Stochastic differential equations in theory of solute transport through inhomogeneous porous media
Stochastic differential equations for solute transport arc constructed from corresponding deterministic transport equations by re-interpreting their physical parameters as random functions of space and time. A partial differential equation for the ensemble-average solute concentration then can be derived from the stochastic transport equation by a cumulant expansion method used in non-equilibrium statistical mechanics. Examples of this approach are given for both conservative and reactive solutes moving through inhomogeneous porous media. The resulting ensemble- average: transport equations are shown to be similar formally to their local-scale, deterministic analogues; but they exhibit additional, field-scale physical parameters arising from correlations among fluctuating, local-scale convective or reactive properties of the solute. Some unresolved conceptual issues attending the interpretation of the ensemble-average solute concentration and the field-scale parameters are discussed briefly.