System-AMG for Fully Coupled Reservoir Simulation with Geomechanics

The consideration of geomechanical effects is becoming more and more important in reservoir simulations. Ensuring stable simulation processes often enough requires handling the entire process with all types of physical unknowns fully implicitly. However, the resulting fully coupled linear systems pose challenges for linear solvers. The number of approaches that can efficiently handle a fully coupled system is extremely limited. System-AMG has demonstrated its efficiency for isothermal and thermal reservoir simulations. At the same time, AMG is known to be a robust and highly efficient linear solver for mere linear elasticity problems. This paper will discuss the combination of the advantages that AMG approaches have for both types of physics. This results in a robust and efficient solution scheme for the fully coupled linear system. The Automatic Differentiation General Purpose Research Simulator (AD-GPRS) is used to produce the Jacobians that are guaranteed to be exact. In a single-phase case, the overall Jacobian matrix takes the form of a constrained linear elasticity system where the flow unknowns serve as a Lagrangian multiplier. In other words, a saddle point system needs to be solved, where the flow and the mechanics problem might come at very different scales. A natural relaxation method for this kind of systems is given by Uzawa smoothing schemes which provide a way to overcome the difficulties that other smoothers may encounter. This approach appears intuitive for single-phase problems, where Gauss-Seidel can be applied in an inexact Uzawa scheme. However, in the multiphase case, incomplete factorization smoothers are required for the flow and transport part. We will discuss the incorporation in an inexact Uzawa scheme, where different realizations are possible, with different advantages and disadvantages. Finally, we propose an adaptive mechanism along with the outer Krylov solver to detect the best-suited realization for a given linear system. In the multiphase case, also the matrix preprocessing, for instance, by Dynamic Row Summing, needs to be considered. However, the process now also needs to reflect the requirements of the Uzawa scheme to be applicable. We demonstrate the performance for widely used test cases as well as for real-world problems of practical interest.