ACOUSTIC LOGGING GUIDED WAVES IN TRANSVERSELY ISOTROPIC FORMATIONS

Transcription

1 ACOUSTIC LOGGING GUIDED WAVES IN TRANSVERSELY ISOTROPIC FORMATIONS by K. J. Ellefsen, C. H. Cheng, and D.P. Schmitt" Earth Resurces Labratry Department f Earth, Atmspheric, and Planetary Sciences Massachusetts Institute f Technlgy Cambridge, MA ABSTRACT We have studied the velcity dispersin f guided waves in transversely istrpic frmatins. Theretical velcity dispersin curves were calculated with elastic cnstants based n labratry and field measurements and cmpared t dispersin curves fr istrpic frmatins having the same vertical p and S-wave velcities. The symmetry axis fr the transverse istrpy was parallel t the brehle. The differences between the phase velcities fr the transversely istrpic and istrpic frmatins depend n the type f wave, its frequency, and the amunt f anistrpy, and can be as high as 7 t percent. The changes in the phase velcity due t changes in the elastic cnstants f the frmatin (cn, C13, C33, C44, and C66) and the bulk mdulus f the brehle fluid (A) vary with frequency. In a hard frmatin, the tube wave's velcity is sensitive t C66 at lw frequencies, t C44 at high frequencies, and t A at all frequencies. The pseud Rayleigh wave is affected by C44 near its cutff frequency and by A at high frequencies. The flexural wave, which is generated by a shear wave lgging tl, is similarly affected by C44 at lw frequencies and by A at high frequencies. As the frmatin becmes sft, the effect f the elastic cnstants upn the phase velcity gradually changes. Like a hard frmatin, the tube wave's velcities in a mderately sft frmatin are primarily affected by C66 and A at lw frequencies, but the influence f C44 is much greater at high frequencies. Since array prcessing methds can accurately estimate the velcity dispersin f the guided waves ver a wide range f frequencies, sme elastic cnstants can be estimated. In a hard frmatin, the refracted P- and S-wave velcities uniquely determine C33 and C44, and an inversin can be used t estimate C66 and A. In a mderately sft frmatin, the refracted P-wave velcity determines C33, the flexural wave frm the 1 Nw at Etudes et Prductins Schlumberger, 26 Rue de la Cavee, Clamart, France

2 76 Ellefsen et al. shear wave lgging tl determines C44, and the tube wave's velcity dispersin can be used t estimate C66 and 'x. INTRODUCTION Knwledge f the mechanical prperties f a frmatin is useful in several areas f hydrcarbn explratin and prductin. In the cntext f elastic wave prpagatin, the mechanical prperties are described by elastic cnstants. These cnstants are currently used t estimate a frmatin's strength, which is imprtant infrmatin in fracturing and preventing cllapse during prductin, and t tie acustic lgging and seismic data, which is necessary fr reservir characterizatin and stratigraphic analysis away frm the brehle. The refracted P- and S-waves frm acustic lgging data are used frequently t determine the bulk and shear mduli, which adequately describe the dynamic behavir f istrpic rcks. Hwever, when a rck is transversely istrpic, which is quite cmmn fr many sedimentary rcks (Thmsen, 1986), the infrmatin frm these tw waves just is nt enugh t cmpletely characterize the mechanical prperties. T vercme this prblem, the guided waves might be used. By applying array prcessing methds t data cllected frm multi-receiver tls, accurate velcity dispersin curves fr the guided waves can be estimated. The imprtant questin is whether the dispersin curves fr the guided waves can be used t estimate the elastic cnstants f a transversely istrpic frmatin. Previus authrs had cnducted numerical studies t determine the effects f a transversely istrpic frmatin upn the acustic lgging wavefrms. White and Tngtaw (1981) studied the waves frm mnple and diple surces and determined hw the frmatin affects the velcities and amplitudes f the refracted waves. They als determined what elastic cnstants affect the tube wave at zer frequency and the pseud-rayleigh wave at its cutff frequency and suggested that this infrmatin culd be used t help determine the elastic cnstants. Chan and Tsang (1983) investigated acustic lgging in radially layered, transversely istrpic frmatins, and their wrk fcused n the refracted waves as White and Tngtaw's (1981) did. The purpse f this paper is t shw hw the phase velcities f the guided waves are affected by the mechanical prperties f transversely istrpic frmatins. Their effects upn the dispersin curves are emphasized by cmparing the curves t thse fr istrpic frmatins having the same vertical P- and S-wave velcities. Nrmalized partial derivatives f the phase velcity with respect t each elastic cnstant are calculated ver a brad range f frequencies. These derivatives shw the relatinship between the dispersin curves and the elastic cnstants and suggest which cnstants can be estimated. The analysis was cnducted fr a hard frmatin and a sft frma-

3 Guided Waves in Anistrpic Frmatins 77 tin, which have typical velcity dispersin curves. METHOD The brehle mdel used t study the prblem f acustic lgging in a transversely istpic medium cnsists f a fluid-filled, infinitely lng, cylindrical hle in an infinite frmatin (Figure 1). The fluid is described by its bulk mdulus (.x) and density (PJ), an istrpic frmatin by tw elastic cnstants (C33 and C44) and density (p), and a transversely istrpic frmatin by five elastic cnstants (en, CI3, C33, C44, and C66) and density (p). The axis f symmetry fr the transverse istrpy is parallel t the brehle. The fluid and frmatin are assumed t be hmgeneus and linearly elastic. The effect f the lgging tl will be ignred t simplify the analysis. Relating the elastic cnstants t varius velcities will help us study the velcity dispersins f the guided waves. In an istrpic frmatin, the P-wave velcity and cnsequently the refracted P-wave velcity are given by vc;,:;rp. Similarly the S-wave velcity and the refracted S-wave velcity are JC44/ p. In a transversely istrpic frmatin, the relatinship between the velcities and the elastic cnstants becmes quite cmplicated (see e.g., White, 1983), but in sme special directins the mathematical relatins simplify. Fr example, the velcities fr vertically prpagating (parallel t the axis f symmetry) P- and S-waves are JC33/ p and JC44/ p, respectively. These velcities als apply t the refracted P- and S-waves in the acustic lgging situatin (White and Tngtaw, 1981). In the hrizntal directin, the P-wave velcity is..;c;;rp, and the S-wave velcity is either JC44/ p r JC661 p depending upn its plarizatin. An imprtant issue is the extent t which the velcity dispersin curves fr the guided waves wuld change if a transversely istrpic frmatin is incrrectly assumed t be istrpic. The elastic mduli in istrpic rcks are determined by the refracted P- and S-waves in hard frmatins and by the refracted P-wave and flexural wave in sft frmatins. In cases where the shear wave lgging tl is nt available and thus there is n flexural wave measurement, frmatin shear wave velcities are smetime btained indirectly frm the tube wave velcities (Chen and Willen, 1984; Stevens and Day, 1986; Cheng and Tksz 1983). With nly this infrmatin, distinguishing between the istrpic and transversely istrpic frmatins is nt pssible. Therefre, an equivalent istrpic frmatin was defined as having the same vertical P- and S wave velcities as the transversely istrpic frmatin. The phase velcity fr guided waves in this equivalent frmatin was calculated using the perid equatin, and the grup velcity using Rayleigh's principle. The phase and grup velcities were calculated ver a brad frequency range fr tube, pseud-rayleigh, flexural, and screw (quadruple) waves in hard and sft, transversely istrpic frmatins and are cmpared t the velcities fr the equivalent

4 78 Ellefsen et al. istrpic frmatins. The hard and sft frmatins are the Green River shale and shale (5000), respectively, fr which the prperties are tabulated by Thmsen (1986). These tw examples were chsen because their velcities exhibit the general characteristics f many hard and sft transversely istrpic frmatins. The effects which the elastic cnstants have upn guided wave phase velcity are shwn by the partial derivatives f velcity with respect t the elastic cnstants. The derivatives are cmputed at cnstant frequency with Rayleigh's principle t avid numerical differentiatin. After nrmalizatin, the derivatives, which will be called sensitivities, have the frm ':; t~ in which c is the phase velcity and m is any elastic cnstant. The sensitivities can be interpreted as the percent change in phase velcity due t a ne percent change in the elastic cnstants. The sensitivities are cmputed fr the hard and sft, transversely istrpic frmatins. RESULTS AND DISCUSSIONS Hard Frmatin The phase and grup velcities fr the guided waves in a transversely istrpic, hard frmatin and an equivalent istrpic frmatin are shwn in Figure 2. In general, the shapes f the tw sets f curves are very similar. The phase and grup velcities f the guided waves in the transversely istrpic frmatin are generally higher than thse fr the equivalent istrpic frmatin, the nly exceptin being the grup velcity f the high frequency pseud-rayleigh wave. The cnsistently higher velcities are caused by the higher rigidity f the transversely istrpic frmatin in the hrizntal directin. The sensitivities fr the guided waves are shwn in Figure 3. Fr the tube wave, the effects f.x and C66 n the phase velcity are imprtant at lw frequencies but diminish as the frequency increases. Althugh the effect f C44 is negligible at lw frequencies, it increases at higher frequencies. By cmparisn, in an istrpic frmatin the effect f C44 is imprtant at lw frequencies and decreases with increasing frequency (Cheng et ai., 1982). Elastic cnstants Cn, C13, and C33 have little influence upn the tube wave's phase velcity. This behavir is similar t the istrpic case in which the frmatin P-wave velcity has little effect n the guided waves. The phase velcities f the pseud-rayleigh, flexural, and screw waves are entirely cntrlled by C44 near their cutff frequencies. This relatinship is expected because the velcity f the refracted S wave, assciated with these guided waves at their cutff frequencies, is determined by C44. As the frequency increases, the influence f C44 diminishes and that f.x increases, the latter becming dminant at high frequencies. The phase velcities are virtually unaffected by the ther elastic cnstants. (

5 Guided Waves in Anistrpic Frmatins 79 Sft Frmatin The phase and grup velcites fr the guided waves in the transversely istrpic, sft frmatin and the equivalent istrpic frmatin are shwn in Figure 4. As in the case f the hard frmatin the shapes f the tw sets f curves are similar, and the velcities in the transversely istrpic frmatin are higher than thse fr the equivalent istrpic frmatin. The differences in the tw sets f velcities are larger than thse in the hard frmatin because the guided waves in a sft frmatin are mre sensitive t the frmatin shear mdulus (rigidity). The sensitivities are shwn in Figure 5. Fr the tube wave, changes in A and C66 greatly affect the phase velcity at lw frequencies, but their influence decreases as frequency increases. The effects f C44 are exactly the ppsite, eventually dminating at high frequencies. The influence f Cn, C13, and C33 n the phase velcity are again quite small. Near the cutff frequencies fr the flexural and screw waves, the phase velcities are cntrlled by C44. As the frequency increases, the effect f C44 diminishes and that f A increases. The ther elastic cnstants have little effect n the phase velcities f these guided waves Similarities exist in the behavir f the guided waves in the hard and sft frmatins. In bth frmatins, the phase velcities are mstly affected by the bulk mdulus f the brehle fluid, A, and the tw frmatin shear mduli, C44 and C66. The ther elastic cnstants, thse mainly assciated with P-wave prpagatin, have little influence ver the guided waves. At high frequencies the tube, flexural and screw waves have similar phase and grup velcities and sensitivities because they behave like a Stneley wave alng a planar, fluid-slid interface. The differences in phase velcities between the transversely istrpic and equivalent istrpic frmatins in ur examples are smetimes as large as percent. Differences f this magnitude generally ccur when the velcity changes between vertically and hrizntally prpagating P- and S-waves are abut fifteen percent. Thmsen's (1986) tabulatin f the elastic cnstants fr sedimentary rcks shws that a large minrity have velcity variatins f this size. Because the phase velcity differences are large enugh t be detected with array prcessing f multi-receiver data, sme f the elastic cnstant might be estimated. In a hard frmatin, the refracted P- and S-waves uniquely determine C33 and C44. The remaining elastic cnstants culd be estimated by inverting the phase velcities at many frequencies. Since the phase velcities f the guided waves are primarily sensitive t A and C66, the inversin wuld nly reslve these tw cnstants well. The sensitivities fr the flexural and screw waves appear similar t that fr the pseud-rayleigh wave tentatively suggesting that the additinal data prvided by these tw waves wuld nt be very helpful in the inversin. In a sft frmatin, the refracted P wave determines

6 80 Ellefsen et al. C33 and a shear lgging tl with its flexural wave wuld establish C44. curves fr the tube wave culd be inverted fr.x and C66. The dispersin CONCLUSIONS I'Ve have studied the prpagatin f tube, pseud-rayleigh, flexural and screw waves in a brehle in a transversely istrpic frmatin. We have als studied the sensitivities f these guided waves t the different elastic cnstants and cmpared them with thse t an istrpic frmatin. It was fund that the tube wave is mre sensitive t the hrizntal shear mdulus (C66) at lw frequencies whereas the psued-rayleigh, flexural and screw waves are sensitive t the vertical shear mdulus (C44), at least near their cutff frequencies. A cmbined inversin f the phase velcity dispersins f tw r mre f these guided waves will allw us t determine bth C44 and C66, and thus the degree f shear wave anistrpy. By the same tken, frmatin shear wave velcities btained indirectly frm tube wave velcities assuming istrpy culd be significantly higher than thse measured by a shear wave tl in a transversely istrpic frmatin-. The degree f P-wave anistrpy cannt be determined frm the phase velcities f these guided waves. ACKNOWLEDGEMENTS This research was supprted by the Full Wavefrm Acustic Lgging Cnsrtium at M.LT. Denis P. Schmitt was supprted by a Funding Members Fellwship at the Earth Resurces Labratry.

7 Guided Waves in Anistrpic Frmatins 81 REFERENCES Chan, A. K. and Tsang, 1., Prpagatin f acustic waves in a fluid-filled brehle surrunded by a cncentrically layered transversely istrpic frmatin, J. Acust. Sc. Am., 74, ,1983. Chen, S.T., and Willen, D.E., Shear wave lgging III slw frmatins; SPWLA Ann. Lgging Symp., Paper DD, Trans. 25th Cheng, C.H., and Tksz, M.N., Determinatin f shear wave velcities in "slw" frmatins; Trans. 24th SPWLA Ann. Lgging Symp., Paper V, Cheng, C.H., Tksz, M.N., and Willis, M.E., Determinatin f in situ attenuatin frm full wavefrm acustic lgs, J. Gephys. Res., 87, , Stevens, J.L., and Day, S.M., Shear velcity lgging in slw frmatins using the Stneley wave; Gephysics, 51, ,1986. Thmsen, 1., Weak elastic anistrpy, Gephysics, 51, , White, J. E., Undergrund Sund, Elsevier Science Publishing Cmpany, Inc., White, J. E. and Tngtaw, C., Cylindrical waves in transversely istrpic media, J. Acust. Sc. Am., 70, , 1981.

8 82 Ellefsen et al. ISOTROPIC FORMATION: ELASTIC MODULI C33, C44 DENSITY P OR TRANSVERSELY ISOTROPIC FORMATION: ELASTIC MODULI C11, C13, C33, C44, C66 DENSITY P BOREHOLE FLUID: BULK MODULUS " DENSITY P Figure 1: Brehle mdel used t study the guided waves.

9 Guided Waves in Anistrpic Frmatins 83 U PHASE VelOCITY rube WAVE 1., GROUP VELOCrTY TUBE WAVE I i: 1.5 '.5 ~ ~ > " '0 1., -... U '... 1; " \ \! " -'-,\ ~ > ~ " w > I PSEUDO-RAYLEIGH WAllE PSEUDO-RAYLEIGH WAVE '0,., FLEXURAL WAVE U URAL WAve \ \\ \, i: '.5 ~ " >, _ '0 1.,,, SCAEWWAV 1.8 \, i! \,! \, \ \ >,,, 1.S \..._---- \ ~ \ '!I '.5 '~.._...-,--- SCREW WAVE '0 0 FREQUENCY (khz) FREOUENCY (khz) Figure 2: Phase and grup velcities fr the tube, pseud-rayleigh, flexural, and screw waves in the transversely istrpic, hard frmatin (slid lines) and the equivalent istrpic frmatin (dtted lines).

10 ( 84 Ellefsen et al. 0., "mlil!U=n,ua" WAv" 0.50r-----nT.!"'="'" TUB" WAvE ~ ""--"""'/;l!;u""''''''''yl"''"'''h'''w'''a'''"'' ' r-----'f"'lecvxural.;;a.""w"'a"vn!e" 0.00.j ,..I:~~=-= , ,F""LEX""'U"""-,,W"Awv"'e' ;: O.O! ;; -in ~ l 0.00L...t.:::::;:::=~::!:::!d 0., ,s"CR'E'i!eu:W"W"'A"ven!"l ;;---..,sc"""R"eW""'W"'A"v"'e' ~ FReCUENCY IkHzl 0.00L -le.:::::~~~~ FREQUENCY (khz) ( --0- Cl' * C,3 C33 ~ C44 C66 ali " Figure 3: Sensitivities fr the tube, pseud-rayleigh, flexural, and screw waves in the transversely istrpic, hard frmatin. Because the sensitivities due t C44, C66, and A are generally much larger than the ther sensitivities, they are gruped tgether and are pltted at a different scale. I.

11 Guided Waves in Anistrpic Frmatins PHASE VELOCITY GROUP velocity TUB.W.VE"" 1.5,ue_W.VE :.-!, '" ~ :> 1.1,., 0 '0 0 '0 1.5,, LEX W. '.5,, ~! \'",. 1.3, \\' 'lexu...w.ve '" " '.3 \ ~ , w \ :> '-... " ", '0 0 I 1.5 " " /: 1,3 " ~ ""'~W'VE..._ ,5 SCAEWWAVE ',3,, \, \,,, , '0 0 '0 FRECUENCY (khz) FREOUENCY (khz) Figure 4: Phase and grup velcities fr the tube, flexural, and screw waves in the transversely istrpic, sft frmatin (slid lines) and the equivalent istrpic frmatin (dtted lines),

12 86 Ellefsen et al. 0, rub.wav , ruiiii!BiE>WWA"V~ > ~ 0.05 ie ~ 0.00 w "' ,0 F X WA.tz::::::=:~ 0.50 l-;;---"'if"l.ex"'uirlu"iiwwa"v~ > ~ 0.05 i< 0.00 i > 0.05 ~ ~ Z 0.00 :Jj '0 ~ p- r SCREW WAVE 0.50 ""'---'::"'"--"'SC"""R"'EW""'W"'A"'V"'"'E.0.05,0 FREQUENCY (khz) '0 FREQUENCY (khz:) ~ C11 * C13 C33 ~ C44 C66 ta A. Figure 5: Sensitivites fr the tube, flexural, and screw waves III the transversely istrpic, sft frmatin.