  Diffusion of particles through an heterogenous obstacle line is modeled as a
two-dimensional diffusion problem with a one--directional nonlinear convective
drift and is examined using two-scale asymptotic analysis. At the scale where
the structure of heterogeneities is observable the obstacle line has an
inherent thickness. Assuming the heterogeneity to be made of an array of
periodically arranged microstructures (e.g. impenetrable solid rectangles), two
scaling regimes are identified: the characteristic size of the microstructure
is either significantly smaller than the thickness of the obstacle line or it
is of the same order of magnitude. We scale up the convection-diffusion model
and compute the effective diffusion and drift tensorial coefficients for both
scaling regimes. The upscaling procedure combines ideas of two-scale
asymptotics homogenization with dimension reduction arguments. Consequences of
these results for the construction of more general transmission boundary
conditions are discussed. We numerically illustrate the behavior of the
upscaled membrane in the finite thickness regime and apply it to describe the
transport of {\rm CO}$_2$ through paperboard.
