Acoustoelastic Effects on Borehole Flexural Waves in Anisotropic Formations under Horizontal Terrestrial Stress Field

摘要

Applying the Stroh theory and based on the works of Hwu and Ting (1989), the complex function solution of stress and displacement fields around an open borehole in intrinsic anisotropic formation under horizontal terrestrial stress field is obtained. For cross-dipole flexural wave propagation along borehole axis, using the perturbation method, the acoustoelastic equation describing the relation between the alteration in phase velocity and terrestrial stress as well as formation intrinsic anisotropy is derived. At last, the numerical examples are provided for both the cases of fast and slow formation where the symmetry axis of a transversely isotropic (TI) formation makes an angle with the borehole axis. The phase velocity dispersion curves of borehole flexural wave and the corresponding velocity-stress coefficient are investigated. Computational results indicate that different from the stressed intrinsic isotropic formation situation, the variation in the phase velocity of flexural wave in stressed intrinsic anisotropic formation is dominated by two factors, one is the intrinsic formation anisotropy itself and the other is the stress-induced anisotropy. The former factor merely causes the borehole flexural wave split while the latter factor induces the dispersion curves intersection for two flexural waves polarized orthogonally. The combined effect of the two factors could strengthen or weaken the phenomenon of crossover for flexural wave dispersion curves. Thus, the dispersion curves of flexural waves may rnnot intersect even under the unequal horizontal terrestrial stress field, whereas it is still possible to observe the crossover of the flexural wave dispersion curves under the equal horizontal terrestrial stress field. The polarized direction of the low-frequency fast flexural wave is no longer consistent with the direction of the maximum horizontal terrestrial stress all the time. Therefore, the crossover of the borehole flexural wave dispersion curves means that the terrestrial stress must exist. On the other hand, we can't exclude the possibility of the existence of terrestrial stress even if the flexural wave dispersion curves do not intersect. Based on the above researches, the method for terrestrial stress inversion from borehole flexural wave dispersion curves obtained by cross-dipole sonic logging in stressed intrinsic anisotropic formation is simply discussed.