POLARIZATION OF FLEXURAL WAVES IN AN
ANISOTROPIC BOREHOLE MODEL
Zhenya Zhu, C. H. Cheng, and M. Nafi Toksoz
Earth Resources Laboratory
Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
Two modes of flexural waves can be generated by a dipole source in an anisotropic
borehole. Their velocities are related to those of the fast and slow shear waves in
the formation. The particle motions and the polarization diagrams of the fast and
slow flexural waves are measured in borehole models made of phenolite materials with
transverse isotropy or orthorhombic anisotropy. The experimental results show that
the particle motion of the fast flexural wave is linear and in the same direction as that
of the fast shear wave in the formation. The polarization direction of the fast flexural
wave coincides with that of the fast shear wave and is independent ofthe direction of the
dipole source. The particle motion of the slow flexural wave is nonlinear and elliptic. Its
polarization direction and variation are dependent on the anisotropic material and the
source direction. This means that the slow flexural wave is a more complicated wave
mode rather than the simple mode where the particle motion generated by a dipole
source is in the direction of the slow shear wave. The polarization characteristics of
the fast flexural wave can be applied to determine the principal axis of an anisotropic
formation by in-line and cross-line logging data.
INTRODUCTION
The rapid development of the dipole direct shear wave logging benefits from the fact that
it can be applied not only in an open borehole, but also in a cased borehole surrounded
by either a hard or a soft formation (Schmitt, 1989; Chen and Eriksen, 1991). It is a very
useful means to explore the azimuthal anisotropy of a formation due to the polarization
of dipole source and the receiver in the plane perpendicular to the borehole axis (Leslie
and Randall, 1992).
3-1
Zhu et al.
The shear velocity in a certain direction of the anisotropic material depends on the
direction of the particle vibration. In general, the shear wave splits into two waves
propagating at different speeds and with particle vibrations perpendicular to each other
(Alford, 1986; Cheadle et al., 1991). The evaluation of anisotropy by shear wave splitting has potential application in the detection of fractures, cracks and other inclusions
(Crampin, 1985; Lo et al., 1986; Winterstein, 1987). The flexural wave generated by
a dipole source in a fluid-filled borehole surrounded by an anisotropic formation also
splits into fast and slow waves whose velocities are related to those of fast and slow
shear waves (Esmersoy, 1990; Zhu et al., 1993; Esmersoy et al., 1994), respectively. The
borehole wave particle motion in anisotropic formations has been observed and analyzed with data recorded in VSP surveys (Leary et aI., 1987; Barton and Zoback, 1988;
Leveille and Seriff, 1989).
The polarization and the particle motion of the fast and slow flexural waves, as
well as the relationship between the flexural and shear waves, are investigated with
anisotropic borehole models in this paper. The experimental results show the different
polarization diagrams of the fast and slow flexural waves and provide the evidence to
explain the received flexural waveforms.
FLEXURAL WAVES IN AN ISOTROPIC BOREHOLE
The measurements are conducted with an isotropic lucite borehole model to check the
dipole transducers and the electronic system, and to compare the results with those in
an anisotropic model. Figure 1 shows a diagram of the lucite model and the measuring
system. The mono/dipole transducers are applied as source or receiver (Zhu et aI.,
1993). The electric signal exciting the source transducer is a 8-pulse. Received signals
are amplified by a wide-band amplifier (20kHz-2MHz). Then they are recorded by a
digital oscilloscope without any electronic filter. The recorded signals can be numerically
filtered during data processing.
When the polarization of the dipole source is fixed in a certain direction and the
spacing between the source and the receiver is constant, the dipole receiver is rotated
and the received waveforms are recorded (Figure 2). Because the borehole and the
dipole transducer are not exactly symmetric under experimental conditions, not only
the flexural wave, but also the high-frequency P-wave are clearly recorded in Figure
2. Their amplitudes vary with the azimuth of the dipole receiver. The P-wave and
flexural wave are completely separated in the time domain of the received waveforms.
We measure their maximum amplitude at different azimuths and show the polarization
in Figure 3, respectively. The arrow indicates the excitation direction of the dipole
source. The main lobes in the polarization diagram are not exactly symmetric. Perhaps
this is caused by two facts: the non-symmetry of the dipole source due to the limitations
of the technology for making the transducer, and the source or receiver is not located
exactly in the center of the borehole. Therefore, only part of the component propagating
with P-wave velocity is received and recorded. The polarization of the P-wave indicates
3-2
Flexural Waves in an Anisotropic Borehole Model
the direction of the dipole source.
Figure 3 shows that the polarization of the P-wave and the flexural wave in the
lucite borehole model is in good agreement with the direction of the dipole source. The
particle motions of the P-wave and flexural wave can be drawn with the recorded wave
traces in Figure 2. Figure 4 draws the particle motions of the P-wave (Figure 4a) and
the flexural wave (Figure 4b), which are picked up from two traces perpendicular to
each other. From Figure 4 it can be seen that the particle motions of both the P-wave
and the flexural wave are almost linear and their direction is in the same direction of
the dipole source.
FLEXURAL WAVES IN AN ANISOTROPIC BOREHOLE
The acoustic fields, both axial and azimuthal, generated by a dipole source are measured
in the borehole models made of anisotropic man-made materials. First we introduce
the acoustic property of the materials and then show the waveforms recorded in our
experiments.
Acoustic Anisotropy of the Modeling Materials
Two kinds of anisotropic phenolite materials are selected in our experiments-(l) phenolite XX-324 with transverse isotropy, and (2) phenolite CE-578 with orthorhombic
anisotropy. These materials are composed of laminated sheets of paper or canvas fabric.
The sizes of the phenolite blocks are 25.4 x 25.4 x 5.1cm3 and 25.4 x 25.4 x 17.8cm3 ,
respectively. The P- and S-velocities along the principal axes of the blocks are measured
with P- and S-wave transducers, respectively. Figure 5 shows the velocity distribution
along the three axes.
Because the two velocities of the shear waves propagating along the Z-axis of the
phenolite XX-324 block have the same speed, the material is transversely isotropic, but
the P-wave velocities along the X-axis and the Y-axis are different. If a borehole is
drilled along the X-axis or the Y-axis, it is an azimuthally anisotropic borehole model.
From Figure 5a, we see that the fast shear velocity here is faster than that in water and
the slow one is slower than that in water. The anisotropy of the shear wave is about
28%.
The anisotropy of phenolite CE-578 is less than that of phenolite XX-324 (Cheadle
et al., 1991). The anisotropies of phenolite CE-578 along the X axis and the Y axis are
9.3% and 3.7%, respectively. All of the shear velocities are very close to the velocity in
water.
Three holes with 1.27 em in diameter are drilled along the X-axis of the phenolite
XX-324, the X-axis, and the Y-axis of the phenolite CE-578, respectively. To eliminate
the boundary influence of the phenolite XX-324 block at the Z-axis, two similar blocks
are glued firmly to the drilled block to form a borehole model with a size of 25.4 x 25.4 x
15.3cm3 .
3-3
Zhu et al.
Axial Acoustic Field in an Anisotropic Borehole
To distinguish acoustic waves generated in the three anisotropic borehole models, array
measurements are performed with monopole and dipole transducers in a water tank.
The received waveforms are recorded, while the source transducer is fixed at a certain
location in the boreholes and the receiver is moved step-by-step along the borehole axis ..
The borehole models are shown in Figure 6. The measuring system is the same one
as that in Figure 1. Because the frequency bands of the electronic source signal (0pulse), the preamplifier, and the oscilloscope are wider than that of received acoustic
waves, the frequency band of the received waveforms mainly depends on the borehole
excitation, the frequency dispersions of the acoustic modes, and the frequency response
of the source and receiver.
Figures 7-9 show the array waveforms received with a monopole and a dipole system in the three borehole models, respectively. There are P-waves, S-waves, pseudoRayleigh waves, and Stoneley waves in the seismograms generated by the monopole.
From the seismograms generated by the dipole transducer, we can see that there are
high-frequency components propagating with P-wave velocity and both fast and slow
flexural waves distinguished by their different velocities. The fast flexural wave is of
high frequency and low amplitude; particularly its amplitude is very small in the phenolite CE-578 model. On the other hand, the slow flexural wave is of low frequency
and high amplitude. We have discussed and compared the difference between the slow
flexural wave and the Stoneley wave (Zhu et al., 1993). They are different acoustic
modes distinguished by their propagating velocity and amplitude variation in azimuths.
Radial Acoustic Field in an Anisotropic Borehole
Two kinds of flexural waves-fast and slow ones-in an azimuthally anisotropic borehole
have been confirmed by the above array measurements. To investigate the polarization
of the fast and slow flexural waves in an anisotropic borehole, we measure the azimuthal
acoustic field generated by a dipole source. While the dipole source is fixed at a certain
azimuth and the spacing between the dipole source and the receiver is also fixed, the
receiver is rotated step-by-step in the borehole and the received signals are digitally
recorded.
Figures 10-12 show the typical seismograms recorded in the phenolite XX-324 model,
the phenolite CE-578 model with an X-direction borehole, and that with a V-direction
borehole, while the polarization direction of the dipole source is located at 30 0 , respectively. In each plot, 40 traces are recorded with the dipole receiver rotating clockwise
with an azimuth increase of 9.72 0 jtrace starting from 00. The aD-direction is that of
the fast shear wave in the anisotropic models. A total of 21 plots is recorded when
the dipole source is located at 00,300,600,900,1200,1500, and 180 0 in the three models.
Only three of the plots are shown here. From these waveforms we can further investigate
the variations of the amplitudes of the fast and slow flexural waves with the azimuths.
3-4
Flexural Waves in an Anisotropic Borehole Model
POLARIZATION OF THE FLEXURAL WAVES IN AN
ANISOTROPIC BOREHOLE
The particle motion diagrams and the polarization of the acoustic waves with different
modes are studied by means of waveforms recorded in the above measurements.
Flexural Waves in the Borehole of Phenolite XX-324
Both the particle motion and the polarization of the flexural waves in a phenolite XX-324
borehole model are studied.
Particle Motion
The fast and slow flexural waves can be selected from the waveforms recorded at two
locations where the directions of the receiver are perpendicular to each other. The
particle motion diagrams can be drawn with the amplitude of the selected waves. Figures
13-16 show the particle motions of the P-wave, and the fast and slow flexural waves,
respectively. The arrows in these figures indicate the direction of the' dipole source
except in Figure 15 where the arrows indicate the direction of the fast shear wave.
Because the compressional wave propagating along the borehole axis is unique, the
amplitude of its particle motion is only related to the direction of the dipole source.
When the polarization of the dipole receiver is in the same direction as that of the
source, the maximum P-wave amplitude should be recorded. Thus the real direction of
the dipole source can be determined by the P-wave particle motion. Figure 13 shows
that the actual source direction determined from the P-wave particle motion is in good
agreement with that indicated by the arrow.
Figure 14 shows the particle motion of the fast flexural wave. Although the direction
of the dipole source indicated by the arrow varies, the direction ofthe particle motion is
rarely changes and is always in the direction of the fast shear wave. However, from these
particle motions we see that the particle motion is not exactly in the same direction as
the fast shear wave. A possible reason is that the vibration direction of the fast flexural
wave is not in the same direction as the geometric principal axis of the model. The
angle difference between them may be around 20°. If this is true, the fast flexural wave
should not be generated, or its amplitude should be at a minimum where the direction
of the dipole source is perpendicular to that of the fast shear wave.
Figure 15 shows the particle motion drawn by means of two fast flexural waveforms
which are selected from the traces where the polarizations of the dipole source and the
receiver are in in-line or are perpendicular to each other (cross-line). In Figure 15 the
arrow indicates the direction of the fast shear wave (Y-axis). From these plots we know
that the direction of the particle motion of the fast flexural wave is almost in agreement
with that of the fast shear wave (Y-axis direction). The real dipole logging tool applied
in oil fields usually has two sets of dipole receivers perpendicular to each other. If
the fast flexural waves are selected from the waveforms received by the two receivers,
3-5
Zhu et al.
similar particle motion can be obtained and the direction of the fast shear wave can be
determined by that of the particle motion.
There is some difference between the directions of the particle motion and the fast
shear wave. If the actual axis of the fast shear wave drifts off the geometric principal
axis about 20°, the resulting particle motion in the direction of the fast shear wave
would be better.
Figure 16 shows the particle motion of the slow flexural wave, which is a non-linear
or elliptic motion. The direction of the elliptic axis varies with that of the dipole source.
Polarization of the Flexural Waves
The polarization diagrams of the fast and the slow flexural waves can be drawn in
a polar coordinate plot by the maximum amplitudes of the waves selected from the
waveforms which are similar to those in Figure 10. Figure 17 shows the polarization
diagrams of the fast and slow flexural waves in the phenolite XX-324 model when the
dipole source is located at 0°,30°,60°,90°,120°,150°, and 180°, respectively. The arrow
indicates the direction of the dipole source. The polarization diagrams are very similar
to the radiation pattern of the dipole source. There is an apparent direction where the
received amplitude is maximum. The polarization of the fast flexural wave is almost
fixed in the 0°-direction (the direction of the fast shear wave or Y-axis). In contrast,
the polarization of the slow flexural wave varies with the direction of the dipole source.
When the polarization of the source rotates counterclockwise from 0° (Y-axis) to 180°,
the polarization of the slow flexural wave rotates counterclockwise from -30° to 150°
approximately. The angle between them is about 30° and does not vary with the
direction of the dipole source. This means that the fast and the slow flexural waves
have different characteristics. The fast one is closely related to the fast shear wave in
the anisotropic formation. The slow one is a more complicated wave than the fast one.
It is related not only to the slow shear wave, but also to the polarization of the dipole
source.
Flexural Waves in the Borehole of Phenolite CE-578
Figures 18-19 show the polarization of the slow flexural waves generated by a dipole
source at various azimuths in the X-axis and the Y-axis borehole models of phenolite
CE-578, respectively. We see that the polarization of the slow flexural wave varies with
the direction of the dipole source and the regular pattern of the variation is different
from that in the borehole of phenolite XX-324. When the direction of the source varies
counterclockwise from 0° to 180° in Figure 18, the polarization of the slow flexural wave
varies clockwise from 60° to 240°. The same regularity of the variation can be see in
Figure 19, but it is not very clear.
3-6
Flexural Waves in an Anisotropic Borehole Model
CONCLUSIONS AND DISCUSSION
Our experimental results show that two modes of the flexural wave can be generated
by a dipole source in an azimuthally anisotropic borehole. The flexural wave at higher
frequency with faster velocity is called a fast flexural wave. The second one is called a
slow flexural wave. Their velocities are close to, but slower than, those of the fast and
the slow shear waves in the anisotropic formation, respectively. The excitation of the
fast flexural wave is related to the formation surrounding the borehole. The amplitude
of the fast flexural wave generated in the borehole of phenolite CE-578 is much smaller
than that of phenolite XX-324.
In the isotropic lucite borehole model, the particle motion of the flexural wave is
linear and its polarization varies completely with the direction of the dipole source.
The polarization and the particle motion direction of the fast flexural wave in an
anisotropic borehole are almost in the same direction of the fast shear wave in the
formation and do not vary with the direction of the dipole source. When the direction
of the dipole source is perpendicular to that of the fast shear wave, hardly any fast
flexural wave is generated. According to this characteristic of the fast flexural wave,
the principal axis of the fast shear wave can be determined by its particle motion from
waveforms received with in-line and cross-line dipole receivers.
The locus ofthe particle motion ofthe slow flexural wave is non-linear. The direction
of the polarization varies with that of the dipole source and is related to the anisotropy
of the formation. Therefore, the slow flexural wave in an anisotropic borehole is a
complicated wave contributed by the waves whose particle motions are in the directions
of both the fast and the slow shear waves.
ACKNOWLEDGMENTS
This study is supported by the Borehole Acoustics and Logging Consortium at M.LT.,
and Department of Energy contract no. DE-FG02-86ER13636.
3-7
Zhu et aL
REFERENCES
Alford, RM., 1986, Shear data in the presence of azimuthal anisotropy, SEG 56th Ann.
Internat. Mtg., Expanded Abstracts, 476-479.
Barton, C., and M. Zoback, 1988, Determination of in-situ stress orientation from borehole guided waves, J. GeophysRes., 93, 7834-7844.
Cheadle, S.P., RJ. Broun, and D.C. Lawton, 1991, Orthorhombic anisotropy: A physical
seismic modeling study, Geophysics, 56, 1603-1613.
Chen, S.T., and A. Eriksen, 1991, Compressional and shear-wave logging in open and
cased holes using a multipole tool, Geophysics, 56, 550-557.
Crampin, S., 1985, Evaluation of anisotropy by shear wave splitting, Geophysics, 50,
142-152.
Esmersoy, C., 1990, Split shear-wave inversion for fracture evaluation, SEG 60th Ann.
Internat. Mtg., Expanded Abstracts, 1400-1403.
Esmersoy, C., K. Koster, M.Williams, A. Boyd, and M. Kane, 1994, Dipole shear
anisotropy logging, SEG 64th Ann. Internat. Mtg., Expanded Abstracts, 1139-1142.
Leary, P.C., YG. Li, and K. Aki, 1987, Observation and modeling of fault-zone fracture
seismic anisotropy, P, SV and SH traveltimes, Geophys. J. Roy. Astr. Soc., 91, 461484.
Leslie, H.D., and C.J. Randall, 1992, Multipole sources in deviated borehole penetrating
anisotropic formations: Numerical and experimental results, J. Acoust. Soc. Am., 91,
12-27.
Leveille, J.P., and A.J. Seriff, 1989, Borehole wave particle motion in anisotropic formations, J. Geophys. Res., g4, 7183-7188.
Lo, T.W., K.B. Coyner, and M.N. Toksiiz, 1986, Experimental determination of elastic
anisotropy of Berea sandstone, Chicopee shale and Chelmsford granite, Geophysics,
51,164-17l.
Schmitt, D.P., 1989, Acoustic multipole logging in transversely isotropic poroelastic
formations J. Acoust. Soc. Am., 86, 2397-242l.
Winterstein, D.R, 1987, Shear waves in exploration: A perspective, SEG 57th Ann.
Internat. Mtg., Expanded Abstracts, 638-64l.
Zhu, Z.Y., C.H. Cheng, and ·M.N. Toksiiz, 1993, Propagation of flexural waves in an
azimuthally anisotropic borehole model, SEG 63rd Ann. Internat. Mtg., Expanded
Abstracts, 68-71.
3-8
Flexural Waves in an Anisotropic Borehole Model
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Figure 1: A lucite borehole model and diagrams of the measuring system.
3-9
Zhu et al.
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3-10
Flexural Waves in an Anisotropic Borehole Model
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Figure 3: The polarizations of the P-wave (dash line) and the flexural wave (solid line)
in the luclte model. The arrow indicates the direction of the dipole source.
3-11
Zhu et al.
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3-12
Flexural Waves in an Anisotropic Borehole Model
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3-13
Zhu et al.
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of phenolite CE-578, and (c) Y-axis of phenolite CE-578.
3-14
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Flexural Waves in an Anisotropic Borehole Model
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in the borehole model of phenolite XX-324. The spacing is 0.5 em/trace.
3-15
Zhu et al.
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3-16
Flexural Waves in an Anisotropic Borehole Model
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3-17
Zhu et al.
XX-30
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Figure 10: Waveforms recorded in the borehole of phenolite XX-324 by fixing the
dipole source at 30° and rotating the dipole receiver with azimuth increasement
of 9.72° /trace starting from the axis of the fast shear wave (0° or Y-axis in Fig. 6).
3-18
Flexural Waves in an Anisotropic Borehole Model
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the dipole source at 30 0 and rotating the dipole receiver with azimuth increasement
of 9.72°/trace starting from the axis of the fast shear wave (00 or Y-axis in Fig. 6).
3-19
Zhu et al.
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Figure 12: Waveforms recorded in the borehole at Y-axis of phenolite CE-578 by fixing
the dipole source at 30 0 and rotating the dipole receiver with azimuth increasement
of 9.72°/trace starting from the axis of the fast shear wave (00 or X-axis in Fig 6).
3-20
Flexural Waves in an Anisotropic Borehole Model
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Figure 13: Particle motion of the P-wave components in the borehole of the phenolite
XX-324. The arrow indicates the direction of the dipole source which is located at
00,300,600,900,1200,1500, and 1800 , respectively.
3-21
Zhu et al.
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Figure 14: The particle motion of the fast flexural wave in the borehole of the phenolite
XX-324. The arrow indicates the direction of the dipole source which is located at
0°,30°,60°,90°,120°,150°, and 180°, respectively.
3-22
_
Flexural Waves in an Anisotropic Borehole Model
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Figure 15: The particle motion of the fast flexural waves received by the dipole receiver
in line and cross line in the borehole of phenolite XX-324. The arrow indicates the
vibration direction of the fast shear wave.
3-23
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Figure 16: The particle motion of the slow flexural wave in the borehole of the phenolite
XX-324. The arrow indicates the direction of the dipole source which is located at
0°,30°,60°,90°,120°,150°, and 180°, respectively.
3-24
Flexural Waves in an Anisotropic Borehole Model
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Figure 17: The polarizations of the fast flexural wave (dash line) and slow flexural wave
(solid line) in the borehole of phenolite XX-324. The dipole source is located at
00,300,600,900,1200,1500, and 180 0, respectively. The arrow indicates the direction
of the dipole source.
3-25
Zhu et al.
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Figure 18: The polarizations of the slow flexural wave in the borehole at X-axis of
phenolite CE-578. The dipole source is located at 0°,30°,60°,90°,120°,150°, and
180°, respectively. The arrow indicates the direction of the dipole source.
3-26
Flexural Waves in an Anisotropic Borehole Model
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Figure 19: The polarizations of the slow flexural wave in the borehole at Y-axis of
phenolite CE-578. The dipole source is located at 00,300,600,900,1200,1500, and
180 0 , respectively. The arrow indicates the direction of the dipole source.
3-27
o
Zhu et al. .
3-28