Streamline 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 simulators' ability to take large timesteps with fewer pressure solutions within an IMPES formulation. However, unlike conventional finite-difference simulators, no clear guidelines are currently available for the choice of timestep for pressure and velocity updates. This has remained largely an uncontrolled approximation, either managed by engineering judgment, or by potentially time-consuming timestep 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 timestep selection during streamline simulation based on three elements. First, we reformulate the equations to be solved by a streamline simulator to include all of the 3D flux termsboth 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, and most importantly, we provide a discrete Courant-Fredrich-Levy (CFL) formulation for the corrector step that leads to a mechanism to ensure numerical stability through the choice of a stable timestep for pressure updates. This discrete CFL formulation now provides us with the same tools for timestep 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 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 role of the transverse flux and our proposed CFL formulation on the accuracy of the solution and on the appropriate choice of timestep 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 timestep within full-field streamline applications.