Effect of local dispersion on solute transport in randomly heterogeneous media

Recently, an exact Eulerian-Lagrangian theory of advective transport in spacetime random velocity fields was developed by one of us. We present a formal extension of this theory that accounts for anisotropic local dispersion. The resultant (conditional) mean transport equation is generally nonlocal in space-time. To assess the impact of local dispersion on the prediction of transport under uncertainty, we adopt a first-order pseudoFickian approximation for this equation. We then solve it numerically by Galerkin finite elements for two-dimensional transport from an instantaneous square source in a uniform (unconditional) mean flow field subject to isotropic local dispersion. We use a higher-order approximation to compute explicitly the standard deviation and coefficient of variation of the predicted concentrations. Our theory shows (in an exact manner), and our numerical results demonstrate (under the above closure approximations), that the effect of local dispersion on first and second concentration moments varies monotonically with the magnitude of the local dispersion coefficient. When this coefficient is small relative to macrodispersion, its effect on the prediction of nonreactive transport under uncertainty can, for all practical purposes, be disregarded. This is contrary to some recent assertions in the literature that local dispersion must always be taken into account, no matter how small.