Stochastic Methods for Flow in Porous Media: Coping with Uncertainties

by Dongxiao Zhang

Overview

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.

• 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

Product Details

ISBN-13:	9780080517773
Publisher:	Elsevier Science
Publication date:	10/11/2001
Sold by:	Barnes & Noble
Format:	eBook
Pages:	368
File size:	9 MB

Read an Excerpt

This book deals with issues of fluid flow in complex geologic environments under uncertainty. The resolution of such issues is important for the rational management of water resources, the preservation of subsurface water quality, the optimization of irrigation and drainage efficiency, the safe and economic extraction of subsurface mineral and energy resources, and the subsurface storage of energy and wastes.

Hydrogeologic parameters such as permeability and porosity have been traditionally viewed as well-defined local quantities that can be assigned unique values at each point in space. Yet subsurface fluid flow takes place in a complex geologic environment whose structural, lithologic and petrophysical characteristics vary in ways that cannot be predicted deterministically in all of their relevant details. These characteristics tend to exhibit discrete and continuous variations on a multiplicity of scales, causing flow parameters to do likewise. In practice, such parameters can at best be measured at selected well locations and depth intervals, where their values depend on the scale (support volume) and mode (instrumentation and procedure) of measurement. Estimating the parameters at points where measurements are not available entails a random error. Quite often, the support of measurement is uncertain and the data are corrupted by experimental and interpretive errors. These errors and uncertainties render the parameters random and the corresponding flow equations stochastic.

The recognition that geology is complex and uncertain has prompted the development of geostatistical methods to help reconstruct it on the basis of limited data. The most common approach is to view parameter values, determined at various points within a more-or-less distinct hydrogeologic unit, as a sample from a spatially correlated random field defined over a continuum. This random field is characterized by a joint (multivariate) probability density function or, equivalently, its joint ensemble moments. The field fluctuates randomly from point to point in the hydrogeologic unit and from one realization to another in probability space. Its spatial statistics are obtained by sampling the field in real space across the unit, and its ensemble statistics are defined in terms of samples collected in probability space across multiple random realizations. Geostatistical analysis consists of inferring such statistics (most commonly the two leading ensemble moments, mean and variance-covariance) from a discrete set of measurements at various locations within the hydrogeologic unit.

Once the statistical properties of relevant random parameters have been inferred from data, the next step is to solve the corresponding stochastic flow equations. This is the subject of the present book. Following a lucid introduction to the theory of correlated random fields, the book details a number of methods for the solution of stochastic flow problems under steady state and transient, single- and two-phase conditions in porous and fractured media. The most common approach is to solve such stochastic flow equations numerically by Monte Carlo simulation. This entails generating numerous equally likely random realizations of the parameter fields, solving a deterministic flow equation for each realization by standard numerical methods, and averaging the results to obtain sample moments of the solution. The approach is conceptually straightforward and has the advantage of applying to a very broad range of both linear and nonlinear flow problems. It however has a number of conceptual and computational drawbacks. The book therefore focuses more heavily on direct methods of solution, which allow one to compute leading statistical moments of hydrogeologic variables, such as fluid pressure and flux, without having to generate multiple realizations of these variables. One direct approach is to write a system of partial differential equations satisfied approximately by leading ensemble moments and to solve them numerically. Though the approach has been known for some time, the book emphasizes its recent application to statistically nonhomogeneous media in which the moments of hydrogeologic parameters, most notably permeability, vary across the field. Such nonhomogeneity may arise from systematic spatial variability of the parameters, proximity to sources and boundaries, and conditioning on measured parameter values. The corresponding partial differential moment equations are derived in a straightforward manner and lend themselves to solution by standard finite difference methods. They form the basis for most applications and computational examples described in the book.

Another direct approach is to write exact or approximate integro-differential equations for moments of interest. Exact integro-differential moment equations have been developed in recent years for steady state and transient flows in saturated porous media and for steady state flow in unsaturated soils in which hydraulic conductivity varies exponentially with capillary pressure head (as well as for advective-dispersive solute transport in random velocity fields). In addition to being mathematically rigorous and elegant due to their exact and compact nature, they are extremely useful in revealing the nonlocal nature of stochastic moment solutions, the effect of information content (scale, quantity and quality of data) on these solutions, the conditions under which nonlocal integro-differential formulations can be localized to yield approximate partial differential moment equations, the nature and properties of corresponding local effective parameters, the relationship between localized moment equations and standard deterministic partial-differential equations of flow (and transport), and the implications of this relationship vis-a-vis the application of standard deterministic models to randomly heterogeneous media under uncertainty. The integro-differential approach relies on Green's functions, which are independent of internal sources and the magnitudes of boundary terms. Once these functions have been computed for a given boundary configuration, they can be used repeatedly to obtain solutions for a wide range of internal sources and boundary terms. The book focuses on the mechanics of how exact integro-differential moment equations are derived, approximated and solved numerically by finite elements. It points out that numerical solutions based on partial-differential and integro-differential moment formulations must ultimately be similar. Computational examples demonstrating the accuracy of the integro-differential approach when applied to complex flow problems in strongly heterogeneous media may be found in the cited literature. The hydrogeologic properties of natural rocks and soils exhibit spatial variations on a multiplicity of scales. Incorporating such scaling in geostatistical and stochastic analyses of hydrogeologic phenomena has become a major challenge. The book provides a brief but useful introduction to this fascinating subject together with some key references, which the reader is encouraged to explore.

Students, teachers, researchers and practitioners concerned with hydrogeologic uncertainty analysis will find much in this book that is instructive, useful and timely.

May 2, 2001
Shlomo P. Neuman
Regents' Professor
Department of Hydrology and Water Resources
University of Arizona, Tucson

Table of Contents

	Foreword	ix
	Preface	xiii
1	Introduction	1
1.1	Stochastic Partial Differential Equations	1
1.2	Saturated Flow with Random Forcing Terms	4
1.2.1	Moment Differential Equations	6
1.2.2	Moment Integral Equations	7
1.2.3	Solutions of Statistical Moments	9
1.2.4	Monte Carlo Simulation Method	10
1.3	Saturated Flow with Random Boundary Conditions	11
1.3.1	Direct Moment Method	12
1.3.2	Direct PDF Method	12
1.3.3	Discussion	15
1.4	Saturated Flow with Random Coefficients	16
1.4.1	Closure Approximation Method	16
1.4.2	Perturbative Expansion Method	19
1.4.3	Green's Function Method	23
1.4.4	Solutions	26
1.4.5	Discussion	28
1.5	Unsaturated Flow in Random Media	30
1.5.1	Moment Differential Equations	30
1.5.2	Solutions and Discussion	33
1.6	Scope of the Book	35
1.7	Exercises	37
2	Stochastic Variables and Processes	40
2.1	Real Random Variables	40
2.1.1	Continuous Random Variables	41
2.1.2	Discrete Random Variables	43
2.1.3	Functions of a Random Variable	44
2.1.4	Statistical Moments	45
2.1.5	Conditional Probability	47
2.1.6	Characteristic Functions and Cumulants	48
2.2	Jointly Distributed Random Variables	49
2.2.1	Joint Distributions	49
2.2.2	Marginal Distributions	50
2.2.3	Functions of Random Variables	51
2.2.4	Statistical Moments	52
2.2.5	Independent Random Variables	53
2.2.6	Conditional Probability and Expectation	54
2.2.7	Joint Characteristic Functions	56
2.2.8	Multivariate Normal and Lognormal Random Variables	58
2.2.9	The Central Limit Theorem	61
2.3	Stochastic Processes and Random Fields	62
2.3.1	Statistics of Stochastic Processes	62
2.3.2	Stationary Stochastic Processes	66
2.3.3	Nonstationary Stochastic Processes	72
2.4	Some Mathematical Tools	82
2.4.1	Dirac Delta Functions	82
2.4.2	Dirac Delta Representation of PDFs	85
2.4.3	Stochastic Continuity and Differentiation	86
2.4.4	Fourier Transform	89
2.4.5	Laplace Transform	92
2.5	Exercises	93
3	Steady-State Saturated Flow	95
3.1	Introduction	95
3.2	Perturbative Expansion Method	96
3.2.1	Moment Partial Differential Equations	98
3.3	Green's Function Method	101
3.3.1	Formal Integrodifferential Equations	101
3.3.2	Perturbed Integrodifferential Equations	107
3.3.3	Alternative Procedure	109
3.4	Numerical Solutions to the Moment Equations	111
3.4.1	Finite Differences	111
3.4.2	Finite Elements	114
3.4.3	Mixed Finite Elements	118
3.4.4	Discussion	121
3.4.5	Illustrative Examples	122
3.5	Analytical Solutions to the Moment Equations	133
3.5.1	Uniform Mean Flow in Stationary Media	133
3.5.2	Flow Subject to Recharge	145
3.5.3	Flow in Linearly Trending Media	148
3.5.4	Convergent Flow	151
3.6	Spectral methods	156
3.6.1	Stationary Spectral Method	156
3.6.2	Nonstationary Spectral Method	161
3.6.3	Application to Flow in Semiconfined Aquifers	165
3.7	Higher-Order Corrections	168
3.8	Space-State Method	173
3.8.1	Space-State Equations	174
3.8.2	Adjoint State Equations	175
3.9	Adomian Decomposition	178
3.9.1	Neumann Series Expansion	178
3.9.2	Discussion	181
3.10	Closure Approximations	182
3.10.1	Hierarchy Closure Approximation	182
3.10.2	Cumulant Neglect Hypothesis	186
3.11	Monte Carlo Simulation Method	187
3.11.1	A Historical Review	188
3.11.2	Methodology	188
3.11.3	Discussion	194
3.12	Some Remarks	195
3.12.1	Application to Gas Flow	195
3.12.2	Conditional Moment Equations	197
3.12.3	Field Theoretic Methods	199
3.13	Exercises	201
4	Transient Saturated Flow	203
4.1	Introduction	203
4.2	Moment Partial Differential Equations	204
4.2.1	Numerical Solution Strategies	207
4.2.2	Illustrative Examples	208
4.3	Moment Integrodifferential Equations	212
4.3.1	Green's Function Representation	213
4.3.2	Transient Effective Hydraulic Conductivity	215
4.4	Some Remarks	217
4.4.1	Adjoint State Equations	217
4.4.2	Laplace-Transformed Moment Equations	218
4.4.3	Quasi-Steady State Flow	218
4.5	Exercises	219
5	Unsaturated Flow	221
5.1	Introduction	221
5.1.1	Governing Equations	221
5.1.2	Constitutive Relations	222
5.1.3	Spatial Variabilities	223
5.2	Spatially Nonstationary Flow	224
5.2.1	Moment Partial Differential Equations	224
5.2.2	Adjoint State Moment Equations	235
5.3	Gravity-Dominated Flow	239
5.3.1	Spectral Analysis	240
5.3.2	Covariance Evaluation	241
5.4	Kirchhoff Transformation	242
5.5	Transient Flow in Nonstationary Media	247
5.5.1	Moment Partial Differential Equations	248
5.5.2	Discussion	258
5.6	Exercises	260
6	Two-Phase Flow	262
6.1	Introduction	262
6.2	Buckley-Leverett Displacement	265
6.2.1	Moment Partial Differential Equations	265
6.2.2	Statistical Moments of Apparent Velocity	268
6.2.3	Solutions	269
6.3	Lagrangian Approach	272
6.3.1	One-Dimensional Flow	272
6.3.2	Two-Dimensional Flow	280
6.3.3	Summary and Discussion	287
6.4	Eulerian Approach	289
6.4.1	Moment Partial Differential Equations	289
6.4.2	Unidirectional, Uniform Mean Flow	293
6.5	Exercises	295
7	Flow in Fractured Porous Media	297
7.1	Introduction	297
7.1.1	Discrete Network Models	297
7.1.2	Dual-Porosity/Double-Permeability Models	298
7.2	Saturated Flow	299
7.2.1	Moment Partial Differential Equations	301
7.2.2	Numerical Solutions	306
7.2.3	Illustrative Examples	307
7.3	Unsaturated Flow	316
7.3.1	Moment Partial Differential Equations	317
7.4	Exercises	324
	References	326
	Index	339

Editorial Reviews

"...presents a comprehensive treatment of stochastic methods for water flow in saturated aquifers and unsaturated soils in a very tutorial and accessible way." —VADOSE ZONE JOURNAL, 2006Excerpt from the Foreword by Prof. Shlomo P. Neuman:"The hydrogeologic properties of natural rocks and soils exhibit spatial variations on a multiplicity of scales. Incorporating such scaling in geostatistical and stochastic analyses of hydrogeologic phenomena has become a major challenge. ...Students, teachers, researchers and practitioners concerned with hydrogeologic uncertainty analysis will find much in this book that is instructive, useful and timely." —Prof. Shlomo P. Neuman

"Dongxiao has clearly put together an excellent intermediate/advanced text on subsurface flow problems, and I highly recommend it as a graduate level text and as a source of information for the researcher and consultant." —PURE AND APPLIED GEOPHYSICS

"...a comprehensive and lucid scientific monograph devoted to uncertainty analysis for a wide range of groundwater flow problems." —JOURNAL OF HYDRAULIC RESEARCH

"This work considers numerous stochastic methods used by researchers for flow description. The methods' positive and negative characteristics are compared and selected methods are shown in examples. ...includes problems at the end of each chapter, making it useful for students and teachers in the field, as well as researchers and practitioners." —BULLETIN AMERICAN METEOROLOGICAL SOCIETY

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