Abstract：Conventional integral methods usually have the singularity problem in solving frequency-dependent elastic modulus of penny shaped fractures,and can only obtain a single normal frequency-dependent modulus.In order to quickly and accurately perform numerical simulation of the penny shaped fracture,based on the Gauss-Lobatto discrete integral,a new approach is proposed.First the second Fredholm integral equation is transformed into a discrete integral in a finite interval.Then,the solving the zero-limit of far field multiple scatter equation and the discrete integral procedure are unified with the high-order approximation.To obtain the frequency-dependent elastic tensor,analytic elastic moduli equations at the high and low frequency limits are respectively obtained with the anisotropic Gassmann equation,the linear slip theory,and the Hudson crack model.Based on the analysis of phase velocity and viscoelastic reflection coefficient,the following understanding are obtained:A.The greater the fracture denty is,the graeter the attenuated peak amplitude is;B.The attenuated peak amplitude has lower frequency when the fracture size or the fluid viscosity becomes greater;C.The incident velocity and attenuation in the vertical direction have greater change than that in the horizontal direction;D.The reflection coefficient of gas-bearing media is significantly larger than that of oil-bearing or water-bearing media.
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