Compaction in a deformable cylindrical porous medium bounded by an elastic impermeable membrane

Hewitt­­­­ et al¹ showed experimentally, and with a one-dimensional theoretical model, that water driven vertically through a packed bed of hydrogel beads causes the medium to compact, reducing porosity and permeability such that the flow rate through the medium plateaus to a constant as the pressure head diverges to infinity. We theoretically investigate a two-dimensional extension of the problem in a tall, thin, axisymmetric cylinder where the radial boundaries of the porous bed are enclosed by a deformable, impermeable boundary with different material properties to the porous bed itself. Flow is driven through the medium by a fixed height of water maintained above the cylinder. We use the small ratio of radial to vertical length scales of the cylinder to find the leading-order flow rate and deformation as a function of the applied pressure head, initial porosity, and the ratio of linear elastic parameters of the porous bed to the bounding membrane, recovering Hewitt et al¹’s model as the membrane becomes much stiffer than the porous matrix. Our model predicts how a compliant bounding membrane augments compactions, but prevents unbounded increase in flow resistance, allowing flow rate to increase monotonically with applied pressure head for a relatively wide range of porous media. This research contributes to a better understanding of flow and deformation coupling in biological soft tissues, such as the brain and human placenta.

¹Hewitt, Duncan R., et al. "Flow-induced compaction of a deformable porous medium." Physical Review E 93.2 (2016): 023116