Simulation of Non-Fickian Transport in Geological Formations With Variable-Scale Heterogeneities

We study solute transport through heterogeneous media by use of continuous time random walk (CTRW) theory. Transport is governed by a joint probability density ψ (s,t), which characterizes tracer particle displacements s with associated times t. Previous work has shown the CTRW theory to be a highly effective transport framework to account for non-Fickian transport in field, laboratory and numerical experiments. Here, we introduce a number of innovations that allow for general solution of the CTRW for arbitrary ψ (s,t) and boundary conditions in 1-3 spatial dimensions. While in many cases, the transition times and distances are governed strictly by the flow field, they can in other cases be strongly influenced by mechanisms such as tracer diffusion into and out of ``stagnant'' zones of the medium and/or adsorption/desorption from the rock surfaces. All of these mechanisms, in addition to the flow field, can be specified, either implicitly or explicitly, in the determination of ψ (s,t), which then can account for a wide range of transport behaviors. By treating unresolved, small-scale heterogeneities (residues) probabilistically with the CTRW formalism, and large-scale heterogeneity variations (trends) deterministically, we develop and solve a Fokker-Planck equation that contains a memory term and a generalized concentration flux term. The advection-dispersion equation is a special case of this equation. The parameters defining these terms are measurable quantities. Our calculations demonstrate long tailing arising (principally) from the memory term, and effects on arrival times that are controlled largely by the generalized concentration flux term. The impact of these extensions to CTRW theory is to provide a means to calculate transport of both passive and sorbing (reactive) tracers in non-stationary geological formations.