Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers

Abstract

The dispersion of a solute plume resulting from unsteady flow in three‐dimensional, heterogeneous porous media was analyzed using stochastic continuum theory. Asymptotic stochastic solutions of the perturbed unsteady flow and solute transport equations were used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three‐dimensional, statistically anisotropic input covariance describing the hydraulic conductivity and an input covariance describing the temporal variability of the mean hydraulic gradient. The flow equation was approximated by neglecting the influence of internal storage due to medium and fluid compressibility (specific storage, S S ≈ 0). The predictive expression for the macrodispersivity tensor is the sum of two components: a spatial variability component identical to previous steady state theory and an unsteady component. Two special cases of unsteady flow were examined: variation only in the magnitude and variation only in the direction of the hydraulic gradient. Gradient magnitude variation produces a slightly larger longitudinal macrodispersivity than does the steady flow case, whereas gradient direction variation produces a significantly larger transverse macrodispersivity. Longitudinal and horizontal transverse macrodispersivities predicted with the unsteady stochastic theory were shown to be of a magnitude similar to observed values from the Borden, Cape Cod, and Columbus tracer experiments.