S. C. Lessoff

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].

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].

A note on the influence of a constant velocity boundary condition on flow and transport in heterogeneous formations

Fluid flow and solute transport in an anisotropic, heterogeneous porous medium with mean flow normal to a constant-flux boundary are considered. The statistical moments of flow and transport variables are determined at second order in log conductivity fluctuation, and they are expressed in terms of the log conductivity variance and integral scales, the mean flow velocity, and the distance from the boundary. The variance of the longitudinal and transverse components of velocity as well as hydraulic head variance and the longitudinal macrodispersivity are analyzed for a bounded medium with axisymmetric, Gaussian log conductivity covariance structure. In this case, all of the moments can be solved by means of a single numerical quadrature. The constant flow boundary increases the variability of head and of flow transverse to the mean flow direction and causes a reduction of the macrodispersivity in a zone adjacent to the boundary. Our results should be useful for the design and testing of numerical models and have important implications for surface infiltration of solutes.

Solute transport in infiltration-redistribution cycles in heterogeneous soils

An analytic model of transient unsaturated infiltration is presented. The model is based on the column conceptualization of flow and transport in unsaturated soil which is expanded here to incorporate repeated infiltration and redistribution stages. The transport of reactive solute is modeled by assuming three mechanisms: advection by gravitational water flow, equilibrium sorption and linear decay. Solutions of the flow and transport equations are derived for multiple infiltration-redistribution cycles and for Dirac and finite pulse solute applications. Expressions are derived for average moisture content and for average concentration regarding the soil saturated conductivity a random value.