Effective Transmissibilities of a Gridblock by Upscaling - Comparison of Direct Methods with Renormalization
- Donald W. Peaceman (Consultant)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 1997
- Document Type
- Journal Paper
- 338 - 349
- 1997. Society of Petroleum Engineers
- 4.3.4 Scale, 5.5 Reservoir Simulation, 5.1.5 Geologic Modeling, 5.5.3 Scaling Methods
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Previous methods for upscaling fine-scale permeability data for use in reservoir simulation produce effective permeabilities for each simulator gridblock. Since trausmissibilities between neighboring gridblocks are required by the simulator, it is better to determine six "half-block transmissibilities" for each gridblock. These can be calculated directly by solving the finite-difference equations for pressure in each of six half-blocks, wherein uniform pressures are applied at two opposite faces and no-flow conditions are applied at the other four faces.
Alternatively, in order to reduce computation time, renormalization has been proposed for calculating effective permeabilities of the gridblocks. These previous proposals ignore anisotropy and require that the elemental blocks and each coarse block be rectangular. In this paper, renormalization is modified to avoid these disadvantages.
Renormalization is most easily implemented if there are N´N´N elemental blocks within each coarse block, where N is a power of 2. A more flexible variation of renormalization is also presented which reduces this restriction somewhat. However, the direct method is much more flexible with regard to the number of elemental blocks in each direction.
While renormalization might be expected to be faster than the direct method, comparison of running times shows that a highly efficient iterative solver for the direct method is somewhat faster than renormalization.
Detailed descriptions of formation properties such as permeability have become more and more available on a fine scale. These detailed descriptions cannot be used directly for reservoir simulation because they would require excessive computer time and memory; hence it is necessary to use relatively coarse gridblocks Accordingly, considerable attention has been given to methods for upscaling fine-scale permeability data to obtain effective permeabilities of the coarse gridblocks.1-5 And, for the most port, the methods have dealt with upscaling to gridblocks that are rectangular parallelepipeds.
Actually, what are required for input to the reservoir simulator are not the effective permeabilities of the coarse blocks. but transmissibilities from each block to each of its neighbors. White and Horne1 and later, King et al.19, seem to be the only authors who considered upscaling to obtain transmissibilities between gridblocks rather than effective permeabilities of entire gridblocks.
Further, commercial simulators are, more and more. incorporating corner-point geometry (CPG), wherein gridblocks can have arbitrary shapes that are defined by the coordinates of the eight corner points of each gridblock. It is the purpose of this paper to discuss the calculation of transmissibility between CPG blocks in the presence of fine-scale permeability heterogeneity.
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