Deformable fractured porous media appear in many geoscience applications. While the extended finite element method (XFEM) has been successfully developed within the computational mechanics community for accurate modeling of deformation, its application in geoscientific applications is not straightforward. This is mainly due to the fact that subsurface formations are heterogeneous and span large length scales with many fractures at different scales. To resolve this limitation, in this work, we propose a novel multiscale formulation for XFEM, based on locally computed enriched basis functions. The local multiscale basis functions capture heterogeneity of th e porous rock properties, and discontinuities introduced by the fractures. In order to preserve accuracy of these basis functions, reduced-dimensional boundary conditions are set as localization condition. Using these multiscale bases, a multiscale coarse-scale system is then governed algebraically and solved. The coarse scale system entails no enrichment due to the fractures. Such formulation allows for significant computational cost reduction, at the same time, it preserves the accuracy of the discrete displacement vector space. The coarse-scale solution is finally interpolated back to the fine scale system, using the same multiscale basis functions. The proposed multiscale XFEM (MS-XFEM) is also integrated within a two-stage algebraic iterative solver, through which error reduction to any desired level can be achieved. Several proof-of-concept numerical tests are presented to assess the performance of the developed method. It is shown that the MS-XFEM is accurate, when compared with the fine-scale reference XFEM solutions. At the same time, it is significantly more efficient than the XFEM on fine-scale resolution, as it significantly reduces the size of the linear systems. As such, it develops a promising scalable XFEM method for large-scale heavily fractured porous media.

可变形的破裂多孔介质出现在许多地球科学应用中。尽管扩展力学有限元方法（XFEM）已在计算力学社区中成功开发出来，可以对变形进行精确建模，但在地球科学应用中的应用并非一帆风顺。这主要是由于以下事实：地下地层是非均质的，跨过大尺度的尺度，并有许多不同尺度的裂缝。为解决此限制，在这项工作中，我们基于本地计算的丰富基函数，为XFEM提出了一种新颖的多尺度公式。局部多尺度基函数捕获了多孔岩石特性的非均质性，以及由裂缝引入的不连续性。为了保持这些基本功能的准确性，将降维边界条件设置为定位条件。然后使用这些多尺度基数，对多尺度粗尺度系统进行代数控制和求解。粗尺度系统由于裂缝而没有富集。这样的公式可以显着降低计算成本，同时，它可以保留离散位移矢量空间的精度。最终，使用相同的多尺度基函数将粗尺度解决方案插值回精细尺度系统。所提出的多尺度XFEM（MS-XFEM）也集成在两级代数迭代求解器中，通过该迭代器可以将误差降低到任何所需的水平。提出了几种概念验证的数值测试，以评估所开发方法的性能。结果表明，MS-XFEM是准确的，与精细参考XFEM解决方案相比。同时，由于它显着减小了线性系统的尺寸，因此在精细分辨率上比XFEM效率更高。因此，它为大型重裂多孔介质开发了一种很有前途的可扩展XFEM方法。