A predictor-corrector formulation for rigorous streamline simulation
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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 their ability to take large time steps with fewer pressure solutions within an IMPES formulation. However, unlike conventional finite difference simulators, no clear guidelines have been available for the choice of time step for pressure and velocity updates. This has remained largely an uncontrolled approximation, either managed by engineering judgment, or by potentially time-consuming time step size sensitivity studies early in a project. This is clearly related to the lack of theoretical understanding of numerical stability and convergence for this solution method. Our results demonstrate a new ability to predict numerical stability for streamline simulation. The analysis of numerical convergence is beyond our current scope. Clearly, both stability and convergence are required for numerical solutions. We will review the pr edictor-corrector streamline formulation recently introduced (SPE 79688, presented at the SPE Reservoir Simulation Symposium, Houston, TX, 3-5 February 2003). This formulation includes the treatment of transverse flux between streamlines, and for the first time provides a rigorous foundation for discussions of numerical stability. We extend those discussions to include the treatment of capillarity and of gravity. In all instances, we will recognize the limits of stability of the calculations through the use of a CFL number, or in the instance of capillarity, a diffusion number. Although the worked examples will all utilize explicit numerical techniques, in most instances, the use of implicit techniques will be obvious. We demonstrate the validity and utility of our approach using a series of numerical experiments in homogeneous and heterogeneous 1/4 five-spot patterns at various mobility ratios. For the discussions of capillarity and of gravity we also provide one-dimensional calculations to better understand our options for their treatment using operator splitting. For these numerical experiments, we pay particular attention to favourable mobility ratio displacements, as they are known to be challenging to streamline simulation. Our results clearly demonstrate the impact of the transverse flux correction on the accuracy of the solution and on the appropriate choice of time step, across a range of mobility ratios. The proposed approach eliminates much of the subjectivity associated with time step selection in streamline simulation, and provides a basis for automatic control of pressure time step within full field streamline applications. Copyright 2005 John Wiley & Sons, Ltd.
