## Operator Splitting Multiscale Finite Volume Element Method
for Two-Phase Flow with Capillary Pressure

**Abstract.**
A numerical method used for solving a two-phase flow problem as found in typical
oil recovery is investigated in the setting of physics-based two-level operator splitting.
The governing equations involve an elliptic differential equation coupled with a parabolic
convection-dominated equation which poses a severe restriction for obtaining fully implicit
numerical solutions. Furthermore, strong heterogeneity of the porous medium over many
length scales adds to the complications for effectively solving the system. One viable approach
is to split the system into three sub-systems: the elliptic, the hyperbolic, and the
parabolic equation, respectively. In doing so, we allow for the use of appropriate numerical
discretization for each type of equation and the careful exchange of information between
them. We propose to use the Multiscale Finite Volume Element (MsFVEM) for the elliptic
and parabolic equations, and a non-oscillatory difference scheme for the hyperbolic equation.
Performance of this procedure is confirmed through several numerical experiments.