This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set of system unknowns comprises both element and face pressures, in addition to the cell saturations, resulting in a $3 \times 3$ block-structured Jacobian matrix with a $2 \times 2$ inner pressure problem. The CPR method is one of the most established techniques for reservoir simulations, but most research focused on solutions for Two-Point Flux Approximation (TPFA)-based discretizations that do not readily extend to our problem formulation. Therefore, we designed a dedicated two-stage strategy, inspired by the CPR algorithm, where a block preconditioner is used for the pressure part with the aim at exploiting the inner $2 \times 2$ structure. The proposed preconditioning framework is tested by an extensive experimentation, comprising both synthetic and realistic applications in Cartesian and non-Cartesian domains.

翻译：本文提出了一种原创预处理子，通过将约束压力残差（CPR）方法与块预处理相结合，高效求解全隐式多相流模型中产生的线性化方程组。该预处理子称为块CPR（BCPR），专为基于拉格朗日乘子的流动模型设计，例如混合杂交有限元（MHFE）近似生成的模型。以基于MHFE的两相流模型原始公式作为BCPR预处理子开发的参考，其中系统未知量集合除单元饱和度外，还包含单元压力与面压力，形成具有$3 \times 3$块结构的雅可比矩阵，其内部为$2 \times 2$压力子问题。CPR方法是油藏模拟中最成熟的技术之一，但现有研究主要集中于基于两点流量近似（TPFA）离散化的求解方案，无法直接扩展至本文问题公式。因此，受CPR算法启发，我们设计了一个专用两阶段策略，其中对压力部分采用块预处理子以利用其内部$2 \times 2$结构。通过大量实验（涵盖笛卡尔与非笛卡尔域中的合成及实际应用）对所提预处理框架进行了测试。