Arthur Moncorgé

Standard solution strategies in reservoir simulators all rely on a global linearization and the solution of large (ill-conditioned) linear systems, particularly for fully implicit formulations. Such solves are expensive, and typically require significant interprocess communication when implemented on a distributed memory architecture. Modern nonlinear domain decomposition (NLDD) is based on adaptive alternation between local and global solves and has proved to be an effective way to reduce the number of global linear solves in serial mode. Herein, we discuss the extension of NLDD to a distributed setting, exemplified by the operational reservoir simulator OPM Flow. We discuss which parts are reused from the standard Newton solver and which modifications are required to deploy the methodology in a standard simulator. We apply our new methods to a series of open benchmark and real-life asset models. In serial, NLDD benefits significantly from using a Gauss–Seidel approach in which subdomains are solved in order of decreasing pressure. This strategy can only be applied effectively within each process in a distributed setting, but not between processes, which potentially poses a challenge to scalability. However, we observe excellent improvement in efficiency of the nonlinear solution phase also in parallel, in particular when slight adjustments are made in the choice of linear solver. Our implementations are freely available as open source, enabling readers to reproduce many of the examples on their own hardware and test the methods on their own models.

We discuss a nonlinear domain-decomposition preconditioning method for fully implicit simulations of multicomponent porous media flow based on the additive Schwarz preconditioned exact Newton method (ASPEN). The method efficiently accelerates nonlinear convergence by resolving unbalanced nonlinearities in a local stage and long-range interactions in a global stage. ASPEN can improve robustness and significantly reduce the number of global iterations compared with standard Newton, but extra work introduced in the local steps makes each global iteration more expensive. We discuss implementation aspects for the local and global stages. We show how the global-stage Jacobian can be transformed to the same form as the fully implicit system, so that one can use standard linear preconditioners and solvers. We compare the computational performance of ASPEN to standard Newton on a series of test cases, ranging from conceptual cases with simplified geometry or flow physics to cases representative of real assets. Our overall conclusion is that ASPEN is outperformed by Newton when this method works well and converges in a few iterations. On the other hand, ASPEN avoids time-step cuts and has significantly lower runtimes in time steps where Newton struggles. A good approach to computational speedup is therefore to adaptively switch between Newton and ASPEN throughout a simulation. A few examples of switching strategies are outlined.

Domain decomposition methods are widely used as preconditioners for Krylov subspace linear solvers. In the simulation of porous media flow there has recently been a growing interest in nonlinear preconditioning methods for Newton’s method. In this work, we perform a numerical study of a spatial additive Schwarz preconditioned exact Newton (ASPEN) method as a nonlinear preconditioner for Newton’s method applied to both fully implicit or sequential implicit schemes for simulating immiscible and compositional multiphase flow. We first review the ASPEN method and discuss how the resulting linearized global equations can be recast so that one can use standard preconditioners developed for the underlying model equations. We observe that the local fully implicit or sequential implicit updates efficiently handle the local nonlinearities, whereas long-range interactions are resolved by the global ASPEN update. The combination of the two updates leads to a very competitive algorithm. We illustrate the behavior of the algorithm for conceptual one and two-dimensional cases, as well as realistic three dimensional models. A complexity analysis demonstrates that Newton’s method with a fully implicit scheme preconditioned by ASPEN is a very robust and scalable alternative to the well-established Newton’s method for fully implicit schemes.