  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.
