Mixed five-point/nine-point finite-difference formulation of multiphase flow in petroleum reservoirs

Nine-point finite-difference formulation of multiphase flow in the x-y pane is well-known to reduce grid-orientation effects. This paper proposes a new implicit pressure, explicit saturation (IMPES) formulation that uses implicit pressures for the customary five points of the x-y grid system, explicit pressures for the four points on the diagonals, and explicit saturations for all nine points. This proposed split-operator scheme produces virtually identical results as the standard nine-point formulation of Yanosik and McCracken, but can be easily implemented in existing five-point simulators without requiring any changes in the commonly used solution algorithms, such as the D-4 Gauss. The use of this new technique should enhance the accuracy of existing large-scale IMPES simulators with little increase in storage requirement or computing time. The method can be used with any set of nine-point coefficients and can be extended to fully implicit simulators.