Flow and Transport in Porous Formations

In the mid-seventies, a new area of research has emerged in subsurface hydrology, namely sto­ chastic modeling of flow and transport. This development has been motivated by the recognition of the ubiquitous presence of heterogeneities in natural formations and of their effect upon transport and flow, on the one hand, and by the vast expansion of computational capability provided by elec­ tronic machines, on the other. Apart from this, one of the areas in which spatial variability of for­ mation properties plays a cardinal role is of contaminant transport, a subject of growing interest and concern. I have been quite fortunate to be engaged in research in this area from its inception and to wit­ ness the rapid growth of the community and of the literature on spatial variability and its impact upon subsurface hydrology. In view of this increasing interest, I decided a few years ago that it would be useful to present the subject in a systematic and comprehensive manner in order to help those who wish to engage themselves in research or application of this new field. I viewed as my primary task to analyze the large scale heterogeneity of aquifers and its effect, presuming that the reader already possesses a background in traditional hydrology. This is achieved in Parts 3, 4 and 5 of the text which incorporate the pertinent material.

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.