effects of formation stress on logging …

EFFECTS OF FORMATION STRESS ON LOGGINGMEASUREMENTS

Xiaojun Huang, Zhenya Zhu, M. Nafi Toks6z, and DanielR. Burns

Earth Resources LaboratoryDepartment of Earth, Atmospheric, and Planetary Sciences

Massachusetts Institute of TechnologyCambridge, MA 02139

ABSTRACT

We show both theoretically and experimentally how stress concentrations affect thevelocity field around a borehole surrounded by a formation with intrinsic ortohombicanisotropy. When Fx = Fy , no extra anisotropy is induced, however, isotropic stressconcentrations are developed in the neighborhood of the borehole. Extra anisotropyis induced only when Fx # Fy , and the level of induced anisotropy is affected by theintrinsic anisotropy of the formation. Experiments show that monopole acoustic wavesare more sensitive to properties in the neighborbood of the borehole than dipole waves.However, only dipole logging can determine the direction of anisotropy. A combinationof monopole and dipole logging may lead to a better investigation of intrinsic as well asinduced anisotropy of the formation.

INTRODUCTION

The elastic velocity of a small-amplitude wave is affected by formation stresses orpre-stresses presumably caused by tectonics and overburden. Traditionally, this phenomenon has been modeled using distributions of microcracks (Eshelby, 1957; Andersonet al., 1974). In recent years, laboratory experiments show that the theory of acoustoelasticity can be used to model stress-induced velocity changes in formations or rocks(Winkler and Liu, 1996; Winkler et ill., 1998). The theory of acoustoelasticity, originally developed in the context of thermoelasticity (Toupin and Bernstein, 1961; Thurstonand Brugger, 1964; Sinha, 1982; Pao and Gamer, 1985), is based on the approach ofa small dynamic field superimposed on a static deformation. The pre-stress is thoughtto change the effective elasticity of a medium. Accordingly, the anisotropic property

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Huang et al.

and elastic velocities of the medium will change. Researchers have applied the theoryof acoustoelasticity to study the influence of pre-stresses on elastic waves propagatingalong a fluid-filled borehole (Norris et al., 1994; Sinha and Kostek, 1996; Liu and Sinha, 2000). It has been proven both numerically and experimentally that a crossover inflexural dispersions is an indicator of stress-induced anisotropy. Crossover in flexuraldispersions has been observed in field data. Using a multi-frequency inversion technique,Huang and Sinha (1999) estimated formation stresses using sonic logging data. Using aphenomenological stress-velocity coupling relation, Tang et al. (1999) developed a theory to explain shear-wave splitting observed in field monopole logging data. A furtherderivation shows Tang's approach agrees with the theory of acoustoelasticity.

A common assumption is that the formation is isotropic, but a good number of underground rocks exhibit considerable intrinsic anisotropy. In this paper, we present atheoretical framework for wave propagation along a fluid-filled borehole surrounded bya pre-stressed formation with intrinsically orthohombic anisotropy. Analytical resultsshow that the existence of a borehole always causes heterogeneous velocity field distribution in the vicinity of a borehole subject to far field stresses. When far-field biaxiallystresses Fx and Fy are not balanced, Le., Fx # Fy , extra anisotropy is induced to theformation which can be detected by both dipole and monopole tools. When Fx # Fv'no extra anisotropy will be induced, but velocities in the formation are generally higher than without stresses. Theoretical results are validated by a repeatable laboratoryexperiment consisting of measuring the intrinsic anisotropy of a granite rock and theanisotropy induced by the stresses perpendicular to the borehole axis. Four nonlinearelastic constants were inverted from shear velocity measurements based on theoretical expressions for formation shear velocities as functions of the magnitude of uniaxialstresses. Our experiments also show that shear head waves received by monopoles aremore sensitive to formation stress variation than flexural waves received by dipole receivers.

EFFECTS OF FORMATION STRESSES ON VELOCITY FIELDAROUND A BOREHOLE: THEORY

Consider a borehole surrounded by an originally homogeneous solid with orthohombicanisotropy where the solid formation is subject to biaxial stresses, Fx and Fy , in thex- and y-directions in the far field (Figure 1). The compressive biaxial stresses arepresumably caused by plate tectonics. The existence of a borehole leads to stress concentration near the borehole. The stress distribution in the vicinity of a borehole is

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Formation Stress on Logging Measurements

given by Timoshenko and Goodier (1982)

a2 3a4 4a2Trr F+(I--)+F-(I+---)cos2e,

7"2 1"'4 r2

a2 3a4

Too = F+ (1 + 2") - r (1 + -4) cos 2e,l' l'

3a4 2a2

Tro F-(1- 7 + 1'2 )sin2e, (1)

Tzr = 0 1

TzO 0,

where F+ = Fx!Fy and F- = Fx;Fy , and a denotes the radius of the borehole. Figure 2shows the static stress concentration around the borehole induced by a compressional uniaxial stress Fx • In the neighbourhood of the borehole, Trr and Too developedcompressional concentration in the y direction and tensile stress concentration in the xdirection, whereas stress concentration of Tzz and Tro are 900 and 450 different, respectively. Stress concentrations happen even when Fx = Fy in the far field (Figure 3). Ifwe denote static displacements caused by the aforementioned formation stresses in 1', e,and z directions as Wr, Wo, and w., respectively, the corresponding static strains in theformation are

ErraWr

= ,ar

EooWr 1 awo-+--l' l' ae '

Ezzawz (2)az

,

Eozawo 1 awz

= az +:;: ae '

ErzaWr awz

= az + ar '

Ero1 aWr awo 1

= ---+-- - -worae ar r·

Because the borehole is very long in the axial or z-direction, the biaxial stresses areperpendicular to axial direction and do not vary in the region of interest. It is reasonableto assume that all cross sections are in the same condition and there is no displacementin the axial direction. w., Ez., Ezr , Ezo, TzT , and Tzo vanish as a result. According tothe theory of acoustoelasticity, the relationship of the static stress and strain complieswith Hooke's Law. There are nine independent linear elastic constants for orthohombicanisotropic solids: Cn, C12, C13, C22, C23, C33, C44, C55, and C66. The normal stress in theaxial direction Tzz is therefore

(3)

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Huang et al.

where

and

Static displacements are obtained by integration of equations (4) and (5),

(4)

(5)

Wr = C12 [+ a2

a4

]- P (r--)-P-(r--)cos21iA r r 3

Cn [+ a2

a4

4a2

]- P (r+--2a)+P-(r--+--4a)cos21iA r r3 r

(6)

(7)

Since it is more convenient to work in Cartesian coordinates later on, we convert staticstresses and displacements to Cartesian coordinates by rotating them by -Ii, and convertIi and pinto tan-1(yjx) and vx2 + y2, respectively, i.e.,

Txxx2 y2 2xy

(8)= x2 + y2 Trr + x2 + y2 Tee - x2 + y2 Tre ,

Tyyy2 x2 2xy

(9)= 2+ 2Trr + 2+ 2Tee + 2+ 2Tre ,X y X Y X Y

Txyxy x2 _ y2

(10)= 2+ 2(Trr - Tee) + 2+ 2Tre,x y x YT xz = Tyz = 0, (11)

x y(12)W x = W r -Wo

vx2 +y2 vx2 +y2Y x

(13)wy = W r +wo ,v x2 +y2 vx2 +y2

W Z O. (14)

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In the presence of biaxial stresses in the propagating medium, equations of motiondescribing small-amplitude waves are written in terms of the modified Piola-Kirchhoffstress tensor of first-line Taj. Introducing Piola-Kirchhoff stresses is necessary in anonlinear formulation that accounts for changes in the surface area and surface normalcaused by a finite deformation of the material. Referring to the statically deformed(intermediate) configuration, the equation of motion and constitutive relation of smallamplitude wave propagating in a pre-stressed medium are (Norris et al., 1994)

(15)

and

(16)

where p is the formation density in the intermediate state, and Caj'Y{3 represents thesecond-order elastic constant. We comply with the convention that a comma followedby an index", or t, denotes differentiation with respect to the corresponding axis or time,respectively. Both the lower case Latin and Greek letters take on the values 1, 2, and3, corresponding to the x, y and z directions, respectively. The Einstein summationconvention for repeated tensor indices is also implied. Using Ca j'Y{3AB to denote thethird-order elastic constant, the effective elastic stiffness tensor Haj'Y{3 can be written as

(17)

where Po is the hydrostatic pressure in a borehole which we have set equal to zero, asthere is no water in the borehole in later experiment, and

9"'i'!{3 = -cai'!{3w~,~ + C",j'Y{3ABEAB + Wa ,LCLi'!{3

+ Wj,MC",M'Y{3 + w'Y,pCa jP{3 + w{3,Qc",j'YQ '

and Ta'Y and EAB are the static stress and strain in the formation given by

and

(18)

(19)

(20)1

EAB = Z(WA,B + WB,A).

We next derive expressions for plan-wave speeds in terms of static stress field. Theresulting expressions for the compressional and shear wave velocities for wave propagating along the z-direction in a medium with orthohombic anisotropy subject to biaxialstress field Fx and Fy in the far-field are given by

pV)(x, y)

+

C133 - C33 C233 - C33C33 + A (C12 T yy - C22T xx) + A (C12 T xx - cllTyy )

C13 C23A(C12Ty y - C22Txx) + A(C12Txx - cllTyy ) , (21)

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Huang et al.

and

(22)

pVt,(X,y)Cl44 - C44 C244 + C44

C44 + A (c12Tyy - C22Txx) + A (C12Txx - cllTyy )

Cl3 C23+ T(c12Tyy - C22Txx) + Th2Txx - cllTyy ). (23)

Substituting properties of an isotropic dry Berea Sandstone block measured by Winkleret al. (1988), equations (21), (22) and (23) yield velocity predictions that are differwithin 5%. Velocity fields in the formation subject to compressional uniaxial stressFx = -IMPa, Fx = -3MPa, biaxial stress Fx = Fy = -IMPa, Fx = Fy = -3MPaand Fx = -4MPaFy = -IMPa are shown in Figures 4, 5, 6, 7, and 8, respectively.Conclusions drawn from the figures are: (1) The velocity field in the originally homogeneous formation becomes heterogeneous when the formation is subject to stresses evenwhen Fx = Fy in the far field; (2) when Fx # Fy, extra anisotropy is induced by stresses, and is much stronger in the vicinity of the borehole. Stress-induced anisotropy canbe detected, together with intrinsic anisotropy in the form of flexural wave splittingin dipole logging or shear head wave splitting in monopole logging; and (3) no extraanisotropy is induced when Fx = F y •

BOREHOLE MODEL AND MEASUREMENTS

Figure 1 shows a granite borehole model of 20 em x 10 em x 10 em. A hole of 1.1 em indiameter is drilled along the long axis (Z axis). To illustrate the induced heterogeneityof the effective elastic stiffness of the rock resulting from stress concentration aroundthe borehole, we selected three locations-A, B, and C-to measure the P- and Svelocities. Locations A and B are close to the borehole and are in X and Y directions,respectively. For each location, a P-wave and two shear wave velocities are measuredwithout applying any stresses. The velocities of the shear wave with polarization at theX axis and at the Y axis are about 2400 m/s and 2700 mis, respectively. They are alittle higher at locations A and B, where they are close to the borehole, indicating theexistence of residual stresses during dilling. There are about 10% intrinsic anisotropy(Figure 1). The diameter of the plane transducers used in our measurements is 1.27em. The velocities measured at locations A and B are point values due to the size ofthe transducers.

Effect of Stress on Borehole Anisotropy

We investigate the effects of a uniaxial stress perpendicular to the borehole axis on therock anisotropy by measuring the shear velocities at locations A, B, and C when a stress

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is applied in the X- or Y-direction. The properties of the rock are listed in Table 1.Figure 9 shows the shear velocities with different polarization and at the three locationswhen the uniaxial stress is in X-direction (Figure 9a) or at the Y-direction (Figure 9b).All velocities increase when the applied stress increases. When the stress is in the Xdirection (the polarization direction of the slow shear wave), the increase of slow shearvelocities is higher than that of fast shear velocities at the three measurement locations.If the stress is at the Y-direction (the polarization direction of the fast shear wave), theincrease of the slow shear velocity is higher than the fast one at location B, but is lowerthan the fast one at location A and C. This phenomenon could be attributed to rockheterogeneity, either intrinsic or caused by drilling. The stress at the Y-direction doesnot change the ratio between fast and slow shear velocities at location C very much.This shows that the stress-induced anisotropy is stronger around the borehole, which isconsistent with theoretical predications. Equations (22) and (23) show that pV§ andpV§y are linear functions of uniaxial stress magnitude Fx or F y , and slopes of pVl and

pV§ with respect to uniaxial stress magnitude are functions of third-order nonlineary

elastic constants C155, C255 and c144, C244, respectively. As a result, these nonlinear elasticconstants can be estimated from laboratory measurements (Table 2).

P Cll C12 C13 C22 C23 C33 C44 C55 C66

(kgjm3 ) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa)2660 32.96 -2.50 0.70 41.92 -0.35 50.10 19.78 16.12 16.36

Table 1: Material properties of the granite.

C155 C255 C144 C244

(GPa) (GPa) (GPa) (GPa)-9030 24874 25882 -15864

Table 2: Nonlinear elastic constants inverted from Vs~,vl,Vf. and Vfv.

Logging Measurement in the Borehole

We performed experiments to measure the monopole and dipole responses in a stressedborehole. The setup is shown schematically in Figure lOa. A pair of transducers with0.9 cm in diameter is applied to generate and receive the monopole or dipole waves inthe borehole. The transducers can be connected as monopole or dipole transducers witha switch (Zhu et al., 1994) One transducer is mounted at a lower section of the boreholeand excited by an electric pulse. The other one moves step by step along the boreholeand records the acoustic waves propagating along the borehole. Figure lOb shows atypical record of the acoustic waveforms when a stress of 5 MPa is at the X-direction.

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Huang et al.

From the waveforms we can see the P-wave, shear waves and Stoneley wave with highamplitude. We can calculate the velocity for each wave from its slope. In the recordshown in Figure lOb, the sampling rate is 0.2 f.lS. When we decrease the sampling rateto 0.1 f.ls, velocities can be calculated more accurately. There are two shear waves attheir arriving area with different velocity. When the stress is in the X- or Y-direction,we record the monopole waveforms and calculate the velocities of the two shear waves.Because the monopole transducer is symmetric with no polarization in the horizontalplane, the directions of the anisotropy induced by stress can not be determined fromthe monopole waveforms.

Figure 11 shows the relationship between stress and shear wave velocities when thestress is in the X-direction (Figure lla) or the Y-direction (Figure lIb), and variesfrom 0 to 6 MPa. The shear velocities measured in the borehole without a stress aredifferent from those measured on the rock block (Figure 1). The fast shear velocity of2680 m/s measured in the borehole is lower than the velocity (2700 m/s or 2760 m/s)measured on the rock block in Figure 1. On the other hand, the slow shear velocity of2540 m/s measured in the borehole is obviously higher than the velocity of 2450 m/sor 2490 m/s measured on the rock block. Theses results may be caused by intrinsicheterogeneity of the rock and/or residual stresses induced by drilling. Figure 11 showsthe variation of fast and slow shear waves with the stress perpendicular to the boreholeaxis. The fast shear velocity increases and the slow shear velocity decreases whenthe stress increases. This is consistent with our theoretical predictions. According tothe theory, compressional stress concentrations develop in the direction perpendicularto the uniaxial stress direction, and tensile stress concentrations develop in the stressdirection (Figure 2). Therefore, the slow shear wave is polarized in the same directionas the uniaxial stress direction, and the fast shear wave is polarized in the perpendiculardirection. Furthermore, equations (2), (22) and (23) show that the fast shear velocityincreases while the slow shear velocity decreases with respect to uniaxial stress. Inparticular, it is clearly shown in the experiment that, when the stress is at the Ydirection, the polarization direction of the fast shear wave, the stress induces strongeranisotropy on the borehole wall.

CONCLUSIONS

Far-field stresses in a plane perpendicular to a borehole axis will induce stress concentrations near the borehole, whether or not Fx = Fy • As a result, the heterogeneous shearwave velocity field is always developed in the vicinity of a borehole. When Fx = F y , noextra anisotropy is induced despite the intrinsic elastic property of the formation. Onthe other hand, extra anisotropy is induced only when Fx # Fy , and the level of inducedanisotropy is affected by the intrinsic anisotropy of the formation. Our experiment alsoshows that monopole acoustic waves are more sensitive to properties in the neighborhood of the borehole than dipole waves, however, only dipole logging can determinethe direction of anisotropy. A combination of monopole and dipole logging may lead to

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better investigation of the intrinsic as well as the induced anisotropy of the formation.

ACKNOWLEDGMENTS

This work was supported by the Borehole Acoustics and Logging/Reservoir DelineationConsortia at the Massachusetts Institute of Technology.

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Huang et al.

REFERENCES

Anderson, D.L., Minster, B., and Cole, D., 1974, The effect of oriented cracks on seismicvelocities, J. Geophys. Res., 19,4011-4105.

Eshelby, J., 1957, The determination of the elastic field of an ellipsoidal inclusion andrelated problems, Proc. Roy. Soc., Ser. A., 241, 376-396.

Huang, X. and Sinha, B.K, 1999, Formation stress determination from borehole acoustic logging: A theoretical foundation, Expanded Abstracts, Soc. Explor. Geophys.,BG2.1, 57-60.

Liu, Q.-H. and Sinha, B.K, 2000, Multiple acoustic waveforms in fluid-filled boreholes inbiaxially stressed formation: A finite-difference method, Geophysics, 100, 1392-1398.

Norris, A.N., Bikash, KS., and Kostek, S., 1994, Acoustoelasticity of solid/fluid composite systems, Geophys. J. Int., 118, 439-446.

Pao, Y.-H. and Gammer, D., 1985, Acoustoelastic waves in orthotropic media, J. Acoust.Soc. Am., 11, 806-812.

Sinha, B., 1982, Elastic waves in crystals under a bias, Ferroelectrics, 41, 61-73.Sinha, B.K and Kostek, S., 1996, Stress-induced azimuthal anisotropy in borehole flex

ural waves, Geophysics, 61, 1899-1907.Tang, X., 1999, Formation stress determination from borehole acoustic logging: A the

oretical foundation, Expanded Abstracts, Soc. Explor. Geophys., BG21.1, 57-60.Thurston, R.N. and Brugger, K, 1964, Third-order elastic constants and the velocity

of small amplitude elastic waves in homogeneously stressed media, Phys. Rev., 33,A1604-161O.

Timoshenko, S.P. and Goodier, J.N., 1982, Theory of Elasticity, McGraw Hill Book Co.,NY.

Toupin, R. and Bernstein, B., 1961, Sound waves in deformed perfectly elastic materials:Acoustoelastic effect, J. Acoust. Soc. Am., 64, 832-837.

Winkler, K W. and Liu, X., 1996, Measurements of third-order elastic constants in rocks,J. Acoust. Soc. Am., 100, 1392-1398.

Winkler, KW., Sinha, B.K, and Plona, T.J., 1998, Effects of borehole stress concentrations on dipole anisotropy measurements, Geophysics, 63, 11-17.

. Zhu, Z., Cheng, C.H., and Toksiiz, M.N., 1994, Experimental study of the flexural wavesin the fractured or cased borehole model, Expanded Abstracts, Soc. Explor. Geophys.,70-73.

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V~=4340m/s

; __..._ V~y= 2700 mls

;--"'-V:y =2760 m/s

; __"'_V:y= 2740 m/s

v:x = 2450 m/s

V~=4340m/s

v~x=2490 rnIs

V~= 4300 mls

A

BFy

Fx~

~

/'

I

II

~ II

~:F~:~

/' I/' r>-+----""L-..-. y

10 em

Fy

Eo

i::l

Figure 1: Borehole model. The formation is of orthohombic anisotropy intrinsicly andsubject to biaxial stresses in the x and y direction. The diameter of the granite boreholemodel used in the experiment is 1.1 em. Footnotes P, Sx and Sy refer to velocities ofthe P-wave and shear waves polarized in x and y directions, respectively.

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Huang et al.

T IIF I when F =-1 MPa F = 0 Te.llF [when F =-1 MPa F = 0rr x x y x x y

-8 0.2

-6 0.50

-4 0

-2 -0.2 -0.5

>- 0 >- -1-0.4

2 -1.5

4 -0.6 -2

6 -2.5-0.8

8-5 0 5 -5 0 5

x x

T .lIF [ when F =-1 MPa F =0 T IIF Iwhen F =-1 MPa F =0r x x y zz x x y

-8 -80.6

-6 -6 0.025

0.4 0.02-4 -4

-2 0.2 -2 0.015

0 00.01

>- 0 >-0.005

2 -0.2 20

4 4-0.4 -0.005

6 6-0.6 -0.01

8 8-5 0 5 -5 0 5

x x

Figure 2: Stress concentration around the borehole induced by compressional uniaxialstress, F x = -IMPa

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Tr/IFxlwhenFx=-1 MPaFy=-1 TailF IwhenF =-1 MPaF=-1x x y-8

-60.5 0.5

-4

-2 -2 0

>- 00 >- 0 -0.5

2-1

4 -0.5 4

6 6 -1.5

8 8-5 0 5 -5 0 5

x x

T,ilFxl when Fx= -1 MPa Fy=-1 TzlIFxlwhenFx=-1 MPaFy=-1

-8 -8

-6 -6 0.015

-4 0.5 -4 0.01

-2 -2 0.005

>- 0 0 >- 0 0

2 2 -0.005

4 -0.5 4 -0.01

6 6 -0.015

8 -1 8-5 0 5 -5 0 5

x x

Figure 3: Stress concentration ar01111d the borehole induced by biaxial stresses, Fx =-IMPaFy = -IMPa.

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Huang et al.

0.95

0.9

1.05

5ox

-5

VSx when Fx=-1 MPa and Fy= 0 MPa

-8 1.1

Vp when F =-1 MPa and F =0 MPax y

-8

-61.03

-4 1.02

-2 1.01

>- 0 >-

2 0.99

4 0.98

6 0.97

8 0.96-5 0 5

x

Vs when F =-1 MPa and F =0 MPay x y

-81.05

-6

-4

>-

0.95

4

6 0.9

8-5 0 5

x

Figure 4: Velocity field in the formation when subject to compressional uniaxial stress,Fx = -lMPa.

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0.9

0.7

1.2

0.6

0.8

1.1

5ox

-55ox

-5

Vp when F =-3 MPa and F =0 MPax y-8 1.1

-6

-4 1.05

-2

>- 0 >-

2 0.954

6 0.9

8

0.9

0.7

0.8

1.1

0.6

5ox

-5

4

2

8

6

-6

-2

-4

Vs when F =-3 MPa and F =0 MPay x y-8

>- 0

Figure 5: Velocity field in the formation when subject to compressional uniaxial stress,Fx = -3MPa.

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Huang et al.

5ox

-5

VSx

when F =-1 MPa and F =-1 MPax y

-81.06

-6

-41.04

-21.02

>, 0

2 0.98

4 0.96

6 0.94

8

Vp when Fx=-1 MPa and Fy=-1 MPa

-8

-6 1.02

-41.01

-2

>, 0

20.99

4

6 0.98

8-5 0 5

x

Vs

when F =-1 MPa and F =-1 MPay x y-8

1.06-6

-41.04

-21.02

>, 0

2 0.98

4 0.96

6 0.94

8-5 0 5

x

Figure 6: Velocity field in the formation when subject to compressional uniaxial stress,Fx = Fy = -IMPa.

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0.8

0.9

5ox

-5

2

4

6

8

VSx when Fx=-3 MPa and Fy=-3 MPa

~ 12

-6

-4

-2

>- 0

5ox

-5

V p when F =-3 MPa and F =-3 MPax y-8

-61.06

-41.04

-2 1.02

>- 0

2 0.98

4 0.96

6 0.94

8 0.92

0.8

0.9

1.2

1.1

5ox

-5

2

4

6

8

Vs when F =-3 MPa and F =-3 MPay x y

-8

-6

-4

-2

>- 0

Figure 7: Velocity field in the formation when subject to compressional uniaxial stress,Fx = F y = -3MPa.

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Huang et al.

0.4

1.2

0.8

0.6

5ox

2

4

VSX

when Fx=-4 MPa and Fy=-1 MPa

-8=="",-6

-4

-2

'"' 0

1.1

0.9

0.95

1.05

0.855o

x-5

4

6

8===

Vp when Fx=-4 MPa and Fy~-1 MPa

-8==",-6

-4

-2

1.2

0.4

0.8

0.6

5ox

-5

2

4

6

8

VSy when Fx=-4 MPa and Fy=-1 MPa

-8

-6

-4

-2

'"' 0

Figure 8: Velocity field in the formation when subject to compressional uniaxial stress,Fx = -4MPaFy = -IMPa.

10-18

2900

~ 2800til--E~

>. 2700:=:<>0CD 2600>~

<IlCD 2500~

en


(a)

··i····· , 1 1 l j ~; .. .. . . .,: : : :.__ -+ Vcy

.. ··· ~ ·;.···::··· ..··..J····::::::::···1..·~···~j=··· · +V;y······· ..

btress at X-axis: • •. ::: .: ) ~ : :...............] ···············r················:···v~~·······

.~~~.;~~J;:~:~~J~:~;:r:::::L::,-:J~!.

___kres~.aLy"axisL .._ _L _

*:t~i;:i~:L~~;.

2400

2900

~ 2800til--E~

>. 2700-'00

CD 2600>~

<IlGl 2500~

en

2400

o

o

1 2 3 4 5 6Uniaxial Stress Fx (MPa)

(b)

1 2 3 4 5 6Uniaxial Stress Fy (MPa)

7

7

Figure 9: Velocities of fast and slow shear waves when the formation is subject touniaxial stress in the (a) x- direction, or in the (b) y- direction. Locations (A, B, C)and directions x and y refer to those in Figure 1.

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Huang et al.

(a)

Water

Fy

, ' ,,.I.,,.,R ,o,,0,,0,

, ,

~::Fx s~':/- -+ -1--

/ , ,/

//

Fx,,~

Granite

Fy

[b]

10

Q)u[!!

8""Eu"''"2- 6OJco."OJC.

(/J 4. ~Q)

.2:Q)uQ)

2a::

o 0.02 0.04lime (m8) 10=20U8

Figure 10: Schematic plot of the experimental setup of (a) monopole logging and (b)typical full waveform record.

10-20


(a)2900

'iii' 2800-E~

~ 2700oo~ 2600

~

IIIGl 2500.s::en

I :.1 :.1................. !,

+ . ·'Fas:tShearW~ve· .

................,1....._ .,,L:::::::::::::j::::=r"h;:::::::,,··'·· .

.................;. : ; ; __..;. j ..

.. .. .. -[- - .... 1II~-:II::L~~:t~~lIIccccc.. Slow Shear Wave

2400o 1 234 5 6

Uniaxial Stress Fx (MPa)7

(b)

_.•_-_ .._ ~ j_ ; ~ ~ j _.._ .

1 234 5 6 7Uniaxial Stress Fy (MPa)

j."'--'.-..="""' ..

.. __ :.1- -- ~

2900

~ 2800Ul-E~

>- 2700-"u0Gl 2600>~

IIIGl 2500.s::en

24000

Figure 11: Velocities of fast and slow shear waves measured from monopole loggingwhen the uniaxial stress is in (a) x direction and (b) y direction.

10-21

Huang et al.

10-22

effects of formation stress on logging …

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