Stochastic Methods for Flow in Porous Media

by Zhang

• ISBN13: 9780127796215
• ISBN10: 0127796215
• Format: Hardcover
• Copyright: 2001-10-03
• Publisher: Elsevier Science

Summary

Stochastic Methods for Flow in Porous Media: Coping with Uncertainties explores fluid flow in complex geologic environments. The parameterization of uncertainty into flow models is important for managing water resources, preserving subsurface water quality, storing energy and wastes, and improving the safety and economics of extracting subsurface mineral and energy resources. This volume systematically introduces a number of stochastic methods used by researchers in the community in a tutorial way and presents methodologies for spatially and temporally stationary as well as nonstationary flows. The author compiles a number of well-known results and useful formulae and includes exercises at the end of each chapter. * As never seen before: * Balanced viewpoint of several stochastic methods, including Greens' function, perturbative expansion, spectral, Feynman diagram, adjoint state, Monte Carlo simulation, and renormalization group methods * Tutorial style of presentation will facilitate use by readers without a prior in-depth knowledge of Stochastic processes * Practical examples throughout the text * Exercises at the end of each chapter reinforce specific concepts and techniques * For the reader who is interested in hands-on experience, a number of computer codes are included and discussed

Table of Contents

Foreword	ix
Preface	xiii
Introduction	1	(39)
Stochastic Partial Differential Equations	1	(3)
Saturated Flow with Random Forcing Terms	4	(7)
Moment Differential Equations	6	(1)
Moment Integral Equations	7	(2)
Solutions of Statistical Moments	9	(1)
Monte Carlo Simulation Method	10	(1)
Saturated Flow with Random Boundary Conditions	11	(5)
Direct Moment Method	12	(1)
Direct PDF Method	12	(3)
Discussion	15	(1)
Saturated Flow with Random Coefficients	16	(14)
Closure Approximation Method	16	(3)
Perturbative Expansion Method	19	(4)
Green's Function Method	23	(3)
Solutions	26	(2)
Discussion	28	(2)
Unsaturated Flow in Random Media	30	(5)
Moment Differential Equations	30	(3)
Solutions and Discussion	33	(2)
Scope of the Book	35	(2)
Exercises	37	(3)
Stochastic Variables and Processes	40	(55)
Real Random Variables	40	(9)
Continuous Random Variables	41	(2)
Discrete Random Variables	43	(1)
Functions of a Random Variable	44	(1)
Statistical Moments	45	(2)
Conditional Probability	47	(1)
Characteristic Functions and Cumulants	48	(1)
Jointly Distributed Random Variables	49	(13)
Joint Distributions	49	(1)
Marginal Distributions	50	(1)
Functions of Random Variables	51	(1)
Statistical Moments	52	(1)
Independent Random Variables	53	(1)
Conditional Probability and Expectation	54	(2)
Joint Characteristic Functions	56	(2)
Multivariate Normal and Lognormal Random Variables	58	(3)
The Central Limit Theorem	61	(1)
Stochastic Processes and Random Fields	62	(20)
Statistics of Stochastic Processes	62	(4)
Stationary Stochastic Processes	66	(6)
Nonstationary Stochastic Processes	72	(10)
Some Mathematical Tools	82	(11)
Dirac Delta Functions	82	(3)
Dirac Delta Representation of PDFs	85	(1)
Stochastic Continuity and Differentiation	86	(3)
Fourier Transform	89	(3)
Laplace Transform	92	(1)
Exercises	93	(2)
Steady-State Saturated Flow	95	(108)
Introduction	95	(1)
Perturbative Expansion Method	96	(5)
Moment Partial Differential Equations	98	(3)
Green's Function Method	101	(10)
Formal Integrodifferential Equations	101	(6)
Perturbed Integrodifferential Equations	107	(2)
Alternative Procedure	109	(2)
Numerical Solutions to the Moment Equations	111	(22)
Finite Differences	111	(3)
Finite Elements	114	(4)
Mixed Finite Elements	118	(3)
Discussion	121	(1)
Illustrative Examples	122	(11)
Analytical Solutions to the Moment Equations	133	(23)
Uniform Mean Flow in Stationary Media	133	(12)
Flow Subject to Recharge	145	(3)
Flow in Linearly Trending Media	148	(3)
Convergent Flow	151	(5)
Spectral methods	156	(12)
Stationary Spectral Method	156	(5)
Nonstationary Spectral Method	161	(4)
Application to Flow in Semiconfined Aquifers	165	(3)
Higher-Order Corrections	168	(5)
Space-State Method	173	(5)
Space-State Equations	174	(1)
Adjoint State Equations	175	(3)
Adomian Decomposition	178	(4)
Neumann Series Expansion	178	(3)
Discussion	181	(1)
Closure Approximations	182	(5)
Hierarchy Closure Approximation	182	(4)
Cumulant Neglect Hypothesis	186	(1)
Monte Carlo Simulation Method	187	(8)
A Historical Review	188	(1)
Methodology	188	(6)
Discussion	194	(1)
Some Remarks	195	(6)
Application to Gas Flow	195	(2)
Conditional Moment Equations	197	(2)
Field Theoretic Methods	199	(2)
Exercises	201	(2)
Transient Saturated Flow	203	(18)
Introduction	203	(1)
Moment Partial Differential Equations	204	(8)
Numerical Solution Strategies	207	(1)
Illustrative Examples	208	(4)
Moment Integrodifferential Equations	212	(5)
Green's Function Representation	213	(2)
Transient Effective Hydraulic Conductivity	215	(2)
Some Remarks	217	(2)
Adjoint State Equations	217	(1)
Laplace-Transformed Moment Equations	218	(1)
Quasi-Steady State Flow	218	(1)
Exercises	219	(2)
Unsaturated Flow	221	(41)
Introduction	221	(3)
Governing Equations	221	(1)
Constitutive Relations	222	(1)
Spatial Variabilities	223	(1)
Spatially Nonstationary Flow	224	(15)
Moment Partial Differential Equations	224	(11)
Adjoint State Moment Equations	235	(4)
Gravity-Dominated Flow	239	(3)
Spectral Analysis	240	(1)
Covariance Evaluation	241	(1)
Kirchhoff Transformation	242	(5)
Transient Flow in Nonstationary Media	247	(13)
Moment Partial Differential Equations	248	(10)
Discussion	258	(2)
Exercises	260	(2)
Two-Phase Flow	262	(35)
Introduction	262	(3)
Buckley---Leverett Displacement	265	(7)
Moment Partial Differential Equations	265	(3)
Statistical Moments of Apparent Velocity	268	(1)
Solutions	269	(3)
Lagrangian Approach	272	(17)
One-Dimensional Flow	272	(8)
Two-Dimensional Flow	280	(7)
Summary and Discussion	287	(2)
Eulerian Approach	289	(6)
Moment Partial Differential Equations	289	(4)
Unidirectional, Uniform Mean Flow	293	(2)
Exercises	295	(2)
Flow in Fractured Porous Media	297	(29)
Introduction	297	(2)
Discrete Network Models	297	(1)
Dual-Porosity/Double-Permeability Models	298	(1)
Saturated Flow	299	(17)
Moment Partial Differential Equations	301	(5)
Numerical Solutions	306	(1)
Illustrative Examples	307	(9)
Unsaturated Flow	316	(8)
Moment Partial Differential Equations	317	(7)
Exercises	324	(2)
References	326	(13)
Index	339