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2 Tensn and Expansn Analyss f Ppe-n-Ppe Rsers: Part A, Theretcal rmulatn Kevn Chuanjan Man, Bn Yue, Adam Szucs, Rcky Theth 2H ffshre nc. Hustn, TX, USA ABSTRACT Ths paper prvdes a mathematcal mdel fr accurate and effcent calculatn f the elngatn f each strng wthn a ppe-n-ppe tp tensned rser system due t gravty, pressure and thermal expansns. The resultng rser system elngatn effect s subsequently derved cnsderng the nteractns amng all rser strngs. n the case where a tensner exsts, ts nnlnear relatnshp between tensn frce and dsplacement can be captured by usng an teratve calculatn methd. Examples shw that the apprach s effcent, and ths mathematcal framewrk s capable f calculatng the rser system tensn as well as dstrbutng tensn t each strng. Wth the prpsed apprach, nner rser ppe pretensn can be determned effcently cnsderng lad cndtns durng the lfe tme f the rser system. KEY WRDS: Rser; Ppe-n-ppe; Pretensn; Tensner; Thermal; Pressure; Expansn NTRDUCTN Ppe-n-ppe Tp Tensned Rser (TTR) systems are wdely used n the ffshre l and gas ndustry. The man feature f a typcal ppe-n-ppe TTR system s a cncentrc nner strng (tubng) prtected by ne r mre prtectve uter strngs (casngs). All strngs are rgdly cnnected t each ther at the tp end f the rser. The bttm ends f casngs are fxed t the subsea wellhead, whereas the bttm end f tubng s fxed t a mud lne tubng hanger r dwnhle packer. Centralzers are typcally used between the uter casng and the nner ppes. nsulatn materals, n the frm f sld, lqud, r gas, exst n the annulus between strngs. Generally, the wall temperature and annulus pressure are hgher fr the nner strngs than the uter strngs. f the strngs were nt cnnected, the free elngatn f nner strng wll be larger than thse f the uter strngs. Hwever, due t the exstence f end cnstrants, pressure and temperature varatns result n the redstrbutn f tensns between the rser strngs. The cmplexty f the prblem s further enhanced by the nner rser pretensn and the rser external frces. The external frces nclude weght, envrnmental lads, and tensner lad. Due t the mprtance and cmplexty f the subject, t s desrable t have a systematc and accurate methdlgy t calculate the tensn dstrbutn acrss all rser strngs. Ths paper develps a mathematcal mdel fr accurate calculatn f the free elngatn f each strng due t gravty, pressure and thermal expansns f a ppe-n-ppe TTR system. The resultng rser system elngatn effect s subsequently derved cnsderng the nteractns f all strngs. Statc calculatns are made assumng the rsers are suspended vertcally frm a flatng structure whch s assumed as fxed. Durng the develpment the theretcal frmulatn, the calculatn cnsders the effects f: Rser temperature; Rser pressure (ncludng end-cap, Pssn effects); Rser weght and stffness; Tensner system stffness. Each rser strng s treated as a sprng wth ts stffness defned by ts length, crss-sectnal area, and Yung s mdulus. The rser strngs are cncentrc and cnnected at bth tp and bttm ends. The bttm end s fxed and the tp end free t strke relatve t the vessel. The tensn frce at the tp end, where the tensn rng cnnects the rser t the tensner system, vares as a functn f tensner strke relatve t the vessel, and s cnsdered n the tensn dstrbutn calculatn. Belw, equatns are derved t prvde a mathematcal mdel fr tensn dstrbutn calculatn. The free expansn equatns are prvded frst; then the equatns fr rser strng expansn wth cnsderng tensner stffness are derved; fnally, the tensn redstrbutn equatns are derved. A generc applcatn f the derved mdel s then prvded as example. The example s prvded t shw Learn mre at

3 hw the mathematcal mdel prvded n ths paper s appled. The example als emphaszes the mprtance f the nner pretensn fr TTR system. nally, cnclusns are prvded. The cmparsn between the mathematcal mdel derved n ths paper and fnte element analyss s presented n anther paper: Tensn and Expansn Analyss f Ppe-n-Ppe Rsers: Part B, nte Element Mdelng. RSER EXPANSN AND TENSN REDSTRBUTN EQUATNS The expansn and tensn dstrbutn equatns are presented n ths sectn. rstly, the free expansn f a sngle-strng rser system s analyzed wth cnsderatn t the thermal and pressure effect. The tensner stffness s then added nt the equatns. The tensn redstrbutn equatns f the rser strngs are subsequently derved cnsderng free rser expansn, rser stffness, and tensner lad. Rser ree Expansn due t Thermal Effect Changes f rser strng temperature frm the undsturbed nstalled rser cndtn result n thermal lads at the surface wellhead. Ths nduces an expansn n the ndvdual rser strng, whch s lmted by the mudlne, surface wellhead bundary cndtns and the verall stffness f the cmbned rser system. Hence, rser effectve tensn s redstrbuted between the strngs t cmpensate fr these restrants n free end thermal expansn. Referrng t a text bk, e.g. (Schaffer et al., 1995), equatns defnng the free thermal expansn due t temperature varatns are gven as fllws: L k T dl k T l (1) T T L k T dl k T l (2) k = ppe ceffcent f thermal expansn L T = uter rser free thermal expansn due t temperature varatn frm undsturbed nstalled cndtn ver uter casng strng length L T l = nner rser free thermal expansn due t temperature varatn frm undsturbed nstalled cndtn ver nner casng strng length T l = change n temperature frm undsturbed nstalled cndtn (.e. befre prductn startup) ver uter casng l strng length T = change n temperature frm undsturbed nstalled cndtn (.e. befre prductn startup) ver nner casng strng length l Rser ree Expansn due t Pressure Effect nternal flud pressure and external flud pressure nduce strans n the ndvdual rser strngs. The expansn f a ppe due t nternal pressure s nt smply the extensn due t the end cap frce. Ttal rser extensn due t pressure s the cmbned effects f bth pressure end cap and Pssn (ballnng) effects. Referrng t AP 2RD and usng Hk s law, equatns defnng expansn due t pressure n a rser are gven as fllws, ( P A P A ) L Lpz EA (3) D L Lp v( P P ) P 2t E (4) ( PD P D ) Lpr v L D D E (5) L pz = rser free expansn due t axal stress L p = rser free expansn due t hp stress L pr = rser free expansn due t radal stress P = rser mean nternal pressure P = rser mean external pressure D = rser utsde dameter D = rser nsde dameter L = rser length E =Yung s Mdulus v = Pssn s Rat (typcally 0.3 fr steel) 2 2 A = crss-sectnal area f rser steel = ( D D ) 4 2 A D 4 2 A D 4 Eq. 3, 4, 5 are used t calculate rser expansn n axal drectn due t end cap pressure, hp stress, and radal stress, respectvely. The axal expansns f hp stress and radal stress represent Pssn s effect and can be cmbned tgether as fllws, ( P A P A ) L Lpp Lp Lpr 2v (6) EA Eq. (3) and Eq. (6) can be cmbned t btan the rser expansn equatn due t pressure, ( P A P A ) L Lp 1 2v (7) EA Eq. (7) can be used fr bth nner rser ppe and uter rser ppe t btan the rser expansn due t pressure. A rser ppe s dvded nt several sectns t accunt fr varatn f ether external r nternal pressures. Eq. (7) can als be derved frm Lamé equatns (Tmshenk, 1976). Spark derved the stran equatn n 1984 startng frm Lamé equatns wth zer effectve stress, whch als prves Eq. (7). Rser ree Expansn due t External Lads External lads n the rser system cnsdered n the calculatn are the fllwng: Tensner lad appled t rser system nner casng rser strng apparent weght durng lck ff t the Learn mre at

4 hanger n the surface wellhead Tubng r wrk/drll strng apparent weghts frm surface BP t reservr durng hung-ff cndtns Addtn f BP, tree, and tensner system related weght abve tensn rng befre runnng tubng r a regular ppe wth unfrm sze, the expansn under tp tensn and weght s gven as fllws, ( 0.5mgL) L L EA (8) = rser tp tensn m = rser mass per unt length g = gravty acceleratn cnstant Tensner System Ether a tensner system r a buyancy tank can be used t tensn a TTR system. The tensner system has nn-lnear frce versus strke. r mst tensner systems, the relatnshp between stffness and strke s als nn-lnear. Dsplacement Changes f Cmbned Rser Strng There are many steps nvlved n rser system nstallatn. The uter rser ppe s nstalled frst wth the rg hk. After the tensner system s engaged t the uter rser, the nner rser s run usng the rg hk and engaged at the nternal teback cnnectr (TBC). The nner rser s then pretensned, and hung n the nner rser casng hanger. The pretensn prcess ccurs durng rser nstallatn and defnes the tensn dstrbutn between rser strngs and strke fr all rser cndtns durng servce. The tensner stffness vares durng the nstallatn prcess as the tensner strkes due t weght and pressure changes. A seres f cmbned rser stretch equatns are prvded n ths sectn wthut and wth cnsderng the tensner stffness. Let T l be tp tensn f uter casng and T l tp tensn f nner rser durng lck-ff the nner rser t the surface wellhead. Then the ttal tp tensn n the tensner system durng lck-ff s, T T T (9) lck l l The cmbned rser extensn durng lck-ff s gven as, L rserl T lck 0.5( m m ) gl k k k = axal stffness f the uter rser k = axal stffness f the nner rser m = uter rser mass per unt length m = nner rser mass per unt length (10) Usng Eq. 8, the expansn f the uter and nner casngs durng lckff are gven as fllws, T 0.5m gl l Ll (11) k T 0.5m gl l Ll (12) k The stretch value f the uter ppe, gven n Eq. (11), s dfferent t the expansn f the rser system, gven n Eq. (10). The dfference between these tw equatns must be cnsdered n rder t accurately calculate the rser strke. The tp tensns f uter and nner rser durng nstallatn affect rser tensn dstrbutn fr peratns. The pretensn prcess durng rser nstallatn ptmzes the rser tensn dstrbutn durng prductn. Pretensn s defned as the ver-pull n the nner rser abve ts submerged weght durng nstallatn. AP des nt drectly defne crtera t determne the pretensn. Hwever, n rder t ptmze the rser respnse, the pretensn s selected such that the base tensns n prmary rser cndtns are balanced between the rser strngs, and cmpressn s mnmzed. The rser stretch due t pressure and thermal effects wth cnsderng tensner stffness s gven as fllws, L k L k P P LrserP (13) k k kten L rserp = rser expansn due t pressure wth tensner L P =uter rser free expansn due t pressure usng Eq. 7 L P =nner rser free expansn due t pressure usng Eq. 7 LT k LT k LrserT (14) k k k ten L rsert = rser expansn due t thermal wth tensner L T =uter rser free expansn due t thermal usng Eq. 1 L T =nner rser free expansn due t thermal usng Eq. 2 Let T tens be tp tensn f the rser durng nrmal peratng cndtn. Smlar t Eq. 10, the rser extensn due t tensner lad s gven as, L rsernr T tens 0.5( m m ) gl k k L rsernr = rser expansn due t tensner lad Rser Tensn Redstrbutn rmula (15) The stretches f rser strngs are lmted by the subsea wellhead, surface wellhead and the verall stffness f the cmbned rser system. nternal and external flud pressure, densty, temperature nduce strans n the ndvdual rser strngs, whch are als lmted by the subsea wellhead and surface wellhead bundary cndtns and the verall stffness f the cmbned rser system. Hence, rser effectve tensn s redstrbuted between the strngs t cmpensate fr these effects and bundary cndtns. Equatns defnng changes n effectve tensn durng the dsplacement f fluds n the rser are gven as fllws, Learn mre at

5 k 1 k 1 f m gl m gl (16) k k 2 k k 2 k 1 k 1 f m gl m gl (17) k k 2 k k 2 Eq. (16) and (17) are used fr the tensn redstrbutn due t weght ncrease fr uter rser nner rser, respectvely. m m = uter rser mass ncrease per unt length = nner rser mass ncrease per unt length The tensn redstrbutn equatns due t pressure effect are gven as fllws, L P k LP k P k LP k k (18) L P k LP k P k LP k k (19) Eq. (24) and (25) are used fr the fnal rser strng tensn due t nternal and external flud pressure, densty, temperature fr bth nner and uter rser. Calculatn Prcedure Nnlnear equatns are used fr tensner system. Therefre, teratns are requred t slve the prpsed mdel n the paper. The calculatn prcedure fr the mdel prvded n ths paper s shwn n gure 1. A pstve tensner lad s frst gven t start the calculatn. Then the tensner strkes and tensner lad are btaned. The current calculated tensner lad s cmpared t the gven tensner lad r prevus strep tensn lad. The teratns are fnshed when the dfference between tw successve calculatns s less than r equal t the cnvergence crtern. Eq. (18) and (19) are used fr the tensn redstrbutn due t pressure effect fr uter rser nner rser, respectvely. The tensn redstrbutn equatns due t thermal effect are gven as fllws, L T k LT k T k LT k k (20) L T k LT k T k LT k k (21) Eq. (20) and (21) are used fr the tensn redstrbutn calculatns due t thermal effect fr uter rser nner rser, respectvely. The tensn redstrbutn equatns due t tensner lad varatn are gven as fllws, k k k T t Ttens (22) k k t tens (23) k = uter rser tp tensn befre tensner lad varatn = nner rser tp tensn befre tensner lad varatn Eq. (22) and (23) are used fr the tensn redstrbutn due t tensner lad varatn fr uter rser nner rser, respectvely. The fnal rser strng tensn due t nternal and external flud pressure, densty, temperature fr uter rser and nner rser are gven as fllw, (24) f f P P T T t t (25) = fnal uter rser tp tensn wth pressure, thermal and weght effect = fnal nner rser tp tensn wth pressure, thermal and weght effect T = Knwn Tensner lad Tcal = Calculated Tensner lad gure 1 lwchart fr Calculatn Prcedure Learn mre at

6 Tensner Stffness (N/m) Elevatn abve MWL (m) Tensner Stffness (N/m) Tensner Lad (te) Tensner Lad (te) EXAMPLE APPLCATN RSER EXPANSN AND TENSN REDSTRBUTN EQUATNS Sftware Mcrsft Excel wth a Macr usng Gal Seek functn s appled fr the example calculatn. Envrnmental Data A summary f the envrnmental data used n the example s gven Table 1. Table 1 Envrnmental data Water depth (m) 1000 Sea water densty (kg/m^3) 1025 Gravty, g (m/sec 2 ) y = x x x x R² = Rser Strke (m) gure 3 Tensner lad vs rser strke prfle fr nstallatn 800 Rser Temperature Rser ppes are classfed as ether ht durng peratng r ambent when t s nt peratng based the lad cndtns. The summer seawater temperature s used when rsers are nt peratng. Rser strng temperature prfles fr the example are shwn n gure 2. Rser Strng Temperature Prfles y = x x x x R² = Rser Strke (m) gure 4 Tensner lad vs rser strke prfle fr peratn E E+06 y = x x R² = E E Temperature (degc) nner Rser Wnter seawater MWL Tensner System uter Casng Summer Seawater Mudlne gure 2 Rser strng temperature prfles Tw sets f stffness parameters fr the tensner system are prvded. The frst set f data prvdes a tp tensn curve fr rser nstallatn. The ther set s fr rser peratn wheren bth uter and nner ppes exst. A furth rder plynmal equatn s appled t estmate the tensner lad vs. tensner strke data, whch s shwn n gure 3 and gure 4, respectvely. Secnd rder plynmal equatns can prvde a gd estmate f the tensner stffness vs. tensner strke, whch are shwn n gure 5 and gure E Tensner Lad (te) gure 5 Tensner lad vs tensn stffness prfle fr nstallatn 3.4E E E E E+06 y = x x R² = E Tensner Lad (te) gure 6 Tensner lad vs tensn stffness prfle fr peratn Learn mre at

7 Rser Stackup A summary f crtcal rser cmpnents fr a generc prductn rser s gven n Table 2. The nn-scaled stack-up sketch fr a rser cnfguratn s shwn n gure 7. Calculatns f the rser expansn start frm base f the taper stress jnt and end at tp f the tensn jnt. The stffness f uter rser and nner rser s N/m and N/m, respectvely. Table 2 Majr rser cmpnent summary Elevatn frm uter Dameter Wall Thckness Cmpnent Mudlne (m) (mm) (mm) Start End uter Tensn Jnt uter Standard Jnts nner Rser Ppe Rser nternal luds A summary f the rser nternal flud prpertes s gven n Table 3. Table 3 Rser nternal flud prperty summary Cndtn Prductn (Nrmal early) Prductn (Nrmal Late) Prductn Shut n Hurrcane Evacuatn nternal fluds nner Ppe uter Ppe lud Densty (kg/m 3 ) nner uter Ppe Ppe Surface pressure (MPa) nner uter Ppe Ppe Brne Gel Brne Gel Brne Gel Brne Gel Drll lr Surface Wellhead Tensner Rd Tensner Rng uter Ppe (Tensn Jnt) nner Ppe uter Ppe (LTSJ) nner Ppe Subsea Wellhead Slp Jnt Surface BP Prductn Tree Surface Wellhead Tensn Jnt nner Rser uter Rser Taper Stress Jnt Subsea Wellhead Cnductr and Casng Man Deck TTR rame Deck MWL = 1000 m Nte: Slp jnt and surface BP are nt nstalled fr nrmal peratng cndtn. gure 7 Rser Stack-up vervew Result Summary The tensn dstrbutns f the nner rser and uter rser fr tw dfferent nner rser pretensns, n (zer) pretensn and 100 te (981 kn) pretensn, are presented n Table 4 and Table 5, respectvely. f the nner rser s nt pretensned, the whle length f the nner rser s n cmpressn durng nrmal peratng cndtn as gven n Table 4. Ths s due t the hgh temperature and hgh pressure n the nner rser. A pretensn must be appled n the nner rser durng nstallatn t ptmze the tensn dstrbutn between nner and uter rser strngs. f the nner rser s pretensned t 100 te durng nstallatn, all rser strngs reman n tensn durng nrmal peratng cndtn as gven n Table 5,. The tensner lads and strkes are pltted aganst the tensner stffness curves n gure 8 t verfy the tensn and strke calculatns. All tensns and strkes match ther respectve tensner stffness curves. Table 4 Rser tensn dstrbutn and strke (pretensn = 0) BP? N N Yes Yes Descrptn Prductn Prductn Prductn Hurrcane (Nrmal early) (Nrmal Late) Shut n Evacuatn uter Rser kn 2,972 2,952 1,980 1,898 Tp Tensn (te) (303) (301) (202) (193) uter Rser Base Tensn nner Rser Tp Tensn nner Rser Base Tensn Cmbned Tp Tensn Cmbned Base Tensn Rser Strke kn 1,946 1, (te) (198) (196) (97) (89) kn (te) (-8) (-6) (16) (27) kn (te) (-40) (-39) (-16) (-6) kn 2,892 2,898 2,137 2,165 (te) (295) (295) (218) (221) kn 1,551 1, (te) (158) (158) (81) (83) m (ft) (1.00) (0.98) (-0.28) (-0.36) Learn mre at

8 Table 5 Rser tensn dstrbutn and strke (pretensn=100 te) BP? N N Yes Yes Descrptn Prductn Prductn Prductn Hurrcane (Nrmal early) (Nrmal Late) Shut n Evacuatn uter Rser kn 2,425 2,405 1,397 1,377 Tp Tensn (te) (247) (245) (142) (140) uter Rser Base Tensn nner Rser Tp Tensn nner Rser Base Tensn Cmbned Tp Tensn Cmbned Base Tensn Rser Strke kn 1,399 1, (te) (143) (141) (38) (36) kn ,032 1,058 (te) (71) (74) (105) (108) kn (te) (39) (41) (73) (75) kn 3,126 3,133 2,428 2,435 (te) (319) (319) (248) (248) kn 1,785 1,782 1,087 1,085 (te) (182) (182) (111) (111) m (ft) (0.21) (0.19) (-1.07) (-1.09) CNCLUSNS A mathematcal mdel fr accurate calculatn f rser strke and tensn dstrbutn due t weght, pressure and thermal expansns f a TTR system has been develped. A nnlnear relatnshp between tensn frce and dsplacement are captured by usng an teratve methd. Examples shw that the apprach s effcent, and ths mathematcal framewrk s capable f calculatng the rser system tensn and tensn dstrbutn n each strng. Wth the prpsed apprach, nner rser (tubng) pretensn can be determned effcently. urthermre, the prpsed mdel can be used t create nput data fr fnte element analyss cnsderng the lad cndtns durng the lfe tme f the rser system. REERENCES Amercan Petrleum nsttute, (2006). Desgn f Rsers fr latng Prductn Systems (PSs) and Tensn-Leg Platfrms (TLPs), Recmmended Practce 2RD. Schaffer, J.P., Saxena, A., Antlvch, S.D., Sanders, T.H., and Warner, S.B. (1995). The Scence and Desgn f Engneerng Materals. RCHARD D. RWN, NC. Spark, C. P. (1984). The nfluence f Tensn, Pressure and Weght n Ppe and Rser Defrmatns and Stresses, J Energy Resurces Technlgy, Vl 106, pp Tmshenk, S., (1976). Strength f Materal, 3 rd Edtn, Part, Kregas, Huntngtn, New Yrk. Yue, B., Man, C. K., Waters, D. (2013). Tensn and Expansn Analyss f Ppe-n-Ppe Rsers: Part B, nte Element Mdelng. Prceedngs f SPE gure 8 Rser Tensner Lad Verfcatn Learn mre at