In this paper, we propose a full finite volume approach to simulate two-phase flows of oil and water in heterogeneous and anisotropic petroleum reservoirs in 2-D. The IMPES procedure is used to solve the coupling between the pressure and the saturation equations. The eliptic pressure equation is discretized by a non-orthodox linear Multi-Point Flux Approximation (MPFA-HD) method capable to handle heterogeneous and highly anisotropic media. Following the MPFA-D (MPFA-Diamond) scheme and some existent non-linear cell centered strategies, the key point in the construction of our scheme is the discretization of the flux across each cell face. First, we construct the one-sided fluxes on each cell independently and then, we integrate both one-sided fluxes over the control surface and finally we express the cell edge fluxes as a convex combination of the one side fluxes, to obtain a unique flux expression. On the other hand, differently from the MPFA-D, in our scheme, fluxes on each cell face are explicitly expressed by one cell centered unknown defined on the cells sharing that face and two auxiliary unknowns defined at two face endpoints that do not necessarily belong to the same face shared by the adjacent cells. These auxiliary vertex unknowns are eliminated by a proper interpolation. To solve the saturation equation, we propose a Modified Flow Oriented Scheme (M-FOS). This flow oriented variant explicitly computes the multidimensional numerical fluxes by using higher order accuracy in space. Besides, for problems with distorted meshes, the proposed formulation takes into account the angular distortion of the computational mesh by means of an adaptive weight, that tunes the multidimensional character of the formulation according to the grid distortion. This strategy diminishes the occurrence of Grid Orientation Effects (GOE). In order to avoid spurious oscillations in higher-order approximations, a recently devised Multidimensional Limiting Process (MLP) is adopted. This strategy guarantees monotone solutions and can be used with any polygonal mesh. Finally, an entropy fix strategy is also employed in order to produce convergent solutions. The performance of our numerical formulation is evaluated by solving some benchmark problems.

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