The multiscale finite-volume (MSFV) method has been developed to solve
multiphase flow problems on large and highly heterogeneous domains
efficiently. It employs an auxiliary coarse grid, together with its
dual, to define and solve a coarse-scale pressure problem. A set of
basis functions, which are local solutions on dual cells, is used to
interpolate the coarse-grid pressure and obtain an approximate
fine-scale pressure distribution. However, if flow takes place in
presence of gravity (or capillarity), the basis functions are not good
interpolators. To treat this case correctly, a correction function is
added to the basis function interpolated pressure. This function, which
is similar to a supplementary basis function independent of the
coarse-scale pressure, allows for a very accurate fine-scale
approximation. In the coarse-scale pressure equation, it appears as an
additional source term and can be regarded as a local correction to the
coarse-scale operator: It modifies the fluxes across the coarse-cell
interfaces defined by the basis functions. Given the closure assumption
that localizes the pressure problem in a dual cell, the derivation of
the local problem that defines the correction function is exact, and no
additional hypothesis is needed. Therefore, as in the original MSFV
method, the only closure approximation is the localization assumption.
The numerical experiments performed for density-driven flow problems
(counter-current flow and lock exchange) demonstrate excellent
agreement between the MSFV solutions and the corresponding fine-scale
reference solutions.