Understanding enhanced seepage through a soft porous material

Mechanical deformations driven by fluid flow through a porous solid are relevant to problems as diverse as cell and tissue mechanics, magma dynamics, and hydrology. In a soft porous medium, for sufficiently small pressure gradients, the flow-rate (seepage) obeys Darcy's law, i.e., is proportional to the pressure gradient. But at large pressure gradients, the flow-rate can be substantially higher. We use direct numerical simulations and theory to understand this phenomenon. We model the soft porous material, as a bed of soft spheres in a hexagonal lattice with random defects. The center of the spheres are held fixed. The region between the spheres is filled with a Newtonian fluid. Our DNS of this system shows that the flow-rate versus pressure gradient relationship in this model can be fit by a quartic polynomial. Next, we construct a theory in two steps. First, we perform lubrication analysis at the pore throat while using Winkler elastic foundation to model the elastic deformation of the spheres. This reveals a nonlinear relationship between pressure gradient and flow-rate across individual micro-channels. Next, we upscale this model to derive a mesoscale relationship between flow-rate and pressure gradient that reproduces the results from DNS.