Timestep Selection During Streamline Simulation via Transverse Flux Correction
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AbstractStreamline simulators have received increased attention 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 are currently 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 understanding of numerical stability and to the lack of error estimates during the solution.We propose a novel approach for time step selection during streamline simulation that is based on three elements. First, we reformulate the equations to be solved by a streamline simulator to include all of the three-dimensional flux terms both aligned with and transverse to the flow directions. These transverse flux terms are totally neglected within the existing streamline simulation formulations. Second, we propose a simple grid-based corrector algorithm to update the saturation to account for the transverse flux. Third, we provide a discrete CFL (Courant-Fredrich-Levy) formulation for the corrector step that leads to a mechanism to ensure numerical stability via the choice of a stable time step for pressure updates. This discrete CFL formulation now provides us with the same tools for time step control as are available within conventional reservoir simulators.We demonstrate the validity and utility of our approach using a series of numerical experiments in homogeneous and heterogeneous !4 five-spot patterns at various mobility ratios. For these numerical experiments, we pay particular attention to favorable 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 streamline simulation, and provides a basis for automatic control of pressure time step within full field streamline applications.
