A general fractal model of flow and solute transport in randomly heterogeneous porous media

Abstract

One of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous. This paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media. We describe the random field using fractional Levy motion (fLm), which more effectively represents a field with a high degree of variability. Understanding flow and solute transport in the fLm field expands the applicability of stochastic theory. Following the work of V. Di Federico and S. P. Neuman (1997) and V. Di Federico et al. (1999), we derive truncated isotropic and anisotropic power variograms for the fLm fields. The results are analogous to but more general than those of fractional Brownian motion (fBm) (V. Di Federico and S. P. Neuman (1997) and V. Di Federico et al. (1999)). For a random field, when the Levy index α in fLm is replaced by a constant of 2, fBm is recovered. This study employs first-order perturbation analyses of flow and solute transport in two-dimensional fLm fields. For the stationary, fBm, and fLm fields, macrodispersion coefficients are compared. The plume is found to grow slowest in the fLm field but will show highest concentration. The proposed fractal model is general in the sense that when the Levy index of fLm α equals 2, the results of flow and solute transport for the commonly used fBm model are recovered. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the results for the stationary exponential or Gaussian model can be well approximated. Since the proposed general fractal model has broader applications than the stationary and fBm models, it simulates flow and solute transport in more different scenarios to yield more accurate modeling results. Copyright 2007 by the American Geophysical Union.