# The breakdown of Darcy's law in a soft porous material

## TL;DR

This study reveals that flow through deformable porous media deviates from Darcy's law, showing nonlinear behavior influenced by material softness, and introduces a universal relation involving pressure difference and shear modulus.

## Contribution

The paper develops a theoretical model explaining the super-linear flow behavior in soft porous materials and confirms a universal relation involving pressure and shear modulus.

## Key findings

- Flow flux is nonlinear and steeper than linear with pressure difference.
- Flow permeability depends on the ratio of pressure difference to shear modulus.
- Theoretical predictions match numerical simulation results.

## Abstract

We perform direct numerical simulations of the flow through a model of a deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, with elastic shear modulus $G$, immersed in a liquid. We use a two-phase approach: the liquid phase is a viscous fluid and the solid phase is modeled as an incompressible viscoelastic material, whose complete nonlinear structural response is considered. We observe that the Darcy flux ($q$) is a nonlinear function -- steeper than linear -- of the pressure-difference ($\Delta P$) across the medium. Furthermore, the flux is larger for a softer medium (smaller $G$). We construct a theory of this super-linear behavior by modelling the channels between the solid cylinders as elastic channels whose walls are made of a material with a linear constitutive relation but can undergo large deformation. Our theory further predicts that the flow permeability is a universal function of $\Delta P/G$, which is confirmed by the present simulations.