Stochastic Methods for Flow in Porous Media: Coping with Uncertainties

Abstract

DONGXIAO ZHANG. Academic Press, San Diego, CA. 2002. Hardcover, 350 pp. $89.95. ISBN: 0-12-779621-5. 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-dim...

Combined Estimation of Hydrogeologic Conceptual Model, Parameter, and Scenario Uncertainty with Application to Uranium Transport at the Hanford Site 300 Area

“…A variety of methods for propagating parameter uncertainty are available, including Monte Carlo simulation, the first-order, second-moment method (Kunstmann et al 2002;Vecchia and Cooley 1987), the stochastic response surface method (Isukapalli et al 1998), and stochastic moment methods (Dagan and Neuman 1997;Zhang 2001). Monte Carlo simulation is the most generally applicable method.…”

“…The stochastic moment methods are appealing because of their potential computational advantage over Monte Carlo simulation. Recent progress in handling conditions that introduce nonstationarities (Zhang 2001) have made these methods more generally applicable.…”

Connectivity metrics for subsurface flow and transport

“…In this framework, the main parameters controlling the degree of heterogeneity were the variance of the logarithm of the hydraulic conductivity and its correlation length. An extremely broad range of important results have been obtained using this model in the stochastic hydrogeology literature [3][4][5][6]. But several authors raised the point that the multi-Gaussian model was too restrictive and could not describe the full range of connectivity patterns that one finds in nature [7][8][9][10].…”

Steady flows driven by sources of random strength in heterogeneous aquifers with application to partially penetrating wells

“…To model such a variability it is customary to adopt the stochastic approach (an exhaustive exposition can be found in the monographs of Zhang 2002;Rubin 2003) which regards the hydraulic conductivity as a stationary RSF. As a consequence, the flow variables will result RSFs as well, and one of the main issue consists into: (1) determining their mean values, and (2) quantifying the associated uncertainty (see e.g., Beran 1968).…”

“…The configuration mostly studied in the past is that of uniform flows in the average (a situation which is ascribed to natural gradient conditions). Mean uniform flows have been investigated experimentally by controlled field tests (e.g., Sudicky 1986; LeBlanc et al 1991), and theoretically by means of numerical codes (a wide review can be found in Zhang 2002;Rubin 2003), as well as of analytical tools (Dagan 1989). There are however many hydrological applications (such as protection zones of wells, pump and treat remediation procedures, pumping tests analysis) of non uniform mean flows.…”