An approximate method for calculating anisotropy parameters and reflectivity of shales with horizontal fractures

Shale is a typical medium of transverse isotropy with a vertical axis of symmetry (VTI), and its strong anisotropy is mainly due to the combined effect of intrinsic anisotropy and that induced by horizontal fractures. To calculate the anisotropy parameters of shale, a physical rock model is built based on Hudson's thin-coin fracture model and Schoenberg's linear-sliding model, and an approximate theoretical calculation method for Thomsen's anisotropy parameters of VTI media with horizontal fractures is proposed. These calculation results using the proposed method confirm that this anisotropy contributed by horizontal fractures cannot be ignored to the overall anisotropy of shale. To simplify Rüger's formula that is an approximate theoretical formula for calculating the anisotropic reflection coefficients of VTI media, a new four-term approximate formula is derived in a standard reflectivity form based on Rüger's and Aki-Richards’ formulas. The simulation results of a VTI theoretical model and logging data of shale reservoirs show that there is only a small difference between the newly derived four-term formula and Rüger's formula for incidence angles less than 40°, and the new four-term formula can correctly reveal the seismic amplitude-versus-offset (AVO) characteristics of VTI media and fully retain the corresponding anisotropic seismic responses. Compared to Rüger's formula, the proposed new formula only has four terms of unknown parameters and can directly decouple Thomsen's anisotropy parameter ε from them, which helps to alleviate the ill-posed problems of simultaneous inversion of multiple parameters and enhance its application potential in seismic inversion of VTI media as shale.