Flow and transport in highly heterogeneous formations: 2. Semianalytical results for isotropic media RID A-2321-2010

[1] 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 sigma(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 sigma(Y)(2) << 1. In particular, it is shown that the first-order longitudinal dispersivity alpha(L) seems to hold for values of the log conductivity variance much larger than expected, up to sigma(Y)(2) approximate to 4. This results from a compensation of errors associated with the first-order approximation. In contrast, for sigma(Y)(2) >> 1, the asymptotic alpha(L) grows exponentially with sigma(Y)(2). The effect of molecular diffusion is considered in a simple manner by introducing a cutoff kappa(C) for the conductivity contrast kappa. The time to reach the asymptotic alpha(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 alpha(L) growing continuously with time. Further analysis of third-order moment of trajectory indicates that for growing values of sigma(Y)(2) the time needed for the plume to become Gaussian can be quite large.