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Continuous Time Random Walk (CTRW) Theory

Groundwater movement in naturally fractured and heterogeneous porous aquifers is highly complex, due to a strongly varying velocity field with multiscale correlation lengths. A key problem is how to describe tracer and contaminant movement in such systems. Realistic quantification of this movement is complicated by the uncertainty in characterization of aquifer properties.

CTRW theory accounts for the often observed non-Fickian (or scale-dependent) dispersion behavior that cannot be properly quantified by using the advection-dispersion equation (ADE). In fact, the ADE is a special, highly restrictive form of the CTRW formulation. The solutions provided here are valid for a wide range of transport regimes and dispersive behaviors.



Groundwater movement in naturally fractured and heterogeneous porous aquifers is highly complex, due to strongly varying velocity fields. A key problem is how to describe tracer and contaminant movement in such systems. Realistic quantification of this movement is complicated by the uncertainty in characterization of aquifer properties.

To date, transport has almost invariably been treated by using some form of the advection-dispersion equation (ADE), or by using related approaches, such as standard particle tracking random walk techniques, that are based on the same assumptions as the ADE. These treatments include deterministic and stochastic approaches. However, as demonstrated frequently in the literature, these approaches often fail to capture contaminant migration even in many "homogeneous" systems. This failure is most clearly evidenced by the finding of "scale-dependent dispersion": contrary to the fundamental assumptions underlying use of the ADE, dispersivity is not constant, and the very nature of the dispersive transport seems to change as a function of time or distance traveled by the contaminant. This scale-dependent behavior (also sometimes referred to as "pre-asymptotic" or "non-Gaussian") is what we shall refer to as "non-Fickian" (or "anomalous") transport. Other evidence of non-Fickian behavior lies in the often observed "unusual" early breakthrough times, and "unusual" long late time tails, in measured breakthrough curves.

The generally accepted explanation for non-Fickian transport is that heterogeneities which cannot be ignored are present at all scales. One approach to address this difficulty is to attempt to resolve the hydraulic conductivity (or velocity) field at a sufficiently high level and apply a numerical code that incorporates the ADE (using either partial differential equation or particle tracking formulations) at the scale of these blocks. However, even highly discretized systems (e.g., with block sizes of the order of 10 cubic meters in large aquifers) have not adequately captured the migration patterns, suggesting that unresolved heterogeneities also exist at these relatively small scales. Maybe somewhat surprisingly, non-Fickian transport has been observed even in small-scale, relatively "homogeneous", laboratory-scale models (Levy and Berkowitz, 2003; Cortis and Berkowitz, 2004). We note, parenthetically, that the all-too-frequent "patch" solution, involving use of a "functional form" for dispersion which allows the dispersivity to change with travel distance or time, is mathematically incorrect, and contradicts the fundamental assumptions used to develop the ADE (Berkowitz and Scher, 1995).

The Continuous Time Random Walk (CTRW) Approach

We mention here briefly the conceptual picture and mathematical language associated with the CTRW approach. Extensive discussion of the entire theory and its application can be found in the references cited in the next section.

We consider the movement of water and chemical species (i.e., tracers) moving through a geological formation. Clearly, heterogeneities occur on a broad range of scales in most geological formations. These heterogeneities can consist of fractures (joints and/or faults), variations in the rock matrix (e.g., grain sizes, mineralogy, layering, and lithology), and/or large-scale geological structures. Under an external pressure gradient, the velocity and flux distributions are determined by the liquid properties and by the structure of the aquifer heterogeneities. Tracer particles (representing the contaminant mass) transported within the water move through the formation via different paths with spatially changing velocities. Different paths are traversed by different numbers of particles. Typically, heterogeneous systems show a broader distribution of velocities than homogeneous systems.

This kind of transport can in general be represented by a joint probability density function, ψ(s,t), which describes each particle "transition" over a distance and direction, s, in time, t. Of course, particle movement occurs along continuous paths; our definition of discrete transitions here refers to a "conceptual discretization" of these paths, which can be made at as high or as low a resolution as desired. Such a function accounts naturally for particle transitions that extend over short and long distances, and over short and long times. Identification of ψ(s,t) lies at the heart of the CTRW theory. Simple asymptotic forms of ψ(s,t) which can exist include exponential decay and power law (algebraic) decay. It can be shown that exponential decay leads to Fickian transport (Margolin and Berkowitz, 2000) and an ADE-like equation. On the other hand, for example, a power law decay whose long time behavior we can approximate as ψ(s,t) ~ t-1-β (with β > 0) can generate non-Fickian transport behavior. Many other forms of ψ(s,t) have also been applied to quantify non-Fickian transport.

The CTRW theory captures a broad range of dispersive transport behaviors. A major advantage of the CTRW theory is that the resulting solutions are robust over a wide range of cases, and require a minimum of fitting parameters. It is important to recognize that the ADE can be derived as a special, limiting case of the CTRW. More specifically, the underlying conceptual picture and mathematical framework of the ADE are valid only under highly restrictive conditions. Details of these conditions, which in simple terms require a very high degree of homogeneity in the hydraulic conductivity, can be found in Berkowitz and Scher (2000), Berkowitz et al. (2002), and Cortis et al. (2004).