Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009 Acoustoelastic Effects on Borehole Flexural Waves in Anisotropic Formations under Horizontal Terrestrial Stress Field Ping’en Li1,2 and Xianyue Su1,3 Abstract: Applying the Stroh theory and based not intersect even under the unequal horizontal on the works of Hwu and Ting (1989), the com- terrestrial stress field, whereas it is still possible to plex function solution of stress and displacement observe the crossover of the flexural wave disper- fields around an open borehole in intrinsic aniso- sion curves under the equal horizontal terrestrial tropic formation under horizontal terrestrial stress stress field. The polarized direction of the low- field is obtained. For cross-dipole flexural wave frequency fast flexural wave is no longer consis- propagation along borehole axis, using the per- tent with the direction of the maximum horizon- turbation method, the acoustoelastic equation de- tal terrestrial stress all the time. Therefore, the scribing the relation between the alteration in crossover of the borehole flexural wave dispersion phase velocity and terrestrial stress as well as for- curves means that the terrestrial stress must exist. mation intrinsic anisotropy is derived. At last, On the other hand, we can’t exclude the possibil- the numerical examples are provided for both the ity of the existence of terrestrial stress even if the cases of fast and slow formation where the sym- flexural wave dispersion curves do not intersect. metry axis of a transversely isotropic (TI) forma- Based on the above researches, the method for tion makes an angle with the borehole axis. The terrestrial stress inversion from borehole flexural phase velocity dispersion curves of borehole flex- wave dispersion curves obtained by cross-dipole ural wave and the corresponding velocity-stress sonic logging in stressed intrinsic anisotropic for- coefficient are investigated. Computational re- mation is simply discussed. sults indicate that different from the stressed in- trinsic isotropic formation situation, the varia- Keyword: Acoustoelastic effect, borehole flex- tion in the phase velocity of flexural wave in ural wave, stressed intrinsic anisotropic forma- stressed intrinsic anisotropic formation is domi- tion, dispersion curve, terrestrial stress inversion nated by two factors, one is the intrinsic forma- method tion anisotropy itself and the other is the stress- induced anisotropy. The former factor merely causes the borehole flexural wave split while the 1 Introduction latter factor induces the dispersion curves inter- The quantitative information of terrestrial stress section for two flexural waves polarized orthog- has an important impact on borehole stability and onally. The combined effect of the two fac- the oilfield production. In the initial stage of oil- tors could strengthen or weaken the phenomenon field development, it will be great significance of crossover for flexural wave dispersion curves. for making oil and gas field development scheme, Thus, the dispersion curves of flexural waves may stability design of borehole wall and well pat- 1 State Key Laboratory for Tubulence and Complex sys- tern optimization if we understand the situation of tem, Department of Mechanics and Aerospace Engineer- stress distribution in exploitation area and reser- ing, College of Engineering, Peking University, 100871, voir local stress. In oilfield production period, the P.R.China 2 variation of abnormal formation stresses may be Institute of Geophysics, China Earthquake Administra- caused by geological condition alteration, oilfield tion, Beijing, 100081, P.R.China 3 Corresponding author. Tel.: +86 010 6275 9378; Fax.: long-term exploitation and other human factors. +86 010 6275 1812. E-mail: [email protected] Making this clear will be beneficial to take timely 174 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009 measures to prevent or reduce the phenomenon of flexural wave dispersion curves can be used as oil well damage which often appears in oilfield. an indicator of existence of terrestrial stress. Ap- At present, the main methods for local stress mea- plying nonlinear acoustoelastic model and pertur- surement in oilfield include the borehole breakout bation integral procedure, considering the uniax- and hydraulic fracturing technique, their costly ial horizontal terrestrial stress and borehole fluid and devastating borehole restrict their application pressure jointly, Cao, Wang, Li, Xie, Liu and Lu in certain extent. (2003), Cao, Wang and Ma (2004) investigated In propagation medium, elastic wave velocities the acoustoelastic effects of Stonely and flexu- are affected by prestresses (terrestrial stress). ral wave as well as the sensitive coefficient and The theory of acoustoelasticity [Pao, Sachse and velocity-stress coefficient. Furthermore, Li, Yin Fukuoda (1984)] was proposed in 1960s. And re- and Su (2006) analyzed the influence of triaxial cently it has been applied to borehole problems, terrestrial stresses on borehole modes. The re- which is the theoretical basis of terrestrial stress sults indicate that for stressed intrinsic isotropic nondestructive examination. Semi-analytical per- formation, it is only the horizontal deviatoric ter- turbation method and pure numerical method are restrial stress will cause the crossover of flexural the two main methods for investigation of the wave dispersion curves, and which is independent borehole modes in complex formation. In 1994, with the horizontal mean terrestrial stress, super- Norris, Sinha and Kostek (1994) first applied the imposed stress and fluid pressure. For cased hole, theory of acoustoelasticity to a prestressed hetero- without considering the terrestrial stress and ap- geneous medium comprising of inviscid fluid and plying perturbation method, Li, Su and Yin (2007) solid parts. Considering the influence of fluid- investigated the flexural wave in anisotropic for- solid interfacial slip, the boundary conditions, mation. The results indicate that because of the constitutive relations and motion equations in the influence of the casing, the flexural wave disper- reference, intermediate and current coordinates sion curves in cased hole in both fast and slow were established, respectively. Then, adopting anisotropic formations all almost tend toward an the perturbation procedure, the first-order pertur- identical Stoneley wave velocity at higher fre- bation in the eigenfrequency of Stonely wave in quency. a borehole with pressurized fluid and prestrained Generally, the perturbation method is only suit- solid in the intermediate coordinate was deduced. able to analyze the dispersion curves of bore- Based on it, Sinha, Kostek and Norris (1995), hole mode. In order to obtain the full-wave field Sinha and Kostek (1996) further studied the dis- in time domain, a purely numerical method has persion alteration of Stonely and flexural waves to be adopted, mainly the finite element method induced by fluid pressure and uniaxial horizon- (FEM) and finite difference method (FDM). For tal terrestrial stress, respectively. Their results general homogeneous anisotropic medium, an indicate that under an uniaxial horizontal terres- effective meshless method based on the local trial stress field, the borehole flexural wave de- Petrov-Galerkin approach is proposed by Sladek pends on the polarization direction, which results J, Sladek V and Atluri (2004) for solution elas- in the dispersion curves intersection for two flex- todynamic problem. Subsequently, Atluri, Liu ural waves polarized orthogonally. Because the and Han (2006) developed a mixed finite differ- intrinsic formation anisotropy merely causes the ence method within the framework of the mesh- flexural dispersion curves split, rather than in- less local Petrov-Galerkin approach for solving tersection, according to the crossover of flexu- solid mechanics problems. Later, a new numer- ral wave dispersion curves, we can distinguish ical algorithm based on meshless local Petrov- the stress-induced anisotropy from the intrinsic Galerkin approach and modified moving least anisotropy of the formation. Subsequently, Win- square method was proposed by Gao, Liu K, Liu kler, Sinha and Plona (1998) proved both by the- Y (2006) for analyzing the wave propagation and ory and experiments that the intersection of the dynamic fracture problems in elastic media. For Acoustoelastic Effects on Borehole Flexural Waves 175 borehole problem, Sinha, Liu and Kostek (1997), slow azimuthal anisotropic formation. The re- Liu and Sinha (2000, 2003) discretized the dy- sults indicate that the variation of flexural wave namic equation of stress-velocity form directly phase velocity is controlled by two factors, one is by applying staggered-grid high-order finite dif- the intrinsic formation anisotropy and the other is ference method. They investigated the dispersion the stress-induced anisotropy. The former factor curves as well as the waveforms in time domain of merely causes flexural wave split while the latter Stonely and flexural wave in fluid-filled borehole factor results in flexural wave dispersion curves in biaxial and triaxial terrestrial stressed forma- intersection. The combined effect of the two fac- tion, using absorbing boundary condition to elim- tors could strengthen or weaken the phenomenon inate the reflection at an artificial finite bound- of crossover for flexural wave dispersion curves. ary. But for purely numerical method in borehole Thus even under an unequal horizontal terrestrial modes problem, it is extremely high