摘要：Petroleum reservoir engineers spend great efforts in the development and production of oil and gas reservoirs by conducting and interpreting the simulation of multiphase flows in porous geological formation.  Meanwhile, environmental scientists use subsurface flow and transport models to investigate and compare for example various schemes to inject and store CO2 in subsurface geological formations, such as depleted reservoirs and deep saline aquifers.  One basic requirement for accurate modeling and simulation of two-phase flow and multiphase flow is to have the predicted physical quantities sit within a physically meaningful range.  For example, the predicated saturation should sit between 0 and 1 while the predicated molar concentration should sit between 0 and the maximum value allowed by the equation of state.  Unfortunately, popular simulation methods used in the industries do not preserve physical bounds.  A commonly used fix to this problem is to simply apply a cut-off operator (say, to the computed saturation) at each time step, i.e., to set the saturation to be zero whenever it becomes negative, and to set it to one whenever it becomes larger than one.  However, this cut-off practice does not only destroy the local mass conservation, but it also damages the global mass conservation, which seriously ruins the numerical accuracy and physical interpretability of the simulation results. Additional desired features include preservation of the spatial discontinuity of saturation arising from different capillary pressure functions, local mass conservation of the numerical solution for both the wetting phase and non-wetting phase, continuity of normal components of numerical fluxes, and unbiasedness of the scheme with respect to the choice of primary variables.

In this talk we present our work on semi-implicit algorithms for two-phase in porous media with multiple capillary pressure functions, one in each subdomain. Our proposed algorithms are locally mass conservative for both phases.  They are able to accurately reproduce the spatial discontinuity of saturation due to different capillary pressure functions, and they correctly ensure that the total velocity is continuous in the normal direction.  Moreover, the new schemes are unbiased with regard to the phases and the saturations of both phases are bounds-preserving (under certain conditions). The proposed semi-implicit algorithms are derived from our novel splitting of variables based on the physics of two-phase flow.  A few interesting examples are presented to demonstrate the efficiency and robustness of the new algorithms.