Hybrid Multiscale Finite Volume method for advection-diffusion equations subject to heterogeneous reactive boundary conditions

Abstract:

We present a hybrid scheme for the coupling of macro and microscale advection-dispersion continuum models for reactive contaminant transport in fractured and porous media. The Multiscale Finite Volume method (MsFV) is employed to approximate the microscale concentration field defined in terms of macroscopic or global degrees of freedom, together with local interpolator and corrector functions capturing microscopic spatial variability. The macroscopic mass balance relations for the MsFV global degrees of freedom are coupled with the macroscopic model, resulting in a global problem for the simultaneous time-stepping of all macroscopic degrees of freedom throughout the domain. We employ a hybrid time-stepping scheme stemming from the local-global decomposition, for which the global time-stepping problem is solved using a larger time-step than the local interpolator and corrector problems.

The hybrid coupling is formulated by applying both micro and macroscale models over overlapping subdomains of the simulation domain, with the overlap denoted as the handshake subdomain Ω^hs, over which continuity of concentration and transport fluxes between models is enforced. Continuity of concentration is enforced by posing a restriction relation between models over Ω^hs. Continuity of fluxes is enforced by prolongating the macroscopic model fluxes across the boundary ∂Ω^hs to microscopic resolution.

We illustrate the proposed hybrid MsFV scheme by solving a contaminant transport problem in fractured media with highly localized homogeneous and heterogeneous reactions.