BOREHOLE EFFECTS ON DOWNHOLE
SEISMIC MEASUREMENTS
by
Chengbin Peng, C.H. Cheng, and M.N. Toksoz
Earth Resources Laboratory
Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
An exact formulation for borehole coupling, which is valid for all frequencies and all
azimuthally symmetric and nonsymrnetric components, is given in this paper. The borehole effects on downhole measurements are studied in detail as functions of frequency,
incidence angle and polarization of an incident wave as well as geophone orientation.
We found that correction of the borehole effect for downhole measurements should be
made for frequencies above 500 Hz in a hard formation. In a soft formation, if the
incidence angle is well away from the resonance angle for a SV incidence, no borehole
correction is needed for frequencies below 300 Hz; while for frequencies above 300 Hz,
the borehole can cause severe problems on downhole measurements. The borehole can
also significantly alter the particle motion direction such that horizontal components
rotation from data itself is unreliable for experiments with frequencies above 1 kHz in
the hard formation and around 500 Hz in the soft formation.
INTRODUCTION
Increasing interest is shown towards the crosshole and VSP surveys at frequencies up
to 1 kHz or more in order to resolve the fine details of structures and lithology between
wells (Bregman et al, 1989; HarriS, 1988; Tura, 1991). However, at frequencies on
the order of 1 kHz, the existence of a fluid-filled borehole has strong influence on the
downhole measurements. Dependent on the frequency and angle of incidence, as well as
the formation properties, the measured displacement on the borehole wall or pressure
at the center of the fluid may be significantly different from that of the incident wave.
Without proper attention to this effect, imaging and inversion techniques that utilize
both the amplitude and phase information of recorded energy may be erroneous because
the borehole is not included as part of the formulation.
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Peng et al.
White (1953) presented a picture of the borehole coupling at zero frequency limit.
As frequency goes to zero, the size of the borehole becomes much smaller than the
wavelength such that the stress caused by an incident elastic plane wave is almost homogeneous at the vicinity (much larger than the borehole radius) around the borehole
if the borehole does not exist. Introduction of a borehole will locally disturb the homogeneous stress field and the change of shape of the borehole can be exactly computed
(Timoshenko and Goodier, 1951). The volume change of the borehole sets up a pressure
inside the fluid in the same way as a piston source does. Schoenberg (1986) developed
a complete theory for the interaction of a plane elastic wave with a fluid-filled borehole
and gave an explicit formuiation for the low-frequency limit. In this theory the elastic
field in the solid and the acoustic field inside the fluid satisfy the corresponding wave
equations and coupling of these two fields is accomplished through the fluid-solid interface boundary conditions. Lovell and Hornby (1990) presented a complete formuiation
for both low and high frequency for the azimuthally symmetric component, so their
formulation can oniy apply to the pressure measurement at the center of the borehole.
They found that for certain angles and for high enough frequencies (10 kHz) marked
resonances in the fluid occur for both shear and compressional incidence.
In this paper, an exact formuiation based on Schoenberg's theory is given for all
frequencies and all azimuthally symmetric and nonsymmetric components. Detailed
studies on the influence of frequency, incidence angle and polarization of incoming
elastic waves (P, SV and SH), as well as the geophone orientation, on the downhole
measurements are given, both for the hard formation (exampled by Berea Sandstone)
and the soft formation (exampled by Pierre Shale).
THEORETICAL FORMULATION
Consider an elastic wave incident on an infinite borehole drilled through a homogeneous
elastic medium with density p, compressional wave speed Ct and shear wave speed {3.
The borehole is filled with fluid which has density Pf and compressional wave speed
Ctf. The radius of the borehole is Tb. When the wave hits the borehole, a transmitted
compressional wave (denoted by displacement fj/) is generated in the fluid, and at the
same time a scattered elastic wave (denoted by US) is produced in the solid. If a three
component geophone is placed in the fluid, uf will be recorded and if it is clamped
against the formation, iJ} + US will be measured, where iJ} denotes the incident wave.
The fluid displacement uf can be expressed in terms of a potential 1/Jf which satisfies
the fluid wave equation as
\l21/Jf
w2
+ 21/J
f = 0,
Ct
(1)
f
where uf = \l1/Jf. The displacement of the scattered wave in the elastic solid can be
Borehole Coupling
represented by three elastic potentials </> (P potential),
potential) as
289
~
(SV potential) and 'I/J (SH
(2)
where
2
\72</> + w2</> = 0
a
w2
\72~ + {32~ = 0
w2
\72'I/J + {32'I/J
= o.
The general solution of these potentials is given by Schoenberg (1986) as
a(;~w) [AoJo(kfr) +2 L~=lin(Ancosne+ A~sinne)Jn(kfr)]
</>f =
</> _
1
- a~\,"J) [BoHg )(kpr)
~
+ 2 L~=lin(Bncosne + B~sin ne)H,\l) (kpr)J
i{32;JW) [CoHa1)(ksr)
'I/J =
+2L~=lin(Cncosne+C~sinne)H,\1)(ksr)]
_{3~SW) [-D~Hal)(ksr) +2 L~=lin(Dnsinne- D~cosne)H,\l)(ksr)] (3)
V
where z dependence and time dependence ei(kzz-wt) is assumed, and k f = w2 / a} - kz2 ,
2/{32_k z 2. Signs of kf, kp, k z are chosen such that
kp = vw 2/a 2 -kz2 and ks =
Im(kp, k., k f ) ~ o.
VW
Given these potentials in (3), the displacement and stress components that are
involved in the boundary conditions can be written as
u rf
a(;~w) [U!o(r)Ao + 2 L~=linU!n(r)(Ancosne + A~sin nell
a~sw) [U!o(r)Bo + 2 L~=linutn(r)(Bncosne + B~sin nell
u</>
r
-
u<r
_{3~SW) [U;o(r)Co + 2 L~=linU$n(r)(Cncosne + C~sinne)]
u'I/J
=
r
-
§~\,"') [U:{;(r)Do + 2 L~=linU%(r)(Dncos ne + D~sin nell
and
_pf = ufrr =
</>
U rr
=
p[a~:;(w) [Rfo(r)A o + 2 L~=lin Rfn(r)(Ancosne + A~sin ne)J
pa;~w) [R</>o(r)Bo + 2 L~=lin R</>n(r)(Bncosne + B~sin nell
290
Peng et al.
e
arr
=
p§~JW) [Reo(r)Co + 2 L::"=li nRen(r)(Cncosne + C~sin nell
'Ij;
=
P,6~JW) [R'Ij;o(r)Do + 2 L::"=lin R'Ij;n(r) (Dncos ne + D~sin nell
arr
and
rP
a re
-
pa~JW) [8r/>o(r)B~ + 2 L::"=linSrPn(r)( -Bnsin ne + B~cos nell
a ere
-
P,6~JW) [8eo(r)C~ + 2 L::"=lin 8 en (r)( -Cnsin ne + C~cosne)]
a'lj;
re
=
PI1~JW) [8'1j;o(r)D~ + 2 L::"=lin 8'1j;n (r)( -Dnsin ne + D~cos nell
and
a'f.
=
pa~Jw) [ZrPo(r)Bo +2L~linZrPn(r)(Bncosne+B~sinne)J
a~z
=
p§~JW)
[Zeo(r)Co
P,6~JW)
[Z'Ij;o(r)D o + 2 L::"=lin Z'Ij;n (r)(Dncosne
a'/!.
+ 2 L::"=linZen(r)(Cncosne + C~sinne)]
+ D~sin nell
where supercripts f, rP, ~ and 'Ij; denote the displacement or stress due to the corresponding potentials. The details of derivation and the coefficients are given in the appendix
A.
The coefficients An, A~, B n , B~, Cn, C~, D n , D~ are determined by applying the continuity of radial displacement and normal stress at the borehole wall
u~(rb +)
+ u~h +) = u!(rn
a~rh+) + a~rh +) = _pi h-)
and vanishing of tangential stresses
a;e h +) +a~e(rb+) = 0
a~zh +)
+ a~.(rb +) = 0
where superscript i denotes the displacement and stress of the incident wave, s denotes
those of the scattered wave and f those in the fluid.
The displacement and stress of incident plane elastic waves (P, SV or SH) are also
expressible in terms of mode summation. For P incidence
u;
- a~~w) [U:O(r)
a;:"
- pa~Jw) [Rpo(r)
P
a re
=
+ 2 L::"=linU;" (r) (cos nv cosne + sin nv sin nell
+ 2 L::"=li n Rpn(r)(cosnv cosne + sin nv sin nell
pa~Jw) [8po(r) +2L::"=lin8Pn(r)(-cosnvsinne+sinnvcosne)]
(4)
\
Borehole Coupling
291
and for SV incidence
-
-,6~\,W) [u,:inr) + 2 I:~=linU;;; (r) (cos nv cos nO + sin nv sin nO)]
PiS'~SW) [Rsvo(r) + 2 I:~=lin RSVn(r) (cos nv cos nO + sin nv sin nO)]
PiS'~SW) [8svo(r) + 2 I:~=lin8svn(r)( - cos nll sin nO + sin nll cos nO)]
PiS'~SW) [Zsvo(r) + 2 I:~=lin ZSVn(r) (cos nv cos nO + sin nv sin nO)]
(5)
and for SH incidence
SH =
Ur
_is'~~W)
ru:aH(r) + 2 I:~=linU!nH (r)( - cos nv sin nO + sin nll cos nO)]
(YSH
rr
=
PiS'~SW) [RsHO(r) + 2 I:~=lin RSHn(r) (- cosnll sin nO + sin nv cos nell
(YSH
rO
=
PiS'~SW) [8sHO(r) + 2 I:~=lin8sHn(r)(cosnv cos nO + sin nv sin nO)]
(YSH
rz
where
1I
PiS'~SW) [ZsHo(r) + 2 I:~=lin ZSHn(r) (- cosnv sin nO + sin nll cos ne)](6)
is the azimuth of the incident wave. Again details are given in the appendix B.
Applying boundary conditions will lead to the following equations for the unknown
coefficients
aU:;' cosnll
= pa RPn cosnv
[ pa ZPncosnv
pa8Pncosnv
SV cosnv
RUrn
PiS' Rsvn cos nll
PiS' ZSVn cos nll
PiS' 8svn cos nll
RUSH
]
I'
rn sin nv
PiS' RSHn sin nll
PiS' ZSHn sin nv
-piS'8sHnsinnv
(7)
P·
Urn
SIn nv
pa RPn sin nll
SV sin nv
Urn
pfJ Rsvn sin nv
PiS' ZSVn sin nll
PiS' 8svn sin nll
_RUSHcosnv
]
I'
rn
-PiS' RSHn cos nll
-PiS' ZSHn cosnv
PiS' 8SHn cos nll
(8)
I'
and
0:'
=
[ paZPnsinnv
pa 8 Pn sin nll
R
I'
where
-U;n(rb)/ p
-R~n(rb)
-Z~n(rb)
-8~nh)
-u!nh)/p
-R1f;n(rb)
-Z1f;n h)
-81f;n (rb)
and the vertical lines in (7) and (8) separate the cases that the incident wave is the
plane P (first column) or SV (second column) or SH (third column) wave.
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Peng et al.
Knowing the coefficients An, A~, B n , B~, Gn , G~, Dn , D~, the displacement both in
the fluid and solid and the pressure inside the fluid can be easily computed. Calculations
are performed with frequency up to 2 kHz and incidence angle 0 ~ 90° for two types
of formation: fast (Berea sandstone) and slow (Pierre shale). The specific parameters
are given in Table 1. Unless otherwise indicated in the text, 11=0 (azimuth of incidence
wave) and e = 0 (azimuth of geophone) are assumed.
COMPRESSIONAL PLANE WAVE INCIDENCE
The borehole distorts not only the amplitude of the incident elastic wave but also the
direction of particle motion, depending on the frequency, incidence angle as well as the
formation properties. In the worst case, the surface wave will be excited such that
the nature of particle motion is changed (from linear motion to elliptic motion). Also
because the problem is nonsymmetric in nature, measurements are also dependent on the
position of the geophone at which experiments are conducted. This section is devoted to
the case of a plane P wave incidence. We will examine the frequency and incidence angle
dependence of the displacement and pressure in the fluid as well as the displacement of
scattered energies in the solid. We will show how different the measurements (corrupted
by the borehole effect) are from the true incoming waves, both in amplitude and particle
motion, and how the measurement may vary as the position of the geophone changes
around the borehole.
Frequency and Incidence Angle Dependence
For an incident P wave with azimuth 11=0 and for a receiver at (r = rb, e = 0), Figure
2 shows the radial and vertical component of the borehole scattered wave, scaled by the
total displacement of the incident wave, I.e., Ilu:(rb)ll/lla'h)11 and Ilu~h)ll/lla'h)ll,
at the solid side of the borehole wall. For both hard and soft formations, the borehole
scattered energy is negligible compared to the incident wave for a frequency below
hundreds of Hertz (500 Hz for hard formation and 250 Hz for soft formation); while
for a frequency above 1 kHz, the scattered energy is a significant portion of geophone
output. For a frequency at 2 kHz, the radial component of the borehole scattered wave
can reach almost 60% of incident total displacement at 90° incidence for both formations
and the vertical component can reach 30% at 45° incidence for the hard formation and
over 40% at 45° incidence and over 60% at grazing incidence for the soft formation. The
radial component of the borehole scattered wave increases with frequency and incidence
angle; the vertical component increases with frequency and incidence angle up to 45°
then decreases with the increase of incidence angle.
Figure 3 shows the radial and vertical component of fluid displacement scaled by the
Borehole Coupling
293
total displacement of the incident wave, i.e., Ilu{(rb)ll/llit'h)11 and Ilu{(rb)ll/llit'h)ll,
at the fluid side of the borehole wall. For the hard formation, the radial component
of fluid displacement is almost independent of frequency except near 90 0 incidence and
increases rapidly with the increase of incidence angle; the vertical component is generally
smaller (less than 20% of the total displacement) than the radial component and shows
a peak at 45 0 incidence. For the soft formation, the radial component shows strong
frequency and incidence angle dependence in a complicated manner: it increases with
frequency when the incidence angle is less than 450 and decreases with frequency when
the incidence angle is greater than 45 0 ; the vertical component increases with frequency
and decreases with incidence angle and unlike the case of the hard formation the peak
is at grazing incident rather than at 45 0 •
Figure 4 shows the pressure (amplitude and phase) at the center of the fluid scaled
by the pressure of the incident P wave ( spherical component of the stress tensor),
I.e., p/lpo, where Po = pa;lw)w 2 (1 - i~:)Jo(kpr) (see Lovell and Hornby, 1990 with
correction), for both the hard and soft formations. For both cases the measured pressure
at the fluid is much less than that if the borehole does not exist, which is due to the
pressure release at the fluid-solid interface, especially for the hard formation. Generally,
increase of frequency and incidence angle will increase the pressure at the fluid. The
phase lag at the soft formation is less than that at the hard formation. It increases
strongly with frequency and shows less dependence on the incidence angle except at the
grazing incidence.
Borehole Reception Pattern
The borehole reception pattern is defined as the ratio in amplitude of what we measure to what we should measure if the borehole does not exist, i.e., Ilpfll/llpoll for
the pressure measurement, Ila/II/IIit'11 for the measurement of fluid displacement and
(liaS +it'ID/IIit'11 for the measurement of solid displacement (geophone output).
Figure 5 shows the borehole reception pattern for pressure at the center of the fiuid at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz. For the hard formation, the reception
pattern shows a main lobe at 90 0 incidence (and is almost independent frequency) for
frequency below 1 kHz. For the soft formation, dependence on incidence angle is very
weak compared to the previous case. At 2 kHz, the curve deflects toward the grazing
incidence rather than the normal incidence.
Figure 6 shows the borehole reception pattern for the fluid displacement at the
borehole wall. For the hard formation, the reception pattern shows little frequency
dependence but strong incidence angle dependence with a main lobe at the 900 of
incidence angle and zero at grazing incidence. For the soft formation, the incidence
angle dependence is much less significant than that in the hard formation for frequencies
294
Peng et al.
below 1 kHz and the amplitude tends to be smaller at 90° incidence than that at grazing
incidence, while for frequencies above 1 kHz, the reception pattern is more complicated
with a valley at 90° incidence and two peaks at 45° and 135° incidences.
Figure 7 shows the borehole reception pattern for the solid displacement at the
borehole wall. For the hard formation, the reception pattern is almost a unit circle with
little frequency dependence. For the soft formation, the pattern looks the same as that
for the hard formation for frequencies below 500 Hz, while for frequencies above 1 kHz,
the amplitude at 90° incidence is significantly less than that at grazing incidence and
frequency dependence is very strong.
Borehole Effect on Particle Motion
Since the displacement is a vector, the borehole distorts not only its amplitude but also
its direction, i.e., particle motion. The estimation of particle motion is very important
for data rotation of downhole 3-component measurements. It is beneficial if we can
understand how and how much the borehole can change the polarization of particle
displacement.
In this paper, inclination deviation is defined as the difference of the inclination
(angle of displacement vector with the borehole axis) of measurement displacement
from that of incident wave; azimuth deviation is defined as the difference of the azimuth
angle of measured displacement from that of incident wave. Another important concept
is the rectilinearity which is a measure of the nature of polarization (Esmersoy, 1984;
Peng and Toksoz, 1991): for a perfect linear polarization the rectilinearity is 1 and for
a perfect circular polarization it is zero.
Figure 8 shows the rectilinearity of displacement at the solid side of the borehole wall
as a function of incidence angle at frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz. For
the hard formation, the rectilinearity is very close to 1 which means the solid motion is
linear, a little bit of the surface mode is excited at 2 kHz near grazing incidence. For the
soft formation the frequency and incidence angle dependence is more obvious, although
the particle motion is dominantly linear except at 2 kHz.
Figure 9 shows the inclination deviation for the displacement at the solid side of
the borehole wall. For the hard formation, the inclination deviation is less than 1°
for frequencies below 1 kHz and 3° at 2 kHz, the deviation is peaked at 45° and 135°
incidence. For the soft formation, the inclination deviation tends to be larger than that
for the hard formation and can reach 6° at grazing incidence at 2 kHz.
Borehole Coupling
295
Effect of Geophone Orientation
Because the problem is nonsymmetric in nature, the displacement and pressure will be
dependent on the azimuth orientation of the geophone or hydrophone with respect to
the incident wave. Figure 10 shows a polar plot of the calculation for a plane P wave
incidence with incidence angle {j = 45° and azimuth angle l/ = O. What we plot in
this figure is the amplitude of solid displacement scaled by that of the incident wave
(reception pattern) as a function of geophone azimuth angle e (0°- 360°). For the
hard formation, the reception pattern is independent of the geophone orientation for a
frequency below 1 kHz, while for a frequency above 1 kHz, it has a large lobe at 180°
azimuth angle (i.e, where the geophone is facing the incident wave which picks up the
strong backscattered wave). For the soft formation the patterns look similar to those for
the hard formation for frequencies below 500 Hz, while for a frequency above 500 Hz,
the reception pattern is significantly less than unity in the forward scattering direction
(0°) and is significantly larger than unity in the backward scattering direction (180°).
Figure 11 shows the inclination deviation of the solid displacement at the borehole
wall as a function of geophone orientation angle e for a plane P wave incidence at
{j = 45°. For the hard formation, the deviation is less than 3° and for the soft formation
it can reach 4° in the forward scattering direction.
Figure 12 shows the azimuth deviation of the solid displacement at the borehole
wall. At low frequencies the deviations are almost zero, but at 2 kHz it can reach 6°
when the geophone is at 90° or 270° azimuth for the hard formation, and 15° when the
geophone is at 45° or 315° azimuth for the soft formation.
SHEAR PLANE WAVE INCIDENCE
The interaction of the shear wave (SV type) with the fluid-filled borehole is more complicated than the case of P incidence. Mode conversion and possible excitation of the
surface wave (e.g., tube wave) occur at the interface, which makes the effect of the
borehole coupling more significant.
Frequency and Incidence Angle Dependence
Figure 13 shows the radial and vertical components of the borehole scattered wave,
scaled by the total displacement of the incident wave, i.e., Ilu:(rb)II/IIUih)11 and Ilu;h)II/IIUi(rb)ll,
at the solid side of the borehole wall with the geophone orientation e = 0). The incident
wave is of SV type plane wave with azimuth l/ = O. For the hard formation, the borehole
scattered energy is negligible at low frequencies, while at high frequencies the scattered
296
Peng et al.
wave can be a significant portion of the total measurement. The radial component of
the borehole scattered wave increases with frequency and decreases with the incidence
angle, while the vertical component of the scattered wave shows a minimum at 45 0 incidence and has two peaks at grazing incidence and normal incidence. For a frequency
around 2 kHz, the radial component of the borehole scattered wave can reach almost
110% of incident total displacement at grazing incidence and the vertical component
can reach 70% at 00 incidence and 50% at normal incidence. For the soft formation,
strong resonances in the fluid occur at low frequencies and certain incidence angle where
the vertical wavenumber (w/ f3 cos 6 ) is equal to the tube wave wavenumber ( w/ CT )
at which the tube wave is excited. At that particular frequency and incidence angle,
the scattered wave can be several times larger than the incident wave. Away from the
resonance region, the frequency and incidence angle dependence is similar to that for
the hard formation.
Figure 14 shows the radial and vertical components of fluid displacement scaled by
the total displacement of incident wave, i.e., Iluth)II/IIiI'h)11 and Ilu{h)II/IIiI'(rb)ll,
at the fluid side of the borehole wall. For the hard formation, the radial component
of fluid displacement is almost independent of frequency except near 0° incidence and
increases rapidly with the decrease of incidence angle; the vertical component is smaller
in magnitude than the radial component and at high frequencies it shows a peak at
0° incidence. For the soft formation, significant fluid resonance is also observed at low
frequencies and certain incidence angle. Away from the resonance region, the radial
component shows complicated frequency dependence: it first decreases with frequency,
reaches a minimum around 1 kHz (depending on the incidence angle) then increases
again. Generally the radial component decreases with the increase of incidence angle.
The vertical component shows almost no frequency when the incidence angle is above
40 0 and increase of incidence angle will decrease its amplitude.
Figure 15 shows the pressure (amplitude and phase) at the center of the fluid scaled
by Po = pf3'(;JWlw2Jo(ksr). For the hard formation and an incidence angle above 40 0
the amplitude shows no dependence on frequency and for an incidence angle below 40°
significant frequency dependence can be observed. The amplitude increases with the
decrease of incidence angle. The phase lag for the hard formation shows strong frequency
and incidence angle dependence. For the soft formation, the pressure resonance is more
significant at low frequencies and certain angles of incidence. Away from the resonance
region, the amplitude is almost independent of frequency for normal incidence. The
phase lag shows much complicated frequency and incidence dependence.
The angle at which resonance occurs can be theoretically predicted at a given frequency. From the tube wave dispersion equation, we can derive the analytical expression
for the tube wave phase velocity at low frequency approximation. Up to the order w2 lnw
(neglecting terms of order w4 ln w or higher), the tube wave phase velocity can be written
Borehole Coupling
297
as
GT=
[1_1
pfa}
(1_ 2,82 _2Pf)(1_2,82)w2n2JnwTb]
2
1 + Pfa;
4 p,82 + Pfa}
a}
P
a
,82
2a
af
(9)
p,8
which agrees with the result of Biot (1952) and White (1983) as
W --->
o.
Figure 16a shows the frequency dependence of the tube wave phase velocity for soft
formation (Pierre shale). For a frequency below 1 kHz, GT > ,8; and for a frequency
above the cut-off, GT < ,8.
The resonance occurs at ~ cos /j = ~, so we have
which has solution only when GT >
low frequency phenomenon.
= cos- 1 L
(10)
GT
,8, which implies that the borehole resonance is a
/j
Figure 16b shows the comparison of the prediction of resonance angle under low
frequency approximation with that of exact solution. Excellent agreement can be found
at frequencies below 400 Hz.
For an plane SH wave incidence, the wave fields inside and outside the borehole
turned out to be much simpler owing to the fact that SH motion is not coupled with
P or SV motion. Figure 17 shows the transverse components of fiuid and scattered
solid displacement at the borehole wall scaled by the total displacement of incident
SH wave, Le., Ilu~(Tb)II/IIUih)11 and Ilu(jh)II/IIUih)ll. For the fast formation, the
displacement in the fiuid is strongly dependent on frequency, but almost independent
of incidence angle at low frequencies. At high frequency the angle dependence is more
significant. At 2 kHz and grazing incidence, the fiuid displacement reaches 160% of the
incident wave. For the soft formation, the displacement in the fiuid depends on both
frequency and incidence angle. Contrary to what we might expect, in this case increase
of frequency will decrease the fluid displacement. There is a peak around 600 Hz and
grazing incidence. The displacement of the scattered wave in the solid increases with
frequency and incidence angle in the fast formation. At 2 kHz and 90 0 incidence it can
reach 70% of the displacement of the incident wave. In the soft formation, the scattered
wave shows similar behaviour as that in the fast formation. At 2 kHz and 900 incidence
it can reach 100% of the displacement of the incident wave.
Borehole Reception Pattern
Figure 18 shows the borehole reception pattern for pressure, i.e., Ilpfll/llpoll, at the
center of fluid at frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for a plane SV wave
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Peng et al.
incidence. For the hard formation, the reception pattern shows two lobes at 45 0 and
135 0 incidence and is almost independent of frequency for frequency below 1 kHz. The
pressure at the center of the fluid is zero at normal incidence and very small at grazing
incidence. For the soft formation, large fluid resonance occurs at 25 0 at 100 Hz and 20°
at 500 Hz. For frequencies above 1 kHz, the fluid resonance disappears. The reception
pattern shows strong and complicated frequency and incidence angle dependence.
Figure 19 shows the borehole reception pattern for the fluid displacement, I.e.,
at the borehole wall for a plane SV incidence. For the hard formation,
the reception pattern shows little frequency dependence but strong incidence angle dependence with a main lobe at the 00 incidence and zero at 90 0 incidence. For the soft
formation, similar to the corresponding pressure reception pattern, strong resonance of
fluid motion occurs at low frequencies and small incidence angle. For frequencies above
1 kHz no resonance occurs, but the reception pattern shows very complicated behaviour.
For example, at 2 kHz, the reception pattern has four lobes. For all the cases, the fluid
motion is zero at 900 incidence.
Ilu/II/IIUiII,
Figure 20 shows the borehole reception pattern for the solid displacement, i.e., (1Ius +
1711)/111711, at the borehole wall for a plane SV incidence. For the hard formation, the
reception pattern is almost a unit circle with little frequency dependence except at 2
kHz. For the soft formation, the resonance is very weak compared to that in the fluid
at low frequencies and small incidence angles. In this case, the reception pattern shows
strong dependence on frequency and incidence angle. Generally the solid displacement
is smaller than that of the incident wave at around normal incidence and is larger around
grazing incidence, especially at high frequencies.
Figure 21 shows the borehole reception pattern for the fluid displacement at the
borehole wall for a plane SH wave incidence. For the hard formation, the reception
pattern is nearly a unit circle except at 2 kHz. For the soft formation, although no
resonance occurs, the reception pattern at low frequencies is quite different from that
at high frequencies. At low frequencies, the pattern is a unit circle; at high frequencies
it has two lobes around 20 0 and 1600 incidence and valleys at both normal and grazing
incidences.
(
Figure 22 shows the borehole reception pattern for the solid displacement at the
borehole wall for a plane SH wave incidence. For the hard formation, the reception
pattern is nearly a unit circle except at 2 kHz. For the soft formation, at frequencies
below 1 kHz the reception pattern is also nearly a unit circle, while at frequencies above
1 kHz three lobes at 200 , 90 0 and 1600 are obvious.
Borehole Coupling
299
Borehole Effect on Particle Motion
Figure 23 shows the rectilinearity of displacement at the solid side of the borehole wall
as a function of incidence angle at frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for
a plane SV incidence. For the hard formation, the rectilinearity is very close to 1 except
at 2 kHz near grazing incidence. For the soft formation the frequency and incidence
angle dependence is more obvious, although the particle motion is dominantly linear
except where the tube wave is excited at low frequencies and where some leaking modes
appear at high frequencies.
Figure 24 shows the inclination deviation for the displacement at the solid side of the
borehole wall for a plane SV incidence. For the hard formation, the inclination deviation
is less than 1° for frequencies below 1 kHz and 3° at 2 kHz. For the soft formation,
the inclination deviation can reach 20° at low frequency near the fluid resonance angle.
For frequencies above 1 kHz, significant deviation of particle motion occurs in the soft
formation.
Effect of Geophone Orientation
Figure 25 shows a polar plot of the calculation for a plane SV wave incidence with
incidence angle 0 = 45° and azimuth angle 1/ = O. Plotted here is the amplitude of solid
displacement scaled by that of the incident wave (reception pattern) as a function of
geophone azimuth angle () (0°- 360°). For the hard formation, the reception pattern is
independent of the geophone orientation for frequencies below 1 kHz, while for frequencies above 1 kHz it has a large lobe at 180° azimuth angle (i.e, the geophone is facing
the incident wave). For the soft formation the pattern looks similar to that for the hard
formation for frequencies below 100 Hz, while for frequencies between 100 - 500 Hz, the
reception pattern is less than unity in the forward scattering direction (0°) and pretty
much the same in the backward direction. For frequencies above 500 Hz, the pattern is
significantly less than unity in the forward direction and significantly larger than unity
in the backward scattering direction (180°).
Figure 26 shows the same polar plot as in Figure 25 except for SH wave incidence.
For the hard formation, the reception pattern for SH incidence is nearly a unit circle
for frequencies below 1 kHz and for a frequency above 1 kHz it also has a large lobe
at 180° azimuth angle. For the soft formation the pattern looks similar to that for the
hard formation for frequencies below 500 Hz, and for a frequency around 1 kHz, the
reception pattern has a large lobe at 180° azimuth, while for frequency around 2 kHz
the main lobes occur at 90° and 270° azimuths.
Figure 27 shows the inclination deviation of the solid displacement at the borehole
wall as a function of geophone orientation angle () for a plane SV wave incidence at
300
Peng et al.
8 = 450 • For the hard formation, the deviation can reach II ° and for the soft formation
it can reach 70° in the forward scattering direction at 2 kHz.
Figure 28 shows the azimuth deviation of the solid displacement at the borehole wall
for a plane SV incidence. At low frequencies the deviation is almost zero, but at 2 kHz
it can reach 100 when the geophone is at 1000 or 2600 azimuth for the hard formation,
and 1000 when the geophone is around 00 azimuth for the soft formation.
(
Borehole Coupling
301
DISCUSSION AND CONCLUSIONS
In this paper, we have presented an exact formulation for borehole coupling based on
Schoenberg's theory, which is valid for all frequency and all azimuthally symmetric
and nonsymmetric components. We have studied the borehole effects on downhole
seismic measurements (both in amplitude and particle motion) as functions of frequency,
incidence angle and polarization of incident wave as well as the geophone orientation.
We found
• For the hard formation and frequency below 500 Hz, the borehole scattered energy
in the solid is less than 10 % in amplitude of the incident wave and the downhole
geophone measurement is nearly not affected by the fluid-filled borehole. Although
frequency plays a less significant role in this case, the pressure measured at the
center of the fluid is strongly dependent on the direction of incidence wave, e.g.,
for a P wave incidence the pressure reception pattern has a big lobe around normal
incidence and for an SY incidence it has two lobes at 45° and 135° incidence.
• For the hard formation and frequency above 500 Hz, especially on the order of 1
kHz, the borehole scattered energy is a significant part of the total measurement of
the downhole geophone for any type of incident wave. The particle motion direction can be different from that of the incoming wave by several degrees depending
on the incidence direction and frequency. The solid displacement measured with
the geophone facing the incoming wave will be noticibly larger than that with the
geophone opposite. The pressure at the center of the fluid shows dependence on
frequency as well as incidence angle.
• For the soft formation and frequency below above 300 - 500 Hertz, the borehole
scattered energy in the solid is also negligible for a P-wave incidence, and in this
case the pressure at the center of the fluid becomes significantly less dependent
on incidence angle as it does in the hard formation. For an SY wave incidence,
significant fluid resonance occurs at certain incidence angles for frequencies below
1 kHz due to the excitation of the tube wave. At these particuiar frequencies and
incidence angles, the solid displacement measured by the downhole geophone and
pressure picked by the downhole hydrophone will be several times larger than those
of the incident wave, the particle motion from geophone measurements differs from
that of the incident wave by as large as 20°.
• For the soft formation and frequency above 1 kHz, the fluid resonance disappears for SY incidence. In this case, the solid displacement and fluid pressure are
strongly dependent on both frequency and incidence angle in a complicated manner. For a P-wave incidence, the displacement in the solid is smaller than that of
the incidence wave at normal incidence and is significantly larger at grazing incidence. Also the measured solid displacement is much larger than the incident wave
302
Peng et al.
when the geophone is facing the incident wave and is smaller when the geophone
is opposite to it.
We conclude that correction of the borehole effect for downhole measurements should
be made for frequencies above 500 Hz in the hard formation. In the soft formation, if the
incidence angle is well away from the resonance angle for an SV incidence, no borehole
correction is needed for frequencies below 300 Hz; for frequencies above 300 Hz, the
borehole can cause severe problems on downhole measurements and proper attention to
this effect should be taken. Since the borehole can significantly alter the particle motion
direction, horizontal components rotation from data itself is unreliable for experiments
with frequencies above 1 kHz and rotation should be done from downhole gyro readings
if possible.
A frequency-wavenumber filter can be designed to remove the borehole corruption
to the measured data so that the true amplitude of the incident wave can be restored.
This can be achieved by inverting the borehole coupling filter using the Wiener-Levinson
algorithm (Robinson, 1967). This will be a subject of further study.
ACKNOWLEDGMENTS
This work was sponsored by the ERL/nCUBE Geophysical Center for Parallel Precessing and the Borehole Acoustics and Logging Consortium at the Earth Resources
Laboratory, MIT. The first author holds an nCUBE fellowship. We are grateful for
helpful discussions with Dr Roger Turpening and Dr Zhenya Zhu.
Borehole Coupling
303
REFERENCES
Biot, M., 1952, Propagation of elastic waves in a cylindrical borehole containing a fluid,
J. Appl. Phys., 23,997-1009.
Bregman, N.D., R.C. Bailey, and C.H. Chapman, 1989, Crosshole seismic tomography,
Geophysics, 54, 200-215.
Esmersoy, C., 1984, Polarization analysis, rotation and velocity estimation in three
component VSP, in Vertical Seismic Profiling, 14B, 236-255.
Harris, J.M., 1988, Cross-well seismic measurements in sedimentary rocks, 58th Ann.
Intemat. Mtg., Soc. Expl. Geophys., Expanded abstracts, 147-150.
Lovell, J. R. and Hornby, B. E., 1990, Borehole coupling at sonic frequencies, Geophysics,
55, 806-814.
Peng, C. and M.N. ToksGz, 1992, Vector seismic array processing with application to
crosshole/VSP data, to be submitted to J. Seismic Exploration.
Robinson, E.A. and S. Treitel, 1980, Geophysical Signal Processing, Prentice Hall, Englewood Clifs, N. J.
Schoenberg, M., 1986, Fluid and solid motion on the neighborhood of a fluid-filled
borehole due to the passage of a low frequency elastic plane wave, Geophysics, 51,
1191-1205.
Timoshenko, S. and J.N. Goodier, 1951, Theory of Elasticity, McGraw-Hill, New York.
Tura, M.A.C, 1991, Application of diffraction tomography to fracture detection, 61st
Ann. Intemat. Mtg., Soc. Expl. Geophys., Expanded abstracts, 836-839.
White, J.E., 1953, Signals in a borehole due to plane waves in the solid, J. Acoust. Soc.
Amer., 25, 906-915.
White, J.E., 1983, Underground Sound: Application of Seismic Waves, Elsevier.
304
Peng et al.
Appendix A.
BASIC EQUATIONS IN CYLINDRICAL
COORDINATE
In terms of four potentials t/>j, t/>,
as
~
and ,p, the displacement and stress can be written
Uj
-
• ot/>j
er or
+ eO-:;:
Ur
-
-+--+-or
oroz roO
ot/>
02~
UO
=
--+----r 00
r oOoz
or
Uz -
lot/>
ot/>
oz
• lot/>j
00
• ot/>j
oz
+ ez
lo,p
1 02~
a,p
+ 02~ _ \72~
OZ2
and
a rr
lot/>
-pw t/>-2pfJ [(-:;:or
2
-
2
1 02t/>
+ r2002
+
02t/>
0 w2
1 o~
1 02~
02~
OZ2)+ oZ(fJ2~+-:;:or + r 2002 + OZ2)
1 o,p
1 02,p
2
+ r 00 - -:;: oroOI
02~
2 02t/>
2pfJ [oroz
aro
2 0 1 ot/>
t/>
0 2 1 o~
~
1 02,p
2pfJ [00(-:;: or - r 2) + ozoO(-:;: or - r 2) + r 2 002
=
0
w2
a rz
1 02,p
+ or(2fJ2~+ OZ2) + 2rozoO I
1 o,p
+ -:;: or
1 w2
02,p
+ 2"(fJ2,p + Oz2)]
and
a zz
=
azO
aOO
=
Except for the displacement and stress components given before, those components
that are not explicitly involved in the boundary conditions are as follows
\
Borehole Coupling
305
W
ar:a ) [Ubo(r)A o + 2 'L':'=linUbn (r)( -Ansin nO + A~cosnO)]
u~ u¢ 0
- a~~w) [Uta(r)Bo + 2 'L':'=linutn(r)( -Bnsin nO + B~cosnO)]
u~ -
- f1~~W) [U~o(r)Go + 2 'L':'=linU~n (r)( -Gnsin nO + G~cosnO)]
u'l/J 0
- f1~~W) [Uta(r)D~ + 2 'L':'= linU't (r)( -Dnsin nO + D~cos nO)]
ar:aW) [U!o(r)A o + 2 'L':'=linufn(r) (Ancos nO + A~sin nO)]
u fz
u¢
=
z
- a~(jV) [U1o(r)B o + 2 'L':'=linu!n (r) (Bncos nO + B~sin nO)]
u~z
_,6~~W) [U;o(r)Go + 2 'L':'=linU;n(r) (Gncos nO + G~sin nO)]
u'l/J
=
z
0
and
¢
pa:Jw) [.c¢o(r)Bo +2 'L':'=lin .c¢n(r)(BncosnO + B~sinnO)]
~
Pf1:JW) [ .c~o(r)Go + 2 'L':'=lin .c~n(r)(GncosnO + G~sin nO)]
(jzz
(j zz
'l/J
0
(j zz
¢
(j 00
-
pa:Jw) [M¢o(r)Bo + 2 'L~lin M¢n(r)(BncosnO + B~sin nO)]
~
p,6~JW) [M~o(r)Go + 2 'L':'=lin M~n(r)(GncosnO + ~sin nO)]
'l/J
p,6~JW) [ M'l/Jo(r)D o + 2 'L':'=lin M'l/Jn (r)(DncosnO + D~sin nO)]
(j 00
(j 00
pa~JW) [N¢o(r)B~ + 2 'L':'=lin N ¢n(r) (-BnsinnO + B~cosnO)]
Pf1:JW) [N~o(r)G~ + 2 'L':'=lin N~n(r)( -Gnsin nO + G~cosnO)]
p,6:JW) [ N 'l/Jo(r)D~ + 2 'L':'=lin N 'l/Jn(r)( -Dnsin nO + D~cos nO)).
Given below are lists of coefficients in the expressions for stress and displacements:
Peng et al.
306
U!n(r) = kfJ~(kfr),
¢
)')
Urn(r)
= kpHn( I(kpr,
Ug¢ (r) = I;, nH~I\kpr),
I;;
n
U!"(r) = ikzH~I)(kpr),
Rfn(r) = -w 2J n (kfr)
R¢n(r) = _[(w 2 Ren(r) =
2{32kz2)H~I)(kpr) + 2{3;;p2 (H~I)' (kpr) _ ~H~I)(kpr))]
2kzZ~2!r [H~I)(ksr) + f.;H~I)' (ksr)
-
k~;r2H~I)(ksr)]
RI.n (r) = -2{32k/n[~1H~I)(ksr)
- -,:LH~I)' (ksr)]
2
s r
lP
l'i;ST
GA._(r) = -2{32kp2n[z;-hH~I)(kpr) - -,:LH~I)' (kpr)]
rvp r2
!pH
Gen(r) =
fl-pT
2kzZ~2(t n[k}r2H~I)(ksr) - f.;H~I)' (ksr)]
Gol'n(r) = -(32ks2[H~I)(ksr)
~
Z¢n ( r ) =
Zener)
+ .;}-H~I)'
(ksr)
l'i;sr
.
2~kzkp{3
2
- k21\H~I)(ksr))
s r
H n(I)' ( kpr)
= _i(k z -t~)ks{32 H~I)' (ksr)
2
Z'lj;n(r) = ikzks{32f;;.H~I\ksr)
£¢n(r) = _[w 2 - 2{32(ka 2 - k/)]H~I)(kpr)
£en(r) = -2{32~(ki - k z 2)H~I) (ksr)
{3
£'lj;n (r) = 0
2
MA. (r) = _(0: 2 - 2(32)k0: 2H~l) (kpr) + 2{32 1;,2 H~l)' (kpr) _ 2{32 kk~ n 2H~l} (kpr)
~
~
pr 2
Men(r) = -2{32~{3 [~H~I)' (ksr) - kk~22 n2H~I)(ksr))
s
s r
Mol'n(r) = 2{32ks2n[k ~ 2H~l}(ksr) - .,LH~I)' (ksr)]
r.p
s r
/'1;ST
T;
PnH2) (kpr)
N¢n(r) = 2i{32
k 2 2k 2
N (r) - i{32 (3 - z .0- n H(I)(k r)
en
ksr
Ie(3
n s
.
2 (1)'(
N 'lj;n ()
r = ~kzks{3
Hn ksr )
Borehole Coupling
-
= J k l - k z 2 and kf = J~ kz 2 •
= k a cos 0 for plane P wave incidence and k z = kf3 cos 0 for plane shear wave inci-
where k a
= ~,
307
kf3
=
W,
kp
=
Jk a
2
-
kz
2
,
ks
kz
dence. 0 is the incidence angle.
Appendix B.
STRESS AND DISPLACEMENT OF INCIDENT
PLANE WAVE
For a plane P wave traveling in the direction ~ = (sin o cos v,sinosinv, coso), where 0
is the incidence angle with respect to the borehole axis and v is the a2imuth angle, the
displacement potential can be written as (omitting ei(kzz-wt) dependence)
- a~~w) exp[ikpr cos(O - v)]
<pp
= _a~~w)
(B-1)
[Jo(kpr) +2 L:::"=lin(cosnvcosnO + sinnvsinnO)Jn(kpr)].
In addition to the displacement and stress components given in (4), we have
u~ = _a~<,w) [USo(r) +2 L:::"=linUSn(r)(-cosnvsinnO + sinnvcosnO)]
- a~~w) [U:O(r)
+ 2 L:::"=linUI,,(r)(cos nv cos nO + sin nv sin nO)].
If the incident plane wave is of SV type, the displacement potential will be
C
(".
sv
=
iltV(w)
w3. I
s:
Slllu
- i
f3
';3(W)
['k srcos (0 -v)1
expz
(B-2)
~ [Jo(ksr) + 2 L:::"=lin(cos nv cos nO + sin nv sin nO)Jn(ksr)1
and
u~v
=
_§~~W)
[U3tY(r)
u~v
=
_§~~W)
[U:cnr) +2L:::"=linU;'';(r)(cosnvcosnO+sinnvsinnO)].
+ 2 L:::"=linUS'; (r) (-
cosnv sinnO + sin nv cos nO)]
Similarly, for a plane SH wave incidence
'l/JSH
f3V(W)
I
.
=
-----wz-~
=
_f3~~W) ~ [Jo(ksr) + 2 L:::"=lin(cosnv cosnO + sin nv sin nO)Jn(ksr)]
Slllu
exp[tksrcos(O - v)]
(B-3)
Peng et al.
308
and
U~H _ _ 13:SW ) [Ug:(r) +2 I::;"=linug:(r)(cosnvcosne + sinnvsinne)]
u~H
_
O.
The coefficients are given below
UeP (r) = Iv" In(kpr),
U!:.(r) = ik.Jn(kpr)
n
~
UeSV
(r) = -liz.-nJn(ksr),
uf: (r) = iksJn(ksr)
n
fesT
Ug:(r) = -k{3J~(ksr),
Rpn(r) = _[(w 2 - 2{32kz 2)Jn (kpr)
+ 2{321v,,2
f;; (J~(kpr)
= -2{32kp2n [z.1
In(kpr) np r2
ZPn(r) = 2i{32k.kpJ~(kpr)
8Pn(r)
UfnH(r) = 0
- -?-In(kpr))]
r.;pT
-!::J~(kpr)]
n-pT
Rsvn(r)
= 2{32k.ks Pn(ksr) + .,LJ~(ksr) - k n;r 2 In(ksr)]
8svn(r)
= -2{32ksk.n[.,LJ~(ksr) - k"""l In(ksr)]
/'l;sT
/l,sT
2
ZSVn(r) = i{32(ks
-
k. 2)J~(ksr)
s
s r2
309
Borehole Coupling
z axis
t
x axis
U
1
k
Figure 1: A fluid-filled borehole (af,Pf) in a solid formation (a,fJ,p). An elastic wave
U' incident on the borehole which causes scattered wave ils in the formation and
fluid motion ilf in the borehole. The borehole radius is Tb. The incidence angle is /j
and azimuth is II. A geophone is placed at radius T and azimuth O.
Peng et a!.
310
Scattered Solid Displacement at the Borehole wall
(scaled by the total displacement of incident P wave)
Horizontal axis =
0.0
Frequency (KHz)
Vertical axis =
Radial
(Fast Fonnation: Berea Sandstone)
0.5
1.0
1.5
2.0
Incidence Angle (degree)
Vertical
(Fast Fonnation: Berea Sandstone)
0.0
0.5
1.0
1.5
:~:O:h-'---'--t---'--t--'-'-~\'T"'"""'-\"""""""\0;'-7\ 5:'~\
0.48
70.
\
60.
50.
0.16
40.
0.08
30.
20.
10,
O,n.L---'-----------.J
Radial
(Slow Fonnation: Pierre Shale)
0.5
1.0
1.5
2.0
0.0
90.0:/-;--c-......,......l.;---';-.......J.......,...\~.......,\~~
~
80.
0.0
Vertical
(Slow Fonnation: Pierre Shale)
0.5
1.0
1.5
2.0
I~-:::~:::=:==::::::::=====t90.0
80.0
os.
~---E-70.0
0.48
60.0
0.40
50,0
0.36
40.0
0.27
0.24
0.18
20.
--.....J:.
0.09
30.0
20.0
0.02
10.0
'-
-4._ _.c:.=--'"""~......-:=~:::.-",':.MCL
Figure 2: The radial and vertical components of the borehole scattered wave scaled by
the total displacement of incident wave for hard (top) and soft (bottom) formations.
The incident wave is a plane P wave with v = O. The geophone is at (r = rb, () = 0)
0.0
311
Borehole Coupling
Fluid Displacement at the Borehole Wall
(scaled by the total displacement of incident P wave)
Horizontal axis =
Frequency (KHz)
Radial
(Fast Formation: Berea Sandstone)
1.5
2.0
0.5
1.0
0.0
90. u
~
80. 070 .v·
",.04_
1.01 ____________
50 .CT
0.80_
0.70
'"
30.V
O.09-~
60.0
(
50.0
0.1.
40.0
30.0
0.09~
0.30
10.u.
70.0
___ 0.'2~ ~
0.50
0.40
20 .CT
0.20
0.10
o.IF
0.0
90. v
Radial
(Slow Formation: Pierre Shale)
0.5
1.0
1.5
2.0
~})
50. li - - - : : : : 0 . 7 0
40..li
.J.
!:l:·o::.6
60. li
0.0
Vertical
(Slow Formation: Pierre Shale)
0.5
1.0
1.5
10.0
0.0
90.0
0.08
80.0
0.20
0.40
~
20.0
2.0
70.0
60.0
0.60
50.0
_o.80-
40.0
.--0.60
30. CT
30.0
0.50--------DAD
20.li
.V
-------------
,-
~
0.60
o
1----0.03---------F
,
0.06
- 80.0
015
40 .li
10. u-:
Vertical
(Fast Formation: Berea Sandstone)
0.0
1.5
2.0
0.5
. 1.0
______0.12 / /
........0.94_
0.89_
70. 0-
Incidence Angle (degree)
I-'-~-'--'-~~.....L.~~--l..,.,~~+90.0
60.li
80. 0-
Vertical axis =
0.20
/20
O"~
r-O.10--........~
20.0
10.0
1 "oj
0.0
------
e
Figure 3: The radial and vertical components of fluid displacement at (r = rb' = 0)
scaled by the total displacement of the incident wave for hard (top) and soft (bottom)
formations. The incident wave is a plane P wave with v = o.
Peng et al.
312
Pressure at the Center of Fluid
(scaled by the pressure of incident P wave)
Horizontal axis
=
Frequency (KHz)
Amplitude
(Fast Formation: Berea Sandstone)
0.5
1.0
1.5
0.0
90.O+-~~--'--'~""""T-'---'-~~';-'-~-'rI
Vertical axis =
Incidence Angle (degree)
Phase
(Fast Formation: Berea Sandstone)
0.0
0.5
1.0
1.5
2.0
I-'--..,......-'-I-~......-'-i-~,........--';-~..,......+90.0
80.
80.0
70.
70.0
60.0
I
40.00 46.00 56( 64C
.1
O.n-L--------------'
Amplitude
(Slow Formation: Pierre Shale)
0.5
1.0
1.5
0.0
2.0
90. rrJ--';-~~'-'-~..,.......L..._'_t~_'_7'~-t-'~Ti
80.
70.
:
\
\
0.0
0.0
Phase
(Slow Formation: Pierre Shale)
0.5
1.0
1.5
2\.00'6.\66
".0:
\
\
""
10.
O ~ '_--'---'-------<:::.........<:::......="""~:==:~::2:~~­
."
~~:~
50.0
40.0
4.00
../ ____
90.0
60.0
~
20.
2.0
I
0.' .. O'\\::
40.
30.
40.0
,'~~'~:
~76 0.\
\
50.0
'-'\
80\ \
,,~
'--_-"-_---"_--''----'----'---'---'-.!L
Figure 4: Pressure at the center of the fluid scaled by Po (see text), the pressure of the
incident P wave, for hard (top) and soft (bottom) formations.
~~:~
10.0
0.0
313
Borehole Coupling
Pressure at center of fluid
0.16
-100 Hz
- - -500 Hz
_. -1000 Hz
0.12
·····2000 Hz
Hard Formation (Berea Sandstone)
0.08
0.04
o
0.04
0.08
0.12
0.16
Pressure at center of fluid
Soft Formation (Pierre Shale)
0.8
0.6
-100 Hz
- - -500 Hz
_. -1000 Hz
..... 2000 Hz
0.4
0.2
o
0.2
0.4
0.6
0.8
Figure 5: Borehole reception pattern for pressure at the center of the fluid at frequencies
100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations.
The incident wave is a plane P wave with v = o.
Peng et al.
314
-100 Hz
- - -500 Hz
_. -1000 Hz
..... 2000 Hz
Fluid side of borehole wall
Hard Formation (Berea Sandstone)
1.2
0.9
0.6
0.3
o
0.3
0.6
0.9
1.2
Fluid side of borehole wall
-100 Hz
Soft Formation (Pierre Shale)
- - -500 Hz
_. -1000 Hz
1.6
1.28
0.96
0.64
0.32
·····2000 Hz
o
0.32
0.64
0.96
1.28
1.6
•
.'
•
••
"",..-
.-'
Figure 6: Borehole reception pattern for fluid displacement at (r = r;, e = 0) at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane P wave with v = O.
Borehole Coupling
315
Solid side of borehole wall
- 1 0 0 Hz
Hard Fonnation (Berea Sandstone)
- - -500 Hz
_. -1000 Hz
1.08
0.864
0.648
0.432
0.216
·····2000 Hz
o
0.216
0.432
0.648
0.864
1.08
Solid side of borehole wall
Soft Formation (Pierre Shale)
1.4
1.225
1.05
0.875
0.7
0.525
0.35
0.175
- 1 0 0 Hz
- - -500 Hz
_. -1000 Hz
·····2000 Hz
o
0.175
0.35
0.525
0.7
0.875
1.05
1.225
1.4
Figure 7: Borehole reception pattern for solid displacement at frequencies 100 Hz, 500
Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. The incident
wave is a plane P wave with 1I = O. The geophone is at (r = rt,rJ = 0).
Peng et al.
316
-100 Hz
Solid side of borehole wall
Hard Fonnation (Berea Sandstone)
- - -500 Hz
- - -1000 Hz
1.002
=_,
-=,-.-,::oo~,....----""=":"-~...',",,'....... 2000 Hz
.>.
t""Oi
~
';:
1ll
a
0.998
..
-II
·
I.
.,
..
\
-
' .
~
.l
·
0.992
0.99
.
/
0.996
tl 0.994
~
-..
~
..
..
··· .
II
..
','
',O. 988 .L..L.l.......I..I.J............u........L1.I~ .......:.u........u..........uJ
o
30
60
90
120
150
Incidence angle (degree)
180
Solid side of borehole wall
-100 Hz
Soft Formation (Pierre Shale)
- - -500 Hz
- - -1000 Hz
..... 2000 Hz
1.02
0.98
-.
.
·
.·
...
....:
.."
>,
.;:
0.96
a"
0.94
tl 0.92
""
--/ . ..· ...----.. .'..
0.9
0.88
•
,
'
II
,
... ..
- -•
'
0.86 ~"""":':'''''''.u.JLL.&''''''L.1.I~""",
..............u..........uJ
o
30
60
90
120
150
180
Incidence angle (degree)
. Figure 8: Rectilinearity for the solid displacement at frequencies 100 Hz, 500 Hz, 1000
Hz and 2000 Hz for hard (top) and soft (bottom) formations. The incident wave is
a plane P wave with v = O. The geophone is at (r = ri,", e = 0).
Borehole Coupling
Solid side of borehole wall
-100 Hz
- --500 Hz
---1000 Hz
.•••• 2000 Hz
Hard Fonnation (Berea Sandstone)
4
3
0;-
...'"on
:::,
'"
,
2
..'.
.1 ".- ........ I.
c
:
0
:c
.;;:
'"
a
'C
- 1
'"c
. .'. .
.
It
•
..
II
-------~.
"I
...,
II
0
'"
. . . . . . . . ....,.
..,
<:i - 3
c
- 4
a
30
60
90
.
... - - - - - - / :
". ..........
:c - 2
.5
-
317
,
'
..
120
.'.
..
.-
..
.
.'
150
180
Incidence angle (degree)
- 1 0 0 Hz
- - -500 Hz
_. -1000 Hz
..... 2000 Hz
Solid side of borehole wall
Soft Formation (pierre Shale)
1
a
0;-
...'"on
:::,
'"
6
II" .. ....
,
c
,-...- -=---....
.,it. - __
2
.51
-;
.;;:
'"c
.
.
•
..
"....
..
..
--~
0'
- 2
'C
.51
-;
c -6
:sc
-
-1a
a
30
60
90
120
150
180
Incidence angle (degree)
Figure 9: Inclination deviation from the incident wave for the solid displacement at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane P wave with v = O. The geophone is at
(r =
= 0).
rt, e
Peng et aJ.
318
Solid side of borehole wall
-100 Hz
Hard Fonnarion (Berea Sandstone)
- - -500 Hz
- - -1000 Hz
••••• 2000 Hz
1.4
1.12
0.84
0.56
0.28
o
0.28
0.56
0.84
.....
.
.•
-1-oO!-..
••• -:';';;0'
••
··
·
..
.".
••
••
••
••
•• -. ;::'\0_......- -
1.12
1.4
Solid side of borehole wall
-100 Hz
Soft Formation (Pierre Shale)
- - -500 Hz
- - -1000 Hz
..... 2000 Hz
1.8
1.5
1.2
0.9
0.6
0.3
o
0.3
0.6
0.9
1.2
1.5
1.8
Figure 10: Effect of geophone orientation on the borehole reception pattern at frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane P wave with {j = 45 and lJ = O. The
geophone is at =
e varies from 00 - 360 0 •
r rt-
(
319
Borehole Coupling
-100 Hz
- - -500 Hz
- - -1000 Hz
••••• 2000 Hz
Solid side of borehole wall
Hard Fonnation (Berea Sandstone)
4
,
.,,,
--
.
,
.-
,
---:
----~
.'
-- --,
,"
,,
t.:--._--_.--:
---------,
.. "..
. .. .
,
"
- 2
""
_1-J......--L-:-':---1--L-~.L....J
o
90
Solid side of borehole wall
5
~c
3
~
2
'"
.~
180
360
-'.
\
-100 Hz
- - -500 Hz
- - -1000 Hz
•.... 2000 Hz
,
•
~
-, .,'
\ /
,..
"...
...
---,:1:.....
-c
270
Geophone orientation (degree)
Soft Fonnation (Pierre Shale)
¥4
•
............--L-':---1--L-.&...J
. ......
,
,,
,
."
,
'. .
..~
....
\'
,
i/'-'
:.
~ 0 E-"'-;';-;;;;';-::..o'rt!'..:;,,;........----~.:.. -=~:;.,.IiOO'=-:.::-;.;-=-
.=~'" - 1
-
- 2 "-'......--"-:-'::.....-"""-:~-~"":-:!-:-'
.....--"~
o
90
180
270
360
Geophone orientation (degree)
Figure 11: Effect of geophone orientation on the inclination deviation at frequencies 100
Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. The
incident wave is a plane P wave with ij = 45 and v = O. The geophone is at r = rte varies from 00 - 3600 .
320
Peng et al.
Solid side of borehole wall
Hard Fonnation (Berea Sandstone)
8
..
6
""...
:::."""
c
.51
.~
2
I
"
:;
- 4
'0
.'
...
"
- - -500 Hz
- . -1000 Hz
..... 2000 Hz
..
", "
....... _
... _ _ .......
........., _----- ...
,
. . ...
. ..
0
0
- 2
.c
... .:
.,
4
-100 Hz
ONE
< -6
- 8
'
0
90
180
.. '
270
360
Geophone orientation (degree)
-100 Hz
---500 Hz
Solid side of borehole wall
Soft Formation (Pierre Shale)
15
"
"...
:::."""
c
.51
.-;;;:
"
'0
-"
.c
E
·s
<
---1000 Hz
••••• 2000 Hz
.,. ...
10
".;..- . . . .
.. • ..
.
."
.------"- .. .
.
..
.
.,
.. ... " ----.~
/!
,. . ..
.. ..
'j,,/
.. ..!
..
l
5
:
/
"0
,.
"',
0
,
,
,
-5
,,
- 10
-15
,
0
90
180
270
,
,
360
Geophone orientation (degree)
Figure 12: Effect of geophone orientation on the azimuth deviation at frequencies 100
Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. The
incident wave is a plane P wave with (j = 45 and v = O. The geophone is at r = rt.
e varies from 00 - 3600 •
321
Borehole Coupling
Scattered Solid Displacement at the Borehole Wall
(scaled by the total displacement of incident SV wave)
Horizontal axis = Frequency (KHz)
Radial
(Fast Fonnation: Berea Sandstone)
0.5
1.0
1.5
2.0
0.0
90'or-~~~========j
80.
Vertical axis = Incidence Angle (degree)
VerticaI
(Fast Fonnation: Berea Sandstone)
0.0
0.5
1.0
1.5
2.0
~~~,-!-,-~""""""'~,.....J,-.)\---""30236
'-+-1-.. : :
70.
0.18
60.
60.0
0.09
0.03
50.
50.0
40.0
40.
~~:
jo.30 0.40
/ /0.50
10.
~
0.71
o.f)"-----'--'--""-''---''''--''''-'''''-''='''''''=
Radial
(Slow Fonnation: Pierre Shale)
0.5
1.0
1.5
2.0
90.or-~......J--.~~==~~=j
0.0
80.
70.
(
Ll....L.._.l.-_...L-L.....L......L...L...:::c.::::....,,;=:r.
Vertical
(Slow Fonnation: Pierre Shale)
0.5
1.0
1.5
0.0
60.
10.0
0.0
2.0
~\
\
\~\~}.oo 90.0
~50
\ \
0.06
---1:.30.0
20.0
0,09
0.06
0.74
------..J-
~~:~
60.0
0.18
50.
50.0
40.
40.0
'----I:
30.
30.0
20.
20.0
0.18
0.06
10.
o.
10.0
1.60
0.0
Figure 13: The radial and vertical components of the borehole scattered wave scaled
by the total displacement of the incident wave for hard (top) and soft (bottom)
formations. The incident wave is a plane SV wave with v = O. The geophone is at
(r = rb, (} = 0)
322
Peng et al.
Fluid Displacement at the Borehole Wall
(scaled by the total displacement of incident SV wave)
Horizontal axis
=
Frequency (KHz)
Vertical axis
Radial
(Fast Formation: Berea Sandstone)
0.5
1.0
1.5
2.0
0.0
90.0t=:==~~0~.0~32=====~===1
=
Incidence Angle (degree)
Vertical
(Fast Formation: Berea Sandstone)
0.0
0.5
1.0
1.5
2.0
I--'-~""""~~.....J-~~--'-~~+
90.0
0.09
80.(}j·-----1o.18---------j
1 - - - - - -10.30- - - - - - - - - 1
70.
n
j
O.4.--------j
1
50.
40.01
0.80- - - - - - - j
60.
30.
1
20.
1 - - - - - 0 . 0 2 - - - - - - - - - [ 80.0
I-----O.OS---------{
1-----0.08----------j: 70.0
1 - - - - -0 . 13---------+ 60.0
1-----===~0.17-===~
1_
0.2050.0
-,0.77
40.0
---~_0.90
30.0
0.51
0.60
10.
1.20
1.50
O.rrL-_-C-_ _"<:::'_~=:':.::.--l
0.0
20.0
0.40
1.05
Radial
(Slow Formation: Pierre Shale)
1.5
0.5
1.0
2.0
10.0
0.80
0.0
Vertical
(Slow Formation: Pierre Shale)
0.5
1.0
1.5
0.0
2.0
90.0
90.
80.
80.0
0.10
70.0
0.30
60.0
-0.60
0.80
50.0
1.26
40.0
~
~~'61
0.70
~
_ _ _ 14.24
1.03
O.
1.40
1.40
1.03
4.61
r.:::;;1
9.6'
30.0
0.80
0.60
~30
6
Figure 14: The radial and vertical components of fluid displacement at (r = r b , e = 0)
scaled by the total displacement of the incident wave for hard (top) and soft (bottom)
formations. The incident wave is a plane SV wave with 1/ = O.
20.0
10.0
0.0
323
Borehole Coupling
Pressure at the Center of Fluid
(scaled (see text), SV wave incidence)
Horizontal axis
= Frequency (KHz)
Vertical axis = Incidence Angle (degree)
Phase
(Fast Fonnation: Berea Sandstone)
Amplitude
(Fast Fonnation: Berea Sandstone)
0.0
1.0
0.5
1.5
2.0
0.0
0.5
90.
0.01
80.
1.0
I
I
I
I
70.
I
-'71. 00
,,
0.08
50.
I
,
,,,
,
,,,
40.
0.t2
30.
,,
,
,
,,
20.
10.
70.0
60.0
50.0
,,
,
,
-150.00
I
0.10
80.0
,
,, ,
,
I
60.
90.0
,
,
,,,
,
I
I
I
0.04
2.0
1.5
I
I
I
,.
.
,
,,
,
,,
,
-- ,
,-
,,
,,
,,
40.0
30.0
.,?:O.OO
20.0
10.0
, •
",
O.
0.0
Amplitude
(Slow Formation: Pierre Shale)
0.0
0.5
1.0
1.5
2.0
90.n+-~~...J....~~...L.....~~'--'-~"""
0.0
Phase
(Slow Formation: Pierre Shale)
0.5
1.0
1.5
I
80·\1]-------0.10-------,
'1-------,0.,,--~-
2.0
f--'--,-;,~-J..~7"1~",","~~~...l-~~-'-t
90.0
I
I
\
I
I
,
\
-176.40-
80.0
,,
\
/
~---t
70.0
60.0
50.0
211.10
40.0
30.0
20.O"j~~~
10.
20.0
10.0
0.0-'-''--''''==--''''-'''''''''''''''--='''''''-=---'
~_ _---.2:~~~=::'::-J.0.0
Figure 15: Pressure at the center of the fluid scaled by Po (see text) for hard (top) and
soft (bottom) formations. The incident wave is a plane BV wave with II = O.
Peng et al.
324
Low Frequency Approximation of Tube Wave Dispersion
in Soft Formation (Pierre Shale)
1000
950
"'
§.
~
.g
g
.,
900
\
850
'" 800
.c
'"
Shear wave
a-
750
0
400
800
1200
1600
2000
Frequency (Hz)
Resonance due to SV Incidence in Soft Formation
90
( Pierre Shale)
80
Q)
70
~
.,
.,
Cl
~
C,
c
.,'"
u
.,c
·u
'"
.5
60
50
40
Exact Solution
30
Low Frequency Approximation
20
10
0
0
400
800
1200
1800
2000
Frequency (Hz)
Figure 16: (top) Frequency of tube wave phase velocity in the soft formation at low
frequency approximation; (bottom) comparison of predicted resonance angle under
low frequency approximation with the exact solution.
325
Borehole Coupling
Transverse Component of Displacement at Borehole Wall
(scaled by the total displacement of incident SH wave)
Horizontal axis
=
Frequency (KHz)
Vertical axis
0.5
1.0
1.5
Incidence Angle (degree)
Fluid
(Slow Formation: Pierre Shale)
Fluid
(Fast Formation: Berea Sandstone)
0.0
=
2.0
90.O+-~-'-t-'--'-..~-'-t-~,..-...J~~"'-1
0.0
0.5
1.0
1.5
2.0
1-'-~.-..L~...............--'-;j~L...r-t;"-rl
90.0
80.
80.0
70.
70.0
60.0
60.
1.01
50.
50.0
1.03
0.87
40.
40.0
1.00
30.0
30.
20.0
20.
1.12
~::::::t 10.0
0.0
10.
o.ni-_.L.--.L.-L...l...-...l...-...l...-L...L.!-L.L-.J
L-_.-L..L._---l.l..LL:L:..-..L=-----l:.
Solid
(Fast Formation: Berea Sandstone)
0.0
0.5
1.0
1.5
Solid
(Slow Formation: Pierre Shale)
2.0
90.o-h-.,....,....,......,......,.....,.......,....-;-';-....,.....,J-"';~-H
0.0
0.5
1.0
1.5
2.0
h-t-t-r'"7'-+-...,.....,;--,........,...~-'--'........,...\-+ 90.0
1.00
80.
80.0
70.
70.0
60.
60.0
50.
50.0
40.
40.0
30.
30.0
20.
20.0
10.0
10.
O.o-L----L--L.-"""=---"':........<:::..,,:::=~
L..J-.l.~(L~~~@5:~'__~~O~.6~. i- 0.0
Figure 17: Transverse components of fluid and solid displacement (scattered) scaled by
the total displacement of the incident wave at (r = rb, = 0). The incident wave is
a plane SH wave with l/ = O.
e
Peng et aJ.
326
Pressure at center of fluid
- 1 0 0 Hz
Hard Formation (Berea Sandstone)
- - -500 Hz
_. -1000 Hz
••••• 2000 Hz
0.12
0.09
.' .....
.
•
0.06
0.03
o
0.03
0.06
0.09
:•
........4"/
......
0.12
Pressure at center of fluid
- 1 0 0 Hz
Soft Formation (Pierre Shale)
- - -500 Hz
- - -1000 Hz
..... 2000 Hz
1
0.8
0.6
0.4
0.2
o
0.2
0.4
0.6
0.8
1
Figure 18: The borehole reception pattern for pressure at the center of the fluid at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane SV wave with v = o.
327
Borehole Coupling
Fluid side of borehole wall
Hard Formation (Berea Sandstone)
2
1.6
1.2
- 1 0 0 Hz
- - -500 Hz
- - -1000 Hz
·····2000 Hz
0.8
0.4
o
0.4
0.8
1.2
1.6
2
Fluid side of borehole wall
Soft Fonnation (Pierre Shale)
4
3
- 1 0 0 Hz
- - -500 Hz
- - -1000 Hz
·····2000 Hz
2
1
o
1
2
3
4
Figure 19: The borehole reception pattern for fluid displacement at (r = r;;,e = 0) at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane SV wave with v = O.
328
Peng et al.
Solid side of borehole wall
-100 Hz
Hard Formation (Berea Sandstone)
- - -500 Hz
_. -1000 Hz
2
1.6
..... 2000 Hz
1.2
0.8
0.4
o
0.4
0.8
1.2
1.6
2
.
•
••
·
Solid side of borehole wall
-100 Hz
Soft Formation (Pierre Shale)
2
1.6
1.2
Ii
1/
,
-:I ... I
- - -500 Hz
- - -1000 Hz
..... 2000 Hz
, :.
0.8
0.4
o
,,
0.4
0.8
1.2
1.6
2
· •.,
.;.:.
\
,
"
\;
Figure 20: The borehole reception pattern for solid displacement at frequencies 100 Hz,
500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. The
incident wave is a plane SV wave with v = O. The geophone is at (r = rt,li = 0).
329
Borehole Coupling
- 1 0 0 Hz
Fluid side of borehole wall
Hard Fonnarion (Berea Sandstone)
1.5
1.2
0.9
,
... -..
-""""
- - -500 Hz
_. -1000 Hz
.. "". 2000 Hz
,
" ,,
,,
0.6
0.3
••
•,
,,
,
:,
,
,,
·
o
0.3
0.6
0.9
1.2
1.5
,
......
Soft Fonnarion (Pierre Shale)
1.4
1.2
0.8
0.6
0.4
0.2
.
,
- 1 0 0 Hz
- - -500 Hz
_. -1000 Hz
Fluid side of horehole wall
1
,,
,
,,
".. ··2000 Hz
···,",, ,.,•
••
,
,,
o
"
0.2
0.4
0.6
0.8
·
•
1
1.2
1.4
Figure 21: The borehole reception pattern for fluid displacement at (r = rb", e = 0) at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane SH wave with 1/ = O.
Peng et al.
330
Solid side of borehole wall
- 1 0 0 Hz
Hard Formation (Berea Sandstone)
- - -500 Hz
- . -1000 Hz
••••• 2000 Hz
1.25
1
0.75
0.5
0.25
o
0.25
0.5
0.75
1
1.25
Solid side of borehole wall
- 1 0 0 Hz
- - -500 Hz
- . -1000 Hz
••••• 2000 Hz
Soft Formation (Pierre Shale)
1.4
1.05
0.7
0.35
:.'. -.
· .•• .
..
•
~
.•
•
o
0.35
0.7
·•
1.05
1.4
Figure 22: The borehole reception pattern for solid displacement at frequencies 100 Hz,
500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. The
incident wave is a plane SH wave with v = O. The geophone is at (r = rt,rJ = 0).
Borehole Coupling
331
Solid side of borehole wall
-100 Hz
Hard Fonnation (BeI'ea Sandstone)
1.1
- - -500 Hz
_. -1000 Hz
••••• 2000 Hz
----;.. ..
_."
.t>
.;:
!l
.5
•
0.9
::
"g
<:>:::
••• 'I.
.
.
..
..
.. .......-. .....
---. . -----
..
'.
,,
0.8
....
O. 7 0!-'-.....,u3.l0~.........
61.J0.....,u..9.L0............1...21.J0L.l-lu..1.L5...
0.........1L.18 0
Incidence angle (degree)
Solid side of borehole wall
-100 Hz
Soft Fonnation (pierre Shale)
1 .1
- - -500 Hz
---1000 Hz
..... 2000 Hz
. .,
,...----.........----=--,.
•
•
..
, ... J
.'
.t>
...
-.1
~.
~ 0.8
<:>:::
0.7
;.,
. '{ I .
.. 1\ • ~
.;: 0.9
S
,
..
1
,:
I
I
I
1
•
,J :
.
\
."
...:
",
.,' It.~ .
~.:: \ -.
/. ..."r.
-0·1
..
'../
I I
I ,
I ,
I 1
"
I, .
I I
1 I
I I
I J
I J
I J
"
O. 6 ~""""':1:".L.LJL.L.I................Lu........L1.&......u.L.L.LJu.a.J
o
30
60
90
120
150
180
Incidence angle (degree)
Figure 23: Rectilinearity for the solid displacement at frequencies 100 Hz, 500 Hz, 1000
Hz and 2000 Hz for hard (top) and soft (bottom) formations. The incident wave is
a plane SV wave with v = O. The geophone is at (r = rt, e = 0).
Peng et aI.
332
-100 Hz
Solid side of borehole wall
Hard Formation (Berea Sandstone)
1
~
c
4
- - -500 Hz
_. -1000 Hz
3
.. ···2000 Hz
2
....
··
·
.'--
--.. "--'-r-
.~
-: - 1
••••-
o
•
'';-2
c
:!
"
•
~"'"
0 E-....-::"=':~...r.-"J!lII--;..'-"-.."...;-::.;-::;._...-
.<;j:
.:
c::i
..
...
- 3
.•••
- 40
~L.L.l.....3~0.......L"':5L;0L.L.l.....9~0.........1
...2LlO..........1~5~0.L"':..1u8 0
Incidence angle (degree)
-100 Hz
- - -500 Hz
- - -1000 Hz
..... 2000 Hz
Solid side of borehole wall
Soft Fonnation (Pierre Shale)
200
·
·· . ..
..
···
". ............
....
·•
'.
150
~
S;, 100
::::,
'"
o
<;j
't"0
--.
'-:~..
.50
c
o
.~.
.:
"
50
.§
-
"
1 00
.. -.
...
<:i. 1 50
c
::
:
:
:
"
....:.
- 20 00UJ
....
30
50
90
120
150
Incidence angle (degree)
180
Figure 24: Inclination deviation from the incident wave for the solid displacement at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane BV wave with lJ = o. The geophone is at
(r = rit, B = 0).
Borehole Coupling
333
Solid side of borehole wall
-100 Hz
Han! Fonnation (Berea Sandstone)
- - -500 Hz
_. -1000 Hz
••••. 2000 Hz
1.4
1.12
0.84
0.56
0.28
o
0.28
0.56
0.84
1. 12
.- .. .'
.'--...."';.;.:.:'"j~
•
··.
·•
··• •
••
•.
...
.
...."'" "1
•
'
1.4
Solid side of borehole wall
-100 Hz
Soft Fonnation (Pierre Shale)
- - -500 Hz
_. -1000 Hz
• •••• 2000 Hz
1.4
... .- ...._-.
.
.......
1.12
0.84
0.56
0.28
o
0.28
0.56
0.84
1.12
•
.--,,•
........
':1
f/
.,
••
:t
•
..
~\.,
.-..,
...\ ,
. ......
.....-4--#
-. ' ........
---
1.4
Figure 25: The effect of geophone orientation on the borehole reception pattern at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane BV wave with {j = 45 and v = O. The
geophone is at =
varies from 00 - 360 0 •
r rt- ()
Peng et al.
334
Solid side of borehole wall
-100 Hz
Hard Fonnation (Berea Sandstone)
- - -500 Hz
_. -1000 Hz
2000 Hz
1.25
..........+...... .... .....
•.
••
..
.
1
0.75
0.5
0.25
o
0.25
0.5
0.75
·••• •
•.
.•.
•.
--.-..............-
'. "...... ...
1.25
~'
Solid side of borehole wall
-100 Hz
- - -500 Hz
- . -1000 Hz
., •-. • .... 2000 Hz
~~:;'=-""":'. ".
Soft Formation (Pierre Shale)
1.5
1.2
0.9
0.6
0.3
o
0.3
0.6
0.9
1.2
•
\
•
\
.
••
"-fl.,-. --..._ _",,"'-.. .... .a •••• _
1.5
Figure 26: The effect of geophone orientation on the borehole reception pattern at
frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom)
formations. The incident wave is a plane SH wave with 8 = 45 and v = O. The
geophone is at =
e varies from 0° - 360°.
r rit-
335
Borehole Coupling
Solid side of borehole wall
-100 Hz
Hard Fonnation (Berea Sandstone)
- - -500 Hz
_. -1000 Hz
••••• 2000 Hz
12
~
10
~
~
8
.§
6
0;:
4
~
..
'"
..=
-=
..." ...
···
,
·. ·
,,·
,
,,
··
·
.
..
..
: a,.:
't:l
.51
2
..· - .- "
0"
.5 0
<:i
- 2
...
,
,,
o
.~:
..
.'..
...
.''
'
'
-
..",
90
180
.: ",
':.~_
",--.,
...
.- --
•
.
'.360
270
Geophone orientation (degree)
Solid side of borehole wall
-100 Hz
Soft Fonnation (Pierre Shale)
- - -500 Hz
_. -1000 Hz
20
~
2
.-..
."
}_..:.::-:.:..........:;:-i
•
1
~
15 - 1 6
,
."
'Z
., ,,
"
•S'
~ -34
=
"
" -5 2
.5
<:i
=
'Z
-
-70
..... 2000 Hz
.' .-
_
10-
.,
.'. ..
··
·...:
,,
.,., ..
::
,
.-oJ·
'
~.
!--''-'--'--:-l:--'--'-......L...'-'--'-...J..-'--'-.a.....J
o
90
180
270
360
Geophone orientation (degree)
Figure 27: The effect of geophone orientation on the inclination deviation at frequencies
100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations.
The incident wave is a plane SV wave with 0 = 45 and v = O. The geophone is at
r=
0 varies from 00 - 3600 •
rt.
Peng et al.
336
Solid side of borehole wall
-100 Hz
Hard Formation (Berea Sandstone)
12
- - -500 Hz
- - -1000 Hz
• •••• 2000 Hz
·· ...
.·
.. ... -- ................
8
o
o
o
o
4
c
~
0
'"
0'_
",,"
.~
'0
-4
~
- 8
.c
----
o
•
..
.. .•·
.
o
o
oN
o
<:
-12
o
90
1·80
o
360
270
Geophone orientation (degree)
Solid side of borehole wall
-100 Hz
Soft Formation (Pierre Shale)
- - -500 Hz
- - -1000 Hz
..... 2000 Hz
11 0
~
~
...• .
.
::=,
::
:~
o
~ 55
~
c
~
.
0 "• /
~
t
'0
.c
0
:
==E - 5 5
..•.
"
...
-".
.. ,--:.:-:- ..
__•• :"-.
.&,,0
'00
..
....".: .."=-:''' /
-....;...
•
~
"00
"
.~
~
<
- 1 1 00~L.-JL.-J.....JL....I---'---'-:-!-":"",,,-I.-I.':'2~70::-'-.........&.:3~6 0
90
180
Geophone orientation (degree)
Figure 28: The effect of geophone orientation on the azimuth deviation at frequencies
100 Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations.
The incident wave is a plane SV wave with 0 = 45 and 1/ = O. The geophone is at
r =
0 varies from 0° - 360°.
rt.