A blended CPR/block preconditioning approach for mixed discretization schemes in reservoir modeling

This work proposes an original preconditioner coupling the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising in fully implicit reservoir simulation flow models. This preconditioner has been specifically designed for Lagrange multipliers-based discretizations, like the Mixed Hybrid Finite Element (MHFE) or the Mimetic Finite Difference (MFD) scheme. Here, we focus on a MHFE-based discretization of the two-phase flow model in porous media, where the phase flux continuity is strongly enforced in the mass balance equation to stabilize the nonlinear solver convergence. While the system unknowns consist of cell and face pressures and cell saturations, the Jacobian matrix, which is non-symmetric, usually large and ill-conditioned, exhibits a 3x3 block structure. The simulator performance in transient simulation essentially pivots on the acceleration capability offered by the preconditioner and the interplay with the linear solver, usually a Krylov subspace method. This is a well-known problem in literature, where the CPR has been established as a robust and efficient tool, but most research focused on solutions for Two-Point Flux Approximation (TPFA)-based discretizations that do not immediately and readily extend to our problem formulation. Therefore, we designed a dedicated multi-stage strategy, inspired by the CPR algorithm, where a block preconditioner is used for the pressure part with the aim at exploiting the inner 2x2 block structure. Approximated decoupling factors of the Jacobian are used to recast the Schur complement. Their computation is performed in parallel, and, at an algebraic level, they are obtained through restriction operators, where the size of the restricted subspaces is either statically or dynamically adapted. A closer inspection reveals that recomputing the decoupling factors from scratch at each nonlinear iteration is not necessary, but they can be updated following the waterfront advancement, thus improving the preconditioner efficiency. The proposed preconditioning framework has been tested with broad experimentation, comprising both synthetic and realistic applications in Cartesian and non-Cartesian domains, highlighting its advantages and weaknesses.