Streamline Tracing on General Triangular or Quadrilateral Grids

Streamline methods have received renewed interest over the past decade as an attractive alternative to traditional finite-difference (FD) simulation. They have been applied successfully to a wide range of problems including production optimization, history matching, and upscaling. Streamline methods are also being extended to provide an efficient and accurate tool for compositional reservoir simulation. One of the key components in a streamline method is the streamline tracing algorithm. Traditionally, streamlines have been traced on regular Cartesian grids using Pollock’s method. Several extensions to distorted or unstructured rectangular, triangular, and polygonal grids have been proposed. All of these formulations are, however, low-order schemes. Here, we propose a unified formulation for high-order streamline tracing on unstructured quadrilateral and triangular grids, based on the use of the stream function. Starting from the theory of mixed finite-element methods (FEMs), we identify several classes of velocity spaces that induce a stream function and are therefore suitable for streamline tracing. In doing so, we provide a theoretical justification for the low-order methods currently in use, and we show how to extend them to achieve high-order accuracy. Consequently, our streamline tracing algorithm is semi-analytical: within each gridblock, the streamline is traced exactly. We give a detailed description of the implementation of the algorithm, and we provide a comparison of lowand high-order tracing methods by means of representative numerical simulations on 2D heterogeneous media.