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

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 intrinsic 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 demands on stress field, flexural wave dispersion curves may the computer resources and its result is hard to be not intersect. Whereas it is still possible to ob- further analysis, which limits its development and serve the crossover of flexural wave dispersion application range. In generally, comparing to the curves under an equal horizontal terrestrial field purely numerical method, the analytic and semi- because of the non-axisymmetry of the stresses analytic method is more suitable for theoretical around the borehole in anisotropic formation, be- study. sides which, the polarized direction of the low- The above researches are all based the assumption frequency fast flexural wave is no longer keep that the formation is an intrinsic isotropic elastic consistent with the direction of the maximum hor- solid, and merely the influence of stress-induced izontal terrestrial stress all the time. anisotropy on borehole mode is analyzed. How- Therefore, in actual cross-dipole sonic logging, if ever, actually, the formation is not isotropic, but the formation is intrinsic anisotropic, a crossover anisotropic in various degrees. The stresses and of the flexural wave dispersion curves means it displacements field around a borehole in aniso- must exist terrestrial stress. On the other hand, we tropic formation is completely different from that can’t exclude the existent possibility of terrestrial in isotropic formation under terrestrial stresses, stress if the flexural wave dispersion curves do not which directly influences the wave field and dis- intersect. Based on the above researches, we sim- persion of borehole mode. Therefore, the study ply discuss the acoustoelastic inversion model and of the acoustoelastic effect on borehole flexu- the method for terrestrial stress inversion from ral wave in intrinsic anisotropic formation un- borehole flexural wave dispersion curves. der terrestrial stress field has important theoretical significance for realization of terrestrial stress 2 Stresses and displacements field around an nondestructive examination in stressed intrinsic open borehole in anisotropic formation un- anisotropic formation through cross-dipole sonic der horizontal terrestrial stress field logging. First of all, we give solutions of the statics prob- In this paper, considering an intrinsic anisotropic lem that an open borehole is in anisotropic for- formation under horizontal terrestrial stress field, mation under horizontal terrestrial stress field. As using the Stroh theory and based on the work of shown in Figure 1, the formation is anisotropic Hwu and Ting (1989), the complex function so- elastic solid and C is its elastic constant. The lutions of stresses and displacements around an ijks horizontal terrestrial stress in x and y direction are open borehole are obtained firstly. Then, ap- S and S respectively. The radius of the bore- plying the nonlinear theory of acoustoelasticity x y hole is a. Without considering the variation of and perturbation integral procedure, the expres- terrestrial stress along the depths, the problem can sion of first-order correction of the phase velocity be simplified as a generalized anisotropic plane for borehole modes is derived. Finally, the nu- strain problem [Ting (1996)], which can be solved merical examples are provided for both fast and by using Stroh theory. Hwu and Ting (1989) first 176 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009 studied the two-dimension problem of the aniso- and let Impα > 0(α = 1,2,3), the correspond- tropic elastic solid with an elliptic hole or rigid in- ing eigenvectors is aα(α = 1,2,3). In equation clusion subjected to a uniform loading at infinity (7), the superscript “T” denotes transpose. a = T applying Stroh theory. Based on their works, the (a1,a2,a3) , Q, R and T are 3 × 3 matrix, their complex function solution for the problem here component respectively is can be obtained easily. Q = C , R = C , T = C (8) Assuming in infinite anisotropic elastic medium, ik i1k1 ik i1k2 ik i2k2 in z plane, the circular hole boundary L is and let z = x +ix = acosψ +iasinψ (1) T 1 2 A =(a1,a2,a3), b =(R + pT)a, (9) Where a is the radius of the circular hole, ψ is a B =(b1,b2,b3) real parameter, x and x are the components of 1 2 the mapping (3) transforms the region outside the the two Cartesian coordinate axis. circular hole in zα plane to the region outside the In zα plane, the corresponding circular hole unit circle in ζα plane. boundary Lα is σ ∞ ε∞ Let ij and ij be the stress and strain at infinity. zα = x1 + pα x2 (2) They are related by the stress-strain laws σ ∞ = ε∞ Considering the conformal mapping [Ting ij Cijks ij (10) (1996)] For generalized plane strain problem, we have the −1 ∞ zα = cα ζα +daζα , (α = 1,2,3) (3) ε = condition 33 0. If strain is known, stressed can be obtained through stress-strain relation and con- where ε∞ = σ ∞ dition 33 0, and vice versa. Assuming that ij a a ∞ cα = (1 −ipα ), dα = (1 +ipα ) (4) and ε are known, writing 2 2 ij ∞ ∞ ∞ ∞ ∞ ∞ 2 2 2 u = x1ε +x2ε , ϕ = x1t −x2t (11) zα + zα −a (1 + pα ) 1 2 2 1 ζα = (5) a(1 −ipα) in which 2 2 2 − zα − zα −a (1 + pα ) ⎛ ⎞ ⎛ ⎞ ζ 1 = (6) ε∞ ε∞ α 11 2 12 a(1 +ipα ) ∞ ∞ ∞ ∞ ε = ⎝ 0 ⎠ = u , ε = ⎝ ε∞ ⎠ = u ∼ (α = , , ) 1 ,1 2 22 ,2 in equations (4) (6), pα 1 2 3 is the three 2ε∞ 2ε∞ complex eigenvalues of the eigenvalue problem ⎛ 13⎞ ⎛ 23⎞ σ ∞ σ ∞ 11 12 + ( + T)+ 2 = ∞ = ⎝σ ∞ ⎠ = −ϕ∞, ∞ = ⎝σ ∞ ⎠ = ϕ∞ Q p R R p T a 0(7)t1 12 ,2 t2 22 ,1 σ ∞ σ ∞ 13 23 (12) y then the solution of displacement u and stress function ϕ can be chosen as the from ∞ ∞ − a x = ε + ε + ζ 1 S u x1 1 x2 2 2Re A ∗ q x (13) ϕ = ∞ − ∞ + ζ −1 x1t2 x2t1 2Re B ∗ q where q is a constant vector to be determined by boundary condition. f (z∗) is the diagonal ma- S y trix Figure 1: An open borehole in anisotropic formation under horizontal terrestrial stress field f (z∗) = diag[ f (z1), f (z2), f (z3)] (14) Acoustoelastic Effects on Borehole Flexural Waves 177

Then, we determine the constant vector q by we have boundary condition. ∞ −1 −1 ∞ ∞ = −ϕ, = − ζ (− + ) At the circular hole boundary L, x1 = acosψ and t1 2 t1 Re B ∗ ,2 B a t2 it1 = ψ x2 asin , thus we have ∞ −1 −1 ∞ ∞ = ϕ, = + ζ (− + ) t2 1 t2 Re B ∗ ,1 B a t2 it1 ∞ ∞ −iψ ∞ ∞ x1ε +x2ε = Re[e a(ε +iε )] 1 2 1 21 ∞ −1 −1 ∞ ∞ (15) ε = u, = ε +Re A ζ B a(−t +it ) ∞ − ∞ = [ −iψ ( ∞ − ∞)] 1 1 1 ∗ ,1 2 1 x1t2 x2t1 Re e a t2 it1 ∞ −1 −1 ∞ ∞ ψ ε2 = u,2 = ε +Re B ζ∗ B a(−t +it ) and in zα plane, we have ζα = ei at the circular 2 ,2 2 1 −1 −iψ hole boundary Lα ,soζα = e . From equations (21) (13) and (15), the displacement uL and the stress where function ϕL at the circular hole boundary L are = [ −iψ ( ε∞ + ε∞ + )] ∂ζ−1 uL Re e a 1 ia 2 2Aq ζ −1 = ∗ − ψ ∞ ∞ (16) ∗ ,1 ϕ = [ i ( − + )] ∂x1 L Re e at2 iat2 2Bq 1 z∗ The traction on the surface of a circular hole van- = 1 − ( + ) 2 2 2 ϕ = a 1 ip∗ z∗ −a (1 + p∗) ishes. Setting L 0 in equation (16), we obtain ∂ζ−1 −1 ∗ 1 −1 ∞ ∞ ζ∗ = q = − B a(t −it ) (17) ,2 ∂x 2 2 1 2 p∗ z∗ Substituting equation (17) into equation (13), we = 1 − ( + ) 2 2 2 obtain the displacement u and stress function ϕ a 1 ip∗ z∗ −a (1 + p∗) respectively as (22) ∞ ∞ − − ∞ ∞ u =x ε +x ε +Re A ζ 1 B 1a(−t +it ) 1 1 2 2 ∗ 2 1 Acquired the stress and strain from equation (21), ϕ = ∞ − ∞ + ζ −1 −1 (− ∞ + ∞) ε = x1t2 x2t1 Re B ∗ B a t2 it1 using stress-strain relation and condition 33 0, σ (18) stress 33 is also gained. Hereto, all the stress, strain and displacement are obtained. For bore- For the problem in this paper, the anisotropic for- hole problem, the cylindrical coordinate is al- mation with a circular borehole is subjected to ways adopted for convenience. Therefore, the horizontal stresses Sx and Sy at infinity, thus we displacement, stress and strain should be trans- have formed into cylindrical coordinate. The specific ∞ ∞ ∞ ∞ ∞ coordinate transformation relation can refer to lit- σ = S , σ = S , σ = σ = σ = 0 (19) 11 x 22 y 12 13 23 erature [Auld (1973)]. According to the stress-strain relation (10) and ε∞ = σ ∞ ε∞ 3 Acoustoelastic equation of borehole mode condition 33 0, the stress 33 and strain 11, ε∞ ε∞ ε∞ ε∞ in anisotropic formation 22, 12, 13 and 23 can be determined. Therefore, σ ∞ ε∞ the stress ij and strain ij are all known. After For a fluid-filled open borehole in anisotropic for- gaining the displacement and stress function from mation under terrestrial stress field, applying the equation (18), the stress and strain can be further theory of acoustroelasticity and perturbation inte- obtained. From the relation ⎛ ⎞ ⎛ ⎞ gral procedure, the first-order perturbation in the ε11 2ε12 phase velocity of borehole mode can be expressed ⎝ ⎠ ⎝ ⎠ ε1 = 0 = u,1, ε2 = ε22 = u,2 as [Norris, Sinha and Kostek (1994)] 2ε13 2ε23 ⎛ ⎞ ⎛ ⎞ (20) m m m ( m )∗ Δv v −v Δω V cˆLγMν uν,M uγ,L dV σ11 σ12 = R = = m m 2 m m ∗ ⎝ ⎠ ⎝ ⎠ v v ω 2ω ρuγ (uγ ) dV t1 = σ12 = −ϕ,2, t2 = σ22 = ϕ,1 R R m m V σ13 σ23 (23) 178 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009 where superscript m refers to the family of guided ity of borehole modes caused by terrestrial stress, wave modes, m = 0andm = 1 denote Stonely we should select the state of anisotropic formation m m and flexural wave, respectively. vR and v are the without terrestrial stress as the unperturbed refer- phase velocity of m family guided wave modes ence state, and the state of formation deformation in the unperturbed reference and biasing stress- under terrestrial stress as the perturbed state. In induced current state. Δω is the variation of this way, the second-order elastic module of for- eigenfrequency ωm between the reference and mation in unperturbed reference state is equal to m ◦ = current state. uγ is the displacement of m fam- that in perturbed state, namely, cLγMν cLγMν , ily guided wave modes in the reference state, and the equation (24) can be simplified. How- superscript “∗” denotes the complex conjugate. ever, before using the equation (23), the solutions Comma “,” represents the partial differential with of displacement field for borehole mode in unper- respect to its corresponding geometric coordinate, turbed state need to be known first. Unfortunately, the summation convention on repeated subscripts for general anisotropic formation, we haven’t the is implied. V is the integral volume, including the complete analytical solution of displacement field fluid in the borehole and the formation around it. till present, which brings us great difficulties to cˆLγMν is the incremental part of equivalent elas- calculate the phase velocity of borehole mode us- tic module of anisotropic formation in terrestrial ing perturbation integral equation (23). In order stress-induced state compared to the second-order to solve this problem, we choose an appropri- ◦ elastic module cLγMν of formation in unperturbed ate isotropic formation without terrestrial stress reference state, namely as the reference unperturbed state, and the state of the intrinsic anisotropic formation under terres- = − ◦ + δ + cˆLγMν cLγMν cLγMν TLM γν cLγKMwν,K trial stress is still selected as the current perturbed +cLKMν wγ,K +cLγMνABEAB (24) state. In this way, we can easily obtain the analytical solution of displacement for borehole mode where cLγMν and cLγMνAB are the second-order and the third-order elastic module tensor of for- in unperturbed state. The next, the key is how to choose the reference isotropic formation in un- mation in current state, respectively, δγν is Kro- perturbed state. We know that for two transverse necker symbol. TLM , EAB and wγ,K are static biasing stresses, strains and static displacement gradi- waves propagation along the borehole axis and ents of the borehole problem, respectively. Ac- polarized orthogonally in general anisotropic for- cording to the acquired complex function solu- mation, their velocities are different. Because the tions previously, the static displacement gradient velocity of the transverse wave in isotropic elastic λ μ can be obtained through derivation to static dis- solid is determined by elastic constant and , placement. In cylindrical coordinate, we have the reference isotropic formation in unperturbed reference state can be chosen from the velocities ∂wr of plane waves propagation along borehole axis in wr,r = = Err, ∂r anisotropic formation [Sinha, Norris and Chang 1 ∂wθ (1994)], namely wθ,θ = +wr = Eθθ, r ∂θ μ = ρ 2 , λ = ρ( 2 − 2 ) qSV VqSV qSV VqP 2VqSV ∂wz 1 ∂wr (26) w , = = E = 0, w ,θ = −wθ , μ = ρV 2 , λ = ρ(V2 −2V 2 ) z z ∂z zz r r ∂θ qSH qSH qSH qP qSH where V , V and V are the velocities of the ∂wθ 1 ∂wz ∂wθ qP qSV qSH wθ, = , w ,θ = , wθ, = , r ∂r z r ∂θ z ∂z three plane wave propagation along the borehole axis in anisotropic formation, respectively. λqSH, ∂wz ∂wr wz,r = , wr,z = , μ and λ ,μ are the elastic constants of ∂r ∂z qSH qSV qSV (25) the reference isotropic formation in unperturbed state when study the qSH and qSV polarized flex- Generally, when using perturbation integral pro- ural wave, respectively, which means that two dif- cedure to compute the alteration in phase veloc- ferent isotropic unperturbed reference models are Acoustoelastic Effects on Borehole Flexural Waves 179 established to gain the displacement solutions of stress, respectively. In equation (28), we have cross-dipole flexural waves polarized in different χ = δ + + ∗ direction in unperturbed state. This choice of the 0 TLM γν cLγKMwν,K cLKMν wγ,K uν,Muγ,L ∗ ∗ unperturbed reference isotropic formation model χ =E u , u +E u , u 1 11 1 1 1,1 22 2 2 2,2 has been fully considered the difference in phase ∗ ∗ ∗ ∗ χ =E u , (2u +u )+u , (2u +u ) 2 11 22 1,1 2,2 3 3 1,1 3,3 velocity for borehole flexural waves polarized or- ∗ ∗ ∗ +E (u , +2u , )u +u , (2u +u ) thogonally in anisotropic formation and makes the 22 1 1 2 2 1,1 3 3 2,2 3,3 ∗ ∗ perturbative correction minimal, which results in χ =2 E (u , u )+E (u , u ) 3 11 3 3 2,2 22 3 3 1,1 high accuracy of perturbed solution [Sinha, Norris ∗ ∗ ∗ χ =E u , u +u , (u +2u ) 4 11 32 3,2 2 3 2,3 3,2 and Chang (1994)]. ∗ ∗ ∗ +E u , u , +u , (u , +2u , ) Because it still hasn’t been found in relevant lit- 22 3 1 3 1 1 3 1 3 3 1 + ( ∗ + ∗ ) eratures about the measured values of the third- 4E12u3,3 u1,2 u2,1 ∗ ∗ order elastic module for anisotropic medium until +4E (u , u +u , u ) 13 1 3 2,2 2 2 3,1 recently, using the known values of the third-order ∗ +4E (u , +u , )u 23 23 3 2 1,1 elastic modules for isotropic medium instead, and ∗ ∗ ∗ χ =(E +E ) u , u +u , (u +2u ) assuming they are identical in different direction. 5 11 22 2 1 2,1 1 2 1,2 2,1 ∗ ∗ ∗ In this way, according to convention, the three in- +E u , u +u , (u +2u ) 11 3 1 3,1 1 3 1,3 3,1 ∗ ∗ ∗ dependent third-order elastic modules are written +E u , (u +2u )+u , u 22 2 3 2,3 3,2 3 2 3,2 as c111, c112 and c123, respectively, and have the ∗ ∗ ∗ +4E (u , +u , )u +u , (u +u ) relation 12 1 2 2 1 1,1 2 2 1,2 2,1 ∗ ∗ ∗ +4E (u , +u , )u +u , (u +u ) 13 1 3 3 1 1,1 3 3 1,3 3,1 1 ∗ ∗ ∗ c144 = (c112 −c123), +4E (u , +u , )u +u , (u +u ) 2 23 2 3 3 2 2,2 3 3 2,3 3,2 ∗ ∗ ∗ ∗ 1 χ =4E u , u +u , (u +u )+u , u c = (c −c ), (27) 6 12 3 2 3,1 2 3 1,3 3,1 1 3 3,2 155 4 111 112 ∗ ∗ +4E (u , +u , )(u , +u , ) 1 13 2 3 3 2 1 2 2 1 c = (c −3c +2c ) + ∗ + ( ∗ + ∗ )+ ∗ 456 8 111 112 123 4E23 u3,1u2,1 u1,3 u1,2 u2,1 u1,2u3,1 (29) Expanding the equation (23), it can be written as In equation (29), the superscript m in displace- Δ Δ +Δ m − m ment components is omitted for convenience, and v = vani vstr = v vR vm vm vm the subscript 1,2,3 corresponds to the r,θ,z in R R R cylindrical coordinate system, respectively. − ◦ ( )∗ Δ cLγMν cLγMν uν,M uγ,L dV vani = V Using relation (27), integration after substituting m ω2 ρ ( )∗ vR 2 V uγ uγ dV the analytical solutions of the static stress, strain Δ vstr = 1 and displacement into equation (28), we have the ∗ vm 2ω2 ρuγ (uγ ) dV acoustoelatic equation of borehole modes as fol- R V lowing χ0dV +c111 χ1dV +c112 χ2dV V V V Δ Δ +Δ m − m v = vani vstr = v vR + χ + χ vm vm vm c123 3dV c144 4dV R R R (30) V V Δv = ani + Sx + Sy m Q Sx Q Sy +c155 χ5dV +c456 χ6dV vR V V (28) where QSx and QSy are called as velocity-stress coefficients corresponding to horizontal terrestrial where Δvani and Δvstr are the variations of the stress Sx and Sy, respectively. They are related phase velocity of borehole guided mode caused to eigenfrequency. From equation (30), noticing by the formation anisotropy and the terrestrial that the last term of equation (24) is related to the 180 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009 third-order elastic module, so QSx and QSy can be formation are listed in Table 1 and Table 2, re- expressed as the summation of the following four spectively. terms For the TI formation as shown in Figure 2, the polarized directions of the plane qSV and SH waves 4 Sx Sx Sx C c C c112 C c Sx = Sx = Sx + 2 111 + 3 + 4 123 propagation along borehole axis parallel with x Q ∑ Qi C1 ◦ i=1 c66 c66 c66 and y axis, respectively, which correspond to 0 ◦ 4 Sy Sy Sy and 90 azimuth in cylindrical coordinate. There- S S C c111 C c112 C c123 QSy = ∑ Q y = C y + 2 + 3 + 4 fore, it refers to the 0◦ and 90◦ azimuthally polar- i 1 c c c i=1 66 66 66 ized flexural waves respectively while using the (31) equations (26) and (28) to calculate. S The validation calculation is given first. When where CSx and C y (i = 1,2,3,4) are called as sen- i i elastic modules C = C = λ +2μ, C = C = sitivity coefficients corresponding to S and S ,re- 11 33 12 13 x y λ , C =(C −C )/2 = μ, the TI formation is spectively. They depend on frequency and merely 44 11 12 degenerated to isotropic one. Setting the param- can be obtained by the numerical method. Ac- eters of the formation and the fluid be identical cording to the equations (30) and (31), the sensi- with that in reference [Li, Yin and Su (2006)], and tivity coefficients do not influence the phase ve- uniaxial horizontal terrestrial stress S = −5MPa, locity of borehole mode directly, but through the x the flexural wave dispersion curves obtained by third-order elastic module, they affect the first- using method in this paper is shown in Figure 3, order correction of phase velocity by velocity- in which curves 2 and 3 are almost consistent with stress coefficient. Therefore, we can investigate those in reference [Li, Yin and Su (2006)]. This the influence of terrestrial stress on phase velocity means the method in this paper is reliable. quantitatively by the velocity-stress coefficient. 4.1 Fast formation 4 Numerical computation and results For fast TI formation, the Figure 4 shows the flex- The influence of stress concentration around a ural wave velocity-stress coefficients correspond- borehole in anisotropic formation on phase ve- ing to Sx and the dispersion curves when the sym- locity of borehole modes are all included in the metry axis of TI formation makes different an- velocity-stress coefficient. The next, we provide a gle ϕ with the borehole axis under the terrestrial numerical example for quantitative analysis of the stress field Sx = −50MPa, Sy = 0. flexural wave dispersion in anisotropic formation Because of the anisotropy induced by stress con- under terrestrial stress. Assuming the formation centration around a borehole and the formation in- is TI elastic solid. A schematic of a borehole and trinsic itself, the formation is not material sym- the global coordinate system xyz are shown in Fig- metry about the borehole axis. In addition the ure 2. The symmetry axis of the formation is in asymmetry of dipole sources, thus the two flex- xz plane and makes an angle ϕ with the borehole ural waves polarized in φ = 0◦ and φ = 90◦ az- axis z. The local Cartesian coordinate x y z is es- imuth are different. From Figure 4, the velocity- tablished in formation, z axis coincides with the stress coefficients in φ = 0◦ azimuth entirely dif- TI symmetry axis and y axis is coincident with fer from that in φ = 90◦. Since the third-order y axis in global coordinate system. Fluid is full elastic modules are much larger than the second- filled in the borehole and the horizontal terrestrial order elastic constants for several magnitudes for stresses in x and y direction are Sx and Sy, respec- the solid formation, the last three velocity-stress tively. coefficients related to third-order modules c111, In computation, we set the fluid compression c112 and c123 are dominated and the first one al- wave velocity in borehole vf = 1500m/s, fluid most can be ignored. The result coincides to equa- 3 ◦ ◦ density ρf = 1000kg/m , and the borehole radius tion (31). With the ϕ change from 0 to 90 ,the = . Sx( = a 0 1016m. The material parameters of the TI curve shape of velocity-stress coefficient Qi i Acoustoelastic Effects on Borehole Flexural Waves 181 py z z′ Formation

x Sx Sx

x′ Fluid

Figure 2: A ﬂuid-ﬁlled borehole in TI formation under terrestrial stress

Table 1: The second-order elastic module and density of TI formation 3 Parameter C11/(GPa) C12/(GPa) C13/(GPa) C33/(GPa) C44/(GPa) ρ(Kg/m ) Fast stratum 37.03 12.91 8.91 27.03 9.06 2600 Slow stratum 21.58 10.18 7.5 16.6 5.3 2560

Table 2: The third-order elastic module of TI formation Parameter C111/(GPa) C112/(GPa) C123/(GPa) Fast stratum -21217 -3044 2361 Slow stratum -21217 -3044 2361

1900 φ 0 =0 1 1800 2 3 1700 φ =900

1600

1500

v/(m/s) 1400

1300

1200

1100 0 5 10 15 Frequency/(KHz)

Figure 3: Flexural wave dispersion when degenerated to isotropic formation under uniaxial terrestrial stress 1 reference state; 2 0◦ azimuth; 3 90◦ azimuth 182 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009

(a1) (a2) 20 QSx 1 25 QSx 2 0 QSx 3 QSx

-1 4

-1 -20

Pa) 5 10 Pa) 10

-40 /(10 S x S /(10 x i x Q S i

1 Q

Q QSx -60 2 -15 QSx 3 Sx -80 Q (ϕ=00) (ϕ=00) 4

-100 -35 04812160481216 0 Frequency/(KHz) (φ=00) Frequency/(KHz) (φ=90 ) (b1) (a3) 00 20 2400 900

0 2200

-1 -20 Pa)

2000 10 -40 /(10 v/(m/s) x S i QSx Q 1 1800 S -60 Q x 2 QSx 3 1600 -80 QSx 4 (ϕ=450) (ϕ=00) 1400 -100 0 4 8 12 16 0481216 0 Frequency/(KHz) Frequency/(KHz) (φ=0 )

Figure 4

2,3,4) is approximately similar, in which the vari- Here, we proved that it still has the characteris- Sx ations of Q2 is the most obvious, especially at tic of dispersion curves intersection for flexural low frequency. From Figures 4a3, 4b3 and 4c3, wave in stressed intrinsic anisotropic formation, a crossover of dispersion curves for two flexural which lays a theoretical foundation for expand- waves polarized orthogonally is clear. And with ing the terrestrial stress nondestructive examina- ϕ increase from 0◦ to 90◦, the frequencies corre- tion method through cross-dipole sonic logging to sponding to the intersection point are of little dif- the condition of intrinsic anisotropic formation. ference and all around 7KHz. It is same with the Under the uniaxial terrestrial stress in y axis di- conclusion of the acoustoelastic effect on bore- rection of Sx = 0, Sy = −50MPa, the Figure 5 hole flexural wave in intrinsic isotropic formation shows the flexural wave velocity-stress coefficient [Sinha and Kostek (1996)], the crossover of flex- related to Sy and the dispersion curves with dif- ural wave dispersion curves in intrinsic anisotro- ferent ϕ. From Figure 5, the curve shape of pic formation is also the unique feature of stress- Sx( = , , ) velocity-stress coefficient Qi i 2 3 4 is of induced anisotropy, which can be used as an indi- slight change when ϕ increase from 0◦ to 90◦. cator of terrestrial stress existence. Although the φ = ◦ Sy In 90 azimuth, the absolute values of Q2 , phenomenon of dispersion curves crossover for S S Q y and Q y at an identical frequency decrease cross-dipole flexural wave has been already ob- 3 4 gradually respectively in low frequency with the served in laboratory and acoustic logging, the the- ϕ increase, in which the decreasing range for the oretical research is still limited to the case of in- absolute value of QSy is the most obvious. In trinsic isotropic formation under terrestrial stress. 2 Acoustoelastic Effects on Borehole Flexural Waves 183

2800 (b2) (b2) 25 00 2600 900 2400 -1 5 2200 Pa)

10 v/(m/s) /(10

x 2000 S i

Q QSx -15 1 1800 QSx 2 QSx 3 1600 0 (ϕ=45 ) QSx 0 4 (ϕ=45 ) -35 1400 0481216 0481216 0 Frequency/(KHz) Frequency/(KHz) (φ=90 ) 25 (c1) (c2) 20

-20 5 -1 -1 Pa) Pa) 10 10 -40 /(10 /(10 x S x S S i -60 Q x i S Q 1 Q Q x QSx -15 1 2 S -80 Q x QSx 2 3 QSx QSx 3 -100 4 S (ϕ=900) Q x (ϕ=900) 4 -120 -35 04812160 4 8 12 16 0 Frequency/(KHz) (φ=0 ) Frequency/(KHz) (φ=900)

(c3) 2600 00 900

2400

2200

v/(m/s) 2000

1800

1600 (ϕ=900) 1400 0481216 Frequency/(KHz) Figure 4: Flexural wave velocity-stress coefﬁcients corresponding to S and dispersion curves in fast TI ◦ ◦ formation when Sx = −50MPa, Sy = 0. (a) φ = 0; (b) φ = 45 ;(c)φ = 90 184 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009

φ = 0◦ azimuth, the alteration in absolute val- persion curves may not be observed. The reason is Sy Sy Sy that the stress-induced anisotropy and the intrinsic ues of Q2 , Q3 and Q4 is not very great with ϕ increase. Comparing the Figure 5a1, 5a2 with anisotropy of formation maybe partially counter- the Figure 4a1, 4a2, respectively, it can be found act each other in this case, which makes the differ- that the velocity-stress coefficients related to Sx in ence in phase velocity of two flexural waves po- φ = 0◦ and φ = 90◦ azimuth are exactly identi- larized orthogonally reduction, especially in high ◦ ◦ cal with coefficients for Sy in φ = 90 and φ = 0 frequency. azimuth. The reason is that the TI formation is material symmetry about borehole axis when ◦ 4.2 Slow formation ϕ = 0 . For two cases that the formation is subjected to an equal stress in x and y axis direc- For slow TI formation in Table 1, the velocities of tion respectively, the stress distribution around a plane qSV and SH wave are all less than the veloc- borehole is completely identical except for the x ity of fluid compression wave in borehole when ϕ and y coordinate interchange. Thus the velocity- increase from 0◦ to 90◦. The flexural wave dis- stress coefficients related to Sx and Sy just ex- persion curves and velocity-stress coefficient are φ = ◦ φ = ◦ change each other in 0 and 90 azimuth. investigated under two different terrestrial stress ϕ = ◦ If 0 , the TI formation is not material sym- fields. metry about the borehole axis, the stress distri- Figure 6 denotes the dispersion curves and bution around a borehole under the two kinds of velocity-stress coefficient related to S in slow TI uniaxial stress condition is completely different. x formation when S = −10MPa, S = 0. Similar Therefore, the velocity-stress coefficients related x y to the case in fast formation, the three velocity- to S and S no longer equal each other, which x y stress coefficients related to the third-order mod- can be seen by comparing Figure 5b, 5c with Fig- ules c , c and c are major. The first one can ure 4b, 4c. We can also find from the Figure 5 111 112 123 be almost neglected. In computation, the third- that the degree of crossover of flexural wave dis- order elastic modules of slow formation is cho- persion curves is quite different with the variation sen as same as that of fast formation, from equa- of ϕ.Whenϕ = 0◦, comparing Figure 5a3 from tion (28)∼(31), we know that the absolute value Figure 4a3, the two group dispersion curves are of velocity-stress coefficient in slow formation is completely identical except exchange in 0◦ and greater than that in fast formation. Under the uni- 90◦ azimuth. That is because the formation is axial terrestrial stress S = −10MPa, the inter- intrinsic symmetry about the borehole axis when x section of dispersion curves comes to more ob- ϕ = 0◦, the phase velocity alteration caused by vious gradually with the ϕ increase form 0◦ and formation intrinsic anisotropy Δv is completely ani 90◦. Moreover, the intersection point moves to same in 0◦ and 90◦ azimuth, while the phase ve- low frequency and the corresponding phase veloc- locity alteration induced by terrestrial stress Δv str ity come to high speed with ϕ increase. exchange in 0◦ and 90◦ azimuth. When ϕ = 45◦, from Figure 5b3, the difference in phase velocity For slow TI formation, under uniaxial terrestrial = = − of two flexural wave is much small at high fre- stress field Sx 0, Sy 10MPa, velocity-stress quency, which makes the crossover of dispersion coefficient related to Sy and dispersion curves of flexural wave are shown in Figure 7. With ϕ curves indistinct. Through local amplification, the ◦ ◦ crossover is still can be seen. When ϕ = 90◦, from increase from 0 to 90 , the variation of three Sy Sy Sy Figure 5c3, the difference in two flexural waves velocity-stress coefficient Q2 , Q3 and Q4 is ob- Sy is the most, and also a crossover of dispersion vious, especially for Q2 . Simultaneously, the curves is the most significant. From above dis- crossover of dispersion curves of two flexural cussion, we find that for intrinsic anisotropic for- waves becomes more distinct gradually. The in- mation, even under the unequal terrestrial stress tersection points are all around 8KHz, and the state, an obvious intersection of flexural wave dis- corresponding phase velocities have no significant difference. Acoustoelastic Effects on Borehole Flexural Waves 185

(a1) (a2) QSy 20 1 25 QSy 2 0 QSy 3 QSy

-1 4 -1 -20 Pa) Pa) 5

10 10 -40 /(10 /(10 y y S S S

i y i Q 1 Q

Q Sy -60 Q -15 2 QSy 3 S -80 Q y 4 0 (ϕ=0 ) (ϕ=00)

-35 -100 0 4 8 12 16 0481216 Frequency/(KHz) (φ=00) Frequency/(KHz) (φ=900)

30 (a3) 0 (b1) S 2400 0 Q y 900 1 QSy 2 QSy 2200 3 10 QSy

-1 4

2000 Pa) 10 /(10 v/(m/s) y S i

1800 Q -10 1600 0 (ϕ=00) (ϕ=45 ) 1400 -30 04812160 4 8 12 16 Frequency/(KHz) Frequency/(KHz) (φ=00)

2600 (b2) (b3) 20 00 900 2400 0

2200 -1

Pa) -20 10 2000 v/(m/s) /(10 y

S S i Q y

Q -40 1 QSy 1800 2 QSy -60 3 QSy 1600 4 0 (ϕ=450) (ϕ=45 ) -80 1400 04812160481216 Frequency/(KHz) (φ=900) Frequency/(KHz) Figure 5 186 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009

30 (c1) (c2) QSy 1 10 QSy 2 QSy 3 10 QSy -10 -1 -1 4 Pa) Pa) 10 10 /(10 /(10 y

y -30 S S i i QSy Q Q 1 -10 QSy 2 -50 QSy 3 QSy 4 0 (ϕ=90 ) (ϕ=900) -30 -70 04812160481216 Frequency/(KHz) (φ=00) Frequency/(KHz) (φ=900)

(c3) 2600 00 900 2400

2200

v/(m/s) 2000

1800

1600

(ϕ=900) 1400 0481216 Frequency/(KHz)

Figure 5: Flexural wave velocity-stress coefﬁcients corresponding to S and dispersion curves in fast TI ◦ ◦ formation when Sx = 0, Sy = −50MPa. (a) φ = 0; (b) φ = 45 ;(c)φ = 90

4.3 Influence of ratio of horizontal terrestrial the crossover of dispersion curves and the terres- stress on crossover of dispersion curves trial stress is more complicated, some new phenomenon appears. From Figure 8a, when η = 0, If it exist horizontal terrestrial stress in x, y axis di- the flexural wave dispersion curves intersect. In rection simultaneously, we let S = ηS ,inwhich y x low frequency, the phase velocity of flexural wave η is the ratio of horizontal terrestrial stress in y at φ = 0◦ is greater. The polarized direction of axis direction to that in x axis direction, ranging low-frequency fast flexural wave coincides with from 0 to 1. The TI formation with ϕ = 90◦ is the direction of maximum horizontal terrestrial chosen as example to study the effect of the hor- stress, namely x axis direction. However, with η izontal terrestrial stress ration on the dispersion increase to 0.4, from Figure 8b, although the flex- curves of flexural waves polarized orthogonally. ural wave split, their dispersion curves do not in- The material parameters are listed in Table 1 and tersect. The polarized direction of low-frequency Table 2. fast flexural wave is still consistent with the di- = − For fast TI formation, letting Sx 50MPa, the rection of maximum horizontal terrestrial stress. η flexural wave dispersion curves with ranging From Figure 8c, when η = 0.8, the crossover from 0 to 1 are plotted in Figure 8. Compar- of dispersion curves can be observed again al- ing with the case of isotropic formation, in in- though it is not obvious. But the polarized direc- trinsic anisotropic formation, the relation between Acoustoelastic Effects on Borehole Flexural Waves 187

100 (a1) (a2) 130 QSx 1 QSx 2 0 S 80 Q x 3 QSx -1 -1 4 Pa) Pa) 30 10 10 -100 /(10 /(10 x x S S i i QSx Q Q 1 -20 QSx -200 2 QSx 3 -70 QSx 4 0 (ϕ=00) -300 (ϕ=0 ) -120 04812160481216 Frequency/(KHz) (φ=00) Frequency/(KHz) (φ=900)

100 (a3) 0 (b1) 1700 0 900

0 1600

-1 1500 Pa) 10 -100 v/(m/s) /(10

x S i Sx Q Q 1400 1 -200 QSx 2 QSx 1300 3 QSx 0 4 (ϕ=0 ) -300 (ϕ=450) 1200 04812160481216 Frequency/(KHz) Frequency/(KHz) (φ=00)

(b2) (b3) 0 1800 0 900 100 QSx 1 1700 QSx 50 2

-1 S Q x 1600 3

Pa) Sx

10 Q 4

0 v/(m/s) /(10 1500 x S i

Q 1400 -50 1300

-100 (ϕ=450) (ϕ=450) 1200 04812160481216 Frequency/(KHz) (φ=900) Frequency/(KHz)

100 (c1) (c2) 100 0 QSx 1 50 QSx 2 -1 -1 QSx -100 3 Pa) Pa) Sx 10 10 Q 4 0 /(10 /(10 x x S S i -200 Sx i Q Q Q 1 QSx 2 -50 QSx -300 3 QSx 4 0 -100 0 (ϕ=90 ) 17 (ϕ=90 ) -400 04812160481216 Frequency/(KHz) (φ=00) Frequency/(KHz) (φ=900)

Figure 6 188 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009

(c3) 0 1800 0 900

1700

1600

v/(m/s) 1500

1400

1300

(ϕ=900) 1200 0481216 Frequency/(KHz) Figure 6: Flexural wave velocity-stress coefficients corresponding to and dispersion curves in fast TI forma- ◦ ◦ tion when Sx = −10MPa, Sy = 0. (a) φ = 0; (b) φ = 45 ;(c)φ = 90 tion of low-frequency fast flexural wave becomes ral wave always coincides with the direction of to y axis direction, which is not the direction of the maximum horizontal terrestrial stress. While maximum horizontal terrestrial stress. It is the when η = 1.0, the flexural wave merely splits and sphere stress state when η = 1.0. In this con- their dispersion curves do not coincide or inter- dition, for isotropic formation, the flexural wave sect. That is the result of combined effect of the dispersion curves do not intersect [Li, Yin and Su formation intrinsic anisotropy and the stress in- (2006)]. Whereas for anisotropic formation, from duced anisotropy, which is also completely differ- Figure 8d, the crossover of dispersion curves be- ent with the case of isotropic formation [Li, Yin comes more obvious. Furthermore, with the fre- and Su (2006)]. quency increase, the azimuth of fast flexural wave Base on the previous analysis, the conclusion can ◦ ◦ changes from 90 to 0 , which is exactly con- be made that in stressed intrinsic anisotropic for- η = trary to the case of 0. From above discus- mation, the alteration in phase velocity of flex- sion, we find that for intrinsic anisotropic forma- ural wave is controlled by two factors, one is tion, the dispersion curves of flexural waves may the intrinsic formation anisotropy and the other not intersect even under the unequal horizontal is the stress-induced anisotropy, which also can terrestrial stress field, whereas it is still possible be seen from equation (30). The two factors all to observe the crossover of flexural wave disper- can cause the flexural wave split, but only the sion curves under the equal horizontal terrestrial second factor result in the intersection of disper- stress field. Moreover, the polarized direction of sion curves. The combined effect of the two fac- the low-frequency fast flexural wave is no longer tors could strengthen or weaken the phenomenon consistent with the direction of the maximum hor- of intersection of flexural wave dispersion curves. izontal terrestrial stress all the time. The above Therefore, the dispersion curves of flexural waves phenomenon is the unique feature of the stressed may not intersect even under the unequal horizon- intrinsic anisotropic formation. tal terrestrial stress field, whereas it is still possi- For slow TI formation, setting Sx = −10MPa, the ble to observe the crossover of the flexural wave influence of η on crossover of flexural wave dis- dispersion curves under the equal horizontal ter- persion curves is shown in Figure 9. Accord- restrial stress field. The polarized direction of the ing to Figure 9, we can find that with η increase low-frequency fast flexural wave is no longer con- from 0 to 0.8, the degree of intersection of dis- sistent with the direction of the maximum hori- persion curves decrease gradually, and the po- zontal terrestrial stress all the time. larized direction of the low-frequency fast flexu- Acoustoelastic Effects on Borehole Flexural Waves 189

100 (a1) (a2) 130 QSy 1 QSy 2 S 0 80 Q y 3 QSy -1

4 -1 Pa) 30 Pa) 10 10 -100 /(10 /(10 y y S S i i Sy Q Q Q -20 1 S -200 Q y 2 QSy -70 3 QSy 4 0 0 (ϕ=0 ) -300 (ϕ=0 ) -120 04812160481216 Frequency/(KHz) (φ=00) Frequency/(KHz) (φ=900)

(a3) 00 (b1) 1700 QSy 0 1 90 100 QSy 2 S 1600 Q y 3 Sy

-1 50 Q 4 Pa)

1500 10

/(10 v/(m/s)

y 0 S i 1400 Q

-50 1300

(ϕ=00) (ϕ=450) 18 -100 1200 04812160481216 Frequency/(KHz) Frequency/(KHz) (φ=00)

Figure 7

5 The acoustoelastic model and method to in- cross-dipole sonic logging in stressed intrinsic an- version terrestrial stress isotropic formation is simply discussed. To determine terrestrial stress from flexural wave The magnitude and the direction of the terrestrial dispersion curves, we assume that the second- stress have an important significance for the oil- order elastic constant of the formation is known, field development and production. The research and the third-order elastic module as well as the of the acoustoelastic effect on borehole mode in- terrestrial stress is unknown quantities to be deter- duced by terrestrial stress is the theoretical foun- mined. The third-order elastic module is also de- dation of the terrestrial stress nondestructive ex- termined while inversion terrestrial stress. A gen- amination using cross-dipole sonic logging. The eral anisotropic solid has 56 independent third- waveform of fast and slow flexural wave in time order elastic modules. While the actual formation domain can be obtained from the 4-component is usually TI elastic solid, the number of the inde- (4-c) data in cross-dipole array acoustic logging. pendent third-order elastic modules is much less Based on it, the dispersion curves of the flexu- than that of general anisotropic media. Thus, we ral waves polarized in two main directions can be can assume the number of the independent third- also calculated. On the basis of the previous re- order elastic modules according to practical researches, the acoustoelastic inversion model and quirements. While discussing the method for ter- method for terrestrial stress inversion from bore- restrial stress inversion, without losing generality, hole flexural wave dispersion curves obtained by we assume it has 6 independent third-order elas- 190 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009

100 (b2) (b3) 00 1700 900

0 1600 -1 Pa) 10 -100 1500 v/(m/s) /(10 y S i QSy Q 1 QSy 2 1400 -200 QSy 3 QSy 4 0 0 (ϕ=45 ) (ϕ=45 ) 1300 -300 04812160481216 Frequency/(KHz) (φ=900) Frequency/(KHz)

100 (c1) (c2) QSy 100 1 QSy 2 QSy 0 3 50 QSy -1

4 -1 Pa) Pa)

10 10 -100

/(10 0 /(10 y y S S i i Sy Q

Q Q 1 QSy 2 -50 -200 QSy 3 QSy 4 (ϕ=900) 0 -100 (ϕ=90 ) -300 04812160481216 0 Frequency/(KHz) (φ=0 ) Frequency/(KHz) (φ=900)

(c3) 00 1700 900

1600

v/(m/s) 1500

1400

0 (ϕ=90 ) 1300 0481216 Frequency/(KHz) Figure 7: Flexural wave velocity-stress coefﬁcients corresponding to and dispersion curves in fast TI forma- ◦ ◦ tion when Sx = 0, Sy = −0MPa. (a) φ = 0; (b) φ = 45 ;(c)φ = 90 Acoustoelastic Effects on Borehole Flexural Waves 191

2800 0 (a) 00 2800 (b) 0 2600 900 900 2600 2400 2400 2200 2200 2000 v/(m/s) v/(m/s) 2000 1800 1800 1600 1600 1400 1400 04812160481216 Frequency/(KHz) (ϕ =900,η =0) Frequency/(KHz) (ϕ =900,η =0.4)

3000 0 (c) 00 3000 (d) 0 0 2800 900 90 2800

2600 2600

2400 2400

2200 2200 v/(m/s) v/(m/s)

2000 2000

1800 1800

1600 1600

1400 1400 04812160481216 0 Frequency/(KHz) (ϕ =900,η =0.8) Frequency/(KHz) (ϕ =90 ,η =1.0) Figure 8: The inﬂuence of ratio of horizontal terrestrial stress on crossover of ﬂexural wave dispersion curves in fast TI formation when φ = 90◦ (a) η = 0; (b) η = 0.4; (c) η = 0.8; (d) η = 1.0

tic modules of the formation here, namely c111, Based on the equation (32), the specific computa- c112, c123, c144, c155 and c456. There are 8 un- tion procedures of the method to inverse terrestrial known quantities to be determined including two stress from flexural wave dispersion curves is: horizontal stresses Sx and Sy. Integration equation (28), we have the acoustoelastic equation as fol- (1) Acquiring the waveform of the fast and slow lowing form flexural main wave from the 4-component data in cross-dipole array acoustic logging, Δ m − m Δ +Δ Δ and then, the dispersion curves of the flex- v = v vR = vani vstr = vani m m m m ural waves polarized in two main directions vR vR vR vR are calculated. Selecting 8 points in disper- CSx c CSx c CSx c + CSx + 2 111 + 3 112 + 4 123 sion curves as the computational points, get 1 c c c 66 66 66 the frequency and flexural wave phase veloc- Sx Sx Sx m C5 c144 C6 c155 C7 c456 ity v for each computational point; + + + Sx (32) c66 c66 c66 (2) Computing the flexural wave phase velocity Sy Sy Sy Sy C c111 C c112 C c123 m + C + 2 + 3 + 4 vR in unperturbed reference state for each 1 c c c 66 66 66 computational point, obtain the alteration in CSx c CSx c CSx c phase velocity caused by terrestrial stress and + 5 144 + 6 155 + 7 456 Sy Δ = Δ + c66 c66 c66 formation intrinsic anisotropy v vstr 192 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009

0 0 1800 (d) 0 1800 (b) 0 0 90 900

1700 1700

1600 1600 v/(m/s) v/(m/s) 1500 1500

1400 1400

1300 1300

04812160481216 0 Frequency/(KHz) (ϕ =90 ,η =0.6) Frequency/(KHz) (ϕ =900,η =0.4) 1900 1900 (c) 00 (d) 00 900 900 1800 1800

1700 1700

1600 1600 v/(m/s) v/(m/s)

1500 1500

1400 1400

1300 1300 0 4 8 12 16 0481216 Frequency/(KHz) (ϕ =900,η =0.8) Frequency/(KHz) (ϕ =900,η =1.0) Figure 9: The inﬂuence of ratio of horizontal terrestrial stress on crossover of ﬂexural wave dispersion curves in slow TI formation when φ = 90◦ (a) η = 0; (b) η = 0.4; (c) η = 0.8; (d) η = 1.0

Δ = m − m vani v vR ; der to get the proper direction of the maximum horizontal terrestrial stress, the forward calcula- (3) Calculating the sensitive coefficients tion is carried out assuming the polarized direc- Sx , Sy,( = ,..., ) Ci Ci i 1 7 and the alteration tion of low-frequency fast flexural wave is the in phase velocity caused by formation direction of the maximum horizontal terrestrial Δ anisotropy itself vani for each computational stress after obtaining the magnitudes of the 2 hor- point using equation (28); izontal terrestrial stresses and the 6 independent (4) According to equation (32), 8 independent third-order elastic modules. If the computed flex- equations describing the variation of flexu- ural wave dispersion curves coincide with those ral wave phase velocity with 8 undetermined of actual acoustic logging, which means the as- quantities can be established. Solving the sumed direction is correct, otherwise, the direc- equation, the 2 horizontal terrestrial stresses tion of the maximum horizontal terrestrial stress and the 6 third-order elastic modules can be is the polarized direction of low-frequency slow obtained. flexural wave.

In intrinsic anisotropic formation, because the 6Conclusion direction of the maximum horizontal terrestrial stress may be inconsistent with the polarized di- Some foundational problems in cross-dipole sonic rection of low-frequency fast flexural wave, in or- logging in stressed intrinsic anisotropic formation Acoustoelastic Effects on Borehole Flexural Waves 193 are studied in this paper. Adopting the Stroh the- Cao, Z. L.; Wang, K. X.; Ma, Z. T. (2004): ory, nonlinear theory of acoustoelasticity and per- Acoustoelastic effects on guided waves in a fluid- turbation method, the influence of the anisotropy filled pressurized borehole in a prestressed forma- of formation itself and induced by terrestrial stress tion. J. Acoust. Soc. Am., vol. 116, pp. 1406- on dispersion curves of borehole flexural waves is 1415. discussed. The results indicate that for stressed Gao, L; Liu, K.; Liu, Y. (2006): Application intrinsic anisotropic formation, the variation of of MLPG method in dynamic fracture problems, flexural wave phase velocity is dominated by the CMES: Computer Modeling in Engineering and formation intrinsic anisotropy and the terrestrial Sciences, 12(3), 181-195. stress-induced anisotropy jointly, which makes Hwu, C. B.; Ting, T. C. T. (1989): Two- the relation between the intersection of flexural dimensional problems of the anisotropic elastic wave dispersion curves and the terrestrial stress solid with an elliptic inclusion, Q. J. Mech. Appl. completely different with the case of intrinsic Math., vol. 42, pp. 553-572. isotropic formation. Even under the unequal horizontal terrestrial stress field, the dispersion curves Li,P.E.;Su,X.Y.;Yin,Y.Q.(2007): Case hole of flexural waves may not intersect, whereas it is flexural modes in anisotropic formations. CMC: still possible to observe the intersection of flexu- Computers Materials & continua, vol. 6, no.2, ral wave dispersion curves under the equal hori- pp.93-102. zontal terrestrial stress field. Moreover, the polar- Li, P. E.; Yin, Y. Q.; Su, X. Y. (2006): Acoustoe- ized direction of the low-frequency fast flexural lastic effects on mode waves in a fluid-filled pres- wave is no longer consistent with the direction of surized borehole in triaxially stressed formations. the maximum horizontal terrestrial stress all the Acta Mechanica Sinica , vol. 22, pp. 569-580. time. Based on the above researches, the terres- Liu, Q. H.; Sinha, B. K. (2000): Multipole trial stress inversion method from flexural wave acoustic waveforms in fluid-filled boreholes in bi- dispersion curves obtained by cross-dipole sonic axially stressed formations: A finite-difference logging in intrinsic anisotropic formation is sim- method, Geophysics, vol. 65, pp. 190-201. ply discussed. Liu, Q. H.; Sinha, B. K. (2003): A 3D cylindrical PML/FDTD method for elastic waves in fluid- Acknowledgement: The project supported by filled pressurized boreholes in triaxially stressed the Special Science Foundation of the Doctoral formations. Geophysics, vol. 68, pp. 1731-1743. Discipline of the Ministry of Education of China Norris, A. N.; Sinha, B. K.; Kostek, S. (1994): (No. 20050001016) Acoustoelasticity of solid/fluid composite sys- tems. Geophys. J. Int., vol. 118, pp, 439-446. References Pao, Y. H.; Sachse, W.; Fukuoka, H. (1984): Acoustoelasticity and ultrasonic measurements of Alturi, S. N.; Liu, H. T.; Han, Z. D. (2006): residual stresses. In: Mason, W. P.; Thurston, R. Meshless local Petrov-Galerkin (MLPG) mixed N. (ed) Physical Acoustics, Academic Press, New finite difference method for solid mechanics, York, pp. 61-143. CMES: Computer Modeling in Engineering and Sinha, B. K.; Kostek, S.; Norris, A. N. (1995): Sciences, 15(1), 1-16 Stoneley and flexural modes in pressurized bore- Auld,B.A.(1973): Acoustic fields and waves in holes. Journal of Geophysical Research,vol. solids. John Wiley and Sons, NewYork 100(B11), pp. 22375-22381. Cao, Z. L.; Wang, K. X.; Li, G.; Xie, R. H.; Liu, Sinha, B. K.; Kostek, S. (1996): Stress-induced J. S.; Lu, X. M. (2003): Dipole flexural waves azimuthal anisotropy in borehole flexural waves. splitting induced by borehole pressurization and Geophysics, vol. 61, pp. 1899-1907. formation stress concentration. Chinese Journal Sinha, B. K.; Liu, Q. H.; Kostek, S. (1996): of Geophysics, vol. 46, pp. 712-718. 194 Copyright c 2009 Tech Science Press CMC, vol.8, no.3, pp.173-194, 2009

Acoustic waves in pressurized boreholes: a ﬁnite difference formulation. Journal of Geophysical Research, vol. 101(B11), pp. 25173-25180. Sinha, B. K.; Norris, A. N.; Chang, S. H. (1994): Borehole ﬂexural modes in anisotropic formations. Geophysics, vol. 59, pp. 1037-1052. Sladek, J.; Sladek, V.; Atluri, S. N. (2004): Meshless local Petrov-Galerkin method in anisotropic elasticity, CMES: Computer Modeling in Engineering and Sciences, 6(5), 477-489. Ting,T.C.T.(1996): Anisotropic elasticity: theory and application. Oxford University Press, Ox- ford. Winkler, K. W.; Sinha, B. K.; Plona, T. J. (1998): Effects of borehole stress concentrations on dipole anisotropy measurements. Geophysics, vol. 63, pp. 11-17.