A stochastic approach to the filling dynamics of a porous medium: full/empty pores duality symmetry and the emergence of Darcy's law

TL;DR

This paper develops a stochastic model for fluid filling in porous media, revealing the emergence of Darcy's law, a diffusive filling dynamics, and a novel symmetry related to pore states, with steady states following a Fermi-Dirac distribution.

Contribution

It introduces a microscopic stochastic framework that links pore-scale dynamics to macroscopic flow laws, highlighting symmetry properties and steady state distributions.

Findings

Derivation of pore-scale dynamical equations

Emergence of Darcy's law from microscopic principles

Prediction of Fermi-Dirac type steady states

Abstract

A stochastic approach to the filling dynamics of an open topology porous structure permeated with a perfectly wetting fluid is presented. From the discrete structure of the disordered voids network with only nearest neighbors links, we derive the "microscopic" (at the pores scale) dynamical equations governing the filling dynamics of the coupled pores and the fluid pressure dynamics. The model yields two fundamental consequences. The first consequence regards the emergence of Darcy's law and the dependence of the predicted permeability with the voids network topology. The second one is the prediction of a diffusive dynamics for the degrees of freedom of the pores filling. These equations exhibit a new type of symmetry manifested by their invariance under the full/empty pores duality transformation jointly with the velocity reversal. Non-trivial steady non-equilibrium pores filling states are also obtained and found to follow a Fermi-Dirac type law. The analogy with the single occupation of lattice sites by fermions is highlighted together with the corresponding hole-particle symmetry.