Homogenizability conditions for multicomponent reactive transport

• We upscale multicomponent reactive transport in porous media.
• We establish sufficient conditions of continuum-scale ADR equations.
• These conditions are summarized in a phase diagram in the (Pe,Da)-space.
• They are verified by numerical simulations of flow through a planar fracture.
• We show numerically that such conditions are necessary and sufficient.

We consider multicomponent reactive transport in porous media involving three reacting species, two of which undergo a nonlinear homogeneous reaction, while a third precipitates on the solid matrix through a heterogeneous nonlinear reaction. The process is fully reversible and can be described with a reaction of the kind A+B⇌C⇌SA+B⇌C⇌S. The system’s behavior is fully controlled by Péclet (Pe) and three Damköhler (Daj,j={1,2,3}) numbers, which quantify the relative importance of the four key mechanisms involved in the transport process, i.e. advection, molecular diffusion, homogeneous and heterogeneous reactions. We use multiple-scale expansions to upscale the pore-scale system of equations to the macroscale, and establish sufficient conditions under which macroscopic local advection–dispersion–reaction equations (ADREs) provide an accurate representation of the pore-scale processes. These conditions reveal that (i) the heterogeneous reaction leads to more stringent constraints compared to the homogeneous reactions, and (ii) advection can favorably enhance pore-scale mixing in the presence of fast reactions and relatively low molecular diffusion. Such conditions are summarized by a phase diagram in the (Pe,Daj)-space, and verified through numerical simulations of multicomponent transport in a planar fracture with reacting walls. Our computations suggest that the constraints derived in our analysis are robust in identifying sufficient as well as necessary conditions for homogenizability.