Boundary integral formulation and two-dimensional fundamental solutions for dynamic behavior analysis of unsaturated soils

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1 Boundry integrl formultion nd to-dimenionl fundmentl olution for dynmic behvior nlyi of unturted oil Pooneh Mghoul, Behrouz Gtmiri, Deni Duhmel To cite thi verion: Pooneh Mghoul, Behrouz Gtmiri, Deni Duhmel. Boundry integrl formultion nd to-dimenionl fundmentl olution for dynmic behvior nlyi of unturted oil. Soil Dynmic nd Erthquke Engineering, Elevier,, 3 (), pp <.6/j.oildyn..5.6>. <hl-66849> HAL Id: hl Submitted on 9 Feb HAL i multi-diciplinry open cce rchive for the depoit nd diemintion of cientific reerch document, hether they re publihed or not. The document my come from teching nd reerch intitution in Frnce or brod, or from public or privte reerch center. L rchive ouverte pluridiciplinire HAL, et detinée u dépôt et à l diffuion de document cientifique de niveu recherche, publié ou non, émnnt de étbliement d eneignement et de recherche frnçi ou étrnger, de lbortoire public ou privé.

2 Boundry Integrl Formultion nd To-Dimenionl Fundmentl Solution for Dynmic Behviour Anlyi of Unturted Soil Pooneh Mghoul, Behrouz Gtmiri,b,*, Deni Duhmel Univerité Pri-Et, UR Nvier, Ecole de Pont, 6 et 8 Avenue Blie Pcl, Cité Decrte, Chmp ur Mrne, Mrne l Vllée, Cedex, Frnce b Deprtment of Civil Engineering, Univerity of Tehrn, P.O. Box , Tehrn, Irn Abtrct. In thi pper the coupled eqution governing the dynmic behviour of unturted oil re derived bed on the poromechnic theory in the frme of the uction-bed mthemticl model preented by Gtmiri (997) nd Gtmiri et l. (998). In thi formultion, the olid keleton diplcement, ter preure nd ir preure re preumed independent vrible. The Boundry Integrl formultion ell fundmentl olution for uch dynmic u p p theory re preented in thi pper for the firt time. The boundry integrl eqution re derived vi the ue of eighted reidul method in y tht permit n ey dicretiztion nd implementtion in Boundry Element code. Alo, the ocited to dimenionl (D) fundmentl olution for uch deformble porou medium ith liner eltic behvior re derived in Lplce trnform domin uing the method of Hörmnder. Finlly, ome numericl reult re preented to ho the ccurcy of the propoed olution. The derived reult re verified nlyticlly by comprion ith the previouly introduced correponding fundmentl olution in eltodynmic limiting ce. Keyord: boundry element method; boundry integrl eqution; fundmentl olution; ingulr behviour; unturted oil; multiphe porou medium; dynmic behviour Introduction

3 Unturted oil re encountered ner the erth urfce here mot of engineering tructure re ultimtely upported. Even though geotechnicl engineering project encounter turted, dry nd unturted oil, mot of the pt tudie hve been done only on turted nd dry oil. Sturted nd dry oil cn become unturted due to eonl vrition. The dynmic behvior of the turted oil h been extenively invetigted [3, 4, 53, 4, 4 mong other]. In the current tte of the knoledge, it could be climed tht behviour of the turted porou medi h been ell undertood. In contrt, the tudy of the dynmic behvior of the unturted porou medi i reltively ne re in the field of geotechnicl erthquke engineering. Accurte meurement of vriou quntitie uch dynmic ter nd ir preure, nd degree of turtion in prtilly turted oil i difficult tk during dynmic loding [43]. Wve propgtion in unturted oil in rid re nd the dynmic repone of uch medi re of gret interet in geophyic, oil nd rock mechnic, nd mny erthquke engineering problem. But, in geomechnic, the behviour of mny medi including more thn to phe i not conitent ith the principle nd concept of clicl turted oil mechnic. Thu, the prediction nd imultion of unturted oil behviour re of gret importnce in mking criticl deciion tht ffect mny fcet of engineering deign nd contruction. An unturted porou medium cn be repreented three-phe (g, liquid, nd olid), or three-component (ter, dry ir, nd olid) ytem in hich to phe cn be clified fluid (i.e. liquid nd g). The liquid phe i conidered to be pure ter contining diolved ir nd the g phe i umed to be binry mixture of ter vpor nd dry ir in non-iotherml ce. The ir in n unturted oil my be in n occluded form hen the degree of turtion i reltively high. At loer degree of turtion, the g phe i continuou. In order to model unturted oil behviour, firt the governing prtil differentil eqution hould be derived nd olved. Becue of the complexity of the governing prtil differentil eqution, ith the exception of ome imple ce, their cloed-form olution re not vilble. Therefore the numericl method, uch the Finite Element Method (FEM) nd the Boundry Element Method (BEM), hould be ued for uch prtil differentil eqution. The FEM h proven to be very effective in olving problem ith bounded domin, prticulrly hen inhomogenitie nd non-liner effect hould be treted. Then, regrding it vt bility in geomechnic, it h been ued in mny code. One of thee code tht h been developed to model the different pect of unturted oil i (θ-stock) progrm hich i ritten by Gtmiri (997) [4]. Thi code h been vlidted by everl ppliction. For the

4 ke of brevity of the text the literture revie h been omitted. An exhutive literture revie h been given in [6,, 3, 8]. For domin of infinite extenion hoever, tndrd finite element dicretiztion led to ve reflection t the edge of the FE meh, hich cn be only prtly eliminted for ome ce, by uing o-clled trnmitting, ilent nd non-reflecting vicou boundrie [37, 5]. Other olution, uch conitent infiniteiml FE cell method, becue of their importnt didvntge of being formulted in trnformed pce, eem cnnot be ued in non-liner dynmic nlyi [3]. The BEM, on the other hnd, i very effective numericl tool for dynmic nlyi of liner eltic bounded nd unbounded medi. The method i very ttrctive for ve propgtion problem, becue the dicretiztion i done only on the boundry, yielding mller mehe nd ytem of eqution. Another dvntge i tht thi method repreent efficiently the outgoing ve through infinite domin, hich i very ueful hen deling ith cttered ve by topogrphicl tructure. When thi method i pplied to problem ith emiinfinite domin, there i no need to model the fr field. In thi method, during the formultion of boundry integrl eqution, the fundmentl olution for the governing prtil differentil eqution hould be firt derived. Indeed, ttempting to olve numericlly the boundry vlue problem for unturted oil uing BEM led one to erch for the ocited fundmentl olution. To the bet of the uthor knoledge, no fundmentl olution exit in the publihed literture for the dynmic modeling of unturted oil o fr, hence the development of BEM model for dynmic behviour of unturted oil i not yet poible. In the turted medi, it eem tht the firt ttempt to obtin fundmentl olution for dynmic poroelticity preented by Burridge nd Vrg (979) [6] for the u i u i formultion ho gve generl olution procedure imilr to tht of Dereieicz (96) []. A inhomogeneity they choe only point force in the olid hich i not ufficient for the uge of uch fundmentl olution in BE formultion. Lter, Norri (985) [4] derived time hrmonic Green function for the me formultion for point force in the olid ell point force in the fluid. He lo obtined explicit ymptotic pproximtion for fr-field diplcement, ell thoe for lo nd high frequency repone. Afterrd, Kyni nd Bnerjee (993) [33] ued olution cheme imilr to tht of Norri (985) [4] nd derived the fundmentl olution in the Lplce trnform domin ell trnient hort-time olution. The Burridge nd Vrg olution obtined for three force, hile thoe of 3

5 Norri, nd Kyni nd Bnerjee hve ued ix vrible (diplcement of the olid keleton nd verge diplcement of the fluid), both of hich eem not to be dequte for ppliction. The firt pproch doe not hve enough vrible nd the econd one h too mny. Mnoli nd Beko (989) [39] hve pointed out the nlogy beteen poroelticity nd thermoelticity. Hoever, thi nlogy i only poible for the ui p formultion. It i lo hon by Bonnet (987) [5] hen he preented the fundmentl olution for the ui p formultion in frequency domin. Further, he concluded tht the ui p formultion i ufficient nd the u i i u formultion i overdetermined. Dominguez (99) [] preented the bic formultion of the frequency domin integrl formultion for dynmic poroelticity in term of olid diplcement nd fluid tre. In nother pper, Dominguez (99) [3] preented boundry element pproch for dynmic poroeltic problem in frequency domin. In thi pper Dominguez ued the equivlence beteen poroelticity nd thermoelticity for obtining the fundmentl olution. Alo Weibe nd Ante (99) [5] eem to be the firt ho obtined time domin to-dimenionl (D) fundmentl olution for the Biot type dynmic poroelticity for the u i i u formultion by neglecting the vicou coupling nd ithout numericl evlution of the kernel function. Without thi retriction Chen (994, b) [8, 9] propoed nother pproximte trnient D nd 3D fundmentl olution for the pecil ce of hort time ell the generl ce for the formultion, hich ere too complicted to be pplied in BE lgorithm. Lter Gtmiri nd Kmlin () [6] hoed tht Chen pproximtion could not be ued in the implified ce of ui olution for the p formultion. They derived other pproximte trnient D fundmentl ui p formultion. Lter Gtmiri nd Nguyen (5) [] propoed cloedform trnient D fundmentl olution for the ui ui p formultion of turted porou medi coniting of incompreible contituent. They hve hon tht their olution i very ccurte olution epecilly in long time. More recently Schnz nd Pryl (4) [48] hve derived dynmic fundmentl olution for deformble olid keleton ith compreible nd incompreible fluid in Lplce trnform domin. By comprion of the to et of the derived fundmentl olution, they hve concluded tht n incompreible model cn only be ued in ve propgtion problem if the hort time behviour i not conidered nd lo if the rtio of the compreion moduli re very inignificnt. Alo, the Green function for continuouly non-homogeneou turted medi obeying Biot dynmic poroeltic theory, p 4

6 hve been derived by Seyrfin et l. (6) [45]. Jut recently, Gtmiri nd Elmi (7) [5] preented n nlyticl olution for the evlution of cttering of ve by circulr cvity in infinite iotropic eltic porou medi. For unturted oil, Gtmiri nd Jbbri (4, b, 5, b) [7, 8, 9, ] hve derived the firt fundmentl olution for the nonliner governing differentil eqution for ttic nd quittic poroeltic medi for both to nd three-dimenionl problem. The correponding thermo-poro-mechnic fundmentl olution for ttic nd qui-ttic problem re, repectively, derived by Jbbri nd Gtmiri (7) [3] (for both to nd three dimenionl problem) nd Gtmiri et l. (9) [4] (for to-dimenionl problem) nd Mghoul et l. (9) [38] (for three-dimenionl problem). Thi pper in conecrted to obtin the boundry integrl eqution nd D fundmentl olution for unturted oil under dynmic loding in order to be ble to model the ve propgtion phenomen in thee medi by BEM. In thi pper firt, the et of fully coupled governing differentil eqution of porou medium turted by to compreible fluid (ter nd ir) ubjected to dynmic loding i obtined. Thee phenomenl formultion re preented bed on the experimentl obervtion nd ith repect to the poromechnic theory in the frme of the uction-bed mthemticl model preented by Gtmiri (997) [4] nd Gtmiri et l. (998) [5]. In thi model, the effect of deformtion on the uction ditribution in oil keleton nd the invere effect re included in the formultion vi uction-dependent formultion of tte urfce of void rtio nd degree of turtion. The liner contitutive l i umed. The mechnicl nd hydrulic propertie of porou medi re umed to be uction dependent. In thi formultion, the olid keleton diplcement, ter preure nd ir preure re preumed independent vrible. Secondly, the Boundry Integrl Eqution (BIE) i developed directly from thoe eqution vi the ue of eighted reidul method for the firt time in y tht permit n ey dicretiztion nd implementtion in numericl code. The ocited fundmentl olution in Lplce trnformed domin i preented by the ue of the method of Hörmnder (963) [9] or Kuprdze (979) [34] for D u p p i formultion of unturted porou medi. A thee olution re the bi of BE formultion their ingulr behvior i lo dicued. In thi ce tht fundmentl olution i knon only in the frequency domin nd it i impoible to obtin the time-dependent fundmentl olution in n explicit nlyticl form 5

7 by n invere trnformtion of the frequency domin reult, the convolution integrl in the BIE cn be numericlly pproximted by ne pproch o-clled Opertionl Qudrture Method developed by Lubich [35, 36]. Finlly, ome numericl reult re plotted tht ho the ccurcy of the propoed olution. The derived reult re verified nlyticlly by comprion ith the previouly introduced correponding fundmentl olution in eltodynmic limiting ce.. Governing eqution In order to hve fully coupled model of unturted oil, the effect of the uction chnge on the keleton deformtion nd on the ter nd ir permebilitie hould be conidered. On the other hnd the influence of tre level nd induced-trin on the degree of turtion nd pore preure diiption mut be tken into ccount ell. The m conervtion eqution of ter nd ir nd the equilibrium eqution of keleton ocited ith ter nd ir flo eqution nd contitutive reltion form complete et of field eqution [4, 5, 3, ].. Bic concept nd kinemtic An unturted porou medium i medium compoed of deformble olid keleton nd porou pce filled by to fluid (ter ( ) nd ir ( )) (Fig.). Soil Prticle Wter Air Figure. Unturted oil cheme 6

8 The interconnected porou pce i the pce through hich fluid m exchnge occur. The remining complementry pce i the mtrix. Hence, the mtrix my be compoed of both olid prt nd diconnected occluded pce, hether turted or not []. The diplcement field i defined by the diplcement of the olid keleton u (or u ) nd the i diplcement of the fluid reltive to the olid fluid (or ). The bolute diplcement of the i U (or U ) i defined in uch y tht the volume of fluid α diplced through unit i re norml to the x direction i i turtion reltive to fluid nd defined in y tht n S U here n i the poroity nd S i i the degree of S n / n;, () S (b) n n S (c)... Skeleton Deformtion The keleton mteril point t time t, fter deformtion of the medium, i expreed by poition vector x = x X, t in Crtein coordinte frme of orthonorml bi in hich X i it initil poition vector. Deformtion grdient of the olid keleton F i defined by F I u () Where I i the econd-order iotropic tenor ith component, here i the Kronecker ij ij delt. The ymbol / x ly tnd for grdient ith repect to x in thi pper nd u i the diplcement vector of the keleton hoe initil nd current poition re X nd x ( u x X ). The Green-Lgrnge trin tenor E hich meure the deformtion chnge function of deformtion grdient F, in the limit of infiniteiml trnformtion reduce to the linerized trin tenor ε : 7

9 t ε u u (3) In infiniteiml trnformtion, the volume dilttion i ritten : ε. u (4) trce ii The obervble volume dilttion of the keleton i due to both the vrition of the porou connected pce nd the volume dilttion of the mtrix. The ltter ill be noted ε, here the ubcript tnd for olid mtrix. If d nd d t denote the volume occupied repectively, by the mtrix, in the reference nd current configurtion, the volume dilttion of the mtrix red t d d d (5) In thi pper the umption of incompreibility of the olid grin i conidered. A conequence t d d nd. Then, the volume dilttion of the keleton i equl to the vrition of the porou connected pce n.... Reltive Flo Vector of Fluid Volume In order to decribe the mteril motion of n unturted porou medium, the movement of the fluid prticle reltive to the initil configurtion of the keleton no need to be pecified. With tht purpoe in mind let dγ be n infiniteiml keleton mteril urfce oriented by unit norml n ( d ritten Γ = n d ). At time t nd per unit of time, α fluid volume q dγ. n d ;, (6) flo through thi urfce dγ hich i folloed in the movement of the contituting keleton prticle. The term q repreent the reltive fluid volume flux per unit of urfce re 8

10 [ L.L.T ]. Since (or ) i the diplcement vector of the fluid reltive to the olid i keleton, the vector x, t i the Eulerin reltive flo vector of fluid volume (ith repect, to the keleton) or Drcy flo velocity for fluid. Next, let v x, t be the fluid prticle velocity reltive to the keleton prticle. From kinemticl point of vie, the velocity, v x, t cn be ritten, v v v ;, (7) Where v x, t i the Eulerin bolute fluid velocity nd it i equl to U., Here, the reltive velocity v x, t cn be defined from the reltive flo vector of fluid volume through the reltion,, n n S n S v v U u (8). M Conervtion Eqution:.. M Conervtion of Solid Skeleton The m of the olid keleton in repreenttive elementry volume cn be ritten (9) M n d t t Where n repreent the volume denity of olid keleton m in the current configurtion; The m conervtion require tht 9

11 d M dt d n d dt t n div n d t t v t t () Where Then e hve div g v g. div v grd g. v ; d d n div dt dt n n v () By the hypothee of the incompreibility of the oil grin ( d / d t ), e obtin d n d t n div v n u () ii, Thi eqution tte tht the deformtion of the olid keleton conit in the rerrngement of the grin only.... M Conervtion of Wter The m of the ter in repreenttive elementry volume cn be ritten (3) M n d n S d t t t t Where n repreent the volume denity of ter m in the current configurtion; The m conervtion require tht d M dt d dt t ns div n S d t t v t n S d t (4)

12 By introducing eqution (7) nd (8) into (4): d M dt n S div n S d t t v (5) t Since eqution (5) mut hold for ny volume, the integrnt mut be zero, giving the t ter fluid m locl blnce eqution commonly clled the ter fluid continuity eqution. By uppoing homogenou medium tht tifie grd it red d d S d n n S n S n S u i, i i, i d t d t d t (6) By the tte eqution of vrition of poroity, eqution (), e obtin n S d dt d S n S u i, i i, i dt (7) The firt prt in the left hnd ide of the eqution (7) cn be ritten d d d p d p C d t d p d t d t (8) Where Then, C d / i the compreibility of ter. d p d p d S C n S n S u i, i i, i dt dt (9) A mentioned lter d S / dt in eqution (9) cn be ritten :

13 d S dt d p p g () dt Where g d d S p p Then finl m eqution tke thi form: S C p C p () i, i ii Where C ng C n S ; C C ng..3. M Conervtion of ir With the me pproch preented for the ter m conervtion, the m conervtion eqution of the ir in repreenttive elementry volume cn be ritten d p d S C n S n S u i, i i, i dt dt () Where C d / ; d p By uing eqution (b), S S, it i obtined d S d S d p p g (3) d t d t dt Then the finl ir m eqution cn be ritten S C p C p (4) i, i ii Where C ng C n S ; C C ng

14 .3 Equilibrium Eqution: The totl eqution of motion for unit element of porou medium cn be ritten (5),, d v d v, d v,. σ g = n S grd. n S grd. v v v v dt dt dt The convective ccelertion term in the bove eqution re neglected becue of numericl difficultie, but reltive ccelertion term re retined [44]. If the reltive ccelertion term of the fluid ( nd ) re omitted, e hve imply. σ g u (6) The equilibrium eqution for the keleton cn be ritten in indicil nottion follo:, p p f u (7) ij ij i i i, j.4 Flo Eqution for the Wter: Bed on generlized Drcy l for decribing the blnce of the force cting on the liquid phe of the repreenttive elementry volume, nmely the grdient of uction, the inerti nd the reitnt force of ter due to it vicoity, the ter velocity in the unturted oil tke the folloing form:, d d, v v, n v k p g v v (8) dt dt 3

15 e d In hich k S S / S u u denote the ter permebility in n unturted oil (here e i the void rtio,,, d nd S re contnt depending on the u oil tudied). By neglecting the convective ccelertion term in the bove eqution, it i obtined (9) n k p g u By omitting the reltive ccelertion term of the ter it i obtined: p u g (3), i k.5 Flo Eqution for the Air: With the me pproch preented for the ter bed on generlized Drcy l, the ir velocity in the unturted oil tke the folloing form: p u g (3), i k d In hich k c / e S i the ir permebility (here i the ir vicoity, e i the void rtio, c nd d re contnt depending on the oil tudied)..6 Contitutive L for the Solid Skeleton: Fully coupled formultion of unturted oil cn be obtined by conidering to contitutive l in order to model the effect of uction on deformtion nd turtion. The firt one i the tre-uction-trin reltion hich conider the effect of uction on trin. The econd 4

16 one i the tre-uction-turtion hich decribe the evolution of turtion under the effect tre level nd vrition of uction [5]. The firt contitutive l of the keleton (tre-uction-trin reltion) cn therefore be defined incrementlly in term of totl net tre chnge : ij ij ijkl kl d p D d (3) In the bove, d p ij ij i the totl net tre increment, D i the tre-trin ijkl reltionhip tenor tht i function of contitutive model of oil. d kl i the trin increment due to totl net tre chnge in unturted oil. It cn be uppoed tht: dε dε dε uc (33) Where dε uc nd d ε re volumetric trin increment due to uction chnge nd totl trin increment, repectively. By introducing eqution (33) into (3), it i obtined: uc ij ij ijkl kl kl d p D d d (34) In thi eqution uc d cn be ritten follo: kl uc uc uc uc d p p D d d D d p p (35) kl kl kl kl kl kl And lo by ubtituting eqution (35) into eqution (34), it i obtined: ij ij ijkl kl ij d p D d F d p p (36) Where uc.. F D D Therefore, by introducing the elticity mtrix D into eqution (36), the contitutive l i ritten : 5

17 ij ij ij kk ij ij d p d d F d p p (37) By integrting thi eqution, (37) cn be ritten p ij ij ij kk ij Fij p p (38).6.. Stte urfce of void rtio A hon before in eqution (36) the uction modulu mtrix F i uc. (39) F D D Where uc D i vector obtined from the tte urfce of void rtio ( e ) hich i function of the independent vrible f, (4) e p p p uc uc (4) d D d p p Where D e (4) e p p uc The elticity mtrix ( D ) cn be preented by uing the bulk modulu nd the tngent modulu,, (43) D D K E D p p p t Where E i tngent eltic modulu hich cn be evluted t 6

18 E E E (44) t l Where E i the eltic modulu in bence of uction nd l (45) E m p p m being contnt, E repreent the effect of uction on the eltic modulu. K i the bulk modulu of n open ytem nd evluted from the urfce tte of void rtio ( e ) e d d p K d v p (46) e p K e e p (47) In order to ure the comptibility ith liner l, the volumetric vrition module mut be defined o tht the volume chnge propertie of unturted oil under increing monotonic vrition of degree of turtion hould be tified: K K L bs / e (48) L Where K i the bulk modulu in bence of uction nd, b re contnt, i the e elling preure nd S tnd for uction ( p p ). For uction equl to zero nd etting, one obtin the bulk modulu in the liner eltic model K L K (49) By uing thi reltion, nd integrting in reltion to the tre nd conidering the influence of the uction, the finl form of tte urfce for void rtio [4] i implified to: 7

19 e e p b p p p / e exp L K (5) It cn be oberved tht ith uction nd tre equl zero, e e. No, by uing the ne formultion of the tte urfce, uc D vector cn be obtined D uc e b p L e p p K e (5).6.. Stte urfce of degree of turtion The econd contitutive l (tre-uction-turtion reltion) hich decribe the evolution of ter turtion under the effect of tre level nd vrition of uction i ritten follo, (5) S f p p p Numerou reltion hve been introduced to define the degree of turtion of unturted oil, but the logrithmic form bed on uction vrition i one of the mot common nd relible one. The exponentil form of the degree of turtion i found here by omitting the dependency to the net tre in the originl eqution [4]: S exp p p (54) In hich i contnt. By uming negtive, one cn ee tht ny incree in uction reult decree in (turted). S nd ny decree in uction reult the pproch of S to one A hon in Figure, thi reltion h good greement ith the expreion of the tte urfce of degree of turtion propoed by Gen et l. (997) [7]. Thi expreion i bed 8

20 Suction (MP) on the experimentl tudie crried out by Villr nd Mrtin (993) [5] nd i obtined by dopting Vn Genutchen model..e+4.e+3 Gen et l. (997) Gtmiri et l. (9).E+.E+.E+.E Degree of turtion Figure. Comprion beteen the tte urfce of degree of turtion propoed by Gtmiri (997) nd Gen et l. (997).7 Summery of the Field Eqution: Then, briefly by neglecting the body force of fluid, the governing eqution re ritten belo for the unknon olid diplcement, ter pore preure nd ir pore preure: Equilibrium eqution:, p p f u (7.rep) ij ij i i i, j Contitutive l for the olid keleton: p ij ij ij kk ij Fij p p (38.rep) M conervtion eqution for ter: S C p C p (.rep) i, i ii 9

21 Flo eqution for ter: i, i i k p u (3.rep) M conervtion eqution for ir: S C p C p (4.rep) i, i ii Flo Eqution for ir: i, i i k p u (3.rep) By introducing (38.rep) into (7.rep), (3.rep) into (.rep) nd (3.rep) into (4.rep), e hve u u F p F p u f (54),,,, S u k u k p C p C p (55),,, S u k u k p C p C p (56),,,.8 Governing Eqution in Lplce Trnformed Domin One of the mot common nd trightforrd method for eliminting the time vrible of prtil differentil eqution i to pply the Lplce trnform. In thi mnner, fter olving the differentil eqution in Lplce trnform domin, one cn obtin the time domin olution by pplying n invere Lplce trnform on the Lplce trnform domin olution [9]. We remember tht [].

22 L f x, t f x, e t f x, t dt (57) n n n n n L f x, t f x, f x, f x,... f x, (57b) And uming i t i t i t u (58) p p t t (58b) Eq (54), (55) nd (56) ill be reduced to u u F p F p u f (59),,,, u k p C (6) p C p,, p (6) u C p k C p,, Where S k nd S k..... We rerite compctly the trnformed coupled differentil eqution ytem Eq (59), (6) nd (6) into the folloing mtrix form: u i f i B p p (6) With the not elf-djoint opertor B : F F i j i j i i k C C j C k C j B (63)

23 With i, j from one to four in to dimenionl problem. In eqution (63), the prtil derivtive,i i denoted by nd i i the Lplcin opertor. ii Bed on thi eqution in the next ection boundry integrl eqution nd fundmentl olution re derived. 3. Boundry integrl eqution To the uthor knoledge, the boundry integrl eqution for dynmic unturted poroelticity hve not yet been obtined. It i imed to chieve to thee integrl eqution t uch level tht it llo ppliction to phyicl meningful problem. The correponding fundmentl olution ill be derived in ection 4. Along ith Boundry Element Method permit n ey dicretiztion nd implementtion in numericl code. To tht end the preent ection i dedicted to the derivtion of et of the boundry integrl eqution for dynmic multiphe poroelticity uing the eighted reidul method. In thi method, the poroeltodynmic integrl eqution i derived directly by equting the inner product of Eq. (59), (6) nd (6), ritten in mtrix form ith opertor mtrix B defined in eqution (63), nd the mtrix of the djoint fundmentl olution G implying tht B G I x (64) to null vector, i.e. u B p p G d ith G G G G U U U 3 4 G G G P P P G G G P P P (65) here the integrtion i performed over domin ith boundry nd vnihing body force nd ource re umed. By thi inner product, eentilly, the error in tifying the governing differentil eqution (59), (6) nd (6), i forced to be orthogonl to The eqution (65) ritten in index nottion become G [47].

24 G G G F G F G u u u p p j, j j j, j, k G G C G C G p u p p 3 j 3 j, 3 j 3 j k G p G u C G p C G p d 4 j 4 j, 4 j 4 j (66) here,, nd i, j, 4. For ech term in Eq (66), integrting by prt tice over the domin uing the theory of Green formul nd uing prtil integrtion, the opertor B i trnformed from cting on the vector of unknon u p p T to the mtrix of fundmentl olution G %. To ho the principl procedure, five exemplry prt of integrl eqution (66) re preented in detil. The remining prtil integrtion for the other prt in integrl eqution (66) cn be performed nlogouly. G u d d d (67) G u n u G n d u G j, j k, k kj, k j, j G u u,, G u d G n d G u d F (67b) j j j d (67c) G p d F G p n d F G p j, j j, k G k G p k G p k G p (67d) p d d d d 3 j 3 j, n 3 j, n 3 j d d d (67e) G u G u n u G 3 j, 3 j 3 j, A hon in Eq. (67) to (67e), the integrl ith one differentition led to the chnge in the ign of the reulting domin integrl hile it remin unchnged in the ce of to integrtion by prt [46]. Thi led the trnformtion of the opertor B into it djoint opertor B. Thi yield the folloing ytem of integrl eqution in index nottion,,, u F p p p n u u n G d k k j kj, k 3 j 4 j j, j, u G G G n G G d p G p G, 3 3, d k p G p G, 4 4, G d k d n j j n n j j n u B i im m j (68) 3

25 in hich F k C C F C k C B (69) By ubtituting Eq (64) into (68) nd uing the property of Dirc delt function x, e rrive t the trnformed dynmic unturted poroeltic boundry integrl repreenttion for the trnformed internl diplcement nd preure given in mtrix form, i.e., S S S u ; U x, ; P x, ; P x, ; t x; W W W c I p ; U x, ; P x, ; P x, ; q x; d A A ga p ; U x, ; P x, ; P x, ; q x; S S S T x, ; Q x, ; Q x, ; u x; W W W T x, ; Q x, ; Q x, ; p x; d A A A T x, ; Q x, ; Q x, ; p x; (7) Where the trction vector, the norml ter flux nd the norml g flux re repectively k, k,, t n u F p p p u u n (7) q k p u n (7b), n q k p u n (7c), n The coefficient c ij h vlue ij for point inide nd zero outide. The vlue of c ij for point on the boundry i determined from the Cuchy principl vlue of the integrl. It i equl to.5 ij for point on here the boundry i mooth. Alo the S T, S Q nd S Q in Eq (7) cn be interpreted the djoint term to the trction vector t, the ter flux q nd the ir flux q 4

26 , U,, T S U S S P S S P S U S S n (7) k k l l l l,,, W W W W W W T U S P S P U U n (7b) k k l l l l,,, A A A A A A T U S P S P U U n (7c) k k l l l l Q k P (73) S S, n W W k P (73b), n Q A A k P (73c), n Q Q k P (74) S S, n W W k P (74b), n Q A A k P (74c), n Q Eq (7) cn be compcted in index nottion for the -D ce folloing ;, ; ;, ; ; j ij i ij i (75) c I u G x t x F x u x d Where t i t q q T, u u p p T nd lo i U P P G U ij P P U P P S S S W W W A A A T Q Q F T ij Q Q T Q Q S S S W W W A A A (76) (76b) With i, j vrie from one to four nd,,k vrie from one to to. 5

27 The time dependent boundry integrl eqution for the unturted oil i obtined by trnformtion to time domin. t c u ; t G t ; x,, t ; x F t ; x, u ; x d I j ij i ij i (77) 4. Fundmentl Solution Here, the fundmentl olution for the unturted poroeltic governing eqution (64) i derived in Lplce trnform domin. Thee olution cn be ued in time-dependent convolution qudrture-bed BE formultion hich need only Lplce trnformed fundmentl olution. The phyicl interprettion of fundmentl olution or kernel of differentil eqution i potentil function x, or on the other hnd, the repone of the medium in the point x to point excittion e in domin ith infinite boundrie hich i Dirc delt function in pce, i.e. nd either Dirc delt function or Heviide tep function i.e. H t in time. Mthemticlly poken fundmentl olution i olution of the eqution B G I x y t here the mtrix of fundmentl olution i denoted by G, the identity mtrix by I nd the mtrix differentil opertor by B. In thi tudy, becue the opertor type of the governing eqution i n ellipticl opertor the explicit D Lplce trnform domin fundmentl olution cn be derived by uing the method of Kuprdze et l. [34] or Hörmnder [9]. The ide of thi method i to reduce the highly complicted opertor given in (63) to imple ell knon opertor. An overvie of thi method i found in the originl ork by Hörmnder [9] nd more exemplry in Reference [38, 4, 6, 7, 8, 9,, 3, 3]. In thi method, in the Lplce trnform domin, the firt tge i to find the mtrix of cofctor co B to clculte the invere mtrix of B I ith B B co / det B B B (78) For the econd tge, e ume tht i clr olution to the eqution 6

28 B I I x B x det det (79) Which give co x (8) B B I Conequently, e get B co (8) G It i knon tht the fundmentl olution hould tify the djoint opertor [49]. A hon before ll the opertor in (63) re elliptic nd not elf-djoint. Therefore, for the deduction of fundmentl olution the djoint opertor B h to be ued: B (8) Where 9,, 3 k, C, C, S,. k, S, k, F, , F, k 3 nd C. 4 At firt folloing formul (78), the determinnt of the opertor to the reult B i clculted. Thi yield det D D D D D D D D D D D D D D D D B D (83) 5 6 here D, D, D nd D re contnt including bove ( z,4) coefficient (Appendix 3 4 z A). 7

29 We cn rerite the bove expreion (83) into the folloing fctorized form [34]: k k 3 4 det B (84) In hich the coefficient,, 3 nd re the root of thi polynomil of order four of 4, hich one of it root i the / tht i relted to the her ve velocity propgting through the medium. The remining prt of eqution (84) i polynomil of order three of, hich hve three root,, 3 4 nd they mut be determined thee hich tify F F C C S F S F k k k k 3 3 C C F C F C 3 4 k k k F C F k 3 C C C C k k S F C C F S F C C F k k k k 3 (85) (85b) 3 4 C C C k k 4 (85c) Thee three root correpond to the three compreionl ve hich re ffected by the degree of turtion nd the ptil ditribution of fluid ithin the medium. Biot demontrte the exitence of to kind of compreionl ve in fully turted medium: the ft ve for hich the olid nd fluid diplcement re in phe, nd the lo ve for hich the diplcement re out of phe. Hoever, hen the pore pce i filled ith to immicible fluid, cpillry force re importnt nd the exitence of third compreionl ve (i.e., econd lo ve) i predicted in the medium [7]. 8

30 From the reltion (85) the coefficient,, 3 4 cn be determined the root of the folloing cubic eqution of unknon (86) Then, by introducing eqution (84) into eqution (78) the mtrix of cofctor The element of thi opertor re preented in Appendix B. Secondly, the clr eqution correponding to (79) become B co i obtined. x (87) 3 4 in hich i n interim opertor, i.e. k k (88) Eqution (88) cn be expreed either of three Eq (89), (9), (9) nd (9): x 3 4 (89) x 3 4 (9) 3 3 x 3 4 (9) 4 4 x 4 3 (9) 9

31 The bove differentil eqution re of the fmilir Helmholtz type. The fundmentl olution of Helmholtz differentil eqution for n only r-dependent fully ymmetric to-dimenionl domin i []: i K r i, i,, 3, 4 (93) By definition of, nd, it i deduced (94) (95) 4 4 Then, (96) Replcing eqution (93) into (96), it i obtined (97) K r K r k k K r 3 K r In hich K r i the modified Beel function of the econd kind of order zero ith the i rgument r x hich denoted the ditnce beteen lod point nd n obervtion point. After ll, e cn determine the component of fundmentl olution tenor by pplying the mtrix of cofctor B co to the clr function hich re: Diplcement cued by Dirc force in the olid. 3

32 (98) G U S K 3 K K K K K 3 4 R K r R K K K 4 3 K r r R K r R K r R K r 3 3 R K 3 r 3 R K r R K r ith R x x r 3 r x x, r, R nd K K S F S F k C C k k F k F k k k k k k K K C C C S C F F C S F C C F k k k k k k k C F F C k F C C F k k k k Wter preure cued by Dirc force in the olid. (99) 3

33 G 3 P S C F F C K r 4 3 Fk F x C F F C 3 K r 3 3 k r F k C F F C 4 K r Fk Air preure cued by Dirc force in the olid. () G 4 P S F C C F K r k F 4 3 F x F C C F 3 K r 3 3 k r k F F C C F 4 K r 4 4 k F Diplcement cued by Dirc ource in the ter fluid. G 3 U W k S k () C S k C S k K r 4 3 S k x C S k C S k 3 K r 3 3 k r k S k C S 4 k C S k K 4 r k S k 3

34 Wter preure cued by Dirc ource in the ter fluid. () G 33 P W k K K r 4 3 K K r K K r With, K C 3 k nd K k S F C k F k k Air preure cued by Dirc ource in the ter fluid. (3) G 43 P W C k S F C K r 4 3 C k S F C 3 K r 3 k k C k S F C 4 K r Diplcement cued by Dirc ource in the ir fluid. 33

35 G 4 U A k S k C S k C S k K r 4 3 S k x C S k C S k 3 K r 3 3 k r k S k C S 4 k C S k K 4 r k S k (4) ith nd K k S k C S k C S k Wter preure cued by Dirc ource in the ir fluid. (5) G 34 P A C k S 4 3 C k S F C k k C F C 3 k S F C 4 K r K K 3 4 r r Air preure cued by Dirc ource in the ir fluid. (6) 34

36 G 44 P A k K K r 4 3 K K r K K r With, K C 3 k nd K S F C k F k k k In the derivtion of the multiphe poroeltodynmic boundry integrl eqution (7) everl bbrevition correponding to n djoint trction or flux re introduced (Eq. 7, 73, 74). At firt, the djoint trction olution i preented. Hoever, due to the extenive only prt re given S C R K 3 r R R 3 4 K r r S C R K 3 r R R 3 4 K r r S S U U,, n n l l l l S C R K r 3 R R K r 3 r S C R K r 4 R R K r 4 r l l S 5,, C r r n K r (7.) Where 35

37 R 4 r r r r n r n r / r 3,,, n,,, n R r r r 4, n,, U n r n C K r C K r C K r C K r S S 3 S 3 S 3 S 3 k, k l l, S, k 5 r n C K r (7.b) n r r, n, W W W C K r C K 3 r 3 C K 3 4 r 4 W W,, r l l l (8.) W W W r r C K r C K r C K r U U n, n, W W W W (8.b) U n n C K r C K r C K r k, k l l n r r, n, A A A C K r C K 3 r 3 C K 3 4 r 4 A A,, r l l l (9.) A A A r r C K r C K r C K r U U n, n, A A A A (9.b) U n n C K r C K r C K r k, k l l The other explicit expreion re S n r n r,, S S S , n, Q C K r C K r C K r r S S S r r C K r C K r C K r () W W W W () Q r C K r C K r C K r, n A A A A () Q r C K r C K r C K r, n S n r n r,, S S S , n, Q C K r C K r C K r r S S S r r C K r C K r C K r (3) 36

38 W W W W (4) Q r C K r C K r C K r, n A A A A (5) Q r C K r C K r C K r, n In hich the coefficient re preented in Appendix C. 5. Singulr behviour A hon in prt 3, the boundry integrl eqution i obtined by moving to the boundry. Then in order to determine the unknon boundry dt, it i necery to kno the behviour of the fundmentl olution hen r x tend to zero, i.e. hen n integrtion point x pproche colloction point. Simple erie expnion of the fundmentl olution ith repect to the vrible r x ho tht the ingulrity of thee olution in the limit r i equl to the eltottic, poro-eltottic or the coutic fundmentl olution (Tble ). W Firt conider G P, the preure due to the Dirc ource in the ter fluid. A 33 r, o doe the rgument of the modified Beel function. The limiting form of K r r i i K r ln r ln ln / r (6) i i i Thu the eqution () reduce to W G P ln / r O r 33 k poroeltottic fundmentl olution (7) A Similrly for G P, it i obtined 44 37

39 A G P ln / r O r 44 k (8) W Second conider G U, Eq. () cn reritten 3 S k W x G U f 3 r 4 k M 4 3 K r S k x M 4 K r k M 4 3 K 4 r 4 (9) Where M k k S k C S k C S f r M r K 4 3 r M 3 4 r K 3 r 3 M 4 3 r K 4 r 4 r In hich K r K r K r r () Therefore it i of interet to find the vlue of f r hen r. Accordingly, e note tht K x x. Thu the numertor nd denomintor of f r tend to zero x r. Thi give u the ce /. We pply L Hôpitl rule nd note tht K x x x lim r f r () 38

40 By ubtituting Eq. () into (9) it i obtined W r S k x 3 ln r 8 () k r G U r O With the me pproch it i obtined A r S k x 4 ln r 8 (3) k r G U r O r F x G P r O r S ln 3 8 (4) k r S r F x 4 ln 8 (5) k r G P r O r W (6) 43 G P O r A (7) 34 G P O r Finlly e dicu G U, hich cn be more compctly expreed S 3 x x S G U h r h r r 4 4 K K K K K r K K K ij K r K K r K r K r (8) 39

41 Where Tble Component Singulrity G U ekly ingulr ( ln / r ) 3 S G U ekly ingulr ( ln / r ) 4 W G U ekly ingulr ( ln / r ) 3 A G P ekly ingulr ( ln / r ) 4 S G P ekly ingulr ( ln / r ) S W P ekly ingulr ( 33 G ln / r ) A P ekly ingulr ( 44 G W P regulr () 43 G A P regulr () 34 G ln / r ) S T hyper ingulr (/r ) W T ekly ingulr ( ln / r ) A T ekly ingulr ( ln / r ) S Q ekly ingulr ( ln / r ) S Q ekly ingulr ( ln / r ) W Q hyper ingulr (/r ) A Q hyper ingulr (/r ) A Q regulr () W Q regulr () h r (9) K K K K r K r r K r r 3 K 3 K 3 4 K 4 K 4 r K r 3 r K r

42 In order to determine the vlue of h r hen r, e tke into ccount tht K x x, thi i the ce /. We pply L Hôpitl rule to Eq. (9), one obtin x h r (3) For r combining Eq. (3) nd (8), nd lo keep in mind K x x x, fter ome lgebric mnipultion, e rrive t ln x x S G U v r O r 8 v r eltottic fundmentl olution 3 4 ln (3) Adjoint fundmentl olution. S r T v r r,, v r n r n,, 4 v r n eltottic fundmentl olution r x S F S F ln r ln r n O r 4 r k k (3) W S T ln r n r r,, n 4 k k ln 4 k r n r r O,, n r (33) A S T ln r n r r,, n 4 k k ln 4 k r n r r O,, n r (34) 4

43 F Q r n r r O r S ln,, n 8 (35) F S ln,, n (36) 8 Q r n r r O r r W n Q O r r, (37) coutic fundmentl olution r A n Q O r r, (38) coutic fundmentl olution A (39) Q O r W Q O r (4) 4

44 5. Anlyticl verifiction of the fundmentl olution Hving derived the fundmentl olution, t thi tge, it i of interet to verify the vlidity of thee olution in ome ht more detil. limiting ce i preented here. Invetigte the olution form k nd k pproch infinity,, nd F pproch zero, to ee if they ould exctly tke the me form eltodynmic fundmentl olution in Lplce trnform domin. 5. Limiting Ce: Eltodynmic Letting k nd k pproch infinity nd, nd F equl zero, the root of the determinnt eqution (85) reduce to to nd e ill hve., 3 nd 4. (4) K K (4) Then G (43) G G 3 j 4 j (44) G 3 4 xx i j G G b i j i j ij C r (45) Where. r C. r C. r K K K C. r C C C. r C. r b K K C C C (46) (47) 43

45 C (48) C (49) Eqution (43) to (45) ho the fundmentl ingulr olution in the Lplce trnform domin for point force in -D olid of infinite extent. Thi limiting ce upport tht the Lplce trnform domin fundmentl olution of dynmic unturted poroelticity for - D ce derived in previou ection re likely to be correct. 6. Viuliztion of ome fundmentl olution Here, ome fundmentl olution re clculted to bring to mind their generl behviour. An unturted oil ith incompreible olid grin i conidered in hich the mteril propertie ere defined in the metric ytem follo: Tble. Mteril dt of n unturted oil (Mechnicl prmeter) K b (-) E (N/m ) K l (-) e (-) b e (-) e (-) σ e (-) n (-) ρ (Kg/m 3 ) Tble 3. Mteril dt of n unturted oil (Wter prmeter) ρ (Kg/m 3 ) K (N/m ) (m/) α (-) S u (-) β (P - ) m uc (-) P (P) Tble 4. Mteril dt of n unturted oil (Air prmeter) ρ (Kg/m 3 ) P tm (N/m ) K (N/m ) b (m ) α (-) μ (N..m - ) P (P)

46 Firtly in Fig. 3, the diplcement in direction due to unit point force in the me direction S U i depicted in 3D veru the ditnce r nd the frequency ω. For thi im, the rel prt of the complex Lplce vrible i et to zero, i.e., i. The bolute vlue of the complex vlued diplcement olution, i.e., the mplitude, i preented. Becue of the ingulrity of the olution, the figure i clipped in the rnge of -. A hon in thi figure, the ingulr behviour for mll vlue of r i roughly independent of the frequency. Alo, for contnt r y from the origin, ve like form decree of mplitude i oberved by increing the frequencie. Thee obervtion re in greement ith turted diplcement fundmentl olution [48] AbU S S Figure 3. Diplcement fundmentl olution b U in direction due to unit point force in the me direction veru frequency, ω nd ditnce, r. Alo, the D viuliztion of the ome fundmentl olution re preented by keeping contnt the ditnce r nd vrying the frequency ω to hve better inight into the behviour of the fundmentl olution. In ddition, ll reult, i.e., the diplcement nd preure reult re normlized to their ingulr behviour preented in ection 4. 45

47 AbU SU ttic FrequencyHz () r =. m SU AbU ttic FrequencyHz (b) r =.5 m S Figure 4. Diplcement fundmentl olution b U normlized ith U veru frequency ω. ttic S In Figure 4, the normlized diplcement fundmentl olution b frequency for to point t r. m nd t r.5 m ditnce from the origin. U i preented veru A comprion beteen Figure () nd Figure (b) ho tht there i difference beteen the rrivl time of the ft compreionl ve. It i evident tht the longer ditnce y, 46

48 the lter rrivl time, i.e., horter frequency. Then in Figure (b) t r.5 m, the ft compreionl ve rrive lter nd influence the horter frequencie. AbPW SPW ttic FrequencyHz () r =. m AbPW SPW ttic FrequencyHz (b) r =.5 m W Figure 5. Wter preure fundmentl olution P b normlized ith W P ttic Next, in Figure 5, the normlized ter preure due to ource in the ter fluid i depicted. A hon, chnge in preure i immeditely in to ce. Therefore, the preure cn not ho trong time, repectively frequency, dependence [48]. 47

49 7. Concluion In thi pper, firtly coupled governing differentil eqution of porou medium turted by to compreible fluid (ter nd ir) ubjected to dynmic loding i preented bed on the poromechnic theory in the frme of the uction-bed mthemticl model preented by Gtmiri [4] nd Gtmiri et l. [5]. After tht, the Boundry Integrl Eqution (BIE) i developed directly from thoe eqution vi the ue of eighted reidul method for the firt time. Finlly, the ocited fundmentl olution in Lplce trnformed domin i preented by the ue of the method of Hörmnder or Kuprdze for D u p p formultion of i unturted porou medi. Alo, the ingulr behviour of the fundmentl olution i tudied in order to be ble to determine the unknon boundry dt. It i oberved tht the ingulrity of thee olution i equl to the eltottic, poroeltotyic or the coutic fundmentl olution. The derived Lplce trnform domin fundmentl olution could be directly implemented in time domin BEM in hich the convolution integrl i numericlly pproximted by ne pproch o-clled Opertionl Qudrture Method developed by Lubich [35, 36] for modelling the trnient behviour of unturted porou medi nd enble one to develop more effective numericl hybrid BE/FE method for olving D nonliner ve propgtion problem in the ner future. 48

50 Appendix A Coefficient of determinnt of B D D D D k k (A.) F F / D (A.) C C S F S F / D (A.3) k k k k 3 F F / D (A.4) 4 D / D 5 S F C C S F C k k k k k C F C F C F C F C k k k (A.5) D C C C S F C C F S F C C F / D (A.6) k k k k k k 6 D D / D 7 / D 8 C C F C C F k k k k F C C F k C C C C C C S F C C F k k k k k k S F C C F k k (A.7) (A.8) D / D k k (A.9) k k k k 9 C C C C C C 49

51 Appendix B The component of cofctor mtrix B co re obtined folloing ij 4 B co C C C C C 8 9 k k C B co C C C C C C i 4 C B co C C C C C C i 4 3 B co C C C C B co C C C C B C C C C C co B C C C C C co co B C C C C C C co B C C C C C C i i (B.) (B.) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) here C 8 9 3, C, C, C, C, C, C, , C , C C C, C, , C, C, , C 4 6 C, C C, C , C, C 8 8 4, , C, , C, C , C C, C, , C C, C , C, C, C, C 4 3, C, 4 9 5

52 C, C C, C , C, C,, C, C 5 3, C, 5 9 C, 5 4 8, C C, nd C C

53 Appendix C C C C C K K 3 4 S K K 4 3 S K 3 K S 3 K 4 K S 4 (C.) (C.) (C.4) (C.5) C S 5 (C.6) C C W S k k 4 3 W S k k k S k C S k C S k k S k C S k C S k (C.7) (C.8) C W 3 S k k C A S k 4 3 C A S k k C A 3 S k k k k S k C S k C S k k S k C S k C S k k S k C S k C S k k S k C S k C S k (C.9) (C.) (C.) (C.) 5

54 C S F 4 3 C F F C Fk (C.3) C S F C F F C Fk (C.4) C S F C F F C Fk (C.5) C C C C C C C S A W C K 4 3 W C K W C K C k S F C k A C k S F C k 3 A C k S F C k 4 3 F 4 3 S F S W F 3 4 F C C F k F F C C F k F F C C F k F F C k S C k 4 3 (C.6) (C.7) (C.8) (C.9) (C.) (C.) (C.) (C.3) (C.4) (C.5) 53

55 C C W F C k S C k W F C k S C k A C K 4 3 A C K A C K (C.6) (C.7) (C.8) (C.9) (C.3) 54

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