A simple stochastic reactive transport model

TL;DR

This paper presents a simple stochastic microscopic model for reactive transport, connecting discrete particle dynamics with classical PDE models, and explaining bimodal concentration peaks observed in solute transport.

Contribution

It introduces a discrete-time Markov chain-based model that converges to a PDE reactive transport model, providing insights into bimodal concentration behavior.

Findings

Model converges to PDE reactive transport model as time steps tend to zero.

Explains bimodal concentration peaks via bimodality of Markov binomial distribution.

Partial densities satisfy PDEs for instantaneous solute injection.

Abstract

We introduce a discrete time microscopic single particle model for kinetic transport. The kinetics is modeled by a two-state Markov chain, the transport by deterministic advection plus a random space step. The position of the particle after $n$ time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to zero linearly, we obtain a random variable $S(t)$, which is closely connected to a well known (deterministic) PDE reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed part of the solute at time $t$ do exist, and satisfy the partial differential equations.