Numerical simulation of 1-D oil and water displacements in petroleum reservoirs using the spectral finite volume method

The study and development of high-order discretization techniques for the modeling and simulation of multiphase flows in petroleum reservoirs are still a challenge from the computational viewpoint due to the difficulty posed by some physical features such as heterogeneity and anisotropy of the medium, that are of paramount importance in this class of applications. In this paper, we present a preliminary investigation of the very high order Spectral Finite Volume Method (SFVM) for the discretization of the saturation equation that describes the 1-D Oil-Water displacement through porous media. In this equation, the hyperbolic term is discretized using a high order SFVM, the capillary term is approximated using a second order CVFD (Control Volume Finite Difference) method and the integration in time is carried out using a third-order Runge-Kutta method. On the other hand, to prevent numerical oscillations near shocks, that are typical from higher order schemes, a slope limiter procedure is used in the reconstruction stage. To evaluate the performance of the SFVM some simple model problems, including capillarity and gravity, are solved, yielding accurate and efficient results. The use of very high order formulations can be of utmost importance, particularly for multidimensional and real large-scale applications with complex wave structures.