Calculation of Constitutive Parameters from Electric and Magnetic Field Measurements in an Anisotropic Medium with a Triaxial Instrument

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1 Intenatonal Jounal of Geoscences, 203, 4, ttp://d.do.og/0.4236/jg Publsed Onlne Janua 203 (ttp:// Calculaton of Consttute Paametes fom lectc agnetc Feld easuements n an Ansotopc edum wt a Taal Instument tan Pekşen Depatment of Geopscal ngneeng, Kocael Unest, Kocael, Tuke mal: etanpeksen@kocael.edu.t Receed August 8, 202; esed Septembe 24, 202; accepted Octobe 23, 202 ABSTRACT A potetcal electc magnetc nducton tenso s consdeed n an ansotopc medum. Te souces ae magnetc dpoles. In suc a medum, consttute paametes can be calculated b combnng electc magnetc feld measuements. Consttute paametes ae not a scala n ts case. Te ae tensos, so paametes ae at least bot oontal etcal components n a unaal medum. Tese calculated paametes fom te feld measuement ae oontal etcal conductt, pemttt, magnetc pemeablt. Opeatng fequenc ange s also qute lage. It s up to 4 G. A potetcal nstument sould measue gadent felds bot electc magnetc tpes as well. Kewods: lectcal Ansotop; Taal easuement; lectc agnetc Dpoles; Inducton Well Loggng. Intoducton Conductt, pemttt, pemeablt of a medum ae known consttute paametes []. Tese paametes ae used to ntepet te subsuface wt dffeent geopscal metods well-loggng tpes. Relate magntudes of tese paametes ae qute dstnct as e well-known popetes of ocks [2-5]. Snce tese paametes seldom a fo most ocks mneals, most scentsts geneall assume delectc pemttt magnetc pemeablt of te medum as fee space fo conenence. In an sotopc medum, consttute paametes ae te same n all dectons; tus, te ae scalas n tat case. owee, n an ansotopc medum tese paametes depend on te dectons, so te must at least be a ecto. In te pesent stud, an ansotopc medum s consdeed; tus, te detemnaton of ts knd medum eques fe paametes n a etcal well. Tese ae oontal etcal conducttes, pemtttes, magnetc pemeablt. Ts knd of ansotop s known as a unaal ansotop. In a Catesan coodnate sstem, consttute paametes ae te same n te dectons, but te ae dffeent n te decton. If conductt alues ae dffeent n te, dectons, ts sot of medum s called as a baal medum. In nducton well loggng, te electc ansotop affects feld measuements. Klen et al. [6] ealed ts effect te stated tat te possble eason fo g eadng could be te electcal ansotop. Obousl, te effects of electcal ansotop make ntepetaton eoneous. To deal wt ts effect elmnate fom te data, te electc ansotop s consdeed. In a wellboe, te feld s also affected fom a elate deaton angle. Tese effects can be ncluded/etacted to/fom te felds b otatng te ule s otaton mat. Zdano et al. [7] studed te tenso well loggng te genealed Doll s dea fo an ansotopc medum. Te sowed tat fom te feld components one can calculate not onl conductt alues (oontal etcal) but also elate deaton beang angles. Te used magna components of magnetc felds. Zang et al. [8] sowed tat usng feld measuements one can detemne elate angles suc as elate dp elate beang as well. In geneal, tee ae tee ndependent coodnate sstems: a well, eat, nstument coodnate sstems. In ts stud, I consde a wole space (a unaal medum) nstument as ae concded wt eac ote s fo te smplct. an scentst [9-3] studed unaal meda due to ts smplct. Tee ae been a few autos studed a baal ansotopc medum, wc can be caacteed b ts conductt n eac dffeent dectons n a Catesan coodnate sstem [4-6]. A e few autos ae studed electc feld, electc magnetc feld measuement togete. Gbenko Zdano [7] consdeed combnaton of electc magnetc felds. Te stated tat te combnaton of te electc magnetc feld measuement ges Copgt 203 ScRes.

2 . PKŞN 59 bette esults wt espect to conductt dstbuton of a well. ee, te dea of combnaton of electc magnetc feld can be etended to an nducton loggng poblem n an ansotopc medum. In ts stud, I combne electc magnetc felds n an ansotopc wole space. Ts s a potetcal nstument. I demonstate tat consttute paametes can be calculated b combnng electc magnetc feld measuement n an ansotopc medum. Paametes ae etcal oontal conducttes, delectc pemtttes, magnetc pemeablt. Tese paametes caactee te feld beaos n a coespondng medum. In geneal, ou am s to measue feld components estmate tose paametes, wc caactee te feld beaos. In te pesent stud, nete elate dp no beang s consdeed n te medum. In geneal, ts potetcal nstument sould measue full electc magnetc feld components te gadent n a well. Zdano [8] gae a defnton of a gadent tpe measuement n an ansotopc medum fo magnetc feld. Te same dea can also be etended to electc feld measuement as well. 2. awell s quatons n an Ansotopc edum oan Ganeo [] studed nducton well loggng n a unaal ansotopc medum. awell s equatons ae gen n suc a medum as: σˆ, (), (2) 0 0 wee s te electc feld (V/m), s te magnetc feld (A/m). s a magnetc dpole moment (Am 2 ). Te conductt tenso n a tansesel sotopc (TI) medum s: 0 0 ˆ 0 0, 0 0 wee s te oontal component s te etcal component of conductt tenso. ae te oontal etcal pemttt of te medum, espectel. Te fee space delectc pemttt s F/m. Te fee space magnetc pemea- 7 blt s μ0 4π 0 /m. Te tansese sotopc medum s ected toug an electomagnetc feld geneated b magnetc dpoles wt unt moments. Te tme depen- t dence s e. f s te fequenc of a souce n et. oan Ganeo [] soled quatons () (2) b usng te et potental. In ts pape, I assume tat te delectc pemttt magnetc pemeablt of te medum ae not fee space. In ts stuaton, electc magnetc feld components can be wtten: (3) ˆ σ π, (4) n m n m σˆ π, (5) wee π s a et ecto s a scala potental. Te medum delectc pemttt magnetc pe meablt ae 0, 0 0, espectel. st fo te elate delectc pemttt, epesents a elate magnetc pemeablt. Supe subscpts depct a magnetc moment ecee decton along te coespondng as; tus, n m can be,,, n quatons (4) (5). In ts pape, I use as te followng notaton: Bold face smbols o caactes wt a at st fo tensos, wle bold face smbols o caactes wtout a at llustate ectos. An caactes wt talc ae a scala. Teefoe, calculatons nole some dad algeba suc as dadcecto dot poduct n quatons (4) (5). Net two sectons nestgate electc magnetc feld components. 3. lectc Inducton Tenso lectc nducton tenso as 9 components n a geneal case. Tee otogonal ecees tansmttes ae elongated wt te coespondng as n a Catesan coodnate sstem. Te souces ae magnetc dpoles ecees ae electomagnetc senso o cols. To dee te electc feld components s a stagtfowad calculaton tat wll not be epeated ee. Reade can fnd all electc components fom [9-2]. Te electc nducton tenso can be epesented n a mat fom as: ˆ, (6) wee supescpts ae magnetc dpole oentatons subscpts ae ecee oentatons. Bea n mnd tat te souces ae magnetc dpoles. Fgue dsplas a potetcal electc magnetc nducton tenso. Fgue 2 sows te beao of electc feld components n a - plane. Fo feld smulaton, te oontal etcal conducttes ae S/m, espectel. Te opeatng fequenc s 20 k. Te feld computaton s conducted wt fee space delectc pemeablt magnetc pemeablt. s a magnetc dpole wt a unt. One can eale tat te smmetcal felds ae te same ampltude, but dffeent sngs. 4. agnetc Inducton Tenso In ts secton, a magnetc nducton tenso s nestgated. As te electc nducton tenso, te magnetc n- Copgt 203 ScRes.

3 60. PKŞN Paametes: Recees n m n m Recees Tansmttes [7,22]. Te magnetc nducton tenso s smmetc. Beao of magnetc feld components s dsplaed n Fgue 3. Te unaal model paametes ae te same as te peous electc feld calculaton. Tanks to commecall aalable 3DX Rt Scanne nducton well loggng nstuments eads a lot of data at dffeent fequences fo aous feld components. Tat means consdeable amount of data ae aalable. Fom 3DX Rt Scanne data, conducttes, elate dp beang angles can be estmated [8]. In Fgue 4, eac components of magnetc nducton tenso ae dsplaed n olumetc 3-D at 20 k. In all panels, magna pats of te magnetc feld ae nomaled b absolute alue of te coespondng components. Te medum conducttes ae: /40 S/m /80 S/m. To able to see all panels, 0. alue used fo off dagonal components. As fo dagonal components, I use alue. Red blue colos sow sgn. Blue used fo negate, wle ed used fo poste alues. Te beaos of components ae qute dffeent fom eac ote s. n m Recees Fgue. A sketc of a taal nducton loggng nstument s n a well. Te pscal medum popetes ae conductt, delectc pemttt, magnetc pemeablt. Te souces ae magnetc dpoles. Recees ae bot magnetc electc feld sensos. Supe subscpts ndcate te decton of tansmtte ecee, espectel (n m can be, ). Conductt delectc pemttt ae tensos, snce te ae pncpal alues at least two alues. Te magnetc pemeablt s assumed a scala. ducton tenso as also 9 components. It can be sown n a mat fom as: ˆ, (7) wee te notaton s te same as te electc nducton tenso. Tee tansmtte ecee cols ae oented along te,, -aes n a Catesan coodnate sstem. Deaton of magnetc feld components s a stagtfowad calculaton tat wll not be epeated ee. agnetc feld of analtc components can be found fom 5. stmaton of at Paametes: Conductt, Delectc Pemttt, agnetc Pemeablt I consde a potetcal mult-component nducton well loggng nstument. As mentoned peousl, te nstument as magnetc dpoles as a souce n te, dectons. Recees sould measue magnetc electc feld components. It s a combnaton of electc magnetc nducton tenso components. It eques some tenso-gadent measuements. Fom ts potetcal electc magnetc nducton tenso-gadent measuement, one can estmate eat (o fomaton) paametes. Tee ae man dffeent fomulas ae gen n Table. In te fst column of te table s a magnetc moment decton, wle te fst ow depcts eat paametes. A smple deaton s gen n Append A. Fom Table, t s eas to see tat tee ae man dffeent fomulas fo estmatng te eat paametes. One can calculate conducttes delectc pemttes n te oontal etcal dectons. Tese fomulas can be useful wen one of tese components s e nose, te ote one can be used fo calculatng te coespondng eat paametes n a pactcal stuaton. Te ma be useful fo ceckng paametes aganst eac ote usng dffeent feld measuements. Ts metod wll obousl ncease te qualt of te fomaton ealuaton. Table as fomulas fo conducttes delectc pemtttes. Append B dees fomulas fo magnetc pemeablt. Tese fomulas ae gen n Table 2. In bot tables, st fo eal magna components of te coespondng feld, espectel. Copgt 203 ScRes.

4 . PKŞN 6 Fgue 2. Real pats of te magnetc feld components ae calculated n an ansotopc medum wt 20 k opeatng fequences at = m. Te oontal etcal conducttes ae /40 /80 S/m, espectel. It s a top ew of a wole space. Fgue 3. Imagna pats of te magnetc feld components ae calculated n an ansotopc medum wt 20 k opeatng fequences at = m. Te oontal etcal conducttes ae /40 /80 S/m, espectel. It s a top ew of te medum. Copgt 203 ScRes.

5 . PKŞN 62 Fgue 4. Volumetc 3-D epesentaton of te magnetc tenso components at alue 0. (off dagonal) (dagonal). Te medum s conducted wt σ = /40 S/m σ = /80 S/m. Blue colos dspla negate, wle ed ones depct poste alues of magna pat of te magnetc feld. Table. agnetc moments ae n te fst column. Te supescpts sow te decton of te magnetc moments. Te est of te fomulas ae appaent conducttes delectc pemtttes. depct fo eal magna components of te coespondng feld. Numecal ample In a unaal medum, I consde as te followng pa- ametes fo numecal calculaton: Te model esponse n a wole space s calculated b a magnetc dpole oented n te decton. lectc magnetc feld components te space deates ae calculated b usng analtc fomulas. Fo consttute paamete calculaton, a goup of te fomulas gen n Table ae used, wc ae on te second ows. All pa- ametes ae elated to a unaal medum estmated, coectl. Fgue 5 dsplas te esult. Note tat tese ae S/m; ; Te fequenc ange fo calculaton begns at 0 k t eps up to 4 G. Te dstance between tansmtte ecees s.6 m as a tpcal nducton T-R dstance. Copgt 203 ScRes.

6 . PKŞN 63 Table 2. agnetc moments ae n te fst column. Te supescpts sow te decton of te magnetc moments. All fomulas n te table can be used fo calculatng appaent magnetc pemeablt. Fgue 5. Consttute paametes ae oontal etcal conductt, elate delectc pemttt magnetc pemeablt of te medum. Te magnetc moment decton s n te decton. On eac panel, sold, dotted, dased lnes sow te esult. Te calculaton done wt te fomulas ae gen on te second ow n Table. Sold lnes llustate oontal alues of bot conductt delectc, wle dotted dased lnes dspla etcal alues of tose paametes. Te fequenc ange s qute lage between at 0 k 4 G. Te medum as conductt alues wt σ = 0. σ = S/m; elate delectc alues wt 27 7 ; elate magnetc pemeablt μ = 255. Te dstance tansmtte ecees s.6 m as a tpcal nducton T-R dstance. onl some smple fomulas n Tables 2. I do not use an knd of optmaton tecnques. owee, te fomulas eque some gadent tpe measuement not onl fo magnetc, but also electc feld. In Fgue 5, dotted dased lnes sow oontal paametes on te fst n te mddle panels. Tee ae two fomulas n Table n te mddle panel. Wen one use te uppe fomula, te esult s negate. Ts s llustated wt dased lnes. Te lowe fomula ges poste alues. Ts s sowed wt dotted lnes. As fo te oontal paametes, te ae depcted wt a sold lne n Fgue 5. As seen fom Table, te eal pats of te fomulas ge conductt alues. Delectc alues can be calculated fom te magna pats of te fo- mulas. 6. Concluson I ae consdeed a potetcal nducton nstument n a unaal medum. In suc a medum, one can calculate consttute paametes suc as conductt, delectc pemttt, magnetc pemeablt. Nowadas, 3DX Copgt 203 ScRes.

7 64. PKŞN Rt Scanne tecnolog s qut matue, but consdeed a suggested potetcal nstument ma be useful fo fomaton ealuaton snce conductt, delectc pemttt magnetc pemeablt alues ae mpotant wt espect to petopscal paametes, because te ae elated to wate satuaton, poost, pemeablt. Te fomulas deeloped te pesent stud ae eact, snce te ae deed fom awell s equatons. I ae not used an knd of appomaton n ode to dee tese paametes. Applcaton of ts metod eques some gadent tpe feld measuement, wc mgt be te most dffcult pat of ts applcaton. RFRNCS [] C. A. Balans, Adanced ngneeng lectomagnetcs, Jon Wle & Sons, Inc., oboken, 989. [2] R. S. Camcael, Pactcal book of Pscal Popetes of Rocks neals, CRC, Boca Raton, 989. [3] F. S. Gant G. F. West, Intepetaton Teo n Appled Geopscs, cgaw-ll Book Compan, New Yok, 965. [4]. N. Nabgan, d., lectomagnetc etods n Appled Geopscs V. I: Teo, SAG Publcaton, Tous Oaks, 987. [5]. S. Zdano G. V. Kelle, Te Geoelectcal etods n Geopscal ploaton, lsee, Amstedam, 994. [6] J. D. Klen, P. R. atn D. F. Allen, Te Petopscs of lectcall Ansotopc Reseos, SPWLA 36t Annual Loggng Smposum, Pas, June 995, Pape. [7]. S. Zdano, D. Kenned. Peksen, Foundatons of Tenso Inducton Well-Loggng, Petopscs, Vol. 42, No. 6, 200, pp [8] Z. Zang, L. Yu, B. Kegsause L. Tabaosk, Detemnaton of Relate Angles Ansotopc Resstt Usng ultcomponent Inducton Loggng Data, Geopscs, Vol. 69, No. 4, 2004, pp do:0.90/ [9] B. I. Andeson, T. D. Babe. G. Lullng, Te Response of Inducton Tools to Dppng Ansotopc Fomatons, SPWLA 36t Annual Loggng Smposum, Pas, June 995, Pape D. [0]. G. Lülng, R. Rostal F. Sa, Pocessng odelng 2 Tools n Dppng, Lamnated Ansotopc Fomatons, SPWLA 35t Annual Loggng Smpo- sum, Tulsa, 9-22 June 994, Pape QQ. [] J.. oan S. C. Ganeo, ffects of Fomaton Ansotop on Resstt Loggng easuements, Geopscs, Vol. 44, No. 7, 979, pp do:0.90/ [2] L. Zong, C. L. Sen, R. Lu,. Btta G. u, Smulaton of T-Aal Inducton Loggng Tools n Laeed Ansotopc Dppng Fomatons, 76t Annual Intenatonal eetng SG, New Oleans, 2006, pp [3] L. L. Zong, J. L, L. C. Sen R. C. Lu, Computaton of Taal Inducton Loggng Tools n Laeed Ansotopc Dppng Fomatons, I Tansactons on Geoscence Remote Sensng, Vol. 46, No. 4, 2008, pp do:0.09/tgrs [4] S. Ganeo, D. Kenned, L. Gao L. San atn, Te Response of a Taal Sonde n a Baal Ansotopc edum, Petopscs, Vol. 43, No. 3, 2002, pp [5] A. G. Nekut, Ansotop Inducton Loggng, Geopscs, Vol. 59, No. 3, 994, pp do:0.90/ [6] N. Yuan, X. C. Ne, R. Lu C. W. Qu, Smulaton of Full Responses of a Taal Inducton Tool n a omogeneous Baal Ansotopc Fomaton, Geopscs, Vol. 75, No. 2, 200, pp do:0.90/ [7] A. Gbenko. S. Zdano, Rgoous 3D Ineson of Tenso lectcal agnetc Inducton Well Loggng Data Inomogeneous eda, 79t Annual Intenatonal eetng SG, ouston, 2009, pp [8]. S. Zdano, etod Appaatus fo Gadent lectomagnetc Inducton Well Loggng, PCT WO 2005/ A [9] X. Lu, D. L. Alumbaug C. J. Wess, Te lectc Felds Cuents Poduced b Inducton Loggng Instuments n Ansotopc eda, Geopscs, Vol. 67, No. 2, 2002, pp do:0.90/ [20]. Peksen, etods fo Intepetaton of Tenso Inducton Well Loggng n Laeed Ansotopc Fomatons, P.D. Tess, Unest of Uta, Salt Lake Ct, [2] T. Wang, Te lectomagnetc Smoke Rng n a Tansesel Isotopc edum, Geopscs, Vol. 67, No. 6, 2002, pp [22]. Rabnoc, L. Tabaosk, B. Cole, L. an de ost. po, Pocessng ult-component Inducton Data fo Fomaton Dps Ansotop, Petopscs, Vol. 47, No. 6, 2006, pp do:0.90/ Copgt 203 ScRes.

8 . PKŞN 65 Append A quaton () can eplctl be wtten a a a 0 0 a 0 0 a 0 0 a, (A.) wee a, a a ae te unt ectos n a Catesan coodnate sstem. ae te components of coespondng felds ete magnetc o electc. Supescpts st fo te magnetc moment decton, wle subscpts st fo te feld dectons. Te fst, te electc magnetc feld components wll be consdeed geneated b an -dected magnetc dpole. Fo ts pupose, I ewte quaton (A.) I ae T a a, (A.2) a a, (A.3) a a. (A.4) quatons (A.2), (A.3) (A.4) allow us to estmate oontal etcal conducttes fom electc magnetc felds. Contnung deaton elds, (A.5), (A.6). (A.7) Te eal pat of quatons (A.5), (A.6) (A.7) ges conductt te magna pat of te same equaton elds to pemttt. Contnung settng up Table, electc-magnetc feld components geneated b a -dected magnetc dpole ges tee moe fomulas. Let us look at quaton () agan, a a a, (A.8) 0 0 a 0 0 a 0 0 a fom quaton (A.8) I can wte as te followng equatons, a a, (A.9) a a, (A.0) a a. (A.) Ten paametes can be estmated as te followng epessons, (A.2), (A.3). (A.4) Conducttes can be calculated fom te eal pat of quatons (A.2), (A.3) (A.4) te magna pat of te same equaton ma be used fo estmatng delectc constant. Fute deaton usng a -dected magnetc dpole ges tee moe fomulas. Let contnue te deaton fom quaton (), Copgt 203 ScRes.

9 66. PKŞN a a a 0 0 a 0 0 a 0 0 a. (A.5) Poceedng wt some e smple algeba allows us to ae ote paametes. Fom quaton (A.5) t s e eas to ae a a, (A.6) a a, (A.7) a a. (A.8) Agan, ang conducttes delectc pemtttes ae eas. Fom te last tee equatons I can dee as te followng epessons:, (A.9), (A.20). (A.2) Fom quatons (A.5), (A.6), (A.7), (A.8), (A.9), (A.0), (A.2), (A.3), (A.4) appaent conductt pemttt can be calculated. Fo ts pupose, te eal pat of te quatons (A.9) (A.20) ae used. oontal etcal conducttes ma be estmated wt, (A.22). (A.23) Smla to peous two fomulas, appaent delectc pemtttes can be, (A.24). (A.25) Fute, consde elate pemttt as 0 0 ewte te last two equatons, I ae Append B 0 0 (A.26). (A.27) Te fomulas fo te magnetc pemeablt estmaton can be deed b usng quaton (2). Rewte quaton (2) eplctl wt dected a magnetc dpole. Bea n mnd tat te magnetc pemeablt s scala. I can wte t as a a a a a a a. (A.28) a a Te fst component of quaton (A.28) wt -dected magnetc dpole wt a unt moment, fom te peous step I can poceed wt Copgt 203 ScRes.

10 . PKŞN 67 a a a, (A.29) keep on deaton, wc elds. (A.30) Fom quaton (A.30), t s eas to ae magnetc pemeablt:. (A.3) It can be deed some fomulas fo te magnetc pe- meablt b usng te same pocedue as on te peous component deaton. a a a (A.32), (A.33). (A.34) Fom te last component of te coespondng ecto, one can get a a a. (A.35), (A.36). (A.37) Ten, ca on te deaton a a a a a a a a a, (A.38) a a a, (A.39), (A.40) elds one moe fomula fo te magnetc pemeablt of te medum, wc s. (A.4) Consdeng te component of te ecto, ten keep on a a a (A.42) wc elds, (A.43) ten t s eas to ae. (A.44) Wen I appl smla pocedue, te est of te mag- netc fomulas ma be deed: a a a, (A.45), (A.46). (A.47) a a a a a a a a a, (A.48) a a a, (A.49) Copgt 203 ScRes.

11 . PKŞN Copgt 203 ScRes. 68, (A.50). (A.5) a a a, (A.52), (A.53). (A.54) a a a, (A.55), (A.56). (A.57)