The XFEM method for Darcy and Stokes problem with heterogeneous media

The classical finite element method can exhibit limitations if the solution is not regular across lines, or surfaces in 3D, that cross the elements of the mesh. For this reason we employ the eXtended Finite Element Method (XFEM) to approximate solutions with jumps or kinks irrespective of the computational grid. This approach is particularly convenient if the discontinuity surfaces are complex or evolve in time. We focus our attention on two problems that are relevant from the point of view of the geophysical applications. We consider Darcy flows in the presence of strongly heterogeneous porous media and employ the XFEM to deal with regions of different permeabilities and in particular to describe pressure and velocity jumps across fractures, represented as immersed interfaces. The XFEM approach can also be beneficial for multiphase flow problems (Stokes problems) where the separated fluids are characterized by different viscosities. A particular attention is dedicated to suitable stabilization techniques for the saddle problems resulting from the discretization of the system