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Diffusion of particles through an heterogenous obstacle line is modeled as atwo-dimensional diffusion problem with a one--directional nonlinear convectivedrift and is examined using two-scale asymptotic analysis. At the scale wherethe structure of heterogeneities is observable the obstacle line has aninherent thickness. Assuming the heterogeneity to be made of an array ofperiodically arranged microstructures (e.g. impenetrable solid rectangles), twoscaling regimes are identified: the characteristic size of the microstructureis either significantly smaller than the thickness of the obstacle line or itis of the same order of magnitude. We scale up the convection-diffusion modeland compute the effective diffusion and drift tensorial coefficients for bothscaling regimes. The upscaling procedure combines ideas of two-scaleasymptotics homogenization with dimension reduction arguments. Consequences ofthese results for the construction of more general transmission boundaryconditions are discussed. We numerically illustrate the behavior of theupscaled membrane in the finite thickness regime and apply it to describe thetransport of {\rm CO}$_2$ through paperboard.

Driven particle flux through a membrane: Two-scale asymptotics of a diffusion equation with polynomial drift