Multiscale preconditioning of Stokes flow in complex porous geometries

Fluid flow through porous media is central to many subsurface (e.g., CO2 storage) and industrial (e.g., fuel cell) applications. The optimization of design and operational protocols, and quantifying the associated uncertainties, requires fluid-dynamics simulations inside the microscale void space of porous samples. This often results in large and ill-conditioned linear(ized) systems that require iterative solvers, for which preconditioning is key to ensure rapid convergence. We present robust and efficient preconditioners for the accelerated solution of saddle-point systems arising from the discretization of the Stokes equation on geometrically complex porous microstructures. They are based on the recently proposed pore-level multiscale method (PLMM) and the more established reduced-order method called the pore network model (PNM). The four preconditioners presented are the monolithic PLMM, monolithic PNM, block PLMM, and block PNM. Compared to existing block preconditioners, accelerated by the algebraic multigrid method, we show our preconditioners are far more robust and efficient. The monolithic PLMM is an algebraic reformulation of the original PLMM, which renders it portable and amenable to non-intrusive implementation in existing software. Similarly, the monolithic PNM is an algebraization of PNM, allowing it to be used as an accelerator of direct numerical simulations (DNS). This bestows PNM with the, heretofore absent, ability to estimate and control prediction errors. The monolithic PLMM/PNM can also be used as approximate solvers that yield globally flux-conservative solutions, usable in many practical settings. We systematically test all preconditioners on 2D/3D geometries and show the monolithic PLMM outperforms all others. All preconditioners can be built and applied on parallel machines.

流体通过多孔介质的流动对于许多地下（例如二氧化碳储存）和工业（例如燃料电池）应用至关重要。设计和操作方案的优化以及相关不确定性的量化，需要在多孔样本的微观孔隙空间内进行流体动力学模拟。这通常会产生大型且条件不佳的线性（化）系统，需要迭代求解器，而预处理对于确保快速收敛至关重要。我们提出了稳健且高效的预处理器，用于加速求解在几何复杂的多孔微观结构上对斯托克斯方程离散化所产生的鞍点系统。它们基于最近提出的孔隙级多尺度方法（PLMM）以及更成熟的称为孔隙网络模型（PNM）的降阶方法。所提出的四种预处理器是整体式PLMM、整体式PNM、块式PLMM和块式PNM。与现有的由代数多重网格方法加速的块预处理器相比，我们表明我们的预处理器更加稳健和高效。整体式PLMM是原始PLMM的一种代数重构，这使其具有可移植性，并易于在现有软件中进行非侵入式实现。类似地，整体式PNM是PNM的一种代数化，使其能够用作直接数值模拟（DNS）的加速器。这赋予了PNM此前所不具备的估计和控制预测误差的能力。整体式PLMM/PNM也可作为近似求解器，产生全局通量守恒的解，可用于许多实际情况。我们在二维/三维几何形状上系统地测试了所有预处理器，并表明整体式PLMM优于所有其他预处理器。所有预处理器都可以在并行机器上构建和应用。