Governing equations for multicomponent porous media flow typically exhibit strong and unbalanced nonlinearities and orders of magnitude variations in time constants. Together with the complexity of field-scale geomodels, with orders-of-magnitude variations in rock properties and cells with high aspect ratios and contorted geometries, this requires implicit discretizations with robust, global nonlinear solvers. We discuss a nonlinear domain decomposition preconditioning method for fully implicit simulations based on an additive Schwarz preconditioned exact Newton method (ASPEN) that is applicable to multicomponent porous media flow simulation. The method efficiently accelerates nonlinear convergence by resolving unbalanced nonlinearities in a local stage and long-range interactions in a global stage. In previous research, we have shown that use of ASPEN can improve robustness and (significantly) reduce the number of global iterations compared with standard Newton. However, each global iteration is more expensive because of the extra work introduced in the local stage.
Herein, we focus on implementation aspects for the local and global stage, including linear and nonlinear solver settings, and scalability. We show how the global-stage Jacobian can be transformed to have the same structure as the fully implicit system, enabling use of highly efficient linear preconditioners, and discuss how the global-stage update can be recast as a correction to the local-stage solutions.
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 such strategies are outlined.