MATHEMATICAL MODELING IN SCIENCE AND ENGINEERING: An Axiomatic Approach -- CONTENTS -- Preface -- 1 AXIOMATIC FORMULATION OF THE BASIC MODELS -- 1.1 Models -- 1.2 Microscopic and macroscopic physics -- 1.3 Kinematics of continuous systems -- 1.3.1 Intensive properties -- 1.3.2 Extensive properties -- 1.4 Balance equations of extensive and intensive properties -- 1.4.1 Global balance equations -- 1.4.2 The local balance equations -- 1.4.3 The role of balance conditions in the modeling of continuous systems -- 1.4.4 Formulation of motion restrictions by means of balance equations -- 1.5 Summary -- Exercises -- References -- 2 MECHANICS OF CLASSICAL CONTINUOUS SYSTEMS -- 2.1 One-phase systems -- 2.2 The basic mathematical model of one-phase systems -- 2.3 The extensive/intensive properties of classical mechanics -- 2.4 Mass conservation -- 2.5 Linear momentum balance -- 2.6 Angular momentum balance -- 2.7 Energy concepts -- 2.8 The balance of kinetic energy -- 2.9 The balance of internal energy -- 2.10 Heat equivalent of mechanical work -- 2.11 Summary of basic equations for solid and fluid mechanics -- 2.12 Some basic concepts of thermodynamics -- 2.12.1 Heat transport -- 2.13 Summary -- Exercises -- References -- 3 MECHANICS OF NON-CLASSICAL CONTINUOUS SYSTEMS -- 3.1 Multiphase systems -- 3.2 The basic mathematical model of multiphase systems -- 3.3 Solute transport in a free fluid -- 3.4 Transport by fluids in porous media -- 3.5 Flow of fluids through porous media -- 3.6 Petroleum reservoirs: the black-oil model -- 3.6.1 Assumptions of the black-oil model -- 3.6.2 Notation -- 3.6.3 Family of extensive properties -- 3.6.4 Differential equations and jump conditions -- 3.7 Summary -- Exercises -- References -- 4 SOLUTE TRANSPORT BY A FREE FLUID -- 4.1 The general equation of solute transport by a free fluid -- 4.2 Transport processes.

4.2.1 Advection -- 4.2.2 Diffusion processes -- 4.3 Mass generation processes -- 4.4 Differential equations of diffusive transport -- 4.5 Well-posed problems for diffusive transport -- 4.5.1 Time-dependent problems -- 4.5.2 Steady state -- 4.6 First-order irreversible processes -- 4.7 Differential equations of non-diffusive transport -- 4.8 Well-posed problems for non-diffusive transport -- 4.8.1 Well-posed problems in one spatial dimension -- 4.8.2 Well-posed problems in several spatial dimensions -- 4.8.3 Well-posed problems for steady-state models -- 4.9 Summary -- Exercises -- References -- 5 FLOW OF A FLUID IN A POROUS MEDIUM -- 5.1 Basic assumptions of the flow model -- 5.2 The basic model for the flow of a fluid through a porous medium -- 5.3 Modeling the elasticity and compressibility -- 5.3.1 Fluid compressibility -- 5.3.2 Pore compressibility -- 5.3.3 The storage coefficient -- 5.4 Darcy's law -- 5.5 Piezometric level -- 5.6 General equation governing flow through a porous medium -- 5.6.1 Special forms of the governing differential equation -- 5.7 Applications of the jump conditions -- 5.8 Well-posed problems -- 5.8.1 Steady-state models -- 5.8.2 Time-dependent problems -- 5.9 Models with a reduced number of spatial dimensions -- 5.9.1 Theoretical derivation of a 2-D model for a confined aquifer -- 5.9.2 Leaky aquitard method -- 5.9.3 The integrodifferential equations approach -- 5.9.4 Other 2-D aquifer models -- 5.10 Summary -- Exercises -- References -- 6 SOLUTE TRANSPORT IN A POROUS MEDIUM -- 6.1 Transport processes -- 6.1.1 Advection -- 6.2 Non-conservative processes -- 6.2.1 First-order irreversible processes -- 6.2.2 Adsorption -- 6.3 Dispersion-diffusion -- 6.4 The equations for transport of solutes in porous media -- 6.5 Well-posed problems -- 6.6 Summary -- Exercises -- References -- 7 MULTIPHASE SYSTEMS.

7.1 Basic model for the flow of multiple-species transport in a multiple-fluid- phase porous medium -- 7.2 Modeling the transport of species i in phase α -- 7.3 The saturated flow case -- 7.4 The air-water system -- 7.5 The immobile air unsaturated flow model -- 7.6 Boundary conditions -- 7.7 Summary -- Exercises -- References -- 8 ENHANCED OIL RECOVERY -- 8.1 Background on oil production and reservoir modeling -- 8.2 Processes to be modeled -- 8.3 Unified formulation of EOR models -- 8.4 The black-oil model -- 8.5 The Compositional Model -- 8.6 Summary -- Exercises -- References -- 9 LINEAR ELASTICITY -- 9.1 Introduction -- 9.2 Elastic Solids -- 9.3 The Linear Elastic Solid -- 9.4 A More on the Displacement Field Decomposition -- 9.5 Strain Analysis -- 9.6. Stress Analysis -- 9.7 Isotropic materials -- 9.8 Stress-strain relations for isotropic materials -- 9.9 The governing differential equations -- 9.9.1 Elastodynamics -- 9.9.2 Elastostatics -- 9.10 Well-posed problems -- 9.10.1 Elastostatics -- 9.10.2 Elastodynamics -- 9.11 Representation of solutions for isotropic elastic solids -- 9.12 Summary -- Exercises -- References -- 10 FLUID MECHANICS -- 10.1 Introduction -- 10.2 Newtonian fluids: Stokes' constitutive equations -- 10.3 Navier-Stokes equations -- 10.4 Complementary constitutive equations -- 10.5 The concepts of incompressible and inviscid fluids -- 10.6 Incompressible fluids -- 10.7 Initial and boundary conditions -- 10.8 Viscous incompressible fluids: steady states -- 10.9 Linearized theory of incompressible fluids -- 10.10 Ideal fluids -- 10.11 Irrotational flows -- 10.12 Extension of Bernoulli's relations to compressible fluids -- 10.13 Shallow-water theory -- 10.14 Inviscid compressible fluids -- 10.14.1 Small perturbations in a compressible fluid: the theory of sound -- 10.14.2 Initiation of motion.

10.14.3 Discontinuous models and shock conditions -- 10.15 Summary -- Exercises -- References -- A: PARTIAL DIFFERENTIAL EQUATIONS -- A.1 Classification -- A.2 Canonical forms -- A.3 Well-posed problems -- A.3.1 Boundary-value problems: the elliptic case -- A.3.2 Initial-boundary-value problems -- References -- B: SOME RESULTS FROM THE CALCULUS -- B.1 Notation -- B.2 Generalized Gauss Theorem -- C: PROOF OF THEOREM -- D: THE BOUNDARY LAYER INCOMPRESSIBILITY APPROXIMATION -- E: INDICIAL NOTATION -- E.1 General -- E.2 Matrix algebra -- E.3 Applications to differential calculus -- Index.