Dynamics of Fluids in Porous Media

796 Pages·1972·37.406 MB·English

by  Bear, Jacob

dynamics of fluids
in porous media
Jacob Bear
Department of Civil Engineering
Technion-Israel Institute of Technology, Haifa
DOVER PUBLICATIONS, INC.
New York
Copyright 8 1972 by American Elsevier Publishing Company, Inc.
All rights reserved
This Dover edition, first published in 1988, is an unabridged, cor-
rected republication of the work first published by the American
Elsevier Publishing Company, Inc., New York, 1972, in its Environ-
mental Science Series.
Library of Congress Cataloging-in-PublicationD ata
Bear, Jacob.
Dynamics of fluids in porous media I by Jacob Bear.
p. cm.
Reprint. Originally published: New York: American Elsevier
Pub. Co., 1972. Originally published in series: Environmental sci-
ence series (New York, 1972- ). With corrections.
Bibliography: p.
Includes index.
ISBN- 13: 978-0-186-65675-5
ISBN- 10: 0-186-65675-6
1. Fluid dynamics. 2. Groundwater flow. I. Title.
TA357.B38 1988
620.1’064-dc19 87-34940
CIP
Manufactured in the United States by Courier Corporation
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www.doverpubIications.com
This book is an attempt to present, in an ordered manner, the theory of dynamics
(actually, also of statics) of fluids in porous media, as applicable to many disciplines
of science and engineering. For some years I have taught courses on flow through
porous media, and have treated this subject as a part of other courses, such as ground
water hydrology, while at the Technion-Israel Institute of Technology, at M.I.T.
where I spent my sabbatical leave (1966-7), and at several other institutions. I have
felt the lack of a suitable textbook on this subject. Ideally, such a text should start
from first principles of fluid mechanics and mechanics of continua, should show the
passage from the microscopic to the macroscopic level of treatment, should emphasize
the special features of porous media, establish the macroscopic theory and then show
how it is applied to cases of practical interest.
It is rather surprising that in spite of its importance in many fields of practical
interest, such as petroleum engineering, ground water hydrology, agricultural
engineering and soil mechanics, so small number of treatises is available on fluids
in porous media. This circumstance is even more surprising in view of the vast amount
of literature published on the subject in a number of scientific and engineering
journals. Although dynamics of fluids in porous media could become an interesting
interdisciplinary course serving several departments, I believe that the relatively
small number of courses offered by universities on the subject is due in part to lack
of a suitable textbook. To overcome this lack I prepared notes for my own classes,
which I present here in the form of a book, hoping that it will serve others in a
similar situation.
The bok is designed primarily for advanced undergraduate students and for
graduates in fields such as ground water hydrology, soil mechanics, soil physics,
drainage and irrigation engineerinq, sanitary engineering, petroleum engineering and
chemical engineering, where flow through porous media plays a fundamental role.
The book, I hope, will also serve the needs of scientists and engineers already active
in these fields, who require a sound theoretical basis for their work. The emphasis in
this book is on understanding the microscopic phenomena occurring in porous media
and on their macroscopic description. The reader is led to grasp the meanings of the
various parameters and coefficients appearing in the macroscopic descriptions of
problems of flow through porous media, and their actual determination, as well as
the limitations and approximations inherent in their description. In each case, the
objective is to achieve a clear formulation of the flow problem considered and a
complete mathematical statement of it in terms of partial differential equations
and a set of initial and boundary conditions. Once a flow problem is stated properly
in mathematical terms, three methods of solution are possible in principle: analytic
xvii
solution, numerical solution aided by high speed digital computers and solution by
means of laboratory models and analogs. All three tools are described in this book.
Typical examples of analytic solutions are scattered throughout the book, but no
attempt is made to present a collection of a large number of solved problems. The
principles of the numerical method of solution are presented, and a detailed descrip-
tion is given of laboratory models and analogs, their scaling and applications.
Mathematics is employed extensively and the reader is expected to have a good
background in advanced engineering mathematics, including such subjects as vector
analysis, Cartesian tensor analysis, partial differential equations and elements of the
theory of functions.
No attempt is made to give a complete citation of all published literature or to
indicate the first author on a particular subject. References selected for citation
are those I think represent a more important point of view, are more appropriate
from the educational point of view or are more readily available for the average
reader.
Obviously a single book, even of this size, cannot include everything related to the
subject treated. Although we consider porous media in general, the discussion is
limited to media with relatively large pores, thus excluding clays and media with
micropores or colloidal-size particles. Similarly, chemical and electrochemicals urface
phenomena are excluded. The discussion is restricted to Newtonian fluids.
With these objectives and limitations in mind, the book starts with examples of
two important porous media: the ground water aquifer and the oil reservoir. An
attempt is made to define porous media, and the continuum approach is introduced
as a tool for treating phenomena in porous media. This requires the definition of a
“representative elementary volume” based on the definition of porosity. Chapter 2
includes a summary of some important fluid and porous media properties. In chapter
3, the concepts of pressure and piezometric head are introduced. Chapter 4 starts with
the definition of velocities and fluxes in a fluid continuum. Then the equations of
conservation of mass, momentum and energy in a fluid continuum are presented,
and using a porous medium conceptual model these equations are averaged to obtain
the basic equations that describe flow through porous media: the equations of
volume and mass conservation, including the equation of mass conservation of a
species in solution (also called the equation of hydrodynamic dispersion), and the
motion equation for the general case of an anisotropic medium and inhomogeneous
fluid. Although the basic equations of motion and of mass conservation are developed
from first principles in chapter 4, chapters 5 and 6 return to these topics, discussing
them from a different point of view, perhaps more suitable for the reader who is
less versed in fluid mechanics. Chapter 5 presents the equation of motion, starting
from its original one-dimensional form (as suggested by Darcy on the basis of exper-
iments), and extending it to three-dimensional flow, compressible fluids and aniso-
tropic media. This chapter also contains a review of theoretical derivations of
Darcy’s law. My objectives in presenting this and similar reviews is to indicate
research methods, such as the use of conceptual and statistical models. A section on
the motion equation at high Reynolds numbers is also included.
In chapter 6, the control volume approach is introduced as a general tool for
developing mass conservation equations. Special attention is devoted to deformable
media. Also included in this chapter is the stream function and its relationship to the
piezometric head. Once the continuity or mass conservation equations have been
established, the next natural step is to consider the initial and boundary conditions.
These are discussed in detail in chapter 7. Special attention is given to the phreatic
surface boundary condition and to its description in the hodograph plane. The second
part of this chapter contains a discussion on various analytic and numerical solution
techniques.
Upon reaching this point, the reader should be able to state a problem of flow
through porous media in terms of an appropriate partial differential equation and
a set of initial and boundary conditions. He should also know the major methods
of solution (analog solutions are discussed in chapter 11).
Chapter 8 deals with the problem of flow in unconfined aquifers. This is a problem
often encountered in ground water hydrology and in drainage. The Dupuit assump-
tions are explained and employed to derive the continuity equations for unconfined
flow. The hodograph method, as a tool for solving two-dimensional, steady phreatic
flow problems, is discussed in detail with many examples. Several linearization
techniques and solutions of the nonlinear equation of unconfined flow are also
presented in this chapter.
In chapter 9 the discussion, hitherto confined to single-phase flow, is extended
to polyphase flow in porous media, a topic of special interest in petroleum engineering.
Starting from the fundamental concepts of saturation, capillary pressure and relative
permeability, the motion and continuity equations are established. The case of
unsaturated flow as treated by soil physicists is presented as a special case of flow
of immiscible fluids, where one of the fluids-the air-is stationary and at constant
pressure. Special cases of interest, dealing with infiltration into soils, are considered
in more detail. A new concept is introduced: that of an abrupt interface as an
approximation replacing the actual transition zone that occurs between two fluids,
whether miscible or immiscible. A detailed discussion is presented on the coastal
interface, of great interest to ground water hydrologists.
Chapter 10 deals with hydrodynamic dispersion. Again, although the fundamental
equation is developed from first principles in chapter 4, a review of several other
theories leading to this equation is presented. Special attention is given to the
coefficient of dispersion and its relationship to matrix and flow characteristics. A
section on heat and mas transfer completes the discussion on hydrodynamic disper-
sion.
Chapter 11 presents the use of models and analogs, both as research tools and as
tools for solving boundary value problems. Following the presentation of a general
method for deriving analog scales, a detailed description is given of the sand box
model, the electric analogs of various types, the Hele-Shaw analogs and the membrane
analog. Recommendations for application are indicated in each case.
In brief, this is the subject matter I have chosen to cover in this book. I have made
an effort to present the information in such a way as to require a minimum of supple-
mentary material, except for those who wish to dig more deeply into the subject.
A large number of problems and exercises is included in this book.
1 should like to express my appreciation to the many individuals who, through
their comments and criticism, have contributed to the completion of this book.
Special thanks are due to Dr. Y. Bachmat, Dr. C. Braester, Mr. E. A. Hefez and
E. Goldshlager, for the help they have given me in reading, discussing and con-
structively criticizing the draft. Thanks are also due to the Department of Civil
Engineering at M.I.T., and especially to Professor C. L. Miller, head of the department,
Professor A. T. Ippen and Professor D. R. F. Harleman, who made it possible for
me to write a large part of this book while spending a most fruitful year as a visiting
professor at M.I.T.
The heaviest burden involved in writing this book was borne by my wife, Siona,
who had to put up with the many inconveniences that are unavoidable when one is
engaged in writing a book. For her constant encouragement to me throughout the
various stages of writing, my hearty gratitude.
I realize that an attempt to represent a systematic account of a theory, such as
I have made here, is bound to have defects. I will accept with gratitude all readers’
suggestions directed toward the improvement of this book.
Haifa, Israel Jacob Bear
1972
Cont ents
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preface xvii
CHAPTER 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction 1
. . . . . . . . . . . . . .
1.1 Aquifers. Ground Water and Oil Reservoirs 1
. . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Definitions 1
. . . . . . . . . .
1.1.2 The Moisture Distribution in a Vertical Profile 2
. . . . . . . . . . . . . . . . . . . .
1.1.3 Classification of Aquifers 4
. . . . . . . . . . . . . . . . . . . . .
1.1.4 Properties of Aquifers 7
. . . . . . . . . . . . . . . . . . . . . . .
1.1.5 The Oil Reservoir 8
. . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Porous Medium 13
. . . . . . . . . . . . . .
1.3 The Continuum Approach to Porous Media 15
. . . . . . . . . . . . . .
1.3.1 The Molecular and Microscopic Levels 15
. . . . . . . . .
1.3.2 Porosity and Representative Elementary Volume 19
. . . . . . . . . . . . . . . . . .
1.3.3 Areal and Linear Porosities 21
. . . . . . . . . . . . . . . . .
1.3.4 Velocity and Specific Discharge 22
. . . . . . . . . . . . . . . . . . . . .
1.3.5 Concluding Remarks 24
CHAPTER 2
. . . . . . . . . . . . . . . . . .
Fluids and Porous Matrix Properties 27
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Fluid Density 27
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Definitions 27
. . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Mixture of Fluids 30
. . . . . . . . . . . . . . . . . . . .
2.1.3 Measurement of Density 31
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Fluid Viscosity 32
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Definition 32
. . . . . . . . . . . . . . . . . . . . .
2.2.2 Non-Newtonian Fluids 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Units 34
. . . . . . . . . . . . . . .
2.2.4 Effect of Pressure and Temperature 34
. . . . . . . . . . . . . . . . . . .
2.2.5 Measurement of Viscosity 35
. . . . . . . . . . . . . . . . . . . . . . . .
2.3 Fluid Compressibility 37
. . . . . . . . . . . . . . . .
2.4 Statistical Description of Porous Media 38
. . . . . . . . . . . . . . . . . . .
2.4.1 Particle-Size Distribution 39
. . . . . . . . . . . . . . . . . . . . .
2.4.2 Pore-Size Distribution 41
. . . . . . . . . . . . . . . . .
2.4.3 Other Statistical Descriptions 42
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Porosity 43
. . . . . . . . . . . . . . . . .
2.5.1 Porosity and Effective Porosity 43
. . . . . . . . . . . . . . . .
2.5.2 Porosity. Structure and Packing 45
. . . . . . . . . . . . . . . . . . . . .
2.5.3 Porosity Measurement 47
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Specific Surface 60
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Definitions 60
. . . . . . . . . . . . . . . .
2.6.2 Measurement of Specific Surface 51
. . . . . . . . . . . . . . . . .
2.7 Matrix and Medium Compressibility 52
CHAPTER 3
. . . . . . . . . . . . . . . . . . . .
Pressure and Piezometric Head 59
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Stress at a Point 59
. . . . . . . . . . . . . . . . . .
3.2 Hydrostatic Pressure Distribution 62
. . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Piezometric Head 63
CHAPTER 4
. . . . .
The Fundamental Fluid Transport Equations in Porous Media 65
. . . . . . . . .
4.1 Particles. Velocities and Fluxes in a Fluid Continuum 65
. . . . . . . . . . . . . . .
4.1.1 Definitions of Particles and Velocities 65
. . . . . . . . . . . . . . . . .
4.1.2 Diffusive Velocities and Fluxes 68
. . . . . . . . . . .
4.1.3 The Eulerian and Lagrangian Points of View 70
. . . . . . . . . . . . . . . . . . .
4.1.4 The Substantial Derivative 7.1
. . . . . . . . . . . . . . . . .
4.2 The General Conservation Principle 74
4.3 Equations of Mass. Momentum and Energy Conservation in a Fluid
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuum 77
. . . . . . . . . . . . . . . . .
4.3.1 Mass Conservation of a Species 77
. . . . . . . . . . . . . .
4.3.2 Mass Conservation of a Fluid System 78
. . . . . . . . .
4.3.3 Conservation of Linear Momentum of a Species 79
o(
. . . . . . .
4.3.4 Conservation of Linear Momentum of a Fluid System 80
. . . . . . . . . . . .
4.4 Constitutive Assumptions and Coupled Processes 82
. . . . . . . . . . . . . . . . . . . .
4.4.1 General Considerations 82
. . . . .
4.4.2 Principles to be Used in Forming Constitutive Equations 84
. . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Coupled Processes 85
. . . . . . . . . . . . . . . . . . . . . .
4.5 A Porous Medium Model 90
. . . . . . . . . . . . . . . .
4.5.1 The Conceptual Model Approach 90
. . . . . . . . . . .
4.5.2 A Model of Flow Through a Porous Medium 92
. . . . . . . . . . . . . . . . . . . . .
4.5.3 Frames of Reference 93
. . . . . . . . . . . . . . . . . . . .
4.5.4 An Averaging Procedure 95
. . . . . . . . . . . . .
4.6 Equations of Volume and Mass Conservation 98
. . . . . . . . . . . . . . . .
4.6.1 Equation of Volume Conservation 98
. . . . . .
4.6.2 Equation of Mass Conservation of a Species in Solution 100
. . . . . . . . . . . . . . . . .
4.6.3 Equation of Mass Conservation 102
. . . . . . . . . . . . . . . . . . . . . . . .
4.7 Equation of Motion 104
. . . . . . . . . . . . . . . . . . . . .
4.8 Tortuosity and Permeability 106
. . . . . . . .
4.8.1 Relationship Between Tortuosity and Permeability 106
. . . . . . . . . . .
4.8.2 Tortuosity and Other Transport Coefficients 112
4.8.3 Formation Factor and Resistivity Index (Amyx 1960) in Reservoir
. . . . . . . . . . . . . . . . . . . . . . . . .
Engineering 113
CHAPTER 5
. . . . . . . . . . .
The Equation of Motion of a Homogeneous Fluid 119
. . . . . . . . . . . . . . . . . . .
5.1 The Experimental Law of Darcy 119
. . . . . . . . . . . . . . . . . . . .
5.2 Generalization of Darcy's Law 122
. . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Isotropic Medium 122
. . . . . . . . . . . . . . . . . . . . . .
5.2.2 Anisotropic Medium 123
. . . . . . . . . . . . . . . . . . . .
5.3 Deviations from Darcy's Law 125
. . . . . . . . . . . . . . . . . . . . . . .
5.3.1 The Upper Limit 125
. . . . . . . . . . . . . . . . . . . . . . .
5.3.2 The Lower Limit 127
. . . . . . . . . . . . . . . . . . . . .
5.3.3 The Slip Phenomenon 128
. . . . . . . . . . . . . . . . . .
5.4 Rotational and Irrotational Motion 129
. . . . . . . . . . . . . . .
5.4.1 The Potential and Pseudopotential 129
. . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Irrotational Flow 131
. . . . . . . . . . . . . .
5.5 Hydraulic Conductivity of Isotropic Media 132
. . . . . . . . . . . .
5.5.1 Hydraulic Conductivity and Permeability 132
. . . . . . . . . . . . . . . . . . . . . .
5.5.2 Units and Examples 136
. . . . . . . . . . . . . . . . . . . . . .
5.6 Anisotropic Permeability 136
. . . . . . . . . . . . . . . . . . . .
5.6.1 The Principal Directions 137
. . . . . . . . . . . . . . . . . . . .
5.6.2 Directional Permeability 143
. . . . . . . . . . . . . . .
5.7 Measurement of Hydraulic Conductivity 148
. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 General 148
. . . . . . . . . . . . . . . .
5.7.2 The Constant Head Permeameter 149
. . . . . . . . . . . . . . . . .
5.7.3 The Falling Head Permeameter 150
. . . . . . . . .
5.7.4 Determining Anisotropic Hydraulic Conductivity 160
. . . . . . . . . . . . . . . . . . . . . . .
5.8 Layered Porous Media 151
. . . . . . . . .
5.8.1 Flow Normal and Parallel to the Medium Layers 151
. .
5.8.2 Equivalent Hydraulic Conductivity of Arbitrarily Directed Flow 155
. . . . .
6.8.3 A Layered Medium as an Equivalent Anisotropic Medium 166
. . . . . . . . . . . . . . . . . . . . . .
5.8.4 Girinskii's Potential 157
. . . . . . . . . . . . . . . . . . . . . . . .
5.9 Compressible Fluids 169
. . . . . . . . . . . . . . . . . . . .
5.10 Derivations of Darcy's Law 161
. . . . . . . . . . . . . . . . . . . .
5.10.1 Capillary Tube Models 162
. . . . . . . . . . . . . . . . . . . . . . .
5.10.2 Fissure Models 164
. . . . . . . . . . . . . . . . . .
5.10.3 Hydraulic Radius Models 165
. . . . . . . . . . . . . . . . . .
5.10.4 Resistance to Flow Models 167
. . . . . . . . . . . . . . . . . . . . . .
6.10.5 Statistical Models 171
. . . . . . . . . . . .
6.10.6 Averaging the Navier-Stokes Equations 173
. . . . . . . . . . . . . . . . . . . . .
6.10.7 Ferrandon's Model 176
. . . . . . . . . . . . . . . . .
5.11 Flow At Large Reynolds Numbers 176
. . . . . . . . . . . . . . . . . . . . . .
5.1 1.1 The Phenomenon 176
. . . . . . . . . . .
5.11.2 Turbulence. Inertial Forces and Separation 177
. . . .
5.11.3 Some Examples of Proposed Nonlinear Motion Equations 182
. . . . . . . . . . . . . . . . . . . .
5.12 Seepage Forces and Stresses 184
. . . . . . . . . . . . . . . . . . . . . . . . .
5.12.1 The Forces 184
. . . . . . . . . . . . . . . . . . . .
5.12.2 Piping and Quicksand 186
CHAPTER 6
. . .
Continuity and Conservation Equations for a Homogeneous Fluid 195
. . . . . . . . . . . . . . . . . . . . . . . .
6.1 The Control Volume 196
. . . . . . . . .
6.2 Mass Conservation in a Nondeformable Porous Matrix 196
. . . . . . . . . . . . . . . . .
6.2.1 The Basic Continuity Equation 196
. . . . . . . . .
6.2.2 Continuity Equation for an Incompressible Fluid 198
. . . . . . . . . .
6.2.3 Continuity Equation for a Compressible Fluid 200
. . . . . . . . . . . . .
6.3 Mass Conservation in a Consolidating Medium 202
. . . . . . . . . . . . . . . . . .
6.3.1 Vertical Compressibility Only 203
.
6.3.2 Extension to Three Phases and to Three-Dimensional Consolidation 208
. . . . . . . . . . . . . . . .
6.3.3 Barometric Efficiency of Aquifers 211
. . . . .
6.4 Continuity Equations for Flow in Confined and Leaky Aquifers 213
. . . . . . . . . . . . . .
6.4.1 The Horizontal Flow Approximation 213
. . . . . . . . . . . . . . . . . . .
6.4.2 Flow in a Confined Aquifer 214
. . . . . . . . . . . . . . . . . . .
6.4.3 Flow in a Leaky Aquifer 216
. . . . . . .
6.4.4 Averaging the Exact Equations over a Vertical Line 218
. . . . . . . . . . . . . . . . .
6.4.5 The Boltzmann Transformation 221
. . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Stream Functions 222
. . . . . . . . . .
6.5.1 Pathlines. Streamlines, Streaklines and Fronts 223
. . . . . . . . .
6.5.2 The Stream Function in Two-Dimensional Flow 225
. . . . . . . .
6.5.3 The Stream Functions in Three-Dimensional Flow 226
6.5.4 The Partial Differential Equations for the Lagrange and Stokes Stream
. . . . . . . . . . . . . . . . . . . . . . . . . .
Functions 230
6.6.6 The Relationships between the Potential and the Stream Functions 233
. . . . . . . . . . . . . . .
6.5.6 Solving Problems in the q-4P lane 235
. . . . . . . . . . . . .
6.6 Flow Nets and Ground Water Contour Maps 236
. . . . . . . . . . . . . . . . . . . . . .
6.6.1 The pt,4 Flow Net 236
. . . . . . . . . . . . . . . .
6.6.2 The Ground Water Contour Map 239
6.7 The Partial Differential Equations Describing Flow of an Inhomogeneous
. . . . . . . . . . . . . . . . .
Incompressible Fluid in Terms of Y 241
. . . . . . . . . . . . . . . . . . . .
6.7.1 Two-Dimensional Flow 241
. . . . . . . . . . . . . . . . . . . . . .
6.7.2 Axisymmetric Flow 243
CHAPTER 7
. . . . . . . . . . . . .
Solving Boundary and Initial Value Problems 247
. . . . . . . . . . . . . . . . . . .
7.1 Initial and Boundary Conditions 248