We formulate and perform an exact Riemann solution of the discontinuous Galerkin method (DGM) for seismic propagation simulation in an anisotropic background medium with a general anisotropic fracture. An orthotropic background medium is considered, a combination of the Riemann flux and DGM is described in detail, and a linear slip (LS) model is incorporated into the DGM. A fracture is considered a geometrical interface across the element on which the LS boundary condition is imposed. Thus, we construct an exact Riemann solver with general anisotropic fractures and derive an analytical Riemann solution with orthotropic cracks. The analytical Riemann solver avoids the numerical errors caused by time discretization at the fracture interfaces. The analytical Riemann flux of the orthotropic fractures is a function of time and is not constant. The developed approach allows for anisotropy of the background model and crack, permits discontinuities in the displacement, and is suitable for arbitrarily irregular fracture structures. Using three numerical examples, the results demonstrate the correctness and validity of our new method and reveal its advantageous performance for complex crack structures.

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