On the stochastic theory of solute transport by unsteady and steady groundwater flow in heterogeneous aquifers

Abstract

In this study the exact and approximate deterministic partial differential equations of the time-space evolution of mean solute concentration, of solute concentration two-point moment and of solute concentration n-point moment for conservative solute transport by unsteady and steady groundwater flow in a heterogeneous aquifer were developed in the real time-space domain under second-order cumulant expansion. These derivations were performed by means of the cumulant expansion method, combined with the calculus for the time-ordered exponential and with the calculus for Lie operator. The mean solute transport equations which describe the time-space evolution of mean conservative solute concentration under transport by unsteady and steady groundwater flows in a heterogeneous aquifer have convective-dispersive forms whose convective and dispersive coefficients are defined precisely in terms of the mean and covariance functions of the pore flow velocity random field. However, owing to the spatial nonuniformity of groundwater flow in heterogeneous porous media a new convective coefficient, besides the standard mean velocity vector, appeared in the derived mean solute transport equations in the case of steady flows. In the case of transport by unsteady groundwater flow in a heterogeneous aquifer, the new convective coefficient (besides the standard mean velocity vector) is due to the non-zero divergence of the pore flow velocity. The fundamental reason for the convective-dispersive form of the mean transport equations is the second-order truncation of the formal cumulant expansion. In the derived mean solute transport equations the macroscopic dispersion coefficient emerges as the time integral of the covariance function of the pore flow velocity. However, both in the case of transport by unsteady groundwater flow and in the case of transport by steady spatially nonstationary-nonuniform groundwater flow the macroscopic dispersion coefficients and convective coefficients vary with spatial locations within the porous medium. Only in the case of transport by steady spatially stationary-nonuniform flow are both the macroscopic dispersion coefficient and the convective coefficient constant with respect to spatial location within the heterogenous aquifer. Therefore, only in the case of conservative solute transport by steady spatially stationary-nonuniform groundwater flow in a heterogeneous aquifer does the mean transport equation apply to all spatial locations within the aquifer with the same parameter values at each time instant.