Mathematical Modeling in Science and

Mathematical and computational modeling (MCM) can be used in a diverse set of applications making it a very appealing and potent modeling tool. This book uses a novel and powerful procedural approach to teaching MCM, the Axiomatic Approach, which permits incorporating in a single model, systems that occur in many different branches of science and engineering. This book focuses on the mathematical models, in which processes to be modeled are expressed as a system of partial differential equations. It introduces a systematic method for constructing such models, which can be applied to any macroscopic physical system. This latter feature of the Axiomatic Approach is very valuable when treating new systems, since it permits formulating the models of previously unknown systems. Using it, many of the systems of most common occurrence in engineering practice are introduced and discussed. The effectiveness of this approach is reflected in the broadness and importance of the subjects treated; they cover a great diversity of topics that are basic in many branches of engineering including: Civil Engineering, Mechanical Engineering, Petroleum Engineering, and Water Resources. These topics include: Flow of fluids and transport of solutes which are free to move in the physical space and where fluids may be restricted to move in a porous medium. The transport of solutes is fundamental in Environmental Engineering Water Resources and Petroleum Engineering since it is the means of predicting contaminant behavior. The porous medium based equations are also used to model Enhanced Oil Recovery which is very important for sustaining the oil supply of the world. Model of static and dynamic elasticity are used in several branches of engineering including Foundation and Seismic Engineering.

ISMAEL HERRERA, PhD. in Applied Mathematics, Brown University, is Distinguished Professor in the Natural Resources Department of the Geophysics Institute at the Universidad Nacional Autónoma de México. He is the Editor of Numerical Methods for Partial Differential Equations and President of the Mexican Society of Numerical Methods in Engineering and Applied Sciences. Dr. Herrera has received the National Science, Mexican Academy of Sciences, and Luis Elizondo Awards, the three most prestigious awards in Mexico granted for scientific achievement.

GEORGE F. PINDER, PhD, has a primary appointment as Professor of Engineering with secondary appointments as Professor of Mathematics and Statistics and Professor of Computer Science at the University of Vermont. He is the author, or co-author, of nine books on mathematical modeling, numerical mathematics, and flow and transport through porous media. He is a recipient of numerous national and international honors and is a member of the National Academy of Engineering.

Preface	p. xiii
Axiomatic Formulation of the Basic Models	p. 1
Models	p. 1
Microscopic and macroscopic physics	p. 2
Kinematics of continuous systems	p. 3
Intensive properties	p. 6
Extensive properties	p. 8
Balance equations of extensive and intensive properties	p. 9
Global balance equations	p. 9
The local balance equations	p. 10
The role of balance conditions in the modeling of continuous systems	p. 13
Formulation of motion restrictions by means of balance equations	p. 14
Summary	p. 16
Exercises	p. 17
References	p. 20
Mechanics of Classical Continuous Systems	p. 23
One-phase systems	p. 23
The basic mathematical model of one-phase systems	p. 24
The extensive/intensive properties of classical mechanics	p. 25
Mass conservation	p. 26
Linear momentum balance	p. 27
Angular momentum balance	p. 29
Energy concepts	p. 32
The balance of kinetic energy	p. 33
The balance of internal energy	p. 34
Heat equivalent of mechanical work	p. 35
Summary of basic equations for solid and fluid mechanics	p. 35
Some basic concepts of thermodynamics	p. 36
Heat transport	p. 36
Summary	p. 38
Exercises	p. 38
References	p. 41
Mechanics of Non-Classical Continuous Systems	p. 45
Multiphase systems i	p. 45
The basic mathematical model of multiphase systems	p. 46
Solute transport in a free fluid	p. 47
Transport by fluids in porous media	p. 49
Flow of fluids through porous media	p. 51
Petroleum reservoirs: the black-oil model	p. 52
Assumptions of the black-oil model	p. 53
Notation	p. 53
Family of extensive properties	p. 54
Differential equations and jump conditions	p. 55
Summary	p. 57
Exercises	p. 57
References	p. 62
Solute Transport by a Free Fluid	p. 63
The general equation of solute transport by a free fluid	p. 64
Transport processes	p. 65
Advection	p. 65
Diffusion processes	p. 65
Mass generation processes	p. 66
Differential equations of diffusive transport	p. 67
Well-posed problems for diffusive transport	p. 69
Time-dependent problems	p. 70
Steady state	p. 71
First-order irreversible processes	p. 71
Differential equations of non-diffusive transport	p. 73
Well-posed problems for non-diffusive transport	p. 73
Well-posed problems in one spatial dimension	p. 74
Well-posed problems in several spatial dimensions	p. 79
Well-posed problems for steady-state models	p. 80
Summary	p. 80
Exercises	p. 81
References	p. 83
Flow of a Fluid in a Porous Medium	p. 85
Basic assumptions of the flow model	p. 85
The basic model for the flow of a fluid through a porous medium	p. 86
Modeling the elasticity and compressibility	p. 87
Fluid compressibility	p. 87
Pore compressibility	p. 88
The storage coefficient	p. 90
Darcy's law	p. 90
Piezometric level	p. 92
General equation governing flow through a porous medium	p. 94
Special forms of the governing differential equation	p. 95
Applications of the jump conditions	p. 96
Well-posed problems	p. 96
Steady-state models	p. 97
Time-dependent problems	p. 99
Models with a reduced number of spatial dimensions	p. 99
Theoretical derivation of a 2-D model for a confined aquifer	p. 100
Leaky aquitard method	p. 102
The integrodifferential equations approach	p. 104
Other 2-D aquifer models	p. 108
Summary	p. 111
Exercises	p. 111
References	p. 115
Solute Transport in a Porous Medium	p. 117
Transport processes	p. 118
Advection	p. 118
Non-conservative processes	p. 118
First-order irreversible processes	p. 119
Adsorption	p. 119
Dispersion-diffusion	p. 121
The equations for transport of solutes in porous media	p. 123
Well-posed problems	p. 125
Summary	p. 125
Exercises	p. 125
References	p. 127
Multiphase Systems	p. 129
Basic model for the flow of multiple-species transport in a multiple-fluid- phase porous medium	p. 129
Modeling the transport of species i in phase ¿	p. 130
The saturated flow case	p. 133
The air-water system	p. 137
The immobile air unsaturated flow model	p. 142
Boundary conditions	p. 143
Summary	p. 145
Exercises	p. 145
References	p. 147
Enhanced Oil Recovery	p. 149
Background on oil production and reservoir modeling	p. 149
Processes to be modeled	p. 151
Unified formulation of EOR models	p. 151
The black-oil model	p. 152
The Compositional Model	p. 156
Summary	p. 160
Exercises	p. 161
References	p. 163
Linear Elasticity	p. 165
Introduction	p. 165
Elastic Solids	p. 166
The Linear Elastic Solid	p. 167
More on the Displacement Field Decomposition	p. 170
Strain Analysis	p. 171
Stress Analysis	p. 173
Isotropic materials	p. 175
Stress-strain relations for isotropic materials	p. 177
The governing differential equations	p. 179
Elastodynamics	p. 180
Elastostatics	p. 180
Well-posed problems	p. 181
Elastostatics	p. 181
Elastodynamics	p. 181
Representation of solutions for isotropic elastic solids	p. 182
Summary	p. 183
Exercises	p. 184
References	p. 186
Fluid Mechanics	p. 189
Introduction	p. 189
Newtonian fluids: Stokes' constitutive equations	p. 190
Navier-Stokes equations	p. 192
Complementary constitutive equations	p. 193
The concepts of incompressible and inviscid fluids	p. 193
Incompressible fluids	p. 194
Initial and boundary conditions	p. 195
Viscous incompressible fluids: steady states	p. 196
Linearized theory of incompressible fluids	p. 196
Ideal fluids	p. 197
Irrotational flows	p. 198
Extension of Bernoulli's relations to compressible fluids	p. 199
Shallow-water theory	p. 200
Inviscid compressible fluids	p. 202
Small perturbations in a compressible fluid: the theory of sound	p. 203
Initiation of motion	p. 204
Discontinuous models and shock conditions	p. 206
Summary	p. 208
Exercises	p. 208
References	p. 210
Partial Differential Equations	p. 211
Classification	p. 211
Canonical forms	p. 213
Well-posed problems	p. 213
Boundary-value problems: the elliptic case	p. 214
Initial-boundary-value problems	p. 214
References	p. 215
Some Results from the Calculus	p. 217
Notation	p. 217
Generalized Gauss Theorem	p. 218
Proof of Theorem	p. 221
The Boundary Layer Incompressibility Approximation	p. 225
Indicial Notation	p. 229
General	p. 229
Matrix algebra	p. 230
Applications to differential calculus	p. 232
Index	p. 235
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Mathematical Modeling in Science and Engineering An Axiomatic Approach

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