A Rigorous Compressible Streamline Formulation for Two and Three-Phase Black-Oil Simulation
Authors: Cheng Hao , Osako Ichiro , Datta-Gupta Akhil , King Michael J. 
Published: December 2006

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 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 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.

Introduction
Streamline simulators have become increasingly popular for high-resolution reservoir simulation using multimillion-cell geologic models. For incompressible or slightly compressible flow and under convection-dominated conditions, streamline models are well known to outperform conventional finite-difference simulation in terms of computational speed. Streamline models can also result in improved accuracy because of subgrid resolution and reduced numerical dispersion and grid-orientation effects (King and Datta-Gupta 1998; Datta-Gupta 2000). To a large extent, the efficiency of the current streamline formulation is a consequence of the incompressibility assumption that allows us to easily and effectively decouple the pressure and saturation calculations during flow simulation. This decoupling has been greatly facilitated by the introduction of the streamline time of flight coordinate (Datta-Gupta and King 1995). Specifically, utilizing the time of flight as the spatial coordinate, the multidimensional saturation calculations are reduced to a series of 1D solutions along streamlines. These 1D solutions can be carried out independently and using relatively large timesteps, as they are not impacted by the underlying geologic grid-based stability limitations. This is the primary advantage of streamline simulation. In addition, for heterogeneity-dominated flow and adverse mobility ratio conditions, the streamlines need to be updated infrequently, leading to further savings in computation time (King and Datta-Gupta 1998; Datta-Gupta 2000).