Flow Through Porous Media at the Percolation Transition

TL;DR

This study investigates fluid flow through a 2D porous medium near the percolation threshold, revealing power-law scaling behaviors and finite-size effects, with implications for understanding complex fluid systems.

Contribution

It provides detailed finite element simulations of flow near the percolation transition, confirming predicted scaling laws and analyzing finite-size effects and energy distributions.

Findings

Flow rate scales as a power law with pressure drop near percolation

Finite-size effects cause rounding of the scaling behavior

Kinetic energy distribution follows a power-law at small energies

Abstract

We study low-Reynolds-number fluid flow through a two-dimensional porous medium modeled as a Lorentz gas. Using extensive finite element simulations we fully resolve the flow fields for packing fractions approaching the percolation threshold. Near the percolation transition, we find a power-law scaling of the flow rate versus the pressure drop with an exponent of $\approx 5/2$, which has been predicted earlier by mapping the macroscopic flow to a discrete flow network [Phys. Rev. Lett. 54, 1985]. Importantly, we observe a rounding of the scaling behavior at small system sizes, which can be rationalized via a finite-size scaling ansatz. Finally, we show that the distribution of the kinetic energy exhibits a power-law scaling over several decades at small energies, originating from collections of self-similar, viscous eddies in the dead-end-channels. Our results lay the foundation for unraveling critical behavior of complex fluids omnipresent in biological and geophysical systems.