Solute transport in heterogeneous formations of bimodal conductivity distribution: 1. Theory

Transport of a conservative solute takes place in a formation made up from a matrix of conductivity K0 and porosity ϑ0 and inclusions of properties K, ϑ. For given inclusions shape, the system is characterized by the two parameters κ = K/K0 and the inclusions volume fraction n. In the past, approximate solutions of the flow and transport problems were obtained under the limit of low variability, i.e., κ − 1 ≪ 1, and arbitrary n [Rubin, 1995]. The present study aims at solving the problem under the opposite limit of a dilute system, i.e., n ≪ 1 and arbitrary κ. We are particularly interested in elongated inclusions (high length/thickness ratio) of high‐permeability contrast to the matrix. Such configurations are related to applications in which lenses or cracks are present in a medium of highly different conductivity (Figure 1). The basic procedure was developed by Eames and Bush [1999] for cylindrical or spherical inclusions, with no porosity contrast. They compute the macrodispersion coefficient, for advective transport past a large number of inclusions located at random. It is based on the solution for the distortion of a material surface of marked particles, moving past an individual inclusion in an unbounded domain and with uniform flow at infinity. In the present study we extend the approach to inclusions of arbitrary porosity and elliptical shape, characterized by the parameter e, the ratio between the small and large axes, with emphasis on e ≪ 1. We present the analytical solution of the flow problem and the procedure, requiring two quadratures, to calculate the macrodispersivity. Analytical solutions are obtained for two particular limits: κ ≪1 and κ ≃ 1. The latter is compared with the limit n ≪ 1 of the solution of Rubin [1995]. The theoretical results are applied to a few cases of hydrological interest [Lessoff and Dagan, this issue].