A STOCHASTIC-MODEL OF REACTIVE SOLUTE TRANSPORT WITH TIME-VARYING VELOCITY IN A HETEROGENEOUS AQUIFER

The cumulant expansion method, used previously by Sposito and Barry (1987) to derive an ensemble average transport equation for a tracer moving in a heterogeneous aquifer, is generalized to the case of a reactive solute that can adsorb linearly and undergo first-order decay. In the process we also generalize the Van Kampen (1987) result for the cumulant expansion of a multiplicative stochastic differential equation containing a time-dependent sure matrix. The resulting partial differential equation exhibits terms with field-scale coefficients that are analogous to those in the corresponding nonstochastic local-scale transport equation. There are also new terms in the third- and fourth-order spatial derivatives of the ensemble average concentration. It is demonstrated that the effective solute velocity for the aqueous concentration, not that for the total concentration (aqueous plus sorbed), is relevant for a field-scale description of solute transport. The field-scale effective solute velocity, dispersion coefficient, retardation factor, and first-order decay parameters, unlike their local-scale counterparts, are time-dependent because of autocorrelations and cross correlations among the random local solute velocity, retardation factor, and first-order decay constant. It is shown also that negative cross correlations between the random tracer solute velocity and the inverse of the local retardation factor may produce both enhanced dispersion and a temporal growth in the field-scale retardation factor. These effects are possible in any heterogeneous aquifer for which a stochastic description of aquifer spatial variability is appropriate.