A locally and globally phase-wise mass conservative numerical algorithm for the two-phase immiscible flow problems in porous media

In this work, we introduce a novel numerical method to solve the problem of two-phase immiscible flow in porous media that is conservative to both phases. In the widely used implicit pressure, explicit saturation (IMPES) scheme, the conservation of mass of both the two phases are summed to form an equation involving the total Darcy's velocity. In the discretization of such an equation it becomes difficult to enforce the conservation of mass of each phase. To guarantee the conservation of mass of both phases locally and hence globally, we introduce a scheme in which the time discretization of the mass conservation equations is considered separately. Cell-centered finite difference (CCFD) methods are adopted for spatial discretization, where the variables of fluid properties (i.e. relative permeability and mobility) are upwinded separately according to the velocity of each phase and not according to the total velocity. Furthermore, this new scheme updates all phase velocities and uses them to update the corresponding phase saturation. In addition, a two-scale of time-splitting methods are adopted for pressure equation and saturation equations to improve the computational efficiency. For the sake of simplicity, we show a number of examples of two-phase system in two-dimensional geometry solved using the new scheme. It is shown that the new scheme is more embracing the physics and it can be more accurate than traditional IMPES scheme, particularly for the cases in which the phase velocities are in opposite direction, and conventional IMPES schemes fail.