Mathematical Modeling in Science and Engineering: An Axiomatic Approach

249 Pages·2012·5.089 MB·English

by  Ismael Herrera, George F. Pinder(auth.)

MATHEMATICAL MODELING
IN SCIENCE AND ENGINEERING
MATHEMATICAL MODELING
IN SCIENCE AND ENGINEERING
An Axiomatic Approach
ISMAEL HERRERA
GEORGE E PINDER
WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Herrera, Ismael.
Mathematical modeling in science and engineering : an axiomatic approach / Ismael Herrera,
George F. Pinder. — 1st ed.
p. cm.
Includes index.
Summary: "This book uses a novel and powerful 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" — provided by publisher.
ISBN 978-1-118-08757-2 (hardback)
1. System analysis — Mathematical models. 2. Science—Mathematical models. 3. Engineering—
Mathematical models. I. Pinder, George Francis, 1942- II. Title.
QA402.H45 2012
501'.51—dc23 2011036329
Printed in the United States of America.
10 9 8 7 6 5 4 3 21
Dedicated to:
The National University of
Mexico (UNAM) where I
learned the paradigms of
mathematical thinking:
generality, clarity and
simplicity;
and to Brown University
where I learned to use them in
the solution of problems of
practical interest.
IHR
My esteemed students,
mentors and colleagues who
have stimulated and nurtured
my interest in the pursuit of
knowledge over these many years.
GFP
CONTENTS
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 properties 9
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 of
continuous systems 13
1.4.4 Formulation of motion restrictions by means of balance
equations 14
1.5 Summary 16
Exercises 17
References 20
vii
ViH CONTENTS
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 mechanics 25
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 mechanics 35
2.12 Some basic concepts of thermodynamics 36
2.12.1 Heat transport 36
2.13 Summary 38
Exercises 38
References 41
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
Exercises 57
References 62
SOLUTE TRANSPORT BY A FREE FLUID 63
4.1 The general equation of solute transport by a free fluid 64
4.2 Transport processes 65
4.2.1 Advection 65
4.2.2 Diffusion processes 65
CONTENTS IX
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
Exercises 81
References 83
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 porous medium 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 medium 94
5.6.1 Special forms of the governing differential equation 95
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 confined
aquifer 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
Exercises 111
X CONTENTS
References 115
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 media 123
6.5 Well-posed problems 125
6.6 Summary 125
Exercises 125
References 127
MULTIPHASE SYSTEMS 129
7.1 Basic model for the flow of multiple-species transport in a
multiple-fluid- phase porous medium 129
7.2 Modeling the transport of species i in phase a 130
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
Exercises 145
References 147
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
Exercises 161
References 163
CONTENTS XI
LINEAR ELASTICITY 165
.1 Introduction 165
'.2 Elastic Solids 166
'.3 The Linear Elastic Solid 167
A More on the Displacement Field
Decomposition 170
'.5 Strain Analysis 171
.6 Stress Analysis 173
'.7 Isotropic materials 175
'.8 Stress-strain relations for isotropic materials 177
.9 The governing differential equations 179
9.9.1 Elastodynamics 180
9.9.2 Elastostatics 180
'.10 Well-posed problems 181
9.10.1 Elastostatics 181
9.10.2 Elastodynamics 181
.11 Representation of solutions for isotropic elastic solids 182
.12 Summary 183
Exercises 184
References 186
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 fluids 199
10.13 Shallow-water theory 200
10.14 Inviscid compressible fluids 202
10.14.1 Small perturbations in a compressible fluid: the theory
of sound 203
CONTENTS
10.14.2 Initiation of motion 204
10.14.3 Discontinuous models and shock conditions 206
10.15 Summary 208
Exercises 208
References 210
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
References 215
SOME RESULTS FROM THE CALCULUS 217
B.l Notation 217
B.2 Generalized Gauss Theorem 218
PROOF OF THEOREM 221
THE BOUNDARY LAYER INCOMPRESSIBILITY
APPROXIMATION 225
INDICIAL NOTATION 229
E.l General 229
E.2 Matrix algebra 230
E.3 Applications to differential calculus 232
235