Speaker
Description
In models of porous media flows, the porosity of the solid matrix is often treated as a static quantity. However, under certain circumstances, such as in soft sedimentary rocks or in magma flows, the porosity of the solid material can evolve under the influence of fluid pressure which can lead to the formation of solitary porosity waves and of higher-porosity channels. We consider a system of nonlinear PDEs for porosity and effective pressure, based on a poroviscoelastic model, which describes such phenomena. We first discuss well-posedness of this PDE problem, which has been established in the literature only for initial porosities of high Sobolev smoothness. We present several results for porosities of low regularity, including cases with jump discontinuities that are of particular interest in geological applications. Then we turn to results on a space-time adaptive numerical method, which is based on a fixed-point scheme inspired by the analysis, combined with a space-time least-squares formulation. This yields an appropriate treatment of discontinuities and enables spatially varying time steps, which are required for efficient approximations of the strongly spatially and temporally localized features of solutions.
