On the ensemble average conservation equations for solute transport in heterogeneous aquifers

In this paper the use of combined cumulant expansion-lie operator theories in the derivation of the conservation equations for solute transport in heterogeneous aquifers is discussed. By means of these theories, starting from the Darcy-scale, the local-scale (vertical size of an aquifer) exact second-order conservation equations of nonreactive solute transport in heterogeneous groundwater aquifers for the mean solute concentration are developed under both unsteady and steady flow conditions. For transport by unsteady flow the general case of a compressible (consolidating) aquifer with compressible fluid is considered. The conservation equations for local-scale solute transport in heterogeneous aquifers for the cases of steady, spatially-nonstationary flow and steady, spatially-stationary flow are obtained as special cases of transport by unsteady flow. The developed local-scale conservation equations for the ensemble average solute concentration are in combined Eulerian-Lagrangian form for transport by unsteady flow. While the forms of these conservation equations are Eulerian, the parameters of these equations are Lagrangian. From an examination of the new macrodispersion expression one sees that the time-ordered covariance in this expression is dependent on the history of the Lagrangian trajectories of the flow field from the initial time of transport to the time of interest. These conservation equations are also nonlocal since they require information on the Lagrangian trajectories of the whole flow field since the beginning of transport. In all of the cases studied a new convection-correction term has emerged in the exact second order conservation equations of solute transport. This term can account for i) the asymmetry of the solute plume with respect to the solute core center, and ii) for the delayed position of the plume core center.