Accelerating Nonlinear Convergence in Reservoir Simulation by Adaptive Relaxation and Nonlinear Domain-Decomposition Preconditioning

Newton methods commonly used for advancing implicit schemes in time often exhibit slow convergence, especially when initial guesses are far from the solution or when there are imbalances in nonlinearities across time and space. A common approach to deal with this is to relax the Newton updates, although this does not always yield good results. Alternatively, nonlinear solvers using local-global domain-decomposition preconditioning strategies have shown greater robustness. However, the observed reductions in iteration count do not always justify the higher computational cost in practical applications. In this work, we compare four different approaches to accelerating the nonlinear convergence of Newton methods. In the first and least invasive method, the convergence is continually monitored throughout the Newton loop to determine adjustments in update relaxation. The convergence monitors are computationally inexpensive and based on quantities that are usually available in commercial reservoir simulators. The second method is a straightforward nonlinear domain-decomposition (NLDD) preconditioning method that enhances the initial guess for Newton’s method by prepending each global Newton iteration with local subdomain solves. To reduce the computational overhead, local solves are adaptively bypassed in subdomains where there are small changes in mobility. The third method adds adaptive relaxation to the global step of the NLDD method. The fourth method uses convergence monitors to adaptively adjust relaxation in Newton and, if necessary, switches on local preconditioning solves depending if convergence issues are determined to be more severe. We investigate the effectiveness of the methods in practical models relevant to oil recovery, CO₂ storage, and geothermal energy storage. Our results show that runtime can be significantly reduced in both single and ensemble models by enhancing the standard Newton’s method with dynamic relaxation. However, the most effective overall approach is NLDD preconditioning, where local solves are adaptively enabled or disabled based on mobility changes.