Numerical evaluation of diffusive and dispersive transport in periodic porous media

At the macroscopic scale, the transport of a fluid component in a porous medium is governed by advective, diffusive, and dispersive fluxes. The two latter fluxes are formulated using a diffusion tensor and a dispersion tensor. The actual value of these tensors depends on the pore geometry and on the magnitude of the advection velocity. Using the double scale expansion technique and provided the problem is homogeneizable, it is well-known that the the diffusion and dispersion tensors can be evaluated from the following procedure. First, a velocity field at the pore scale is obtained by solving a Stokes-like problem. Then, this velocity field is used in a vector-valued advection-diffusion equation, still posed at the pore scale, and from which solution the the diffusion and dispersion tensors are evaluated. Since we are interested in 3D pore geometries for which analytical solutions are not available, we use the finite element method to approximate the pore scale velocity field and to solve the vector-valued advection-diffusion equation. Three strategies for approximating the pore scale velocity field are proposed and analyzed. The first uses nodal Lagrange finite elements with Galerkin/Least Squares stabilization. The second uses mixed Crouzeix-Raviart/P_0 finite elements. The third uses the previous mixed finite elements and then projects the velocity field onto a vector space of divergence-free fields using Brezzi-Douglas-Marini finite elements. We perform numerical simulations for cubic and centered cubic networks of spheres, with several values of the porosity. We also vary the velocity magnitude to study its effect on the diffusion and dispersion tensors.Presenters: name: Tardif d'Hamonville, Pierre affiliation: CERMICS - ENP