A control-theoretic approach to adaptive time stepping in reservoir simulations and its impacts on efficiency and robustness

• Adaptive time stepping is central to the performance of numerical simulators solving time-dependent partial differential equations • Using a control-theoretic approach to suggest time-step sizes provides a solid foundation for a complete time-stepping procedure • Improving the adaptive time stepping in reservoir simulations can significantly improve both efficiency and robustness Adaptive time stepping is a central feature of most simulators solving time-dependent differential equations. Ideally, the time-stepping procedure should contribute to efficient and robust simulations. In this work, we first present a general overview of the adaptive-time-stepping design problem which is expected to help guide researchers and practitioners in constructing time-stepping procedures for their specific applications. Next, we provide a synthesis of the theory related to a control-theoretic approach to adaptive time stepping, and explain how this may form some of the parts of an adaptive-time-stepping procedure by helping to propose time-step sizes and decide whether executed time steps should be accepted or rejected. We combine our design framework with the control-theoretic approach, and formulate adaptive-time-stepping algorithms for reservoir simulations; a little-investigated application area for control theory. When doing so, we construct controllers which use the relative change in pressure and saturation as a proxy for the temporal discretisation error to propose new time steps, with the goal of keeping the relative change close to a target value. We then apply this complete time-stepping algorithm to real-field black-oil reservoir models using the industry-standard reservoir simulator OPM Flow. The simulator and all the tested reservoir models are open-source. By testing various versions of the novel time-stepping algorithm in the context of reservoir simulations, we arrive at a preferred third-order controller which seems to work well for several models. Our experiments also reveal that the choice of time-stepping algorithm has a notable impact on the efficiency of the simulator, and we observe impactful differences in robustness.