A Coupled Time Domain Random Walk Approach for Transport in Media Characterized by Broadly-distributed Heterogeneity Length Scales

TL;DR

This paper introduces a coupled time domain random walk method for modeling solute transport in heterogeneous media with broad distributions of heterogeneity scales, accurately capturing late-time concentration behavior.

Contribution

It develops a novel analytical framework linking medium geometry to transport dynamics, enabling efficient coarse-grained and fine-scale concentration reconstructions.

Findings

Accurately reproduces late-time concentration variance scaling.

Reconstructed fine-scale concentration matches direct numerical simulations.

Demonstrates effectiveness in power-law media with heavy-tailed distributions.

Abstract

We develop a time domain random walk approach for conservative solute transport in heterogeneous media where medium properties vary over a distribution of length scales. The spatial transition lengths are equal to the heterogeneity length scales, and thus determined by medium geometry. We derive analytical expressions for the associated transition times and probabilities in one spatial dimension. This approach determines the coarse-grained solute concentration at the interfaces between regions; we derive a generalized master equation for the evolution of the coarse-grained concentration and reconstruct the fine-scale concentration using the propagator of the subscale transport mechanism. The performance of this approach is demonstrated for diffusion under random retardation in power-law media characterized by heavy-tailed lengthscale and retardation distributions. The coarse representation preserves the correct late-time scaling of concentration variance, and the reconstructed fine-scale concentration is essentially identical to that obtained by direct numerical simulation by random walk particle tracking.