The Stokes-Brinkman equations model fluid flow in highly heterogeneous porous media. In this paper, we consider the numerical solution of the Stokes-Brinkman equations with stochastic permeabilities, where the permeabilities in subdomains are assumed to be independent and uniformly distributed within a known interval. We employ a truncated anchored ANOVA decomposition alongside stochastic collocation to estimate the moments of the velocity and pressure solutions. Through an adaptive procedure selecting only the most important ANOVA directions, we reduce the number of collocation points needed for accurate estimation of the statistical moments. However, for even modest stochastic dimensions, the number of collocation points remains too large to perform high-fidelity solves at each point. We use reduced basis methods to alleviate the computational burden by approximating the expensive high-fidelity solves with inexpensive approximate solutions on a low-dimensional space. We furthermore develop and analyze rigorous a posteriori error estimates for the reduced basis approximation. We apply these methods to 2D problems considering both isotropic and anisotropic permeabilities.

Stokes-Brinkman方程可对高度非均质多孔介质中的流体流动进行建模。在本文中，我们考虑具有随机渗透率的Stokes-Brinkman方程的数值解，其中假定子域中的渗透率是独立的并且在已知区间内均匀分布。我们采用截断的锚定方差分析和随机配置，以估计速度和压力解的矩。通过仅选择最重要的方差分析方向的自适应程序，我们减少了准确估计统计矩所需的并置点数。但是，即使是较小的随机尺寸，并置点的数量仍然太大，无法在每个点上执行高保真度求解。通过在低维空间上使用便宜的近似解来近似昂贵的高保真解，我们使用简化的基础方法来减轻计算负担。我们进一步开发和分析严格的后验误差估计，以简化基数近似。我们将这些方法应用于同时考虑各向同性和各向异性渗透率的二维问题。