Summary
Streamline 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 3D saturation equation into 1D 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 very often 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 many 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 through use of pressure updates and remapping of saturations. Our proposed approach preserves the 1D 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 timesteps without grid-based stability limits.
We demonstrate the validity and practical utility of our approach using synthetic and field examples and comparison with a commercial finite-difference simulator. A comparison of the number of pressure solutions and the CFL numbers for the streamline and finite-difference simulation indicates that our proposed compressible streamline approach is likely to offer substantial computational advantage.
