1 3 J Braz. Soc. Mech. Sci. Eng. DOI 10.1007/s40430-017-0795-8 TECHNICAL PAPER Numerical simulation of 1-D oil and water displacements in petroleum reservoirs using the spectral finite volume method G. Galindez-Ramirez 1 · M. R. A. Souza 1 · D. K. E. Carvalho 1 · P. R. M. Lyra 1 Received: 12 October 2016 / Accepted: 13 April 2017 © The Brazilian Society of Mechanical Sciences and Engineering 2017 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. Keywords Petroleum reservoirs · Two-phase flows · Finite volume method · High order method 1 Introduction Over the past several years, CFD research groups have been developing alternative formulations for different class of applications. One intensive research area is the devel- opment of high-order numerical methods for the simula- tion of two phase flows in heterogeneous porous media by finite volume approximations. Some high-order schemes were implemented and tested for second-order accuracy using a MUSCL (Monotonic Upstream-Centered Scheme for Conservation Laws) reconstruction, including a limit- ing procedure to get a TVD (total variation diminishing) scheme [18, 19]. Other schemes were applied to petroleum reservoir simulation, including the Mass-Weighted Upwind Scheme (MWU), the Streamline-Based Upwind Scheme, and a TVD-scheme which uses second order interpolation functions [21, 22]. These schemes are at most second-order accuracy in regions of smooth solutions and first-order close to shocks. However, as the reconstruction technique of these methods relies on using neighboring elements around the current element for polynomial reconstruction in each time step. These methods were found to be com- putationally very expensive, for orders of accuracy greater than two [12, 13]. This fact motivated our investigation of the spectral finite volume method, which was proposed by Wang [39] as a suitable alternative to k-exact high-order methods such as high order finite volume (HOFV) [18], the Abstract The study and development of high-order dis- cretization techniques for the modeling and simulation of multiphase flows in petroleum reservoirs are still a chal- lenge from the computational viewpoint due to the dif- ficulty posed by some physical features such as heteroge- neity 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 discretiza- tion of the saturation equation that describes the 1-D Oil- Water displacement through porous media. In this equa- tion, 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 capillar- ity and gravity, are solved, yielding accurate and efficient Technical Editor: Celso Kazuyuki Morooka. * D. K. E. Carvalho dkarlo@uol.com.br; dkarlo101@gmail.com G. Galindez-Ramirez ggalindezr@gmail.com M. R. A. Souza souza2105@gmail.com P. R. M. Lyra prmlyra@padmec.org 1 Mechanical Engineering Department, Federal University of pernambuco, Av. da arquiterura, Recife, PE 50740-550, Brazil