Numerical investigation of a new class of models of Darcy-scale flows with flow-dependent permeability

The Darcy model for flows in porous media is hugely popular among researchers and practitioners yet there are many problems where the classical Darcy model is not efficient and accurate as it gives rise to manifestly nonphysical singularities. We aim to investigate numerically a new class of mathematical models that allow for handling nonphysical singularities while preserving the advantages of the classical Darcy model. The introduced dependence of the permeability of the porous matrix on the flow that passes through it makes it necessary to compute the flow field and the permeability field simultaneously, and we therefore develop a novel numerical method to compute the solution to a strongly nonlinear system of PDEs arising in the problem. Our approach allows one to take characteristics of the flow geometry into account in numerical solution and we demonstrate the predictive potential of the generalized Darcy model through numerical tests.