Yielding to percolation: a universal scale

TL;DR

This paper introduces a universal scale for the critical pressure gradient needed to initiate flow of yield-stress fluids in porous media, validated through simulations and data, advancing the understanding of non-linear fluid flow in complex geometries.

Contribution

A new universal scale based on variational energy principles is derived for yield-stress fluid percolation, extending Darcy law applicability to non-linear rheological flows.

Findings

The universal scale accurately predicts critical pressure gradients across various obstacle geometries.

Numerical simulations confirm the scale's validity for different obstacle topologies.

The approach generalizes flow initiation criteria for yield-stress fluids in complex porous structures.

Abstract

A theoretical and computational study analysing the initiation of yield-stress fluids percolation in porous media is presented. Yield-stress fluid flows through porous media are complicated due to the non-linear rheological behaviour of this type of fluids, rendering the conventional Darcy type approach invalid. A critical pressure gradient must be exceeded to commence the flow of a yield-stress fluid in a porous medium. As the first step in generalising the Darcy law for yield-stress fluids, a universal scale based on the variational formulation of the energy equation is derived for the critical pressure gradient which reduces to purely geometrical feature of the porous media. The presented scaling is then validated by both exhaustive numerical simulations (using an adaptive finite element approach based on the augmented Lagrangian method), and also the previously published data. The considered porous media is constructed by randomised obstacles with various topologies; namely, square, circular and alternatively polygonal obstacles which are mimicked based on Voronoi tessellation of circular cases. Moreover, computations for the bi-dispersed obstacle cases are performed which further demonstrate the validity of the proposed universal scaling.