Multiscale Extended Finite Element Method for Deformable Fractured Porous Media

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.