  We develop a locally conservative, finite element method for simulation of
two-phase flow on quadrilateral meshes that minimize the number of degrees of
freedom (DoFs) subject to accuracy requirements and the DoF continuity
constraints. We use a mixed finite element method (MFEM) for the flow problem
and an enriched Galerkin method (EG) for the transport, stabilized with an
entropy viscosity. Standard elements for MFEM lose accuracy on quadrilaterals,
so we use the newly developed AC elements which have our desired properties.
Standard tensor product spaces used in EG have many excess DoFs, so we would
like to use the minimal DoF serendipity elements. However, the standard
elements lose accuracy on quadrilaterals, so we use the newly developed direct
serendipity elements. We use the Hoteit-Firoozabadi formulation, which requires
a capillary flux. We compute this in a novel way that does not break down when
one of the saturations degenerate to its residual value. Extension to three
dimensions is described. Numerical tests show that accurate results are
obtained.
