Introduction to Modeling of Transport Phenomena in Porous Media

The main purpose of this book is to provide the theoretical background to engineers and scientists engaged in modeling transport phenomena in porous media, in connection with various engineering projects, and to serve as a text for senior and graduate courses on transport phenomena in porous media. Such courses are taught in various disciplines, e. g. , civil engineering, chemical engineering, reservoir engineering, agricultural engineering and soil science. In these disciplines, problems are encountered in which various extensive quantities, e. g. , mass and heat, are transported through a porous material domain. Often the porous material contains several fluid phases, and the various extensive quantities are transported simultaneously throughout the multiphase system. In all these disciplines, management decisions related to a system's development and its operation have to be made. To do so, the 'manager', or the planner, needs a tool that will enable him to forecast the response of the system to the implementation of proposed management schemes. This forecast takes the form of spatial and temporal distributions of variables that describe the future state of the considered system. Pressure, stress, strain, density, velocity, solute concentration, temperature, etc. , for each phase in the system, and sometime for a component of a phase, may serve as examples of state variables. The tool that enables the required predictions is the model. A model may be defined as a simplified version of the real (porous medium) system that approximately simulates the excitation-response relations of the latter.

A General Theory.- 1 The Porous Medium.- 1.1 Definition and Classification of Porous Media.- 1.1.1 Definition of a porous medium.- 1.1.2 Classification of porous media.- 1.1.3 Some geometrical characteristics of porous media.- 1.1.4 Homogeneity and isotropy of a porous medium.- 1.2 The Continuum Model of a Porous Medium.- 1.2.1 The need for a continuum approach.- 1.2.2 Representative Elementary Volume (REV).- 1.2.3 Selection of REV.- 1.2.4 Representative Elementary Area (REA).- 1.3 Macroscopic Values.- 1.3.1 Volume and mass averages.- 1.3.2 Areal averages.- 1.3.3 Relationship between volume and areal averages.- 1.4 Higher-Order Averaging.- 1.4.1 Smoothing out macroscopic heterogeneity.- 1.4.2 The hydraulic approach.- 1.4.3 Compartmental models.- 1.5 Multicontinuum Models.- 1.5.1 Fractured porous media.- 1.5.2 Multilayer systems.- 2 Macroscopic Description of Transport Phenomena in Porous Media.- 2.1 Elements of Kinematics of Continua.- 2.1.1 Points and particles.- 2.1.2 Coordinates.- 2.1.3 Displacement and strain.- 2.1.4 Processes.- 2.1.5 Material derivative.- 2.1.6 Velocities.- 2.1.7 Flux and discharge.- 2.1.8 Gauss’ theorem.- 2.1.9 Reynolds’ transport theorem.- 2.1.10 Green’s vector theorem.- 2.1.11 Pathlines, transport lines and transport functions.- 2.1.12 Velocity potential and complex potential.- 2.1.13 Movement of a front.- 2.2 Microscopic Balance and Constitutive Equations.- 2.2.1 Derivation of balance equations.- 2.2.2 Particular cases of balance equations.- 2.2.3 Constitutive equations.- 2.2.4 Coupled transport phenomena.- 2.2.5 Phase equilibrium.- 2.3 Averaging Rules.- 2.3.1 Average of a sum.- 2.3.2 Average of a product.- 2.3.3 Average of a time derivative.- 2.3.4 Average of a spatial derivative.- 2.3.5 Average of a spatial derivative of a scalar satisfying ?2G = 0.- 2.3.6 The coefficient T?*.- 2.3.7 Average of a material derivative.- 2.4 Macroscopic Balance Equations.- 2.4.1 General balance equation.- 2.4.2 Mass balance of a phase.- 2.4.3 Volume balance of a phase.- 2.4.4 Mass balance equation for a component of a phase.- 2.4.5 Balance equation for the linear momentum of a phase.- 2.4.6 Heat balance for a phase and for a saturated porous medium.- 2.4.7 Mass balance in a fractured porous medium.- 2.4.8 Megascopic balance equation.- 2.5 Stress and Strain in a Porous Medium.- 2.5.1 Total stress.- 2.5.2 Effective stress.- 2.5.3 Forces acting on the solid matrix.- 2.6 Macroscopic Fluxes.- 2.6.1 Advective flux of a single Newtonian fluid.- 2.6.2 Advective fluxes in a multiphase system.- 2.6.3 Diffusive flux.- 2.6.4 Dispersive flux.- 2.6.5 Transport coefficients.- 2.6.6 Coupled fluxes.- 2.6.7 Macrodispersive flux.- 2.7 Macroscopic Boundary Conditions.- 2.7.1 Macroscopic boundary.- 2.7.2 The general boundary condition.- 2.7.3 Boundary conditions between two porous media in single phase flow.- 2.7.4 Boundary conditions between two porous media in multiphase flow.- 2.7.5 Boundary between two fluids.- 2.7.6 Boundary with a ‘well mixed’ domain.- 2.7.7 Boundary with fluid phase change.- 2.7.8 Boundary between a porous medium and an overlying body of flowing fluid.- 3 Mathematical Statement of a Transport Problem.- 3.1 Standard Content of a Problem Statement.- 3.1.1 Conceptual model.- 3.1.2 Mathematical model.- 3.2 Multicontinuum Models.- 3.3 Deletion of Nondominant Effects.- 3.3.1 Methodology.- 3.3.2 Examples.- 3.3.3 Concluding remarks.- B Application.- 4 Mass Transport of a Single Fluid Phase Under Isothermal Conditions.- 4.1 Mass Balance Equations.- 4.1.1 The basic equation.- 4.1.2 Stationary rigid porous medium.- 4.1.3 Deformable porous medium.- 4.2 Boundary Conditions.- 4.2.1 Boundary of prescribed pressure or head.- 4.2.2 Boundary of prescribed mass flux.- 4.2.3 Semipervious boundary.- 4.2.4 Discontinuity in solid matrix properties.- 4.2.5 Sharp interface between two fluids.- 4.2.6 Phreatic surface.- 4.2.7 Seepage face.- 4.3 Complete Mathematical Model.- 4.4 Inertial Effects.- 5 Mass Transport of Multiple Fluid Phases Under Isothermal Conditions.- 5.1 Hydrostatics of a Multiphase System.- 5.1.1 Interfacial tension and capillary pressure.- 5.1.2 Capillary pressure curves.- 5.1.3 Three fluid phases.- 5.1.4 Saturation at medium discontinuity.- 5.2 Advective Fluxes.- 5.2.1 Two fluids.- 5.2.2 Two-phase effective permeability.- 5.2.3 Three-phase effective permeability.- 5.3 Mass Balance Equations.- 5.3.1 Basic equations.- 5.3.2 Nondeformable porous medium.- 5.3.3 Deformable porous medium.- 5.3.4 Buckley-Leverett approximation.- 5.3.5 Flow with interphase mass transfer.- 5.3.6 Immobile fluid phase.- 5.4 Complete Model of Multiphase Flow.- 5.4.1 Boundary and initial conditions.- 5.4.2 Complete model.- 5.4.3 Saturated-unsaturated flow domain.- 6 Transport of a Component in a Fluid Phase Under Isothermal Conditions.- 6.1 Balance Equation for a Component of a Phase.- 6.1.1 The dispersive flux.- 6.1.2 Diffusive flux.- 6.1.3 Sources and sinks at the solid-fluid interface.- 6.1.4 Sources and sinks within the liquid phase.- 6.1.5 Mass balance equation for a single component.- 6.1.6 Variable fluid density and deformable porous medium.- 6.1.7 Balance equations with immobile liquid.- 6.1.8 Fractured porous media.- 6.2 Boundary Conditions.- 6.2.1 Boundary of prescribed concentration.- 6.2.2 Boundary of prescribed flux.- 6.2.3 Boundary between two porous media.- 6.2.4 Boundary with a body of fluid.- 6.2.5 Boundary between two fluids.- 6.2.6 Phreatic surface.- 6.2.7 Seepage face.- 6.3 Complete Mathematical Model.- 6.4 Multicomponent systems.- 6.4.1 Radionuclide and other decay chains.- 6.4.2 Two multicomponent phases.- 6.4.3 Three multicomponent phases.- 7 Heat and Mass Transport.- 7.1 Fluxes.- 7.1.1 Advective flux.- 7.1.2 Dispersive flux.- 7.1.3 Diffusive flux.- 7.2 Balance Equations.- 7.2.1 Single fluid phase.- 7.2.2 Multiple fluid phases.- 7.2.3 Deformable porous medium.- 7.3 Initial and Boundary Conditions.- 7.3.1 Boundary of prescribed temperature.- 7.3.2 Boundary of prescribed flux.- 7.3.3 Boundary between two porous media.- 7.3.4 Boundary with a ‘well mixed’ domain.- 7.3.5 Boundary with phase change.- 7.4 Complete Mathematical Model.- 7.5 Natural Convection.- 8 Hydraulic Approach to Transport in Aquifers.- 8.1 Essentially Horizontal Flow Approximation.- 8.2 Integration Along Thickness.- 8.3 Conditions on the Top and Bottom Surfaces.- 8.3.1 General flux condition on a boundary.- 8.3.2 Conditions for mass transport of a single fluid phase.- 8.3.3 Conditions for a component of a fluid phase.- 8.3.4 Heat.- 8.3.5 Conditions for stress.- 8.4 Particular Balance Equations for an Aquifer.- 8.4.1 Single fluid phase.- 8.4.2 Component of a phase.- 8.4.3 Fluids separated by an abrupt interface.- 8.5 Aquifer Compaction.- 8.5.1 Integrated flow equation.- 8.5.2 Integrated equilibrium equation.- 8.6 Complete Statement of a Problem of Transport in an Aquifer.- 8.6.1 Mass of a single fluid phase.- 8.6.2 Mass of a component of a fluid phase.- 8.6.3 Saturated-unsaturated mass and component transport.- References.- Problems.