Introduction to Modeling of Transport Phenomena in Porous Media

Theory and Applications of Transport in Porous Media

Series Editor: JACOB BEAR Technion - Israel Institute of Technology, Haifa, Israel

1. Horia I. Ene and Dan Polisevski: Thermal Flow in Porous Media, 1987. ISBN 90-277-2225-0

2. Jacob Bear and Arnold Verruijt: Modeling Groundwater Flow and Pollution, 1987. ISBN 1-55608-014-X

3. G. I. Barenblatt, V. M. Entov and V. M. Ryzhik: Theory of Fluid Flows Through Natural Rocks, 1990. ISBN 0-7923-0167-6

4. Jacob Bear and Yehuda Bachmat: Introduction to Modeling of Transport Phenomena in Porous Media, 1990. ISBN 0-7923-0557-4.

Volume 4

Introduction to Modeling of Transport Phenomena in Porous Media

by

Jacob Bear

Albert and Anne Mansfield Chair in Water Resources, Department o/C,ivil Engineering, Technion -Israel Institute o/Technology, Haifa, Israel

and

Yehuda Bachmat

Hydrological Service, Minsitry 0/ Agriculture, Jerusalem, Israel

Kluwer Academic Publishers Dordrecht / Boston / London

Library of Congress Cataloging in Publication Data

Bear, Jacob. Introduction to modeling of transport pheno~ena in porous media I

Jacob Bear, Yehuda Bachmat. p. cm. -- (Theory and applications of transport in porous

media; v. 4) Includes bibliographical references.

1. Porous materials--Permeabi lity--Mathematical models. 2. Transport theory--Mathematical models. I. Bachmat, Y. II. Title. III. Series. TA418.9.P6B43 1990 530.4'25--dc20 89-26707

ISBN-13: 978-0-7923-1106-5 DOl: 10.1007/978-94-009-1926-6

e-ISBN-13: 978-94-009-1926-6

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Kluwer Academic Publishers incorporates the publishing programmes of D, Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.

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Softcover reprint of the hardcover I st edition 1990

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Contents

Preface Xl

List of Main Symbols xvii

A General Theory 1

1 The Porous Medium 3 1.1 Definition and Classification of Porous Media 4

1.1.1 Definition of a porous medium 4 1.1.2 Classification of porous media 6 1.1.3 Some geometrical characteristics of porous media 8 1.1.4 Homogeneity and isotropy of a porous medium 13

1.2 The Continuum Model of a Porous Medium 14 1.2.1 The need for a continuum approach 14 1.2.2 Representative Elementary Volume (REV) 16 1.2.3 Selection of REV 16 1.2.4 Representative Elementary Area (REA) 29

1.3 Macroscopic Values 30 1.3.1 Volume and mass averages 31 1.3.2 Areal averages 34 1.3.3 Relationship between volume and areal averages 34

1.4 Higher- Order Averaging 37 1.4.1 Smoothing out macroscopic heterogeneity 37 1.4.2 The hydraulic approach 38 1.4.3 Compartmental models 39

1.5 Multicontinuum Models 40 1.5.1 Fractured porous media 40 1.5.2 Multilayer systems 42

vi CONTENTS

2 Macroscopic Description of Transport Phenomena in Porous Media 43 2.1 Elements of Kinematics of Continua 44

2.1.1 Points and particles 44 2.1.2 Coordinates 44 2.1.3 Displacement and strain 46 2.1.4 Processes 47 2.1.5 Material derivative 47 2.1.6 Velocities 53 2.1.7 Flux and discharge 55 2.1.8 Gauss' theorem 56 2.1.9 Reynolds' transport theorem 59 2.1.10 Green's vector theorem 62 2.1.11 Pathlines, transport lines and transport functions 62 2.1.12 Velocity potential and complex potential 67 2.1.13 Movement of a front 70

2.2 Microscopic Balance and Constitutive Equations 72 2.2.1 Derivation of balance equations 72 2.2.2 Particular cases of balance equations 78 2.2.3 Constitutive equations 88 2.2.4 Coupled transport phenomena 99 2.2.5 Phase equilibrium 108

2.3 Averaging Rules 115 2.3.1 Average of a sum 116 2.3.2 Average of a product 116 2.3.3 Average of a time derivative 117 2.3.4 A verage of a spatial derivative 119 2.3.5 A verage of a spatial derivative of a scalar satisfying

\12G = 0 124 2.3.6 The coefficient T~ 129 2.3.7 A verage of a material derivative 131

2.4 Macroscopic Balance Equations 132 2.4.1 General balance equation 132 2.4.2 Mass balance of a phase 135 2.4.3 Volume balance of a phase 136 2.4.4 'Mass balance equation for a component of a phase 137 2.4.5 Balance equation for the linear momentum of a phase 138 2.4.6 Heat balance for a phase and for a saturated porous

medium 140

CONTENTS vii

2.4.7 Mass balance in a fractured porous medium 145 2.4.8 Megascopic balance equation 147

2.5 Stress and Strain in a Porous Medium 149 2.5.1 Total stress 150 2.5.2 Effective stress 153 2.5.3 Forces a.cting on the solid matrix 159

2.6 Macroscopic Fluxes 162 2.6.1 Advective flux of a single Newtonian fluid 162 2.6.2 Advective fluxes in a multiphase system 179 2.6.3 Diffusive flux 189 2.6.4 Dispersive flux 196 2.6.5 Transport coefficients 210 2.6.6 Coupled fluxes 226 2.6.7 Macrodispersive flux 228

2.7 Macroscopic Boundary Conditions 230 2.7.1 Macroscopic boundary 232 2.7.2 The general boundary condition 236 2.7.3 ijoundary conditions between two porous media in

single phase flow 239 2.7.4 Boundary conditions between two porous media

in multi phase flow 248 2.7.5 Boundary between two fluids 251 2.7.6 Boundary with a 'well mixed' domain 255 2.7.7 Boundary with fluid phase change 257 2.7.8 Boundary between a porous medium and an overlying

body of flowing fluid 261

3 Mathematical Statement of a Transport Problem 3.1 Standard Content of a Problem Statement

3.1.1 Conceptual model 3.1.2 Mathematical model

3.2 Multicontinuum Models 3.3 Deletion of Nondominant Effects

3.3.1 Methodology 3.3.2 Examples 3.3.3 Concluding r.emarks

263 263 263 264 266 268 269 274 285

viii CONTENTS

B Application 287

4 Mass Transport of a Single Fluid Phase Under Isothermal Conditions 289 4.1 Mass Balance Equations 289

4.1.1 The basic equation 289 4.1.2 Stationary rigid porous medium 293 4.1.3 Deformable porous medium 298

4.2 Boundary Conditions 312 4.2.1 Boundary of prescribed pressure or head 312 4.2.2 Boundary of prescribed mass flux 313 4.2.3 Semipervious boundary 314 4.2.4 Discontinuity in solid matrix properties 315 4.2.5 Sharp interface between two fluids 316 4.2.6 Phreatic surface 318 4.2.7 Seepage face 321

4.3 Complete Mathematical Model 322 4.4 Inertial Effects 323

5 Mass Transport of Multiple Fluid Phases Under Isothermal Conditions 327 5.1 Hydrostatics of a Multiphase System 328

5.1.1 Interfacial tension and capillary pressure 328 5.1.2 Capillary pressure curves 339 5.1.3 Three fluid phases 351 5.1.4 .Saturation at medium discontinuity 355

5.2 Advective Fluxes 357 5.2.1 Two fluids 357

. 5.2.2 Two-phase effective permeability 359 5.2.3 Three-phase effective permeability 366

5.3 Mass Balance ~quations 369 5.3.1 Basic equations 370 5.3.2 Nondeformable porous medium 371 5.3.3 Deformable porous medium 376 5.3.4 Buckley-Leverett approximation 378 5.3.5 Flow with interphase mass transfer 381 5.3.6 Immobile fluid phase 390

5.4 Complete Model of Multiphase Flow 391 5.4.1 Boundary and initial conditions 391

CONTENTS

5.4.2 Complete model 5.4.3 Saturated-unsaturated flow domain

6 Transport of a Component in a Fluid Phase Under

ix

396 398

Isothermal Conditions 399 6.1 Balance Equation for a Component of a Phase 401

6.1.1 The dispersive flux 401 6.1.2 Diffusive flux 405 6.1.3 Sources and sinks at the solid-fluid interface 406 6.1.4 Sources and sinks within the liquid phase 413 6.1.5 Mass balance equation for a single component 417 6.1.6 Variable fluid density and deformable porous medium 423 6.1. 7 Balance equations with immobile liquid 424 6.1.8 Fractured porous media 428

6.2 Boundary Conditions 430 6.2.1 Boundary of prescribed concentration 430 6.2.2 Boundary of prescribed flux 430 6.2.3 Boundary between two porous media 431 6.2.4 Boundary with a body of fluid 431 6.2.5 Boundary between two fluids 432 6.2.6 Phreatic surface 433 6.2.7 Seepage face 433

6.3 Complete Mathematical Model 434 6.4 Multicomponent systems 436

6.4.1 Radionuclide and other decay chains 437 6.4.2 Two multicomponent phases 438 6.4.3 Three multicomponent phases 441

7 Heat and Mass Transport 449 7.1 Fluxes 450

7.1.1 Advective flux 450 7.1.2 Dispersive flux 7.1.3 Diffusive flux

7.2 Balance Equations 7.2.1 Single fluid phase 7.2.2 Multiple fluid phases 7.2.3 Deformable porous medium

7.3 Initial and Boundary Conditions 7.3.1 Boundary of prescribed temperature

455 456 459 459 461 468 470 471

x

7.3.2 Boundary of prescribed flux 7.3.3 Boundary between two porous media 7.3.4 Boundary with a 'well mixed' domain 7.3.5 Boundary with phase change

7.4 Complete Mathematical Model 7.5 Natural Convection

CONTENTS

471 471 472 472 473 474

8 Hydraulic Approach to Transport in Aquifers 481 8.1 Essentially Horizontal Flow Approximation 482 8.2 Integration Along Thickness 485 8.3 Conditions on the Top and Bottom Surfaces 488

8.3.1 General flux condition on a boundary 489 8.3.2 Conditions for mass transport of a single fluid phase 490 8.3.3 Conditions for a component of a fluid phase 491 8.3.4 Heat 492 8.3.5 Conditions for stress 493

8.4 Particular Balance Equations for an Aquifer 494 8.4.1 Single fl.uid phase 494 8.4.2 Component of a phase 498 8.4.3 Fluids separated by an abrupt interface 499

8.5 Aquifer Compaction 504 8.5.1 Integrated fl.ow equation 504 8.5.2 Integrated equilibrium equation 508

8.6 Complete Statement of a Problem of Transport in an Aquifer 514 8.6.1 Mass of a single fluid phase 514 8.6.2 Mass of a component of a fluid phase 515 8.6.3 Saturated-unsaturated mass and component transport 516

References 517

Problems 522

Index 541

Preface

The main purpose of this book is to provide the theoretical background to engineers and scientists engaged in modeling transport phenomena in porous media, in connection with various engineering projects, and to serve as a text for senior and graduate courses on transport phenomena in porous media. Such courses are taught in various disciplines, e.g., civil engineering, chemical engineering, reservoir engineering, agricultural engineering and soil science. In these disciplines, problems are encountered in which various extensive quantities, e.g., mass and heat, are transported through a porous material domain. Often the porous material contains several fluid phases, and the various extensive quantities are transported simultaneously throughout the multiphase system. In all these disciplines, management decisions related to a system's development and its operation have to be made. To do so, the 'manager', or the planner, needs a tool that will enable him to forecast the response of the system to the implementation of proposed management schemes. This forecast takes the form of spatial and temporal distributions of variables that describe the future state of the considered system. Pressure, stress, strain, density, velocity, solute concentration, temperature, etc., for each phase in the system, and sometime for a component of a phase, may serve as examples of state variables.

The tool that enables the required predictions is the model. A model may be defined as a simplified version of the real (porous medium) system that approximately simulates the excitation-response relations of the latter. The first, and, perhaps, the most important step in the modeling process is the construction of a conceptual model of the system and of its behavior. This step includes a verbal description of the system's composition, the physical phenomena that take place and the mechanisms that govern them, as well as the relevant properties of the medium in which they occur, all subject to the required output of the model. The description takes the form of a set of assumptions, subjectively selected by the modeler, to express (in words) his understanding and approximation of the real system and the

xi

xii PREFACE

processes taking place within it, for the purpose of providing management with information in a particular case of interest. Because the model is a subjective, simplified version of the real system, no unique model exists for a given porous medium system. Different sets of simplifying assumptions, each suitable for a particular task of management, will result in different models.

The next step, often referred to as mathematical modeling of the problem, involves the representation of the conceptual model in the form of mathe­matical relationships. The solution of the mathematical model, a task that requires an appropriate algorithm, yields the required forecast.

In principle, mathematical models of problems can be stated and solved at the microscopic level, in terms of values of variables specified at points in­side each of the phases, whether fluid or solid, that occupy a porous medium domain. Obviously, this approach is not feasible, since it is impossible to describe in detail the geometry of the interphase boundaries, and to observe and measure quantities at this level. Therefore, the mathematical model of a transport problem is simplified by transforming it from the microscopic level, to a macroscopic, or continuum one, where it is formulated in terms of measurable variables that are averages of microscopic quantities.

The main objective of this book is to present, in a systematic way, a methodology for constructing mathematical models of transport problems in a porous medium domain on the basis of the continuum approach.

The book is .divided into two parts: Part A, consisting of three chapters, presents the general theory of modeling transport phenomena in a porous medium domain. The first chapter presents the continuum approach, defines microscopic and macroscopic levels of description, introduces the concept of spatial averaging and defines some macroscopic characteristics of porous media and their constituents. Chapter 2 continues to develop this approach by presenting a description of transport phenomena at the microscopic level, and by deriving averaging rules for transforming the microscopic level of description of any transport problem to a macroscopic one, in which all state variables and parameters are macroscopic quantities. The various laws that govern the macroscopic fluxes of extensive quantities in a porous medium domain, are also derived. Thermodynamic concepts and quantities are introduced whenever necessary. Finally, by adding macroscopic initial and boundary conditions, a complete mathematical statement, or mathematical model, of any transport problem at the macroscopic level of description is obtained. This is discussed in Chapter 3.

In Part B, the general theory presented in Part A is applied to specific

PREFACE xiii

problems of transport of mass and volume of a phase, mass of a component and heat in single and multiphase fluid systems in a porous medium domain. The appropriate models are developed at the macroscopic level.

The mathematical models developed in this part are applicable to prob­lems such as water flow and transport of pollutants in aquifers and in the unsaturated zone, flow of oil, water and gas in petroleum reservoirs, ra­dioactive waste disposal in deep geological formations, land subsidence due to pumping from aquifers, heat storage in aquifers and solute transport in reactors in the chemical industry. Only a limited number of basic models are considered. However, the examples presented should also enable the reader to develop models for other applications.

In each case, the discussion leads to the construction of a complete math­ematical statement of the problem, in terms of appropriate macroscopic state variables and parameters. Problems in which several extensive quantities, such as mass and heat, are transported simultaneously, are also discussed. The hydraulic approach, which, under certain circumstances, is a very useful approximation for treating transport phenomena in relatively thin layers, is also presented.

By mastering the material presented in this book, one should be able to formulate the complete mathematical statement of any problem of transport in any porous medium domain as a well-posed problem. Explicitly one should be able to carry out the following tasks:

(a) Formulate the conceptual model that underlies the mathematical de­scription of the various transport phenomena in the considered porous medium domain.

(b) Express the flux of any extensive quantity of any phase in terms of macroscopic state variables and transport coefficients.

(c) Formulate the closed set of equations that is required in order to solve any given transport problem in a specified porous medium domain, given the necessary source functions and constitutive relations.

(d) Formulate the initial and boundary conditions of the given transport problem, that are necessary and sufficient to guarantee a unique solu­tion of the equations of (c). These equations, together with initial and boundary conditions constitute the complete mathematical model of the problem.

Although we have repeatedly reiterated that our main objective is to present a mathematical description of a physical system and processes of

xiv PREFACE

interest, the amount of mathematical background required of the reader is minimal: basic calculus, some elements of vector and tensor analysis and some elements of probability theory. Throughout the book, it is emphasized that mathematical symbols and relationships are employed only as a compact and often simplified (or approximate) tool for the quantitative description of the physical reality of a given problem. The need to understand the meaning of the various coefficients that appear in the (macroscopic) equations and to estimate their values when solving a specific problem, are continuously emphasized.

The book contains problems that should help the reader to understand and utilize the subject matter. Some of the problems introduce expansions of the presented material.

If this is what we have in mind in writing this book, why do we call it an INTRODUCTION TO .... ? It is to remind ourselves and the reader that the material presented in this book is just the first, albeit important and essential, body of knowledge required for solving a problem of transport in a porous medium domain. Our primary objective is to enable the reader to formulate a problem of transport of any extensive quantity in any porous medium domain which he may encounter, as a well-posed mathematical model. The methods and algorithms for the actual solution of the mathe­matical model, once it has been formulated, in a specific case of practical interest, are beyond the scope of this book.

The frontiers of knowledge related to modeling transport in porous media are continuously being pushed foreword. Pressed by urgent needs to provide solutions for problems of importance to society, a large number of scientists, in a variety of disciplines, are currently engaged in research that continuously contributes to the understanding and quantitative description of transport phenomena in porous media and to the solution of models that describe them. Examples of such problems are groundwater pollution by hazardous wastes, produced by industry and by agricultural activities, contamination from repositories for radioactive wastes, improved techniques for enhanced oil and gas production and storage of energy in aquifers. Of special interest is the research done on physico-chemical and biological processes that take place within phases and among them, especially in connection with organic pollutants, and remediation methods for their removal from the subsurface. Concurrently, research continues on methodologies of utilizing field data for solving field problems. These include design of observation networks, re­gionalization of point data, system identification (including parameter es­timation), taking into account spatial heterogeneity of the porous medium

PREFACE xv

and stochasticity of the processes involved. In addition, efforts are made to improve model solving techniques. Much effort is devoted to methods for coping with the various aspects of uncertainty associated with models, from uncertainty in the processes that take place, through uncertainty in the spatial distribution of model coefficients and scale effects. Research is also conducted on the description of transport phenomena when the con­tinuum approach is not applicable. As emphasized above, all these topics are not considered in this book. There is no doubt that the results of this research, combined with advanced methods for the actual solution of field problems, will obviously, in the future, expand and modify the form of the models presented here. However, the basic concepts of modeling, continuum considerations and balances will always underlie any model formulation.

Large parts of this book are the outcome of an ongoing research program on transport phenomena in porous media, conducted by us at the Technion­Israel Institute of Technology and at the Hydrological Service, Ministry of Agriculture. We wish to thank the Fund for the Promotion of Research at the Technion for the financial support that made this research possible. Thanks are due to our students at many universities for their challenging comments.

Jacob Bear and Yehuda Bachmat Israel, 1989

List of Main Symbols

a As subscript, symbol denoting air. a Dispersivity of porous medium. aijkl Component of a. aL Longitudinal dispersivity of isotropic porous medium. aT Transversal dispersivity of isotropic porous medium. Ao Area of Ao. Ao Domain of Representative Elementary Area, of area Ao. Aoa Domain of a-phase in Ao.

b A constant vector. B Thickness of thin porous medium domain. Ba Formation volume factor of an a-phase. S Segment of a domain's boundary. Balance operator.

c As subscript, symbol denoting characteristic value. c~ Concentration of ,-component in a-phase (= p~ = mass

of component per unit volume of phase). Cr Hydraulic resistance of a semi pervious layer (=ratio of

thickness to hydraulic conductivity). Cfjkl Component of fluid's viscosity coefficient. Cf}kl Component of elasticity tensor. Cp Specific heat of fluid at constant pressure. Cs Specific heat of solid at constant strain. Cv Specific heat of fluid at constant volume.

Ct:. Coefficient (= ft~/~fIN=l)'

d* Length characterizing macroscopic heterogeneity. Dw Moisture diffusivity. D~ Coefficient of dispersion of E in a-phase. D~ Coefficient of dispersion of mass of ,-component in a-phase.

xvii

xviii LIST OF MAIN SYMBOLS

'D~ Coefficient of molecular diffusion of ")'-component in a-phase. 'D~"i Coefficient of molecular diffusion of ")'-component in a

porous medium. D~h Coefficient of hydrodynamic dispersion of a ")'-component in

an a-phase (= 'D~"i + D~).

e Void ratio (=Uov/Uos). Density of an extensive quantity, E. e~ Density of E~ (=E per unit volume of a-phase). E Young's modulus of elasticity. Eu Euler number. E~ An extensive quantity, E, of ")'-component in a-phase

( e.g., E = m,m"i,M,H).

f As subscript, symbol denoting fluid. f!""'{3 Rate of transfer of E from the a-phase to the j3 one,

fr F

F F Fr :F

9 G Gijk ... GE Gr

h h

across their common microscopic interface, per unit volume of porous medium. As subscript, symbol denoting fractures. Concentration of a component on the solid surface (as mass of component per unit mass of solid matrix). An equation of a surface, F(x, t) = O. Body force. Froude number. Force exerted on a solid matrix by the fluid.

Gravity acceleration. As subscript, symbol denoting gas. Molar free energy. A tensorial quantity. A state variable of an E continuum. Grashof number.

Elevation of water table of a phreatic aquifer. Enthalpy. As subscript, symbol denoting hydrodynamic dispersion. An oriented distance of length h. Capillary !lressure head. Capillary rise. Relative humidity. Heat. As superscript, symbol denoting heat.

LIST OF MAIN SYMBOLS xix

-tE-Y J '" J*E

J~h Jm

Jr JH JtE~

J"Y

.J

Lvap

L*

U nit tensor. Specific internal energy of a-phase.

Microscopic diffusive flux of El with respect to E2 (= el(VEl - V E2)). Microscopic conductive heat flux (= pJ(VH - vm)). Microscopic diffusive flux of total mass (== jmU = p(vm - V)). Microscopic diffusive mass flux of I-component ( == jmU = p"Y(V"Y - V)). Total microscopic flux of E~ (= e~VE~). Dispersive flux of E. Sum of diffusive and dispersive fluxes of E in a-phase. Macroscopic diffusive mass flux (== jm Cl ). Flux due to hydrodynamic dispersion of mass (Jm + J*m). Macroscopic conductive heat flux (== jH Cl

).

Total macroscopic flux of E~. Macroscopic diffusive flux of I-component (j"YCl). Jacobian. Hydraulic gradient .

Permeability. Effective permeability of a-phase. Relative permeability of a-phase, for an isotropic porous medium. Degradation rate constant of I-component in a-phase. Hydraulic conductivity. Partitioning coefficient.

Characteristic size of REV. Characteristic size of REA.

Magnitude of f as determined by considering porosity. Also f0Jx, ftL Latent heat of vaporization. Characteristic size of domain.

Mass; also as superscript. Mass of I-component of a-phase. Molecular weight. Superscript denoting momentum. Molecular weight of I-component. Momentum of a-phase. Mobility of an a-phase.

xx LIST OF MAIN SYMBOLS

Maf3 Mobility ratio of an a and j3 fluid phases.

n Porosity. As subscript, symbol denoting nonwetting fluid. nA Areal porosity. neff Effective porosity. it Estimate of n. N Number of items. N Rate of accretion on a phreatic surface.

P Pressure. Pc Capillary pressure. pV Vapor pressure. POI Pressure in a-phase. pix Partial pressure of i-component in a-phase. pb As subscript, symbol denoting porous block. pm As subscript, smbol denoting porous medium. P Probability. Pe Peclet number. Pr Prandtl number.

qa Specific discharge of a-phase. qr Specific discharge of a-phase, relative to the solid. qm Mass weighted specific discharge. QE Discharge of E.

r An oriented distance. R Universal gas constant. Rate of recharge of aquifer. Radius of REV. Rix Solubility of i-component in a-phase. RC As subscript, reservoir conditions. Ra Rayleigh number. Re Reynolds number. Ri Richardson number. n Symbol for a domain, or region.

s Specific entropy. Length. As subscript, symbol denoting solid. S Entropy. Aquifer storativity. SC As subscript, standard (or stock tank) conditions. So Specific storativity of porous medium. Sr Specific retention.

LIST OF MAIN SYMBOLS

Sy Specific yield. Sa Saturation of a-phase (e.g., a = a,w). Sa{3 Area of Sa{3-surface (similarly, Saa, etc.). S Areal domain of surface area S. So Surface surrounding Uo. Soa Surface surrounding Uoa . Saa The a - a-surface on So (similarly, S(3(3). Sa(3 Surface of contact of a-phase with all other phases

(denoted by 13) within Uo •

t Time. T Temperature. Aquifer transmissivity. T* Tortuosity.

U a Velocity of surface (e.g., of Sa(3)' U Volume. Volume of domain U. UE Volume occupied by a quantity E. Uo Volume of domain of REV (Uo ).

U Domain of volume U. Similarly, domains Uoa , Uos, etc., of volumes Uoa , Uos, etc., respectively.

v As subscript, symbol indicating void space. Specific volume of mass (=11 p).

vE Specific volume of E (= lie). V Velocity. V s Velocity of solid. Va Volume weighted velocity of a-phase (= VUa).

VEJ Velocity of E~-continuum. V~ Velocity of ,-component of a-phase.

w As subscript, symbol denoting water, or wetting fluid. w Displacement.

x Horizontal coordinate. x Position vector. x = X-Xo' x, Position vector of point at microscopic level. Xo Position vector of centroid of REV. Xoa Position vector of centroid of Uoa .

XXI

xxii LIST OF MAIN SYMBOLS

X Mole fraction of ,-component.

y Horizontal coordinate.

z Vertical coordinate (positive upward). A complex variable. Z Compressibility factor.

Greek letters

(3

(3p (3R /3r , ,(x) ,a(x) ,a(3 rE ;;(

rSE

Dij

~

~a E

£

c c

As subscript, symbol for an a-phase. Transfer coefficient of E. As subscript, symbol for a (3-phase. A symbol for all other phases, except a). Coefficient of fluid compressibility at constant pressure. Coefficient of rock compressibility. Coefficient of fluid compressibility at constant temperature. As superscript, symbol denoting a ,-component. Characteristic function of void space. Characteristic function of a-phase. Interfacial tension between a and (3-phases. Rate of production of EJ, per unit mass of phase. Rate of production of E on a boundary. Components of Kronecker delta. Characteristic distance from solid surface to fluid within void space. Hydraulic radius of a-phase. Small error. Volumetric strain. First invariant of c. Strain tensor. Rate of strain. Components of c. Dilatation of the solid matrix. Dilatation of the solid phase. Coefficient of thermoelasticity. Contact angle. Volumetric fraction of a-phase. Fluid content. Fraction of a - a-surface in So. Coefficient of radioactive decay (= liT, T = mean half life).

LIST OF MAIN SYMBOLS xxiii

Aa Thermal conductivity of a-phase. A coefficient of an a-phase . .x~ Coefficient of thermal conductivity of a-phase in a porous medium. AH Combined thermal conductivity in the fluid and the solid,

in a saturated porous medium. A *H Combined thermal dispersion in the fluid and thermal conductivity.

in the fluid and the solid, in a saturated porous medium. A~ Lame's constant of elastic solid. P,a Dynamic viscosity of an a-phase. p,'Y Chemical potential of a I-component of a phase. p,~ Lame's constant of elastic solid. v Poisson's ratio. Va Kinematic viscosity of an a-phase (= P,a/Pa).

Va Outward unit vector to Ua (3 on Sa(3.

e Material coordinates of a particle. eE Material coordinates of an E-particle. 11" Extensive quantity associated with a surface. Pa Mass density of a-phase. p'J, Mass density of I-component in a-phase. p'J,mol Molar concentration of I-component in a-phase

(= number of moles of I, per unit volume of a-phase). a Stress tensor. a~ Effective stress (= a~). ~a(3 Specific area of Sa(3. Similarly, ~aa for Saa, etc. T Shear stress. T'Y Correlation coefficient of I(X). <p Piezometric head. if>E Velocity potential of an E-continuum. <i>E Potential of an E-flux field. x( B) Function of moisture content. 'lj;'J, Specific value of EJ (i.e., quantity of E of I, per unit mass of a-phase). 'lj; Suction, or matric suction. \Ii Stream function. \liE E-transport function. wJ Mass fraction of I-component in a-phase. wE Complex potential of E-continuum.

xxiv LIST OF MAIN SYMBOLS

Special symbols

Average, volume average, or phase average of ( .. ) (=Jo JuJ .. ) dU). Intrinsic phase average of ( .. ) (= -u1 ];u ( .. ) dU).

00< 0

G Deviation of G from its intrinsic phase average, Ga., over an REV. Cov(.) Covariance of (.). Var(.) Variance of (.). E(.) Expected value of (.). "..-a.[3 ( .. ) -a. ( .. ) (.1 () G

A verage of ( .. ) over the Sa.[3-surface.

Average of ( .. ) over a-phase in REA.

Average of ( .. ) over thickness of thin domain.

Deviation of ( .. ) from (.1. Material derivative of G.

V·A Divergence of a vector A ( == div A). V A Gradient of a scalar A (== grad A). DtV Material derivative of ( .. ), as observed by the E-continuum. ( .. ) Transpose of ( .. ).