Numerical study of Darcy's law of yield stress fluids on a deep tree-like network

TL;DR

This study uses an exact numerical approach to analyze the flow of yield stress fluids in deep, tree-like porous networks, confirming theoretical predictions for large structures beyond previous moderate-sized tests.

Contribution

It introduces an adapted algorithm for precise simulation of flow in extensive tree-like structures, extending the validation of asymptotic predictions to much larger systems.

Findings

Confirmed asymptotic predictions for large trees

Validated non-linear flow behavior in deep porous networks

Extended previous moderate-size simulations to thousands of generations

Abstract

Understanding the flow dynamics of yield stress fluids in porous media presents a substantial challenge. Both experiments and extensive numerical simulations frequently show a non-linear relationship between the flow rate and the pressure gradient, deviating from the traditional Darcy law. In this article, we consider a tree-like porous structure and utilize an exact mapping with the directed polymer (DP) with disordered bond energies on the Cayley tree. Specifically, we adapt an algorithm recently introduced by Brunet et al. [Europhys. Lett. 131, 40002 (2020)] to simulate exactly the tip region of branching random walks with the help of a spinal decomposition, to accurately compute the flow on extensive trees with several thousand generations. Our results confirm the asymptotic predictions proposed by Schimmenti et al. [Phys. Rev. E 108, L023102 (2023)], tested therein only for moderate trees of about 20 generations.