The finite volume techniques for solving fluid flow problems can be broadly classified in cell center and cell vertex methodologies. The conventional finite volume techniques belong to the former class, since the elements coming from the grid generator are used as control volumes for performing the balances of the physical quantities, like mass, momentum and energy. The majority of the available methods in the literature uses this type of approach for solving multiphase flows of oil, water and gas in porous media. In petroleum reservoir simulation it is common for the conventional finite volume method to be linked to corner-point grids. This paper presents an Element-based Finite Volume Method (EbFVM), whereby the control volumes are constructed using parts of the elements, generating polygonal meshes in which mass, momentum and energy conservation are enforced. Polygonal meshes considerably reduce the number of unknowns of the linear system when compared with conventional finite volume methods. Three-dimensional hybrid grids are employed for the solution of oil-water flows in a porous media resembling a petroleum reservoir using the IMPES, sequential and fully implicit approaches. The analytical solution of the 1D Buckley-Leverett problem is used for evaluation purposes, and numerical solutions for 2D and 3D problems using unstructured grids are carried out to demonstrate the generality of the method and for comparing the robustness, convergence rate and CPU time of the IMPES and Fully Implicit solutions. Memory usage and convergence rate are also presented for the solution of 3D problems using tetrahedral grids in a cell-center and cell-vertex methodologies.

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