Chapter II Seismoelectric Theory

Transcription

1 Chapter II Seimelectric Thery II.1 Seimelectric Gerning Equatin Electrkinetic eect are aciated with the rie electric r electrmagnetic repne due t relatie change luid phae int lid phae [7]. In the earth, electrkinetic eect may ccur in luid-aturated pru rck / material. The eect i a reult a nanmeter-cale eparatin charge in which a bund charge i ixed t the urace the lid grain i balanced by a diuiely ditributed ree cuntercharge / in in the mbile layer adjacent luid (igure II.1). Figure II.1. Electrkinetic eect induced by prpagatin eimic wae in hmgeneu material creating a micrcpic charge eparatin caued by lw treaming current. The electric duble layer i repnible r treaming current at the grain cale [9]. Thi charge eparatin i called the electric duble layer. Micrcpically, thee eect are due t adrptin negatiely charged in in the rck matrice that in turn caue the exce ree pitie in in rck luid. In ther wrd, the mlecular charge eparatin ccur at matrix-luid bundarie are becming electric duble layer [18]. The eparatin i caued by prpagatin eimic wae in rck that gie rie t relatie diplacement in carrying luid with repect t the rck matrix, enabling treaming electric current t lw. Thi

2 current will prduce urther eparatin in macrcpic cale which becme the urce induced electric ield r electrmagnetic wae radiatin. Prpertie the ield are inluenced by electrchemical prpertie the matrix-luid bundary and mbility in in luid, type luid (e.g. water, air, r il), type matrix (clatic r carbnate), and mechanical and tructural prpertie the rck (elatic mduli, prity, permeability, and aturatin). The theretical backgrund the eimelectric phenmena caued by the electrkinetic eect i decribed by Pride [] wh btain a general et equatin that gern the cupled eimic and electrmagnetic repne a pru material. Auming an i t e ω time dependence all ield and diplacement, the gerning equatin are r t D r ω ρ u = P + τ + ω ρ w (II.1) c t r D r r r T rt τ = G( u + u 3 ui ) (II.) P c 1 B u = U p B B α w (II.3) iω w k( ω) η ( ) + = L ω p ω ρ u + ( ) ( ) J L ω σ ω E (II.4) H = J iωε E (II.5) r E = iωμ H (II.6) Equatin (II.1), (II.), and (II.3) are Bit' equatin r the lid diplacement u and iltratin diplacement w (the relatie luid-grain diplacement), where equatin (II.4) explained the luid lw iω w and electrical current J tranprt equatin the Darcy and Ohm law, and equatin (II.5) and (II.6) are nnethele the Maxwell' equatin r the electric and magnetic ield E and H D. The deiatric tre tenr t t D t t τ i deined a τ = τ + I where t τ i P C the ttal bulk tre tenr acting n the material, and bulk) preure, while I t i the identity matrix. P C i the cnining (r ttal

3 Next, phyical parameter cntained in the gerning equatin are explained. There are three prelatic cntant: (1) the undrained bulk mdulu (Gamann mdulu), deined a the rati the cnining preure change t the ample dilatatin r a ealed ample; () Skemptn ceicient U B, deined a the rati the luid preure t the cnining preure increment under the p ame undrained cnditin; and (3) the Bit and Willi cntant α, deined a the rati cnining preure t luid-preure increment, under the cnditin that the ample lume de nt change. A general exact relatin between thee three prelatic mduli iα B = 1 U, where i the drained bulk mdulu. Under the pecial retrictin that the grain are itrpic and hmgeneu within the pru material, luid-ubtitutin relatin can be tated a α = 1, (II.7) and B = U α + φ α ( ), (II.8) =, (II.9) 1 Bα where i the luid bulk mdulu, and i the bulk mdulu the lid grain material. Bit mduli C and M, which are ued in the deriatin eimic t electric traner unctin, can be derie rm preiu relatin a C = B U and M = B U α. S in cmplete rm, C and M are written a + D C =, 1+ D (II.1) M 1, φ 1+ D (II.11) [ r with D = ( 1 φ) ], (II.1) φ while ther Bit mdulu H and can be derie a 4 H = G + G, (II.13) 3 and U can be expanded t

4 U = r + φ + 1+ D ( 1+ φ ) D. (II.14) Other parameter tated in the gerning equatin are the grain ramewrk hear mdulu G, the bulk denity ρ, the pre luid denity ρ, the angular requencyω, the electrical cnductiityσ, the electrical permittiity ε, hydraulic permeability k, the magnetic permeability μ, medium prity φ, and the luid icityη. Hweer, the mt undamental parameter cntrlling all the electrkinetic cupling i the electrkinetic cupling ceicient L. It decribe the relatin between grain-luid mtin and the electric ield. I L i et t zer, the gerning equatin will decuple int rdinary Bit' and Maxwell' equatin. Since the gerning equatin are written in requency dmain,, me parameter are al requency dependence equatin []. They are, the analytic exprein r the requency dependence relatin the three pru-media tranprt ceicient L ( ω), k ( ω), and σ ( ω) denity ~ ρ ( ω), and eectie electrical permitiity ( ω), and deined parameter eectie ε ~ : ~ ~ ω m d 3 d L( ω ) = L i 4 i (II.15) ωt Λ δ EM ( ) ω ω k ω = k 1 i i (II.16) ω t m ω t φσ C + ( ) = + em C σ ω 1 α σ Λ ~ ρ ~ ε ( ω) i = ω k η ( ω) ( ω) ( ω) ε ( ω) + σ ( ω) ρ ( ω) L ( ω) 1. (II.17) (II.18) i = ~. (II.19) ω The mt releant parameter in thee exprein i the angular tranitin requency,

5 φ η ωt = (II.) α k ρ which eparate the lw-requency icu lw and the high-requency inertial lw. With alue n the rder eeral kilhertz r ten kilhertz, the tranitin requency 5 ω t π = 1 Hz i well abe the 1 Hz t 1 khz eimic requency band, which i deired. Therere, a typical cnditin rck and il where ω << ωt i aumed. T cmplete the decriptin parameter, the tatic part the electrkinetic cupling ceicient i deined a L = ~ φ ε κ ζ d 1, (II.1) α η Λ While the ret are the icu kin depth 1 { ( )} δ = η ωρ, (II.) the dimeninle number where ( Λ ) ( ) m = φ α k, (II.3) Λ i a weighted lume-t-urace rati the pre pace, the lid cnductiity σ, medium turtucity α, luid cnductiityσ, zeta ptential ζ, teady-lw permeability k, Debye lenght d ~, and the electrmigratin and electrmtic cnductance are C and C repectiely. Explicit deinitin d ~, σ, σ, Cem and C are derie by Pride [, 1] in rm d m em ~ ε κ kt =, (II.4) e z C φ σ = σ, (II.5) α m ( b+ + σ = e z C b ), (II.6) C C em ~ ezς = 4 d e z Cm b+ ch 1, (II.7) kt ( ε κ ς ) ~ 3 1 i d ~ 1 = P, (II.8) dη P δ

6 and P ~ 16kT d Cm e zς = ch 1, (II.9) ε κ ς kt where kt explained the thermal energy, C m explained the alt / inic cncentratin, e and z explained the electrn charge and number ree electrn, and and b are the inic mbility catin and anin, repectiely. b + Parameter and ariable ued in thi thei, alng with their name and repectie unit meaurement, are hwn in Table II.1. Table II.1. Phyical Parameter ued in thi thei Symbl Meaning Unit Symbl Meaning Unit B Skemptn ceicient C Electrical urce term A/m α α Bit-Willi cntant Turtucity C m Salt cnentratin ml/liter δ Dirac delta unctin E Electric Field V/m δ Slw wae diuie kin depth F r Seimic urce term Pa δ EM kin depth m Hydraulic urce m/ ε Electrical F/m term permitiity F G Frmatin actr Lame - hear mdulu Pa ε ε Fluid electrical permitiity Electrical permittiity acuum H Magnetic ield A/m ς Zeta ptential V I t Tenr identity η Fluid icity Pa J Current denity A/m λ Waelength m k Dynamic m λ U Undrained Pa permeability Lame cntant m F/m F/m

7 k Steady lw (dc) permeability m μ Magnetic permeability Drained bulk Pa μ Permeability H/m mdulu acuum Fluid bulk mdulu Pa ρ Bulk denity kg/m 3 r Frame bulk mdulu Pa U L Slid (grain) bulk mdulu Undrained bulk mdulu (Gamann mdulu) Electrkinetic cupling ceicient M Surce mment J Pa Pa H/m ρ Fluid denity kg/m 3 ρ Slid denity kg/m 3 ~ ρ Eectie luid inertia (C m)/(n ) σ Bulk electric cnductiity σ Fluid electric cnductiity p Fluid preure Pa t τ Ttal bulk tre tenr p c Cnining preure Pa t D τ Deiatric tre tenr q Slid-luid relatie m/ τ Itrpic tre elcity tenr () t Dimeninle urce φ Prity unctin t Time ω Angular requency u Slid diplacement m ω t Tranitin angular requency p P-wae elcity m/ w Slid-luid relatie m diplacement kg/m 3 S/m S/m Pa Pa rad/ Hz

8 Nw, the tw type electrkinetic cupling that preent in the tranprt law equatin (II. 4) are decribed. Seimic wae generate a rce p + ω ρ u that, in additin t driing a Darcy luid iltratin ( k η )( p + ω ρ u), al tranprt the diue charge the duble layer relatie t the bund charge n L p + ω ρ u. the grain urace, reulting in a treaming electric current ( ) Such generatin an electric current rm an applied luid-preure gradient i knwn a electriltratin and i repnible r -called eimelectric phenmena. Cnerely, when an applied electric ield E act n a pru material, in additin t drie a cnductin current gien byσ E, it al act a a bdy rce n the exce charge the diue duble layer, reulting in a net luid iltratin gien by L E. Such generatin a luid iltratin rm an electric ield i knwn a electr-mi, and it i repnible r -called electreimic phenmena. I the cupling ceicient L were et t zer, there wuld be cmplete decupling between the prelatic and electrmagnetic repne ield. By uing a plane-wae lutin the gerning equatin, Pride and Haarten [] derie the elementary diplacement and EM ield (eigen repne) an itrpic and hmgeneu whlepace. Uing their reult, Garambi and Dietrich [5] expre the electric ield E a a unctin the eimic diplacement u and demntrate that lw-requency apprximatin thee relatinhip may be written in the rm r 1 E = σ ε ρ κ η ζ 1 ρ ρ C H u t (II.3) Equatin (II.3), called a the eimic t electric ield (eimelectric) traner uctin in lw-requency regime; i alid in the diuie regime, i.e., when cnductin current dminate diplacement current. Thi cnditin i expreed by the relatinhip σ ( ω) >> ω ε ( ω), where ( ω) ε i the dynamic bulk dielectric permittiity the pru material. And, uing thi equatin, eimelectric repne rm ariu D mdel will be carried ut. (Fr mre inrmatin n the traner unctin deriatin, ne can reer t the Appendix A).

9 II. Type Seimelectric Repne Three ditinct type eimelectric phenmena hae been bered in the ield, beginning with the initial experiment cnducted by Thmpn [7] and cntinuing with experiment by Martner and Spark [17], Lng and Rier [16], Thmpn and Git [6], Butler et al. [], Garambi and Dietrich [5], and the mt recent by Haine [9]. Deelpment the releant eimelectric thery by Pride [], Pride and Haarten [], and Haarten and Pride [8] create better undertanding the phyic behind the bered phenmena and ugget that the eimelectric methd culd pride ueul new inrmatin abut thin layer and lw prpertie. The irt type eimelectric repne ccur when cmpreinal wae traeling thrugh a luid-aturated pru material create a luid preure gradient that induce pre luid lw [9]. Due t the electric duble layer that exit at the grain-luid bundary, the pre luid mement tranprt a mall amunt electric charge relatie t the electric charge n the adjacent grain. The trength and plarity the electric duble layer arie with grain cmpitin and luid chemitry, but ten (uch a r quartz grain) the grain urace charge i negatie and the luid pitie. The net lw charge relatie t the grain i a treaming electrical current that create a charge eparatin within a cmpreinal wae, at the cale the eimic waelet. Thi charge eparatin ha an aciated electric ield which we reer t a the ceimic ield that i cupled with a cmpreinal (P) wae [] (Figure II.). Thi i the mt cmmnly and mt eaily bered eimelectric phenmenn. It dier rm the hrizntal cmpnent grund acceleratin nly by a requency-independent calar becaue the cle relatin between grain/luid acceleratin and the ceimic ield.

10 Figure II.. Electrkinetic cnerin caued by eimic wae prpagatin in hmgeneu pru medium, generating ceimic ield. a). Ceimic ield recrded n the urace ater eimic wae prpgate underneath the receier. b). Cnnectin between eimic wae and ceimic ield. The ecnd mechanim ccur when P-wae encunter an interace in material prpertie (elatic, chemical, lw-prpertie). The charge eparatin in the wae i diturbed (Figure II.3), cauing aymmetry in the charge ditributin, and reulting in what can be apprximated a an cillating electric diple with it dminant cntributin cming rm the irt eimic Frenel zne [5, 8, 9]. In ther wrd, the entire regin the irt Frenel zne act a a dik ertical electric diple. Since the electrmagnetic waelength at eimic requencie are typically enrmu cmpared t the depth the interace, inductin eect can be neglected, and the electric ield generated at the interace can be mdeled a being quai-tatic. The reulting electric ield ditributin i that a quai-tatic diple. Thi ield (Figure II.3), called the interace repne, can be meaured almt immediately at the Earth' urace ince the trael-time electrmagnetic radiatin i negligible cmpared with that eimic wae ( EM ~ 1 5 P ). Unlike

11 the ceimic ield, which cntain inrmatin nly abut the material immediately urrunding the receier diple, the interace repne can pride ueul new uburace inrmatin. In particular, the interace repne ccur een r ery thin layer (~ 1 - m), uch a thin racture in lid rck, r a thin impermeable layer in an aquier r reerir. Figure II.3. Interace bundary repne r electrkinetic cnerin at an interace happen when eimic wae cre the layer bundary and act a the irt Frenel zne generating electric ield repne. Due t it diple behair, the electric ield prpagatin, generated rm thi mechanim, i ater than a tw-way trael eimic wae. Other type eimelectric repne, and nt a cncern thi thei, i predicted by Pride and Haarten [] and bered by Haine [9]. Thi repne named the direct ield, a it can be cnidered a analgu t the eimic direct wae. A directed pint urce (e.g. a ledgehammer impact, a pped t an itrpic explin) create an enhanced luid preure n ne ide the impact pint (beneath the hammer trike plate) and a decreaed preure n the ther. The luid-preure ditributin equilibrate, which reult in a charge eparatin due t the electrkinetic mechanim preiuly dicued. The charge eparatin at a ertical impact pint will hae a trng ertical diple cmpnent and the

12 aciated electric ield emanating rm the regin impact i what we call the direct ield. Similar t the interace repne, the direct ield hw the amplitude pattern a diple and reered plarity n ppite ide the ht pint. It ccur at the time the urce impule, and cntinue until the il ha relaxed t it riginal tate. II.3 Identiying Seimelectric Repne Seimelectric methd can ptentially pride aluable inrmatin abut imprtant uburace target that can nt be btained uing ther methd, including inrmatin abut change in lw prpertie [9]. The eimelectric interace repne i created at change in lw prpertie, een r layer t thin t be imaged by relectin eimic methd. The interace repne i enitie t ther prpertie, i.e. electric duble-layer trength, rigidity, etc., and i a weak ignal. Fr the cae tw hrizntal layer, a eimelectric data (eimelectrgram), cllected with gemetry imilar t cnentinal eimic recrding, include imultaneu recrding the interace repne rm uburace layer and ceimic ield (Figure II.5). The direct ield may al be recrded alng with the ther tw rm eimelectric repne, but thee ccur nly r a brie perid ater time zer d nt preent a prblem in urey targeting layer deeper than a ew meter. The ceimic ield and interace repne are cmparable magnitude when the recrding antenna are within a eimic waelength the interace. Hweer, ince the interace repne ield i eentially a diple, it magnitude i decreaing a the ditance r receier rm the interace increaed and can thu be many rder magnitude maller than the ceimic ield, epecially a target depth increae. Ceimic energy therere repreent a rmidable rm cherent urce-generated nie that can bcure large part the eimelectric recrd. S althugh the eimelectric methd ha great prmie, ne mut better undertand eimelectric eect and data cllectin bere the methd will be iable.

13 Figure II.5. A typical image ynthetic r recrded eimelectrgram cntaining three pattern electric ield: the ceimic ield, interace repne, and direct ield [9].