Flow and transport in highly heterogeneous formations: 2. Semianalytical results for isotropic media

Abstract

Flow and transport of nonreactive solutes in heterogeneous porous media is studied by adopting a multi‐indicator model of permeability structure. The porous formation is modeled as a collection of blocks of uniform permeability K implanted at random in a matrix of constant conductivity K 0 . The multi‐indicator model leads to simple semianalytical solutions based on the self‐consistent argument, which are valid for a high degree of heterogeneity. The methodology is applied to isotropic formations to derive a few statistical moments of the velocity field and of the solute particles trajectory as functions of time and of the log conductivity variance σ Y 2 . Along the common models of aquifer permeability distribution the distribution of Y = ln K is assumed to be normal. All the semianalytical results degenerate in the well‐known first‐order results when σ Y 2 ≪ 1. In particular, it is shown that the first‐order longitudinal dispersivity α L seems to hold for values of the log conductivity variance much larger than expected, up to σ Y 2 ≈ 4. This results from a compensation of errors associated with the first‐order approximation. In contrast, for σ Y 2 ≫ 1, the asymptotic α L grows exponentially with σ Y 2 . The effect of molecular diffusion is considered in a simple manner by introducing a cutoff κ C for the conductivity contrast κ. The time to reach the asymptotic α L grows with the log conductivity variance. It is thus observed that transport in highly heterogeneous formations can be characterized by a very prolonged, non‐Fickian stage, with dispersivity α L growing continuously with time. Further analysis of third‐order moment of trajectory indicates that for growing values of σ Y 2 the time needed for the plume to become Gaussian can be quite large.