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

Abstract

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.

New stochastic model for dispersion in heterogeneous porous media: 2. Gaussian plume transmission across stepwise velocity fluctuations

“…The previous article [15] showed that the stochastic dispersion model gives diffusive dispersion (with a constant dispersivity) for a constant drift velocity, in agreement with the advection-dispersion equation (ADE). On the other hand, when the drift velocity increases or decreases linearly, a time-dependent dispersivity was found.…”

“…The quantity X n (t) defined above is a generalisation of the peak position X(t) previously encountered in [15]. Introducing the abbreviated notation 1-33 for Eq.…”

“…In [15] a time lapse function w was introduced, that gives the time taken for a fluid element to move from a source point x 0 to a target point x in a variable drift velocity u(x), by the equation…”

“…The details of this modelling approach and its relationship to previous work, including other types of stochastic models for porous dispersion, was discussed in the previous article [15] and is not repeated here. The reader is also referred to the previous paper for occasional reference to equations derived there.…”

“…Inherent in the idea of a piecewise representation, is to use the solute concentration that exits from one interval of the representation, as the input value on the entrance boundary of the next region. Thus the consideration of initial value problems that was the main interest in the case of the infinite domain problems considered before [15], has to be extended to boundary value problems in this context. Section 2 below carries this through for the case of advection, and shows that for deterministic transport the solutions obtained in an unbounded problem can be rigorously combined by appropriate treatment of boundary values to piecewise velocity profiles.…”