A new upwind flux for a jump boundary condition applied to 3D viscous fracture modeling

We present a discontinuous Galerkin (DG) algorithm with nonconformal meshes to simulate 3D elastic wave propagation in heterogeneous media with arbitrary discrete fractures. In our method, the fractures are not limited to be planar, single, and lossless, but can be curved, intersecting, and viscous. In contrast to the exact volumetric modeling for the extremely thin layer, explicitly treating an individual fracture as a geometry surface (i.e., an imperfect contact interface) requires the jump condition for displacement/velocity, but the continuity of traction vector on the fracture interface. A new upwind flux is proposed to weakly impose this jump boundary condition in the DG framework. This flux guarantees the stability and accuracy of the DG schemes to model arbitrary fractures. Unlike conventional Riemann solvers applied to continuous media, this solution involves an evolutionary update on the Godunov states. Besides this, no extra computational cost is added. In addition, we can extend the fracture interface into a perfectly matched layer to mimic an infinitely large fracture. Quantitative comparisons of the waveforms between our algorithm and an independent finite element code demonstrate the accuracy and efficiency of our algorithm.