Anomalous diffusion in porous media

Abstract

In this paper, an incompressible single phase and single component flow in a porous media presenting a non-Fickian behaviour is studied. The model is composed by a parabolic equation for the pressure, with homogeneous Dirichlet or Neumann boundary conditions, coupled with a mass conservation equation for the concentration, a transport equation for the mass flux and by Darcy’s law for the velocity. The transport equation for the mass flux is established assuming that this quantity at a certain point and at a certain time, depend on the concentration gradient in neighbour points (both in time and space). In order to numerical validate this approach, an IMEX finite element method is proposed to solve the coupled system of equations. The qualitative behaviour of the physical unknowns is illustrated and its dependence on the memory effect is discussed.