We present a novel hybrid (or multiphysics) algorithm, which couples
pore-scale and Darcy descriptions of two-phase flow in porous media.
The flow at the pore-scale is described by the Navier?Stokes equations,
and the Volume of Fluid (VOF) method is used to model the evolution
of the fluid?fluid interface. An extension of the Multiscale Finite
Volume (MsFV) method is employed to construct the Darcy-scale problem.
First, a set of local interpolators for pressure and velocity is
constructed by solving the Navier?Stokes equations; then, a coarse
mass-conservation problem is constructed by averaging the pore-scale
velocity over the cells of a coarse grid, which act as control volumes;
finally, a conservative pore-scale velocity field is reconstructed
and used to advect the fluid?fluid interface. The method relies on
the localization assumptions used to compute the interpolators (which
are quite straightforward extensions of the standard MsFV) and on
the postulate that the coarse-scale fluxes are proportional to the
coarse-pressure differences. By numerical simulations of two-phase
problems, we demonstrate that these assumptions provide hybrid solutions
that are in good agreement with reference pore-scale solutions and
are able to model the transition from stable to unstable flow regimes.
Our hybrid method can naturally take advantage of several adaptive
strategies and allows considering pore-scale fluxes only in some
regions, while Darcy fluxes are used in the rest of the domain. Moreover,
since the method relies on the assumption that the relationship between
coarse-scale fluxes and pressure differences is local, it can be
used as a numerical tool to investigate the limits of validity of
Darcy's law and to understand the link between pore-scale quantities
and their corresponding Darcy-scale variables.