变形裂隙多孔介质的多尺度扩展有限元方法
Multiscale Extended Finite Element Method for Deformable Fractured Porous Media
摘要：
可变形裂隙多孔介质在许多地学应用中出现。虽然扩展有限元（XFEM）已成功地在计算力学界的变形精确建模，其应用在自然地球科学的应用是不简单的。这主要是由于地下地层是异质的，并且跨越大的长度尺度，具有不同尺度的许多裂缝。在这项工作中，我们提出了一种新的多尺度制定XFEM，基于局部计算的基函数。局部多尺度基函数捕捉裂缝引入的非均匀性和不连续性。局部边界条件被设置为遵循降维系统，以保持基函数的准确性。使用这些多尺度的基础上，一个多尺度的粗尺度系统，然后治理代数和解决，其中没有丰富的裂缝存在。这种公式允许显着的计算成本降低，在同一时间，它保留了离散位移向量空间的精度。最终使用相同的多尺度基函数将粗尺度解插值回细尺度系统。建议的多尺度XFEM（MS-XFEM）也集成在一个两阶段的代数迭代求解器，通过它可以实现任何所需的水平的误差减少。几个概念验证的数值试验，以评估所开发的方法的性能。结果表明，MS-XFEM是准确的，当与细尺度参考XFEM解决方案。同时，它在精细尺度分辨率上比XFEM更有效。因此，它开发了第一个可扩展的XFEM方法的大规模严重裂缝多孔介质。
英文摘要：
Deformable fractured porous media appear in many geoscience applications. While the extended finite element (XFEM) has been successfully developed within the computational mechanics community for accurate modeling of the deformation, its application in natural 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. In this work, we propose a novel multiscale formulation for XFEM, based on locally computed basis functions. The local multiscale basis functions capture the heterogeneity and discontinuities introduced by fractures. Local boundary conditions are set to follow a reduced-dimensional system, in order to preserve the accuracy of the basis functions. Using these multiscale bases, a multiscale coarse-scale system is then governed algebraically and solved, in which no enrichment due to the fractures exist. 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 such, it develops the first scalable XFEM method for large-scale heavily fractured porous media.
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