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Stochastic methods for flow and solute transport in heterogeneous porous media have become well received by the hydrology community in the last two to three decades. While several books on stochastic analysis for flow problems have been published in the last few years, this is the first book that concentrates on stochastic methods for subsurface contaminant transport. This book consists of 10 chapters by 16 authors from 12 organizations.
In Chapter 1, Govindaraju first gives a brief description of the motivation for using stochastic methods in subsurface contaminant transport problems. He then introduces some basic concepts and terminology of stochastic theories. This chapter is especially useful for those who do not have background in probability theories. In Chapter 2, Rajaram first describes the spectral representation of random fields, followed by a step by step derivation of the specific discharge spectrum, which is a basis for estimating macrodispersion in heterogeneous porous media. Expressions for macrodispersivities are formulated in some detail using both Eulerian and Lagrangian approaches. The theoretical models are validated by comparing dispersivity predictions against both field data and numerical simulations. Although macrodispersion is an important quantity to characterize solute spreading, in some cases certain features of the concentration distribution, such as fluctuation and dilution, may not be well described by macrodispersion alone. Kapoor and Kitanidis present their studies on the concentration fluctuations and dilution in Chapter 3. They begin with some fundamental concepts of seepage velocity and local dispersion and discuss the possibility of enhanced spreading due to heterogeneity of the medium properties. Stochastic flow and transport models are illustrated through an example.
In Chapter 4, Harter presents a comprehensive stochastic analysis of transport of reactive solutes. Following a brief description on subsurface contaminant sources, Harter introduces the reactive transport models, which include various types of sorption and transformation, such as sorption and radioactive decay. He then introduces several different approaches for predicting reactive contaminant transport in physically and chemically heterogeneous porous media. Effective velocity and macrodispersion are presented analytically for a number of specific cases, including spatially variable linear equilibrium sorption and nonlinear sorption with kinetic processes in heterogeneous media. In some cases, the effective retardation coefficients are also given. Harter also provides detailed formulation and discussion of stochastic analysis of breakthrough curves in terms of travel time moments. The collection of analytical results available in the literature is found useful.
Chapter 5, by Ginn, describes the formulation of multicomponent reactive transport processes using the streamtube ensemble method, which reduces the three-dimensional reactive transport simulation into a series of one-dimensional reactive transport along the streamtubes and represents the solute movement in terms of travel time and its moments by averaging streamtubes using stochastic averaging.
In Chapter 6, Das et al. present a detailed description of the theory and application of time moment analysis for studying reactive solute transport in one-dimensional homogeneous soils. Beginning with a brief history of the moment methods, they define time moments and some related functions on the basis of the probability theory, and then relate the probability density functions of the travel time to concentration distributions. The chapter then provides detailed derivations of time moments, using Aris' method of moments, for the physical and chemical nonequilibrium transport models, followed by an example for computing moments from experimental data. The authors also demonstrate applications of the method for estimating parameters of the transport equations, effective transport parameters, and breakthrough curves.
The cumulant expansion method is an alternative to study solute transport in heterogeneous porous media. One advantage of this technique is that it can be used to derive the exact second-order ensemble average for conservation equations of solute transport. In Chapter 7, Kavvas and Wu combine the cumulant expansions, time-ordered exponential operators, and Lie operator in deriving ensemble average for transport conservation equations under both unsteady flow and steady, spatially nonstationary flow fields. By comparing with Monte Carlo simulations this approach is found to give correct mean concentrations. The validity of this approach would be enhanced if the two-point moment equations (yielding concentration variance predictions) were solved and examined.
In Chapter 8, Serrano describes semigroup and decomposition methods for solving stochastic contaminant transport problems. For deterministic permeability fields with random initial and boundary conditions, the semigroup method can be directly applied. When the permeability fields are stochastic, Serrano shows that Adomain decomposition methods may be used to approximate the semigroup operator. He then illustrates the method using several cases, including convection–dispersion, scale-dependent transport, and non-Fickian transport. This chapter focuses on solute dispersion and mean concentration prediction but does not discuss the uncertainty associated with the mean prediction.
In Chapter 9, Graham addresses the extended Kalman filter for estimating hydrogeochemical parameters by recursively incorporating new measurements into the models. This method has been applied to several field-scale models, including estimation of transmissivity and recharge in the Floridan aquifer, solute transport modeling in the Borden tracer experiment, and residual NAPL distribution at Hill Air Force Base at Ogden.
Chapter 10, by Moroni and Cushman, presents a three-dimensional particle tracking velocimetry method (PTV), for quantitatively describing three-dimensional trajectories of solute particles in porous media that are homogeneous at large scale but heterogeneous at the pore scale, under non-Fickian dispersion. In particular, using their theory, they investigate the applicability of classical stochastic models for conservative solute transport.
Overall, this is an excellent book for those who are interesting in learning various stochastic techniques for predicting solute spreading in heterogeneous porous media. The book not only includes some basic concepts for those who do not have a background in stochastic theories, but also presents advanced topics pertinent to stochastic analysis of solute transport in heterogeneous subsurface environments. The stochastic techniques presented in the edited volume represent the various schools of thought. The book would be more helpful to the readership if the pros and cons of these methods were critically evaluated. Another point that we wish the book addressed better is the uncertainty associated with concentration predictions. This is not necessarily a weakness of the book. Rather, it reflects the state of the art in the field at the time the book was written.
National Laboratory, Hydrology and Geology Group (EES-6), Los Alamos, NM; Dongxiao Zhang, The University of Oklahoma
* Corresponding author (zhiming{at}lanl.gov).
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