| Foreword | p. ix |
| Preface | p. xiii |
| Introduction | p. 1 |
| Stochastic Partial Differential Equations | p. 1 |
| Saturated Flow with Random Forcing Terms | p. 4 |
| Moment Differential Equations | p. 6 |
| Moment Integral Equations | p. 7 |
| Solutions of Statistical Moments | p. 9 |
| Monte Carlo Simulation Method | p. 10 |
| Saturated Flow with Random Boundary Conditions | p. 11 |
| Direct Moment Method | p. 12 |
| Direct PDF Method | p. 12 |
| Discussion | p. 15 |
| Saturated Flow with Random Coefficients | p. 16 |
| Closure Approximation Method | p. 16 |
| Perturbative Expansion Method | p. 19 |
| Green's Function Method | p. 23 |
| Solutions | p. 26 |
| Discussion | p. 28 |
| Unsaturated Flow in Random Media | p. 30 |
| Moment Differential Equations | p. 30 |
| Solutions and Discussion | p. 33 |
| Scope of the Book | p. 35 |
| Exercises | p. 37 |
| Stochastic Variables and Processes | p. 40 |
| Real Random Variables | p. 40 |
| Continuous Random Variables | p. 41 |
| Discrete Random Variables | p. 43 |
| Functions of a Random Variable | p. 44 |
| Statistical Moments | p. 45 |
| Conditional Probability | p. 47 |
| Characteristic Functions and Cumulants | p. 48 |
| Jointly Distributed Random Variables | p. 49 |
| Joint Distributions | p. 49 |
| Marginal Distributions | p. 50 |
| Functions of Random Variables | p. 51 |
| Statistical Moments | p. 52 |
| Independent Random Variables | p. 53 |
| Conditional Probability and Expectation | p. 54 |
| Joint Characteristic Functions | p. 56 |
| Multivariate Normal and Lognormal Random Variables | p. 58 |
| The Central Limit Theorem | p. 61 |
| Stochastic Processes and Random Fields | p. 62 |
| Statistics of Stochastic Processes | p. 62 |
| Stationary Stochastic Processes | p. 66 |
| Nonstationary Stochastic Processes | p. 72 |
| Some Mathematical Tools | p. 82 |
| Dirac Delta Functions | p. 82 |
| Dirac Delta Representation of PDFs | p. 85 |
| Stochastic Continuity and Differentiation | p. 86 |
| Fourier Transform | p. 89 |
| Laplace Transform | p. 92 |
| Exercises | p. 93 |
| Steady-State Saturated Flow | p. 95 |
| Introduction | p. 95 |
| Perturbative Expansion Method | p. 96 |
| Moment Partial Differential Equations | p. 98 |
| Green's Function Method | p. 101 |
| Formal Integrodifferential Equations | p. 101 |
| Perturbed Integrodifferential Equations | p. 107 |
| Alternative Procedure | p. 109 |
| Numerical Solutions to the Moment Equations | p. 111 |
| Finite Differences | p. 111 |
| Finite Elements | p. 114 |
| Mixed Finite Elements | p. 118 |
| Discussion | p. 121 |
| Illustrative Examples | p. 122 |
| Analytical Solutions to the Moment Equations | p. 133 |
| Uniform Mean Flow in Stationary Media | p. 133 |
| Flow Subject to Recharge | p. 145 |
| Flow in Linearly Trending Media | p. 148 |
| Convergent Flow | p. 151 |
| Spectral methods | p. 156 |
| Stationary Spectral Method | p. 156 |
| Nonstationary Spectral Method | p. 161 |
| Application to Flow in Semiconfined Aquifers | p. 165 |
| Higher-Order Corrections | p. 168 |
| Space-State Method | p. 173 |
| Space-State Equations | p. 174 |
| Adjoint State Equations | p. 175 |
| Adomian Decomposition | p. 178 |
| Neumann Series Expansion | p. 178 |
| Discussion | p. 181 |
| Closure Approximations | p. 182 |
| Hierarchy Closure Approximation | p. 182 |
| Cumulant Neglect Hypothesis | p. 186 |
| Monte Carlo Simulation Method | p. 187 |
| A Historical Review | p. 188 |
| Methodology | p. 188 |
| Discussion | p. 194 |
| Some Remarks | p. 195 |
| Application to Gas Flow | p. 195 |
| Conditional Moment Equations | p. 197 |
| Field Theoretic Methods | p. 199 |
| Exercises | p. 201 |
| Transient Saturated Flow | p. 203 |
| Introduction | p. 203 |
| Moment Partial Differential Equations | p. 204 |
| Numerical Solution Strategies | p. 207 |
| Illustrative Examples | p. 208 |
| Moment Integrodifferential Equations | p. 212 |
| Green's Function Representation | p. 213 |
| Transient Effective Hydraulic Conductivity | p. 215 |
| Some Remarks | p. 217 |
| Adjoint State Equations | p. 217 |
| Laplace-Transformed Moment Equations | p. 218 |
| Quasi-Steady State Flow | p. 218 |
| Exercises | p. 219 |
| Unsaturated Flow | p. 221 |
| Introduction | p. 221 |
| Governing Equations | p. 221 |
| Constitutive Relations | p. 222 |
| Spatial Variabilities | p. 223 |
| Spatially Nonstationary Flow | p. 224 |
| Moment Partial Differential Equations | p. 224 |
| Adjoint State Moment Equations | p. 235 |
| Gravity-Dominated Flow | p. 239 |
| Spectral Analysis | p. 240 |
| Covariance Evaluation | p. 241 |
| Kirchhoff Transformation | p. 242 |
| Transient Flow in Nonstationary Media | p. 247 |
| Moment Partial Differential Equations | p. 248 |
| Discussion | p. 258 |
| Exercises | p. 260 |
| Two-Phase Flow | p. 262 |
| Introduction | p. 262 |
| Buckley-Leverett Displacement | p. 265 |
| Moment Partial Differential Equations | p. 265 |
| Statistical Moments of Apparent Velocity | p. 268 |
| Solutions | p. 269 |
| Lagrangian Approach | p. 272 |
| One-Dimensional Flow | p. 272 |
| Two-Dimensional Flow | p. 280 |
| Summary and Discussion | p. 287 |
| Eulerian Approach | p. 289 |
| Moment Partial Differential Equations | p. 289 |
| Unidirectional, Uniform Mean Flow | p. 293 |
| Exercises | p. 295 |
| Flow in Fractured Porous Media | p. 297 |
| Introduction | p. 297 |
| Discrete Network Models | p. 297 |
| Dual-Porosity/Double-Permeability Models | p. 298 |
| Saturated Flow | p. 299 |
| Moment Partial Differential Equations | p. 301 |
| Numerical Solutions | p. 306 |
| Illustrative Examples | p. 307 |
| Unsaturated Flow | p. 316 |
| Moment Partial Differential Equations | p. 317 |
| Exercises | p. 324 |
| References | p. 326 |
| Index | p. 339 |
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