Exact Riemann solver of the discontinuous Galerkin method for seismic wave propagation in fractured media — II: Viscoelastic background with viscous fractures

In this paper, a new exact Riemann solver was derived and established for the discontinuous Galerkin method (DGM) to simulate seismic wave propagation in viscoelastic media with viscous fractures. Background materials and the fractures were anisotropic. The proposed approach broke through the limitations of traditional lossless isotropic fractures and achieved, for the first time, the coupling of dual essential features: viscosity and anisotropy. The spring-mass (SP) rheological model was used to describe the viscosity property of the fracture, noting that the linear-slip (LS) model was a special case of the SP model. Using the discontinuous Rankine–Hugoniot jump condition, the SP model was incorporated into the Riemann flux across fracture interfaces. In addition, the analytical Riemann flux with viscous fractures was formulated. The Riemann flux without fractures, applied to homogeneous or heterogeneous materials, was a constant, whereas the fractured Riemann flux involved an evolutionary update with time function. Three numerical experiments were used to validate that the developed approach can effectively characterize the viscosity of fractures.