BOREHOLE EFFECTS ON DOWNHOLE SEISMIC MEASUREMENTS

BOREHOLE EFFECTS ON DOWNHOLESEISMIC MEASUREMENTS

by

Chengbin Peng, C.H. Cheng, and M.N. Toksoz

Earth Resources LaboratoryDepartment of Earth, Atmospheric, and Planetary Sciences

Massachusetts Institute of TechnologyCambridge, MA 02139

ABSTRACT

An exact formulation for borehole coupling, which is valid for all frequencies and allazimuthally 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 bemade for frequencies above 500 Hz in a hard formation. In a soft formation, if theincidence angle is well away from the resonance angle for a SV incidence, no boreholecorrection is needed for frequencies below 300 Hz; while for frequencies above 300 Hz,the borehole can cause severe problems on downhole measurements. The borehole canalso significantly alter the particle motion direction such that horizontal componentsrotation from data itself is unreliable for experiments with frequencies above 1 kHz inthe hard formation and around 500 Hz in the soft formation.

INTRODUCTION

Increasing interest is shown towards the crosshole and VSP surveys at frequencies upto 1 kHz or more in order to resolve the fine details of structures and lithology betweenwells (Bregman et al, 1989; HarriS, 1988; Tura, 1991). However, at frequencies onthe order of 1 kHz, the existence of a fluid-filled borehole has strong influence on thedownhole measurements. Dependent on the frequency and angle of incidence, as well asthe formation properties, the measured displacement on the borehole wall or pressureat 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 utilizeboth the amplitude and phase information of recorded energy may be erroneous becausethe borehole is not included as part of the formulation.

288 Peng et al.

(1)

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 thewavelength 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 boreholeif 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 pressureinside the fluid in the same way as a piston source does. Schoenberg (1986) developeda complete theory for the interaction of a plane elastic wave with a fluid-filled boreholeand gave an explicit formuiation for the low-frequency limit. In this theory the elasticfield in the solid and the acoustic field inside the fluid satisfy the corresponding waveequations and coupling of these two fields is accomplished through the fluid-solid interface boundary conditions. Lovell and Hornby (1990) presented a complete formuiationfor both low and high frequency for the azimuthally symmetric component, so theirformulation 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) markedresonances in the fluid occur for both shear and compressional incidence.

In this paper, an exact formuiation based on Schoenberg's theory is given for allfrequencies and all azimuthally symmetric and nonsymmetric components. Detailedstudies on the influence of frequency, incidence angle and polarization of incomingelastic waves (P, SV and SH), as well as the geophone orientation, on the downholemeasurements 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 homogeneouselastic 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 speedCtf. The radius of the borehole is Tb. When the wave hits the borehole, a transmittedcompressional wave (denoted by displacement fj/) is generated in the fluid, and at thesame time a scattered elastic wave (denoted by US) is produced in the solid. If a threecomponent geophone is placed in the fluid, uf will be recorded and if it is clampedagainst 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 satisfiesthe fluid wave equation as

w2

\l21/Jf + 21/Jf = 0,Ctf

where uf = \l1/Jf. The displacement of the scattered wave in the elastic solid can be

Borehole Coupling 289

represented by three elastic potentials </> (P potential), ~ (SV potential) and 'I/J (SHpotential) as

(2)

where2

\72</> + w2</> = 0a

w2

\72~ + {32~ = 0

w2

\72'I/J + {32'I/J = o.

The general solution of these potentials is given by Schoenberg (1986) as

</>f = a(;~w) [AoJo(kfr) +2 L~=lin(Ancosne+ A~sinne)Jn(kfr)]

</> _ - a~\,"J) [BoHg1)(kpr) + 2 L~=lin(Bncosne + B~sin ne)H,\l) (kpr)J

~ i{32;JW) [CoHa1)(ksr) +2L~=lin(Cncosne+C~sinne)H,\1)(ksr)]

'I/J = _{3~SW) [-D~Hal)(ksr) +2 L~=lin(Dnsinne- D~cosne)H,\l)(ksr)] (3)

where z dependence and time dependence ei(kzz-wt) is assumed, and kf = Vw2 / a} - kz2 ,

kp = vw2/a2 -kz2 and ks = VW2/{32_kz2. Signs of kf, kp, kz are chosen such that

Im(kp, k., k f ) ~ o.

Given these potentials in (3), the displacement and stress components that areinvolved in the boundary conditions can be written as

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~sinnell

_{3~SW) [U;o(r)Co + 2 L~=linU$n(r)(Cncosne + C~sinne)]

- §~\,"') [U:{;(r)Do + 2 L~=linU%(r)(Dncos ne + D~sin nell

u fr

u</>r

u<r

u'I/J =r

and

_pf = ufrr

</>Urr

=

=

p[a~:;(w) [Rfo(r)Ao + 2 L~=linRfn(r)(Ancosne + A~sin ne)J

pa;~w) [R</>o(r)Bo+ 2 L~=linR</>n(r)(Bncosne + B~sin nell

290

e =arr

'Ij; =arr

and

arP -re

ae -re

a'lj; =reand

Peng et al.

p§~JW) [Reo(r)Co +2 L::"=linRen(r)(Cncosne + C~sin nell

P,6~JW) [R'Ij;o(r)Do + 2L::"=linR'Ij;n(r) (Dncos ne + D~sin nell

pa~JW) [8r/>o(r)B~+2 L::"=linSrPn(r)(-Bnsin ne + B~cos nell

P,6~JW) [8eo(r)C~+ 2 L::"=lin8 en(r)( -Cnsin ne + C~cosne)]

PI1~JW) [8'1j;o(r)D~ + 2 L::"=lin8'1j;n (r)( -Dnsin ne + D~cos nell

a'f. = pa~Jw) [ZrPo(r)Bo +2L~linZrPn(r)(Bncosne+B~sinne)J

a~z = p§~JW) [Zeo(r)Co + 2 L::"=linZen(r)(Cncosne + C~sinne)]

a'/!. P,6~JW) [Z'Ij;o(r)Do + 2 L::"=linZ'Ij;n (r)(Dncosne + 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 appendixA.

The coefficients An, A~, B n, B~, Cn, C~, Dn, 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+) = _pih-)

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 denotesthose of the scattered wave and f those in the fluid.

The displacement and stress of incident plane elastic waves (P, SV or SH) are alsoexpressible in terms of mode summation. For P incidence

u; - a~~w) [U:O(r) + 2 L::"=linU;" (r) (cos nv cosne + sin nv sin nell

a;:" - pa~Jw) [Rpo(r) + 2 L::"=linRpn(r)(cosnv cosne + sin nv sin nell

a P = pa~Jw) [8po(r) +2L::"=lin8Pn(r)(-cosnvsinne+sinnvcosne)]re

\

(4)

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:~=linRSVn(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:~=linZSVn(r) (cos nv cos nO + sin nv sin nO)] (5)

USH =r

(YSH =rr

(YSH =rO

(YSHrz

and for SH incidence

_is'~~W) ru:aH(r) + 2 I:~=linU!nH (r)( - cos nv sin nO + sin nll cos nO)]

PiS'~SW) [RsHO(r) + 2 I:~=linRSHn(r) (- cosnll sin nO + sin nv cos nell

PiS'~SW) [8sHO(r) + 2 I:~=lin8sHn(r)(cosnv cos nO + sin nv sin nO)]

PiS'~SW) [ZsHo(r) + 2 I:~=linZSHn(r) (- cosnv sin nO + sin nll cos ne)](6)

where 1I 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 unknowncoefficients

and

[

aU:;' cosnll= pa RPn cosnv

pa ZPncosnvpa8Pncosnv

[

UP·0:' rn SIn nv

= pa RPn sin nllpaZPnsinnvpa 8 Pn sinnll

RUSV cosnvI' rn

PiS' Rsvn cos nllPiS' ZSVn cos nllPiS' 8svn cos nll

R USV sin nvI' rn

pfJ Rsvn sin nvPiS' ZSVn sin nllPiS' 8svn sin nll

RUSH sin nv ]I' rn

PiS' RSHn sinnllPiS' ZSHn sin nv

-piS'8sHnsinnv

_RUSHcosnv ]I' rn

-PiS' RSHn cos nll-PiS' ZSHn cosnvPiS' 8SHn cos nll

(7)

(8)

where-U;n(rb)/p-R~n(rb)

-Z~n(rb)

-8~nh)

-u!nh)/p-R1f;n(rb)-Z1f;nh)-81f;n (rb)

and the vertical lines in (7) and (8) separate the cases that the incident wave is theplane P (first column) or SV (second column) or SH (third column) wave.

292 Peng et al.

Knowing the coefficients An, A~, Bn, B~, Gn, G~, Dn, D~, the displacement both inthe fluid and solid and the pressure inside the fluid can be easily computed. Calculationsare performed with frequency up to 2 kHz and incidence angle 0 ~ 90° for two typesof formation: fast (Berea sandstone) and slow (Pierre shale). The specific parametersare given in Table 1. Unless otherwise indicated in the text, 11=0 (azimuth of incidencewave) 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 thedirection of particle motion, depending on the frequency, incidence angle as well as theformation properties. In the worst case, the surface wave will be excited such thatthe nature of particle motion is changed (from linear motion to elliptic motion). Alsobecause the problem is nonsymmetric in nature, measurements are also dependent on theposition of the geophone at which experiments are conducted. This section is devoted tothe case of a plane P wave incidence. We will examine the frequency and incidence angledependence of the displacement and pressure in the fluid as well as the displacement ofscattered energies in the solid. We will show how different the measurements (corruptedby the borehole effect) are from the true incoming waves, both in amplitude and particlemotion, and how the measurement may vary as the position of the geophone changesaround 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), Figure2 shows the radial and vertical component of the borehole scattered wave, scaled by thetotal 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 boreholescattered energy is negligible compared to the incident wave for a frequency belowhundreds of Hertz (500 Hz for hard formation and 250 Hz for soft formation); whilefor a frequency above 1 kHz, the scattered energy is a significant portion of geophoneoutput. For a frequency at 2 kHz, the radial component of the borehole scattered wavecan reach almost 60% of incident total displacement at 90° incidence for both formationsand the vertical component can reach 30% at 45° incidence for the hard formation andover 40% at 45° incidence and over 60% at grazing incidence for the soft formation. Theradial component of the borehole scattered wave increases with frequency and incidenceangle; 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


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 componentof fluid displacement is almost independent of frequency except near 900 incidence andincreases rapidly with the increase of incidence angle; the vertical component is generallysmaller (less than 20% of the total displacement) than the radial component and showsa peak at 450 incidence. For the soft formation, the radial component shows strongfrequency and incidence angle dependence in a complicated manner: it increases withfrequency when the incidence angle is less than 450 and decreases with frequency whenthe incidence angle is greater than 450

; the vertical component increases with frequencyand decreases with incidence angle and unlike the case of the hard formation the peakis at grazing incident rather than at 450

•

Figure 4 shows the pressure (amplitude and phase) at the center of the fluid scaledby the pressure of the incident P wave ( spherical component of the stress tensor),

I.e., p/lpo, where Po = pa;lw)w2(1 - i~:)Jo(kpr) (see Lovell and Hornby, 1990 withcorrection), for both the hard and soft formations. For both cases the measured pressureat the fluid is much less than that if the borehole does not exist, which is due to thepressure 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. Thephase lag at the soft formation is less than that at the hard formation. It increasesstrongly with frequency and shows less dependence on the incidence angle except at thegrazing 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 forthe 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 atfrequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz. For the hard formation, the receptionpattern shows a main lobe at 900 incidence (and is almost independent frequency) forfrequency below 1 kHz. For the soft formation, dependence on incidence angle is veryweak compared to the previous case. At 2 kHz, the curve deflects toward the grazingincidence rather than the normal incidence.

Figure 6 shows the borehole reception pattern for the fluid displacement at theborehole wall. For the hard formation, the reception pattern shows little frequencydependence but strong incidence angle dependence with a main lobe at the 900 ofincidence angle and zero at grazing incidence. For the soft formation, the incidenceangle 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 grazingincidence, while for frequencies above 1 kHz, the reception pattern is more complicatedwith 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 theborehole wall. For the hard formation, the reception pattern is almost a unit circle withlittle frequency dependence. For the soft formation, the pattern looks the same as thatfor 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 andfrequency dependence is very strong.

Borehole Effect on Particle Motion

Since the displacement is a vector, the borehole distorts not only its amplitude but alsoits direction, i.e., particle motion. The estimation of particle motion is very importantfor data rotation of downhole 3-component measurements. It is beneficial if we canunderstand how and how much the borehole can change the polarization of particledisplacement.

In this paper, inclination deviation is defined as the difference of the inclination(angle of displacement vector with the borehole axis) of measurement displacementfrom that of incident wave; azimuth deviation is defined as the difference of the azimuthangle of measured displacement from that of incident wave. Another important conceptis 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 fora perfect circular polarization it is zero.

Figure 8 shows the rectilinearity of displacement at the solid side of the borehole wallas a function of incidence angle at frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz. Forthe hard formation, the rectilinearity is very close to 1 which means the solid motion islinear, a little bit of the surface mode is excited at 2 kHz near grazing incidence. For thesoft formation the frequency and incidence angle dependence is more obvious, althoughthe particle motion is dominantly linear except at 2 kHz.

Figure 9 shows the inclination deviation for the displacement at the solid side ofthe 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 thatfor the hard formation and can reach 6° at grazing incidence at 2 kHz.

Borehole Coupling

Effect of Geophone Orientation

295

Because the problem is nonsymmetric in nature, the displacement and pressure will bedependent on the azimuth orientation of the geophone or hydrophone with respect tothe incident wave. Figure 10 shows a polar plot of the calculation for a plane P waveincidence with incidence angle {j = 45° and azimuth angle l/ = O. What we plot inthis 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 thehard formation, the reception pattern is independent of the geophone orientation for afrequency 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 thestrong backscattered wave). For the soft formation the patterns look similar to those forthe 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 boreholewall 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 formationit can reach 4° in the forward scattering direction.

Figure 12 shows the azimuth deviation of the solid displacement at the boreholewall. 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 thegeophone 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 thesurface wave (e.g., tube wave) occur at the interface, which makes the effect of theborehole 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 incidentwave is of SV type plane wave with azimuth l/ = O. For the hard formation, the boreholescattered 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 ofthe borehole scattered wave increases with frequency and decreases with the incidenceangle, while the vertical component of the scattered wave shows a minimum at 450 incidence and has two peaks at grazing incidence and normal incidence. For a frequencyaround 2 kHz, the radial component of the borehole scattered wave can reach almost110% of incident total displacement at grazing incidence and the vertical componentcan 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 wherethe 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 theresonance region, the frequency and incidence angle dependence is similar to that forthe hard formation.

Figure 14 shows the radial and vertical components of fluid displacement scaled bythe 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 componentof fluid displacement is almost independent of frequency except near 0° incidence andincreases rapidly with the decrease of incidence angle; the vertical component is smallerin magnitude than the radial component and at high frequencies it shows a peak at0° incidence. For the soft formation, significant fluid resonance is also observed at lowfrequencies and certain incidence angle. Away from the resonance region, the radialcomponent shows complicated frequency dependence: it first decreases with frequency,reaches a minimum around 1 kHz (depending on the incidence angle) then increasesagain. Generally the radial component decreases with the increase of incidence angle.The vertical component shows almost no frequency when the incidence angle is above400 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 400

the amplitude shows no dependence on frequency and for an incidence angle below 40°significant frequency dependence can be observed. The amplitude increases with thedecrease of incidence angle. The phase lag for the hard formation shows strong frequencyand incidence angle dependence. For the soft formation, the pressure resonance is moresignificant at low frequencies and certain angles of incidence. Away from the resonanceregion, the amplitude is almost independent of frequency for normal incidence. Thephase 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 expressionfor 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

GT=

Borehole Coupling

as

af [1_1 pfa} (1_ 2,82 _2Pf)(1_2,82)w2n2JnwTb]

1 + Pfa; 4 p,82 +Pfa} a} P a2

,82 2ap,8

which agrees with the result of Biot (1952) and White (1983) as W ---> o.

297

(9)

Figure 16a shows the frequency dependence of the tube wave phase velocity for softformation (Pierre shale). For a frequency below 1 kHz, GT > ,8; and for a frequencyabove the cut-off, GT < ,8.

The resonance occurs at ~ cos /j = ~, so we have

/j = cos- 1 L (10)GT

which has solution only when GT > ,8, which implies that the borehole resonance is alow frequency phenomenon.

Figure 16b shows the comparison of the prediction of resonance angle under lowfrequency approximation with that of exact solution. Excellent agreement can be foundat frequencies below 400 Hz.

For an plane SH wave incidence, the wave fields inside and outside the boreholeturned out to be much simpler owing to the fact that SH motion is not coupled withP or SV motion. Figure 17 shows the transverse components of fiuid and scatteredsolid displacement at the borehole wall scaled by the total displacement of incidentSH wave, Le., Ilu~(Tb)II/IIUih)11 and Ilu(jh)II/IIUih)ll. For the fast formation, thedisplacement in the fiuid is strongly dependent on frequency, but almost independentof incidence angle at low frequencies. At high frequency the angle dependence is moresignificant. At 2 kHz and grazing incidence, the fiuid displacement reaches 160% of theincident wave. For the soft formation, the displacement in the fiuid depends on bothfrequency and incidence angle. Contrary to what we might expect, in this case increaseof frequency will decrease the fluid displacement. There is a peak around 600 Hz andgrazing incidence. The displacement of the scattered wave in the solid increases withfrequency and incidence angle in the fast formation. At 2 kHz and 900 incidence it canreach 70% of the displacement of the incident wave. In the soft formation, the scatteredwave shows similar behaviour as that in the fast formation. At 2 kHz and 900 incidenceit 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 thecenter of fluid at frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz for a plane SV wave

298 Peng et al.

incidence. For the hard formation, the reception pattern shows two lobes at 450 and1350 incidence and is almost independent of frequency for frequency below 1 kHz. Thepressure at the center of the fluid is zero at normal incidence and very small at grazingincidence. For the soft formation, large fluid resonance occurs at 250 at 100 Hz and 20°at 500 Hz. For frequencies above 1 kHz, the fluid resonance disappears. The receptionpattern shows strong and complicated frequency and incidence angle dependence.

Figure 19 shows the borehole reception pattern for the fluid displacement, I.e.,Ilu/II/IIUiII, 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 900 incidence. For the softformation, similar to the corresponding pressure reception pattern, strong resonance offluid motion occurs at low frequencies and small incidence angle. For frequencies above1 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 fluidmotion is zero at 900 incidence.

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, thereception pattern is almost a unit circle with little frequency dependence except at 2kHz. For the soft formation, the resonance is very weak compared to that in the fluidat low frequencies and small incidence angles. In this case, the reception pattern showsstrong dependence on frequency and incidence angle. Generally the solid displacementis smaller than that of the incident wave at around normal incidence and is larger aroundgrazing incidence, especially at high frequencies.

Figure 21 shows the borehole reception pattern for the fluid displacement at theborehole wall for a plane SH wave incidence. For the hard formation, the receptionpattern is nearly a unit circle except at 2 kHz. For the soft formation, although noresonance occurs, the reception pattern at low frequencies is quite different from thatat high frequencies. At low frequencies, the pattern is a unit circle; at high frequenciesit has two lobes around 200 and 1600 incidence and valleys at both normal and grazingincidences.

Figure 22 shows the borehole reception pattern for the solid displacement at theborehole wall for a plane SH wave incidence. For the hard formation, the receptionpattern is nearly a unit circle except at 2 kHz. For the soft formation, at frequenciesbelow 1 kHz the reception pattern is also nearly a unit circle, while at frequencies above1 kHz three lobes at 200

, 900 and 1600 are obvious.

(

Borehole Coupling

Borehole Effect on Particle Motion

299

Figure 23 shows the rectilinearity of displacement at the solid side of the borehole wallas a function of incidence angle at frequencies 100 Hz, 500 Hz, 1000 Hz and 2000 Hz fora plane SV incidence. For the hard formation, the rectilinearity is very close to 1 exceptat 2 kHz near grazing incidence. For the soft formation the frequency and incidenceangle dependence is more obvious, although the particle motion is dominantly linearexcept where the tube wave is excited at low frequencies and where some leaking modesappear at high frequencies.

Figure 24 shows the inclination deviation for the displacement at the solid side of theborehole wall for a plane SV incidence. For the hard formation, the inclination deviationis 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 softformation.

Effect of Geophone Orientation

Figure 25 shows a polar plot of the calculation for a plane SV wave incidence withincidence angle 0 = 45° and azimuth angle 1/ = O. Plotted here is the amplitude of soliddisplacement scaled by that of the incident wave (reception pattern) as a function ofgeophone azimuth angle () (0°- 360°). For the hard formation, the reception pattern isindependent 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 facingthe incident wave). For the soft formation the pattern looks similar to that for the hardformation for frequencies below 100 Hz, while for frequencies between 100 - 500 Hz, thereception pattern is less than unity in the forward scattering direction (0°) and prettymuch the same in the backward direction. For frequencies above 500 Hz, the pattern issignificantly less than unity in the forward direction and significantly larger than unityin 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 circlefor frequencies below 1 kHz and for a frequency above 1 kHz it also has a large lobeat 180° azimuth angle. For the soft formation the pattern looks similar to that for thehard formation for frequencies below 500 Hz, and for a frequency around 1 kHz, thereception pattern has a large lobe at 180° azimuth, while for frequency around 2 kHzthe main lobes occur at 90° and 270° azimuths.

Figure 27 shows the inclination deviation of the solid displacement at the boreholewall 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 wallfor a plane SV incidence. At low frequencies the deviation is almost zero, but at 2 kHzit 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

DISCUSSION AND CONCLUSIONS

301

In this paper, we have presented an exact formulation for borehole coupling based onSchoenberg's theory, which is valid for all frequency and all azimuthally symmetricand nonsymmetric components. We have studied the borehole effects on downholeseismic 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 energyin the solid is less than 10 % in amplitude of the incident wave and the downholegeophone measurement is nearly not affected by the fluid-filled borehole. Althoughfrequency plays a less significant role in this case, the pressure measured at thecenter 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 normalincidence 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 1kHz, the borehole scattered energy is a significant part of the total measurement ofthe downhole geophone for any type of incident wave. The particle motion direction can be different from that of the incoming wave by several degrees dependingon the incidence direction and frequency. The solid displacement measured withthe geophone facing the incoming wave will be noticibly larger than that with thegeophone opposite. The pressure at the center of the fluid shows dependence onfrequency as well as incidence angle.

• For the soft formation and frequency below above 300 - 500 Hertz, the boreholescattered energy in the solid is also negligible for a P-wave incidence, and in thiscase the pressure at the center of the fluid becomes significantly less dependenton incidence angle as it does in the hard formation. For an SY wave incidence,significant fluid resonance occurs at certain incidence angles for frequencies below1 kHz due to the excitation of the tube wave. At these particuiar frequencies andincidence angles, the solid displacement measured by the downhole geophone andpressure picked by the downhole hydrophone will be several times larger than thoseof the incident wave, the particle motion from geophone measurements differs fromthat 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 arestrongly 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 ofthe 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 geophoneis opposite to it.

We conclude that correction of the borehole effect for downhole measurements shouldbe made for frequencies above 500 Hz in the hard formation. In the soft formation, if theincidence angle is well away from the resonance angle for an SV incidence, no boreholecorrection is needed for frequencies below 300 Hz; for frequencies above 300 Hz, theborehole can cause severe problems on downhole measurements and proper attention tothis effect should be taken. Since the borehole can significantly alter the particle motiondirection, horizontal components rotation from data itself is unreliable for experimentswith frequencies above 1 kHz and rotation should be done from downhole gyro readingsif possible.

A frequency-wavenumber filter can be designed to remove the borehole corruptionto 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-Levinsonalgorithm (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 ResourcesLaboratory, MIT. The first author holds an nCUBE fellowship. We are grateful forhelpful discussions with Dr Roger Turpening and Dr Zhenya Zhu.

Borehole Coupling

REFERENCES

303

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 threecomponent 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 tocrosshole/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-filledborehole 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, 61stAnn. 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 CYLINDRICALCOORDINATE

In terms of four potentials t/>j, t/>, ~ and ,p, the displacement and stress can be writtenas

Uj• ot/>j • lot/>j • ot/>j

- er or + eO-:;: 00 + ez oz

ot/> 02~ lo,pUr - -+--+--

or oroz roO

lot/> 1 02~ a,pUO = --+-----

r 00 r oOoz or

Uz - ot/> + 02~ _ \72~oz OZ2

and

2 2 lot/> 1 02t/> 02t/> 0 w2 1 o~ 1 02~ 02~arr - -pw t/>-2pfJ [(-:;:or + r2002 + OZ2)+ oZ(fJ2~+-:;:or+ r 2002 + OZ2)

1 o,p 1 02,p+r 2 00 - -:;: oroOI

2 02t/> 0 w2 02~ 1 02,parz 2pfJ [oroz + or(2fJ2~+ OZ2) + 2rozoO I

2 0 1 ot/> t/> 02 1 o~ ~ 1 02,p 1 o,p 1 w2 02,paro = 2pfJ [00(-:;: or - r 2) + ozoO(-:;: or - r2) + r 2 002 + -:;: or + 2"(fJ2,p + Oz2)]

and

azz =

azO

aOO =

\

Except for the displacement and stress components given before, those componentsthat are not explicitly involved in the boundary conditions are as follows

Borehole Coupling

u~ - ar:aW) [Ubo(r)Ao + 2 'L':'=linUbn (r)( -Ansin nO + A~cosnO)]

u¢ - - a~~w) [Uta(r)Bo + 2 'L':'=linutn(r)( -Bnsin nO + B~cosnO)]0

u~ - - f1~~W) [U~o(r)Go + 2 'L':'=linU~n (r)( -Gnsin nO + G~cosnO)]

u'l/J - - f1~~W) [Uta(r)D~ + 2 'L':'= linU't (r)( -Dnsin nO + D~cos nO)]0

uf ar:aW) [U!o(r)Ao + 2 'L':'=linufn(r) (Ancos nO + A~sin nO)]z

u¢ = - a~(jV) [U1o(r)Bo+ 2 'L':'=linu!n (r) (Bncos nO + B~sin nO)]z

u~ _,6~~W) [U;o(r)Go + 2 'L':'=linU;n(r) (Gncos nO + G~sin nO)]z

u'l/J = 0z

and

¢ pa:Jw) [.c¢o(r)Bo +2 'L':'=lin .c¢n(r)(BncosnO + B~sinnO)](jzz

~ Pf1:JW) [ .c~o(r)Go + 2 'L':'=lin .c~n(r)(GncosnO + G~sin nO)](j zz

'l/J 0(j zz

¢pa:Jw) [M¢o(r)Bo + 2 'L~lin M¢n(r)(BncosnO + B~sin nO)](j00 -

~ p,6~JW) [M~o(r)Go + 2 'L':'=lin M~n(r)(GncosnO + ~sin nO)](j00

'l/J p,6~JW) [ M'l/Jo(r)Do + 2 'L':'=lin M'l/Jn (r)(DncosnO + D~sin nO)](j00

305

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:

306 Peng et al.

U!n(r) = kfJ~(kfr),

¢ (I)')Urn(r) = kpHn (kpr,

Ug¢ (r) = I;, nH~I\kpr),n I;;

U!"(r) = ikzH~I)(kpr),

Rfn(r) = -w2Jn(kfr)

R¢n(r) = _[(w2 - 2{32kz2)H~I)(kpr) + 2{3;;p2 (H~I)' (kpr) _ ~H~I)(kpr))]

Ren(r) = 2kzZ~2!r [H~I)(ksr) + f.;H~I)' (ksr) - k~;r2H~I)(ksr)]

RI. (r) = -2{32k/n[~1H~I)(ksr) - -,:LH~I)' (ksr)]lPn s r 2 l'i;ST

GA._(r) = -2{32kp2n[z;-hH~I)(kpr) - -,:LH~I)' (kpr)]!pH rvp r2 fl-pT

Gen(r) = 2kzZ~2(t n[k}r2H~I)(ksr) - f.;H~I)' (ksr)]

Gol'n(r) = -(32ks2[H~I)(ksr) + .;}-H~I)' (ksr) - k21\H~I)(ksr))~ l'i;sr s r

( ) . 2 (I)' (Z¢n r = 2~kzkp{3 Hn kpr)

Zener) = _i(kz2-t~)ks{32 H~I)' (ksr)

Z'lj;n(r) = ikzks{32f;;.H~I\ksr)

£¢n(r) = _[w2 - 2{32(ka2 - k/)]H~I)(kpr)

£en(r) = -2{32~(ki - kz2)H~I) (ksr){3

£'lj;n (r) = 0

MA. (r) = _(0:2 - 2(32)k0:2H~l) (kpr) + 2{32 1;,2 H~l)' (kpr) _ 2{32 kk~2

n2H~l} (kpr)~ ~ pr2

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

N¢n(r) = 2i{32T;PnH2) (kpr)

k 2 2k 2N (r) - i{32 (3 - z .0-nH(I)(k r)en - ksr Ie(3 n s

N () . 2 (1)'( )'lj;n r = ~kzks{3 Hn ksr


where ka = ~, kf3 = W, kp = Jka2

- kz2

, ks = Jkl- kz2 and kf = J~ -kz2•

kz = ka cos 0 for plane P wave incidence and kz = kf3 cos 0 for plane shear wave incidence. 0 is the incidence angle.

Appendix B. STRESS AND DISPLACEMENT OF INCIDENTPLANE WAVE

For a plane P wave traveling in the direction ~ = (sin o cos v,sinosinv, coso), where 0is the incidence angle with respect to the borehole axis and v is the a2imuth angle, thedisplacement potential can be written as (omitting ei(kzz-wt) dependence)

<pp - a~~w) exp[ikpr cos(O - v)] (B-1)

= _a~~w) [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

Csv = iltV(w) I ['k (0 )1 (B-2)(". w3. s: expz srcos -vSlllu

- if3

';3(W) ~ [Jo(ksr) + 2 L:::"=lin(cos nv cos nO + sin nv sin nO)Jn(ksr)1

and

u~v = _§~~W) [U3tY(r) + 2 L:::"=linUS'; (r) (- cosnv sinnO + sin nv cos nO)]

u~v = _§~~W) [U:cnr) +2L:::"=linU;'';(r)(cosnvcosnO+sinnvsinnO)].

Similarly, for a plane SH wave incidence

f3V(W) I .'l/JSH = -----wz-~ exp[tksrcos(O - v)] (B-3)

Slllu

= _f3~~W) ~ [Jo(ksr) + 2 L:::"=lin(cosnv cosnO + sin nv sin nO)Jn(ksr)]

308

and

Peng et al.

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), UfnH(r) = 0

Rpn(r) = _[(w2 - 2{32kz2)Jn(kpr) + 2{321v,,2 (J~(kpr) - -?-In(kpr))]f;; r.;pT

8Pn(r) = -2{32kp2n [z.1 In(kpr) - -!::J~(kpr)]np r2 n-pT

ZPn(r) = 2i{32k.kpJ~(kpr)

Rsvn(r) = 2{32k.ksPn(ksr) + .,LJ~(ksr) - kn; In(ksr)]/'l;sT s r 2

8svn(r) = -2{32ksk.n[.,LJ~(ksr) - k"""l In(ksr)]/l,sT s r 2

ZSVn(r) = i{32(ks2 - k. 2)J~(ksr)

Borehole Coupling

z axis

t

309

1U

k

x axis

Figure 1: A fluid-filled borehole (af,Pf) in a solid formation (a,fJ,p). An elastic waveU' incident on the borehole which causes scattered wave ils in the formation andfluid 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.

310 Peng et a!.

Scattered Solid Displacement at the Borehole wall(scaled by the total displacement of incident P wave)

Horizontal axis = Frequency (KHz) Vertical axis = Incidence Angle (degree)

Radial(Fast Fonnation: Berea Sandstone)

0.5 1.0 1.5 2.0

Vertical(Fast Fonnation: Berea Sandstone)

0.5 1.0 1.50.00.0

:~: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

0.36

0.27

0.18

0.09

0.02

2.0

I~-:::~:::=:==::::::::=====t90.0

80.0

~---E-70.0

60.0

50,0

40.0

--.....J:. 30.0

20.0

10.0

'- -4.__.c:.=--'"""~......-:=~:::.-",':.MCL 0.0

Vertical(Slow Fonnation: Pierre Shale)

0.0 0.5 1.0 1.5

0.40

0.24

Radial(Slow Fonnation: Pierre Shale)

0.5 1.0 1.5 2.00.090.0:/-;--c-......,......l.;---';-.......J.......,...\~.......,\~~

80. ~ os.

0.48

20.

Figure 2: The radial and vertical components of the borehole scattered wave scaled bythe 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)


Fluid Displacement at the Borehole Wall(scaled by the total displacement of incident P wave)


Vertical(Fast Formation: Berea Sandstone)

0.5 . 1.0 1.5

70.0

60.0

50.0

40.0

30.0

20.0

10.0!:l:·o::.6 .J. 0.0

2.0I-'-~-'--'-~~.....L.~~--l..,.,~~+90.01----0.03---------F, 0.06 - 80.0,- -------------

O.09-~______0.12 //015 (

~ 0.1.

___ 0.'2~ ~0.09~

0.0

Radial(Fast Formation: Berea Sandstone)

0.5 1.0 1.5 2.0u~ ",.04_

0- 1.01 ____________

.v· ........0.94_

.li 0.89_

.CT 0.80_

.li 0.70

0.60

'".V 0.50

.CT 0.40

0.30

.u. 0.20

0.10

.IF

0.090.

80.

70

60

50

40

30

20

10

o

2.090.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

10.0

0.0

Vertical(Slow Formation: Pierre Shale)

0.5 1.0 1.5

0.08

0.20

0.40

0.60

_o.80-

/201 "oj

0.0

Radial(Slow Formation: Pierre Shale)

0.5 1.0 1.5 2.0v

~})0-

0-

li

---::::0.70~li

.li .--0.60

CT 0.50---------

li DAD

O"~u-: 0.20

r-O.10--........~------.V

0.090.

80.

70.

60.

50.

40.

30.

20.

10.

o

Figure 3: The radial and vertical components of fluid displacement at (r = rb' e= 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.

312 Peng et al.

Horizontal axis =

Pressure at the Center of Fluid(scaled by the pressure of incident P wave)

Frequency (KHz) Vertical axis = Incidence Angle (degree)

O.n-L--------------'

Amplitude(Fast Formation: Berea Sandstone)

0.5 1.0 1.5

Phase(Fast Formation: Berea Sandstone)

0.5 1.0 1.5 2.0I-'--..,......-'-I-~......-'-i-~,........--';-~..,......+90.0

80.0

70.0

60.0

I 50.0

.1 40.040.00 46.00 56( 64C

,'~~'~:0.0

0.00.090.O+-~~--'--'~""""T-'---'-~~';-'-~-'rI

80.

70.

Amplitude(Slow Formation: Pierre Shale)

0.5 1.0 1.5 2.0

4.00

Phase(Slow Formation: Pierre Shale)

0.5 1.0 1.5 2.0

\\

I 90.0

".0:2.00'6.\66 ~~:~60.0

50.0

40.0

80\ \ ~~:~'-'\ ,,~ 10.0

'--_-"-_---"_--''----'----'---'---'-.!L 0.0

0.00.090. rrJ--';-~~'-'-~..,.......L..._'_t~_'_7'~-t-'~Ti

80. \ \ ~76 0.\70. \ \

: 0.' .. O'\\::40. ""~30.

20.

10. ../ ____

O ~ '_--'---'-------<:::.........<:::......="""~:==:~::2:~~."

Figure 4: Pressure at the center of the fluid scaled by Po (see text), the pressure of theincident P wave, for hard (top) and soft (bottom) formations.


Pressure at center of fluid

Hard Formation (Berea Sandstone)

0.16

0.12

0.08

0.04

o0.04

0.08

0.12

0.16


Soft Formation (Pierre Shale)

0.8

0.6

0.4

0.2

o0.2

0.4

0.6

0.8

-100 Hz

- - -500 Hz_. -1000 Hz

·····2000 Hz

-100 Hz

- - -500 Hz_. -1000 Hz

..... 2000 Hz

Figure 5: Borehole reception pattern for pressure at the center of the fluid at frequencies100 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.

314 Peng et al.

Fluid side of borehole wall


1.2

0.9

0.6

0.3

o0.3

0.6

0.9

1.2

-100 Hz

- - -500 Hz_. -1000 Hz

..... 2000 Hz

-100 Hz

- - -500 Hz_. -1000 Hz

·····2000 Hz

"",..- .-'•.'•••

1.61.28

0.960.640.32

o0.32

0.64

0.961.28

1.6



Figure 6: Borehole reception pattern for fluid displacement at (r = r;, e = 0) atfrequencies 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.


-100 Hz

- - -500 Hz_. -1000 Hz

·····2000 Hz

Solid side of borehole wall

Hard Fonnation (Berea Sandstone)

1.080.864

0.648

0.432

0.216

o0.216

0.432

0.6480.864

1.08

Solid side of borehole wallSoft Formation (Pierre Shale)

1.41.225

1.050.875

0.70.525

0.350.175

o0.175

0.350.525

0.70.875

1.051.225

1.4

-100 Hz

- - -500 Hz_. -1000 Hz

·····2000 Hz

Figure 7: Borehole reception pattern for solid displacement at frequencies 100 Hz, 500Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. The incidentwave is a plane P wave with 1I = O. The geophone is at (r = rt,rJ = 0).

316 Peng et al.

-100 Hz

- - -500 Hz

-- -1000 Hz

-=,-.-,::oo~,....----""=":"-~...',",,'....... 2000 Hzt""Oi .>. -.. .,=_,. -

.. -II '.

·· ./ ~.l \. ~.· .· .····· .I. .. II ..

',' ',-0.99

0.992



1.002

0.998~';:1ll 0.996

atl 0.994

~

180O. 988 .L..L.l.......I..I.J............u........L1.I~.......:.u........u..........uJ

o 30 60 90 120 150Incidence angle (degree)



-- ./

•.."

1.02

0.98

>, 0.96-.;:.." 0.94atl 0.92"" 0.9

0.88

·....···.

...:. ...

. ---- .. ., .'... '

II --•, .. ... '

-100 Hz

- - -500 Hz

-- -1000 Hz..... 2000 Hz

0.86 ~"""":':'''''''.u.JLL.&''''''L.1.I~""",..............u..........uJo 30 60 90 120 150 180

Incidence angle (degree)

. Figure 8: Rectilinearity for the solid displacement at frequencies 100 Hz, 500 Hz, 1000Hz and 2000 Hz for hard (top) and soft (bottom) formations. The incident wave isa plane P wave with v = O. The geophone is at (r = ri,", e= 0).


30 60 90 120 150 180Incidence angle (degree)



4

0;-3

'"... 2on'":::,c0:c a'".;;:

'"'C - 1c0:c - 2'".5<:i - 3c- - 4a

.'.., . .It ..'..1 ".- ........ I.

: • .. II

-------~.

-100 Hz

- --500 Hz

---1000 Hz.•••• 2000 Hz

"I ... ------/:". .......... . ..II ........ ....,. .-. .. '. ., .. .. ., .

, .'' ..


Soft Formation (pierre Shale)1 a

0;-6'"...on

'":::,c 2

.51-;.;;:'" - 2'Cc

.51-;

- 6c:sc-- 1 aa

-100 Hz

- - -500 Hz_. -1000 Hz

..... 2000 Hz

.II" .. .... ..

, ..,it. - __ ".... •. .. --.... ..,-..--=- --~ 0'


Figure 9: Inclination deviation from the incident wave for the solid displacement atfrequencies 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 = rt, e= 0).

318 Peng et aJ.


Hard Fonnarion (Berea Sandstone)

-100 Hz

- - -500 Hz-- -1000 Hz••••• 2000 Hz

••• -:';';;0'-1-oO!-........•.

••···••·••••••...

".•• -. ;::'\0_......--

1.4

1.12

0.84

0.56

0.28

o0.28

0.56

0.84

1.12

1.4



1.8

1.51.2

0.90.6

0.3o

0.30.6

0.91.2

1.51.8

-100 Hz

- - -500 Hz-- -1000 Hz..... 2000 Hz

(

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. Thegeophone is at r= rtevaries from 00

- 3600•


. ---------,:--._--_.--:.. ".. . .. " ..


Hard Fonnation (Berea Sandstone)4

, .,,, .--- ---:----~

-100 Hz

- - -500 Hz-- -1000 Hz••••• 2000 Hz .,

.'",,,

, -, --t- ,

•"" ......

- 2 _1-J......--L-:-':---1--L-~.L....J......--L-':---1--L-.&...Jo 90 180 270 360

Geophone orientation (degree)

." .......,,,,

,

,i/'-':.

-100 Hz

- - -500 Hz-- -1000 Hz•.... 2000 Hz¥ 4 -'.

~ 3 \c , •

~ 2 ~

'" '/ '.~ -, \ . ..~- ., ,. . .-c "... ... ---- ....,:1:..... \'~ 0 E-"'-;';-;;;;';-::..o'rt!'..:;,,;........----~.:.. -=~:;.,.IiOO'=-:.::-;.;-=-

'".=~ - 1-

Solid side of borehole wallSoft Fonnation (Pierre Shale)

5

- 2 "-'......--"-:-'::.....-"""-:~-~"":-:!-:-'.....--"~o 90 180 270 360Geophone orientation (degree)

Figure 11: Effect of geophone orientation on the inclination deviation at frequencies 100Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. Theincident wave is a plane P wave with ij = 45 and v = O. The geophone is at r = rtevaries from 00 - 3600 .

320 Peng et al.

90 180 270 360Geophone orientation (degree)

........

-100 Hz

- - -500 Hz

-. -1000 Hz..... 2000 Hz

....... _----..,

,... .' .. '

. ..: .... .. ", .. ..' ",

I ....... _ ... __ ....... "0

Solid side of borehole wallHard Fonnation (Berea Sandstone)

8

6

"" 4...""":::. 2c

.51 0

.~" - 2'0.c:; - 4EON

- 6<- 8

0

90 180 270 360Geophone orientation (degree)

-100 Hz

---500 Hz

---1000 Hz••••• 2000 Hz

• ...-----... . .,

" .~ /!,. . ..'j,,/ !. ,, ., ,. ...

...,..

,...,..,..

.., .. .. .l ".;..- ........ . "

: / "0 ,.. "', "------ ..

" 1 0

"...""" 5:::.c

.51-; 0.;;:"'0.c - 5-"E·s - 1 0<

- 1 50



15

Figure 12: Effect of geophone orientation on the azimuth deviation at frequencies 100Hz, 500 Hz, 1000 Hz and 2000 Hz for hard (top) and soft (bottom) formations. Theincident wave is a plane P wave with (j = 45 and v = O. The geophone is at r = rt.evaries from 00

- 3600•


0.09

0.03

2.0

~~~,-!-,-~""""""'~,.....J,-.)\---""30236'-+-1-.. ::

0.1860.0

50.0

40.0

---1:.30.0

(

0,09 20.0

10.0

Ll....L.._.l.-_...L-L.....L......L...L...:::c.::::....,,;=:r. 0.0

Scattered Solid Displacement at the Borehole Wall(scaled by the total displacement of incident SV wave)


Radial VerticaI(Fast Fonnation: Berea Sandstone) (Fast Fonnation: Berea Sandstone)

0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.50.0

90'or-~~~========j80.

70.

60.

50.

40.

~~: jo.30 0.4010. / /0.50

~ 0.71o.f)"-----'--'--""-''---''''--''''-'''''-''='''''''=

10.0

0.18 20.00.06

2.0

~\\ \~\~}.oo 90.0

\ \ ~50 0.74 ------..J- ~~:~0.06 0.18 60.0

50.0

40.0'----I:

30.0

Vertical(Slow Fonnation: Pierre Shale)

0.0 0.5 1.0 1.52.0

1.60

0.06

Radial(Slow Fonnation: Pierre Shale)

0.5 1.0 1.50.0

90.or-~......J--.~~==~~=j

80.

70.

60.

50.

40.

30.

20.

10.

o. 0.0Figure 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)


80.0

70.0

60.0

50.0

40.0

30.0

0.40 20.00.51

10.00.60

0.800.0

2.0I--'-~""""~~.....J-~~--'-~~+ 90.0

1-----0.02---------[I-----O.OS---------{1-----0.08----------j:

1-----0.13---------+1-----===~0.17-===~1_ 0.20-

Vertical(Fast Formation: Berea Sandstone)

0.0 0.5 1.0 1.5

1.20

1.05

Radial(Fast Formation: Berea Sandstone)

0.5 1.0 1.5 2.0

j O.4.--------j

1 0.80-------j

1 ---~_0.90

0.0

90.0t=:==~~0~.0~32=====~===10.09

80.(}j·-----1o.18---------jn 1------10.30---------1

70.

60.

50.40.01 -,0.77

30.

20.

10.1.50O.rrL-_-C-__"<:::'_~=:':.::.--l

Radial Vertical(Slow Formation: Pierre Shale) (Slow Formation: Pierre Shale)

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.090. 90.0

80. 80.00.10

0.30 70.0

60.0-0.60

0.80

1.2650.0

40.0

~30.0

0.70 ~~'61 0.80 20.0~ 9.6' 0.60

1.03___ 14.24

~30 10.01.40 4.61

1.40 1.03 r.:::;;1 6O. 0.0

Figure 14: The radial and vertical components of fluid displacement at (r = rb,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.


Amplitude(Fast Fonnation: Berea Sandstone)

Horizontal axis =

Pressure at the Center of Fluid(scaled (see text), SV wave incidence)

Frequency (KHz) Vertical axis = Incidence Angle (degree)

Phase(Fast Fonnation: Berea Sandstone)

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.090. I I 90.0

0.01 I ,I I

80. I I 80.00.04 I ,

I ,70. I ,

70.0I,

I,

60.,

60.00.08 -'71.00 ,, ,, ,

50.,

50.0I -150.00I ,

0.10 , ,40. , , , , 40.0, , ,, , ,30. 0.t2 , , , ,

30.0,, , .,?:O.OO, ., , ,-20. . - 20.0, , -, ,, ,-, ,10.

, , 10.0•,",

O. 0.0

211.10

Phase(Slow Formation: Pierre Shale)

0.5 1.0 1.5 2.0f--'--,-;,~-J..~7"1~",","~~~...l-~~-'-t 90.0

I II I\ I 80.0\ ,, ,\ / 70.0-176.40- ~---t

60.0

50.0

40.0

30.0

20.0

10.0

~__---.2:~~~=::'::-J.0.0

0.02.0

Amplitude(Slow Formation: Pierre Shale)

0.5 1.0 1.5

20.O"j~~~

10.

0.0-'-''--''''==--''''-'''''''''''''''--='''''''-=---'

'1-------,0.,,--~-

0.090.n+-~~...J....~~...L.....~~'--'-~"""

80·\1]-------0.10-------,

Figure 15: Pressure at the center of the fluid scaled by Po (see text) for hard (top) andsoft (bottom) formations. The incident wave is a plane BV wave with II = O.

324 Peng et al.

Low Frequency Approximation of Tube Wave Dispersion

in Soft Formation (Pierre Shale)1000

950

"'§.900

~.gg 850.,'"'" 800.ca-

7500

\Shear wave

400 800 1200 1600 2000Frequency (Hz)

Resonance due to SV Incidence in Soft Formation90

80Q) 70~Cl 60.,~

50.,C,

40c

'"., 30uc., 20

'"·u 1 0.50

0

( Pierre Shale)

Exact Solution

Low Frequency Approximation

400 800 1200 1800 2000

Frequency (Hz)

Figure 16: (top) Frequency of tube wave phase velocity in the soft formation at lowfrequency approximation; (bottom) comparison of predicted resonance angle underlow frequency approximation with the exact solution.


Transverse Component of Displacement at Borehole Wall(scaled by the total displacement of incident SH wave)


Fluid(Fast Formation: Berea Sandstone)

0.5 1.0 1.5 2.0

0.87

1.00

1.12

Fluid(Slow Formation: Pierre Shale)

0.5 1.0 1.5 2.01-'-~.-..L~...............--'-;j~L...r-t;"-rl 90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

~::::::t 10.0L-_.-L..L._---l.l..LL:L:..-..L=-----l:. 0.0

0.0

1.031.01

0.090.O+-~-'-t-'--'-..~-'-t-~,..-...J~~"'-1

80.

70.

60.

50.

40.

30.

20.

10.

o.ni-_.L.--.L.-L...l...-...l...-...l...-L...L.!-L.L-.J

Solid(Slow Formation: Pierre Shale)

0.5 1.0 1.5 2.0h-t-t-r'"7'-+-...,.....,;--,........,...~-'--'........,...\-+ 90.0

1.00 80.0

70.0

60.0

50.0

40.0

30.0

20.0

10.0

L..J-.l.~(L~~~@5:~'__~~O~.6~. i- 0.0

0.02.0

Solid(Fast Formation: Berea Sandstone)

0.5 1.0 1.50.090.o-h-.,....,....,......,......,.....,.......,....-;-';-....,.....,J-"';~-H

80.

70.

60.

50.

40.

30.

20.

10.

O.o-L----L--L.-"""=---"':........<:::..,,:::=~

Figure 17: Transverse components of fluid and solid displacement (scattered) scaled bythe total displacement of the incident wave at (r = rb, e= 0). The incident wave isa plane SH wave with l/ = O.

326 Peng et aJ.



0.12

0.09

0.06

0.03

o0.03

0.06

0.09

0.12

......' .•

:•........4"/. .....

-100 Hz

- - -500 Hz_. -1000 Hz

••••• 2000 Hz



1

0.8

0.6

0.40.2

o0.2

0.40.6

0.8

1

-100 Hz

- - -500 Hz

-- -1000 Hz..... 2000 Hz

Figure 18: The borehole reception pattern for pressure at the center of the fluid atfrequencies 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.




21.61.2

0.80.4

o0.4

0.81.21.6

2

-100 Hz

- - -500 Hz-- -1000 Hz

·····2000 Hz


Soft Fonnation (Pierre Shale)

4

3

2

1

o1

2

3

4

-100 Hz

- - -500 Hz

-- -1000 Hz

·····2000 Hz

Figure 19: The borehole reception pattern for fluid displacement at (r = r;;,e = 0) atfrequencies 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.



21.6

1.2

0.80.4

o0.4

0.81.2

1.6

2

.•••·

-100 Hz

- - -500 Hz_. -1000 Hz

..... 2000 Hz

Solid side of borehole wallSoft Formation (Pierre Shale)

21.6

1.2

0.80.4

o0.4

0.81.2

1.6

2

Ii1/

I I-: ... ,, :.

•· .,.;.:.\ ,"\;

-100 Hz

- - -500 Hz

-- -1000 Hz

..... 2000 Hz

,,

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. Theincident wave is a plane SV wave with v = O. The geophone is at (r = rt,li = 0).


-100 Hz

- - -500 Hz_. -1000 Hz

.. "". 2000 Hz


Hard Fonnarion (Berea Sandstone)

1.5

1.2

0.9

0.60.3

o0.30.60.91.2

1.5

Fluid side of horehole wall

Soft Fonnarion (Pierre Shale)

1.41.2

10.80.60.40.2

o0.20.40.60.8

11.21.4

,...-..

,-"""" " ,,

,,•••,,,·,:,,,,..,,,,,,

......,

-100 Hz

- - -500 Hz_. -1000 Hz

".. ··2000 Hz

,",· ,· ,, .· ,•••,,"

•·

Figure 21: The borehole reception pattern for fluid displacement at (r = rb", e= 0) atfrequencies 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.

330 Peng et al.



1.251

0.75

0.5

0.25

o0.25

0.5

0.751

1.25

-100 Hz

- - -500 Hz-. -1000 Hz••••• 2000 Hz



1.4

1.05

0.7

0.35

o0.35

0.7

1.05

1.4

.'.: -.· .• ••~ ...•.

•

·•

-100 Hz

- - -500 Hz-. -1000 Hz••••• 2000 Hz

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. Theincident wave is a plane SH wave with v = O. The geophone is at (r = rt,rJ = 0).



Hard Fonnation (BeI'ea Sandstone)1.1

-100 Hz

- - -500 Hz_. -1000 Hz

••••• 2000 Hz

••• 'I.

----;.. ..•_.".t>.;:

!l.5 0.9::"g<:>:::

0.8

......

. .......-----. ....... -----....

'. , , ....

O. 7 0!-'-.....,u3.l0~.........61.J0.....,u..9.L0............1...21.J0L.l-lu..1.L5...0.........1L.18 0



Soft Fonnation (pierre Shale)1 .1

., ", ..-0·1 .,' It.~ .

~ : \ /. ...".: -. r.. .. I, ..'../ I I

1 II II JI JI J

"

-100 Hz

- - -500 Hz

---1000 Hz..... 2000 Hz

,...----.........----=--,..• •

'{, ... J .'-.1 , ;.,. I .

.. 1\ • ~ •~. 1 ,J : ...,: \ ."

I I ...:I 1I II ,I ,I 1

"0.7

.t>.;: 0.9....S~ 0.8<:>:::

O. 6 ~""""':1:".L.LJL.L.I................Lu........L1.&......u.L.L.LJu.a.Jo 30 60 90 120 150 180


Figure 23: Rectilinearity for the solid displacement at frequencies 100 Hz, 500 Hz, 1000Hz and 2000 Hz for hard (top) and soft (bottom) formations. The incident wave isa plane SV wave with v = O. The geophone is at (r = rt, e= 0).

332 Peng et aI.

.."·····



4

-100 Hz

- - -500 Hz_. -1000 Hz

3 .. ···2000 Hz12 :!~ ....c •.: ~"'" .'--<;j 0 E-....-::"=':~...r.-"J!lII--;..'-"-.."...;-::.;-::;._...-.~ -- .."--'-r--: - 1 ••••-o •'';-2 ... .-.:c::i - 3 •••c- - 4 0~L.L.l.....3~0.......L"':5L;0L.L.l.....9~0.........1...2LlO..........1~5~0.L"':..1u8 0


Solid side of borehole wallSoft Fonnation (Pierre Shale)

200

-100 Hz

- - -500 Hz

-- -1000 Hz..... 2000 Hz...

"..". ................

'.· "···········--.o •'-:~.. :.. :

-. :. :.. :" .. ....:

150~S;, 100

'"::::,50

.§<;j

't"0 .50co.~. 1 00.:<:i. 1 50c-- 20 00UJ ....


Figure 24: Inclination deviation from the incident wave for the solid displacement atfrequencies 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).



Han! Fonnation (Berea Sandstone)-100 Hz

- - -500 Hz_. -1000 Hz

••••. 2000 Hz.'...."';.;.:.:'"j~.' ---.- ..

•.·•··•·····• ••••....'....."'" "1

1.4

1.12

0.84

0.56

0.28

o0.28

0.56

0.84

1. 12

1.4


Solid side of borehole wall -100 Hz

- - -500 Hz_. -1000 Hz

• •••• 2000 Hz.-....... ........_-..--,,- ........• •

':1••f/.,:t•..~\.,...\. ,-..,

. .....-4--#......-. --' ........

1.4

1.12

0.84

0.56

0.28

o0.28

0.56

0.84

1.12

1.4

Figure 25: The effect of geophone orientation on the borehole reception pattern atfrequencies 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. Thegeophone is at r= rt- () varies from 00

- 3600•

334 Peng et al.



1.25

1

0.75

0.5

0.25

o0.25

0.5

0.75

-100 Hz

- - -500 Hz_. -1000 Hz

.........+...... ..... 2000 Hz. ....•.

••....

·• ••••..•.

•.'. --.-.........~'"......... .....-

1.25



1.5

1 . 2

0.9

0.6

0.3

o0.3

0.6

0.9

1 . 2

1 .5

-100 Hz

- - -500 Hz-. -1000 Hz

• •.... 2000 Hz., -.~~:;'=-""":'. ".

•\•\ .••"-fl., _ _",,"'--. --... . .--.. ...

a ••••_

Figure 26: The effect of geophone orientation on the borehole reception pattern atfrequencies 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. Thegeophone is at r= rit- e varies from 0° - 360°.




1 2

'.-

-100 Hz

- - -500 Hz_. -1000 Hz

••••• 2000 Hz

-

.". .. ., .· .· .· ., ., ., .· .· '· .· ', ., .. ', ., ., .· .· '· '· .... .. ..: a,.: ", .: ",0" .~: ",--., ':.~_ •· - ." - ... .- -- ...

~ 1 0

~~ 8~

.§ 6..0;: 4'"'t:l

=.51 2...5 0<:i=- - 2 o 90 180 270

Geophone orientation (degree)360

360

-100 Hz

- - -500 Hz_. -1000 Hz

..... 2000 Hz.'.-

10-.,.'.

.-.. ."}_..:.::-:.:..........:;:-i1 _

,."

•

,,.,::,

··. '-oJ

..·····...:,,,., ...~.

- 7 0 !--''-'--'--:-l:--'--'-......L...'-'--'-...J..-'--'-.a.....Jo 90 180 270




20

~ 2

~15 - 1 6'Z

"•S'~ -34="'Z".5 - 5 2<:i=-

Figure 27: The effect of geophone orientation on the inclination deviation at frequencies100 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 atr = rt. 0 varies from 00

- 3600•

336 Peng et al.

90 1·80 270Geophone orientation (degree)


Hard Formation (Berea Sandstone)1 2

-100 Hz

- - -500 Hz

-- -1000 Hz• •••• 2000 Hz

360

o

o

··•..o

0'_ •

",," ----...

o.o

...oo

oo ....-- ...............

·····.....4

8

c~ 0'".~'0 -4.c

~oN - 8<:

- 1 2o

'00

-100 Hz

- - -500 Hz-- -1000 Hz..... 2000 Hz



11 0 ~

•:~....o 0

~ . "~ 55 : ...~ : -".

::=, : .. ,--:.:-:- ..c • / ••:"-.~ 0" __ .&,,0 ..

.~ ..•. -....;... /....".: "t .. =-:''''0 •.c ~

== -5 5 "E 00

.~ "< ~

- 1 1 00~L.-JL.-J.....JL....I---'---'-:-!-":"",,,-I.-I.':'2~70::-'-.........&.:3~6090 180


Figure 28: The effect of geophone orientation on the azimuth deviation at frequencies100 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 atr = rt. 0 varies from 0° - 360°.

Documents

BOREHOLE EFFECTS ON DOWNHOLE SEISMIC MEASUREMENTS