Transport in disordered media with spatially nonuniform fields

Abstract

The theoretical treatment of transport in a disordered system in the presence of a system-wide force field F(x) or spatially varying macroscopic velocity field v(x) is developed in the framework of continuous time random walk (CTRW). The physical basis of CTRW and related fractional derivative equations relies on a mapping of the aggregate of transition rates w(s,s'), between sites s and s', in the Master equation describing the system kinetics, onto a joint probability distribution function psi(s,t). This distribution is calculated from the ensemble average of a position-dependent functional of w(s,s'); the procedure is effective when the scale of heterogeneities is much smaller than the system size. However, statistical homogeneity does not hold in the presence of large heterogeneities, which control the macroscopic v(x), or in the case of an interaction of F(x) with the transition rates. The transport equation, incorporating large-scale heterogeneity, involves the use of a local ensemble average to obtain a position-dependent psi(s,t;x); this determines a memory function, M(t;x), which is convoluted with the advection-dispersion operator. A prototype transport equation for a system with statistical inhomogeneity is developed as an integrodifferential equation. It is solved numerically for particles migrating with a steady-state Darcy velocity v(x , determined for different permeability fields and boundary conditions. The nature of the solutions as a function of key transport parameters (e.g., a characteristic time tc) is explored, and solutions are also compared to those of the advection-dispersion equation for v(x) and to a laboratory experiment. This transport equation is in contrast to the fractional Fokker-Planck equation, which is based on a decoupling of F(x) or v(x) with the transition rates w(s,s'). Further, an analytic expression for the effect of a variance of the ensemble average on the solution of the CTRW transport equation is derived.