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Spatial variability and the uncertainty in characterizing the flow domain play an important role in the transport of contaminants in porous media: they affect the pathlines followed by solute particles, the spread of solute bodies, the shape of breakthrough curves, the spatial variability of the concentration, and the ability to quantify any of these accurately. This chapter briefly reviews some basic concepts which we shall later employ for the analysis of solute transport in heterogeneous media, and also points out some issues we shall address in the subsequent chapters. Our exposition in chapters 8-10 on contaminant transport is built around the Lagrangian and the Eulerian approaches for analyzing transport. The Eulerian approach is a statement of mass conservation in control volumes of arbitrary dimensions, in the form of the advection-dispersion equation. As such, it is well suited for numerical modeling in complex flow configurations. Its main difficulties, however, are in the assignment of parameters, both hydrogeological and geochemical, to the numerical grid blocks such that the effects of subgrid-scale heterogeneity are accounted for, and in the numerical dispersion that occurs in advection-dominated flow situations. Another difficulty is in the disparity between the scale of the numerical elements and the scale of the samples collected in the field, which makes the interpretation of field data difficult. The Lagrangian approach focuses on the displacements and travel times of solute bodies of arbitrary dimensions, using the displacements of small solute particles along streamlines as its basic building block. Tracking such displacements requires that the solute particles do not transfer across streamlines. Since such mass transfer may only occur due to pore-scale dispersion, Lagrangian approaches are ideally suited for advection-dominated situations. Let us start by considering the displacement of a small solute body, a particle, as a function of time. “Small” here implies that the solute body is much smaller than the characteristic scale of heterogeneity. At the same time, to qualify for a description of its movement using Darcy’s law, the solute body also needs to be larger than a few pores. The small dimension of the solute body ensures that it moves along a single streamline and that it does not disintegrate due to velocity shear.

In this chapter we consider the diffusive and mixing properties of fluids moving in heterogeneous porous media by means of Eulerian velocity fields. We shall discuss here the basic principles of advection and dispersion in heterogeneous media from an Eulerian perspective. The common theme for the various methods we shall explore is the treatment of the concentration as an SRF. The first approach models the concentration through its statistical moments, such as the expected value and the variance, and computes them through a set of differential equations, an equation for each statistical moment. The second approach is MC based, and it computes an ensemble of physically plausible realizations of the concentration field, which can then be used for computing the statistics of the concentration, and from there the probability of events such as the concentration exceeding a threshold value at specified locations and times. Let us emphasize that stochastic modeling of contaminant transport is not just about the effects of media heterogeneity. There is also room for stochastic modeling in uniform media if there is uncertainty with regard to the media's parameters. Prediction with parameter error is discussed in chapter 13. Let us consider the case of passive solutes which are injected into a fluid body at rest.

The potential for human activities to adversely affect the environment has become of increasing concern during the past three decades. As a result, the transport and fate of contaminants in subsurface systems has become one of the major research areas in the environmental/hydrological/earth sciences. An understanding of how contaminants move in the subsurface is required to address problems of characterizing and remediating soil and groundwater contaminated by chemicals associated with industrial and commercial operations, waste-disposal facilities, and agricultural production. Furthermore, such knowledge is needed for accurate risk assessments; for example, to evaluate the probability that contaminants associated with a chemical spill will reach an aquifer. Just as importantly, knowledge of contaminant transport and fate is necessary to design “pollution-prevention” strategies. A tremendous amount of research on the transport of solutes in porous media has been generated by several disciplines, including analytical chemistry (chromatography), chemical engineering, civil/environmental engineering, geology, hydrology, petroleum engineering, and soil science. This research includes the results of theoretical studies designed to pose and evaluate hypotheses, the results of experiments designed to test hypotheses and investigate processes, and the development and application of mathematical models useful for integrating theoretical and experimental results and for evaluating complex systems. While much of the previous research has focused on transport of nonreactive solutes, it is the transport of “reactive” solutes that is currently receiving increased attention. Reactive solutes are those subject to phase-transfer processes (e.g., sorption, precipitation/dissolution) and transformation reactions (e.g., biodegradalion). Of special interest in the field of contaminant transport is so-called nonideal transport. In the most general sense, nonideal transport can be described as transport behavior that deviates from the behavior that is predicted using a given set of assumptions. A homogeneous porous medium and linear, instantaneous phase transfers and transformation reactions are the most basic set of assumptions for ideal solute transport in porous media. As discussed in a recent review, transport of reactive contaminants is often nonideal (Brusseau, 1994). The potential causes of nonideal transport include rate-limited and nonlinear mass transfer and transformation reactions, as well as spatial (and temporal) variability of material properties.