Robust Streamline Tracing Using Intercell Fluxes in Locally Refined and Unstructured Grids
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AbstractA robust, fast, and easy to implement streamline tracing algorithm for use in polygonal cells based upon cell face fluxes has been developed. This problem arises naturally as a post processing step for finite difference or finite volume reservoir simulators, where intercell fluxes are known but intracell velocities need to be interpolated and streamline trajectories calculated. Our proposed solution relies upon a refinement of the polygonal cells and provides a velocity field which is locally conservative, allows analytic calculation of the streamline trajectories, and does not require knowledge of the sub-cell permeability tensor or transmissibilities. Previous industry work on polygonal cells has used the corner velocity interpolation (CVI) method. We extend these results to support convex, nearly degenerate and non-convex polygons as may arise for complex grid geometries, and grids with local grid refinement. We also simplify the CVI construction through the use of a lower order velocity interpolation approach using sub-cell refinement. This allows an analytic calculation of streamline trajectories and time of flight on the sub-cells, instead of the numerical calculation required by CVI. The sub-cell refinement construction is very similar to the method of Prvost. The advantage of the proposed approach lies in the fact that it is solely based on the cell face fluxes and avoids the use of sub-cell permeabilities or transmissibilities. Instead it uses a "Minimum Velocity Variance" construction to obtain the smoothest possible velocity field. It is locally conservative and allows an exact analytic solution for streamline trajectories and time of flight.In our study, eight different velocity interpolation schemes are evaluated. Each approach has identical normal velocities on the polygonal cell faces, but different interpolated velocities. The locally conservative schemes provide better streamline densities but introduce some degree of tangential velocity discontinuity at the sub-cell boundaries. The schemes with continuous velocity are not conservative. Higher order velocity models which are both continuous and conservative may also be used but are not robust for velocity reconstruction when utilized with lower order flux boundary conditions. The schemes are studied using single cell models and a quarter 5-spot case discretized using a polygonal (PEBI) grid.
