Simplified numerical model for clarifying scaling behavior in the intermediate dispersion regime in homogeneous porous media

TL;DR

This paper introduces a simplified numerical model to clarify the non-linear dispersion behavior in homogeneous porous media's intermediate regime, revealing it as a phase transition driven by pore-scale transport mechanisms.

Contribution

The study presents a turbulence-free, fractality-free model that explains the non-linearity as a phase transition, introducing a new order parameter for better understanding of dispersion regimes.

Findings

The intermediate dispersion regime is a phase transition between diffusive and ordered transport.

Taylor dispersion is not the primary factor in the non-linear behavior.

A new ratio of solute in advective vs. diffusive channels acts as an order parameter.

Abstract

The dispersion of solute in porous media shows a non-linear increase in the transition from diffusion to advection dominated dispersion as the flow velocity is raised. In the past, the behavior in this intermediate regime has been explained with a variety of models. {We present and use a simplified numerical model which does not contain any turbulence, Taylor dispersion, or fractality. With it, we show that the non-linearity in the intermediate regime nevertheless occurs. Furthermore,} we show that that the intermediate regime can be regarded as a phase transition between random, diffusive transport at low flow velocity and ordered transport controlled by the geometry of the pore space at high flow velocities. This phase transition explains the first-order behavior in the intermediate regime. A new quantifier, the ratio of the amount of solute in dominantly advective versus dominantly diffusive pore channels, plays the role of `order parameter' of this phase transition. Taylor dispersion, often invoked to explain the supra-linear behavior of longitudinal dispersion in this regime, was found not to be of primary importance. The novel treatment of the intermediate regime paves the way for a more accurate description of dispersion as a function of flow velocity, spanning the whole range of P\'eclet numbers relevant to practical applications, such as ground water remediation.