Solute transport in heterogeneous formations of bimodal conductivity distribution: 2. Applications

The theoretical results of Dagan and Lessoff [this issue] are applied to three types of media (Figure 1): horizontal lenses submerged in a homogeneous matrix, sparse cracks of random orientation in a matrix of contrasting permeability, and channels of high permeability at the surface of a homogeneous medium. These discrete features are modeled as sparse elliptical inclusions of arbitrary conductivity. The longitudinal macrodispersivity is determined by the methodology of part 1 as function of the parameters characterizing the medium: the conductivity ratio κ, the anisotropy ratio of the ellipsis e, the porosity ratio ϑ/ϑ0, and the volume fraction n ≪ 1 or the fracture number per unit volume. Unlike existing stochastic continuum solutions that are first order in the logconductivity variance, the model developed here applies for an arbitrary permeability variance. This is of great advantage in media with high conductivity contrasts between the matrix and the inclusions. Simple results are obtained for inclusions of low conductivity that lead to high macrodispersivity values that are underpredicted by the first-order continuum approach. In contrast, the presence of thin and highly conductive cracks leads to a finite longitudinal macrodispersivity that depends mainly on their length and the number density. An attempt is made to compare the present approach with the numerical simulations of Desbarats [1990].