Spatial Error and Convergence in Streamline Simulation
- E. Jimenez (Texas A&M University) | K. Sabir (Texas A&M University) | A. Datta-Gupta (Texas A&M University) | M.J. King (BP America)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Simulation Symposium, 31 January-2 February, The Woodlands, Texas
- Publication Date
- Document Type
- Conference Paper
- 2005. Society of Petroleum Engineers
- 5.4.1 Waterflooding, 5.5 Reservoir Simulation, 5.1.2 Faults and Fracture Characterisation, 5.5.7 Streamline Simulation, 5.1.8 Seismic Modelling, 5.3.1 Flow in Porous Media, 4.3.4 Scale, 4.1.2 Separation and Treating, 5.6.5 Tracers, 5.1 Reservoir Characterisation
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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 due to 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 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 only be resolved 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.
Streamline simulation has developed rapidly over the last ten years within the oil industry1-9. Unlike the earlier streamtube calculations, which date back to the 1930?s, streamline simulators have dispensed with the explicit construction of volume elements (the tubes) and replaced them with calculations along lines. Each line may be thought of as tracing out the center of a streamtube, with the velocity obtained from a numerical finite difference calculation. In contrast, within a streamtube, fluid velocity is obtained from the volumetric flux per unit area, where the area must be calculated explicitly as part of the streamtube construction. With streamlines the geometry is implicit, making it simple to perform calculations in three dimensions. To leading order, streamline simulation appears as a sum of one dimensional simulations, and so calculations in one, two or three dimensions are essentially equivalent. It is this ease of formulation which has transformed the class of problems which we can study with streamline simulation. Where streamtube calculations emphasized two dimensional sweep and pattern floods, streamline simulation has been applied to the full range of multiphase and multi-component physical and chemical processes in three dimensions10,11.
Although streamline simulators have received wide-spread attention over the past decade, no systematic study has been performed to understand the sources of spatial error and their convergence properties. The spatial discretization in streamline simulation generally takes two forms. First, a transverse discretization of the domain in terms of streamlines. The number of streamlines used during simulation determines the degree of transverse resolution. Second, a longitudinal discretization along streamlines for numerical solution of the transport equations along streamlines. In fact, much of the computational advantage of streamline models comes from the decomposition of the 3-D saturation calculations into these 1- D calculations along streamlines. There is an additional zeroth order truncation error associated with faulted grids, which will also be discussed.
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