A Rigorous Compressible Streamline Formulation for Two- and Three-Phase Black-Oil Simulation
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AbstractStreamline simulators have received increased attention in the petroleum industry because of their ability to effectively handle multimillion cell detailed geologic models and large simulation models. The efficiency of streamline simulation has relied primarily on the decoupling of the 3-D saturation equation into 1-D equations along streamlines using the streamline time of flight as the spatial coordinate. Until now, this decoupling has been strictly valid for incompressible flow. Applications to compressible flow have generally lacked strong theoretical foundations and for the most part yielded mixed or unsatisfactory results.In this paper for the first time we generalize streamline models to compressible flow using a rigorous formulation while retaining much of its favorable characteristics. Our new formulation is based on three major elements and requires only minor modifications to existing streamline models. First, we introduce an effective density for the total fluids along the streamlines. This density captures the changes in the fluid volume with pressure and can be conveniently and efficiently traced along streamlines. Thus, we simultaneously compute time of flight and volume changes along streamlines. Second, we incorporate a density-dependent source term in the streamline saturation equation to account for compressibility effects. Third, the effective density, fluid volumes and the time-of-flight information are used to incorporate cross-streamline effects via pressure updates and remapping of saturations. Our proposed approach preserves the 1-D nature of the saturation calculations and all the associated advantages of the streamline approach. The saturation calculations are fully decoupled from the underlying grid and can be carried out using large time steps without grid-based stability limits.We demonstrate the validity and practical utility of our approach using synthetic and field examples and comparison with both commercial finite difference and streamline simulators. The synthetic examples involve waterflooding in a -five spot pattern under undersaturated conditions and also three phase flow with both free and solution gas. Our results show close agreement with the finite difference simulator in terms of water-oil and gas-oil ratio histories for an extended period of time. The field example is from a highly heterogeneous carbonate reservoir in West Texas and includes multiple patterns consisting of 11 injectors, 31 producers and over 30 years of production history. Our proposed formulation results in significant improvement in performance prediction over current commercial streamline simulators.
