Modelling flow-driven pore closure of weakening poroelastic media

Abstract

Poroelastic materials, such as polymer tissue scaffolds, porous rocks, and hydrogels, can weaken due to interactions between the solid skeleton and chemical species in the interstitial fluid. We develop a mathematical model for a poroelastic material to provide fundamental mechanistic insight into how weakening the material can affect the time-varying mechanics of the system. Our model couples large-deformation poroelasticity with an advection-diffusion equation for the solute. Furthermore, we introduce a decay equation for the material stiffness, whose rate of decay depends on the solute concentration. In this way, we describe a three-way coupling between poroelastic deformation, weakening of the skeleton and transport of solute through the material. We exploit numerical and analytical techniques to reveal the flow-driven uniaxial compression of a weakening poroelastic material and determine parameter regimes for which weakening the material facilitates pore closure at the downstream boundary. We identify parameter regimes in which (1) a steady state is attained without pore closure, (2) pore closure occurs at a finite time or (3) the pores close instantaneously; we uncover case (2) through the introduction of weakening into the system. We provide insights into the relationship between the differing behaviours and the separation between the timescales of the system. For systems with slow weakening, we derive a leading-order approximation for the time of pore closure, treating the ratio of the timescales of poroelastic relaxation and weakening as a small parameter, and investigate the accuracy of this approximation and the new behaviours that arise when these timescales become comparable.