  We aim at understanding transport in porous materials including regions with
both high and low diffusivities. For such scenarios, the transport becomes
structured (here: {\em micro-macro}). The geometry we have in mind includes
regions of low diffusivity arranged in a locally-periodic fashion. We choose a
prototypical advection-diffusion system (of minimal size), discuss its formal
homogenization (the heterogenous medium being now assumed to be made of zones
with circular areas of low diffusivity of $x$-varying sizes), and prove the
weak solvability of the limit two-scale reaction-diffusion model. A special
feature of our analysis is that most of the basic estimates (positivity,
$L^\infty$-bounds, uniqueness, energy inequality) are obtained in $x$-dependent
Bochner spaces.
