Approximation of asymptotic dispersivity of conservative solute in unsaturated heterogeneous media with steady state flow

Abstract

Abstract. The relation between the heterogeneity of hydraulic properties and effective asymptotic transport is studied for microscopically heterogeneous but macroscopically homogeneous unsaturated media with steady flow. Heterogeneity is described by the scaling of the hydraulic functions 0 (½m) and K(0). The study is based on an analytical approximation of asymptotic dispersivity under the assumption that matric potential is spatially constant. For the special case of a water-saturated medium the result of the stochastic continuum theory is recovered. When applied to several published numerical simulations, the approximated asymptotic dispersivities are found to agree well with the numerical values. By exploring the heterogeneous media for various hydraulic states, the approximation resolves some apparent inconsistencies found in the simulations. The cases considered span the entire range from perfect to zero correlation between scaling factors of matric potential and hydraulic conductivity. It is demonstrated that this correlation is crucial for the behavior of asymptotic dispersivity with changing flow rate. Since weak to moderate correlations are often found in soils, this result has significant implications for solute transport through heterogeneous soils. In this work we aim at analyzing the quantitative effect of hydraulic heterogeneity on asymptotic transport by an analytical approximation. The applicability of the approximation is evaluated for the cases considered in the numerical studies mentioned above. In contrast to the stochastic continuum approach of Yeh et al. [1985a, b] and Russo [1993], which assumes a spatially constant water content, our analytical approximation is based on the assumption of gravity flow, that is, spatially constant matric potential. This assumption is motivated by the continuity of the matric potential across any boundary which results from its gradient being a driving force of water flux. Finally, simple expressions for the asymptotic dispersivity for arbitrary parameterization of hydraulic functions, flow rates, and characteristics of heterogeneity are obtained.In the following we will often refer to the results of a particular numerical simulation. For better readability, we introduce the abbreviations given in Table 1 instead of using full citations. 2.Theoretical Framework Description of Heterogeneity by ScalingOur approach for the approximation of the asymptotic dispersivity is based on the scaling of the hydraulic variables to account for the spatial variability of the soil hydraulic functions. These variables are the matric potential era, the water content 0, and the hydraulic conductivity K, which is considered as isotropic. For simplicity we restrict our analysis to the scaling of ½m and K. However, it is straightforward to include the scaling of 0 or that of chemical parameters.Let The simulations are ordered with increasing complexity of the underlying heterogeneity. For simulations S1, S2, and S3 the correlation between scaling of matric potential and hydraul...

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