Table Of ContentFundamentals of Transport Phenomena
in Porous Media
NATO ASI Series
Advanced Science Institutes Series
A Series presenting the results of activities sponsored by the NATO Science
Committee, which aims at the dissemination of advanced scientific and technological
knowledge, with a view to strengthening links between scientific communities.
The Series is published by an international board of publishers in conjunction with the
NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical and D. Reidel Publishing Company
Physical Sciences Dordrecht and Boston
0 Behavioural and Martinus Nijhoff Publishers
Social Sciences DordrechUBoston/Lancaster
E Applied Sciences
F Computer and Springer-Verlag
Systems Sciences Berlin/Heidelberg/New York
G Ecological Sciences
Series E: Applied Sciences - No. 82
Fundamentals of
Transport Phenomena
in Porous Media
edited by
Jacob Bear
Department of Civil Engineering
Technion-Israel Institute of Technology
Haifa 32000, Israel
M. Yavuz Corapcioglu
Department of Civil Engineering
University of Delaware
Newark, Delaware 19716
USA
1984 Martinus Nijhoff Publishers
Dordrecht I Boston I Lancaster
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on Mechanics of Fluids in Porous
Media, Newark, Delaware, USA, July 18-27, 1982
library of Congress Cataloging in Publication Data
RATO Advanced Stu~ Institute on Mechanics ot Fluids in
Porous Media (1982 : lIewark, Del.)
Fundamentals ot transport phenomena in porous media.
(RATO advanced science institutes series. Series E,
Applied sciences ; no. 82)
"Proceedings ot the RATO Advanced St~ Institute on
Mechanics ot Fluids in Porous Media, Newark, Delaware,
U.S.A., July 18-27. 1982n~.p. verso.
Includes index.
1. Groundwater flov-Congresses. 2. Porous materials
--Congresses. 3. Transport theory-Congresses.
1. Bear. Jacob. II. Corapcioglu. M. Yavuz.
III. Series.
TC176.N39 1982 620.1'064 84-10178
ISBN-13: 978-94-009-6177-7 e-ISBN-13: 978-94-009-6175-3
001: 10.1007/978-94-009-6175-3
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Copyright © 1984 by Martinus Nijhoff Publishers. Dordrecht
Softtcover reprint of the hardcover 15t edition 1984
v
PREFACE
This volume contains the lectures presented at the NATO
Advanced Study Institute that took place at the University of
Delaware, Newark, Delaware, July 18-27, 1982. The purpose of this
Institute was to provide an international forum for exchange of
ideas and dissemination of knowledge on some selected topics in
Mechanics of Fluids in Porous Media.
Processes of transport of such extensive quantities as mass
of a phase, mass of a component of a phase, momentum and/or heat
occur in diversified fields, such as petroleum reservoir engineer
ing, groundwater hydraulics, soil mechanics, industrial filtration,
water purification, wastewater treatment, soil drainage and irri
gation, and geothermal energy production. In all these areas,
scientists, engineers and planners make use of mathematical models
that describe the relevant transport processes that occur within
porous medium domains, and enable the forecasting of the future
state of the latter in response to planned activities. The mathe
matical models, in turn, are based on the understanding of
phenomena, often within the void space, and on theories that re
late these phenomena to measurable quantities. Because of the
pressing needs in areas of practical interest, such as the develop
ment of groundwater resources, the control and abatement of
groundwater contamination, underground energy storage and geo
thermal energy production, a vast amount of research efforts in
all these fields has contributed, especially in the last t~o
decades, to our understanding and ability to describe transport
phenomena. In recent years, these research efforts have been
significantly accelerated, attracting scientists from many
disciplines. The practical needs of solving boundary value
problems in heterogeneous domains, irregular boundaries, coupled
phenomena and multiple dependent variables, led to the develop
ment o~ a variety of powerful numerical techniques. The reali
zation that real fields are highly heterogeneous and that the
degree of heterogeneity depends on the scale of the problem, led
to the introduction of stochastic concepts as an additional tool
for the description of phenomena.
Accordingly, the objectives of the Institute were to review
and discuss recent advances in research on transport phenomena in
vi
porous media, with special emphasis on the frontiers of knowledge
in this area and on a unified approach by scientists coming from
different disciplines. To enable in depth discussions, mainly
four topics were included: the fundamentals of transport proc
esses, porous medium deformation, the stochastic approach and
numerical methods.
Accordingly, the first part of this volume is devoted to
the development of the fundamental equations of mass, momentum
and energy in a porous medium, where the latter is viewed as a
continuum. Two methods are presented for obtaining these equations
in the case of multiphase (and possibly multicomponent) flow in a
(possibly) deformable porous medium: the mixture theory and a
theory based on volume averaging. In the latter case, the
averaging is over a representative elementary volume of the porous
medium, of the equations of balance of the considered extensive
quantities, written at the microscopic level, i.e., at a point
within an individual phase present in the system. When this
approach is applied to the equation describing momentum balance,
and subject to certain assumptions, including low Reynolds number,
Darcy's law is obtained. As the Reynolds number is increased,
the flow regime changes until full turbulence is reached. This
phenomenon - as it. actually occurs within the pore space - is
demonstrated by using advanced laser monitoring and photographing
techniques. Multiphase flow problems are introduced by a dis
cussion on infiltration. The multi-media approach of handling
heterogeneities in aquifer systems - applicable to both leaky
aquifers as well as to fractured porous media - is demonstrated
by considering the hydraulics of wells in heterogeneous porous
media.
In the second part, theories on porous medium deformability
under saturated and unsaturated soil conditions are reviewed, with
special attention focused on the subject of consolidation. One of
the major reviewed topics is Biot's theory which underlies many of
the presentations. Various constitutive equations representing non
elastic behavior are presented and discussed. A methodology is
presented for dealing with the problem of land subsidence and
horizontal displacements due to pumping from an aquifer.
The usual continuum approach and the macroscopic equations
derived by employing it may fail to describe transport phenomena
when dealing with large scale problems in aquifers due to the
large variability in transport and storage properties of the con
sidered domains. The solution of transport phenomena in domains.
with such heterogeneities, which always exist and on which infor
mation is available only in statistical forms, requires a special
method of treatment - the stochasic approach. Two papers are
presented which represent a frontier in research efforts on this
important subject.
vii
Once a mathematical model - or a well-posed boundary value
problem - is established as a satisfactory representation of a
process, its solution for cases of practical interest usually
calls for a numerical technique~ This is due to the heterogeneity
of the domain, the irregularity of its boundaries, non-linearity
of the problem and other fa~tors which preclude the possibility of
an analytical solution. Several advanced numerical methods,
especially finite element ones, and a comparison between them, are
presented. Of special interest is the conjugate gradient method
which facilitates the treatment of very large problems. The
method is demonstrated for problems of flow and land subsidence.
A number of persons deserve special appreciation for their
valuable help in the organization of this meeting. They include
Dr. A. F. Gaudy, Jr., Chairman of the Department of Civil Engi
neering of the University of Delaware, for permitting the organ
izers to utilize the Department's facilities, Ms. Connie Weber for
secretarial help, including the typing of an endless number of
letters, Ms. Janice Holton, of the Department of Continuing Edu
cation of the University of Delaware, for arranging the housing
and conference f;acilities, and to the graduate students, S. Mathur
and A. Haridas, of; the University of Delaware f;or their help at
various stages.
We greatly appreciate the financial support of NATO
without which this Institute would not be possible. We are also
grateful to the authors for accepting the invitation to lecture
and to prepa;r-e written papers, and to all the participants for
their contributions during the discussions.
We hope that this volume will be of; value also to others
in this f;ield who have not attended the meeting.
July, 1982
J. Bear M. Y. Corapcioglu
ix
TABLE OF CONTENTS
Preface v
Part 1. FUNDAMENTALS OF TRANSPORT PROCESSES 1
J. Bear and Y. Bachmat
Transport Phenomena in Porous Media - Basic Equations 3
R.M. Bowen
Porous Media Model Formulations by the Theory
of Mixtures 63
R.G. Carbonell and S. Whitaker
Heat and Mass Transfer in Porous Media 121
A. Dybbs and R.V. Edwards
A New Look at Porous Media Fluid Mechanics -
Darcy to Turbulent 199
M. Vauclin
Infiltration in Unsaturated Soils 257
T.D. Streltsova
Well Hydraulics in Heterogeneous Porous Media 315
Part 2. DEFORMATION OF POROUS MEDIA 347
A. Verruijt
The Theory of Consolidation 349
M.Y. Corapcioglu
Land Subsidence - A. A State-of-the-Art Review 369
x
M.Y. Corapcioglu and J. Bear
Land Subsidence - B. A Regional Mathematical Model 445
for Land Subsidence due to Pumping
D. Kolymbas
Anelastic Deformation of Porous Media 499
D.G. Fredlund
Consolidation of Unsaturated Porous Media 525
H.B. Kingsbury
Determination of Material Parameters of Poroelastic
Media 579
R.L. Schiffman
A Bibliography of Consolidation 617
Part 3. STOCHASTIC APPROACH TO POROUS MEDIA 671
L.W. Gelhar
Stochastic Analysis of Flow in Heterogeneous Porous
Media 673
G. de MarsHy
Spatial Variability of Properties in Porous Media:
A Stochastic Approach 719
Part 4. NUMERICAL MODELLING TECHNIQUES 771
T.N. Narasimhan
Formulation of Numerical Equations 773
R.W. Lewis and P.M. Roberts
The Finite Element Method in Ponms Media Flow 805
W.G. Gray
Comparison of Finite Difference and Finite Element
Methods 899
G. Gambolati and A. Perd on
The Conjugate Gradients in Subsurface Flow and
Land Subsidence Modelling 953
Subject Index 985
PART 1
FUNDAMENTALS OF
TRANSPORT PROCESSES
