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 _kε$k_{\epsilon }$ of the porous medium is associated to a scalar function which only depends on the third variable, assuming that _kε$k_{\epsilon }$ (resp. the inverse of _kε$k_{\epsilon }$) converges to some measure _λ∗${\lambda }_{\ast }$ (resp. λ^∗${\lambda }^{\ast }$). 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 _kε$k_{\epsilon }$, which correspond to stratified porous media, and give a numerical test for a 1D model.