In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

翻译：本文针对多孔介质中的单相流动问题，对时空MORe DWR（基于对偶加权残差误差估计的模型降阶）框架进行了扩展与深化。具体而言，研究问题为包含矢量位移（地质力学）与达西流动压力方程耦合的比奥系统。MORe DWR方法引入了一种目标导向的自适应增量式本征正交分解（POD）降阶模型（ROM）。在模拟过程中，对降阶目标泛函的误差进行估计；若该估计值超过预设阈值，则动态扩充POD基函数。该方法可减少模拟多孔介质所需全阶模型求解的总次数，实现对关注量的稳健估计，并生成适用于当前问题的降阶基函数。本文采用时空伽辽金离散格式：空间方向使用泰勒-胡德单元，时间方向采用分段常数间断伽辽金法（其与向后欧拉格式具有等价性）。通过经典的二维曼德尔基准问题与三维地基承载力算例验证了该方法的高效性。