Accurate Streamline Tracing on Three-Dimensional Tetrahedral and Hexahedral Grids

Abstract

Abstract Modern reservoir simulation grids are geometrically complex (distorted or unstructured grids) and often populated with tensor permeabilities. The solution of the flow problem on these advanced grids requires accurate finite volume methods such as the multipoint flux approximation (MPFA) schemes. Existing streamline tracing algorithms do not preserve the accuracy of the MPFA solutions and therefore represent a source of errors for the streamline method. Very recently, MPFA methods have been related to mixed finite element methods (MFEM) on 3D tetrahedral and hexahedral grids. This link equips MPFA with a new mathematical framework that we exploit to design a new streamline tracing method. Our approach was already used successfully in 2D and this paper presents the extension to 3D of the previous work by the authors. In the new method, the velocity field is interpolated from the MPFA subfluxes using mixed finite element velocity shape functions. The streamlines are then integrated to arbitrary accuracy. The method is the natural extension of Pollock's (1988) tracing method for general tetrahedral or hexahedral grids. The new method presents two major advantages over the streamline tracing methods developed for MPFA by Prévost et al.1 and Hægland et al.2 First, our algorithm provides more accurate streamlines and second, the direct interpolation of the velocity from the MPFA subfluxes avoids the expensive flux postprocessing used by the other methods. After a detailed description of the theory and implementation of our streamline tracing method, we use challenging test cases to compare its accuracy and efficiency to other existing streamline tracing methods.