Physically consistent numerical methods for flows in porous media

Simulation of multi-phase flow in porous media has wide applications such as petroleum fields and water resources. In this talk we will introduce physically consistent numerical methods for the simulation of incompressible and immiscible two-phase flow in heterogeneous porous media with capillary pressure. The new algorithm is unbiased and locally mass conservative for both of phases. We will also introduce an efficient energy stable numerical method for the Maxwell-Stefan-Darcy two-phase flow model in porous media, which can preserve multiple important physical properties of the model. Moreover, a fully discrete scheme for the stochastic Stokes-Darcy equations with multiplicative noise and its numerical analysis will also be discussed.