Spatial Error and Convergence in Streamline Simulation

Abstract

Streamline models are routinely used for waterflood optimization and management and are being extended to more complex processes (e.g., compositional simulation). Despite these new developments, no systematic study has examined the underlying numerical spatial and temporal discretization errors in streamline simulation and their convergence. Such studies are a prerequisite to determining the optimal density of streamlines during simulation and ensuring the resulting accuracy of the solution.In this paper, we first examine transverse spatial errors (e.g., errors resulting from the number or placement of streamlines). We provide an analytic proof and a numeric demonstration of the order of spatial convergence of the mass-balance discretization error. Both global and local calculations are performed, and they demonstrate the impact of stagnation regions on the order of convergence. A second transverse error arises for faulted grids, where lack of flux continuity at cell faces can lead to incorrect trajectories. These trajectory errors are of zeroth order and can be resolved only by introducing additional degrees of freedom into the streamline velocity model. Longitudinal spatial errors also arise and are associated with the inaccurate calculation of time of flight across cells.We show that the commonly used algorithm for corner-point cells leads to inaccurate time-of-flight calculations for stratigraphic grids, depending upon aspect ratio. We provide a simple and exact means of calculating the time of flight for arbitrary corner-point cells, or unstructured grids, in two or three dimensions, for either compressible or incompressible flow. Finally, using this new time-of-flight formulation, we analyze a series of cross-sectional finite-difference simulations to identify grid-orientation errors in the numerical calculation of flux and spatial error.

Robust streamline tracing using inter‐cell fluxes in locally refined and unstructured grids

“…This was later extended to corner point cells by other authors [Cordes and Kinzelbach, 1992;Prévost et al, 2001;Jimenez et al, 2007] on the basis of the lowest-order RaviatThomas (RT 0 ) space and an isoparametric mapping. We will call this the Extended Pollock Method (EPM).…”

Velocity interpolation and streamline tracing on irregular geometries

“…15 and the solution can be found analytically. Variants by Cordes and Kinzelbach [7], Prevost et al [24] and Jimenez et al [14] are often used for grids with general hexahedral cells.…”

Mixed finite elements for solving 2-D diffusion-type equations

“…This is a semianalytical method that recovers the exit point of a streamline and the time of flight in an element of the mesh by assuming that each component of the velocity field varies linearly inside the element. The method, initially developed for rectangles, was extended to irregular grids [Prevost et al, 2002;Jimenez et al, 2005] by using the Piola transform of the velocity vector from the reference element to the physical one.…”