Higher Order Modes in Acoustic Logging While Drilling
Shihong Chi, Zhenya Zhu, Rama Rao, and M. Nafi Toksöz
Earth Resources Laboratory
Dept. of Earth, Atmospheric and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02142
Abstract
In multipole acoustic logging while drilling (LWD), the fundamental modes dominate recorded waveforms.
Higher order modes may also appear and complicate the processing of LWD data. In dipole LWD measurements,
the dipole tool mode is often not well separated from the flexural mode. This makes the shear wave measurement
more difficult.
We conducted theoretical and numerical analysis on dipole LWD logging responses. We found that hexapole
mode may be present in the dipole waveforms. Laboratory dipole data show the presence of hexapole mode, which
approaches asymptotically to the formation shear wave velocity. This observation supports our conclusion. We may
make use of these higher order modes for accurate determination of formation shear wave velocity.
Introduction
Wireline multipole acoustic logging tools directly measure the formation shear wave velocity. LWD acoustic
logging tools function similarly and the fundamental modes dominate recorded waveforms. However, the large tool
body occupies most of the borehole. A LWD tool excites tool modes and the borehole wave modes. The flexural
mode dominates the dipole LWD wavefields in a borehole. The screw modes dominate the quadrupole wavefields.
Tool mode removal remains a challenge for processing LWD data, particular for dipole tools. Formation shear
velocity may not be correctly determined due to the mode contamination.
Other modes also appear in LWD data. Tang et al. (2003) observed non-quadrupole (monopole and dipole)
wave modes in quadrupole data. These wave modes complicate the processing and interpretation of LWD data.
However, we may also take advantage of these modes and improve the accuracy of formation shear wave velocity
measurement.
We conduct theoretical and numerical analysis of multipole excitations and receiver responses of an LWD
tool in a fluid-filled borehole. We assume the borehole penetrates a slow formation. We discuss the implications of
higher order modes on the wavefields.
Multipole source and waveform construction
We use 2n monopole source to construct a multipole source of order n (Kurkjian and Chang, 1986). The
adjacent point sources are 180 degree out of phase (Figure 1).
A
B
Figure 1: Schematic of a dipole source constructed using monopole sources
The resulting potential in the frequency-axial wavenumber domain is given by:
∞
φ = ∑ ε ( 2 j −1) n I ( 2 j −1) n (kα r0 )K ( 2 j −1) n (kα r ) cos((2 j − 1)nθ )
(1)
j =1
where
In
and
Kn
are the modified Bessel functions of the first and second kind of order n, θ is the azimuth, r0 is the
outer radius of an LWD tool, α is borehole fluid velocity,
⎧1 if n = 0 , and
ω2 .
2
ε =
n
⎨
⎩2 if n ≠ 0
kα = k z −
α2
Radius r0 is large enough for an LWD tool that we may have to consider the second term in equation (1). We focus
on dipole tools. The dipole source can be approximated by
φ = I 1 (kα r0 )K 1 (kα r ) cos(θ ) + I 3 (kα r0 )K 3 (kα r ) cos(3θ )
(2)
The radiation conditions require that the reflected wavefield potential be finite at the borehole axis. Therefore,
it can be written as
φ sca = A1 I 1 (kα r ) cos(θ ) + A3 I 3 (kα r ) cos(3θ )
(3)
In practice, the dipole tool response is constructed by subtracting responses from two receiver arrays A and B
(Figure 1). Using equations (2) and (3), we obtain the response potential at arrays A and B:
φ A = 2 I 1 (kα r0 )K 1 (kα r ) cos(θ ) + 2 I 3 (kα r0 )K 3 (kα r ) cos(3θ )
+ 2 A1 I 1 (kα r ) cos(θ ) + 2 A3 I 3 (kα r ) cos(3θ )
(4)
and
φ B = 2 I 1 (kα r0 )K 1 (kα r ) cos(θ + π ) + 2 I 3 (kα r0 )K 3 (kα r ) cos(3θ + 3π )
+ 2 A1 I 1 (kα r ) cos(θ + π ) + 2 A3 I 3 (kα r ) cos(3θ + 3π )
(5)
Then we obtain the dipole measurement at (r0, 0, z) as follows:
2
φdipole = φ A − φB = 4 I1 (kα r0 )K1 (kα r0 ) + 4 I 3 (kα r0 )K 3 (kα r0 )
+ 4 A1 I1 (kα r0 ) + 4 A3 I 3 (kα r0 )
(6)
The I3 terms are energy coming from the hexapole mode. From theoretical dispersion analysis, Hexapole
mode measures formation shear wave velocity at cut-off frequency and is not influenced by the presence of the tool.
Therefore, the higher order mode may help determine the formation shear wave velocity.
Numerical modeling of multipole responses
We focus on dipole LWD response only. Equation (6) represents the receiver responses. We expect to see
dipole and hexapole modes in this second order approximation. We use a slow formation model described in Table
1. Figure 2 compares the dipole and hexapole responses. The maximum amplitude of the flexural mode (red line) is
about 2 ~ 3 time that of the hexapole mode (black line). However, we see that contribution from the second term is
still significant. If the dipole and hexapole sources fire simultaneously, the total response (blue line) is quite
different from that of dipole source alone (red line) (Figure 3). From this modeling result, it can be seen that the
hexapole mode makes distinguishable contribution in dipole LWD measurements. We also draw similar conclusions
for quadrupole LWD measurements, but the higher modes are much weaker than the dipole mode.
Laboratory observation
We also conducted laboratory measurements using a scaled LWD tool (Zhu et al., 2004). Figure 4 shows the
dipole waveforms in a lucite (slow formation). We use a frequency semblance method to process the waveform data
and obtain the dispersion curves for tool flexural, borehole flexural, and borehole hexapole modes (Zhu et al., 2004).
Then we compute the theoretical dispersion curves for each mode and overlay them on top of the numerical results
to identify each mode (Figure 5). Theoretical dispersion curve reasonably matches the frequency semblance
generated dispersion (Figure 5). The hexapole components have very high coherence and measure the formation
shear wave velocity. Time domain semblance does not resolve the hexapole. Frequency domain analysis of this type
shows that it is advantageous to use the measurable energy in higher order modes. This observation confirms our
theoretical prediction that higher order modes may contribute to LWD measurements. In addition, if one measures
the velocity from the hexapole mode and make corrections according to dipole dispersion, one may overestimate the
velocity.
Conclusion
Theoretical expansion of multipole sources in LWD application shows that higher order modes can change the
characteristics of the receiver responses. In dipole LWD, the hexapole energy is half or one-third of the pure dipole
energy. The true dipole response primarily represents the joint contribution from both modes. Our laboratory
measurement clearly shows the hexapole mode reaches shear wave velocity of a slow formation. We may make
better use of these higher order modes in determining formation shear wave velocity. We need to be careful not to
overestimate formation velocity using dipole dispersion when hexapole mode exists.
Acknowledgements
This work is supported by the Earth Resources Laboratory Borehole and Acoustic Logging Consortium and
the Founding Members of the Earth Resources Laboratory.
3
References
1. X. M. Tang, D. Patterson, V. Dubinsky, C.W. Harrison, A. Bolshakov, Logging-while-drilling shear and
compressional measurements in varying environments, 44th annual SPWLA Symposium, Paper 2003 II.
2. Zhenya Zhu, Rama Rao, Daniel R. Burns, M. Nafi Toksöz, Experimental studies of multipole logging with scaled
borehole models, MIT Earth Resources Laboratory, Industry Consortia, Annual Report, 2004.
3. Kurkjian, A. L., and Chang, S. K., 1986, Acoustic multipole sources in fluid-filled boreholes, Geophysics, 51,
148-163.
4
rock
2000
1000
2.0
Vp (m/s)
Vs (m/s)
Density (g/cm3)
Inner radius (m)
Outer radius (m)
Borehole radius (m)
Near offset (m)
Receiver spacing (m)
tool
5940
3220
7.84
0.024
0.092
0.108
1.37
0.15
fluid
1500
0
1.0
Table 1: Model parameters used for the numerical simulation described in Figures 2 and 3.
2.4
2.2
2
2
Offset (m)
Offset (m)
2.2
1.8
1.6
1.6
1.4
0
1.8
1.4
0.5
1
1.5
2
2.5
3
3.5
4
Time (ms)
Figure 2. Comparison of dipole and hexapole
contribution to synthetic LWD dipole
measurements. The black and red lines denote
hexapole and dipole waveforms, respectively.
1.2
0
0.5
1
1.5
2
2.5
Time (ms)
3
3.5
4
Figure 3. Pure dipole vs. mixed dipole and
hexapole synthetic LWD measurements. The
red and blue lines denote the waveforms for the
pure dipole and mixed pole modes,
respectively.
5
Figure 5. Dispersion analysis of the dipole
waveforms in Figure 4.
Figure 4. Dipole waveforms measured using a
scaled LWD tool.
6