Flow and Transport in Porous Formations

1 Mathematical Preliminaries: Elements of Probability Theory and Random Functions.- 1.1 Random variables. Statistical moments.- 1.2 Joint probability distributions. Conditional probability. Multivariate normal distributions.- 1.3 Random functions. Stationarity. Isotropy.- 1.4 Differentiation and integration of random functions Microscale and integral scale.- 1.5 Differentiation of random discontinuous functions.- 1.6 Spectral methods.- 1.7 Random functions of stationary increments.- 1.8 Conditional Gaussian probability and interpolation by kriging.- 1.9 Spatial averages of random functions.- 1.10 The ergodic hypothesis.- 2 The Laboratory Scale (Homogeneous Media).- 2.1 Introduction.- 2.2 Geometry of porous media and space averaging.- 2.2.1 Geometry of porous media.- 2.2.2 Space averages and macroscopic variables.- 2.3 The microscopic equations of flow and transport.- 2.4 Averaging of derivatives of microscopic variables.- 2.5 Macroscopic variables and macroscopic equations of mass and energy conservation.- 2.5.1 Definition of macroscopic variables.- 2.5.2 The macroscopic equation of state and of mass conservation.- 2.5.3 The macroscopic equation of conservation of energy.- 2.5.4 The macroscopic equation of solute mass conservation.- 2.6 The macroscopic equations of conservation of momentum.- 2.7 The constitutive equation of heat transfer (effective heat conductivity).- 2.7.1 Definitions and experimental evidence.- 2.7.2 Theoretical derivation of the constitutive equation and of bounds of effective conductivity.- 2.7.3 Evaluation of the effective heat conductivity with the aid of models of porous media.- 2.8 The constitutive equation of mass transfer (effective diffusion coefficient).- 2.9 Darcy’s law.- 2.9.1 Definitions and experimental evidence.- 2.9.2 Theoretical derivation of Darcy’s law.- 2.9.3 Derivation of permeability with the aid of models.- 2.9.4 Generalizations of Darcy’s law.- 2.10 Convective-diffusive transport (hydrodynamic dispersion).- 2.10.1 Definitions and experimental evidence.- 2.10.2 The Taylor-Aris theory of dispersion in a tube.- 2.10.3 Saffman’s (1960) model of dispersion in porous media.- 2.10.4 Some limitations and generalizations of the equation of dispersion.- 2.11 Summary of macroscopic equations of water flow.- 2.11.1 Rigid solid matrix, incompressible and homogeneous fluid.- 2.11.2 Rigid matrix, incompressible but nonhomogeneous fluid.- 2.11.3 Deformable elastic matrix, homogeneous and elastic fluid.- 2.12 Summary of macroscopic equations of solute and heat transfer.- 2.12.1 Solute transport.- 2.12.2 Heat transport.- 2.13 Flow and transport boundary conditions.- 2.13.1 Flow condition at an impervious boundary.- 2.13.2 Boundary between two homogeneous porous bodies.- 2.13.3 Boundary with free fluids.- 2.13.4 A free-surface (water-table, phreatic surface).- 2.13.5 Boundary conditions for solute and heat transport.- 3 Water Flow at the Local (Formation) Scale.- 3.1 Introduction.- 3.2 The heterogeneous structure of aquifers at the local (formation) scale.- 3.2.1 A few field findings.- 3.2.2 Statistical representation of heterogeneous formations and their classification.- 3.2.3 A few examples of covariances Cy(r).- 3.2.4 Statistical properties of the space average $$\bar y$$.- 3.2.5 Effect of parameters estimation errors and summarizing comments.- 3.3 General formulation of the direct problem and of the equations of flow.- 3.3.1 General statement of the direct problem.- 3.3.2 A few general observations on the stochastic problem.- 3.4 The effective hydraulic conductivity.- 3.4.1 Steady uniform flow: general statement and absolute bounds.- 3.4.2 Small perturbation, first-order approximation of Kef.- 3.4.3 The self-consistent approach.- 3.4.4 Effective conductivity of a two-phase formation.- 3.4.5 Influence of boundary on effective conductivity.- 3.4.6 The influence of nonuniformity of average flow.- 3.4.7 Effective conductivity and storativity in unsteady flow through compressible formations.- 3.5 Solutions of the mean flow equations (examples of exact solutions).- 3.5.1 General.- 3.5.2 Illustration of exact solutions of a few classes of flow.- 3.6 Solutions of the mean flow equations (approximate methods).- 3.6.1 The method of singularities.- 3.6.2 Linearization of the free-surface condition.- 3.6.3 How through layered formations of large conductivity contrast.- 3.6.4 The shallow-water flow approximation (Dupuit-Forcheimer-Boussinesq).- 3.7 Second-order statistical moments of the flow variables.- 3.7.1 Introduction.- 3.7.2 Steady, uniform in the average, flow in unbounded formations. First-order approximation.- 3.7.3 The effect of nonlinearity of logconductivity variance upon head covariances.- 3.7.4 The effect of boundaries on head covariances.- 3.7.5 Specific discharge covariances.- 3.7.6 The effect of space averaging upon specific discharge variance.- 3.7.7 The effect of parameters estimation errors.- 4 Solute Transport at the Local (Formation) Scale.- 4.1 Introduction.- 4.2 Afew field findings.- 4.3 The conceptual model.- 4.3.1 General.- 4.3.2 The statistics of fluid particles displacements (the Lagrangian framework).- 4.3.3 The statistics of particles displacements (the Eulerian framework).- 4.3.4 The concentration expected value.- 4.3.5 The concentration variance.- 4.3.6 The concentration spatial moments.- 4.4 A few numerical simulations of solute transport in heterogeneous formations.- 4.5 Transport through stratified formations.- 4.5.1 Introduction.- 4.5.2 Steady flow parallel to the bedding.- 4.5.3 Flow tilted with respect to the bedding.- 4.6 Transport informations of three-dimensional heterogeneous structures.- 4.6.1 Introduction.- 4.6.2 General formulation of the first-order solution.- 4.6.3 Time dependent “macrodispersivity” (first-order approximation, high Pe).- 4.6.4 Asymptotic, large travel time, “macrodispersivity” (first-order solution).- 4.7 Two-dimensional transport and comparison with afield experiment.- 4.8 Effects of nonlinearity and unsteadiness.- 4.8.1 Effect of nonlineanity in ?y2 upon transport.- 4.8.2 Unsteady and nonuniform mean flow.- 4.9 Transport of reactive solutes. Effect of parameters estimation errors..- 4.9.1 Transport of reactive solutes.- 4.9.2 The effect of parameters estimation errors.- 5 Flow and Transport at the Regional Scale.- 5.1 Introduction.- 5.2 Analysis of field data and statistical characterization of heterogeneity.- 5.2.1 A few field findings.- 5.2.2 Definition of hydraulic properties at the regional scale.- 5.2.3 Statistical representation of heterogeneity.- 5.3 Mathematical statement of the direct problem.- 5.4 Effective properties and the solutions of the equations of mean flow.- 5.4.1 Effective transmissivity and storativity (confined flow, unconditional probability).- 5.4.2 Effective transmissivity (unconfined flow, unconditional probability).- 5.4.3 A few solutions of the mean flow equations.- 5.5 Second-order statistical moments of the flow variables. The effect of conditioning.- 5.5.1 Introduction.- 5.5.2 First-order approximation for steady flow (unconditional probability).- 5.5.3 Solution of the direct problem in steady flow by conditional probability.- 5.5.4 The effect of boundaries on head covariances (unconditional probability).- 5.5.5 Effects of nonlinearity in ?y2 and of unsteadiness.- 5.6 The inverse (identification) problem.- 5.6.1 Introduction.- 5.6.2 Mathematical statement of the identification problem and the structure of its solution.- 5.6.3 Stochastic identification of aquifer parameters by first-order approximation.- 5.6.4 Stochastic nonlinear approach based on a numerical solution (Carrera and Neumann, 1986).- 5.7 Transport at the regional scale.- 5.7.1 General.- 5.7.2 Transport from “non-point sources”.- 5.7.3 Transport from “point sources”.- 5.8 Modeling transport by travel time approach.- 5.8.1 General.- 5.8.2 One-dimensional transport.- 5.8.3 Two-dimensional transport.