A family of advection schemes is presented for application to element-based finite-volume methods. In these schemes, flow skewness is accounted for by means of a mass flux ratio in a way that positivity is guaranteed. These features assure that the discretization of the advection term does not introduce nonphysical spatial oscillations and reduces to some extent the adverse influence of the grid orientation on the numerical solutions. By means of an analysis of the associated truncation error, the directional properties of the advection schemes are investigated. This analysis and some application examples suggest that one particular scheme in the family yields numerical solutions with minimal influence of the grid topology.

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This paper addresses the key issue of calculating fluxes at the control-volume interfaces when finite-volumes are employed for the solution of partial differential equations. This calculation becomes even more significant when unstructured grids are used, since the flux approximation involving only two grid points is no longer correct. Two finite volume methods with the ability in dealing with unstructured grids, the EbFVM-Element-based Finite Volume Method and the MPFA-Multi-Point Flux Approximation are presented, pointing out the way the fluxes are numerically evaluated. The methods are applied to a porous media flow with full permeability tensors and non-orthogonal grids and the results are compared with analytical solutions. The results can be extended to any diffusion operator, like heat and mass diffusion, in anisotropic media.

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.