Multiphase Flow and Transport Processes in the Subsurface

1. Introduction.- 1.1 Problem classification.- 1.2 Problem formulation and exact definition of the subject.- 1.2.1 Application of the different models.- 1.2.2 Remarks on the term model.- 1.2.3 Objective and structure of this book.- 2. Fundamental principles of conceptual modeling.- 2.1 Preliminary remarks.- 2.1.1 General remarks.- 2.1.2 Definitions and fundamental terms.- 2.2 System properties.- 2.2.1 Mass and mole fractions.- 2.2.2 Density.- 2.2.3 Viscosity.- 2.2.4 Specific enthalpy, specific internal energy.- 2.2.5 Surface tension.- 2.2.6 Specific heat capacity.- 2.3 Phase state, phase transition, phase change.- 2.3.1 Phase state.- 2.3.2 Phase transition, phase change.- 2.4 Capillarity.- 2.4.1 Microscopic capillarity.- 2.4.2 Macroscopic capillarity.- 2.4.3 Capillarity in fractures.- 2.5 Hysteresis.- 2.6 Definition of different saturations.- 2.7 Relative permeability.- 2.7.1 Permeability.- 2.7.2 Relative permeability at the micro scale.- 2.7.3 Relative permeability at the macro scale.- 2.7.4 Relative permeability-saturation relation in fractures.- 2.7.5 Fracture-matrix interaction.- 2.8 Pressure and temperature dependence of porosity.- Mathematical modeling.- 3.1 General balance equation.- 3.1.1 Preconditions and assumptions.- 3.1.2 The Reynolds transport theorem in integral form.- 3.1.3 Derivation of the general balance equation.- 3.1.4 Initial and boundary conditions.- 3.1.5 Choice of the primary variables.- 3.2 Continuity equation per phase.- 3.2.1 Time derivative.- 3.3 Momentum equation and Darcy's law.- 3.3.1 General remarks.- 3.3.2 Darcy's law of single-phase flow.- 3.3.3 Generalization of Darcy's law for multiphase flow.- 3.4 General form of the multiphase flow equation.- 3.4.1 Pressure formulation.- 3.4.2 Pressure-saturation formulation.- 3.4.3 Saturation formulation.- 3.4.4 Mathematical modeling for three-phase infiltration and remobilization processes.- 3.5 Transport equation.- 3.5.1 Basic transport equation.- 3.5.2 Transport in a multiphase system.- 3.5.3 Description of the mass transfer between phases.- 3.5.4 Multicomponent transport processes in the gas phase.- 3.6 Energy equation.- 3.7 Multiphase/multicomponent system.- 4. Numerical modeling.- 4.1 Classification.- 4.1.1 Problem and special solution methods.- 4.1.2 Fundamentals of discretization.- 4.1.3 Conservative discretization.- 4.1.4 Weighted residual method.- 4.2 Finite element and finite volume methods.- 4.2.1 Spatial discretization.- 4.2.2 Choice of element types.- 4.2.3' Galerkin finite element method.- 4.2.4 Sub domain collocation - finite volume method.- 4.2.5 Time discretization.- 4.3 Linearization of the multiphase problem.- 4.3.1 Weak nonlinearities.- 4.3.2 Strong nonlinearities.- 4.3.3 Handling of the nonlinearities.- 4.3.4 Example: Linearized two-phase equation.- 4.4 Discussion of the instationary hyperbolic (convective) transport equation.- 4.4.1 Classification of hyperbolic differential equations.- 4.4.2 A linear hyperbolic transport equation.- 4.4.3 A quasilinear hyperbolic transport equation - Buckley-Levereit equation.- 4.4.4 Analytical solutions for the Buckley-Lev ereit problem.- 4.5 Special discretization methods.- 4.5.1 Motivation.- 4.5.2 Upwind method - finite difference method.- 4.5.3 Explicit upwind method of first order - Fully Upwind.- 4.5.4 Multidimensional upwind method of first order.- 4.5.5 Explicit upwind method of higher order - TVD techniques.- 4.5.6 Implicit upwind method of first order - Fully Upwind.- 4.5.7 Petrov-Galerkin finite element method.- 4.5.8 Additional remarks on conservative discretization.- 4.5.9 Flux-corrected method.- 4.5.10 Mixed-hybrid finite element methods.- 5. Comparison of the different discretization methods.- 5.1 Discretization.- 5.1.1 Finite element Galerkin method.- 5.1.2 Sub domain collocation finite volume method (box method).- 5.2 Boundedness principle - discussion of a monotonic solution.- 5.3 Comparative study of the different methods in homogeneous porous media.- 5.3.1 Multiphase flow without capillary pressure effects - Buckley-Lev ereit problem.- 5.3.2 Multiphase flow with capillary pressure effects - McWhorter problem.- 5.4 Heterogeneity effects.- 5.5 Comparative study of the methods for flow in heterogeneous porous media.- 5.6 Five-spot waterflood problem.- 6. Test problems - applications.- 6.1 DNAPL-Infiltration.- 6.2 LNAPL-Infiltration.- 6.3 Non-isothermal multiphase/multicomponent flow.- 6.3.1 Heat pipe.- 6.3.2 Study of bench-scale experiments.- 7. Final remarks.