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4067VEMENTS OF SUBMARINE PIPELINES CLOSE TO PLATFORMS

Andrew C. Palmer, University of Manchester;T.S. Ling, Total Oil Marine Limited

Copyrlght 1981 Offshore Technology ConferenceThis paper was presented at the 13th Annual OTC in Houston, TX, May 4-7, 1981. The material is subject to correction by the author. Permission to copy Is restricted to an abstract of not morethan 300 words.

ABSTRACTAn analytical model of expansion movements a t the

ends of pipel ines is developed. A comparison withmeasurements on two North Sea pipelines shows that theanalysis is consistent with observed behaviour, and ca nbe used to assess th e resu l t s of co rr ect ive ac ti on. Analternative mechanism, that of creep deformation incorrosion coa ting , is analysed brief ly.INTRODUCTION

Expansion due to changes in temperature andinternal pressure can p roduce substantia l movements a tth e ends of submarine pipelines l At platforms, thesemovements are important b e c a ~ s e they can overstressr i sers and elbows, and bring the pipe into contact withthe platform i t se l f .

The paper beg in s by describing th e mechanisms thatgive r ise to expansion movements, and goes on to ananalysis that predicts how much movement i s to beexpected. The resu l t s are compared with measurements ontwo North Sea pipelines. In a few instances, anothermechanism may occur, and the movement may be due tocreep deformation in th e corrosi9n coating : th is wil lbe ana ly sed brief ly.MOVEMENTS AT THE END OFA PIPELINE

Consider a s t raight submarine pipeline connectedto a platform r iser (Fig.la). The r i se r passes throughc lamps on the platform, and then has a 900 elbow. At ashor t d i st ance from the platform, the pipeline reachesth e bottom, and from then on i s continuously in contactwith i t .

I t i s helpful to begin by considering why thepipel ine should tend to move. The operating temperatureand pressure are h ig he r t ha n the temperature andpressure when the pipe was t ied in . Because th etemperature i s higher , t he p ip e li ne tends to expand.Far from the platform, the expansion i s constrained byfrict ion between t he p ip e li ne and the sea bottom, amlongitudinal expansion s tresses are se t up. At thepla t form, however, t he p ip e li ne i s only sl ightlyconstrained (by the ver t ical leg of the r iser , which i srelat ively f lexible) , and there it can expand freelyReferences and i l lus t rat ions a t end of paper

17

and move towards the piatform:Alterat ions o f p re ss ur e also cause movements.Close to the elbow, in the horizontal leg, thelongi tudinal st ress is tens i le , am the combinationof circumferential and longitudinal s t ress induces alongitudinal tensi le s t r a in , and therefore alongitudinal movement. Far from the platform, on the

other hand, longitudinal movement is prevented byf r ic t ion on the b ottom : th ere the strain is zero andt he long itud ina l s t ress is no t the same as it is closeto the elbow.

I t follows that both temperature and pressurechanges induce movements. At a distance from thepJ.atform, f r ic t ion prevents these movements, but itdoes not do so close to the platform. The movementsoccur within a t ransit ion region whose length dependson the l imit ing f r ic t ional force between the bottomand the p ipe l i ne : i f f r ic t ion i s large, the t ransit iorregion i s shor t and the movements are s ma ll, b ut i ffrict ion i s small the movements are larger.

I f the operating temperature and pressure arereduced, the movement towards the platform i s reversed.only par t of the original movement re turns , and thereremains a residual movement towards the pla t form, eveni f t he p res su re and temperature are returned to th ei rt ie- in values. This i s because f r ic t ion always opposesmotion, so that when the temperature is reduced thef r ic t ional forces do not return to zero , b ut par t ia l lyreverse, holding t he p ip el in e in i t s extended positionand preventing it from sl ipping back.ANALYSIS

The ideal izat ions used in the analysis are thosecustomary in pipeline engineering, and the errors theyintroduce wil l almost alw ays be negligible i n p r ac ti ce .They are :(1) that the pipe remains elas t ic , am t ha t i t sma te ri al p roper ti es a re described by Young 1 s modulus E,Poisson's rat io v an d l inear thermal expansioncoeffic ient ll .(2) that the pipe can be treated as a s t ra i gh t t h in walled circular tube of thickness t and mean radius R(defined as (outs ide d iamete r - t.

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Longitudinal movements are confined to a length z , th edistance from the platform to th e (imaginary) 'anchorpoint ' beyond which no movement occurs. Beyond th i s ,EL i s zero, and socrL vpR/t Ea8 (4)

vpR/t Eet8 l exp(-x/A) in x Z (5)The longitudinal st ress between th e platform and th eanchor point i s statical1.y determinate. F ig . 1.b showsthe forces that act on a segment of th e pipel ine andi t s contents between sect ion across the r iser j u s tabove th e e lbow and a ver t ical sect ion a t a distance xfrom the platform; x i s less than z. At the r ight-handend, 21TRtcrL i s t he long i tud ina l force in the pipe wall ,and 1TR2p th e longitudinal force on t he cont en ts . Att he s ec ti on above the e1.bow, the only horizontal forceis the shear force S, which is negligible. OVer th elength x, the pipeline i s moving towards the platform,and so th e bottom exerts on th e pipe a force f peruni t l eng th , d i re c ted away from th e platform. Sincethe segment and i t s contents are in equi l ibr ium, th eresu l tan t horizontal force on it must be zero, and so

(3) tha t the l imit ing longitudinal force f per uni tlength, between the pipel ine and th e bottom, isuniform along t he l engt h, independent of the d is tancemoved, and the same for ei ther direction of motion.(4) tha t when the l ine was t ied in , i t s temperature wasthe same as t ha t of the s ea wate r d ur in g subsequentoperation, that i t s in ternal pressure was negligible ,and there was no cold spr in g.(S) tha t the force differences as soc ia ted wit h thelongi tud ina l pressure grad ien t are negligible overthe length of pipel ine tha t takes p art in the movement(6) tha t the temperature of the l ine is not necessari lyuniform, but can be represented by an exponentialfunction of distance from t he p l at fo rm , so t ha t

8 (x) = 81 exp (-x/A) (1 )where 8 (x) i s the temperature difference between thepipel ine and the water, a t a distance x from theplatform, 8 i s th e difference a t the plat form, and Ai s a decay length over which the temperature differencefa l ls to l ie (0.369) of i t s i n i t i a l value. This assumeddistribution corresponds to the steady s tate reached i ff luid f lows a long th e pipel ine away from th e platforma t a uniform r a t e , and t he o ve ra ll heat t ransfercoeff ic ien t is independent of time and temperature.A n eg ativ e v alu e o f A represents f low towards theplatform, and a zero value represents uniformtemperature.(7) that the shear force in the v e rt ic a l r is e r le g i snegligible by compari son with other forces in thesystem.

The longitudinal s t rain E: and s t ress cr , theci rcumferent ial st ress crH and Lthe t e m p e r a t u ~ e r ise 8a re r el at ed by the stress-s train- temperature relat ion1."'L = E(O"L - VO"H) + eta (2 )

and the change in ci rcumferent ial s t ress to thepressure p.bYcrH = pR/t

o = fx + 21TRtcrL - 1TR2pcrL = ~ R / t - fx/21TRt in x < Z

(3 )

(6)(7 )

18

The length Z over which movanent occurs can be found .f.rom the condition that crL is continuous a t z, and so,by equating the values of crL in equations (5) and (7)z is the solution ofZ = (1TR2p / f ) {1 - 2v+ 2E : l t exp(-Z!A)} (8 )

I f th e temperature is uniform, th is reduces toZ = (1TR2p/f ) ( I - 2v + 2Ea8l t/pR) (9)The displacement u, posit ive away from the platform,

i s r el at ed t o the longitudinal strain byE:L '" du/dx (1.0)

and ca n be determined by substi tu ting (7 ) in to (2)and then integrat ing (10)'. At the platform, themovement !::. i s!::. = ~ : E : L ( X ) dx

a8 l A{l-exp (-z/A) ) + ~ { { ~ - V ) p R z / t - f z2/41TRt}(11)

and i f the temperature i s uniform!::. = 1TRE (a8 l )2 t / f { 1 + E:8l r ' ~ - V ) } 2 (12)I t should be noted tha t th e temperature effect and thepressure e ff e ct i n te r ac t in a nonlinear manner the

t o t a l expansion movement i s not th e sum of themovement tha t would be i nduced by pressure alone andthe movement tha t would be i nduced by temperaturealone.I f the temperature and pressure are reduced, asegment of the pipeline moves away from the platform,

and on tha t segment th e f r i c t iona l force acts towardsthe platform. I f the temperature i s uniform bothb e f o r e and after a t empe r a t u r e r e d u c t i o n f r a n a1. t o a2and t he p r es su r e i s simultaneous1.y raiuced from PI toP2, analysis by the method described above shows t ha treversed movement occurs over a distance y , 1.ess thanz, given byy = t{ ( ~ - v ) (Pl-P2)1TR2 + Eet1TRt(8 l -8 2 )} (13)tha t a t the platform em the reverse movement !::.sd is2lIsd = f/1TRtE (14)and that t h e long it ud ina l s t ress i s

~ P 2 R / t + fx/21TRt in x < YcrL = ~ l R / t - v (PrP2)R/t + Ea{8 r 82) - fx/21TRt

iny

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used to find th e force needed to move i t . In thisinstance f i s taken as 1500 N/m (153 kg/m), whichcorresponds to a coefficient of 0.9 and a gas-f i l ledsuhnerged weight of 1670 N/m (170 kg/m), but inorder to check the influence of the choice ofcoefficient a Second se t of calculations was made withf equal to 1330 N/m.is determined by

(18)The temperature decay length

A = mc /gpwhere m i s the mass flow r a t e , cp the gas s pe c if ic h ea tand g the rate of heat t ransfer :from t he p ip e li ne , peruni t l engt h p er uni t temperature difference from thesurrounding water : it i s assumed tha t th e thermalresis tance between the gas and th e s teel pipe i s smallby comparison with tha t between th e s teel and the waterThe estimated value of g is 100 W/m degC. The decaylength A is found to be several kIn, even a t low flowrates .

6l exp(-x/Al) to 62exp(-x/A2)' for instance. A segmento f p ip el in e close to the platform then moves away fromit , but another intermediate segment moves towards theplatform, because of interact ion between the nonuniform distribution of thermal st rain and a reductionin a xia l compressive force tha t follows from th er ev er sa l o f movement. Analysis then leads to coupleddifferent ial equations tha t have to be i nt egr a ted s tepby step. However, in many pract ical instances oftemperature and pressure reductions during operat ion,th e le ng th o f p ip el in e influenced by reversed movementis small enough for the simple idealization of uniformtemperature to be a reasonable one.

I t i s sometimes necessary to reduce expansionmovements by increasing t h e r e si s tance to movementaf te r t h e p ipe l ine ha s gone into operat ion. T his can bedone by backfi l l ing g ra ve l o r crushed roock over thepipel ine. An important pract ical case i s the followingsequence

and t he addi ti on a l movement a t the platform ist.ad = ( ~ - v ) (P3-PJ.)RV/Et+ a(63A3(J.-exp(-v/A3)-61Al (l-exp(-v/A l - 2 7 f ~ t E 1 : s: F ( ~ ) dx (17)

COMPARISON BETWEEN FIELD MEASUREMENTS AND THEORYThe F ri gg g as pipeline 1 a t treatment platform TPlin th e North Sea provides an unusually favourableopportunity for comparison between observation andanalysis. The r i se r i s in a ver t i ca l sha f t i ns id e th econcrete platform; the shaf t i s usually dry, but i snormally flooded in win te r. A f te r an elbow a t th ebottom of th e r i se r , th e pipel ine passes through a sealand leaves the platform through a hori zon t al t unne l.Access to the elbow through the shaf t permits accuratemeasurements of pipel ine movement, without th e need to

rely on divers.The analy si s r equ ir e s data on th e platform

p re ss ur e, t he gas tempera tu re, and th e pipe properties,a l l readily available and SUbjec t to l i t t l e uncertainty.I t also r equ ir es t he l imit ing longi tud inal f r ict ionalforce f per unit length, and th e t empera ture decaylength A. These two quan ti ti es a r e more dif f icul t toest imate, and it was therefore desi rable to invest igateth e sensi t ivi ty of the r es ul ts to th e assumed values.Since t h e p ipe li ne was not bu ri ed , i t s l imit ingf r i c t iona l resis tance could be estimated by multiplyingi t s suhnerged weight by a longitudinal f r ict ioncoefficient. Experience in bottom pu ll i n st a ll a ti o n ofindicates tha t i f a pipeline ha s been in place fo rsome time, a coefficient of between 0.8 and 1 should be

(m)oc cur s

34303780

zdistance overwhich movement

1.0841.1921.035

.movementa tplatformem)

30, conditions were as follows :l3 l bars (12.8 MN/m2)38 0c28 Mcm/day (990 MMscf/d)

8.7 kIn8.7 kIn

16, conditions were as follows2144 bars (14.4 MN/m )40.3 c30 Mcm/day

On Septemberpressuretemperatureflow rate

pressuretemperatureflow rate

On November

calculatedf 1500 N/m, Af = 1330 N/m, Aobserved

The combination of pressure, temperature and flowra te th at would be expected to induce th e maximummovement occurred on on e o f th re e days in the fa l l of1979, and each wil l be examined in turn.

The sea tempera ture .i s taken as SoC, and so th etemperature r ise 61 i s 33 dege. Fig. 3 is a graph ofcalculated movement a t the platform as a function of ,fo r the al ternat ive assumed values of f . The estimatedvalue of A is 8.7 kIn. The calculated and observedmovements are as follows :

The agreement between observed and calculated valuesis good.On October 22, cond it ion s w:ere as followspressure 133. bars (13.0 MN/m2)temperature 39.5 cflow ra te 25 Mcm/day

and the estimated value of A was 7.6 kIn. The calculatedmovement i s 1.114 m for 1500 N/m longi tudinal f r ic t ion ,and the movement observed was 0.970 m. I t is not clearwhy the observed movement was 0.065 m less than onSeptember 30, but the most l ikely explanation is tha tthe system temperature had not fully returned to i t ssteady-state distribution af t e r a br ie f shutdown onOctober 21..

and th e es tima ted va lue of A was 9.5 kIn. The calculatedmovement was 1.272 m, again fo r 1500 N/m longitudinalf r ic t ion , and the observed movement was 1.090 m.

The analytical model ca n be tested fur ther by

+ 2 ~ R t E { ( 6 3 e x P ( - v / A 3-6 1exp (-v/A l )} (16

pressure PI ' temperature 6l exP(-x/Al)anchoring backfil l is placed over th e l i ne ,and i nc rea se s t he l im i ti ng r e si st ance tolongi tudinal movement by F(x) per uni t lengththe pressure is increased to P3 and th etemperature to 63exp(-x/A3) > 6l exp(-x/Al)

is instal led, th e le ng th v over whichoccurs is th e s olu tio n o f th e

thenf i r s t

thenAfte r th e anchorfur ther movementequation

(x)dx

19

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comparison between observed and calculated movementsduring a shutdown. A br ief shutdown last ing 12 hourstook place on November 14 1979, and th e observedreverse movement was 0.230 m af t e r 12 hours. Thepressure drop was not measured, but was estimated to be4.0 MN/m2 (41 bars) af t e r 12 hours. An estimate of therate of temperature drop was made by assuming th e heatt ransfer rate to be 100 W/m degC, th e same as when gasis flowing (which i s reasonable, since most of thethermal resistance i s associated with the concrete,rather than with heat t ransfer a t the inside wall); th ethermal capacity of the pipel ine an d th e ga s i sestimated to be 1.32 MJ/m de g C. After th e 12 hours,the estimated fa l l in temperature ~ 1 - e 2 i s 33.7 degC,and so th e temperature ha d fallen from i t s i n i t i a lvalue of 400 C almost to the sea temperature of SoC. Incalculat ing the longitudinal movement, it was assumedthat f was 1500 N/m, and that the effect of variationof temperature with distance from th e platform wasnegligible, a reasonable assumption since the distanceover which reversed movement occurs i s only 1500 m. Thecalculated reverse movement was 0.337 m. The agreementwith the measured value is quite gbod, in the l i gh t ofth e sensit ivi ty of th e resu l t to th e changes oftemperature and pressure, neither of which was measureddirect ly.After an evaluation of the movements observed a tthe platform, it was decided to take action to reduceth e movements that would follow future increases inoperating temperature and pressure. In the early monthsof 1980, crushed rock backfi l l was placed on a numberof sections of the pipel ine close to th e platform, aspar t of a wider program of span correct ion. Thespecif ied cover above th e pipe i s 1 m. Taking therock par t ic le specific gravi ty as 2.7, and the in-placevoids rat io as 0.6 (porosity 0 .38) , the submerged uni tweight is 10.3 kN/m3 (1050 kg/m3 ) . The estimated

a d d ~ t ~ o n a L L o n g ~ t u d ~ n a L r e s ~ s t a n c e F L4.L kN/mthe pipel ine moves through the rock (against the extrafrict ion generated by th e rock 's weight) and 12.7 kN/mi f th e rock above th e pipel ine i s carried along withi t . However, since the calculation of F involves anumber of uncertain factors (among them the s ta te ofs t ress in th e rock above the pipeline, and th e extentto which arching ca n t ransfer the weight of the rockabove th e l ine to th e rock on ei ther s ide) , it wasdecided to adopt a lower value of 7 kN/m fo r designpurposes.

A further shutdown on March 22 made it possibleto confirm the effect iveness of th e backfi l l . At thattime, rock had been placed over two sections, one of464 m (from pk 360.136 to 360.600) and one of 93 m(from pk 359.623 to 359.716); th e platform tunnelentrance i s a t pk 361.068. Fig'. 4 shows th e movementsand temperature and pressure changes that occurred.The observed movements ca n be compared with thosecalculated under three al ternat ive assumptions, thatth e addit ional longitudinal resistance F generated bythe backf i l l i s 14 kN/m, that it i s 7 kN/m, an d that iti s zero. The comparison confirms that the presence ofthe backfi l l does reduce th e movements signif icant ly,but the resu l t s are not suff icient ly sensit ive to thevalue of F fo r it to be possible to make anindependent estimate. Another comparison ca n be madeby calculating th e ' forward' movements af t e r r es ta r ta t midnight on March 22/23. The calculated movementsare 0.114 m i f F i s 14 kN/m, 0.118 m i f F i s 7 kN/m,and 0.163 m i f F i s zero, in th e f i r s t 5 hours ofoperation, while the observed movement was 0.125 m.

20

Loekenl ha s described a second instance ofsubmarine pipeline movement, in the 36-inch (914.4 romEkofisk-Emden ga s l ine a t platform R. Stephens an dRawlins2 describe work on the creep movement of anunspecified 'p ipeline E I , but a comparison between thepapers strongly suggests that they are describing thesame l ine as Loeken. The submerged weight i s notavai lable , but ca n be estimated a t 1860 N/m (190 kg/m)in operat ion. The f i r s t 400 m from the platform areunburied, and so in this sect ion the longitudinalresistance i s 1680 N/m i f the frict ion coeff ic ien t i s0.9. Beyond that there i s 2 to 3 m cover of sand. Theaddit ional resistance generated by th is cover i s hardto est imate, but Loeken suggests a value of 44 kN/m fo1.8 m cover an d a so i l frict ion angle of 400 Withinth e period covered by the reported movement data , themaximum movement away from th e platform shouldcorrespond to mid-April 1978, when the maximum pressurwas 11.7 MN/m2 (1700 Ib/ in2 ) and the temperaturereached 420 c (l080 F). The calculated end movement i splotted in Fig. 5 , as a function of the uncertainlongitudinal resistance F in the buried sect ion. Theobserved movement i s somewhat less than the calculatedmovement, unless F i s as low as 10 kN/m (1 tonne/m;700 l b / f t ) . The resu l t i s sensi t ive to the amount ofcover between 400 and 800 m from the platform, an d10 kN/m may be a reasonable value, part icular ly i farching i s signif icant or i f the cover i s less thanintended.CREEP BETWEEN A PIPELINE AND ITS WEIGHT COATING

Some instances of thermal movement may be due todeformation of bituminous corrosion coating, whichmay soften when the pipeline temperature r i ses as itgoes into operat ion, and wil l then deform as a viscousflu id . That would allow the pipe to expand even thoughi t s concrete coating remained statiQIlary, by sheard e f o r m a t ~ o n w ~ t h ~ the c o r r o s ~ o n c o a t ~ n g , and t h ~ smechanism i s on e explanation of movements whichcontinue to increase af t e r the pipeline i s in operatioI t i s important to make clear that it i s not the onlyor most l ikely explanation of such movements : in mospipelines, the operating pressure, temperature and flowrate are progressively increased af ter s ta r t -up , an dquite modest increases can produce substantialaddit ional movements.

A complete analysis of creep deformation i scomplicated, because the corrosion coating ha s acomplex rheological behavior close to i t s softeningtemperature, may be non-Newtonian, and is stronglytemperature-sensitive. However, a simplified modelthrows l igh t on the factors that govern creep, andallows on e to determine whether or not it might beimportant.Imagine a pipeline subject to thermal expansionalone, so that the pressure ef fec t i s negligible. The

temperature increase is e ( x , ~ ) a t time and distancex from th e platform. Th e displacement of the pipe i t set se l f i s u { x , ~ ) , an d the displacement of the concretecoating i s v ( x , ~ ) , so u- v i s the relat ive movementbetween the pipe an d the coating. The thickness of thecorrosion coating i s h, and i t s material is ideal isedas a l inear viscous f luid with viscosity n a t the pipetemperature. The ear l ier assumption that the pipe isthin-walled i s extended to include the coating, sothat the radius of the coating i s identified with themean radius R. Longitudinal forces in the concrete areassumed to be negligible b y comparison with those inthe s t ee l . Fig.6 shows schematically the forces that

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act on th e di f fe rent parts of an element dx. At thelef t-hand end of the element, the thermal s t raincomponent is a6 and th e t o t a l longi tudinal s t rain i sau/ax, an d so the longi tudinal st ress is E(au/ax - a6),from (2), an d the longitudinal force i s th is s t ressmultiplied by the wall cross-section 2 ~ R t . At the r ightend, th e longi tudinal force i s different , because au/axand a6 wil l in general have different values. Theshear force on the outer surface of the pipe i s theproduct of th e viscosi ty n, the veloci ty gradient

( a u / a ~ - a v / a ~ ) / h an d the area 2 ~ R d x . Since iner t iaterms are negligible, longi tudinal equilibrium of thepipe element gives th e governing dif feren t ia l equationau av (a 2u a6 )a:r .- a:r = (Eth/n) ~ - a ax (19)

The group Eth/n has the dimensions of (length)2/time,an d is a di f fus iv i ty , analogous to thermal diffusivi tyin heat transfer theory an d coeffic ient ofconsolidation in so i l mechanics. Stephens an d Rawlins2derived th e same group, in a ra ther different way.

A general solution i s complex, because of th estrong dependence of viscosi ty on temperature. I t isuseful to examine a simplified problem. Imagine tha tthe temperature 6 is rapidly raised to 61' an d thenheld uniform along the pipel ine an d constant with time,so tha t o for < 06 ( x , ~ ) (20)61 fo r > 0which is an ideal izat ion of rapid s tart-up a t a highflow rate . The r ise in temperature wil l be followed byan immediate expansion, because of s l ip between th econcrete coating an d the bottom. The amount of movementis governed by the analysis described ear l ier , becauserelat ive movement between the pipe an d the concretecannot occur instantaneously, since tha t would imply anin f in i te veloci ty gradient in the corrosion coating. I tfollows t ha t , immediate af ter the temperature r i se ,

- I T R t ~ e ) 2 E ( 1 - x / z ) 2 / f for x < zv(x,O+) = u(x,O+) = (21)o for x > zwhere z = 2 ~ R t a 6 E / f (22)Once th e instantaneous motion has taken place, viscouscreep begins. Intuit ion suggests that creep wil l relaxth e longitudinal forces, and tha t the force between theconcrete coating and the bottom wil l tend to fa l lra ther than r ise (at leas t in th e zone tha t slippedin i t i a l ly ) . I f th is i s so, we ca n conjecture tha t therewil l be no further movement of the concrete, so that

= 0 for > 0 (23)an d then, since 6 is uniform, th e governing(19) becomes th e diffusion equation equation

a u / a ~ (Eth/n)a 2u/ax2 (24)The i n i t i a l condition is (21), an d the boundarycondition a t th e end x equal to zero i s given by th econdition tha t the longi tudinal s t ress a t the end bezero, an d i s

a6 for > 0 (25)The solution of (24) subjec t to these i n i t i a l an dboundary conditons is elementary, and is

21

u(x, t)= - 2a6 { ~ e x p (_x2 / 4 K ~ ) - ~ e r f c (x2 / 4 K ~ ) ~ } 21 (X) { (x-x' ) 2 (x+x') }+ --- u(x ' ,0 ) exp - - - - - exp -" - - -( ~ K ~ ) ~ 0 4 K ~ 4 K ~dx ' (26)

whereK (Eth/n) (27)

At th e end, the displacement u (0 , a t time i su ( O , ~ ) = -2a6 ( K ~ / ~ ) I : ! - ( ~ K ~ ~ Z 2 ) ' i s: 1 _ ~ ) 2 e x p { _ ~ : ~ 2 }

(28)where A i s the in i t ia l displacement a t the end, from(21). The f i r s t term in this expression becanesdominant as increases, and corresponds to thecontinued expansion of the pipe through the coating,while the second term decays from A to zero, an drepresents the redis tribution of the i n i t i a l movement.

This analysis can be generalised to includepressure , and to allow for temperature gradients. I tha s unfortunately not been possible to compare it witha case of movement tha t i s known to be du e to viscouscreep in the corrosion coating.CONCLUSIONS

Observed movements of s u b m a r i n e " ~ i p e l i n e s areshown to be consistent with an elas t ic / f r ic t ionalexpansion analysis (which is well known in a simpleform). The analys is correct ly predicts reversedmovement during shutdowns, and th e response to backfi l lintended to increase longitudinal resis tance. Othereffects , such as th e relaxation of ' snaking' andre la t ive movement in the corrosion coating, maysometimes be s ignificant, but it is not necessary toappeal to them to explain th e movement of the Friggpipel ine.NOTATIONEfFhpRt\ lVXYzat:.e:nKA6'V

Young's modulusl imit ing longi tudinal f r ic t ion per uni t lengthadditional limiting longi tudinal f r ict ion providedby backfi l lcoating thicknesspressuremean radiuswall thicknessdisplacement of pipedisplacement of concrete coatingdistance from platformdistance over which reversed movement occursduring shutdo"W11distance over which expansion movement occursl inear thermal expansion coeffic ientmovement a t platforms t rainviscosi ty(diffusia:ity)I:!decay length for exponential dis tribution oftemperaturePoisson's rat iointegrat ion variable

(J s t ress

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ACKNOWLEDGEMENTThe authors thank Total Oil Marine Limited forpermission to publish this paper.

a:w(j )a:

CLAMP

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REFERENCES1. Loeken, P.A. 'The "creep" on the Ekofisk-Emden 36"gas pipeline ' , Proceedings, 12th Annual Offshore

Technology Conference, Houston, 1980, paper OTC3783, 393-40l.2. Stephens, H.G. and Rawlins, C.E. 'Axial movement of

warm buried pipe l ines ' , Proceedings, Interpipe 1980Houston, 146-161.

Fig. 1 - DEFINITION SKETCH (a ) geometry (b) forces on a segment.

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DISTANCE FROM PLATFORMFig. 2 - STRESS, STRAIN AND MOVEMENT AT THE END OF A PIPELINESolid l ines represent condition after a reduction in temperatureand pressure, dashed l ines condition before; numbers refer toequations in text .

Fig. 3 - RELATION BETWEEN CALCULATED MOVEMENT AND TEMPERATURE DECAY LENGTH.

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