.19) − Hwell (3.2)
600
H (m)
800
400
E (k
)
N (km
m)
700
7700
7600 800
7500
Figure 3.11 Plot of the observed heads and of the planar trend.
Step B: Immediacy of observed wells The immediacy or closeness of each pair of wells s and t was measured by two quantities, their separation δrs,t and corresponding azimuth αs,t : 1
δrs,t = [(E·wells − E·wellt )2 + (N·wells − N·wellt )2 ] 2
N·wells − N·wellt αs,t = atan E·wells − E·wellt
(3.3)
Care must be exercise when computing αs,t to avoid indeterminate answers. Step C: Parameters optimisation The residual head δHi at each grid node i – defined by its coordinates Ei and Ni – was estimated by the inverse distance squares algorithm taking only into account the observed residual heads δHj at P nearby wells – defined by its coordinates Ej and Nj – around the estimation node i. This algorithm required four parameters to be optimised: a) b) c)
The quantity P of neighboring wells around estimation nodes. One constant scale factor A to adapt the value of the distance squared. Two constant parameters B and C respectively related to the eccentricity and the orientation of the domain anisotropy.
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Data analysis
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The resulting algorithm expression is: j ∈ cluster of P nearby wells around estimation point i
δHj =
1+A(δri, j )2
1 sin(αi, j +c)2 B2 + 1 cos(αi, j+C )2 B2
This expression can be concisely written as: δHj =
δHj
1
j ∈ cluster of P nearby wells around estimation point i 1+A(δri,j )2
1
(3.4)
1 sin(αi, j +C)2 B2 + 1 cos(αi, j+C )2 B2
(wi, j δHk )
(3.5)
j ∈ cluster of P nearby wells around estimation point i
The averaging weights wi, j of the above approximant are calculated by: 1
1 + A(δri, j )2 wi, j =
1 2 + 2 cos(αi, j + C) B 1
sin(αi, j +
C)2 B2
(3.6)
j ∈ cluster of P nearby wells around estimation point i
×
1
1 + A(δri, j )
1 2 + 2 cos(αi, j + C) B 1
sin(αi, j +
C)2 B2
It easy to understand that the averaging weights wi,j are between zero and one. In addition, their total sum is always unity. The values rapidly fall with separation (see fig. 3.12): 1
Weight
0.1 0.01
1103 1104
5104
0
1105
1.5105
2105
Separation between observation and estimation points (m)
Figure 3.12 Weight value declines as radial separation increases. Graph plotted after optimisation of the four parameters P, A, B and C.
To optimise the four parameters P, A,B and C,a tractable approach was used to minimise each parameter separately, going from simple to successively more complex expressions. This approach leads to some loss of mathematical rigour, but it is not too compromising.
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The objective function was the sum of all squared interpolation errors between observed and estimated residuals at all observation points precisely, i.e. at all the wells. As the first parameter to be optimised was A, the sum of squared errors A to be minimised was: A =
m
(estimated dhs − observed dHs )2
(3.7)
s=0
To estimate the residual heads at points t based on the observed heads at all points s, except at the target well, the following expression was used: m 1 (s = t) δH s 1 + A(δrs, t )2 s=0 δHt = (3.8) m 1 2 s=o 1 + A(δrs, t ) However, to enhance the influence of the neighbourhood of the estimation points, the squared errors sum A was modified as follows:
m average δr 2 2 A = (estimated dHs − observed dHs ) average drs s=0
(3.9)
Squared error sum
In this expression,“average δr’’ corresponds to the average of the separations of all pairs of wells s and t, as defined above, and “average δrs ’’ corresponds to the average of the separations of the well s and all the wells t. The graphical expression of the A value that minimises the squared error sum A is shown bellow (see fig. 3.13):
2.05106 2106 1.95106
A 158.094109
1.9106 1.85106 0
1107
2107 Parameter A
3107
4107
Figure 3.13 Variation of the squared errors sum A with the parameter A. The A value that minimises this sum was found by a non-linear optimisation method (conjugate gradient in this case).
The next parameters to be optimised were B and C. These parameters control the inherent anisotropy of the observed values concurrently due to the geometric pattern of the sampling points and to the geologic heterogeneity of the monitored domain. However, to estimate residual heads at points t based on the observed heads at all points s, except at
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Data analysis
103
the target well, a more complex expression was adopted, including not only the separation δrs, t but also the corresponding azimuth αs, t : m (s = t) s=0 1 + A(δrs,t )2
δHt =
m s=0 1 + A(δrs, t )2
δHs ·
1 1 sin(αi,j + C)2 B2 +
1 cos(αi, j + C)2 B2
1 1 sin(αi, j + C)2 B2 +
1 cos(αi, j + C)2 B2
(3.10)
To enhance the influence of the observations in the far field and eventually detect an existing anisotropic pattern, the squared errors sum BC was again modified as follows:
m average δrs 2 2 (estimated dHs − observed dHs ) (3.11) BC = average dr s=0
Squared error sum
Compared to the last expression for A it is important to note that in the above expression for BC the ratio [average δrs ]/[average δr] is inverted. The graphical expressions of the B and C values that simultaneously minimise the squared error sum BC are shown bellow (see fig. 3.14): 1.5106 1.45106 1.4106 1.3510
B 0.163
6
1.3106
0
2
4 Parameter B
6
8
Squared error sum
1.38106 1.36106
C 26.029 deg
1.34106 1.32106
0
100
200
300
Parameter C
Figure 3.14 Variation of the squared errors sum BC with the parameters B and C. The B and C values that concomitantly minimise this sum were found by a non-linear optimisation method (conjugate gradient in this case). Optimum values can also be found graphically after painstaking trials.
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Anisotropy ellipse 120
90
60
2
7.7106
30
150 1 180
0
0 7.6106
330
210 240
300
270
7.5106
Best fit ellipse Anisotropy factor
1 sin(α C) B 2
2
1 cos (α C)2 B2
6.5105 7105 7.5105
Figure 3.15 Best fit ellipse that reflects the inherent anisotropy of the observed values. It is not mathematically possible to separate what is due to the geometric pattern of the sampling points from what is due to the geologic heterogeneity of the monitored domain.
The polar graph of the optimised anisotropy ellipse is shown above (see fig. 3.15). However, this plot may give a false impression of the effects of the detected anisotropy. In fact, as the anisotropy factors multiply the distance squared in the denominator of the algorithm’s expression, their effects on the grid values are inverted. In other words, the grid-based contours “shrink in’’ and “stretch out’’ in the opposite directions of the major and minor axis of the best fit ellipsis, respectively. Besides the influence of the sample pattern, a qualified hydrogeologist may infer additional geological influences on the detected anisotropy. The last parameter to be optimised was the quantity P of neighbouring wells around estimation points. Then, applying the same objective function (by now defined by the optimised parameters A, B and C) for each cluster of points around the estimation points and successively increasing P from the nearest observation well to those less closer, it was possible to clearly see that eight (8) was the best number (see fig. 3.16).
Squared error sum
1.54106 1.52106
P values Best P value
1.5106 P8
1.48106 1.46106 1.44106 1.42106
0
5
10
15
20
25
P (-)
Figure 3.16 Variation of the squared errors sum P with the quantity P of neighbouring wells around estimation points. Eight (8) determines the first minimum.
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D a t a a n a l y s i s 105
After griding all residuals using the completely optimised algorithm, corresponding piezometric levels were found by adding to each inferred residual the related deterministic trend value. To reduce the “bull’s-eye’’ aspect inevitably associated with inverse distance algorithms, a smoothing filter may be applied to the grid nodes (i, j).
3.4 Handling of flow rate data Flow rate data from springs, watercourses, pumping wells etc. deserve a similar treatment to that applied to hydraulic head data. However, it is sometimes possible to draw practical conclusions using hydrology techniques when, for instance, using extreme value distributions. In fact, one of the most annoying accidents during gallery developments for underground mining is sudden water inrushes before cementation is done. Example 3.2 shows how to adapt hydrologic techniques to analyse flow rate data.
Example 3.2 Fig. 3.17 shows a typical but relatively small inrush observed a few weeks after the gallery head crossed an apparently sealed fracture in dolomite land.
Figure 3.17 Relatively small inrush observed a few days after completing the gallery excavation in dolomite land (Vazante underground mine, State of minas Gerais, Brazil).
The coordinates of the location of the significant inrushes, their flow rates and dynamic hydraulic heads, measured immediately after water irruption, were systematically registered during the excavations of the development galleries forVazante mine, State of Minas
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Number of observed events (-)
Gerais, Brazil, which is located in dolomite land. As these inflows were not laminar, the measured discharges were normalised by the square root of the hydraulic head: L3 /T/L1/2 or L2.5 /T (see fig. 3.18).
1103 100 10 1 1
10
1103
100 2.5
Normalised inrush (m
/hr)
Figure 3.18 Histogram for 266 normalised inrushes (m2.5 /hr) systematically registered from December 1982 to December 2002.
It was possible to fit these data to a Gumbel extreme values distribution defined by: G(A,B) = (1 − e−e
A·ln(Q)+B
)
(3.12)
Cumulative frequency ( % )
In the above equation A and B are the fitted parameters and Q corresponds to the normalized inrush. Minimization of squared error gave A = 0.316 and B = −0.114 (see fig. 3.19). 100
50 Gumbel distribution Cumulative frequency of observed events
0 1103
0.01
0.1
1
10
100
1103
1104
Normalized inrush (m2.5/s)
Figure 3.19 Fitted Gumbel distribution of extreme values to the observed events.
Consider a gallery segment of length B as a baseline. If P is the probability of an occurrence of a normalised inrush along B less than Q, the probability of an occurrence of an inrush exceeding this value is 1−P. Now consider the length L of the whole gallery defined by the product kB. Then the probability of an occurrence along L of a specific flow less than Q is (1−P)kB . The probability of an occurrence greater than Q is [1−(1−P)kB ]. In the case of extreme flows, this complementary probability measures the expected risk (see fig. 3.20).
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Cumulative risc of one inrush (%)
Data analysis
107
100
< 0.1 m^2.5/hr < 0.3 m^2.5/hr < 1 m^2.5/hr < 3 m^2.5/hr < 10 m^2.5/hr < 30 m^2.5/hr
50
0 10
1103
100
1104
Gallery length (m)
Figure 3.20 Expected risk of at least one inrush. Six curves defined from 0.1 m2.5 /hr to 30 m2.5 /hr during the excavation of more 10000 m of galleries.
This type of analysis can also be made considering different lithologies and depths.
3.5 Handling of hydraulic conductivity data 3.5.1
Pr elim i n a r i e s
As previously discussed, the size of a pseudo-continuous subsystem depends on the observation scale. At the selected scale, a preliminary estimate of the eigenvalues and eigenvectors of the hydraulic conductivity of a subsystem may be inferred from its lithologic and structural geologic features. In some cases, previous experience from similar places may also allow first assessments. However, to get useful results from numerical models, these parameters must be adjusted by solving inverse problems, applied either to large-scale field tests or to comprehensive monitoring data. It is important to always keep in mind that the hydraulic conductivity of a rock mass subsystem is, in fact, a scale dependent property to be applied to a fictitious pseudocontinuous subsystem. Its eigenvalues and eigenvectors range over many orders of magnitude from place to place. Furthermore, they reflect the intrinsic assumptions concerning anisotropy and boundary conditions of the analytical model employed in their determination. In other words, different models applied to the same tested volume may yield different, and sometimes quite dissimilar, values. Moreover, as these closed solutions are normally applied to ideal homogeneous and anisotropic domains, never encountered in nature, their results lose meaning as the heterogeneity of the test space increases. In view of these observations, the modeller must exercise judgment, considering costs and time limits, to appropriately decide what type of approach best fits his needs. For an example of how the choice of approach affects the result, see table 3.2 concerning the results of a seven days test involving two very deep thermal wells, 540 m apart, performed for the Brazilian Department of Mineral Production in 2000: well P (E 422.547 m, N 6.966.167 m, Z 436.8 m, 700 m of total depth, 160 m of bottomfilter length, artesian discharge of 320 m3 /h, external diameter 121/4 to 8 3/8 ) and well T (E 421.957 m, N 6. 966.273 m, Z 423.26 m, 680 m of total depth, 140 m of
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Fractured rock hydraulics
Table 3.2 Example of how the choice of the analytical solution affects computed test results. Analytical solution
Confined sandstones Confined sandstones Basalt cap average permeability (m/s) transmissivity (m2 /s) storativity (-)
Copper-Jacob: steady-state solution 1.018·10−3 Hantush: transient solution for 3.616·10−4 leaking aquifers
1.282·10−4
1.485·10−7
bottom-filter length, artesian discharge of 268.95 m3 /h, external diameter 121/4 to 8 3/8 ). Both wells traverse a 540 m basalt cap and penetrate the confined sandstone (Mercosur aquifer, structured by Triassic-Jurassic sandstones confined by Cretaceous basalt flows, spreading over 1,194,000 km2 and including the Paraná Bassin and part of the Chaco-Paraná Bassin).
3.6 Hydraulic transmissivity and connectivity 3.6.1
P r elim i n a r i e s
The effective hydraulic transmissivity of discontinuities and their hydraulic connectivity at a chosen observation scale deserve attention when modelling a pseudo-continuous network for a fractured rock mass. 3.6.2
H yd r au l i c c o n d u c t i vi t y a p p r a i s al
The hydraulic transmissivity of a fracture may be partially blocked by secondary mineralisation, chemical weathering or other kinds of infillings. A crude appraisal of the areal extent of these obstructions may result from detailed fieldwork on bare outcrops or on excavated slopes. Usually, it is assumed that statistics derived from a planar support, as a planar discontinuity, may approximately match those resulting from several random linear supports on that same planar support. Then, the relative areal extent κ of the fracture blockage may be estimated from exposed fracture traces on outcrops by the division of the sum of the lengths of clogged segments δlc by the sum of the lengths of all traces δl, i.e.: κ = δlc / δl. This implies that for each group of open fractures observed on undisturbed borehole core samples the apparent average spacing ea roughly results from the product of the true average spacing e by the relative areal extent of κ obstructions, i.e. ea = κ·e. Therefore, for partially blocked fractures, i.e. for 0 < κ < 1, their average transmissivity T may be approximated by the Maxwell (1 − κ) κ) mixture rule: T = T0 · Tκ , where T0 and Tκ are, respectively, the non obstructed and obstructed fracture transmissivity. From these considerations, it follows that it is very difficult to directly appraise the true hydraulic transmissivity of fractures at a chosen observation scale. To complicate the problem, one must remember that in most cases seepage along partially opened fractures may be partially concentrated on anastomosed preferential small conduits within the fracture plane.
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D a t a a n a l y s i s 109
Figure 3.21 Oriented plug for permeability tests extracted from a large diameter whole-core sample.
Modelling strategies to evaluate the hydraulic conductivity of intact rock or rock masses depend on the desired observation scale as exemplified in the following sections. 3.6.2.1 H y d ra u l i c tes ts a t “c or e s a m p l e’’ s c a le The production and the storage of the oil reserves on fractured and porous sandstone reservoirs depend on the amount of connections between several groups of small discontinuities a few centimetres apart one another. Indeed, the commercial extraction of the hydrocarbons trapped in the interstitial pores of the porous matrix may be held back if these small fractures are almost clogged by secondary mineralisation. However, if these small fractures remain pervious and interconnected, extraction may be very efficient and effective. To have an idea of how much such fractures can improve exploitation, the average transmissivity of the relevant fracture groups may be measured by permeability tests made under different confined pressures on small samples (plugs) extracted from large diameter cores from prospecting wells (see fig. 3.21). If these fractures are partially blocked, the nature and areal extent of their sealing may be judged by visual inspection. The matrix permeability and corresponding effective porosity may be concomitantly measured in the laboratory. If the rock matrix is anisotropic, the evaluation of its hydraulic conductivity tensor requires many measurements of the oriented permeability Kd (n) on many “plugs’’ from the same whole core along different orientations. In fact, Kd (n) along a given direction defined by its unit vector n is related to the tensor |K| by the relationship Kd (n) = 1/[nT ·|K|−1 ·n]. Measuring more than six directional conductivities in many directions allows the determination of the components of |K| by inverse methods (Hsieh and Neuman, 1985). 3.6.2.2 H y drau l i c tes ts a t “b or eh ol e i n teg r a l c o r e’’ s c a le Integral sampling methods have been described in several papers (Rocha, M. and Barroso, M., 1971). The method consists of drilling a rock core already stiffened by a
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Fractured rock hydraulics
coaxially oriented rigid rod previously cemented in a smaller inner hole to ensure the integrity of the sampled material. Example 3.3 shows how the anisotropic character of the permeability of rock masses can be approximately determined from water pressure tests and information about the structural attitude and the nature of their open and connected discontinuities as revealed by integral samples (Rocha, M. and Franciss, F. O., 1977).
Example 3.3 The foundation of the AguaVermelha Dam, in the State of São Paulo, Brazil, was investigated with help of 13 HW integral samples and 128 water pressure tests. The average eigenvalues and anisotropy of the permeabilities estimated are summarised below and the corresponding eigenvectors are depicted in fig. 3.22.
Compact basalt
Eigenvalue m/s
K max /K min
group CB1 2.95E-05 16 eigenvalues 2.81E-05 1.81E-06 group CB2 8.51E-07 7 eigenvalues 7.76E-07 1.28E-07 group CB3 4.78E-07 6 eigenvalues 4.26E-07 7.41E-08 group CB4 3.16E-07 27 eigenvalues 3.01E-07 1.19E-08 group CB5 6.30E-06 675 eigenvalues 6.19E-06 9.33E-09 group CB6 Impervious
Vesicular amygdaloidal Eigenvalue basalt m/s group VA1 eigenvalues group VA2 eigenvalues group VA3 eigenvalues group VA4 eigenvalues group VA5
3.89E-06 6 3.31E-06 6.03E-07 6.60E-06 55 5.62E-06 1.20E-07 1.69E-06 1070 1.69E-06 1.58E-09 4.78E-07 24 4.57E-07 1.97E-08 Impervious
N 30
30
60
E
KIII
120
120 KI
N 30
60
KIII
W
KIII E
120
120
W
KII
120
150
150
E 120
KI 150
30
60
60 KII
K max /K min
group BB1 3.31E-06 12 eigenvalues 3.16E-06 2.69E-07 group BB2 1.00E-07 2 eigenvalues 5.25E-07 5.25E-08 group BB3 2.13E-07 18 eigenvalues 2.04E-07 1.20E-08 group BB4 Impervious
30
60
60 KII
150
Eigenvalue m/s
N
30
W
Basaltic K max /K min breccia
150
KI
150
S
S
S
(a)
(b)
(c)
Figure 3.22 Spread of the eigenvectors of the compact basalt group (a), vesicular-amygdaloidal basalt group (b) and basaltic breccias group (c).
For each test length L, the hydraulic conductivity tensor is evaluated by taking into consideration all open and supposedly connected fractures inside and around the volume L3 .
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To tentatively minimise the sampling bias (Terzaghi, 1965), it is strongly recommended to include all fractures occurring above the top and below the bottom of the integral core that satisfy the condition (L + δ L)·cos(θ) < L/2, where θ is the fracture dip (see fig. 3.23). Fracture included in the sample
N 30
θ
30
60
L
60
δL
6
KIII 0.18 10
m/s
W
KI 0.16 10
L
4
m/s
E
L 120
120
150
KII 0.13 104 m/s
150
S
L
(2) Fracture not included in the sample
– (1)
(3)
0.1 – 0.5 mm 0.5 – 1.0 mm 1.0 – 5.0 mm
Figure 3.23 Left side: criteria for including fractures above and bellow the sampling volume. Right side: Schmidt diagram of fracture poles from an integral core as well as the associated hydraulic conductivity tensor.
To evaluate the hydraulic conductivity tensor for each test interval L, associated with i fractures (i = 1, 2, 3. . . N) within the sampling volume L3 , one must proceed as follows: Measure the average dip θi , dip orientation λi and the hydraulic aperture ei of each fracture i. b) For each fracture i, roughly evaluate its transmissivity Ti (derived from its hydraulic aperture ei and from the character of existing infillings) and the corresponding equivalent hydraulic conductivity ki (≈Ti /L). As eigenvalues are tentatively calibrated later, based on the water pressure tests results, reliable relative estimates are much more important than absolute values. c) Compute the components of the hydraulic conductivity: sin(λi )2 + cos(θi )2 cos(λi )2 −sin(θi )2 cos(λi ) sin(λi ) −sin(θi ) cos(θi ) cos(λi ) 2 2 2 2 ·|Ki | = ki −sin(θi ) cos(λi ) sin(λi ) sin(θi ) cos(λi ) + cos(θi ) −sin(θi ) cos(θi ) sin(λi ) a)
−sin(θi ) cos(θi ) cos(λi )
−sin(θi ) cos(θi )sin(λi )2
sin(θi )2 (3.13)
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d)
Compute the resulting tensor |K| by the summation rule: (k11,1 + k21,1 + · · · + k11,1 + · · · + kN1,1 ) (k11,2 + k21,2 + · · · ) (k11,3 + · · · ) |K| = (k12,1 + k22,1 + · · · ) (k12,2 + k22,2 + · · · ) (k12,3 + · · · ) (k13,1 + · · · ) (k13,2 + · · · ) (k13,3 + · · · ) (3.14)
e) f)
Compute the eigenvalues Kj and the corresponding eigenvectors αzj of the resulting second order tensor |K|. Determine the homogeneous transformation factor f to convert pressure tests results from an anisotropic to an isotropic condition (a homogeneous transformation guarantees volume invariance for elementary subsystems thus preserving source strengths in water pressure tests): 13 13 3 3 2 ! (αzj ) f = Kj (3.15) Kj j=1 j=1
g)
As the borehole radius r almost equals the geometric mean of the transformed borehole cross-section, compute the average permeability Kmean for each water pressure test, H being the test pressure:
Q fL k·mean = ln 1.6 (3.16) 2πfLH r
h)
For each test, multiply the previously estimated eigenvalues Kj by a calibrating factor: ρ=
kmean 13 3 " Kj
(3.17)
j=1
These correcting factors ρ may depart more than one order from unity.Thus, to get reasonable results, the first estimate of the transmissivity of each fracture observed in an integral sample in step b) must be made carefully but in relative terms, i.e. in comparison to the transmissivity of the most prominent observed fracture in the sample.
3.6.2.3 H y dra u l i c tes ts at “c l u s ter of b or e h o le s’’ s c a le Traditional pumping tests may be adapted to measure the anisotropic character of pervious media (see, for example, Hantush, 1956). For a confined volume of a fractured rock mass, taken as quasi-homogeneous but anisotropic, there is an elegant inverse solution to evaluate its equivalent hydraulic conductivity and specific storage (Hsieh and Neuman, 1985). Example 3.4, abridged from Appendix 5B, “Using a multiple-borehole test to determine the hydraulic conductivity tensor of a rock mass’’, in Rock Fractures and Fluid
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Flow: Contemporary Understanding and Applications, National Academic Press, Washington, 1996, exemplifies this methodology (Hsieh and Neuman, 1985).
Example 3.4 The hydraulic conductivity of a sample volume of Oracle granite, north of Tucson, Arizona, USA, was estimated with help of multiple pumping tests in three boreholes H-2, H-3 and H-6, drilled in a triangular pattern and separated by distances between 7 to 11 m (see fig. 3.24). 0
N
5
10
Scale (m)
H-6
H-3
H-2
Figure 3.24 Location of the boreholes drilled to perform the 3D tests.
To minimise the influence of the boundary conditions, as a rule located far from the test domain, only short transient pumping (or injection) tests are performed. In each test, water is pumped from a packer-bounded interval of one borehole and corresponding head drops are measured on other packer-bounded intervals of the remaining boreholes (see fig. 3.25). Six or more six short transient tests give sufficient observation data to write a set of mathematical equations relating to the sought hydraulic parameters. Between any pumping point i and any observation point j prevails a relationship involving the distance Rij between these two points, the time t, the pumping rate Q, the head drop hij , the directional conductivity Kij , the determinant of the hydraulic conductivity tensor |K| and specific storage Ss :
·hij =
Q(Kij )
1 2 1
erfc
4πRij (|K|) 2
2
(Rij ) Ss 4Kij t
12
(3.18)
Solving this equation system (usually by a weighted least squares method) gives the six components of the second-order symmetric and positive-definite tensor that best describes the anisotropic conductivity K of the tested volume and the specific storage Ss . If the tested rock mass volume can be taken as a statistically homogeneous but anisotropic pervious medium, a three-dimensional plot of the square root of the directional hydraulic diffusivity versus direction should depict an ellipsoid, as shown in fig. 3.26. The eigenvalues and corresponding eigenvectors of the best-fit K are shown fig. 3.27.
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N
0
5
Sca
H-6 H-3
le (m
10
)
H-2
Observation interval
Point O
Point C
Pumping interval
Observation interval Point B
Figure 3.25 Spatial relationship of one of the several short transient pumping tests performed. Water is pumped from the packer-bounded interval O of borehole H-6 and corresponding head drops are measured in the packer-bounded intervals B and C of boreholes H-2 and H-3. N
E
Figure 3.26 Best-fitted ellipsoid of the square root of the directional hydraulic diffusivities estimated from transient pumping tests results.
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Eigenvectors (lower hemisphere projection) 90 120
60
150
30
180
0
330
210
300
240 270 Latitude 60 Latitude 30 Equator
K max = 1.6E-07 m/s; bearing 75 N; plunge 39 K int = 6.9E-08 m/s; bearing 247 N; plunge 51 K min = 2.2E-08 m/s; bearing 342 N; plunge 4
Figure 3.27 Estimated eigenvalues and eigenvectors of the anisotropic average hydraulic conductivity of Oracle granite rock masses. Most probable specific storage is 5.1E-06 m−1 .
3.6.2.4 H y d ra u l i c tes ts a t “a q u i fer’’ s c a l e Having in mind the obvious limitations of extending its applicability to other scales and flow regimes, the methodology suggested by Hsieh and Neuman may be adapted to measure the average hydraulic conductivity of delimited volumes within larger and almost homogeneous units of a flow domain. These results may be extrapolated to the whole domain by appropriate geostatistical models. One of the main advantages of this methodology is that it takes into account, implicitly and collectively, all complex and 3D-varying fractures characteristics that are hard to identify and parameterise by field work and core inspection, such as true shape, persistence, hydraulic transmissivity, interconnectivity, etc. Example 3.5 exemplifies one variant of this methodology.
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Example 3.5 A 60◦ dip tabular ore body is traversed by two major structures (see fig. 3.28). The direction and dip of neighbouring discontinuities of the host rock mass are closely related to the attitudes of the ore body and these two major structures. Lightly inclined original WT
Tabular 60 dip ore body 600 400
3000 2000 Inclined impervious boundary
1000
Two subparallel and subvertical major faults
0 1000
2000
0
200 0
2000
Figure 3.28 Perspective of the tabular ore body and the two major structures. The impervious bottom together with the slightly inclined water table defines the main boundaries of the natural flow domain.
The footwall of the tabular ore body and the two major structures are highly permeable. Their intercrossings behave as very efficient line drains able to induce the local gravitational lowering of the originally lightly inclined water table (see fig. 3.29). Lowered WT by gravitational drainage
600 400
3000 Two subparallel and subvertical drainage axis
2000 Dipping impervious boundary
1000
0
0 1000
200
2000
0
2000
Figure 3.29 Contour map of the lowered water table based on the readings of 16 piezometers.
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Assuming an infinite and anisotropic pervious medium, Hsieh and Neuman gave a closed steady-state solution to determine the head drop at an arbitrary point k induced by a line source (or line drain) of length 2Li centered at the origin i of a Cartesian coordinate system (see fig. 3.30). Z ∆Ri,k vector components
Li vector components
∆Xi, k
Li
X
Li
LiY
∆Ri,k
∆Yi, k ∆Zi, k
LiZ Y
∆Ri,k : distance between points i and k
X
Line source (or line drain) of length 2Li centred at origin i of coordinate system
Figure 3.30 Coordinate system referred to a line drain of length 2Li .
For more than one line drain Li placed at i different locations (i = 1, 2, 3. . .), the resultant head drop Hk at an arbitrary point k may be calculated by: 12 RTi,k A Ri,k RTi,k A Li RTi,k A Li +2 T +1 + T + 1 LiT A Li Li A Li Li A Li Qi ln 1 2 T T T Ri,k A Li Ri,k A Li Ri,k A Ri,k −2 T +1 + T −1 LiT A Li Li A Li Li A Li ·Hk = 1 8π(LiT A Li ) 2 i (3.19) The other variables in this formula have the following meanings: – – – –
Ri,k : distance between the i line drain center and an arbitrary point k described by its Cartesian components (Xi,k , Yi,k , Zi,k ). Qi : the total drainage rate percolating to the line drain i. A: the adjoint matrix A corresponding to |K| K−1 , where |K| is the determinant of the average hydraulic conductivity tensor K. Li : the Cartesian components of the half length Li of the i line drain.
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For the situation depicted, the existing field data, 16 piezometers readings and two almost steady-state discharge measurements at the bottom of the two line drains (0.99 and 0.48 m3 /s), provided enough data to fit an approximation to the average hydraulic conductivity tensor K of the surrounding rock mass. As the closed solution was deduced for an infinite domain, the image method was applied. However, only the two most influencing image drains were added to the problem: one for the WT boundary and one for the impervious bottom boundary. The quadratic error sum concerning the difference between the calculated and the measured head drops was minimised by the Levenberg-Marquadt method. However, to start the search algorithm at the neighbourhood of the lowest minimum of the quadratic error sum 7-D space, the first approximation of K was based on the geometry and the known hydraulic characteristics of the tabular ore body and the major structures. The resulting best-fit average K was: 2.855 · 10−5 −1.33 · 10−5 −2.97 · 10−6 m −5 (3.20) K= 1.28 · 10−5 2.44 · 10−6 · s −1.33 · 10 −2.97 · 10−6 2.44 · 10−6 1.442 · 10−6 The corresponding eigenvalues and eigenvectors are depicted in fig. 3.31. Eigenvectors (lower hemispher projection) 90 120
60
150
30
180
0
330
210
300
240 270
Latitude 60 Latitude 30 Equador K max = 3.655E-5 m/s; bearing 120 N; plunge 6.2 K int = 5.343E-6 m/s; bearing 211 N; plunge 9.5 K min = 8.973E-7 m/s; bearing 357 N; plunge 78.5
Figure 3.31 Best-fit average K. Besides a pronounced change in the first approximation eigenvectors, the optimised minimum quadratic error sum was 71 times lower than its value for the starting tensor eigenvalues.
Recycling the image method based on the best-fit solution did not improve appreciably its value.
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10 Piezometers Dynamic correlation: 0.85 a 0.90 Dynamic correlation: 0.90 a 0.95 Dynamic correlation: 0.95 a 1.00
Y (km)
8
6
4
2 2
4
6
8
10
12
14
16
X (km)
Figure 3.32 Hydraulic interconnection between piezometers as suggested by their mutually dynamic correlation coefficient.
3.6.3
H yd r a ul i c c o n n e c t i vi t y a p p r a i s al
Poor hydraulic connectivity between apparently persistent discontinuities may also hamper groundwater flow. Indeed, a model exclusively based on the geometry of the discontinuities without taking into consideration the extent of their hydraulic connectivity may overestimate the hydraulic behaviour of the rock mass. Normally, intercrossed major discontinuities (topographic trace lengths of more than a few kilometres) are mutually connected and are not blocked by much smaller ones. Modelling strategies to appraise the hydraulic connectivity of discontinuities are exemplified below. 3.6.3.1 D y na m i c c or r el a ti on s of WT ti m e s e r ie s Preferential corridors of groundwater movement normally induce similar and quasisimultaneous response of water table measurements in piezometers located on their flow paths. Therefore, for any pair of piezometers that behaviour may be approximately judged by the dynamic correlation of the fluctuation of their water table time series. As comparisons of time ordered data are essential to get realistic results, one must use non-parametric correlation measures such as, Spearmann’s or Kendall’s’ τ tests. High dynamic correlation values may imply high hydraulic connectivity. These tests must be performed, one by one, for all pairs of piezometers under analysis. Results from testing the dynamic correlation of 172 × 172 pairs of piezometers more or less erratically scattered over an area of 200 km2 are mapped in fig. 3.32. 3.6.3.2
Fi l t e ri n g WT c on tou r m a p s
Water table contour grids may be filtered by different types of algorithms (low-pass, high-pass, contrast amplification, etc.) to enhance some of their peculiarities. A simple
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6000 N (m)
abs
hi1,j hi1,j hj1 hi,j1 4hi,j hi,j
4000
4000
6000
8000
10000
12000
14000
16000
18000
E (m)
Figure 3.33 A normalised Laplacian operator (as tagged on this figure) applied to the hydraulic head contour map (100 m × 100 m grid) of fig. 3.32 may enhance the anisotropic trends of fracture hydraulic connectivity (some of them marked out by straight lines).
filter applied to a node calculates a weighted average of its hydraulic head and the hydraulic heads at its neighbouring nodes. The size and shape of the neighbourhood and the specific weights used will have an impact on the effect of each type of filter. If there is enough data, and it is reasonably well distributed, a simple normalised Laplacian operator may reveal some fracture hydraulic connectivity trends as shown in fig. 3.33.
3.7 Modelling hydrogeological systems 3.7.1
C on ce p t s
An arbitrary bounded rock mass may be viewed as an isolated geological system formed by an assemblage of interrelated lithologic and structural features, from minute to huge scales, making up an integrated whole, containing around 90 to 99% of solid matter and a small proportion of fluid matter in the remaining space. If its fluid phase is in focus, it is labelled a hydrogeological system, and this is generally too complex to be fully appreciated. However, simplified models may describe some of its main aspects and behaviour from different points of view. For example, a “groundwater management model’’ may be formulated to inform decisions about cost-effective operation granting environmental safety. A numerical simulation devised to predict groundwater flow under different boundary conditions requires two paired models. The first model, called “conceptual’’ and necessarily conceived before the second, explains the various modes of occurrence and properties of the fluid phase by means of verbal, iconic and quantified descriptors based on the interpretation of many kinds of field and laboratory investigations, such as geological mapping, borehole core samples, geochemical indicators, geophysical parameters, hydrological data, etc. The second one, the “mathematical’’ model, results from the applications of the principles of hydraulics to the conceptual model. It describes, in purely mathematical terms, the physical and chemical processes involving the percolation of the interconnected and unrestrained fluid phase, including its
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D a t a a n a l y s i s 121
carrying capacity. It is utilised to clarify the past and to predict the future behaviour of the fluid phase for different scenarios. Its retrospective and predictive ability must be always measured against some reference space-time values and if detected errors surpass predefined levels, both models must be improved in a cyclic way or replaced by better representations. However, depending on the required levels of accuracy, the improved models may become increasingly sophisticated and complex. Construction of a conceptual model is an inductive process involving inferences and correlations from a finite set of field observations and investigations. It is not a deterministic outcome of an encoded procedure, 100% correct, as with a mathematical model. Its degree of realism depends on the inherent complexity of the hydrogeological system as well as on the economic constrains on time and energy required to build it. It may be more or less acceptable according to the needed output and level of detail. In contrast, the mathematical model is a formal deductive process, presumably well formulated, whose reliability is entirely reliant on the premises of the conceptual model. Consequently, it is simple to recognise that the conceptual construct is by far the most important part of a numerical simulation process. Conceptual models vary in how the main structural features that convey groundwater are treated, how their properties are estimated and how their hydraulic performances are described. Conductive fractures and conduits within a subsystem may be modelled as a network of separated but interconnected discontinuities or grouped together in a single entity. Their hydraulic properties may derive from the integration of small-scale statistical data or from the interpretation of large-scale tests results. Corresponding hydraulic behaviour may be portrayed for each discontinuity or for groups of fractures, or by a combination of both sorts of descriptions. A well organized and commented anthology on the subject can be found in Rock Fractures and Fluid Flow, National Academic Press, Washington, 1966. 3.7.2
Gu id el i n e s t o c o n c e p t u a l m o d e l s
Generally, hard rock masses are inhomogeneous at many scales and their conceptual models, depending on the type of the numerical approach to be followed, may be difficult to construct. However, conceptual models for pseudo-continuous solutions compliant to permeabilities described by second order tensors are relatively simple to implement for most practical applications. There are no fixed rules to build them but only a few guidelines. Indeed, taking into account all comments and observations presented in Chapter 1, one may proceed as follows: a)
b)
c)
If a dependable topographic map is not available at a scale judged appropriate for the intended simulation, including watercourses, ponds, springs, seeps and other hydrogeological features of interest, compile cartographic and surveys data to generate the model base map, preferably as a suitable digital elevation model. If not available or not adequate, produce a hydrogeological map compliant to the compiled base map relying on existing geological and hydraulic information properly supplemented by fieldwork, geophysics, geochemistry, borehole sampling, hydraulic field tests, etc. To cope with the frequent scarcity of data far from the area of interest of the simulation, the model boundaries are preferably placed on “natural’’ Dirichlet
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d)
e)
f)
g)
h)
i)
j)
Fractured rock hydraulics
and Neumann conditions, such as lakes, rivers, practically impervious geological contacts or deep impervious rock masses. Agree on the network parameters and on the cell dimensions to accommodate the details of the numerical simulation and subdivide the flow domain into distinct units having almost similar hydraulic properties based on their hydrogeological features and properties. For the intact rock matrix within each cell or group of cells, evaluate the absolute or relative average 3D permeability tensor based on laboratory tests, published data or experience. For the fractured rock mass within each cell or group of cells, evaluate the absolute or relative average 3D permeability tensor taking into account the properties of the intact rock matrix and of the minor discontinuities of the rock mass. As before, this evaluation can rely on field tests, published data or experience. Field tests, as those proposed by Hshieh et al., takes into account the hidden contributions of all unseen discontinuities, without exception, and implicitly integrate the effect of the intact rock matrix. All major discontinuities whose trace length exceeds around a quarter to a half of the smaller horizontal dimension of the base map will be considered in the next step. Based on geological and structural features, field tests, published data, experience etc., evaluate the absolute or relative average 3D permeability tensor of the major discontinuities not included in previous steps. It is important to keep in mind that their eigenvalues and eigenvectors must implicitly translate their hydraulic transmissivity and interconnectivity at the simulation scale. To take into account the influence of major discontinuities, apply the summation rule (see Chapter 1, section 1.1.6.2, Fractures and conduits) to explicitly integrate the 3D permeability tensor of each cell or group of cells. Calibrate the 3D permeability tensors of the grid cells by optimising the numerical simulation based on known time series of hydraulic head values at observation points in the flow domain and its boundaries as well as on known time series of discharge values at sources, sinks and installed wells. This calibration becomes easier if based on relative rather than absolute previously inferred values. Recalibrate the simulation based on a time series of the gross mass balance of the flow domain, but only if properly known.
To assure a relative consistency between all these evaluations and calibrations, it is important to make use of the same premises and choice criteria for all sensible parameters of the whole model. After a few progressive adjustments to factual data, the calibrated simulation may replicate the probable groundwater flow with some dependability. In case of disagreements between the calibration results and the factual observations of an experienced hydrogeologist, decisions about the conceptual model should be inclined to the interpretation of the hydrogeologist.
References Bohling, G., 2005, “Kriging’’, Kansas Geological Survey, C&PE 940, October. Franciss, F. O., 1994, First Underground Refrigerated LPG Caverns in Brazil, Eurock 94, Rock Mechanics in Petroleum Engineering, Delft, Netherlands.
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D a t a a n a l y s i s 123 Hantush, M. S., 1956, “Analysis of Data from Pumping Tests in Anisotropic Aquifers’’, Journal of Geophysics Research, no. 72. Hsieh, P.A. and Neuman, S.P., 1985, “Field determination of the three-dimensional hydraulic conductivity tensor of anisotropic media’’, Water Resources Research, Vol. 21, no. 11. Kosakowski, G., “Introduction to Applied Geostatistics’’, Waste Management Laboratory, Paul Scherrer Institut, 5200 Villigen PSI, Switzerland. Press, W. H. et alia, 1994, Linear Prediction and Linear Predictive Coding, Chapter 13, in Numerical Recipes, Cambridge University Press, 2nd Ed. Rocha, M. and Barroso, M., 1971, “A method of integral sampling of rock masses’’, Rock Mechanics, Vol. 3, 1. Rocha, M. and Franciss, F. O., 1977, “Determination of Permeability in Anisotropic Rock Masses from Integral Samples’’, Rock Mechanics, Vol. 9/2–3. Sophocles, J. O., 1992, “Yule-Walker algorithm and Burg’s method in Optimum Signal Processing’’, Macmillan. Terzaghi, R., 1965, “Sources of errors in joint surveys’’, Geotechnique. 15: 287–303.
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Chapter 4
Finite differences
4.1 Preliminaries Actual groundwater flow systems concerning fractured rock masses are generally inhomogeneous, irregularly anisotropic and particularised by mixed boundary conditions. Therefore, the numerical simulation of the spatial and temporal variation of their hydraulic heads by a single polynomial approximant may bear unacceptable errors. However, fragmenting these complex systems into simple adjoining subsystems, each one described by its own polynomial approximant, may reduce the solution errors to acceptable levels. Evidently, these restricted “piecewise’’ approximants, properly allocated for each subsystem, must best fit the connecting boundary conditions to the adjacent subsystems. Finite difference and finite element methods make use of this approach. In the last decades, the decrease in size and cost concurrent with the increase of speed and memory of commercial computers as well as the development of user friendly programming languages has induced most universities, research centres and independent consultant engineers, geologists, geophysicists and hydrogeologists to explore the possibilities of numerical techniques more frequently. Indeed, working out practical problems requiring the solution of partial differential equations applied to relatively complex domains may only be acceptably realistic using numerical techniques. Among them, finite difference algorithms are very attractive since they are simple and intuitive as well as suited to spreadsheet programming. There is a vast literature on this subject including technical papers and textbooks. The reader is encouraged to gain more knowledge on this subject (Bear, 1979; Wang and Anderson, 1982; Lam, 1994; Press et al., 2007). However, given the introductory character of this book, only essential concepts and consistent finite difference algorithms are discussed in this chapter, including 3D applications for fractured rock masses.
4.2 Finite difference b asics 4.2.1
Differ e n c e e q u a t i o n s
The derivatives occurring in partial differential equations may be replaced by finite differences, thus converting these partial differential equations into simpler difference equations. As a result, continuity and boundary condition differential equations may be substituted by equivalent difference equations.
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Figure 4.1 Example of rough discretisation of a horizontal aquifer.
Note that any 3D problem domain can be discretised, i.e. partitioned into rectangular cuboids defined by the size of their edges and the coordinates of their geometrical centres (see fig. 4.1). After this process, continuity difference equations may be applied at the centres of all cuboids within the problem domain. Additional supplementary difference equations may be formulated consisting of specified heads and/or flows at all cuboids fitted to the boundaries of the problem domain (Dirichlet and Neumann conditions). These equations, taken simultaneously, generate a set of algebraic equations whose numerical treatment gives the approximate solution of the partial differential equation at all cuboids’ centres of the problem domain. A similar argument applies to 2D or 1D problems. 4.2.2
Fin it e d i f f e r e n c e s
Numerical values of a continuous function may only be known at regularly spaced points in space. Even with this restriction, its first-order and second-order partial derivatives can be estimated by appropriate finite differences at these points. As an example, consider part of the x-trace of the sloping water table of a horizontal aquifer divided into four cells, 100 m wide (see fig. 4.2). The water table elevation, taken as equivalent to the hydraulic head h by Dupuit’s assumption, is only described at abscissas 150 m, 250 m, 350 m and 400 m. However, based on simple geometry, it is possible to estimate the first and the second-order partial x-derivatives of h at the cell boundaries and cell midpoints respectively. Denoting by δx the 100 m equal intervals, the first-order partial x-derivative at the cell boundary at xi−1/2 = 200 m, just in the middle of xi−1 = 150 m and xi = 250 m, may be estimated by:
hi+1 − hi ∂ h + O(δx2 ) = (4.1) ∂x i+ 1 δx 2
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Head h (m)
540
530
(xi1 150 m, h 534.68 m)
(xi 250 m, h 532.72 m) (xi1 350 m, h 530.3 m)
100
200
300 x (m)
Continuous head h
400
Discrete head h values at cell mid-point
500
Cell boundary
dh/dx (m/m)
Figure 4.2 Trace of the sloping water table of a horizontal aquifer on an x-cross-section divided into four cells, 100 m wide. Three generic contiguous points are defined by (xi−1 = 150 m, h = 534.68 m), (xi = 250 m, h = 532.72 m) and (xi+1 = 350 m, h = 530.30 m). 0.015 x i
1 2
200 m, h 0.020 x
x i
0.02
1 2
300 m, h 0.024 x x i
0.025 100
200
300 x (m)
3 2
400
400 m, h 0.019 x
500
Numerical first derivative at cell boundary
Figure 4.3 x-∂h/∂x plot of the numerical of the first-order partial derivatives at cell boundaries xi−1/2 = 200 m, xi+1/2 = 300 m and xi+3/2 = 400 m.
The additional term O(δx2 ) in equation 4.1 stands for the approximation error. By a similar rule, the first-order partial x-derivative at xi+1/2 = 300 m, just in the middle of xi = 250 m and xi+1 = 350 m, may be estimated by:
∂ h ∂x
i− 12
=
hi − hi−1 + O(δx2 ) δx
(4.2)
For the above expressions (equation 4.1 and equation 4.2), called central approximation formulas, the error is proportional to δx2 . Consequently, dividing by two the interval δx, i.e. making new δx = δx/2, would decrease the truncation error to a quarter of the previous one. The x-∂h/∂x plot of the numerical first-order partial derivatives at the cell boundaries 200 m, 300 m and 400 m is depicted in fig. 4.3. The second-order partial x-derivative at the cell boundary 300 m, in the middle of xi = 250 m and xi+1 = 350 m, may also be estimated by a central approximation:
∂2 h ∂x2
= i
∂ ∂ h ∂x ∂x
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i
hi+1 − hi hi − hi−1 − δx δx = + O(δx2 ) δx
(4.3)
d[dh/dx]/dx (m/m/m)
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Fractured rock hydraulics
1·104 5·105
xi 250· m, 2 h 0.000037 x 2
0 5·105 100
200
xi1 350 m, 2 h 0.000058 x 2
300 x (m)
400
500
Numerical second derivative at cell mid point
Figure 4.4 x-∂2 h/∂x2 plot of the numerical second-order partial derivatives at cell mid-point xi = 250 m and xi+1 = 350 m.
As before, the term O(δx2 ) corresponds to the order of magnitude of the approximation error. The x-∂2 h/∂x2 plot of the numerical second-order partial derivatives at cell midpoints 250 m and 350 m is shown in fig. 4.4. In general, an n-order finite difference equation applied at a point xi may be formally deduced from a backward or a central or a forward Taylor series expansion about this point by ignoring the terms containing higher-order derivatives. The truncation error is proportional to the order of magnitude of the first term of the discarded part of the series. For a first-order finite difference equation, the truncation error is O(δx) for backward or forward expansions and O(δx2 ) for a central expansion. The first-order and second-order partial derivatives approximations estimated by appropriate finite differences, as discussed above, can be applied to any continuous parameter over the problem domain. However, the numerical algorithm must be constructed by keeping in mind that the hydraulic conductivity is implicitly considered a continuous parameter in the general continuity equation. As an important consequence, the hydraulic conductivity must be assumed as a continuous property, even if highly variable, to assure dependable algorithms. For an example of this possibility see fig. 4.5 which shows one of the results obtained from a numerical sensitivity analysis devised to simulate the head decay during the downward percolation of water through a layer of laminated sediments.
4.2.3
Differe n c e e q u a t i o n s f o r s t e ady-s t at e s y s t e m s
Finite difference equations that are algebraically equivalent to the continuity differential equations for steady-state systems can also be directly derived. As an example, consider a rectangular cuboid centered at point i defined by edges δx, δy and δz (see fig. 4.6). Assuming constant density ρ, the Poisson equation applied to a 1D steady-state system all along the x-axis can be written as:
∂ ∂ k h =Q ∂x ∂x
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(4.4)
Finite differences
1
129
0.01 1·104 1·105
0.5
1·106
K (m/s)
H (m)
1·103
1·107 0
0
0.2
0.4
0.6
0.8
1·108
L (m) Head (m)
Permeability (m/s)
Figure 4.5 One of the numerical results of a sensitivity analysis of the head decay associated with the vertical percolation of water in an encapsulated integral core sample, measuring 1 m length, extracted from a thick layer of laminated sediments. Heads are referred to the left y-axis. Jagged vertical permeabilities are referred to the right y-axis (log scaled). The permeability of each 1 cm thick lamina varied randomly between 10−8 and 10−2 m/s.
z
i1
i x
i1 y
Figure 4.6 Cuboid with edges δx, δy and δz centered at point xi . Water enters left area δyδz at point xi−1/2 and leaves right area δyδz at point xi+1/2 under hydraulic gradients (hi − hi−1 )/δx and (hi+1 − hi )/δx, respectively.
Considering the hydraulic conductivity k as a continuous variable, the left-hand side of equation 4.4 can be expanded by the product rule and the expression takes the form:
∂ ∂ ∂ ∂ k h +k h =Q (4.5) ∂x ∂x ∂x ∂x Consider now three generic contiguous points: xi−1 , xi and xi+1 . To reduce the truncation error and assure a stable algorithm, the finite difference approximation for the first term of the left-hand side of the equation must be applied to the central point xi . This may be done by taking the average of the approximations at midpoints xi−1/2 and xi+1/2 .
∂ ∂ ∂ ∂ + k h k h
∂x ∂x ∂x ∂x ∂ ∂ i− 12 i+ 12 k h = (4.6) + O(δx2 ) ∂x ∂x 2 i
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Then, the equivalent finite difference equation for above expression is:
∂ ∂ k h ∂x ∂x
i
ki+1 − ki hi+1 − hi ki − ki−1 hi − hi−1 + δx δx δx δx = + O(∂x2 ) 2 (4.7)
By the same arguments, the finite difference equation for the second product of the left-hand side of the equation is:
k
∂ ∂ h = ki ∂x ∂x
hi+1 − hi δx
−
hi − hi−1 δx
δx
+ O(δx2 )
(4.8)
Simplifying and collecting terms, the finite difference equivalent for the continuity equation (equation 4.4) is:
ki+1 + ki 2
hi+1 − hi δx
−
ki + ki−1 2
hi − hi−1 δx
δx
=Q
(4.9)
Multiplying both members of this expression by the subsystem volume, and taken into account that δδyδz = δS and δxδyδz = δV, yields:
ki+1 + ki 2
hi+1 − hi δx
−
ki + ki−1 2
hi − hi−1 δx
δS = Q δV
(4.10)
This expression has a clear physical meaning. The left-hand side measures the difference between the entering and leaving discharges along the x-axis through the two opposite faces of area δS. The right-hand side measures the amount of water volume Q · δV added or drained by an occasional source (or sink) of strength Q per unit volume. It is important to keep in mind that this algorithm considers the hydraulic conductivity as a continuous variable, but only valued at the cell centres. In this case, the hydraulic conductivity at midpoints xi−1/2 and xi+1/2 must be estimated by an arithmetic mean. However, if one assumes that the hydraulic conductivity takes constant values inside each cell, forming a series circuit along the x-axis, it is recommended to approximate the hydraulic conductivity at midpoints xi−1/2 and xi+1/2 by a harmonic mean (Bear, 1972). A simple exercise to compare the numerical effect of these two types of mean is shown in fig. 4.7. Considering the premises that support the pseudo-continuous concept for random fractures and, implicitly, for assuming averages between cell midpoints, the author has a preference for the arithmetic mean based on his modelling experience.
© 2010 Taylor & Francis Group, London, UK
Finite differences
K assumed continuous but only defined at interval midpoints
K assumed constant in each interval (series circuit) 0.1
0.1
0.4
1105
0.2
1103
0.6 0.4
1105
0.2 1107
0 0
0.2
0.4
0.6
0.8
1107
0
1
0
0.2
0.4
L (m) Head (m)
0.6
0.8
L (m)
Permeability (m/s): max K/min K 10
Head (m)
K assumed continuous but only defined at interval midpoints
Permeability (m/s): max K/min K 10
K assumed constant in each interval (series circuit) 0.1
0.1
0.6 0.4
1105
0.2
1103
0.6 0.4
1105
K (m/s)
110
3
H (m)
0.8 K (m/s)
0.8
0.2 1107
0 0
0.2
0.4
0.6
0.8
1107
0
1
0
0.2
0.4
L (m) Head (m)
0.6
0.8
L (m)
Permeability (m/s): max K/min K 100
Head (m)
K assumed continuous but only defined at interval midpoints
Permeability (m/s): max K/min K 100
K assumed constant in each interval (series circuit) 0.1
0.1 3
110
0.6 0.4
1105
0.2
H (m)
0.8 K (m/s)
0.8
1103
0.6 0.4
1105
K (m/s)
H (m)
K (m/s)
0.6
H (m)
0.8 1103
K (m/s)
H (m)
0.8
H (m)
131
0.2 1107
0 0
0.2
0.4
0.6
0.8
1
1107
0 0
0.2
L (m) Head (m)
Permeability (m/s): max K/min K 1000
0.4
0.6
0.8
L (m) Head (m)
Permeability (m/s): max K/min K 1000
Figure 4.7 These three pairs of L × H plots give an idea about the influence of the type of mean, arithmetic or harmonic, used in the finite difference algorithm. Compare the results of the numerical simulation of the head decay for a steady upward percolation of water from beginning to end of a homogenous soil column with height L = 1 m under a constant head of H = 1 m. The soil permeability is 10−6 m/s in all cases except for a 10 cm layer in the column centre that is N times more pervious than the rest of the soil. The left and right plots correspond respectively to the arithmetic and harmonic means for N equal to 10, 100 and 1000, as indicated in the plots. For N = 10, there is no appreciable difference between the left and right plots. For N = 100 or 1000 the left plots show negligible decays for the central cell, as expected, while the right plots remain practically unchanged.
4.2.4
Differ e n c e e q u a t i o n s f o r u n s t e ady-s t at e s y s t e m s
For unsteady-state systems, a 1D general continuity equation system all along the x-axis can be written as:
∂ ∂ ∂ k h = SV h +Q ∂x ∂x ∂t The term SV is the specific volumetric storage.
© 2010 Taylor & Francis Group, London, UK
(4.11)
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H(x,0) constant
1.5 H(x,t) surface 1 H (m)
n·t 1 n – ·t 2
(n1)·t 0.5
Drain 0
0
xi1 xi
10
8 xi1
6 4
X (m
)
2
20
in)
t (m
30
Figure 4.8 Transient surface H(x, t) estimated by a Crank-Nicolson scheme. The first-order partial t-derivative ∂h/∂t is approximated at point xi and at time step n + 1/2, i.e. just in the middle of the t-steps n and n + 1. The finite expression for the space continuity is taken as the arithmetic average of the finite expressions at point xi at t-steps n and n + 1.
As with the space-variable x, the time variable t can also be discretised into many steps of equal time-intervals δt counted from a starting time t0 . Then, the product n · δt measures the time interval (t − t0 ). The first-order partial t-derivative ∂h/∂t occurring in the right-hand side of the governing equation 4.11 may be approximated at point xi and at time step n + 1/2, i.e. just in the middle of the time steps n and n + 1, then:
SV
∂ h ∂t
n+ 12
+ Q = SV
(hi )n+1 − (hi )n + Q + O(δt2 ) δt
(4.12)
In this equation, the pseudo-exponents n and n + 1 stand for the values of the head hi at the moments (n)·δt and (n + 1)·δt (see fig. 4.8). The finite expression for space-continuity at xi , on the left-hand side of the equation 4.12 is undefined at the t-step n + 1/2. However, this value may be taken as the arithmetic average of the corresponding finite expressions at time steps n and n + 1. Then,
© 2010 Taylor & Francis Group, London, UK
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for each t-step n + 1/2, the equivalent finite difference equation for continuity is:
(hi+1 )n − (hi )n ki+1 + ki ki + ki−1 (hi )n − (hi−1 )n − ··· 2 δx 2 δx 1
δS
2 ki+1 + ki (hi+1 )n+1 − (hi )n+1 ki + ki−1 (hi )n+1 − (hi−1 )n+1 + − 2 δx 2 δx n+1 n (hi ) − (hi ) · · · SV δt = (4.13) δV +Q As before, the pseudo-exponents stand for the t-steps n and n + 1. This algorithm is unconditionally stable and second-order accurate in time and space. It generates for each t-step a set of simultaneous equations that must be solved to find the head values h corresponding to that t-step. This clever solution was jointly proposed by John Crank (1916–2006), a British mathematical physicist partly dedicated to numerical solutions of partial differential equations, and Phyllis Nicolson (1917–1968), also a British mathematician associated with the development of the Crank-Nicolson scheme in mid twentieth century.
4.2.5
Differ e n c e e q u a t i o n s f o r b o u n dar y c o ndi t i o ns
If the finite difference equations are applied to the interior nodes of a steady-state groundwater system for which all the heads at the exterior nodes, i.e. at the boundary nodes, are already specified (Dirichlet boundary conditions), then the number of unknown heads and algebraic equations are the same. However, the number of unknowns exceeds the number of algebraic equations if the discharge rates at one or more exterior nodes are specified as an alternative for heads (Neumann boundary conditions). In this case, supplementary equations are needed to solve the problem. For example, consider the Neumann prescription applied to a boundary point xi−1 . In this case, the hydraulic exit gradient ∂h/∂xi−1 at that point can be approximated by a second-order accurate backward difference:
∂ 3 · hi−1 − 4 · hi + hi+1 = h + O(δx2 ) (4.14) ∂x i−1 2 · δx Recall that a quadratic interpolation polynomial collocated at three successive and collinear points xi−1 , xi and xi+1 separated by equal intervals δx yields this same algorithm. In equation 4.14, the heads hi−1 , hi and h1+1 refer to the successive three collinear points xi−1 , xi and x1+1 . These points are located along an interior normal to the boundary at xi−1 . The corresponding Neumann condition is: k
3hi−1 − 4h1 + hi+1 δxδz = −qi−1 2δx
© 2010 Taylor & Francis Group, London, UK
(4.15)
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Fractured rock hydraulics
Then, it is possible to express hi−1 in terms of qi−1 , hi and hi+1 : hi−1
1 δx qi−1 = 4hi − hi+1 − 2 3 δxδz k
(4.16)
Replacing hi−1 occurring in all algebraic equations for the left-hand expression of equation 4.16 equates the number of equations and unknowns. After solving this set for the interior heads, the remaining border heads hi−1 are directly calculated by the Neumann condition. For unsteady simulations, both conditions may be time-dependent values, if this is the case. There are other ways to apply boundary conditions, including approximated solutions for refined and non-regular boundaries (see, for example, Lam, 1994). 4.2.6
S imu l t a n e o u s d i f f e r e n c e e q u a t i o ns
4.2.6.1 P re l i m i n a r i es For vast groundwater systems, described by inhomogeneous and anisotropic subsystems, full of different kinds of wells, drains, natural springs and sinks, particularised by many types of boundary and initial conditions, the resulting set of simultaneous difference equations may involve several thousands of unknowns. Solving it, according to their hydraulic properties and boundary and initial conditions, yields the head values at the centres of the discrete cuboids within the problem domain or fitted to its borders. Given the introductory character of this book, only two iterative techniques, known as the Jacobi and Gauss-Seidel methods will be considered next. Carl Gustav Jacob Jacobi (1804–1851), Johann Carl Friedrich Gauss (1777–1855) and Philipp Ludwig von Seidel (1821–1896) were outstanding and inspired nineteenth century German mathematicians. These methods are easily programmable with the help of commercial mathematical and graphical software. See some specialised sites: www.office.microsoft.com/excel, www.ibm.com/software/lotus, www.wolfram.com, www.ptc.com, www.mathworks.com. 4.2.6.2 Gau s s-Sei d el i ter a ti v e r ou ti n e Example 4.1 provides an easy way to grasp the essence of the Gauss-Seidel iteration routine.
Example 4.1 In this 2D elementary exercise, groundwater flows within a homogeneous, isotropic, horizontal and confined sandstone aquifer trapped by an upper and a bottom layer of impervious shale. In addition, two almost parallel and vertical diabase dykes, practically impervious, isolate a 1000 m wide sandstone band between them. A piece of that band, measuring 1000 m × 1000 m in plan view, may be taken as a simple groundwater system to be numerically simulated (see fig. 4.9).
© 2010 Taylor & Francis Group, London, UK
Finite differences
135
Vertical almost impervious diabase dyke Impervious upper layer of shale
Vertical almost impervious diabase dyke
Confined sandstone aquifer
Impervious bottom layer of shale
Figure 4.9 Confined sandstone aquifer, “sandwiched’’ between an upper and a lower shale layers. Two parallel and vertical impervious diabase dykes, one in the front and the other in the rear of the perspective, isolate the aquifer laterally.
Now, consider a 2D square lattice defined by 20 × 20 squares of 50 m size (see fig. 4.10). Denoting by hi,j the head value at the centre (i, j) of each square and by δx its side dimension, the finite difference equation is simply: hi, j =
qi ,j 2 1 hi−1,j + hi+1 ,j + hi ,j−1 + hi ,j+1 + δx 4 T
(4.17)
This equation applied at the 400 interior nodes of the 2D square lattice generates a set of simultaneous equations characterised by a band diagonal matrix for the head coefficients. In this case, the Gauss-Seidel method may be used with success. The heads along the left and right sides of the grid, respectively defined by x1 = 0 m and x21 = 1000 m, are prescribed by Dirichlet (see fig. 4.11). The upper and lower sides of the grid, defined by y1 = 0 m and y20 = 1000 m, are no-flow borders except at point (x18 = 850 m, y1 = 0 m) where 50 m3 /hr of water leaks through a breach in the front dyke. The sandstone constant transmissivity is 3.5 · 10−4 m2 /s. Moreover, 150 m3 /hr and 75 m3 /hr of water are respectively pumped from and injected into the aquifer at points (x8 = 350 m, y14 = 650 m) and (x16 = 750 m, y10 = 450 m). The effect of the pumping or the injection well is simulated by a uniform withdrawal or accretion on top of the cell equal to the pumping or injection rate divided by the top area δx2 . For the breach in the front dyke, the uniform leakage corresponds to the total leak divided by the lateral area δxδz. To start a Gauss-Seidel iteration, an initial inexact solution array for all unknown heads hi,j must be advanced. There are many ways to do this. For example, it is possible to fit a surface h0 (x, y) to the specified boundary heads. Other approach is to find a simplified
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Fractured rock hydraulics
1000
800
Qpump 150
m3 hr
600 Y (m)
Qinjection 75
m3 hr
400 Tsandstone 3.5 104
m2 s
200
Qleak 75
0
0
200
400
600
m3 hr 800
1000
X (m) Grid centers
Impervious upper border (Neumann)
Pumping well
Impervious lower border (Neumann)
Injection well
Pervious left border (Dirichlet)
Border leak
Pervious right border (Dirichlet)
Figure 4.10 2D square lattice formed by 400 squares of 50 m size. To each grid centre (i, j), defined by coordinates (xi , yj ), corresponds a generic head hi ,j . Four adjacent heads are described by four triplets: (xi−1 , yj , hi−1 ,j ), (xi+1 , yj , hi+1 ,j ), (xi , yj−1 , hi ,j−1 ) and (xi , yj+1 , hi ,j+1 ). Subscripts i and j run from 1 to 21.
solution employing an analytical method. Then, for all interior points (i, j) where qi = 0, except for the pump point (8, 14), injection point (16, 10) and near border points along rows (i, 2) and (i, 20), the general finite difference equation gives the head value hi,j : hi,j =
1 (hi−1,j + hi+1,j + hi,j−1 + hi,j+1 ) 4
(4.18)
The heads h8,14 at the pump point (8, 14) and h16,10 at injection point (16, 10) are given by: (Qpump )8,14 1 h8,14 = h7,14 + h9,14 + h8,13 + h8,15 + (4.19) 4 T
© 2010 Taylor & Francis Group, London, UK
Finite differences
137
Dirichlet heads (m)
700
Left border Right border
600
500
400
0
200
400
600
800
1000
Y (m)
Figure 4.11 Specified head along the left and right borders (Dirichlet boundary conditions).
h16,10 =
(Qinjection )16,10 1 (h15,10 + h17,10 + h16,9 + h16,11 ) + 4 T
(4.20)
In these expressions, Qpump and Qinjection are negative and positive quantities, respectively. For the near border points along rows (i, 2) and (i, 20), the general expressions for hi,j include unknown heads along the boundaries (i, 1) and (i, 21). At these border points, the corresponding exit gradients may be approximated by three-point formulas:
∂ −hi,3 + 4hi,2 − 3hi,1 = h + O(δy2 ) ∂y i,1 2δy (4.21)
3hi,21 − 4hi,20 + hi,19 ∂ 2 = h + O(δy ) ∂y i,21 2δy Now, for impervious boundaries, all exit gradients normal to the no-flow borders are zero. Then, equating these approximations for ∂h/∂y to zero at points (i, 1) and (i, 21) it is possible to express hi,1 and hi,21 in terms of the interior heads, except for the leak at point (18, 1): 1 (4hi,2 − hi,3 ) 3 1 = (4hi,20 − hi,19 ) 3
hi,1 = hi,21
(4.22)
Substituting the right-hand side of these expressions for hi,1 and hi,21 the general finite difference equation at points along the rows (i, 2) and (i, 20) removes these unknowns, except near the leak point (18, 1): 1 1 hi,2 = hi−1,2 + hi+1,2 + (4hi,2 − hi,3 ) + hi,3 4 3 (4.23) 1 1 hi−1,20 + hi+1,20 + hi,19 + (4hi,20 − hi,19 ) hi,20 = 4 3
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Fractured rock hydraulics
Simplifying and collecting: ·hi,2 = ·hi,20
3 3 1 hi−1,2 + hi+1,2 + hi,3 8 8 4
(4.24)
3 3 1 = hi−1,20 + hi+1,20 + hi,19 8 8 4
At the leak point (18, 1), the Neumann condition states that: (Qleak )18,1 −h18,3 + 4 h18,2 − 3 h18,1 δxδz = − 2δy k
(4.25)
Taking into account that δx = δy and that k · δz =T, this equation simplifies to: −h18,3 + 4h18,2 − 3h18,1 = −2
(Qleak )18,1 T
In this expression, Qleak is also a negative quantity, from which: 2(Qleak )18,1 1 4h18,2 − h18,3 + h18,1 = 3 T
(4.26)
(4.27)
Substituting for h18,1 the general finite difference equation at point (18, 2) removes this unknown: 2(Qleak )18,1 1 1 h18,2 = (4.28) h17,2 + h19,2 + 4h18,2 − h18,3 + + h18,3 4 3 T Simplifying and collecting: h18,2 =
3 3 1 1 Q.leak 18,1 h17,2 + h19,2 + h18,3 + 8 8 4 4 T
(4.29)
Having expressed all interior heads in terms of their interior neighbours, the iteration procedure starts from the initial solution estimate [hi,j ]0 , running from i = 1 to 21 and from j = 2 to 20 (because columns (1, j) and (21, j) are Dirichlet prescribed) as follows: •
At all interior points (i, j), except at the pump point (8, 14), injection point (16, 10) and near border points along rows (i, 2) and (i, 20): (hi,j )k+1 =
•
1 (hi−1,j )k+1 + (hi+1,j )k + (hi,j−1 )k+1 + (hi,j+1 )k 4
(4.30)
At pump point (8, 14) and injection point (16, 10): (Qpump )8,14 1 k+1 k+1 k k+1 k = + (h9,14 ) + (h8,13 ) + (h8,15 ) + (h8,14 ) (h7,14 ) 4 T (h16,10 )k+1 =
(Qinjection )16,10 1 (h15,10 )k+1 + (h17,10 )k + (h16,9 )k+1 + (h16,11 )k + 4 T (4.31)
© 2010 Taylor & Francis Group, London, UK
Finite differences
•
139
At all near border points (i, 2) and (i, 20), except near the leak point (18, 1): (hi,2 )k+1 = k+1
(hi,20 ) •
3 3 1 (hi−1,2 )k+1 + (hi+1,2 )k + (hi,3 )k 8 8 4
(4.32)
3 3 1 = (hi−1,20 )k+1 + (hi+1,20 )k + (hi,19 )k+1 8 8 4
Near the leak point (18, 1): (h18,2 )k+1 =
3 3 1 1 (Qleak )18,1 (h17,2 )k+1 + (h19,2 )k + (h18,3 )k + 8 8 4 4 T
(4.33)
In the above iteration formulas, the pseudo-exponents k or k + 1 stand respectively for the approximations of the head values hi,j after iteration or k + 1. Moreover, it is take for granted that these systematic computations are performed in such a way that iterations k + 1 for hi−1,j and hi,j−1 are always calculated before the iteration k + 1 for hi,j , thus avoiding the need for extra storage.
1000
49
0
515
900
h 545 m h 540 m h 535 m h 530 m h 525 m h 520 m h 515 m h 510 m h 505 m h 500 m h 495 m h 490 m h 485 m h 480 m h 475 m h 470 m h 465 m h 460 m h 455 m h 450 m h 445 m h 440 m h 435 m h 430 m h 425 m h 420 m h 415 m
800 700
515
Y (m)
490
600 500
49
0
46
300
5
490
400
515
200 100 0 0
100
200
300
400
500 X (m)
600
700
800
900
Figure 4.12 Equipotentials distribution for the area modelled in example 4.1.
© 2010 Taylor & Francis Group, London, UK
1000
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Fractured rock hydraulics
Iteration stops when the greatest relative change after iteration k + 1, calculated by [((hi,j )k+1 − (hi,j )k )/(hi,j )k ], taking into account all iterated values hi,j or only some specific and critical points, is smaller than a prescribed limit ε: (hi,j )k+1 − (hi,j )k <ε (4.34) max (hi,j )k At the end of the iteration procedure, all head values along the border points (i, 1) and (i, 21), except for the leak point, are calculated by the Neumann condition: hi,1 = hi,21
1 (4hi,2 − hi,3 ) 3
(4.35)
1 = (4hi,20 − hi,19 ) 3
At the leak point (18, 1): 2(Qleak )18,1 1 h18,1 = 4h18,2 − h18,3 + 3 T
(4.36)
The resulting contour heads are shown in fig. 4.12.
The Gauss-Seidel iteration procedure is in fact an organised improvement of a nineteenth century method credited to Jacobi where, unlike Gauss–Seidel, old values (hi−1,j )k and (hi,j−1 )k are never replaced by newly iterated values within the same iteration loop, as formally pointed out in the general Jacobi iterative formula: (hi,j )k+1 =
qi,j 2 1 (hi−1,j )k + (hi+1,j )k + (hi,j−1 )k + (hi,j+1 )k + δx 4 T
(4.37)
For most practical applications, both methods converge. However, in the author’s personal experience, there are cases for which Jacobi converges and Gauss-Seidel does not (Varga, 1962). 4.2.6.3 C ra nk-N i c h ol s on i ter a ti v e r ou ti n e Example 4.2 provides an easy way to learn how to apply the Crank-Nicholson iterative routines for each t-step of an unsteady simulation.
Example 4.2 In this elementary exercise consider the transient uplift pressure relief under a 10 m wide airport concrete pavement as shown in fig. 4.13. Any point of the contact joint between the pavement and subbase may be located by its abscissa x, varying from 0 m to 10 m. To apply a finite difference algorithm, that joint may be discretised into 20 equal cells, each one measuring δxi = 0.5 m, where the subscript i
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Finite differences
141
10 m
1m
Figure 4.13 Schematic cross-section of a 10 m wide airport concrete pavement. A pervious and continuous contact joint demarcates the interface between the pavement base and the cement-bound subbase. Two longitudinal drains at its extremities assure the uplift pressure relief. The initial water pressure after an exceptional inundation and before full drain operation is 1.5 m.
runs from 1 to 20. A maximum uplift pressure (hi )0 = 1.5 m at all cell nodes takes place after an exceptional inundation and before full drain operation at time t0 . This defines the initial conditions of the problem. Immediately after the start of the full drainage, the two longitudinal drains permanently keep the uplift pressure equal to zero at both extremities of the pavement, identified by cells δx1 and δx20 . This typifies two Dirichlet boundary conditions. Denoting by (hi )n the uplift pressure at the node i of each cell δxi at the t-step tn = t0 + n · δt, the appropriate Crank-Nicolson finite difference equation is: (hi−1 )n−1 − 2 · (hi )n−1 + (hi+1 )n−1 2 = FN · (hi−1 )n − 2 · (hi )n + (hi+1 )n + 2
(hi )n − (hi )n−1
(4.38)
The non-dimensional factor FN is called the Fourier number and is defined by: FN =
T · δt SC · δx2
(4.39)
Now, the pseudo-exponents n and n + 1 denote the t-step order instead of the iteration cycle index as in the Gauss-Seidel method. The variables T, Sc , δx and δt stand respectively for contact transmissivity, contact storativity, grid cell length and time interval. Applying the finite difference equation at the centre of each contact cell generates a set of 20 simultaneous linear equations that has to be solved for each t-step after t0 either by an iterative routine, easily programmable, or by a more computer-efficient non-iterative method. Keeping always the heads at both edge cells equal to zero, i.e. (h1 )1 = (h20 )1 = 0, the iterative routine formula determines the heads (hi )1 for the first t-step t1 = t0 + δt: (hi )1 =
1 FN · [(hi−1 )0 − 2 · (hi )0 + (hi+1 )0 + (hi−1 )1 + (hi+1 )1 ] + 2 · (hi )0 · 2 1 + FN
© 2010 Taylor & Francis Group, London, UK
(4.40)
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Fractured rock hydraulics
Note that for the Crank-Nicolson numerical scheme, a low Fn number is not required for stability, however it is required for numerical accuracy. The first iteration stops when the greatest relative change after the last iteration cycle is smaller than a prescribed limit ε. For the consecutive t-steps tn = tn−1 + n · δt defined by n = 2, 3, 4 . . . the extreme values must always be equal to zero, i.e. (h1 )n = (h20 )n = 0. Then, the iteration formulas for the subsequent iterative routines are: (hi )2 =
1 FN · [(hi−1 )1 − 2 · (hi )1 + (hi+1 )1 + (hi−1 )2 + (hi+1 )2 ] + 2 · (hi )1 · 2 1 + FN
(hi )3 =
1 FN · [(hi−1 )2 − 2 · (hi )2 + (hi+1 )2 + (hi−1 )3 + (hi+1 )3 ] + 2 · (hi )2 · 2 1 + FN
............................................................................. ............................................................................. ............................................................................. (hi )n+1 =
(hi )n =
1 FN · [(hi−1 )n − 2 · (hi )n + (hi+1 )n + (hi−1 )n+1 + (hi+1 )n+1 ] + 2 · (hi )n · 2 1 + FN
1 FN · [(hi−1 )n−1 − 2 · (hi )n−1 + (hi+1 )n−1 + (hi−1 )n + (hi+1 )n ] + 2 · (hi )n−1 · 2 1 + FN (4.41)
If correctly applied, the Gauss-Seidel method may be use in each iterative routine to find (hi )n at the n t-step. Considering T = 1/3 · 10−6 m2 /s, Sc = 2 · 10−5 , δx = 0.5 m and a time interval δt = 1 s, the corresponding Fourier number is FN = 0.667. After performing 190 iterative routines, the resulting x-h curves for t-steps of 0 s, 20 s . . . 160 s are shown in fig. 4.14. The resulting x-t-h surface corresponding to the uplift pressure decay is shown in fig. 4.15.
h (m)
1.5 1 0.5 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
x (m) Initial head Head after 20 s
Head after 40 s Head after 80 s
Head after 160 s
Figure 4.14 Plot of x-h curves for t-steps of 0 s, 20 s . . . 160 s. Note that for homogeneous parameters this problem is symmetric. This means that it could be solved considering only half the pavement and prescribing zero-flow at x5 (Neumann boundary condition).
© 2010 Taylor & Francis Group, London, UK
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143
1.5
h (m)
1
0.5
0
3 2
2 4
x (m 6 )
1 8
in)
t (m
0
Figure 4.15 Plot of the x-t-h surface for t-steps continuously running from 0 s to 190 s.
4.3 Finite differences algorithms for fractured rock masses 4.3.1
Pr elim i n a r i e s
Finite differences algorithms for general anisotropic models are simple to deduce from basic concepts and, for practical purposes, may reasonably simulate the hydraulic behaviour of anisotropic rocks including their discontinuities even if geometrically complex. This section presents some 3D finite difference algorithms to solve direct problems regarding groundwater flow through randomly fractured rock masses. Related 2D and 1D algorithms are easily derived from these 3D solutions. Whenever possible, it is convenient to reduce an intricate fractured rock mass system into an assemblage of quasi-homogeneous subsystems having quasi-uniform characteristics and dominant anisotropy. Then, depending on the desired details, each quasi-homogeneous subsystem may be further subdivided into smaller subsystems but preserving common properties. To refine even more, it is always possible to individualise properties for each elementary subsystem. It is important to keep in mind before applying finite difference algorithms that, as any system is part of a larger one, its boundary conditions must replicate the flow
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Fractured rock hydraulics
regime connecting both systems to preserve mass input-output balance. Wrong assumptions may produce an apparently reliable pattern of equipotentials but lead to incorrect predictions of hydraulic gradients and flow rates. Besides acceptable boundary and initial condition hypotheses, practical solutions of direct problems require a sound knowledge of the spatial distribution of the hydraulic anisotropic transmissivity [T] or conductivity [k].
4.3.2
S t ea dy-s t a t e s o l u t i o n s
4.3.2.1 D u pu i t’s a p p r ox i m a ti on 4.3.2.1.1
U N C O N F I N E D F L OW
Assume the following premises: • • •
Pervious media anisotropic and inhomogeneous referred to an orthogonal frame xyz (axis z: zenithal). Subsystems or cells defined by upright prismatic columns but having gentle inclined bases (see fig. 4.16). Hydraulic gradients estimated by finite differences (this is equivalent to approximating local hydraulic heads by a second order interpolation polynomial collocated at the centre of mass of the middle subsystem and of the adjacent subsystems). Consider the following list of symbols:
• • • • • •
i, j: indices of the coordinate (xi , yj ) of the central point of the middle cell (vertical z-axis passes through this point). i − 1: index one step to the west of i. i + 1: index one step to the east of i. j − 1: index one step to the south of j. j + 1: index one step to the north of j. [k] (LT−1 ): anisotropic hydraulic conductivity, defined by the second order tensor:
k=
•
kxx kyx
kxy kyy
J (−): local hydraulic gradient, defined by the vector:
J J= x Jy
•
(4.42)
(4.43)
q (LT−1 ): local specific discharge, defined by the vector:
qx q= qy
qxx + qxy = qyx + qyy
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(4.44)
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Previous WT Water accretion ω Free water table WT δH Saturated column q Impervious bottom
H
Datum B
δxi
Figure 4.16 Subsystem defined by a vertical parallelepiped, of which the base is an inclined parallelogram.
• • • • • • • • • • •
qx (LT−1 ): component of the specific discharge q along the x-direction resulting from the addition of the two parallel subcomponents qxx + qxy . qy (LT−1 ): component of the specific discharge q along the y-direction resulting from the addition of the two parallel subcomponents qyx + qyy . kxx (LT−1 ): component of [k] that responds for the discharge vector qxx parallel to the x-axis and is induced by the hydraulic gradient component Jx . kxy (LT−1 ): component of [k] that responds for the discharge vector qxy parallel to the x-axis and is induced by the hydraulic gradient component Jy . kyy (LT−1 ): component of [k] that responds for the discharge vector qyy parallel to the y-axis and is induced by the hydraulic gradient component Jy . kyx (LT−1 ): component of [k] that responds for the discharge vector qyx parallel to the y-axis and induced is by the hydraulic gradient component Jy . It is important to note that as kxy = kyx this implies qxy = qyx . B (L): base elevation that may have a variable but relatively gentle inclination; referred to an arbitrary horizontal datum. H (L): hydraulic head measured from the base elevation B; if measured from the arbitrary horizontal datum: Hdatum = B + H. δx (L): base dimension of the prismatic column along the x-direction. δy (L): base dimension of the prismatic column along the y-direction. Qx (L3 T−1 ): discharge across the vertical x-cross-section: Qx = qx · H · δy.
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• • •
Fractured rock hydraulics
Qy (L3 T−1 ): discharge across the vertical y-cross-section: Qy = qy · H · δx. Qx,y (L3 T−1 ): sum of discharges Qx and Qy : Qx,y = Qx + Qy . ω (LT−1 ): vertical unit-area recharge.
Assuming constant water density, it is possible to deduce the finite difference algorithm by the direct application of the continuity principle. Consider the coordinates (xi , yj ) of the central point of the prismatic cell conveniently defined by the indices (i, j). Steady flows Qi and Qj continuously traverse the orthogonal cross-sections H · δyi and H · δxj of this cell. Total Qi,j flow is Qi,j = Qi + Qj and according to the Darcy’s law may be calculated by: ∂ (B + H) ∂x (kxy )i,j 1 i,j · Hi,j · · (kyy )i,j 1 ∂ (B + H) ∂y i,j
Qi,j = −
δy 0
0 (kxx )i,j · δx (kyx )i,j
(4.45)
Equation 4.45 can be expanded as: ∂ B (k ) · δy (k ) · δy ∂x i,j xy i,j xx i,j · · · (k ) · δx (k ) · δx · ∂ yy i,j yx i,j B 1 ∂y i,j ·
Qi,j = −Hi,j · 1 ∂ H (kxx )i,j · δy (kxy )i,j · δy ∂x i,j + ·
(kyx )i,j · δx (kyy )i,j · δx ∂ H ∂y i,j
(4.46)
Recalling the derivation rules and substituting H2 with V, the non-linear terms can be “linearised’’. Then, equation 4.46 takes the form:
∂ (Vi,j ) · B (k ) · δy (k ) · δy ∂x i,j xy i,j xx i,j
· · · (k ) · δx (k ) · δx · ∂ 1 yy i,j yx i,j (Vi,j ) 2 · B 1 ∂y i,j ·
Qi,j = − 1 ∂ 1 V · ∂x (kxx )i,j · δy (kxy )i,j · δy 2 i,j
+ · (kyx )i,j · δx (kyy )i,j · δx 1 ∂ · V 2 ∂y i,j
1 2
(4.47)
In the immediate neighbourhood of the central prismatic cell at (i, j) are four prismatic cells located at (i − 1, j) and (i + 1, j) plus (i, j − 1) and (i, j + 1). The four adjacent faces are defined at the middle intervals (i + 1/2, j) and (i − 1/2, j) plus (i, j + 1/2) and (i, j − 1/2). To approximate the discharges Qi+1/2,j , Qi−1/2,j , Qi,j+1/2 and Qi,j−1/2
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147
that traverse these faces by difference equations, one must apply the rules discussed in section 4.2.2 of this chapter. Now, continuity requires for each cell at (i, j) that:
Qi+ 1 ,j 2
−Qi− 1 ,j + Qi,j+ 1 2 2
−Qi,j− 1 − ωi,j · δx · δy = 0 2
(4.48)
Approximating these discharges by their corresponding finite difference equations in equation 4.48, collecting and solving for Vi,j , one obtains an appropriate algorithm for a Gauss-Seidel iteration routine: (Vi,j )k+1 = [(ADU )i,j ]−1 · [(BDU )i,j ]k
(4.49)
In the above equation, k stands for the iteration cycle. For each cell (i, j), the term [ADU ]i,j remains constant and is calculated by:
1 1
2 · δx 2 · δx 1 (ADU )i,j = (CRU )i,j · + (CLD )i,j · · 1 1 1 2 · δy
(4.50)
2 · δy
The coefficients [CRU ]i,j and [CLD ]i,j may be obtained from arithmetic means involving the adjacent cells:
(kxx )i+1,j + (kxx )i,j
2 (CRU )i,j = (k ) yx i+1,j + (kxy )i,j 2
(kxx )i,j + (kxx )i−1,j
2 (CLD )i,j = (k ) + (k ) yx i,j yx i−1,j 2
· δy · δx
· δy · δx
· δy 2 (kyy )i,j+1 + (kyy )i,j · δx 2
(kxy )i,j+1 + (kxy )i,j
(kxy )i,j + (kxy )i,j−1 2 (kyy )i,j + (kyy )i,j−1 2
· δy · δx
(4.51)
(4.52)
Other types of means may be used. However, as pointed out in section 4.2.3, the arithmetic mean better fit pseudo-continuous solutions for fractured rock masses.
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Fractured rock hydraulics
For each cell (i, j) the term {[BDU ]i,j }k changes after each k iteration until the solution stabilises. It is calculated by:
1 (Vi+1,j )k 2 · δx (CRU )i,j · . . . 1 (Vi,j+1 )k · k 2 δy 1 (BDU )i,j = · k 1 1 (Vi−1,j ) 2 · δx + (C ) · LD i,j 1 (Vi,j−1 )k · 2 δy (Bi+1,j )k − (Bi−1,j )k 1 k 2 · δx 1 · (CC )i,j · . . . V2 · + i,j (Bi,j+1 )k − (Bi,j−1 )k 1 2 · δy + ωi,j · δx · δy
(4.53)
Note the term [V1/2 ]i,j in the second expression of the right hand side. Boundary conditions must also be written in terms of V instead of H. For each cell (i, j) the term [CC ]i,j of equation 4.53 remains constant and is calculated by: (kxx )i,j · δy (kxy )i,j · δy (CC )i,j = (4.54) (kyx )i,j · δx (kyy )i,j · δx After solving the problem, the solution in V is transformed back to H, i.e. H = V1/2 . Example 4.3, a simple exercise, helps to understand the use of the algorithm.
Example 4.3 Imagine a pervious sandstone layer, 250 m thick, gently dipping over an almost impervious shale layer in an intertropical zone. Subvertical fractures split the upper layer in many large mosaics (see fig. 4.17). The pervious upper layer is subjected to copious and persistent rains that attain more than 2000 mm/year and last for several months during the seasonal rainfall. Almost 10 to 20% of the total rainfall infiltrates deeply into this layer during the wet months. Due to the intense chemical attack in this hot climate, soluble elements of the weathered cap are progressively removed and transported by the groundwater flow to natural exit springs at the base of the upper layer. If these subvertical fractures are not very pervious, compared to the permeability of the remaining rock mass, these springs are dispersed around the sandstone-shale contact periphery, mostly in the re-entrant forms of the relief. However, due to long-term weathering these fractures become progressively eroded, more pervious and “capture’’ an ever growing part of the groundwater flow. By progressive backward erosion along the fracture traces, springs gradually concentrate at
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Finite differences
149
the lowest points of the subvertical fractures. This geodynamic process has positive feedback. To analyse the effect of the contrast of the fracture transmissivity to the rock mass hydraulic conductivity, it was judged convenient to simulate the steady-state behaviour of the groundwater flow for different conditions.
8000 620 670
N (m)
6000
645
695
4000
720
2000
0 0
2000
4000
6000
8000
10000
12000
14000
E (m)
Figure 4.17 Gentle dipping pervious dominant sandstone layer 250 m thick over an almost impervious shale layer. Subvertical fractures split the upper layer in many large mosaics.
The main input data for this problem are: •
•
Boundary conditions: A very pervious subvertical fracture delimits the north boundary of the cap layer. Suppose that three piezometers located in this fracture gave a seasonal series of WT measurements during the wet season. Then, it was possible to approximate an average Dirichlet condition at this border (see fig. 4.18). Dirichlet conditions for the remaining boundaries were taken at each point (i, j) as having the variable ground elevation (top of the shale layer). Average unit-area vertical recharge: – An exceptional continuous recharge was simulated: w = 14.964 mm/day
•
Hydraulic conductivity of the top layer: – Eigenvalues: 3 × 10−6 m kR = 1 × 10−6 s
© 2010 Taylor & Francis Group, London, UK
(4.55)
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Fractured rock hydraulics
WT level (m)
900
800
700
600
0
5 103
1 104
1.5 104
X (m)
Figure 4.18 Approximate Dirichlet condition at the very pervious subvertical fractures at the north border of fig. 4.17.
– Eigenvectors: Direction of max kR : along 106◦ clockwise from the y-axis. Direction of min kR : perpendicular to max kR . – Hydraulic conductivity of any cell (i, j) referred to the xyz frame:
2.848 × 10−6 (k R )i,j = −5.299 × 10−7 •
•
−5.299 × 10−7 1.152 × 10−6
m s
(4.56)
Fracture data: Numerical simulations were made for four cases (discussed in more detail later in this example). Data concerning the fractures for case 1 are condensed in table 4.1. Model discretisation: The model was discretised into 140 × 80 = 11,200 prismatic cells. The subvertical fractures were also discretised so as to best fit these cells. It is important to note that it is always possible to discretised curved traces for Dupuit’s approach or fracture surfaces for a true 3D model. However, in these cases the transmissivity tensor varies locally with the trace xy-slope or the geometry of the surface elements and the modeller has more work to do. The equivalent hydraulic conductivity of each cell of the sandstone layer was estimated according to the methodology explained in Chapter I. Therefore, the diagonal and off-diagonal components of the resulting equivalent hydraulic conductivity for each cell (i, j) were estimated by the formulas below. In these formulas, the indices (i, j) are purposely suppressed for ease of reading: – Diagonal terms:
Kxx = (kR )xx + Kyy = (kR )yy +
9 n=0
9 n=0
© 2010 Taylor & Francis Group, London, UK
[(kF )xx ]n .
(4.57)
[(kF )xx ]n .
(4.58)
Finite differences
Table 4.1
Case 1 fracture number.
Fracture start Fracture and end number coordinates X (m) 0 1 2 3 4 5 6 7 8 9 10
151
Start End Start End Start End Start End Start End Start End Start End Start End Start End Start End Start End
1737 11760 6518 4125 905.3 10580 4450 10570 12790 11040 5821 7250 10580 10130 13480 1735 9388 7724 9359 9195 9400 9566
Y (m) 8400 8400 8397 4245 6985 7226 1786 7211 2444 6257 8400 1234 7255 8400 3537 8404 4070 1134 4056 842.37 5324 8400
– Off-diagonal terms: 9 [(kF )xx ]n . Kxy = (kR )xy + Kyy = Kxy
Slope of the in the xy plane fracture trace (◦ positive counterFracture clockwise) length (m)
Isotropic fracture transmissivity (m2 /s)
0.00
10100
2.73E-03
60.26
4837
1.31E-03
1.18
9702
2.63E-03
41.52
8147
2.21E-03
−65.23
4295
1.16E-03
−79.00
7335
1.99E-03
−65.56
1208
3.27E-04
−22.55
12780
3.46E-03
60.46
3448
9.33E-04
86.53
3306
8.95E-04
86.31
3106
8.41E-04
(4.59)
n=0
(4.60)
In these formulas, index R stands for rock mass. It was assumed that the hydraulic conductivity of the rock mass implicitly includes the effect of the minor discontinuities. Index F stands for fracture and sub-index n for the fracture number for a total of 10 fractures (see table 4.1). To get better results, the extremities of the parallel segments pertaining to the same fracture were joined by extra cells so as to improve continuity (see fig. 4.19).
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Fractured rock hydraulics
8000
y (m)
6000
4000
2000
0 0
1·104
5000 x (m)
Figure 4.19
•
Final outline of the cells containing fractures. Shades are proportional to the geometric mean of the eigenvalues of each cell (numerical values are not shown). Note the junction of the extremities of the parallel segments pertaining to the same fracture by extra cells.
Simulation results: Simulations were run for four cases considering different values for fracture transmissivities as presented in table 4.2. Note for cases 1 to 3 the very high contrast between the fracture transmissivities and the rock mass hydraulic conductivity.
Table 4.2 Simulated cases classified by the maximum and minimum ratio between fracture transmissivity (m2 /s) and the geometric mean of the sandstone eigenvalues (m/s). Relation Relation [maximum fracture transmissivity (m2 /s)]/ [minimum fracture transmissivity (m2 /s)]/ Case [geometric mean of kR eigenvalues (m/s)] [geometric mean of kR eigenvalues (m/s)] 0 1 2 3
1 2.015 · 103 m 1.832 · 104 m 1.832 · 105 m
1 190.526 m 1.732 · 103 m 1.732 · 104 m
Simulation results are presented in fig. 4.20, fig. 4.21, fig. 4.22 and fig. 4.23. For cases 2 and 3 it was necessary to modify the parameters resulting from equation 4.59 to guarantee a stable algorithm (see discussion on page 155).
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1330 m 8000
1270 m 115 0
105 0 9 85 50 0
0 125
50 10
500
1150 m
76 5
6000
0 85 0 95
12 50
50
11
1090 m 1030 m
1050
500 785
4000
970 m
950
y (m)
1210 m 0 75
1150
910 m 850 m
50 10
50
850 750
0
12
95
790 m
2000 1150 850 95 0
730 m 1050
670 m 610 m
0 0
2000
4000
6000
8000
10000
12000
14000
550 m
x (m)
Figure 4.20 Case 0 – Groundwater flow pattern without fractures to allow comparison of the effect of growing transmissivity values. Compare with fig. 4.21, fig. 4.22 and fig. 4.23.
1210 m
8000
11
6000
95 0 765 500 85 0
105
0
1150 m 1090 m
0
75
1030 m
0
50
85
50 11
0
4000
910 m
1050
850 m
85
0
950
850 950 1050
y (m)
970 m 75
1150
2000
95 8500
790 m
750
730 m 1050
670 m 610 m
0 0
2000
4000
6000
8000
10000
12000
14000
550 m
x (m)
Figure 4.21 Case 1 – Groundwater flow pattern for the data presented in table 4.1. The maximum and minimum ratio between the fracture transmissivity (m2 /s) and the geometric mean of the sandstone eigenvalues (m/s) was, respectively, 2.015 · 103 m and 190.526 m.
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Fractured rock hydraulics
1030 m
8000 850
970 m 0 750
850
65
6000
75
950
0 95
85 0 950
790 m 730 m
950
y (m)
850 m
0 85
850
950
0 75
4000
910 m
0
2000 850
670 m
95
0
610 m 0 0
2000
4000
6000
8000
10000
12000
14000
550 m
x (m)
Figure 4.22 Case 2 – Groundwater flow pattern considering the same data presented in table 4.1 but now for ratios respectively equal to 1.832 · 104 m and 1.732 · 103 m, i.e. 1000 times more pervious than case 1.
8000
1000 m 780
780
780780
780
0
780
880 m
880 780
820 m
780
4000 780
N (m)
680
68
68
6000
940 m
0
880
0
0
680
980
2000
760 m
88
88
880
700 m
880 780
780
640 m 0 0
2000
4000
6000
8000
10000
12000
14000
580 m
E (m)
Figure 4.23 Case 3 – Groundwater flow pattern considering the same data presented in table 4.1 but now for ratios respectively equal to 1.832 · 105 m and 1.732 · 104 m, 1000 times more pervious than case 2. In this figure the fracture xy-traces are located in their geographic correct position to show the effect of the bias introduced by locating each fracture in the middle of the prismatic cells.
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Now, it is time to consider the question of the stability of the algorithm condensed in equation 4.49. In effect, as the contrast between the hydraulic transmissivity of the discontinuities and the hydraulic conductivity of the rock mass, implicitly including minor discontinuities, grows, the finite algorithm may become unstable depending on the cell dimensions, the number of fractures and, most of all, on the interplay between the eigenvectors of the rock mass hydraulic conductivity kR and the eigenvectors of the fracture hydraulic transmissivities kF . Without a doubt, instabilities are only caused by the off-diagonal terms [kR ]xy and [kF ]xy and arise at the neighbourhood of cells signalled by heavily contrasted hydraulic conductivities. In example 4.3, they arose for cases 2 and 3. Results presented in Figure 4.20, case 0, and Figure 4.20, case 1, correspond to simulations constructed without any stability problem. Fortunately, without appreciable harm evidenced by a smoothing effect on the equipotential contours near the critical cells, these instabilities are easily cured. In fact, after computing Kxy by equation 4.59, it is sufficient to test and substitute for Kxy the maximum value between [kR ]xy and Kxy :
selected_Kxy = max[(kR )xy , Kxy ].
(4.61)
It is important to recall that the off-diagonal terms of the hydraulic conductivity tensor of each cell, i.e. [kR ]xy and [kF ]xy , are usually small if compared to the corresponding diagonal terms. In addition, [kR ]xy and [kF ]xy may be positive or negative quantities totally independent of each other. For that reason, equation 4.61 must be applied to relative values, not for absolute values. Sometimes both terms, i.e. [kR ]xy for the rock mass and [kF ]xy for individual fractures, may be positive or negative and sometimes one of them is positive and the other negative. In fact, for a group of fractures within the same cell, the resulting summation [kF ]xy for all fractures of that group may be very low, sometimes lower than [kR ]xy , despite the fact that absolute values of the individual terms [kF ]xy of each fracture may occasionally be very high. Therefore, [kR ]xy and the sum Kxy = {[kR ]xy + [kF ]xy } may be of the same order but set apart by too many dissimilar eigenvectors. Finally, it must be said that the anisotropic character of the resulting tensor, mostly dependent on the numerical values of the tensor diagonal elements, is relatively preserved by the recommended correction. Then, by observing the stability criterion condensed in equation 4.61, it is possible to analyse an intricate group of vertical fractures. As an example, consider the 20 intercrossed fractures shown in fig. 4.24. To enhance the effect of the fracture transmissivities, the rock mass intrinsic permeability kR was considered isotropic and set to an extremely low value: 10−30 m/s. On the other hand, the maximum and minimum fracture transmissivities were set very high: 8.749 · 1010 m2 /s and 1.174 · 107 m2 /s, respectively. As these values differ by more than 1040 from kR one may be tempted to set it to zero. This is not a good idea, as it leads to numerical overflow. Fig. 4.25 shows the contour plot of the simulated head distribution and the border of the impervious base. Fig. 4.26 shows the contour plot of the total discharge Q = QE + QN that traverses each cell.
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Fractured rock hydraulics
1.104
8000
N (m)
6000
4000
2000
0 0
4
1·10 E (m)
5000
4
2·104
1.5·10
Figure 4.24 Final outline of 20 cells containing fractures. Shades are proportional to the geometric mean of the eigenvalues of each cell (numerical values are not shown). To get better results, the extremities of the segments pertaining to the same fracture were joined by extra cells. Cell dimensions: 200 m × 200 m. 1.104 8000
N
6000 4000
2000 0 0 5000 1·104 E
1.5·104 2·104
Figure 4.25 Contour plot of the simulated head distribution. Note the border of the inclined impervious base. Average water column above base was 245 m.
4.3.2.1.2
C O N F I N E D F L OW
Assume the same premises as for unconfined flow. Consider also the same list of symbols with the following modifications: • • •
Hc (L): thickness of the confined layer measured from the base elevation B. Qx (L3 T−1 ): discharge across the vertical x-cross-section: Qx = qx · Hc · δy. Qy (L3 T−1 ): discharge across the vertical y-cross-section: Qy = qy · Hc · δx.
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Finite differences
157
10000
8000
N (m)
6000
4000
2000
0 0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
E (m)
Figure 4.26 Contour plot of the total discharge Q = QE + QN that traverses each cell containing fractures. Numerical values not plotted.
•
(LT−1 ): unit-area vertical leakage or infiltration at the top and/or at the base of the confined layer: both proportional to the head difference between the confined layer and the base of the overlying layer and/or the top of the underlying layer.
For constant water density, the total flow Qi,j across a prismatic cell estimated by the Darcy’s law is: ∂ (B + H) ∂x (kxy )i,j 1 i,j · (Hc )i,j · · (kyy )i,j 1 ∂ (B + H) ∂y i,j
Qi,j = −
δy 0
0 (kxx )i,j · (kyx )i,j δx
(4.62)
Recall that B + H = B + (HC + P/ρg). Continuity requires for each cell:
Qi+ 1 ,j 2
−Qi− 1 ,j + Qi,j+ 1 2 2
−Qi,j− 1 − · δx · δy = 0 2
(4.63)
Approximating discharges by the corresponding finite difference equations in equation 4.63, collecting and solving for Hi,j , one obtains an appropriate algorithm for a GaussSeidel iteration routine: (Hi,j )k+1 = [(ADC )i,j ]−1 · [(BDC )i,j ]k In the above equation, k stands for the iteration cycle.
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(4.64)
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Fractured rock hydraulics
For each cell (i, j), the constant term [ADC ]i,j is calculated by:
1 1
δx δx 1 (ADC )i,j = (HC )i,j · (CRU )i,j · + (CLD )i,j · · 1 1 1 δy
(4.65)
δy
For each cell (i, j) the varying term {[BDC ]i,j }k is calculated by:
1 (Hi+1,j )k 1 (Hi−1,j )k 2 · 2 · δx δx 1 [(BDC )i,j ]k = (HC )i,j · (CRU )i,j · ) · +(C ... . LD i,j 1 (Hi,j+1 )k 1 (Hi,j−1 )k 1 · · 2 δy 2 δy (Bi+1,j )k − (Bi−1,j )k 2 · δx 1 +(HC )i,j · (CC )i,j · · + i,j · δx · δy (Bi,j+1 )k − (Bi,j−1 )k 1 2 · δy (4.66) The coefficients (CRU )i,j , (CLD )i,j and (CC )i,j keep the same meaning as defined for unconfined flow. 4.3.2.2 3 D a l g or i th m s Assume the same premises as before. Consider now the complementary list of symbols: • • • • • • • •
•
i, j, k: indices of the coordinate (xi , yj , zk ) of the central point of the middle cell (vertical z-axis pass through this point). i − 1: index one step to the west of i. i + 1: index one step to the east of i. j − 1: index one step to the south of j. j + 1: index one step to the north of j. k − 1 (−): index one step to the nadir of k. k + 1 (−): index one step to the zenith of k. [k] (LT−1 ): anisotropic hydraulic conductivity, defined by the second order tensor: kxx kxy kxz k = kyx kyy kyz (4.67) kzx kzy kzz J (−): local hydraulic gradient, defined by the vector: Jx J = Jy Jz
© 2010 Taylor & Francis Group, London, UK
(4.68)
Finite differences
159
ITERATION CYCLE
SLAB 0 SLAB 1
SLAB n SLAB n1
SLAB N
Figure 4.27 Iterative routine cycles are sequentially and orderly performed from the start to the end of the slab pile until convergence. In every cycle, after the iteration routine for slab n has finished, its output at the interface n/n + 1 is entered as an input for slab n+1 before starting its iteration routine. Cycle order may be reversed.
•
q (LT−1 ): local specific discharge, defined by the vector: qxx + qxy + qxz q = qyx + qyy + qyz qzx + qzy + qzz
• • • • • • • • • • • • •
(4.69)
qr (LT−1 ) and r = i or j or k: component of the specific discharge q along the r-direction resulting from the addition of the three parallel subcomponents: qri + qrj + qrk . krs (LT−1 ) and r or s = i or j or k: component of [k] that responds for the discharge vector qrs parallel to the r-axis and induced by the hydraulic gradient component Js . It is important to note that as krs = ksr this implies qrs = qsr . B (L): base elevation that may have a variable but relatively gentle inclination, referred to an arbitrary horizontal datum Hdatum . H (L): hydraulic head measured from the arbitrary horizontal datum Hdatum . δx (L): base dimension of the cuboid cell along the x-direction. δy (L): base dimension of the cuboid cell along the y-direction. δz (L): height dimension of the cuboid cell, measured along the zenithal z-direction. Qx (L3 T−1 ): horizontal discharge across the vertical x-cross-section: Qx = qx · δy · δz. Qy (L3 T−1 ): horizontal discharge across the vertical y-cross-section: Qy = qy · δx · δz. Qz (L3 T−1 ): vertical discharge across the horizontal z-cross-section: Qz = qz · δx · δy. Qx,y,z (L3 T−1 ): sum of all discharge components: Qx,y,z = Qx + Qy + Qz . ω (LT−1 ): vertical unit-area recharge at the phreatic surface. (T−1 ): unit-volume sink or source within the cuboid cell.
© 2010 Taylor & Francis Group, London, UK
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Fractured rock hydraulics
Modelling a true 3D simulation is more laborious than resorting to Dupuit’s approach. Therefore, it only must be used if really necessary. Usually, a true 3D simulation requires splitting the system domain into N+1 slabs that are parallel to the slightly inclined base. These slabs are labelled from 0 to N. Consequently, two adjacent slabs are labelled n and n+1. Assuming constant water density, it is possible to deduce for each slab, recognised by its own label, its proper finite difference algorithm. In addition, for the lowest slab, labelled by 0, and for the highest slab, labelled by N, it is also possible to deduce the subsidiary algebraic equations for the boundary conditions. Then, iterative routine cycles are sequentially and orderly performed from the start to end of the slab pile until convergence. In every cycle, after finishing the iteration routine for slab n, its output at the interface n/n + 1 is entered as an input for slab n + 1 before starting the iteration routine for slab n + 1 (see fig. 4.27). Example 4.4 helps to understand the use of the algorithm. Each problem may require a different approach and the best way to understand 3D modelling principles is by doing this simple exercise.
Example 4.4 A slightly inclined layer of very weathered and very pervious calcareous sandstone, 10 m thick on average, resting on a relative impervious and also inclined base, was contaminated upstream of a remnant of a weathered vertical fault, more pervious than the surrounding rock mass (see fig. 4.28 for more information). To select the best pumping layout, many possibilities were simulated varying the number of pumping wells, the pumping strength, the well locations etc. This example shows the result of one of these simulations (and not the one considered to be the best). The model covered a volume of surface area of 100 m × 100 m divided into three slabs with the following initial arrangement before pumping: – – –
Upper slab: thickness: 2.5 m; elevation 95.0 m at the top and 92.5 m at the “upper slab-mid slab’’ interface; initial thickness δz = 2.5 m. Mid slab: thickness: 5.0 m; elevation 92.5 m at the “upper slab-mid slab’’ interface and 87.5 m at the “mid slab-bottom slab’’ interface; initial thickness δz = 5.0 m. Bottom slab: thickness: 2.5 m; elevation 82.5 m at the “mid slab-bottom slab’’ interface and 85.0 m at the base; initial thickness δz = 2.5 m.
Each slab was divided into 20 × 20 = 100 cells. The quasi-horizontal cross-section of each cell measured 5 m × 5 m, i.e. δx = 5 m and δy = 5 m. However, to easily converge to the final WT configuration, the thickness δz of each slab was progressively contracted to best fit the changing WT boundary condition until convergence. The estimated eigenvalues for the hydraulic conductivity tensor of the weathered calcareous sandstone, including minor discontinuities, were: 1.5 × 10−5 m keigenvalues = 1 × 10−5 (4.70) s 7.5 × 10−6
© 2010 Taylor & Francis Group, London, UK
Finite differences
161
Natural WT elevation before pumping (m)
100 90 Hazardous waste
80 70
95
m
WT elevation: cross-section AB
50 95
WT elevation
Y (m)
60
m
40
95 94.96 94.92 94.88 94.84 94.8
30
0
20
40
60
Length from A to B
20 10 0 0
10
20
30
40
50
60
70
80
90
100
Figure 4.28 Groundwater flows from top-left corner to bottom-right corner. The trace of the remnant of a vertical fault, about 30 times more pervious than the calcareous sandstone, was located downstream of the hazardous waste and allowed faster transport of contaminants along its trace besides the advective transport within the rock mass. In one of the layouts simulated, three heavy pumps, with their filters just traversing the middle slab, were located as indicated in this figure. These pumping filters 5 m long were simulated by tube transmissivities having a vertical permeability 10 times that of the rock mass. Their pumping rates are also indicated. Note the effect of the fault on the equipotentials contouring before pumping (slice AB). Note also the undesirable graphical distortion of the equipotential contouring at the neighbourhood of highly contrasted cells. This distortion may be reduced using a finer grid.
Referred to the xyz frame adopted, the transformed hydraulic conductivity was:
kCS
1.425 × 10−5 = −2.234 × 10−6 −1.357 × 10−7
−2.234 × 10−6 8.323 × 10−6 −4.054 × 10−7
−1.357 × 10−7 m −4.054 × 10−7 s 9.925 × 10−6
Corresponding eigenvalues and eigenvectors are shown in fig. 4.29.
© 2010 Taylor & Francis Group, London, UK
(4.71)
162
Fractured rock hydraulics
90 120
60
150
30
180
0
330
210
240
300 270
60 Latitude 30 Latitude Equator K max K min K int
Figure 4.29 Eigenvalues and eigenvectors of the anisotropic hydraulic conductivity of the calcareous sandstone slabs.
For the fault, the strike transmissivity was taken as 30 times the maximum eigenvalue of the calcareous sandstone. Its dip transmissivity was taken as 80% of the strike transmissivity. Their values were: Tstrike_transmissivity = 4.5 × 10−4 Tdip_transmissivity
m2 s
m2 = 36 × 10−4 s
(4.72)
Referred to the xyz frame adopted the transformed transmissivity for the vertical fault was: 0 1.191 × 10−4 −1.985 × 10−4 2 m Tfault = −1.985 × 10−4 3.309 × 10−4 (4.73) 0 s 0 0 3.6 × 10−4 The average vertical accretion ω simulated corresponded to an infiltration of 3.5 mm/day.
© 2010 Taylor & Francis Group, London, UK
Finite differences
163
To solve this problem in a simpler way, one has to deduce three finite difference algorithms for the continuity condition applied to the upper, middle and bottom slabs. In addition, top and base boundary conditions require two more finite difference algorithms. There are many alternative ways to do this, but the solution presented below gave satisfactory results. See fig. 4.30 to grasp the main points assumed.
WT δz 2.5 m
δz 5.0 m
Hmid
δz 2.5 m Hbase
Figure 4.30 Initial thicknesses of slabs. Heads are calculated at the base, middle of the intermediate slab and at the phreatic.Thickness δz of each slab at each cell was progressively contracted by the same percentage to best fit the changingWT boundary condition until convergence.
Continuity for the mid slab (index k): Consider the central point of the cuboid cell referred to the coordinates (xi , yj , zk ) conveniently defined by the indices (i, j, k). The sum of the three discharges Qi , Qj and Qk traversing the three median cross-sections is Qi,j,k = Qi + Qj + Qk . The total discharge Qi,j,k is calculated by: ∂ H ∂x
i,j,k 1 0 ∂ H 0 · · 1 ∂x i,j,k 1 δx · δy ∂ H ∂x i,j,k
Qi,j,k
(kxx )i,j,k = (kyx )i,j,k (kzx )i,j,k
(kxy )i,j,k (kyy )i,j,k (kzy )i,j,k
(kxz )i,j,k δy · δz (kyz )i,j,k · 0 0 (kzz )i,j,k
0 δx · δz 0
(4.74) The immediate neighbourhoods of the cell cuboid at (i, j, k) are respectively located at (i − 1, j, k) and (i + 1, j, k) plus (i, j − 1, k) and (i, j + 1, k) plus (i, j, k − 1) and (i, j, k + 1). Then, its six adjacent faces are defined at mid-intervals (i + 1/2, j, k) and (i − 1/2, j, k) plus (i, j + 1/2, k) and (i, j − 1/2, k) plus (i, j, k − 1/2) and (i, j, k + 1/2). To approximate the discharges Qi+1/2,j,k , Qi−1/2,j,k , Qi,j+1/2,k , Qi,j−1/2 , Qi,j,k+1/2 and Qi,j,k−1/2 that traverse these six
© 2010 Taylor & Francis Group, London, UK
164
Fractured rock hydraulics
faces by difference equations, one must again apply the rules discussed in section 4.2.2 of this chapter. As done before, continuity requires for each cell at (i, j, k) that: Qi+ 12 ,j,k − Qi− 12 ,j,k . . . = 0. + Qi,j+ 12 ,k − Qi,j− 12 ,k · · · + Qi,j,k+ 12 − Qi,j,k− 12 − i,j,k · ∂x · ∂y · ∂z
(4.75)
Approximating these discharges by corresponding finite difference equations in equation 4.48, collecting and solving for Hi,j,k , one obtains the appropriate iterative algorithm: (Hi,j,k )n+1 = [[(A3D )i,j,k ]n ]−1 · [(B3D )i,j,k ]n
(4.76)
In the above equation, n now stands for the iteration cycle. For each cuboid cell (i, j, k), the term (A3D )i,j,k now changes for each n iteration because the thickness δz is constantly adjusted until the algorithm converges. This term is calculated by:
(A3D )i,j,k = [(CRUT )i,j,k
δy · δzi,j,k + (CLDB )i,j,k ] · 0 0
0 δx · δzi,j,k 0
1 δx 0 1 1 · 1 0 · δy 1 δx · δy 1 δzi,j,k (4.77)
The coefficients (CRUT )i,j,k and (CLDB )i,j,k may be obtained from arithmetic means involving, respectively, the adjacent cells:
(CRUT )i,j,k
(k ) xx i+1,j,k + (k xx )i,j,k (kyx )i,j+1,k + (k yx )i,j,k (kzx )i,j,k+1 + (k zx )i,j,k 2 2 2 (kxy )i+1,j,k + (k xy )i,j,k (kyy )i,j+1,k + (k yy )i,j,k (kzy )i+j,k+1 + (k zy )i,j,k = 2 2 2 (k ) (k (k + (k ) ) + (k ) ) + (k ) xz i+1,j,k yz i,j+1,k zz i,j,k+1 xz i,j,k yz i,j,k zz i,j,k 2
© 2010 Taylor & Francis Group, London, UK
2
2
(4.78)
Finite differences
(CLDB )i,j,k
(k ) xx i−1,j,k + (k xx )i,j,k 2 (k ) xy i−1,j,k + (k xy )i,j,k = 2 (k ) xz i−1,j,k + (k xz )i,j,k
(kyx )i,j−1,k + (k yx )i,j,k 2 (kyy )i,j−1,k + (k yy )i,j,k 2 (kyz )i,j,k−1 + (k yz )i,j,k
2
165
(kzx )i,j,k−1 + (k zx )i,j,k 2 (kzy )i+j−1,k + (k zy )i,j,k 2 (kzz )i,j,k−1 + (k zz )i,j,k
2
2 (4.79)
For each cell (i, j, k) the term [(B3D ) solution stabilises. It is calculated by:
i,j,k ]
n
also changes after each n iteration until the H
(B3D )i,j,k = (CRUT )i,j,k
δy · δz i,j,k · 0 0
0 δx · δzi,j,k 0
δy · δzi,j,k +(CLDB )i,j,k · 0 0
0 δx · δzi,j,k 0
i+1,j,k
δx 0 1 Hi,j+1,k 0 · · 1 · · · δy δx · δy 1 Hi,j,k+1 δzi,j,k H i−1,j,k δx 0 1 Hi,j−1,k 0 · · 1 · · · δy δx · δy 1 Hi,j,k−1 δzi,j,k
+ i,j,k · δx · δy · δzi,j,k
(4.80)
Note that δzi,j,k has indices in both expressions to emphasise its changing character. Continuity for the upper slab (index k + 1): The algorithm for the upper slab was deduced by the same principles used for the mid slab with two exceptions: –
–
First, to get the head WT on the phreatic surface, a virtual upper slab 2.5 m thick was placed above the model. However, only a half-thickness of 2.5 m was effectively considered for computing. Second, for obvious reasons, the vertical component of the total flow through the phreatic was considered null.Then, the corresponding hydraulic gradient was set equal to zero.
The appropriate iterative algorithm is: (WTi,j,k+1 )n+1 = [[(ATOP )i,j,k+1 ]n ]−1 · [(BTOP )i,j,k+1 ]n
© 2010 Taylor & Francis Group, London, UK
(4.81)
166
Fractured rock hydraulics
The terms (ATOP )i,j,k+1 and (BTOP )i,j,k+1 are respectively calculated by:
δy ·
(ATOP )i,j,k+1
= (CRUT )i,j,k+1 ·
δzi,j,k+1 2 0 0
1 δx 1 δzi,j,k+1 1 δx · 0 · · 1 · · · 2 δy 1 0 δx · δy 0 0
0
δzi,j,k+1 δy · 0 2 δzi,j,k+1 + (CLDB )i,j,k+1 · 0 δx · 2 0
·
0 0 δx · δy
0
1 δx 1 δy 1 δzi,j,k+1
1 · 1 1 (4.82)
(BTOP )i,j,k+1
= (CRUT )i,j,k+1 ·
δy ·
δzi,j,k+1 2 0 0
δx ·
0
0
δzi,j,k+1 2
0
0
WT i+1,j,k+1 δx 1 WT i,j+1,k+1 · 1 ··· · 1 δy
δx · δy
0
WT i−1,j,k+1 δzi,j,k+1 δy · 0 0 δx 2 1 WT i,j−1,k+1 δz i,j,k+1 + (CLDB )i,j,k+1 · · · 1 .. 0 0 δx · δy 1 2 middle_Hi,j,k 0 0 δx · δy δzi,j,k+1
+ i,j,k+1 · δx · δy ·
δzi,j,k+1 + ωi,j · δx · δy 2
(4.83)
Continuity for the bottom slab (index k − 1): Continuity for the bottom slab was deduced in the same way as for the top slab with the difference that the no-flow boundary was the base itself. The appropriate iterative algorithm is: (Hi,j,k−1 )n+1 = [[(ABOTTOM )i,j,k−1 ]n ]−1 · [(BBOTTOM )i,j,k−1 ]n
© 2010 Taylor & Francis Group, London, UK
(4.84)
Finite differences
167
The terms (ABOTTOM )i,j,k−1 and (BBOTTOM )i,j,k−1 are respectively calculated by:
(ABOTTOM )i,j,k−1
δzi,j,k−1 δy · 2 = (CRUT )i,j,k−1 · 0
0 δx ·
0
δy ·
+ (CLDB )i,j,k−1 ·
1 0 δx 1 1 · · 1 ··· 0 δy 1 1 δx · δy δzi,j,k
δzi,j,k−1 2 0
δzi,j,k−1 2 0 0
δx ·
0
0
δzi,j,k−1 2
0
1 δx 1 1 · · 1 1 δy
δx · δy
0
0 (4.85)
(BBOTTOM )i,j,k−1
δzi,j,k−1 δy · 2 = (CRUT )i,j,k−1 · 0 0
δx ·
0
0
δzi,j,k−1 2
0
+ i,j,k · δx · δy ·
δx ·
0
δx · δy
0
0
δzi,j,k−1 2
0
0
bottom )i+1,j,k
δx (Hbottom )i,j+1,k 1 · · 1 · · · δy 1 (Hmiddle )i,j,k+1 δzi,j,k (H bottom )i−1,j,k δx 1 · (Hbottom )i,j−1,k · 1 · · · δy 1
δzi,j,k δy · 2 + (CLDB )i,j,k−1 · 0 0
(H
δx · δy
δzi,j,k−1 2
0 (4.86)
Impervious base boundary condition (index k − 1): Phreatic and impervious boundary conditions are a bit more complicated. For the impervious base, the scalar product at point (i, j, k − 1) of its exterior normal and the specific discharge at this same point must be zero, as expected: zi+1,j,k−1 − zi−1,j,k−1 − · ui 2 · δx (qx )i,j,k−1 · ui zi,j+1,k−1 − zi,j−1,k−1 − · uj · (qy )i,j,k−1 · uj = 0 2 · δy (qz )i,j,k−1 · uk uk
© 2010 Taylor & Francis Group, London, UK
(4.87)
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Fractured rock hydraulics
In the above equation, the vectors ui , uj and uk stand for the unit vectors (versors) of the Cartesian axes xyz. Developing this condition gives the appropriate iterative algorithm: (Hi,j,k−1 )n+1 = [[(ANB )i,j,k−1 ]n ]−1 · [(BNB )i,j,k−1 ]n
(4.88)
The terms (ANB )i,j,k−1 and (BNB )i,j,k−1 are respectively calculated by:
(ANB )i,j,k−1
(BNB )i,j,k−1
zi+1,j,k−1 − zi−1,j,k−1 − 2 · δx (k ) xx i,j,k−1 zi,j+1,k−1 − zi,j−1,k−1 (kyx )i,j,k−1 = − · 2 · δy (kzx )i,j,k−1
1 zi+1,j,k−1 − zi−1,j,k−1 − 2 · δx (kxx )i,j,k−1 zi,j+1,k−1 − zi,j−1,k−1 = − · (kyx )i,j,k−1 2 · δy (kzx )i,j,k−1 1
(kxy )i,j,k−1 (kyy )i,j,k−1 (kzy )i,j,k−1
(kxy )i,j,k−1 (kyy )i,j,k−1 (kzy )i,j,k−1
(kxz )i,j,k−1 (kyz )i,j,k−1 · − (kzz )i,j,k−1
0 0 3 2 · δzi,j,k
(4.89)
H i+1,j,k−1 − Hi−1,j,k−1 − 2 · δx (kxz )i,j,k−1 Hi,j+1,k−1 − Hi,j−1,k−1 − (kyz )i,j,k−1 · 2 · δy (kzz )i,j,k−1 Hi,j,k+1 + 4 · Hi,j,k − 2 · δzi,j,k (4.90)
It is important to observe that the vertical gradient Ji,j,k−1 at the impervious base was calculated by the three-point formula, involving the heads WTi,j,k+1 , Hi,j,k and Hi,j,k−1 . WT elevation for steady-state pumping (m) 100
90 Hazardous waste
80
95.4 m 95 m
70
Y (m)
94 .2 m
1 m3/hr
94.6 m
m .2 94
60
94.2 m
1.5 m3/hr 50
1 m3/hr
93.8 m m .2 93
93.4 m
40 94.2 m
93 m
30 92.6 m
20 92.2 m
10
0 0
10
20
30
40
50 X (m)
60
70
80
90
100
Figure 4.31 Contour map of the depressed WT showing the influence of the very pervious fault as a sort of a vertical and very narrow groundwater buffer.
© 2010 Taylor & Francis Group, London, UK
Finite differences
169
Phreatic boundary condition (index k + 1): For the WT the scalar product at point (i, j, k − 1) of its exterior normal and the specific discharge at this same point must equate to the vertical accretion ωi,j : WT i+1,j,k+1 − WTi−1,j,k+1 − · ui 2 · δx (qx )i,j,k+1 · ui 0 · ui WTi,j+1,k+1 − WTi,j−1,k+1 (qy )i,j,k+1 · uj 0 · uj − · u j · − = 0. 2 · δy (q ) w · u · u z i,j,k+1 k i,j k uk
(4.91)
Developing that condition gives the appropriate iterative algorithm: (WTi,j,k+1 )n+1 = [[(ANT )i,j,k+1 ]n ]−1 · [(BNT )i,j,k+1 ]n
(4.92)
Elevation 92 m: head distribution (m) 100
90 Hazardous waste
80
95.4 m 95 m
70 2m 94.
1 m3/hr 94. 2m
50
94.6 m 94.2 m
1.5 m3/hr
1 m3/hr
93.8 m
93 .2 m
Y (m)
60
93.4 m
40 94.4
m
93 m
30 92.6 m
20 92.2 m
10
0 0
10
20
30
40
50 X (m)
60
70
80
90
100
Figure 4.32 Contour map of equipotential at elevation 92 m. As the 3D xyz grid was progressively deformed until equilibrium, H-data at elevation 92 m was interpolated without difficulty from the grid (x, y, z, H) values.
© 2010 Taylor & Francis Group, London, UK
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Fractured rock hydraulics
The terms (ANT )i,j,k+1 and (BNT )i,j,k+1 are respectively calculated by:
(ANT )i,j,k+1
WT i+1,j,k+1 − WTi−1,j,k+1 − 2 · δx (k ) xx i,j,k+1 WTi,j+1,k+1 − WTi,j−1,k+1 · (kyx )i,j,k+1 − = (k ) 2 · δy zx i,j,k+1 1
(kxy )i,j,k+1 (kyy )i,j,k+1 (kzy )i,j,k+1
(kxz )i,j,k+1 (kyz )i,j,k+1 · (kzz )i,j,k+1
0 0 3 − 2 · δzi,j,k−1 (4.93)
Elevation 89 m: head distribution (m) 100
90 Hazardous waste
80
95.4 m 95 m
50
1
93 .4 m
1 m3/hr
94.2 m
92.4 m
.4 94
Y (m)
60
94.6 m
.4 m 94
m
70
93.8 m
m3/hr
93.4 m m .4 93
93 m
40 94.4
92.6 m
m
30
92.2 m 91.8 m
20 91.4 m
10
0 0
10
20
30
40
50
60
70
80
90
100
X (m)
Figure 4.33 Contour map of equipotential at elevation 89 m. H-data interpolated without difficulty from the grid (x, y, z, H) values. (BNT )i,j,k+1
WT i+1,j,k+1 − WTi−1,j,k+1 − 2 · δx (k ) (k ) (k ) WT xx i,j,k+1 xy i,j,k+1 xz i,j,k+1 i+1,j,k+1 − WTi−1,j,k+1 (kyx )i,j,k+1 (kyy )i,j,k+1 (kyz )i,j,k+1 WTi,j+1,k+1 − WTi,j−1,k+1 − · · · − · 2 · δx 2 · δy (kzx )i,j,k+1 (kzy )i,j,k+1 (kzz )i,j,k+1 WTi,j+1,k+1 − WTi,j−1,k+1 · = −4 · (H)i,j,k + (H)i,j,k−1 − 2 · δy 2 · δzi,j,k−1 0 1 + 0 −ωi,j (4.94)
© 2010 Taylor & Francis Group, London, UK
Finite differences
171
Elevation 86 m: head distribution (m) 100
90
Hazardous waste
80
94.8 m
95.4 m 95 m 93.8
94 .8 m
70
m
94.6 m
1 m3/hr
94.2 m
92.8 m 92.8 m
1.5 50
m3/hr
m 94.8
Y (m)
60
93.8 m
1 m3/hr 93.4 m
40
3.8
m
9
93 m
94.8 m
92.6 m
30 94.8
20
92.2 m
m
91.8 m
10
0 0
10
20
30
40
50 X (m)
60
70
80
90
100
Figure 4.34 Contour map of equipotential at elevation 86 m. H-data interpolated without difficulty from the grid (x, y, z, H) values.
It is important to observe that the vertical gradient Ji, j,k+1 at WT was again calculated by the three-point formula, involving the heads WTi, j,k+1 ,Hi, j,k and Hi, j,k−1 . These five conditions produce a system of simultaneous algebraic equations to be solved iteratively. It must be noted that more sophisticated algorithms can be deduced if judged convenient. The results obtained for this example are summarised in the following figures: – – – – – – –
Fig. 4.31: Depressed WT for steady-state pumping. Fig. 4.32: Contour map of equipotential at elevation 92 m. Fig. 4.33: Contour map of equipotential at elevation 89 m. Fig. 4.34: Contour map of equipotential at elevation 86 m. Fig. 4.35: Contour map of total flow at the inclined base. Fig. 4.36: Contour map of total flow at the mid slab. Fig. 4.37: 3D perspective of the pumping system.
© 2010 Taylor & Francis Group, London, UK
172
Fractured rock hydraulics
Impervious bottom: head distribution (m) 100
90
Hazardous waste
80
95.2 m
70 94 .4 m
60
m 94.4
1 m³/hr
94.8 m 94.4 m
Y (m)
1.5 m³/hr 50
94 m 93 .4 m
1 m³/hr
93.6 m
40 94.4
m
93.2 m
30 92.8 m
20 92.4 m
10
0 0
10
20
30
40
50 X (m)
60
70
80
90
100
Figure 4.35 Contour map of equipotential at the inclined base, reflecting the proximity of the lower end of the 5 m pump filter.
Mid slab: 3D cell discharge distribution (m³/hr) 100
90
Hazardous waste
80
1 m3/hr 0.9 m3/hr
70 0.8 m3/hr 0.7 m3/hr
60 Y (m)
0.6 m3/hr 50
0.5 m3/hr 0.4 m3/hr
40
0.3 m3/hr 30
0.2 m3/hr 0.1 m3/hr
20 0 m3/hr 10
0 0
10
20
30
40
50 X (m)
60
70
80
90
100
Figure 4.36 Contour map of total flow traversing the cuboids cell of the mid slab, entering and leaving by its six faces. Note the efficiency of the pumping near the fault by reversing its natural flow and contributing to mitigating and reversing the contaminant transport.
© 2010 Taylor & Francis Group, London, UK
Finite differences
173
0
20 40 60 80
Figure 4.37 3D perspective of the pumping system to show its efficiency. All data reported to the middle of the intermediate slab: head values are not shown. Streamlines are somehow incorrect because anisotropy was not considered in the calculations. Half close isosurface enveloping the three pumps corresponds to head 93.75 m.
z (m)
95
90
85 0
100 80 20
40
60 x (m)
40 20
80 100
60
m)
y(
0
Figure 4.38 Perspective of the self-adjusted grid as to satisfy all continuity and boundary requirements (deformed scales).
As pointed out, the 3D grid was successively adapted to correspond to all requirements (see fig. 4.38). Finally, it must be noted that instabilities did not occur in this exercise. If it happens, the cure is the same as recommended for the Dupuit’s approach in example 4.3.
© 2010 Taylor & Francis Group, London, UK
174
Fractured rock hydraulics
Drawdown (m)
600
595
590
585 20
40
60
80
100
120
Day
Figure 4.39 The theoretical transient underground mine dewatering in an anisotropic pervious media was roughly approximated by a closed solution for a finite linear drain calibrated by five monitored points.
4.3.3 Tr a n s i e n t s o l u t i o n s Transient solutions follow traditional procedures, either explicitly or implicitly. For example, the explicit algorithm for a confined aquifer modelled by Dupuit’s approach is: [(Hc )i,j ]n+1 = [(Hc )i,j ]n · · ·
− (Hi,j )n δx (CRU )i,j · n n (Hi,j+1 ) − (Hi,j ) δy (H )n − (H n i,j i−1,j )
δx [(Hc )i,j ]n · 1 1 δt −(CLD )i,j · (H )n − (H n · · · · · · · + · · ) 1 i,j i,j−1 (SC )i,j δx · δy δy B − B i+1,j i−1,j 2 · δx +(CC )i,j · Bi,j+1 − Bi,j−1 2 · δy + i,j · δx · δy (4.95)
(H
i+1,j )
n
As a first tentative attempt to get a stable explicit algorithm, the Fourier coefficient FNi,j must be restricted to: 0 < FNi,j =
1 1 δt < · (SC )i,j δx · δy 2
© 2010 Taylor & Francis Group, London, UK
(4.96)
Finite differences
175
In any case, a suitable Crank-Nicholson scheme may also be used, as explained in section 4.2.6.3. But now, different and variable Fourier coefficients FNi,j must translate the complexity of inhomogeneous and anisotropic systems and the available experience on this matter, specifically for fractured rocks, is scarce. On the other hand, calibration resorting to observed temporal data series, besides being sometimes difficult if there are too many conflicting data, may be not reliable for time spans other than the monitored. In practice, if time predictions are really needed, a simple time-dependent model may approximately answer current questions for a restricted time span (see fig 4.39). For true 3D simulation the problem is still more complex.
References Bear, J., 1972, Dynamics of Fluids in Porous Media, American Elsevier, New York, NY. Lam, C.Y., 1994, Applied Numerical Methods for Partial Differential Equations, Prentice Hall. Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. 2007, Numerical Recipes: The Art if Scientific Computing, Cambrige University Press, third edition. Varga, R. S., 1962, Matrix Iterative Analysis, Prentice Hall. Wang, H. F. and Anderson, M. P., 1982, Introduction to Groundwater Modeling – Finite Difference and Finite Element Methods, W. H. Freeman and Company.
© 2010 Taylor & Francis Group, London, UK
