A multipoint flux approximation with diamond stencil finite volume scheme for the two-dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid-grid method

Two-phase flows of oil and water in naturally fractured reservoirs can be described by a system of nonlinear partial differential equations that comprises of an elliptic pressure equation and hyperbolic saturation equation coupled through the total velocity field. Modeling this problem is a great challenge, due to the complexity of the depositional environments, including inclined layers and fractures with different sizes and shapes, and random spatial distribution. In this work, to solve the pressure equation, we adopted a cell-centered finite-volume method with a multipoint flux approximation that uses the "diamond stencil" (MPFA-D) coupled with a hybrid-gridmethod (HyG) to deal with the fractures. The classical first-order upwind method was used to solve the saturation equation, in its explicit and implicit versions. The MPFA-D is a very robust and flexible formulation that is capable of handling highly heterogeneous and anisotropic domains using general polygonal meshes. In the strategy developed in this work, the mesh that discretize the domain must fit the spatial position of the fractures, so that they are associated to the control surfaces-as (n - 1)D cells-therefore, the calculation of the fluxes in these control surfaces is dependent on the pressures on fractures and on the adjacent volumes. InHyG, the fractures are expanded to nD in the computational domain. The proposed formulation presented quite remarkable results when compared with similar formulations using classical full pressure support and triangle pressure support methods, or even the with MPFA-D itself when the fractures are treated as nD geometric entities.