New stochastic model for dispersion in heterogeneous porous media: 1. Application to unbounded domains

A new model of solute dispersion in porous media that avoids Fickian assumptions and that can be applied to variable drift velocities as in non-homogeneous or geometrically constricted aquifers, is presented. A key feature is the recognition that because drift velocity acts as a driving coefficient in the kinematical equation that describes random fluid displacements at the pore scale, the use of Ito calculus and related tools from stochastic differential equation theory (SPDE) is required to properly model interaction between pore scale randomness and macroscopic change of the drift velocity. Solute transport is described by formulating an integral version of the solute mass conservation equations, using a probability density. By appropriate linking of this to the related but distinct probability density arising from the kinematical SPDE, it is shown that the evolution of a Gaussian solute plume can be calculated, and in particular its time-dependent variance and hence dispersivity. Exact analytical solutions of the differential and integral equations that this procedure involves, are presented for the case of a constant drift velocity, as well as for a constant velocity gradient. In the former case, diffusive dispersion as familiar from the advection–dispersion equation is recovered. However, in the latter case, it is shown that there are not only reversible kinematical dispersion effects, but also irreversible, intrinsically stochastic contributions in excess of that predicted by diffusive dispersion. Moreover, this intrinsic contribution has a non-linear time dependence and hence opens up the way for an explanation of the strong observed scale dependence of dispersivity.