An advanced fully-implicit solver for heterogeneous porous media based on foam-extend

Multiphase flow in porous media is present in many engineering applications, including hydrogeology, oil recovery, and CO2 sequestration. Accurate predictions of fluid behavior in these systems can improve process efficiency while mitigating environmental and health risks. Commercial simulators and open source software, such as the porousMultiphaseFoam repository based on the OpenFOAM framework, have been developed to model this type of problem. However, simulating heterogeneous porous media with heterogeneous porosity and permeability distributions poses significant numerical challenges. We introduce coupledMatrixFoam, an OpenFOAM-based solver designed for enhanced numerical stability and robustness. coupledMatrixFoam integrates the Eulerian multi-fluid formulation for phase fractions with Darcy's law for porous media flow, applying a fully implicit, block-coupled solution for pressure and phase fractions. The solver is based on foam-extend 5.0, leveraging the latest fvBlockMatrix developments to improve computational efficiency. This approach enables a significant increase in time step sizes, particularly in cases involving capillary pressure effects and other complex physical interactions. This work details the formulation, implementation and validation of coupledMatrixFoam, including comparisons with porousMultiphaseFoam that uses a segregated approach, to assess performance improvements. Additionally, a scalability analysis is conducted, demonstrating the solver's ability for high-performance computing (HPC) applications, which are essential for large-scale, real-world simulations.

多孔介质内的多相流广泛存在于诸多工程应用场景中，涵盖水文地质、石油开采以及二氧化碳封存等领域。对这类系统内流体流动行为的精准预测，既能提升工艺效率，又可降低环境与健康风险。目前已开发出商用模拟器与开源软件，例如基于OpenFOAM框架的porousMultiphaseFoam代码仓库，用于对此类问题进行建模。然而，针对孔隙度与渗透率分布均呈非均质性的多孔介质开展模拟，仍面临显著的数值计算挑战。本文提出coupledMatrixFoam——一款基于OpenFOAM的求解器，旨在提升数值计算的稳定性与鲁棒性。该求解器将用于相分数计算的欧拉多流体公式（Eulerian multi-fluid formulation），与描述多孔介质流动的达西定律（Darcy's law）相整合，并针对压力与相分数采用全隐式块耦合求解方案。本求解器基于foam-extend 5.0开发，通过利用最新的fvBlockMatrix技术提升计算效率。该方案可大幅增大允许使用的时间步长，尤其在涉及毛管压力效应及其他复杂物理交互作用的场景中效果显著。本文详细阐述了coupledMatrixFoam的公式推导、代码实现与验证过程，其中包括与采用分离式求解方案的porousMultiphaseFoam的对比实验，以评估其性能提升效果。此外，本文还开展了可扩展性分析，证明该求解器适配高性能计算（HPC）应用场景，而这类场景对于大规模真实世界模拟而言至关重要。