Streamline Tracing on General Tetrahedral and Hexahedral Grids in the Presence of Tensor Permeability Coefficients

Abstract

Modern reservoir simulation models often involve geometrically complex (distorted or unstructured) three-dimensional grids populated with full-tensor permeability coefficients. The solution of the flow problem on such grids demands the use of accurate spatial discretizations such as the multipoint flux approximation (MPFA) finite volume schemes or mixed finite element methods (MFEM). The extension of the streamline method to modern grids therefore requires a streamline tracing algorithm adapted to these advanced discretizations. In this paper, we present a new algorithm to trace streamlines from MPFA or MFEM on general tetrahedral or hexahedral grids and in the presence of full-tensor permeabilities. Our approach was already used successfully in 2D and this paper presents the extension to 3D of the previous work by the authors. The method is based on the mathematical framework of MFEM. Since MPFA schemes have recently been interpreted as MFEM, our streamline tracing algorithm is also applicable for the MPFA finite volume methods. Using the mixed finite element velocity shape functions, the velocity field is reconstructed in each grid cell by direct interpolation of the MFEM velocity unknowns or MPFA subfluxes. An integration of the velocity field to arbitrary accuracy yields the streamlines. The method is the natural extension of Pollock’s (1988) tracing method to general tetrahedral or hexahedral grids. The new algorithm is more accurate than the methods developed by Prévost (2003) and Haegland et al. (2007) to trace streamlines from MPFA solutions and avoids the expensive flux postprocessing techniques used by both methods. After a description of the theory and implementation of our streamline tracing method, we test its performance on a full-field reservoir model discretized by an unstructured grid and populated with heterogeneous tensor coefficients.