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In this work is presented a numerical formulation for reservoir simulation in which the element-based finite-volume method (EbFVM) is applied to the discretization of the differential equations that describe macroscopic multiphase flow in petroleum reservoirs. The spatial discretization is performed by means of quadrilateral unstructured grids, which are adequate for representing two-dimensional domains of any complexity in an accurate and efficient manner. Although mass conservation is enforced over polygonal control volumes constructed in a vertex-centered fashion, media properties are assigned to the primal-grid quadrilateral elements. In this way, non-homogeneous full tensor permeabilities can be handled straightforwardly. Piecewise bilinear shape functions are used for approximating the main variables in the differential equations. The exception is the advection term in the saturation equation, which is approximated by means of a twodimensional positivity-preserving upwind scheme. Numerical results without noticeable grid orientation effects were obtained using this type of approximation, even for the most adverse cases with high mobility ratios and piston-type displacements. Additionally, some simple problems with known analytical solution were solved in order to assess the accuracy of the method. We show that the approximation of the pressure field is second-order even for non-homogeneous anisotropic media. Finally, the ability for solving fluid displacements in faulted reservoirs of complex geometry was tested with a synthetic problem.

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