Preconditioning for Lagrange Multipliers-Based Reservoir Flow Simulation

Being able to reproduce the complex flow dynamics that take place in the subsurface is a key requirement for efficient hydrocarbon recovery and other field operations, including hydrogen storage and CO₂ sequestration. Reservoir simulators are, therefore, essential assets for the energy industry, especially in the midst of the energy transition. Running a reservoir simulation involves solving sequences of large-size systems of linear equations, usually by means of iterative Krylov subspace solvers. This is typically one of the most time- and resource-consuming tasks in a simulation and is largely affected by the robustness and efficiency of the preconditioner supplied along with the Krylov solver itself. Moreover, the introduction of advanced discretization schemes, such as the Mixed Hybrid Finite Element (MHFE) method, to improve the accuracy of the model outcome, challenges the performance of existing solving techniques. One of the most established preconditioners for reservoir simulation applications is the Constrained Pressure Residual (CPR) method, which was designed taking the standard Two-Point Flux Approximation (TPFA) scheme as a reference. However, extending the conventional CPR algorithm to systems of equations originating from MHFE-based discretizations does not prove to be a robust approach, given the different structure of the matrix blocks. In this dissertation, building on the success of the CPR method, we develop an original preconditioning technique in which we adapt the conventional CPR algorithm to the MHFE matrix structure by introducing a block preconditioner for the pressure subproblem; the resulting approach is thus denoted as Block CPR (BCPR). The original aspect in the design of the block preconditioner lies in the way in which the Schur complement is approximated, i.e., by using explicit inexact versions of the Jacobian decoupling factors. This inner tool, within the global BCPR algorithm, has been denoted as Explicit Decoupling Factor Approximation (EDFA). The extensive experimental phase, first on the EDFA method alone and then on the BCPR preconditioner, proved the robustness and flexibility of these preconditioning tools even on challenging realistic applications.