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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
