For many years there have been formulations considered for modeling single phase ow on general hexahedra grids. These include the extended mixed nite element method, and families of mimetic nite di erence methods. In most of these schemes either no rate of convergence of the algorithm has been demonstrated both theoret- ically and computationally or a more complicated saddle point system needs to be solved for an accurate solution. Here we describe a multipoint ux mixed nite element (MFMFE) method [5, 2, 3]. This method is motivated from the multipoint ux approximation (MPFA) method . The MFMFE method is locally conservative with continuous ux approximations and is a cell-centered scheme for the pressure. Compared to the MPFA method, the MFMFE has a variational formulation, since it can be viewed as a mixed nite element with special approximating spaces and quadrature rules. The framework allows han- dling of hexahedral grids with non-planar faces by applying trilinear mappings from physical elements to reference cubic elements. In addition, there are several multi- scale and multiphysics extensions such as the mortar mixed nite element method that allows the treatment of non-matching grids . Extensions to the two-phase oil-water ow are considered. We reformulate the two- phase model in terms of total velocity, capillary velocity, water pressure, and water saturation. We choose water pressure and water saturation as primary variables. The total velocity is driven by the gradient of the water pressure and total mobility. Iterative coupling scheme is employed for the coupled system. This scheme allows treatments of di erent time scales for the water pressure and water saturation. In each time step, we rst solve the pressure equation using the MFMFE method; we then Center for Subsurface Modeling, The University of Texas at Austin, Austin, TX 78712; firstname.lastname@example.org. yCenter for Subsurface Modeling, The University of Texas at Austin, Austin, TX 78712; email@example.com. 1 solve the saturation using discontinuous Galerkin (DG) method by taking multiple small time steps within the large time step. In addition, the MFMFE method allows e cient computations of the total and capillary velocity since the method gives the local velocity approximation in terms of surrounding pressure degrees of freedom. Both theoretical and computational results are discussed and presented. Exten- sions to advection-di usion equations and non-Newtonian polymer ooding  are also discussed.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-F1-032-04
Acknowledgements: A portion of this research was supported by the U.S. Department of Energy, Office of Science, Officeof Basic Energy Sciences. The Center for Frontiers of Subsurface Energy Security (CFSES) is a DOEEnergy Frontier Research Center, under Contract No. DE-SC0001114. The authors gratefully acknowledgethe financial support provided by the NSF-CDI under contract number DMS 0835745. GuangriXue is supported by Award No. KUS-F1-032-04, made by King Abdullah University of Science andTechnology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.