  In this paper, we study the numerical approximation of a coupled system of
elliptic-parabolic equations posed on two separated spatial scales. The model
equations describe the interplay between macroscopic and microscopic pressures
in an unsaturated heterogeneous medium with distributed microstructures as they
often arise in modeling reactive flow in cementitious-based materials. Besides
ensuring the well-posedness of our two-scale model, we design two-scale
convergent numerical approximations and prove a priori error estimates for the
semidiscrete case. We complement our analysis with simulation results
illustrating the expected behaviour of the system.
