Cell-centred Higher Resolution Finite-volume Total Velocity Vt and Va Formulations on Structured and Unstructured grids

Y.Xie [email protected], [email protected] Novel cell-centred finite-volume formulations are presented for two-phase flow with gravity and capillary pressure on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity and capillary pressure flux operators. Two formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both the Vt and the Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However when gravity is present, flow regimes are identified where the Va method becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first order single point upstream weighting.