Adaptive Timestepping, Linearization, and A Posteriori Error Control for Multiphase Flow of Immiscible Fluids in Porous Media with Wells

Abstract

Summary This work focuses on the development of adaptive timesteps, stopping criteria, and error control strategies for reservoir simulations with fully implicit (FIM) solvers. Using a rigorous error control framework, an adaptive time selector combined with nonlinear stopping criteria is used to control nonlinear iterations as well as to balance accuracy and robustness for challenging nonlinear simulations. In reservoir simulation, efficiently solving a system of nonlinear equations arising from the FIM method can be computationally burdensome for complex recovery processes. Theoretically, an FIM reservoir simulator has no stability limit on the timestep size. In practice, standard Newton’s method often fails to converge for large timestep sizes and must therefore cut the timestep multiple times to achieve convergence, resulting in a large number of unnecessary iterations. Another cause of nonlinear convergence issues is the presence of wells, which are often presented as singular point/line sources strongly coupled to the reservoir model, posing additional restrictions on the timestep choice. Here, we use a posteriori error estimators to avoid unnecessary nonlinear iterations and timestep cuts when solving immiscible multiphase flow. First, we estimate error components (e.g., spatial, temporal, and nonlinear) and then apply these to balancing criteria, providing us with dynamic and adaptive strategies to control timestep and nonlinear iterations. The error estimators are fully and locally computable, inexpensive to use, and target the various error components, including well singularities. The method provides an adaptive criterion for stopping the nonlinear iteration process whenever the linearization error does not significantly affect the overall error. Simultaneously, timesteps are adapted to maintain a constant size of the temporal discretization error with respect to the total error. Altogether, this avoids using unnecessary linearization iterations, wasteful timestep cuts, and too small timesteps. To demonstrate the effectiveness of these adaptive features, we present results for a suite of cases, covering both standard benchmarks and conceptual problems incorporating highly heterogeneous media with multiple wells. The proposed timestep selector cooperates with the new stopping criteria to improve nonlinear solver performance and increases robustness for cases with high nonlinearity. Perhaps most important, the adaptive features ensure balanced temporal and spatial errors while maintaining sufficiently small nonlinear errors, which ensures solution accuracy by accurately reproducing saturation fronts, production plateau, and breakthrough times.