Seismic waves in poroviscoelastic media: A tutorial INTRODUCTION 3

Transcription

1 Semc wave n porovcoelatc meda: A tutoral Edted and partally retranlated by Pat F. Daley ABSTRACT porovcoelatc meda Intally, t hould be mentoned that th report a hghly edted redrat, modcaton and partal retranlaton o a paper by Frenkel (944) whch wa wrtten n Ruan and tranlated by Cheng (97). A the ttle o the orgnal paper dealt wth emoelectrc theory, ome o the orgnal text ha been deleted, ome moved to Appendx A, and addtonal text and ormulae have been added. The wrter o th apologze to the orgnal author and tranlator or th (poble) mue o ther academc reearch, but t, compared wth other mlar work, explan n a arly uncomplcated manner the mplementaton o Darcy law nto the elatodynamc equaton, producng, at leat n prncple, the equaton o moton or compreonal ( P ) and hear ( S ) wave propagatng n a porovcoelatc medum. Other paper and text on th topc wll have to be conulted or more numercally pecc apect o th problem. However, a reaonable undertandng o the content o th report hould make uch undertakng much eaer. All equaton have been re-derved and typographc error corrected, and pont that may requre urther nvetgaton are annotated n ootnote. Appendx B ha been added, n whch all o the parameter ued n the text are dened, a the notaton o the orgnal tranlaton ha been moded to conorm to recent work dealng wth th topc. Fnally, t hould be noted that a the orgnal paper wa everely moded, th a prelmnary report and ubject to change to mprove t readablty or to ntroduce addtonal relevant content. INTRODUCTION 3 The proceng o emc data by geocentt baed almot excluvely on elatodynamc theory. For the general area o emologcal tudy, th a reaonable choce. However, or thoe dealng wth the hydrocarbon exploraton and data proceng related to th endeavor, the queton o whether a more relevant and accurate theory hould be ncorporated n ther work, peccally that o porovcoelatcty. A the ba o hydrocarbon recovery rom reervor at depth nvolve the low o a lud through a porou medum, whch recovered and brought to the earth urace, would ndcate that elatotodynamc theory alone mght be ound lackng when ued n explanng the dynamc o th proce. Porovcoelatc theory would be the more Department o Geocence, Unverty o Calgary: pdaley@ucalgary.ca The theory o moton o a lud n a porou medum baed on Darcy law doe not take nto account the act that the partcle comprng the dry porou medum can be elatcally compreed and extended and aume that the external orce and hydrotatc preure act only on the lud whch occupy the pore. 3 Lnearzed elatodynamc theory decrbng emc wave propagaton n an otropc homogeneou medum pecc to the aorementoned medum type wth no theoretcal provon, even n the nhomogeneou extenon, or the extence o a tatc lud phae, much le a lud phae that ha a low rate wthn n the medum meaured relatve to the otropc old. CREWES Reearch Report Volume 0 (008)

2 Daley correct choce. A the core currcula at pot econdary nttuton n the geocence are degned to provde a general treatment o emc wave propagaton, the ncluon o th type o pecalzed topc mot oten ha to be delegated to a pot graduate program. The development o porovcoelatc theory preented here drected at the geocentt. Startng wth bac elatodynamc theory, Darcy law ntroduced to contruct equaton governng the propagaton o emc wave n a porou medum wth a lud occupyng the pore. It by no mean an all ncluve expoton o the porovcoelatc problem, but rather ha been degned to provde an ntroducton o the topc whch hould allow or more pecc problem n th area to be condered. DRY POROUS MEDIUM A porou medum, aumng a two phae ytem, characterzed, rom the pont o vew o elatc properte, a the propagaton o wave n each o t two conttuent; the old and lud phae. The propagaton o each phae partally dependent on the propagaton o the other. Let t ntally be aumed that the lud phae totally abent o that the volume occuped by the pore empty. The elatc properte o the medum may then be decrbed, rom a macrocopc pont o vew by equaton related to wave propagaton n an elatodynamc old, whch wll taken here to be otropc. The quantte τ = δ λθ + μe () are the component o the elatc tre tenor, (,,,3) and δ = 0, k The related quantte e u = + xk τ = where δ = or = k u = are the component o partcle dplacement vector, dependent on the Cartean coordnate x ( =,, 3). Lettng θ = u = u x dene the relatve change o the eectve volume o the dry porou medum. Alo, λ and μ are Lamé coecent, whch pecy the elatc properte o the dry medum, whch a porou elatc medum wth empty pore. From equaton () the expreon or the component o the elatc orce actng upon a unt volume o the dry porou medum are (Ak and Rchard, 980) are the component o the tran tenor, where (,, 3) u x τ θ u u Φ = = + + In vector notaton the above become k λ μ μ. k xk x k xk x k xk k (3) () CREWES Reearch Report Volume 0 (008)

3 porovcoelatc meda ( λ μ) θ μ Φ = + + u (4) In the macrocopc theory o a porou elatc medum, only thoe dtance that are large when compared to the dmenon o the pore 4, and uch volume that contan a large number o pore wthn a reaonable volume o the old medum are condered. The preence and degree o poroty accounted or by the coecent φ 5 whch equal to the rato o the volume o pore, V to the total (macrocopc) volume occuped by the porou medum, that, V = V + V, where V the volume actually lled by the old conttuent. The actual denty o the old conttuent wll be denoted by ρ and the mean (macrocopc or eectve) denty o the porou medum by. Reerrng thee quantte to the volume V and V + V or a unt ma reult n A a conequence ρ =, =. V V + V (5) V V = ρ = ρ. V V V V + + Employng the denton o φ (6) = ρ φ. (7) The change n the volume o the porou medum cont o two part:. the change n volume o a unt ma o the old phae, Δ V, and. the change n the volume o the pore aocated wth th unt ma, Δ V. For the cae o mall tran 6, whch are routnely condered n elatodynamc theory, thee quantte can be taken to be proportonal to one another, o that Δ V = α Δ V (8) where α ome proportonalty coecent 7, whch together wth the poroty φ pece the mechancal properte o the dry porou medum. Wth the atance o th parameter, t poble to expre the varaton o the degree o poroty o the porou 4 Probably more accurately, a dtrbuton o pore ze that doe not volate the macrocopc theory. 5 Th an overly cautou manner o ntroducng poroty. In what ollow, φ poroty. 6 Small tran mply mall deormaton, whch the oundaton o lnearzaton, a well a other related approxmaton to the tate equaton derved rom phycal prncple. 7 Although th may eem arly traghtorward, α a very mportant quantty, whch not etmated correctly, can caue large naccurace n any related computaton. CREWES Reearch Report Volume 0 (008) 3

4 Daley medum due to t deormaton. Ung the denton o φ or a unt ma, the ollowng reult whch lead to Snce then ( Δ +Δ ) ΔV V V V V ΔV V ΔV Δ φ = = V V V V V V Δφ φ V V ΔV ΔV V V α = = ΔV V + V V V + V V V + V φ = = = V φ φ ( α) φ + Δ φ = ΔV ( V + V ) From another perpectve, accordng to the denton o the quantty θ = u t ollow that. 8. (9) (0) () () ( α ) ΔV Δ V +Δ V + ΔV θ = = = V V + V V + V Thu the ubequent relatonhp between Δ φ and θ obtaned a. (3) φ( + α) Δ φ = θ. (4) ( + α ) In the cae o hgh poroty, the coecent α mut be mall compared wth unty, o that the compreon or expanon o the porou medum realzed prmarly a the reult o the compreon or expanon o t pore. Wth a decreae n poroty, α mut ncreae. It natural to aume, however, that or value o φ not equal to zero, the product φ( + α) mut be maller than unty,.e., + α < φ. The orce actng on a old body and characterzed by the tre tenor τ can be dvded nto a preure, p = ( 3)( τ + τ + τ33 ) and hear tree, peced by the tenor 8 Compare th to what termed tortuoty. 4 CREWES Reearch Report Volume 0 (008)

5 porovcoelatc meda τ τ δ τ τ δ. (5) = = + p 3 In the abence o hear tree, the deormaton o the body reduced to ether a mple expanon or compreon, wth the preure beng dened by the ormula ( 33 ) p = λθ + μ e + e + e τ = λ+ μ θ 3 3 whch ollow rom equaton () together wth the denton o θ. Thu (6) where b 3 θ = p (7) b = λ + μ the bulk modulu 9 o the dry porou medum, and a a reult, takng equaton (4) nto conderaton, φ( + α) ( + α ) Δ φ = b p. (8) Hence, the condton + α < φ hold, the compreon o the dry porou medum mut be accompaned by a decreae n poroty, wherea or the cae + α > φ the oppote would be true. The ollowng relaton wll alo be needed n what ollow e = δ σ p + ϑτ (9) whch are obtaned by employng equaton () or the quantte e. The coecent σ and ϑ are dened by the ormulae μ ( + ) λ+ σ = 3 p, ϑ =. μ λ μ μ (0) The abence o hear tree characterzed by the relatonhp τ = τ = τ 33 = p whch together wth τ = τ 3 = τ 3 = 0 0, requre that equaton (9) reduce to equaton (7) n th cae. SATURATED POROUS MEDIUM To th pont a dry porou medum ha been condered. Now aume that all o the pore are totally lled wth a lud, whch can low reely wthn the pore pace. The queton that are : What wll be the nluence o the lud phae under uch a tuaton on the macrocopc properte o the porou medum? 9 See the denton o compreblty n Appendx B. 0 τ ymmetrc, o that τ = τ = τ = 0 alo hold n th cae. 3 3 CREWES Reearch Report Volume 0 (008) 5

6 Daley In order to reman n equlbrum the lud phae mut, n the abence o external orce, be ubjected to the ame hydrotatc preure, p, at all pont n the connected pace ormed by the pore. Th preure alo exerted on the old keleton o the porou medum. The reultng deormaton o th keleton mut be proportonal to the varaton o the volume o the old phae n the ame rato a that o the lud phae n the ollowng manner ΔV ΔV = = p () V V where the actual (eectve) bulk modulu o the old phae. It ollow that the entre (macrocopc) volume o the porou medum mut vary at the ame rato Δ V = θ = p () V and that the poroty percentage mut reman unaltered. Conequently Δ φ = 0. (3) Comparng equaton (7) and () ndcate that the hydrotatc preure, p, equvalent to the compreon o the porou medum, produced by t, o that the preure n the old gven by whch uch that t maller than b p = p (4) p becaue > b. 3 Equaton () requre an addtonal condton to complete t peccaton. Th gven by Δρ = p (5) ρ relatng the true denty o the lud (a well a the bulk modulu o the lud) wth the hydrotatc preure and repreent the approxmate orm o the equaton o tate o the lud at a contant temperature. 4 What meant here that all o the pore pace eectve. That, there are no pore pace that are not connected to another pore pace. Taken to the lmt, all pore pace are connected n ome ahon. I all the pore pace not eectve, the ollowng dervaton are quetonable wthout ome correcton actor that compenate or the unconnected pore pace. Agan ee tortuoty. Agan ee the denton o compreblty n Appendx B. 3 ( 3) 4 > = λ + μ. b the bulk modulu o the lud. The denton o compreblty gven n Appendx B. 6 CREWES Reearch Report Volume 0 (008)

7 porovcoelatc meda It mut be kept n mnd that the ma o the lud, ρ V, llng the pore n a certan part o the old keleton, V,, generally peakng, a varable quantty 5, when compared to the old conttuent, ρ V, whch reman contant. The preure p and p are totally derent n ther orgn and nature, and a uch are completely ndependent o one another. The total varaton o the macrocopc volume o the porou medum, due to ther combned acton, equal to the um o the varaton (elatodynamc), due to each o them beng condered eparately. Addng the expreon (7) and () reult n θ = p p. (6) b Th ormula vald n the pecal cae o the abence o hear tree n the old keleton o the rock. Gven that thee tree are abent when the lud ha no relatve velocty (moton) to the rock (lud at ret), the tre tenor n the rock reduce to, n the general cae, to the um o equaton (9) and the tenor ( 3 ) p δ. Th lead to the relatonhp e = δ σ p p + ϑτ. (7) 3 The above equaton, together wth equaton (5) pecy the deormaton o the lud and old phae a uncton o the tree. To derve the equaton o moton n the old, the tree mut be expreed a uncton o deormaton (tran). Dene ϕ =Δ ρ ρ uch that ϕ (not to be mtaken or φ ) become a characterzaton o the lqud phae n a mlar manner a θ doe or the old phae. Replacng p n p = ϕ equaton (7) may be rewrtten n the orm complance wth equaton (5) ( ) e δ ϕ = δ σ p + ϑτ. (8) 3 τ = δ λθ e ϕ + μ δ 3 ϕ (9) or nce λ+ μ = 3 b b τ δ = λθ ϕ + μe Thee equaton together wth the equaton o tate o the lud phae (30) 5 That, the lud compreble. I t were ncompreble then the ma o the lud would not be varable. CREWES Reearch Report Volume 0 (008) 7

8 Daley p = ϕ (3) allow or the determnaton o the elatodynamc volume orce on the old n the cae when the quantte τ and p vary arbtrarly wthn the medum. In clac elatodynamc theory, the vector component o the elatc orce, wth reerence to ome unt volume o the medum, are gven by Φ = τ. (3) k xk In the problem beng condered here, the above expreon reer only to the old phae, whch contaned n a unt volume o the medum. In the preence o a hydrotatc preure, a unt volume o the old alo acted upon by the orce p, whch dtrbuted between the lud and old phae accordng to the rato o ther repectve volume, whch, φ ( φ). Equaton or the component o Φ may now be wrtten a τ p Φ = φ x x ( ). (33) k k The orce actng on the lud phae per unt volume o the old ( ) Φ = φ. (34) Subttutng equaton (33) and (34) nto the expreon (30) and (3) reult n (or the old) b u u k ϕ Φ = λθ ϕ + μ + + ( φ ) xk k xk x k xk x b u ϕ = ( λ+ μ) θ ϕ + μ + ( φ) xk k xk x or n vector notaton For the lqud phae In the next ecton, law. ( ) p (35) Φ = λ + μ θ + μ u + φ ϕ. (36) Φ and Φ = φ ϕ. (37) ( ) Φ wll be related through the ntroducton o Darcy 8 CREWES Reearch Report Volume 0 (008)

9 porovcoelatc meda EQUATIONS OF MOTION IN A POROUS MEDIUM: THE INTERACTION BETWEEN SOLID AND FLUID PHASES From the theory o hydrodynamc, the mean velocty o the low o a lud, whch completely occupe the pore n a porou old, under the addtonal aumpton o abolute rgdty o the old keleton, determned by Darcy equaton (Crat and Hawkn, 99) k k v = ( p ) ( p ) η + F = η + F (38) where η the vcoty, k the permeablty and F denote the external orce actng on the lud contaned wthn a unt volume o the porou medum. The permeablty proportonal to the poroty, φ, and the average value o a cro ecton o the pore n the porou medum. Th would ndcate that the relatonhp k = contant φ (39) vald, where a lnear dmenon n the porou medum. To complete the peccaton o the problem, a n a tandard hydrodynamc problem, a contnuty equaton requred. Th gven by + ( v ) = 0 (40) where = φρ the mean (eectve) denty o the lud n ome macrocopcally mall regon, that contan a ucent pore denty to provde an adequately accurate denton o. 6 Equaton (38) reer to the cae o teady low. For varable low, t replaced by 6 In ome o the lterature on th topc, n the econd term on the let de o equaton (40), replaced by ρ.o that the contnuty equaton ha the orm φ + ( ρ v ) = 0. It can be hown that n th orm t contradct the law o conervaton o ma o the lud, when compared to the denton o mean macrocopc velocty o low, v. Th crcumtance o no conequence a long a the lqud aumed to be ncompreble or the old keleton o the porou medum taken to be abolutely rgd. (In the latter cae, the denton o v mut be omewhat altered.) In the cae o a deormable keleton, the ue o the equaton contaned n th ootnote wll lead to erroneou reult. CREWES Reearch Report Volume 0 (008) 9

10 Daley The hgher order term ( v ) v η = p + F v. (4) k v ha been neglected n the above equaton. Equaton (4) nexact 7 becaue o the abence o the actor φ precedng the gradent o the preure, p. Introducng th actor, the corrected equaton or the moton n the lud obtaned a v η = φ p + F v (4) κ where the permeablty ha been replaced by the poroty normalzed permeablty dened a κ k φ = = contant (43) whoe ntroducton enure Darcy law wll be ated or the cae o teady low o the lud. Equaton (4) may be generalzed to the cae when the deormablty and the moblty o the old keleton become mportant, n the propagaton o elatc vbraton.. Here, the abolute velocty o the lqud, v, mut be replaced by t velocty relatve to the old phae, v v. The quantty v, where, v = u (u beng partcle dplacement) denote the mean macrocopc velocty o the partcle n the old phae at ome arbtrary pont wthn the porou medum. The relatve velocty connected to the rcton orce actng on the lud n a unt volume o the porou medum through the ormula F = η ( v v ) (44) κ the old phae beng acted on by the lud phae n a unt volume by an equal but oppote orce, F = F. Replacng v n equaton (4) by v v and p by ϕ reult nally n v η = φ ϕ+ F ( v v ). (45) κ In the abence o external orce equaton (45) may be wrtten a 7 Both o the normalzaton κ = k φ and = φ ρ caue th ncontency. 0 CREWES Reearch Report Volume 0 (008)

11 porovcoelatc meda ρ v η = ( ) t ϕ v v. (46) κφ The equaton o moton o the old phae n the general cae o a relatve moton o the lud can be wrtten a v η = Φ + F + ( v v ) (47) κ A n elatodynamc theory, velocty condered to be a uncton o tme and the patal coordnate o a old partcle, o that v the peccaton o acceleraton. In the lud phae, v not the exact expreon or the correpondng quantty. Rather, the exact expreon n th cae t + ( ) v v v, whch nclude a term that wa prevouly dropped. In practce, however, the moton o the lud o low that the addton o the extra term doe not play gncant role, a t neglgbly mall. It = φ ρ, n hould alo be mentoned that the eectve denty o the old phae, equaton (47) reer to the untreed tate and mut a a conequence mut be aumed to be a contant quantty. Subttutng expreon (36) or orce, Φ n equaton (47) produce, n the abence o body v η = ( b ) ( ) t λ + μ θ + μ u+ + φ ϕ + v v. (48) κ Equaton (45) or (46) and (48) are uncton o the quantte: u, v, ρ, ϕ and φ ( v = u,θ = u and = contant ). The quantte ρ and ϕ are related by the expreon ( 0) ( 0 ) ( ) ρ = ρ ϕ (49) where ρ the average denty o the lud. It eectve denty related to the velocty v through the contnuty equaton (40). The varaton o the poroty may be expreed n term o θ a φ( α) ( α ) Δ φ = θ ϕ (50) + Th relaton obtaned rom equaton (7) θ replaced by θ = p, whch n b term o the old phae preure and a ha been hown n the dervaton o equaton (30), equal to θ ( ) ϕ.. Thu there are ve equaton n ve unknown uch that the equaton o moton n the porou old are ully determned. CREWES Reearch Report Volume 0 (008)

12 Daley COMPRESSIONAL WAVE PROPAGATION IN A POROUS MEDIUM In th ecton compreonal wave propagaton nvetgated, under the aumpton o mall vbraton n the porou medum, whch allow or the lnearzaton o the equaton o moton. Wth th uppoton, the mall coecent o quantte o nteret may be replaced by ther value n an untreed medum, a done n clacal elatodynamc and hydrodynamc problem. The quantte n queton nclude: u, v, v, θ, ϕ and Δ φ. For the analy o compreonal wave the operator " " appled to the equaton o moton. The ollowng ormula wll be ued together wth the equaton θ v = = u (5) φ ρ Δφ ϕ v = = = + (5) φ ρ φ whch ollow rom equaton (0) and n t lnearzed orm, relatng equaton (3) wth equaton (50), can be wrtten a θ ϕ v = ( β ) + β (53) where the ollowng ntermedate varable have been ntroduce to reduce the complexty o notaton wthn the problem β =, β = + ( β ) φ ( + α). (54) Applyng the operator " " to both de o the lnearzed equaton (48) and utlzng the prevou equaton n th ecton, the ollowng reult θ E b η ϕ θ = θ + φ ϕ β β +. (55) 8 κ In a mlar manner equaton (43) become ϕ β η ϕ β θ θ = ϕ β βρ κ β. (56) Equaton (55) and (56) contan, at leat n prncple, the peccaton o the propagaton o compreonal vbraton n a aturated porou medum. Beore proceedng to poble oluton method, a plane wave oluton wll be condered n the next ecton. 8 Recall that λ + μ = E Young modulu or an elatc medum (dry porou medum). CREWES Reearch Report Volume 0 (008)

13 PLANE COMPRESSIONAL WAVES porovcoelatc meda Rather than attempt to olve the mot general cae o propagaton o compreonal wave n a porou medum, a plane wave oluton wll be condered (wth a dampng coecent or vcoty term). Aume a plane wave type o oluton or the quantte θ and ϕ o the orm ( ) wt qx e (57) where t tme and the drecton o propagaton may be arbtrarly choen. Let th wt ( qx) drecton be the x drecton o the plane wave oluton e. Here, ω π = the requency o the vbraton and q π the complex wave number n the drecton o wave propagaton 9. It equal, n the abence o dampng, to the recprocal o the wave length, λ. The rato ω q the generally complex valued velocty o wave propagaton. Under thee condton, the derental equaton (55) and (56) reduce to a ytem o two lnear algebrac equaton or the ampltude θ and ϕ, a 0 and E ηβ ηβ ξ + θ + εξ ϕ = 0 κω κω (58) β ηβ η θ + ξ + ϕ = 0 (59) β κ βω βρ κ βω where ξ = q ω = WP and ε = ( φ b ). Th quantty, the propagaton velocty o the wave, determned rom the oluton o the quadratc equaton E ηβ η ξ + ξ + κω β ρ κ βω μβ β ηβ εξ = 0 κ ω β κ β ω whch repreent the compatblty condton o equaton (59). Ater ome manpulaton the above quadratc equaton reduce to (60) 9 q ω the complex lowne n the drecton o propagaton. 0 ε ( φ ) = See equaton (A.3). b A quadratc equaton mple two velocte, thee are the at and low compreonal wave velocte. For the ytem o equaton Aθ = 0, A a matrx, θ a vector, to have a oluton, det [ A ] mut be equal to zero. CREWES Reearch Report Volume 0 (008) 3

14 Daley E E β η β b ξ + + ε E+ ξ β ρ βρ β κω β η + + = 0 κω The two root o equaton (6) wll not be gven here a they are ealy obtaned numercally. 3 It hould be noted that or large value o the parameter, ζ = η κ one o the root correpond to a wave wth a very mall dampng actor (at compreonal) whle the other, to a very large dampng actor (low compreonal). Wave o the econd type may be dcult to detect, however, recent paper have ndcated ther extence (See or example, Couy and Bourbe,984). An approxmate determnaton o the value o ξ, correpond to a wave o the rt knd. Approxmatng ξ n a ere n term o the power o the mall parameter, ωκ η = χ, reult n ξ = ξ + χξ + (6) 0 Subttutng th truncated ere nto equaton (6) and equatng the coecent o the varou power o χ, tartng wth χ produce (6) β b E+ ξ0 + = 0 β E E β β b ξ0 + + ε ξ0 E+ ξ+ = 0 β ρ βρ β β E E β β b ξξ ε ξ E+ ξ= 0 β ρ βρ β β etc. The rt o thee equaton lead to (63) (64) (65) W P0 = ( β β ) ( ) E+ + b (66) a ξ = W where W ndcate velocty and the ubcrpt the type o wave and order 0 P0 o approxmaton. 3 Equaton (6) ha two root correpondng to the at and low compreonal wave. However, the expreon or thee two compreonal wave velocte, obtaned by olvng the quadratc, are complex and conequently o lmted ueulne. For th reaon, the analy that ollow, whch provde an approxmaton to the at compreonal wave, wll be purued a the reultng expreon provde a uable pont o reerence rom whch numercal experment may be undertaken to acertan the aect o varyng pecc quantte on the velocty and dampng actor. 4 CREWES Reearch Report Volume 0 (008)

15 porovcoelatc meda Inertng the expreon or ξ 0 = WP nto equaton (64) reult n the ollowng rt 0 order correcton E 3 E β ξ0 + + ε ξ0 + ξ0 β ρ βρ β χξ = χ. (67) + Wth a rt order accuracy wth repect to χ or the complex propagaton velocty o the (at) compreonal wave, W P, determned by the ormula that W P ξ ξ ξ0 ξ0 = ξ = ξ + χ ξ + χ (68) W W P 0 = + (69) P 0 κ ωwp ξ η where ξ gven by equaton (67). Inertng th expreon n the exponent o the actor exp ( ωt qx) = exp ( ωt ωx WP ) and wrtng the later n the orm exp ( ωt ωx WP ) δx, where δ the dampng coecent o the wave per unt 0 length, the ollowng expreon or th coecent obtaned E β E + + ε W W κω β ρ β βρ δ = ηw P0 P0 ( + ) 3 P0 The dampng coecent ha been hown to be proportonal to the quare o the vbraton =. Th the ame a or the cae o the propagaton o compreonal wave n an ordnary vcou lqud. requency, that to ω ( π ) In concluon, conder the lmtng cae o wave propagaton n a medum wth vanhng poroty. In th cae, φ 0, o that a a conequence, = ρ and = 0. Alo, κ 0 5, β = φ( + α ) whch ollow rom equaton (50), β = and b =. Under thee condton, δ vanhe and W P 0 reduce to E ρ - the tandard expreon or the compreonal wave propagaton n an otropc elatc old medum. (70) 4 A appear n the denomnator o the magnary part o th equaton. It doe not appear n Frenkel. 5 k = contant φ, κ k φ =. CREWES Reearch Report Volume 0 (008) 5

16 Daley Equaton (33) enable the determnaton o the varaton o th velocty wth an ncreae n the number and ze o pore, whch are aumed to be lud aturated. An eental role played here by the decreae o the elatc modulu b o the old keleton. Th topc not dealt wth here a t complex and hould be the topc o an ndependent tudy. PROPAGATION OF SHEAR WAVES IN A POROUS MEDIUM The equaton governng the propagaton o hear wave n a porou medum can be obtaned rom the undamental equaton o moton, (45) and (48), by applyng the operator " " to them. Introducng the notaton, Ω = v and Ω = v, whch are the angular velocte o the old keleton and o the lud phae, repectvely, reult n Ω η = μ Ω + Ω Ω Ω κ η = ( Ω κ Ω ). (7) I a plane wave oluton, whch propagate n the (arbtrary) x drecton t qx e ω, aumed, the above equaton reduce to the lnear equaton η ( μξ ) Ω = ( Ω Ω ) κω (7) Ω = η ( Ω Ω ) κω (73) where ξ = q ω and q and ω have been prevouly dened. Elmnatng Ω and rom equaton (73) reult n the ollowng equaton or ξ =. χ μξ (74) From th t ollow that μξ = + or ξ = + + χ + χ μ μ Thu, n the rt approxmaton wth repect to χ ( ) Ω. (75) + ξ = χ (76) μ μ Th ormula how that the hear wave propagate n a lud lled porou medum wth the velocty 6 CREWES Reearch Report Volume 0 (008)

17 porovcoelatc meda W S0 wth the dampng coecent μ + ωκ = = + WS μ μ χ (77) ωκ ωκ δ = = μ η η ( + ) W S 0 It can be een rom equaton (78), that a n the cae o compreonal wave, the dampng coecent proportonal to the quare o the crcular requency, ω. CONCLUSIONS Equaton o partcle moton have been derved or a porou medum whoe pore are lud aturated. Flud low wthn the pore o the old keleton reult n rcton between the two phae. Th aect the perceved (meaured) value o many o the quantte decrbng the total medum n uch a manner that they may der when compared to reult obtaned ung elatodynamc theory. Employng porovcoelatc theory n emc exploraton and data proceng hould, baed on the dervaton preented here, produce a more accurate decrpton o the phycal procee and parameter nvolved n hydrocarbon recovery rom reervor at depth wthn the earth. Granted, a number o aumpton and approxmaton have been made n the coure o the nvetgaton. However, t dcult to juty not ung at leat ome apect o the theory preented when dealng wth emologcal problem related to lud lled porou meda, whch are a reaonable approxmaton o hydrocarbon reervor. What are conpcuou by ther abence are reerence to other text and paper on th topc. A urvey o the lterature produce numerou text and paper on th ubject. A th work ha been ndcated a beng tutoral n nature, wth the pecc ntent o ntroducng the topc o emc wave propagaton n a porovcoelatc medum, t wa thought that a th report eentally el contaned, the ltng o thee ctaton would not contrbute, and pobly be a detrment. A ubequent report, whch under preparaton, deal wth numercal oluton poblte. It contan numerou reerence applcable to the general topc o wave propagaton n a porovcoelatc medum. REFERENCES Ak. and Rchard P.G., 980. Quanttatve Semology, W.H. Freeman and Company, San Francco. Couy, O. and Bourbe, T., 984, Propagaton de onde acoutque dan le mleux poreux ature, Rev. Intr. Fr. Petrole, 39, no., (n French). Crat, B.C. and Hawkn, M.S. 99, Appled Petroleum Reervor Engneerng, Prentce Hall, Englewood Cl, N.J. Frenkel, J., 944, On the theory o emc and emoelectrc phenomena n a mot ol, Journal o Phyc, 3, No. 4, 30-4 (n Ruan). [Edtor o Englh tranlaton: Alexander Cheng, 97.] (78) CREWES Reearch Report Volume 0 (008) 7

18 Daley APPENDIX A: A SPECIAL CASE AND A SOLUTION OF THE EQUATIONS OF MOTION BASED ON THE METHOD OF SUCESSIVE APPROXIMATIONS An mportant pecal, or rather lmtng cae, correpondng to an extremely large η κ value o the parameter (an extreme mallne o pore) wll be condered rt. Dvdng equaton (55) and (56) by th parameter and notng that the quantte θ and ϕ mut have nte value, the ollowng expreon obtaned relatng them n th ntance β ϕ = θ (A) β Th relatonhp ndcate that the two velocte v and v are dentcal. That, there no relatve moton o the lud wth repect to the old. Under the condton (A), equaton (55) reduce to where whle equaton (56) become θ E β = + ε θ β ε = φ b (A) (A3) ϕ β β ρ = ϕ (A4) The later equaton contradct equaton (A), nce the uncton θ and ϕ mut be connected by the relaton (A), unle the velocty o the propagaton o wave, gven by equaton (A) ( E ) + ( εβ β ) whch concde wth the wave velocty dened by equaton (A4) - ρ β β. Proceedng to the next topc n th Appendx, t can be een that when the parameter η κ tend to nnty, the derence o the velocte v v or ther η κ or ( ) dvergence tend to zero n a manner uch that t product wth th parameter reman nte. eepng th n mnd, th derence wll be repreented n the orm o a ere n power o the mall parameter κη= ζ, uch that 8 CREWES Reearch Report Volume 0 (008)

19 porovcoelatc meda β 3 ϕ = θ + ζψ+ ζ ψ + ζ ψ3+ β where ψ, ψ, are ome unknown uncton wth nte value. (A5) Beore ubttutng th expreon nto equaton (55) and (56) t mut be noted that n olvng thee equaton by the method o ucceve approxmaton the uncton θ mut alo be expanded n a power ere n ζ, o that and a a conequence θ = θ + ζθ + ζ θ + (A6) 0 β β β ϕ = θ0 + ζ θ+ ψ + ζ θ + ψ +. (A7) β β β Ater ubttutng thee expreon nto equaton (55) and (56) and equatng le power o ζ n both o them, a ytem o equaton obtaned or the determnaton o the uncton θ k and ψ k. In the zero order approxmaton (term not contanng the parameter ζ ) and makng ue o the notaton n equaton (A) reult n θ 0 E β β = ε θ ψ β (A8) θ 0 β β ψ = θ 0 + (A9) ρ β Multplyng the rt o thee equaton by and the econd by and then addng them, reult n the ollowng expreon or θ 0 beng obtaned 0 + = E+ ε + φ θ0 θ β β β β Upon ntroducng the denton o ε lead to 0 b + = E + θ0 t β θ β. (A0) Th equaton decrbe wave whch are propagated wthout dampng at velocty (A) CREWES Reearch Report Volume 0 (008) 9

20 Daley W P0 = ( β β ) ( ) E+ + b. (A) To obtan the next term n the approxmaton expreon (A6) and (A7) mut be ubttuted n equaton (55) and (56), and equate the rt order term n ζ to obtan the ytem o equaton θ E β β ψ = ε θ + + ε ψ+ (A3) β θ ψ β β ψ + β = θ + ψ (A4) ρ β Multplyng the rt o thee by and the econd by and addng, the ollowng relaton between θ and ψ or θ and ϕ reult θ β = E ε θ t β ρ ψ β + ε + ψ t ρ. (A5) Introducng equaton (A) θ β ψ ε + φ + + W P θ 0 = + ψ t t or by vrtue o the denton o ε (equaton (A3)) (A6) ( ) θ β ψ + + b W P θ 0 = + ψ t t. (A6) Comparng th equaton wth equaton (A) there ollow, among other thng, that the rght hand de mut be orthogonal to the uncton θ 0. For the determnaton o θ, the uncton ψ n (A6) mut be replaced by t expreon n θ 0 ung one o equaton (A8) or (A9). Thu θ θ E β t + 0 W P θ 0 = ε θ + 0 β ( ) + θ E + β b 0 ε θ0 β + β Th proce can be contnued to obtan equaton o hgher order. (A7) 0 CREWES Reearch Report Volume 0 (008)

21 APPENDIX B: NOTATION porovcoelatc meda What ollow are denton o the parameter ued n th report. Addtonal upport n th may be ound on and related lnk. compreblty bulk modulu See, and. (B) b e k = + u xk u x elatc tran tenor. (B) [ ] E Young' modulu = λ+ μ (tenle tre)/(tenle tran) N/m. A meaure o the tne o a gven materal. φ, Δφ poroty - the percentage o pore volume that can contan lud. Eectve poroty exclude olated pore and reer only to the connected pore volume n a rock that contrbute to lud low. Total poroty the total pore volume o o the rock. Δφ change n poroty. (B3) (B4) F external orce actng on the old phae n a unt volume. (B5) F external orce actng on the lud phae n a unt volume. (B6) η F = v v κ volume o the porou medum. F rcton orce actng on the lud n a unt = F an equal but oppote orce actng on the old phae due to lud rcton n a unt volume. (B7) (B8) μ Lamé' coecent n the elatc lmt. (B9) mmble - the nablty o two lud to mx to orm orm a homogeneou mxture; ol and water are mmble lud. k permeablty, the ablty or meaurement o a rock' ablty to tranmt lud, meaured n darce or mllldarce. (Relatve and abolute.) (See κ.) (B0) (B) CREWES Reearch Report Volume 0 (008)

22 Daley b,, bulk modulu - the rato or percent change n volume to the change o preure appled to a lud or rock. ( = λ+ ( 3 ) μ). b bulk modulu o the dry old. actual bulk modulu o the old - ( > b). bulk modulu o the lud. Invere the related compreblty coecent. (B) λ Lamé' coecent n the elatc lmt. (B3) p =, preure n the old phae and lud phae. (B4) τ = δ λθ + μe tre tenor or an elatc medum. (B5) ( φ ) Tortuoty, oten dened a: T = + (B6) th u component ( =,, 3) o the partcle dplacement vector. V, V, V volume. V = V + V where V the volume actually lled by the old phae and V by the lqud phae n a porou medum. the volume occuped (B7) (B8) ΔV, ΔV, ΔV varaton o volume - total, porou medum, lud. (B9) δ u partcle dplacement n the porou medum. (B0) v = u partcle velocty n the old phae. (B) v mean low velocty o the lud phae. (B) α proportonalty contant whch together wth poroty, φ, pece the mechancal properte o the dry porou medum. ( Δ V = αδ V ) ( α = b ) β = φ ( + α ) (B3) (B4) β = + ( β ) (B5) exponental dampng uncton aumng plane wave ncdence. (See χ.) (B6) CREWES Reearch Report Volume 0 (008)

23 ε = φ ϕ Δρ b porovcoelatc meda (B7) = (B8) ρ τ ( Φ ) componento elatc orce n a unt volume Φ = k xk or the old phae. (B9) φ p orce actng o the lud phae n a unt volume. ( ) ( ) Φ Φ = (B30) =, eectve denty - denty o the rock matrx wth pore ( ). Eectve lud denty ( ). (B3) χ a parameter ntroduced to pecy the dampng o a wave' (B3) ampltude, gven plane wave propagaton. exp( ( δ * x) ) κ = k φ poroty normalzed permeablty. (See k.) (B33) λ wave length. (B34) η vcoty - a property o lud that ndcate ther retance to low, dened a the rato o hear tre to hear rate. Meaured n poe (dyne-ec/cm ) or centpoe - /00 o a poe. One centpoe equal one mllpacal-ec. Vcoty mut have a tated or undertood hear rate n order to have meanng. (B35) θ = u where u partcle dplacement n the old phae. (B36) ρ ( ) =, gran denty - the denty o a rock wth no poroty ( ρ ). Flud denty ( ρ ). ( 43) μ ( + μ ) λ + σ = μ λ (B37) (B38) ϑ = (B39) μ = ω π ω the crcular requency related to the vbratonal requency. ( vcoty/permeablty [eectve]) (B40) ζ = η κ (B4) CREWES Reearch Report Volume 0 (008) 3

24 Daley χ = ( ωη ) κ (B4) θ and ϕ dlaton o old and lud phae, θ = u (B43) 4 CREWES Reearch Report Volume 0 (008)