The Mathematics of Fluid Flow Through Porous Media

Standards Information Network (Verlag)
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  • erschienen am 8. Juni 2021
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  • 224 Seiten
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Master the techniques necessary to build and use computational models of porous media fluid flow

In The Mathematics of Fluid Flow Through Porous Media, distinguished professor and mathematician Dr. Myron B. Allen delivers a one-stop and mathematically rigorous source of the foundational principles of porous medium flow modeling. The book shows readers how to design intelligent computation models for groundwater flow, contaminant transport, and petroleum reservoir simulation.

Discussions of the mathematical fundamentals allow readers to prepare to work on computational problems at the frontiers of the field. Introducing several advanced techniques, including the method of characteristics, fundamental solutions, similarity methods, and dimensional analysis, The Mathematics of Fluid Flow Through Porous Media is an indispensable resource for students who have not previously encountered these concepts and need to master them to conduct computer simulations.

Teaching mastery of a subject that has increasingly become a standard tool for engineers and applied mathematicians, and containing 75 exercises suitable for self-study or as part of a formal course, the book also includes:

  • A thorough introduction to the mechanics of fluid flow in porous media, including the kinematics of simple continua, single-continuum balance laws, and constitutive relationships
  • An exploration of single-fluid flows in porous media, including Darcy's Law, non-Darcy flows, the single-phase flow equation, areal flows, and flows with wells
  • Practical discussions of solute transport, including the transport equation, hydrodynamic dispersion, one-dimensional transport, and transport with adsorption
  • A treatment of multiphase flows, including capillarity at the micro- and macroscale

Perfect for graduate students in mathematics, civil engineering, petroleum engineering, soil science, and geophysics, The Mathematics of Fluid Flow Through Porous Media also belongs on the bookshelves of any researcher who wishes to extend their research into areas involving flows in porous media.

Myron B. Allen, is Professor Emeritus of Mathematics at the University of Wyoming in Laramie, Wyoming, USA. He is the author of Continuum Mechanics: The Birthplace of Mathematical Models and co-author of the first and second editions of Numerical Analysis for Applied Science.

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John Wiley & Sons Inc
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978-1-119-66387-4 (9781119663874)
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Myron B. Allen, is Professor Emeritus of Mathematics at the University of Wyoming in Laramie, Wyoming, USA. He is the author of Continuum Mechanics: The Birthplace of Mathematical Models and co-author of the first and second editions of Numerical Analysis for Applied Science.
Preface xiii

1 Introduction 1

1.1 Historical setting 1

1.2 Partial di_erential equations (PDEs) 2

1.3 Dimensions and units 5

1.4 Limitations in scope 6

2 Mechanics 9

2.1 Kinematics of simple continua 9

2.1.1 Referential and spatial coordinates 9

2.1.2 Velocity and the material derivative 12

2.2 Balance laws for simple continua 13

2.2.1 Mass balance 14

2.2.2 Momentum balance 16

2.3 Constitutive relationships 20

2.3.1 Body force 21

2.3.2 Stress in uids 22

2.3.3 The Navier-Stokes equation 23

2.4 Two classic problems in uid mechanics 25

2.4.1 Hagen-Poiseuille ow 26

2.4.2 The Stokes problem 28

2.5 Multiconstituent continua 30

2.5.1 Constituents 30

2.5.2 Densities and volume fractions 32

2.5.3 Multiconstituent mass balance 35

2.5.4 Multiconstituent momentum balance 37

3 Single-Fluid Flow Equations 39

3.1 Darcy's law 39

3.1.1 Fluid momentum balance 41

3.1.2 Constitutive laws for the uid 41



3.1.3 Filtration velocity 44

3.1.4 Permeability 45

3.2 Non-Darcy ows 46

3.2.1 The Brinkman law 46

3.2.2 The Forchheimer equation 48

3.2.3 The Klinkenberg e_ect 49

3.3 The single-uid ow equation 50

3.3.1 Fluid compressibility and storage 51

3.3.2 Combining Darcy's law and the mass balance 52

3.4 Potential form of the ow equation 52

3.4.1 Conditions for the existence of a potential 53

3.4.2 Calculating the scalar potential 54

3.4.3 Piezometric head 56

3.4.4 Head-based ow equation 57

3.4.5 Auxiliary conditions for the ow equation 59

3.5 Areal ow equation 61

3.5.1 Vertically averaged mass balance 62

3.5.2 Vertically averaged Darcy's law 65

3.6 Variational forms for steady ow 66

3.6.1 Standard variational form 67

3.6.2 Mixed variational form 68

3.7 Flow in anisotropic porous media 70

3.7.1 The permeability tensor 70

3.7.2 Matrix representations of the permeability tensor 71

3.7.3 Isotropy and homgeneity 74

3.7.4 Properties of the permeability tensor 74

3.7.5 Is permeability symmetric? 77

4 Single-Fluid Flow Problems 81

4.1 Steady areal ows with wells 81

4.1.1 The Dupuit-Thiem model 81

4.1.2 Dirac _ models 85

4.1.3 Areal ow in an in_nite aquifer with one well 88

4.2 The Theis model for transient ows 91

4.2.1 Model formulation 91

4.2.2 Dimensional analysis of the Theis model 92

4.2.3 The Theis drawdown solution 95

4.2.4 Solving the Theis model via similarity methods 97

4.3 Boussinesq and porous medium equations 102

4.3.1 Derivation of the Boussinesq equation 104


4.3.2 The porous medium equation 107

4.3.3 A model problem with a self-similar solution 108

5 Solute Transport 115

5.1 The transport equation 115

5.1.1 Mass balance of miscible species 116

5.1.2 Hydrodynamic dispersion 117

5.2 One-dimensional advection 121

5.2.1 Pure advection and the method of characteristics 122

5.2.2 Auxiliary conditions for _rst-order PDEs 125

5.2.3 Weak solutions 126

5.3 The advection-di_usion equation 128

5.3.1 The moving plume problem 129

5.3.2 The moving front problem 131

5.4 Transport with adsorption 135

5.4.1 Mass balance for adsorbate 136

5.4.2 Linear isotherms and retardation 137

5.4.3 Concave-down isotherms and front sharpening 138

5.4.4 The Rankine-Hugoniot condition 141

6 Multiuid Flows 147

6.1 Capillarity 148

6.1.1 Physics of curved interfaces 148

6.1.2 Wettability 152

6.1.3 Capillarity at the macroscale 154

6.2 Variably saturated ow 157

6.2.1 Pressure head and moisture content 157

6.2.2 The Richards equation 159

6.2.3 Alternative forms of the Richards equation 161

6.2.4 Wetting fronts 163

6.3 Two-uid ows 164

6.3.1 The Muskat-Meres model 164

6.3.2 Two-uid ow equations 166

6.3.3 Classi_cation of simpli_ed ow equations 167

6.4 The Buckley-Leverett problem 170

6.4.1 The saturation equation 170

6.4.2 Welge tangent construction 173

6.4.3 Conservation form 178

6.4.4 Analysis of oil recovery 179

6.5 Viscous _ngering 182


6.5.1 The displacement front and its perturbation 184

6.5.2 Dynamics of the displacement front 187

6.5.3 Stability of the displacement front 188

6.6 Three-uid ows 190

6.6.1 Flow equations 191

6.6.2 Rock-uid properties 193

6.7 Three-uid fractional ow analysis 195

6.7.1 A simpli_ed three-uid system 195

6.7.2 Classi_cation of the three-uid system 197

6.7.3 Saturation velocities and saturation paths 199

6.7.4 An example of three-uid displacement 202

7 Flows With Mass Exchange 207

7.1 General compositional equations 208

7.1.1 Constituents, species, and phases 208

7.1.2 Mass balance equations 210

7.1.3 Species ow equations 211

7.2 Black-oil models 213

7.2.1 Reservoir and stock-tank conditions 213

7.2.2 The black-oil equations 214

7.3 Compositional ows in porous media 217

7.3.1 A simpli_ed compositional formulation 217

7.3.2 Conversion to molar variables 218

7.4 Fluid-phase thermodynamics 220

7.4.1 Flash calculations 221

7.4.2 Equation-of-state methods 222

Appendices 225

A Dedicated Symbols 227

B Useful Curvilinear Coordinates 229

B.1 Polar coordinates 229

B.2 Cylindrical coordinates 230

B.3 Spherical coordinates 233

C The Buckingham Pi theorem 235

C.1 Physical dimensions and units 235

C.2 The Buckingham theorem 236


D Surface Integrals 239

D.1 De_nition of a surface integral 239

D.2 The Stokes theorem 241

D.3 A corollary to the Stokes theorem 241

Bibliography 244

Index 259


1.1 Historical Setting

The mathematical theory of fluid flows in porous media has a distinguished history. Most of this theory ultimately rests on Henry Darcy's 1856 engineering study [43], summarized in Section 3.1, of the water supplies in Dijon, France. A year after the publication of this meticulous and seminal work, Jules Dupuit [49], a giant among early groundwater scientists, recognized that Darcy's findings implied a differential equation. This observation proved to be crucial. For the next 75 years or so, the subject grew to encompass problems in multiple space dimensions-hence partial differential equations (PDEs)-with major contributions emerging mainly from the groundwater hydrology community. Pioneers included Joseph Boussinesq [25, 26], Philipp Forchheimer [53, 54], Charles S. Slichter [136], Edgar Buckingham [30], and Lorenzo A. Richards [129].

Interest in the mathematics of porous-medium flows blossomed as oil production increased in economic importance during the early twentieth century. Prominent in the early petroleum engineering literature in this area are works by P.G. Nutting [110], Morris Muskat and his collaborators [104-107, 159, 160], and Miles C. Leverett and his collaborators [29, 95-97]. Between 1930 and 1960, mathematicians, groundwater hydrologists, petroleum engineers, and geoscientists made tremendous progress in understanding the PDEs that govern underground fluid flows.

Today, mathematical models of porous-medium flow encompass linear and nonlinear PDEs of all major types, as well as systems involving PDEs having different types. The analysis of these equations and their numerical approximations requires an increasing level of mathematical and computational sophistication, and the models themselves have become essential design tools in the management of underground fluid resources.

From a philosophical perspective, credit for these advances belongs to scientists and engineers who clung tenaciously-often in the face of skepticism on the part of more "practically" oriented colleagues-to two premises. The first is that the key to effective modeling resides in careful mathematical reasoning. While this premise seems platitudinous, at any moment in history some practitioners believe that their science is too inherently messy to justify fastidious mathematics. On the contrary, the need for painstaking logical inferences from premises and hypotheses is arguably never greater than when the data are complicated, confusing, or hard to obtain.

The second premise is more subtle: In the absence of good data, sound mathematical models are essential. Far from outstripping the data, mathematical models tell us what data we really need. Moreover, they tell us what qualitative properties we can expect in predictions arising from a given input data set. They also reveal how properties of the data, such as its spatial variability and uncertainty, affect the models' predictive capabilities. If the required data cannot in principle be acquired, if the qualitative properties of the model conflict with the empirical evidence, or if the model cannot, in principle, provide stable predictions in the face of heterogeneity and uncertainty, then we must admit that our understanding is incomplete.

1.2 Partial Differential Equations (PDEs)

Most realistic models of fluid flows in porous media use PDEs, "the natural dialect of continuum science" [62], written at scales appropriate for bench- or field-scale observations. In practical applications, these equations are complicated. They are posed on geometrically irregular, multidimensional domains; they often have highly variable coefficients; they can involve coupled systems of equations; in many applications they are nonlinear. For these reasons, we must often replace the exact PDEs by arithmetic approximations that one can solve using electronic machines.

The practical need for computational methods notwithstanding, a grasp of the analytic aspects of the PDEs remains an important asset for any porous-medium modeler. What types of initial and boundary conditions yield well-posed problems? Do the solutions obey a priori bounds based on the initial or boundary data? Do the numerical approximations respect these bounds? Does the PDE tend to smooth or preserve numerically problematic sharp fronts as time advances? Do shocks form from continuous initial data?

In the first half of the twentieth century, pioneering numerical analysts Richard Courant, Kurt Friedrichs, Hans Lewy, and John von Neumann-all immigrants to the United States-recognized that one cannot successfully "arithmetize analysis" [23] without understanding the differential equations. Designing stable, convergent, accurate, and efficient approximations to PDEs requires mathematical insight into the equations being approximated. A visionary 1947 consulting report [152] by von Neumann, developing the first petroleum reservoir simulator designed for a computer, illustrates this principle.

This book aims to promote this type of insight. We examine PDE-based models of porous-medium flows in geometries and settings simple enough to admit analysis without numerical approximations but realistic enough to reveal important structures.

From a mathematical perspective, the study of fluid flows in porous media offers fertile ground for inquiry into PDEs more generally. In particular, this book employs many broadly applicable concepts in the theory of PDEs, including:

  1. Mass and momentum balance laws
  2. Variational principles
  3. Fundamental solutions
  4. The principle of superposition
  5. Similarity methods
  6. Stability analysis
  7. The method of characteristics and jump conditions.

Where possible, the narrative introduces these topics in the simplest possible settings before applying them to more complicated problems.

Topic 1, covered in Chapter 2, deserves comment. Few PDE texts at this level discuss balance laws in the detail pursued here. However, it is hard to build intuition about porous-medium flows without knowing the principles from which they arise. The balance laws furnish those principles. On the other hand, a completely rigorous study of balance laws for fluids flowing in porous media would require a monograph-length treatment in its own right. Chapter 2 reflects an attempt to weigh the importance of fundamental principles against the need for a concise explanation of how the governing PDEs emerge from basic laws of physics. The references offer suggestions for deeper inquiry.

We frequently refer to PDEs according to a classification system inherited from the algebra of quadratic equations. The utility of this system becomes more apparent as one becomes more familiar with examples. For now, it suffices to review the system for second-order PDEs in two independent variables having the form


Here, , , and are functions of the independent variables and , which we can replace with and in time-dependent problems; is the unknown solution; and denotes a function of five variables that describes the lower-order terms in the PDE.

The highest-order terms determine the classification. Thediscriminant of Eq. (1.1) is , which is a function of . Equation (1.1) is

  • hyperbolic at any point of the -plane where ;
  • parabolic at any point of the -plane where ;
  • elliptic at any point of the -plane where .

Extending this terminology, we say that a first-order PDE of the form

is hyperbolic at any point where .

Exercise 1.1 Verify the following classifications, where and are real-valued with :

Mathematicians associate the wave equation with time-dependent processes that exhibit wave-like behavior, the heat equation with time-dependent processes that exhibit diffusive behavior, and the Laplace equation with steady-state processes. These associations arise from applications, some of which this book explores, reinforced by theoretical analyses of the three exemplars in Exercise 1.1. For more information about the classification of PDEs, see [65, Section 2-6].

1.3 Dimensions and Units

In contrast to most texts on pure mathematics, in this book physical dimensions play an important role. We adopt the basic physical quantities length, mass, and time, having physical dimensions , , and , respectively. All other physical quantities encountered in this book-except for one case involving temperature in Chapter 7 -are derived quantities, having physical dimensions that are products of powers of , , and .

For example, the physical dimension of force arises from Newton's second law , where denotes mass and denotes acceleration:

Analyzing the physical dimensions of quantities that arise in physical laws can yield surprisingly powerful mathematical results. Subsequent chapters exploit this concept many times.

Physical laws such as require a way to assign numerical values to the physical quantities involved. We do this by comparison with standards, a process called measurement. For example, to assign a...

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