We consider a nonlinear degenerate problem modeling the transport  of a nonaqueous-phase liquid (NAPL) inside a saturated porous medium. We establish the uniqueness of a weak solution and study its asymptotic behavior as t tends to +∞. We also prove that a complex structure of a porous medium provides matching conditions between different rock heterogeneities. We finally consider a special choiceof permeabilities in one and two dimensional cases and present some numerical experiments using finite volume methods.

transport of NAPLs, Leverett’s model, existence and unicity, asymptotic behavior, matching conditions, numerical experiments.