Synopsis
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 So
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