Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines, for which the rock physics model is indispensable. However, up to now, multitudinous analytical models focus primarily on the cracked rock with the isotropic background, while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth. Hence, we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy, respectively, by forty groups of numerical experiments. Based on the conclusion from the experiments, the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background. Further, we comprehensively analyzed, according to the developed model, effects of the crack dip angle, background anisotropy, filling fluid and crack density on the effective elastic properties of the cracked rock. The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P- and SH-wave velocities, whereas they have relatively small effects on the SV-wave velocity anisotropy. Moreover, the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters, while its effects on shear coefficients depend on the crack dip angle. The increasing crack density reduces velocities of the dry rock, and decreasing rates of the velocities are affected by the crack dip angle. By comparing with exact numerical results and experimental data, it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients. Moreover, the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.