Temporal Markov Processes for Transport in Porous Media: Random Lattice Networks

TL;DR

This paper introduces discrete temporal Markov models to accurately simulate non-Fickian transport in random porous media, demonstrating improved efficiency and applicability over previous models, especially in unstructured networks and low-velocity scenarios.

Contribution

The study develops a novel discrete temporal Markov modeling approach that captures non-Fickian dispersion in porous media with fewer assumptions and extends applicability to unstructured networks.

Findings

Models accurately reproduce spreading behavior in porous networks.

Discrete temporal Markov models outperform correlated CTRW in efficiency.

Extension of state space improves predictions in low-velocity regimes.

Abstract

Monte Carlo (MC) simulations of transport in random porous networks indicate that for high variances of the log-normal permeability distribution, the transport of a passive tracer is non-Fickian. Here we model this non-Fickian dispersion in random porous networks using discrete temporal Markov models. We show that such temporal models capture the spreading behavior accurately. This is true despite the fact that the slow velocities are strongly correlated in time, and some studies have suggested that the persistence of low velocities would render the temporal Markovian model inapplicable. Compared to previously proposed temporal stochastic differential equations with case specific drift and diffusion terms, the models presented here require fewer modeling assumptions. Moreover, we show that discrete temporal Markov models can be used to represent dispersion in unstructured networks, which are widely used to model porous media. A new method is proposed to extend the state space of temporal Markov models to improve the model predictions in the presence of extremely low velocities in particle trajectories and extend the applicability of the model to higher temporal resolutions. Finally, it is shown that by combining multiple transitions, temporal models are more efficient for computing particle evolution compared to correlated CTRW with spatial increments that are equal to the lengths of the links in the network.