Mathematical Modeling in Science and Engineering: An Axiomatic Approach

A powerful, unified approach to mathematical and computationalmodeling in science and engineering

Mathematical and computational modeling makes it possible topredict the behavior of a broad range of systems across a broadrange of disciplines. This text guides students and professionalsthrough the axiomatic approach, a powerful method that will enablethem to easily master the principle types of mathematical andcomputational models used in engineering and science. Readers willdiscover that this axiomatic approach not only enables them tosystematically construct effective models, it also enables them toapply these models to any macroscopic physical system.

Mathematical Modeling in Science and Engineering focuseson models in which the processes to be modeled are expressed assystems of partial differential equations. It begins with anintroductory discussion of the axiomatic formulation of basicmodels, setting the foundation for further topics such as:

Mechanics of classical and non–classical continuous systems

Solute transport by a free fluid

Flow of a fluid in a porous medium

Multiphase systems

Enhanced oil recovery

Fluid mechanics

Throughout the text, diagrams are provided to help readersvisualize and better understand complex mathematical concepts. Aset of exercises at the end of each chapter enables readers to puttheir new modeling skills into practice. There is also abibliography in each chapter to facilitate further investigation ofindividual topics.

Mathematical Modeling in Science and Engineering is idealfor both students and professionals across the many disciplines ofscience and engineering that depend on mathematical andcomputational modeling to predict and understand complexsystems.

Preface xiii

1 AXIOMATIC FORMULATION OF THE BASIC MODELS 1

1.1 Models 1

1.2 Microscopic and macroscopic physics 2

1.3 Kinematics of continuous systems 3

1.3.1 Intensive properties 6

1.3.2 Extensive properties 8

1.4 Balance equations of extensive and intensive properties9

1.4.1 Global balance equations 9

1.4.2 The local balance equations 10

1.4.3 The role of balance conditions in the modeling ofcontinuous systems 13

1.4.4 Formulation of motion restrictions by means of balanceequations 14

1.5 Summary 16

2 MECHANICS OF CLASSICAL CONTINUOUS SYSTEMS 23

2.1 One–phase systems 23

2.2 The basic mathematical model of one–phase systems 24

2.3 The extensive/intensive properties of classical mechanics25

2.4 Mass conservation 26

2.5 Linear momentum balance 27

2.6 Angular momentum balance 29

2.7 Energy concepts 32

2.8 The balance of kinetic energy 33

2.9 The balance of internal energy 34

2.10 Heat equivalent of mechanical work 35

2.11 Summary of basic equations for solid and fluid mechanics35

2.12 Some basic concepts of thermodynamics 36

2.12.1 Heat transport 36

2.13 Summary 38

3 MECHANICS OF NON–CLASSICAL CONTINUOUS SYSTEMS 45

3.1 Multiphase systems 45

3.2 The basic mathematical model of multiphase systems 46

3.3 Solute transport in a free fluid 47

3.4 Transport by fluids in porous media 49

3.5 Flow of fluids through porous media 51

3.6 Petroleum reservoirs: the black–oil model 52

3.6.1 Assumptions of the black–oil model 53

3.6.2 Notation 53

3.6.3 Family of extensive properties 54

3.6.4 Differential equations and jump conditions 55

3.7 Summary 57

4 SOLUTE TRANSPORT BY A FREE FLUID 63

4.1 The general equation of solute transport by a free fluid64

4.2 Transport processes 65

4.2.1 Advection 65

4.2.2 Diffusion processes 65

4.3 Mass generation processes 66

4.4 Differential equations of diffusive transport 67

4.5 Well–posed problems for diffusive transport 69

4.5.1 Time–dependent problems 70

4.5.2 Steady state 71

4.6 First–order irreversible processes 71

4.7 Differential equations of non–diffusive transport 73

4.8 Well–posed problems for non–diffusive transport 73

4.8.1 Well–posed problems in one spatial dimension 74

4.8.2 Well–posed problems in several spatial dimensions 79

4.8.3 Well–posed problems for steady–state models 80

4.9 Summary 80

5 FLOW OF A FLUID IN A POROUS MEDIUM 85

5.1 Basic assumptions of the flow model 85

5.2 The basic model for the flow of a fluid through a porousmedium 86

5.3 Modeling the elasticity and compressibility 87

5.3.1 Fluid compressibility 87

5.3.2 Pore compressibility 88

5.3.3 The storage coefficient 90

5.4 Darcy′s law 90

5.5 Piezometric level 92

5.6 General equation governing flow through a porous medium94

5.6.1 Special forms of the governing differential equation95

5.7 Applications of the jump conditions 96

5.8 Well–posed problems 96

5.8.1 Steady–state models 97

5.8.2 Time–dependent problems 99

5.9 Models with a reduced number of spatial dimensions 99

5.9.1 Theoretical derivation of a 2–D model for a confinedaquifer 100

5.9.2 Leaky aquitard method 102

5.9.3 The integrodifferential equations approach 104

5.9.4 Other 2–D aquifer models 108

5.10 Summary 111

6 SOLUTE TRANSPORT IN A POROUS MEDIUM 117

6.1 Transport processes 118

6.1.1 Advection 118

6.2 Non–conservative processes 118

6.2.1 First–order irreversible processes 119

6.2.2 Adsorption 119

6.3 Dispersion–diffusion 121

6.4 The equations for transport of solutes in porous media123

6.5 Well–posed problems 125

6.6 Summary 125

7 MULTIPHASE SYSTEMS 129

7.1 Basic model for the flow of multiple–species transport in amultiple–fluid– phase porous medium 129

7.2 Modeling the transport of species i in phase a130

7.3 The saturated flow case 133

7.4 The air–water system 137

7.5 The immobile air unsaturated flow model 142

7.6 Boundary conditions 143

7.7 Summary 145

8 ENHANCED OIL RECOVERY 149

8.1 Background on oil production and reservoir modeling 149

8.2 Processes to be modeled 151

8.3 Unified formulation of EOR models 151

8.4 The black–oil model 152

8.5 The Compositional Model 156

8.6 Summary 160

9 LINEAR ELASTICITY 165

9.1 Introduction 165

9.2 Elastic Solids 166

9.3 The Linear Elastic Solid 167

9.4 More on the Displacement Field Decomposition 170

9.5 Strain Analysis 171

9.6 Stress Analysis 173

9.7 Isotropic materials 175

9.8 Stress–strain relations for isotropic materials 177

9.9 The governing differential equations 179

9.9.1 Elastodynamics 180

9.9.2 Elastostatics 180

9.10 Well–posed problems 181

9.10.1 Elastostatics 181

9.10.2 Elastodynamics 181

9.11 Representation of solutions for isotropic elastic solids182

9.12 Summary 183

10 FLUID MECHANICS 189

10.1 Introduction 189

10.2 Newtonian fluids: Stokes′ constitutive equations 190

10.3 Navier–Stokes equations 192

10.4 Complementary constitutive equations 193

10.5 The concepts of incompressible and inviscid fluids 193

10.6 Incompressible fluids 194

10.7 Initial and boundary conditions 195

10.8 Viscous incompressible fluids: steady states 196

10.9 Linearized theory of incompressible fluids 196

10.10 Ideal fluids 197

10.11 Irrotational flows 198

10.12 Extension of Bernoulli′s relations to compressible fluids199

10.13 Shallow–water theory 200

10.14 Inviscid compressible fluids 202

10.14.1 Small perturbations in a compressible fluid: the theoryof sound 203

10.14.2 Initiation of motion 204

10.14.3 Discontinuous models and shock conditions 206

10.15 Summary 208

A: PARTIAL DIFFERENTIAL EQUATIONS 211

A. 1 Classification 211

A.2 Canonical forms 213

A.3 Well–posed problems 213

A.3.1 Boundary–value problems: the elliptic case 214

A.3.2 Initial–boundary–value problems 214

B: SOME RESULTS FROM THE CALCULUS 217

B.l Notation 217

B.2 Generalized Gauss Theorem 218

C: PROOF OF THEOREM 221

D: THE BOUNDARY LAYER INCOMPRESSIBILITY APPROXIMATION225

E: INDICIAL NOTATION 229

E.l General 229

E.2 Matrix algebra 230

E.3 Applications to differential calculus 232

Index 235

ISMAEL HERRERA, PhD. in Applied Mathematics, BrownUniversity, is Distinguished Professor in the Natural ResourcesDepartment of the Geophysics Institute at the Universidad NacionalAutónoma de México. He is the Editor of Numerical Methodsfor Partial Differential Equations and President of the MexicanSociety of Numerical Methods in Engineering and Applied Sciences.Dr. Herrera has received the National Science, Mexican Academy ofSciences, and Luis Elizondo Awards, the three most prestigiousawards in Mexico granted for scientific achievement.

GEORGE F. PINDER, PhD, has a primary appointment asProfessor of Engineering with secondary appointments as Professorof Mathematics and Statistics and Professor of Computer Science atthe University of Vermont. He is the author, or co–author, of ninebooks on mathematical modeling, numerical mathematics, and flow andtransport through porous media. He is a recipient of numerousnational and international honors and is a member of the NationalAcademy of Engineering.