Macrodispersivity Tensor for Nonreactive Solute Transport in Isotropic and Anisotropic Fractal Porous Media: Analytical Solutions

Abstract

Using spectral stochastic theory, macrodispersivity tensors are developed for one‐, two‐, and three‐dimensional isotropic and anisotropic fractal porous media. Since natural geologic aquifers are always bounded, we introduce the concept of a maximum length scale ((Lmax). In the one‐dimensional case, the asymptotic (long‐time) longitudinal macrodispersivity (A∞) is proportional to Lmax2, and in the multidimensional cases, A∞ ∝ Lmax. In the multidimensional cases, as the fractal dimension increases, A∞ decreases, because a larger fractal dimension means the tortuosity of solute particles is larger, leading to more mixing in the transverse direction, which in turn reduces longitudinal spreading. The transverse macrodispersivities are shown to be analogous to traditional spectral stochastic results. In the multidimensional cases, Lmax is shown to be controlled by the aquifer thickness. As a result, ergodic conditions may be reached in relatively thin aquifers, allowing the use of the asymptotic macrodispersivity expressions obtained here for single realizations.