On solute transport in a heterogeneous porous formation under saturated and unsaturated water flows

Abstract

The results of a recent study of solute transport in a hypothetical scale‐heterogeneous soil under transient unsaturated flow conditions (Russo, this issue) suggested that the time‐dependent components of the spatial covariance tensor, based on a single realization of the solute concentration in unsaturated flow, are in relatively good agreement with the components of the spatial covariance tensor, based on the ensemble‐average solute concentration derived by Dagan (1984), for solute transport under steady saturated flow. Furthermore, the asymptotic value of the longitudinal component of the dispersivity tensor λzz, estimated from the horizontally averaged solute concentration in the unsaturated flow domain, was found to be in good agreement with the asymptotic λzz derived by Dagan (1982, 1984) and Gelhar and Axness (1983) for saturated flow. These findings are explored in a general Lagrangian formulation, using small‐perturbation, first‐order approximations. The results of the analysis of the velocity field in the unsaturated flow domain of the hypothetical scale‐heterogeneous soil of Russo (this issue) suggest that there was good agreement between the result of the Lagrangian kinematic analysis and the result of the analysis of the spatial moments of the concentration distribution, in agreement with the stochastic theory of transport in saturated flow. This is explained by the finding that under unsaturated flow conditions there is an increase in the variability of the hydraulic conductivity K (and concurrently an increase in the variability of the velocity Vz), accompanied by a decrease in the correlation scale of K (and concurrently a decrease in the correlation scale of Vz) relative to the saturated conditions. The increase in the variability of the unsaturated conductivity stemmed from the variability in the water content θ and the strong nonlinear dependence of K on θ, while the accompanying decrease in its correlation scale stemmed from the negative correlation between F = log Ks and θ, which persisted for a relatively large separation distance in the longitudinal vertical direction. Consequently, the product between the conductivity variance and its correlation scale is of similar magnitude for both saturated and unsaturated flow regimes. The results suggest that in spite of the complexity of unsaturated flow, as compared with saturated flow, it might be amenable to a similar analysis.