Numerical study of diffusion on a random-mixed-bond lattice.

Diffusion on lattices with random mixed bonds in two and three dimensions is reconsidered using a random walk (RW) algorithm, which is equivalent to the master equation. In this numerical study the main focus is on the simple case of two different transition rates W(1),W(2) along bonds between sites. Although analysis of diffusion and transport on this type of disordered medium, especially for the case of one-bond pure percolation (i.e., W(1)=0 ), comprises a sizable subliterature, we exhibit additional basic results for the two-bond case: When the probability p of W(2) replacing W(1) in a lattice of W(1) bonds is below the percolation threshold p(c) , the mean square displacement r(2) is a nonlinear function of time t . A best fit to the lnr[(2) vs ln t plot is a straight line with the value of the slope varying with p,Delta,d , where Delta identical with W(2)/W(1) and d is the dimension, i.e., r(2) proportional, variant t(1+eta(p,Delta,d)) with eta>0 for Delta>1 . In other terms, all the diffusion (D identical with(r)(2)/2t proportional, variant t(eta)) is anomalous superdiffusion for p1 for d=2,3 . Previous work in the literature for d=2 with a different RW algorithm established an effective diffusion constant D(eff) , which was shown to scale as (p(c)-p)(1/2) . However, the anomalous nature (time dependence) of D(t) becomes manifest with an expanded regime of t , increased range of Delta , and the use of our algorithm. The nature of the superdiffusion is related to the percolation cluster geometry and Lévy walks.

Transport behavior of coupled continuous-time random walks.

The origin of anomalous or non-Fickian transport in disordered media is the broad spectrum of transition rates intrinsic to these systems. A system that contains within it heterogeneities over multiple length scales is geological formations. The continuous time random walk (CTRW) framework, which has been demonstrated to be an effective means to model non-Fickian transport features in these systems and to have predictive capacities, has at its core this full spectrum represented as a joint probability density psi(s,t) of random space time displacements (s,t) . Transport in a random fracture network (RFN) has been calculated with a coupled psi(s,t) and has subsequently been shown to be approximated well by a decoupled form psi(s,t)=F(s)psi(t) . The latter form has been used extensively to model non-Fickian transport in conjunction with a velocity distribution Phi(xi),xi identical with 1v, where v is the velocity magnitude. The power-law behavior of psi(t) proportional to (-1-beta), which determines non-Fickian transport, derives from the large xi dependence of Phi(xi) . In this study we use numerical CTRW simulations to explore the expanded transport phenomena derived from a coupled psi(s,t) . Specifically, we introduce the features of a power-law dependence in the s distribution with different Phi(xi) distributions (including a constant v) coupled by t=s(xi) . Unlike Lévy flights in this coupled scenario the spatial moments of the plumes are well defined. The shapes of the plumes depend on the entire Phi(xi) distribution, i.e., both small and large xi dependence; there is a competition between long displacements (which depend on the small xi dependence) and large time events (which depend on a power law for large xi). These features give rise to an enhanced range of transport behavior with a broader scope of applications, e.g., to correlated migrations in a RFN and in heterogeneous permeability fields. The approximation to the decoupled case is investigated as a function of the nature of the s distribution.