Mathematical Problems in Engineering / 2013 / Article / Tab 1

Research Article

Challenges in the Application of Fractional Derivative Models in Capturing Solute Transport in Porous Media: Darcy-Scale Fractional Dispersion and the Influence of Medium Properties

Table 1

Comparison of the TFDM (2) and the standard CTRW model reviewed by Berkowitz et al. [9]. Note that the standard CTRW defines various empirical memory functions, and the memory function listed below is the most commonly used one (which is a truncated power law similar to the memory function used in the TFDM (2a) and (2b)). In the CTRW model, the parameter denotes the ratio of the two characteristic times and , and is the incomplete Gamma function.

ComparisonTFDM (2a) and (2b)The standard CTRW [9]

Physical theory behind the modelScaling limit of CTRWThe general master equation
Form of the memory function
Number and value of parameters to capture sub-diffusion3 ( ), with 4 ( ), with
Mechanism for sub-diffusionDiffusionSlow advection
Modeling super-diffusionApplicable with N/A
Modeling the mixed diffusionApplicable with and N/A
Multidimensional extensionMultiscaling index [12]No multiscaling index
Spatial variability of transport and can vary in space and are spatial averages

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