Asymptotic analysis of a transport model of organic material through a stratified porous medium: Concentration effect

We consider the transport through capillarity of an organic material inside a porous medium, using Leverett’s model. We first prove an existence result for a weak solution of this nonlinear evolution problem, using a regularization process. We then describe the asymptotic behavior of the solution, when the permeability kεkε of the porous medium is associated to a scalar function which only depends on the third variable, assuming that kεkε (resp. the inverse of kεkε) converges to some measure λ∗λ∗ (resp. λ∗λ∗). We use Γ-convergence arguments in order to describe this asymptotic behavior. We finally characterize the asymptotic behavior of the problem, considering special choices of the permeability kεkε, which correspond to stratified porous media, and give a numerical test for a 1D model.