Finite Difference Scheme Based on the Lebedev Grid for Seismic Wave Propagation in Fractured Media

Fractures in underground media are mostly vertical and orthogonal. Based on the assumption of long wavelengths, fractures can be considered infinitely thin planes embedded in a homogeneous medium. The fracture interface satisfies the conditions of displacement discontinuity and stress continuity, i.e. a linear slip boundary. The finite difference method is typically used to simulate the propagation of seismic waves at the fracture interface. In this study, a new finite difference scheme is proposed based on the velocity-stress equation, which can be used to simulate the propagation of seismic waves in vertical and orthogonal fracture media. The new finite difference scheme more closely resembles the conditions of actual vertical and orthogonal fractures. The Lebedev grid was adopted, and no interpolation was required for the calculation, which improved the accuracy. The numerical simulation of the new finite difference scheme reveals its long-term stability and accuracy. This scheme can be used for the analysis of reservoir fracture delineation. In addition, an explicit slip interface condition and the new finite difference scheme were used to comparatively model fractured media. Virtually identical results were obtained, which verifies the effectiveness of this model. Finally, based on the boundary conditions of linear slip, an improved Zoeppritz equation was established, and the effect of fracture compliance on the reflection and transmission coefficients was studied. This scheme can be used for the analysis of reservoir fracture traps.