A rigorous compressible streamline formulation for black oil and compositional simulation

Abstract

In this study for the first time we generalize streamline models to compressible flow using a rigorous formulation while retaining most of its computational advantages. Our new formulation is based on three major elements and requires only minor modifications to existing streamline models. First, we introduce a relative 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/composition conservation equation to account for compressibility effects. Third, the relative 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 conservation calculations and all the associated advantages of the streamline approach. The conservation calculations are fully decoupled from the underlying grid and can be carried out using large time steps without gridbased stability limits. We also extend the streamline simulation to compositional modeling including compressibility effects. Given the favorable computational scaling properties of streamline models, the potential advantage for compositional simulation can be even more compelling. Although several papers have discussed compositional simulation formulation, they all suffer from a major limitation, particularly for compressible flow. All of the previous works assume, either explicitly or implicitly, that the divergence of total flux along streamlines is negligible. This is not only incorrect for compressible flow but also introduces inconsistency between the pressure and conservation equations. We examine the implications of these assumptions on the accuracy of compositional streamline simulation using a novel and rigorous treatment of compressibility. We demonstrated the validity and practical utility of our approach using synthetic and field examples and comparison with a finite difference simulator. Throughout the validation for compositional model, we found out the importance of finer segments discretizations along streamlines. We introduce optimal coarsening of segments to minimize flash calculations on each segment while keeping the accuracy of finer segments.