A boundary-integral framework to simulate viscous erosion of a porous medium

Abstract

We develop numerical methods to simulate the fluid-mechanical erosion of many bodies in two-dimensional Stokes flow. The broad aim is to simulate the erosion of a porous medium (e.g. groundwater flow) with grain-scale resolution. Our fluid solver is based on a second-kind boundary integral formulation of the Stokes equations that is discretized with a spectrally-accurate Nystrom method and solved with fast-multipole-accelerated GMRES. The fluid solver provides the surface shear stress which is used to advance solid boundaries. We regularize interface evolution via curvature penalization using the $\theta$-$L$ formulation, which affords numerically stable treatment of stiff terms and therefore permits large time steps. The overall accuracy of our method is spectral in space and second-order in time. The method is computationally efficient, with the fluid solver requiring $\mathcal{O}(N)$ operations per GMRES iteration, a mesh-independent number of GMRES iterations, and a one-time $\mathcal{O}(N^2)$ computation to compute the shear stress. We benchmark single-body results against analytical predictions for the limiting morphology and vanishing rate. Multibody simulations reveal the spontaneous formation of channels between bodies of close initial proximity. The channelization is associated with a dramatic reduction in the resistance of the porous medium, much more than would be expected from the reduction in grain size alone.