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Hydrogeological variables of a groundwater system, for example, the hydraulic head and contaminant concentration, vary with space and time. The variability is due to spatial heterogeneities of geological materials and temporal variations of the internal and external input to and output from the groundwater system, such as groundwater recharge, evapotranspiration, and base flow to streams. These spatial heterogeneities and temporal variations are difficult if not impossible to characterize. As a result, modeling of fluid flow and solute transport in the subsurface environment is prone to errors and uncertainties. Yet, modeling of fluid flow and solute transport in soils and aquifers has traditionally been carried out almost exclusively with deterministic approaches.
A large volume of research has been published, and various theories of subsurface flow and solute transport have been developed based on stochastic (as opposed to deterministic) methods since the publication of a seminal paper by Freeze (1975). As a result, our knowledge and understanding of fluid flow and solute transport in complex heterogeneous soils and aquifers have been significantly enhanced and improved. However, applications of stochastic theories and approaches in real world problems have been limited, and they have not become a routine tool in hydrological modeling (Dagan, 2002). In a forum organized for the journal Stochastic Environmental Research and Risk Assessment (Zhang and Zhang, 2004), a series of short articles written by some of the most prominent researchers in this area of research tackle the question: Why aren't stochastic hydrogeological approaches more widely used in real-world applications? Several participants in the forum pointed out that lack of background and training and tools is one of the main reasons. I think that the book, Stochastic Methods for Flow in Porous Media: Coping with Uncertainties, written by Dongxiao Zhang, provides an excellent training tool for teaching necessary skills needed in filling the gap between the theory and application.
Dongxiao's book covers a broad spectrum of stochastic methods for both stationary and nonstationary flow problems in saturated aquifers and unsaturated soils with numerous tutorial examples and useful exercises. Chapter 1 of the book introduces some of the basic concepts in stochastic subsurface hydrology and presents the moment differential and integral equations, direct moment and PDF method, and solutions for the cases of one-dimensional flow with random forcing terms, boundary conditions, and parameters. The closure difficulties and the way to close the system at low orders were introduced in this chapter. Chapter 2 reviews the basics of stochastic variables and processes, which I found very useful for teaching the stochastic subsurface hydrology course to those who are unfamiliar with this topic. Chapter 3 deals with more general steady-state flow problems in three-dimensional aquifers. This is the most comprehensive chapter as the topic has been extensively researched during the last two decades. Most important contributions on this topic are cited here. The chapter presents several ways of deriving the moment equations. It describes almost all methods commonly used by researchers in stochastic subsurface hydrology to solve stochastic flow equations, including analytical, numerical, Green's function, and spectral methods. as well as the Monte Carlo simulations. A significant part of this chapter is also devoted to numerical methods, including the finite difference, finite element, and mixed finite element methods. Other methods mentioned in this chapter are stationary and nonstationary spectral methods, higher-order corrections to the perturbation expansions, Adomian decomposition, and renormalization group methods. Chapter 4 focuses on transient flow in three-dimensional aquifers and presents the moment partial differential and integro-differential equations and some of the solution methods introduced in previous chapters. Chapter 5 discusses unsaturated flow problems, which are perhaps most relevant to Vadose Zone Journal readers. The chapter provides a comprehensive review of research in this area. Both steady and transient problems are discussed. Chapter 6 is devoted to two-phase flows. Eulerian and Lagrangian approaches are presented, and solutions for special cases are provided. The last chapter presents moment equations and their numerical solutions to saturated and unsaturated flow in fractured media.
One shortcoming of this book—a reflection of a shortcoming of the research in this area in the past 30 yr—is that it does not address temporal variability. Uncertainties in subsurface flow modeling may be caused by both spatial variability of aquifer properties and temporal variability of internal and external sources/sinks and boundary conditions. It is this reviewer's opinion that more emphasis and research effort should be made in coping with uncertainty introduced by the temporal variability.
Overall, Dongxiao's book presents a comprehensive treatment of stochastic methods for water flow in saturated aquifers and unsaturated soils in a very tutorial and accessible way. The book will serve as an excellent textbook for a graduate course on stochastic subsurface hydrology and as a useful reference for hydrologists who are interested in this topic. I have introduced this book to several of my Ph.D. students and postdoctoral researchers. I highly recommend it to those who want to gain knowledge on this subject.
University of Iowa, 121 TH, Dep. of Geoscience, Iowa City, IA 52242
(you-kuan-zhang{at}uiowa.edu)
REFERENCES
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