We aim at understanding transport in porous materials including regions with both high
and low diffusivities. For such scenarios, the transport becomes structured (here: 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^inf-bounds, uniqueness, energy
inequality) are obtained in x-dependent Bochner spaces.
Keywords: Heterogeneous porous materials, homogenization, micro-macro transport,
two-scale model, reaction-diffusion system, weak solvability.