Large-Scale Reservoir Simulatios Part I. a Priori Estimates

High performance computing technology offers the petroleum industry the ability to solve previously prohibitive large-scale reservoir problems. In July 1999, our group, in cooperation with the Petroleum Exploration and Development Institute of Daqing Oil Field, China, ran a million-gridblock-scale reservoir simulation on DAWN 2000, which is a home-made supercomputer, and a loosely coupled PC cluster seperately. The parallel computing methods that we used derived from the domain decomposition methods with no overlap. The next goal of our group is to solve reservoir simulations with millions of gridblocks on parallel machines. Unfortunately, it seems that the original computing method is not scalable enough. We believe that the reason is rather geologic than mathematical. As the simulating area becomes larger and larger, the geologic faults will be more and more complicated. Therefore, the non-matching grids on the interfaces of the substructures will be increasing largely, and possessing entirely different properties. This will inevitably lead to the poor performance of the original computing methods. The purpose of this paper is trying to find an effective way to remove as many of the geologic non-matching grids as possible from the interfaces. The operator split method, not a very new technique, proposed by Douglas and Dupont[JD71], can solve this problem. Because, for quite a few reservoir problems, the reservoir Ω can be taken to be unions of right prisms. Or, mathematically, Ω = ∪Ωi, where Ωi = Ωxy × [0, li], Ωxy ⊂ R. When only upright wells are available, the original reservoir problem can be divided into an xy-direction, two-dimensional problem and a z-direction, onedimensional problem in some of the subdomains. So, the geologic non-matching grids on the interfaces can be greatly reduced. For a detailed introduction of operator split method, see [JD71] and [Mar90]. Generalizations of this method to parabolic problems on nonrectangular regions were presented by Hayes [Hay81]. Special treatments for convection-diffusion problems, parabolic and hyperbolic equations were considered by Krishnamachari, Hayes and Russell[SHR89] (without theoretical analysis), Bramble, Ewing and Li[BEL89], Bialecki and Fernandes[BF93], and Fernandes and Fairweather [FF91]. Applications of these methods to problems in fluid flow, physics of semiconductors and elastoplastic dynamics were described by Hayes and Krishnamachari [HK84], Berezin and Yanenko[BY84], and Migual, Pinsky and Taylor[MPT83]. The main purpose of using operator split method here is to reduce the geologic non-matching grids on the interfaces, instead of saving the memory costs and the