In the last 20 years many discretization schemes have been developed to approximate Darcy fluxes on polyhedral cells in heterogeneous anisotropic porous media. Among them, we can distinguished cell based approaches like the Two Point Flux Approximation (TPFA) or the Multi Point Flux Approximation (MPFA) schemes, face based approaches like the family of Hybrid Mimetic Mixed (HMM) methods and nodal based discretizations like the Vertex Approximate Gradient (VAG) scheme.  They all have their own drawbacks and advantages which typically depend on the type of cells and on the anisotropy of the medium.
 
During the ongoing PhD of Laurence Beaude, we propose a new strategy to combine these discretizations on arbitrary subsets of cells or faces in order to adapt the choice of the discretization scheme in different parts of the mesh.  In this approach, cell based discretizations are considered as face based discretizations for which the face or subface unknowns can be eliminated. We consider two types of strategies to couple face based and cell based discretizations. The first one is based on a partition of the cells, each cell using either nodal or face unknowns. The second approach is based on a partition of the faces, each face using either face or nodal unknowns. In both cases, the coupling is performed using a node to face interpolation operator which must be chosen to ensure both the consistency, the coercivity and the limit conformity properties of the combined discretization. The convergence analysis is performed in the gradient scheme framework and convergence is proved for arbitrary cell or face partitions of the mesh. For face partitions, an addtitional stabilization is required to ensure the coercivity while for cell partitions no additional stabilization is needed.

Illustration of 2D mesh using Cartesian and simplectic cells where it is possible to apply a nodal based scheme on the simplectic cells and a face based scheme on the Cartesian cells.

The efficiency of our approach is first tested for single phase flow problems on 3D meshes using a combination of the HMM and VAG discretizations. Then, it is studied for two-phase compositional non-isothermal Darcy flows on a simplified 2D cut of the Bouillante geothermal reservoir using a combination of the TPFA and VAG discretizations on different types of meshes.

The model and its numerical validation have been presented at the CMWR conference in Saint-Malo, France in june 2018 and an abstract has been accepted for oral presentation at the next ECMOR conference in september 2018 in Barcelona.