Wave-Induced Pwp Acting on a Buried Pipeline

7/29/2019 Wave-Induced Pwp Acting on a Buried Pipeline

1/14

CHAPTER240

WAVE-INDUCEDPOREPRESSUREACTINGONABURIEDSUBMARINEPIPELINE

WaldemarMAGDA

ABSTRACTTheesponseofasandyeabedourfacewaterwaves,with special

emphasistowave-inducedexcessporewaterpressureoscillationsisstudiedhereinelationoheverticaltabilityofsubmarineburiedpipelines.hemainobjectofthepeperistopresentastudyofthedistributionpaternoftheporewaterpressureactingaroundthepipeline,andtocalculatetheseepageforce,theup-liftorceparticularly,affectinghepipelnestability,underheassumptionofcompressibleboththeporefluidandsoilskeleton,forthecaseofanarbitraryseabeddepthaswellasfortheinfinitethicknessofthesubsoil.

INTRODUCTIONGenerally,heroblemssociatedwithuriedubmarineipelinese-

pends,onhewaterandwaveconditions.hewaveclimatplaysaverym-portantoleandcannfluencehenteractionbetweenhesubmarineburiedpipelineandhesurroundingoi lsignificantly.npractice,pipelineocatednwaterdepthsupo0 mareburied,whilsthecovermustaveahicknessrangingrom0.5o1.0m,dependinguponhewaterdepthandhecoveringmaterial.

Submarinepipelinesburiedinaseabedareanengineeringmeansoftrans-portforcrudeoilandnaturalgasfrom"off-shore" ilfieldsontoaland.Whenwavespassoverapermeablesandyseabed,porewaterpressureiscontinuously

* Ph.D.,ResearchEngr.UniversityofHannover,Sonderforschungsbereich205,Appelstr.A,3000Hannover1 ,Germany(onleavefromheMarineCivilEngineeringDepartment,HydroengineeringFaculty,echnicalUniversityofGdansk,Poland)

3135


2/14

3136 COASTALENGINEERING992 inducedwithint.mongallenvironmentaloadsusuallyconsiderednoff-shore"pipelinesdesign,thewave-inducedporewaterpressureplaysoneof themostmportantole.hemostriticalproblemndetermininghestabilityofapipelineburiedinpermeableoilsunderwaveloadingishepredictionoftheporewaterpressuresnheoilnavicinityofapipelineDursthoffandMazurkiewicz,1985).Anexcessoftheporewaterpressurecancauseinstabilityofaseabed,liquefactionof theuppersandlayerandthenfloatationwhichcaneveneadoafailureofasubmarinepipeline.hewave-inducedexcessporewaterpressuredevelopedinavicinityof aburiedpipelineisconsideredasaoneofthemainpartsinadesignprocedure.Thewave-inducedupliftorceactingonhepipelinescomparableohedisplacedwaterweightfhepipelineslocatednheporewaterpressureboundaryayerandannadequatedesigncancauselotationofpipelineand,subsequently,aneadocostlyailures.Therefore,tsessentialomproveourknowledgeonhenteractionmongwaves,seabedandasubmarinepipeline.

-fto Figure Definitionsketchfortheupliftorceanalysis.

Its veryomplexndchallengingaskoefineroperlyhewave-inducedexcessporewaterpressureieldaroundasubmarinepipelineburiedinaporousseabed.anyesearchessimplyfiedheproblemassumingboththeporousmediumandporewaterincompressible.UnderthisassumptionLaietal . (1974),LiuandO'Donnell1979)ndLennon1985)nvestigatedhis


3/14

SUBMARINEBURIEDPIPELINES137problemusinganumericalanalysis.iuandO'Donnell1979)consideredwodifferentypesofwavesactingontheseabed,namely,monochromaticandsoli-tary,andintroducedthe integralequationmethodtosolvetheresultingintegralequation.nanumericalsolutionproceduredevelopedbyLennon1985)hepressuredistributiononhepipelinewascalculatedusingalsoheboundaryintegralequationmethodBIEM).Employingconformalmappingechniques,MacPherson1978)andMcDougaletal.1988)presentedanalyticalsolutionsforthecaseofaninfinitedepthoftheseabed,whereasMonkmeyeretal.1983)developedasolutionusingso-calledimagepipe"methodwhich,comparingtotheformer,canbeapplicablealsotoasoillayerofafinitehickness.

Thecommonfeaturenhestudiesmentionedaboveshatheeffectofcompressibilityofboththeporewaterandporousmediumwas neglected.More-over,someesearchersshowedhatheresadifferencebetweenheoreticallycomputedvaluesofporewaterpressureandthoseobservedinexperiments.nlaboratorystudiesonthestabilityofburiedpipelines,Philipsetal.1979)con-cludedthatpotentialtheorydidnotgenerallygiveanaccuraterepresentationoftheransmissionofwave-inducedpressuresthroughthesand,whencomparingtothetestesults.

Reporteddifferencesbetweentheoreticalandexperimentalresultscanhavethreemainreasons,namely:

-heheoriesarebasedontheDarcymodelandthereforetheydonotcon-tainallmportantsoil/waterparametersincompressibleporewaterandnondeformablesoilskeletonareassumed),-oundaryconditionsappliedntocomputationarenotealistic,peciallywhencomparingwithlaboratorytestsenvironment(seabedlayerofafinitethickness),

-aluesofparametersusedincalculationsarenotexactlythesamelikethese'in-situ'whichaccompanylaboratoryinvestigations.

Theproposedmethodofcalculationisbasedon:-heorewaterressureheorywherehemainoilndporeluida-rametersareconsideredandainitesandbedayersystemsakenntoaccount,-heimagepipe'heorywhichsableosolvebothheupperatheseabottom)andtheloweratfinitedepthoftheseabedlayer)boundarycon-ditions,andalsotheboundaryconditioninducedbypresenceofapipeline(perturbationorscatteringeffect).Anmplementationofcertainoilandporewaterparameters,.g.om-

pressibilityandpermeability,leadsnottotheLaplaceequation,whichdependsonlyonageometryoftheproblem,buttothestorageequation,whichismuchmorecomplexinform.UsingMadsen's(1978)generalsolutionofthisequation,ananalyticalsolutionforafinitethicknessof theseabedlayerhasbeenderivedandverifiedqualitativelynnumerousarge-scaleaboratoryexperimentsnabigwave-flume,andquantitativelyinsmall-scalelaboratory tests(Magda,1989,


4/14

3138OASTALENGINEERING992 1991).heseestsenabledostudynfluencesofsinglesoil/waterparameterchangesonhecharacteroftheporewaterpressuredampingwithinaporousmedium. greatattentionhasbeenputomodellingandcontrollingdifferentdegreesofsaturationwhichsofaspecialnterestorcoastalandidalareaswhere,becauseofacontinuouswatertablemovementandwave-breakingzones,thesedimentisnotandcannotbetreatedasasaturatedmedium.

MATHEMATICALFORMULATIONOFTHEPROBLEMIntroducingapipeline-likestructurentoasoilbody,tsnotoeasyo

deriveasolutiontothegoverningequationforflowofacompressibleporefluidinacompressibleporousmedium(e.g.ivenby:Madsen,1978;Yamamotoetal.,1978).herefore,aftersomemathematicalmanipulations,andpresentingthesolutionnermsofthepore-waterpressureandeffectivestresses,anewformofthegoverningequationcanbeobtainedOkusa,1985):

v!(v J-^)'=" wherepishewave-inducedexcessporewaterpressure,shecoefficientofconsolidation,tisthetime,andVistheLaplacianoperator.Thecoefficientofconsolidation,c,canbedefinedfortheunsaturatedsoilas

n 2u+ K 2G(l-fx)where7isheunitweightoftheporefluid,kistheisotropiccoefficientofsoilpermeability,nisheporosityof theporousbed,\iishePoisson'sratio,K,is thebulkmodulusofwater,andGisheshearmodulusofsoil.romthistseasilyeenhathesolutionofEq.1)canbeformulatedasamixedsolutionofboththeLaplaceequation

V2p=0 3)andtheconsolidationdiffusion)equationV-I|- 4 ,inwodimensions.thasobepointedhatometimese.g.iuandSun,1987)hesimplificationofthesolutionohegoverningflowequationisgoingtoofarand,duetothetotaleliminationofthesoildisplacements,theproblemiseducedonlyoheconsolidationequation.owever,hecorrectolutionhasobereatedasasumofthegeneralsolutionsohelastwodifferentialequationsofthesecondorder.


5/14

SUBMARINEBURIEDPIPELINES139SOLUTIONMETHOD

Assuminghathewave-inducedhydrodynamicpressureatheseabedsdescribedbyheperiodicfunction

p=Pgexp[i(ax ut)] 5)wherea=2ir/Lshewavenumber,Lshewaveength,>=2TT/Tsheangularvelocity,Tishewaveperiod,andPQshepressureamplitudeatheseabed,anddueoinearityoftheabovementionedcomponentequations,alltheunknownsintheproblemconsidered(amongothers:hewave-inducedporepressure)areperiodicwithaandw.Then,thewave-inducedporepressurepis representedby

Pf(z)exp[i(ax cot)] 6)where(z)s unctionf nly.ntroducinghisntoEqs.3)nd4),thegeneralsolutionsepresentedbyhesumofthesolutionsromhewofollowingdifferentialequations

%-{?-$)'-> ' Becausehegoverningequationsarelinear,hewave-inducedstressescanbeobtainedbysuperposing,aspreviouslyindicatedbyYarnamoto1981)andOkusa(1985a).ThegeneralsolutionsfifEq.7)andf%ofEq.8)are

/i=CLexp(az)+DLexp(-az)9)ftCcexp(nz)+Doexp(-Kz)10)

whereCL,CC,DL,DCreintegralconstantsdependingonheboundarycon-ditionsand

/ 2a?11)Forhecaseofinfinitelyhickhomogeneoussediment,hewave-inducedporepressures,stresses,anddisplacementsmustendozeroas >oo.Therefore(Okusa,1985):

CL+Cc= 12)


6/14

3140 COASTALENGINEERING1992_ 2(l+ /i)B3(1-g)

CL" +2/i*-JB- +2/iB-B13)wheretheSkempton'sporepressurecoefficientBsdefinedas*-' + )

whereashevolumecompressibilityofthesediment,3shevolumecom-pressibilityoftheporefluid.

Now,usingacomplementarywaveloadingmethod,.e.wowaveshavingthesamephaseanddifferentamplitudesCx,andCc)areassumedforsolvingtheLaplaceequationandconsolidationequationseparately,onecanwrite

p=CLxL+G0xC (15)whereLandCdenotevaluesobtainedfromthesolutionsofLaplaceequationandconsolidationdiffusion)equation,espectively,assumingforbothofthemaunitamplitudeoftheinducinghydrodynamicpressurewaveatheseabed.

Aontributionfhearticularomponents,uppliedyheolutionsofLaplaceequationndconsolidationequation,oheotalolutionofheproblemisillustratedinTab..

DegreeofSaturation CL Go S=1.00 0.998 0.002S=0.99 0.855 0.145S=0.98 0.748 0.252S=0.97 0.665 0.335S=0.96 0.598 0.402S=0.95 0.544 0.456

Table ContributionofsinglecomponentsfromtheLaplaceequationandconsolidationequation)nhetotalsolution,withegardtodifferentsaturationconditions.

Ithasto bestressedoncemorethatthe influenceofpartlysaturatedseabedconditionsispredominantfortheinvestigatedcaseoftheupliftorceactingonaburiedsubmarinepipeline.Therefore,thebothsolutionstotheLaplaceequa-tionandconsolidationequationhavetobealwaysakenintoaccountsimulta-neously.oconfirmanexistenceofpartlysaturatedconditionsnanaturalenvironment,ameasuringcampaignwasconductedonNorderneyIslandGer-many).Aftersamplingandstatisticalanalysisofthemeasuredandcalculated


7/14

SUBMARINEBURIEDPIPELINES 3141results,themeanvalueofthedegreeofsaturation wasfoundtobe0.975(MagdaandDavidov,1990).

Fig. howsasetofporepressureprofileswithdepthcalculatedorhesingularolutions,.g.heLaplacendonsolidationroblems,onsideredseparately,comparingthemwithasolutionforthecompoundedproblemofthecompressiblefluidflowthroughcompressiblemedia.

vQ

Degreeofsaturation,S=0.97

-0,2 0 ,2,4,6 Porepressure,p/Po- ]

0,8

Figure2 Comparisonofdifferentsolutionsfortheporepressuredistributionwithdepth.

Asimilaranalysiscanalsobeperformedforhecaseofafinitehicknessoftheseabedayer.heformulasdescribingcoefficientsCL,CD,DI,,DDre,however,muchmorecomplicated.

ThesolutiontotheLaplaceequation,Eq.3),fortheboundaryconditionsproblemcreated byafinitethicknessofthe seabedlayerandapipe-likestructureembededinthesoilsediment,isnottrivialbutdoesnotbringanytroubles.Asdocumentedintheintroduction,itspossibletoobtainthissolutionusing,forexample,oneoftheeportedconformalmappingtechniques.

Itisnotaneasytasktosolvetheconsolidationpartialdifferentialequation,Eq.4),inheCartesiancoordinatesystemfortheidenticaltotheabovemen-tionedboundaryconditionsproblem.Therefore,toovercomethedifficulties,thesolutionmethodpresentedbelowisbasedonthecylindricalcircular-cylinder)coordinatesystem.Theonsolidationquation,lsoknownasiffusionorheatonductionequation,isconsidered.tcanbepresentedingeneralformas


8/14

3142OASTALENGINEERING992 VV= 16)

Thesolutionofanyofthescalarequationstike:heLaplaceequation,thePoissonequation,hediffusionequation,hewaveequation,hedampedwaveequation,ransmissioninequation,ndheecotrwaveequationmaybereducedtoasolutionofthescalarHelmholtzequation,oritsspecialcase-theLaplaceequationMoonandSpencer,971).orheconsolidationequation(16),et

U(ui)T(i) 17)whereUisafunctionofthespacecoordinatesandTisafunctionof timeonly.Substitutionntoheconsolidationequationallowsheseparationoftheimepart,giving

YU+KU=0 18a)+K2h2T=0 186)at

whereKsheseparationconstant.ThesolutionoftheHelmholtzequation18a)dependsonhespacevari-

ablesandtheboundaryconditions,andwillbedifferentforeachproblem.Theequationintime(18b),however,isindependentofthecoordinatesystem.Thusthesolutionoftheconsolidationequationisalways

< p=U(u1,u2,us)e-K2hh 19)Geometryoftheproblem,i.e.ircularpipeburiedinaseabed(seeFig.1),

advicesousethecircular-cylindercoordinatesu\=r 0


9/14

SUBMARINEBURIEDPIPELINES143 at 1dit o A x_ ___ .Y+;*+{K-*)R=02 2 a )

+A 20=O 226)whereR, reunctionsfr,,espectively,nd nd reeparationconstants.heseequationsreolvedorRand0,ndhesolutionoftheHelmholtzequationhasafollowingform

U(r,0)=R(r)Q(0) 23)Differentialequation22b)hasafollowinggeneralsolution

0(0)=acosA0+/3sinA0 24)Forhegoverningproblem, saharmonicunctionof withaperiod7r, therefore,0musthavehesameeature.tspossibleonlywhen sep-resentedbyanntegernumber.yimitingheangeofvaluesof onlyopositiveones(A=0,1,2,...,n,...)bothfunctionsQ{0)andR{r)canbewrittenaccordinglyas

Qo(0),i(0),@2(0),...,en0),... ; R0(r),R1(r),R2(r),...,Rn(r),... (25)Inhisway,aninfinitesystemofsolutionsorEq.23)sobtainedwhichnowcanbewrittenas

oo U(r,0)=T[ancosn0+/3nsinn0]Rn(r)26)

re=0Eq.22a)canbeconsideredasheBesselequationwhichngeneralform

canbewrittend2W 1dW , , , , , ,,, _.-TT+--J-+(ji+q2-sw)W= 27)aw* waw

ThegeneralseriessolutionofEq.27)maybewritten,ors integer,W=AJ,(ti,q,w)+BJ-s(n,q,w)28)

Theseeriesrevalideverywherenheinitecomplexplane.f =,ninteger,J-ssnolongerindependentofJ,andthegeneralsolutionofEq.27)is

W=AJn(ii,,q,w)+Byn(n,q,w)29)


10/14

3144OASTALENGINEERING992 whereJndynreheBesselwaveunctionsofheirstandsecondkind(alsocalledheWeberfunction),espectively.ffi=0thisishecase)tcanbeconcludedthatheBesselfunctionsdegenerateandEq.29)becomes

W=AJn(qw)+Byn(qw) 30)IntroducingheHankelunctionsi.e.heBesselunctionsofhehird

kind,whichreinearombinationsfheBesselunctionsfheirstndsecondkinds)

H \qw)=J{qw)+iyn{qw)31)H \qw)=J(qw)-iyn{qw)32)

where:WWreheHankelunctionsofheirstandsecondkind,e-spectively,andofordern,thegeneralsolutionofEq.27)maybealsowritten(MoonandSpencer,1971):

W=AH%\qw)+BH(Z\qw)33)ComparingnowEq.22a)andEq.27),andreplacingWbyRandqwbyer,onehas

R=AH (KT)+BH (KT) 34)Two-dimensionalHelmholtzquation18a),escribingiffraction,fter

transformationntohepolarcoordinatessystemgetsaformwhichsknownasheBesselequation,hesolutionofwhich,intwo-dimensionalscatteringbylocalizedobjectsinaseaof constantdepthcanbeconstructedbysuperpositionofthefollowingtermsMei,1989):

(35)H {m . n O H\KT)) \osn0

Becauseof theasymptoticbehaviouroftheHankelfunctions

{K:!}"(=)V*


11/14

SUBMARINEBURIEDPIPELINES145whereidenotestheimaginaryunit,andcomparingitwithEq.6),hesepara-tionconstantcanbeexpressedby

K=M 38)V ^vInfact, sacomplexnumberandcanbepresentedinageneralformas

K=va+ib where a=0 and 6=> 39)Thiscanalsobewritten

K=a1+ib' 40)Comparisonofthelastwoexpresionsshowsthat

a',b'>1 when 6>1 and 0


12/14

3146 COASTALENGINEERING992

0,5

^43 - * * f t ,Q 1,5 2,5

VAy/4y//v/jy/st$i

0,2,4,6,8Porepressure,p/Po- ] Figure3 Definitionsketchfortheupliftorceanalysisinfluenceof

differentsaturationconditions).

0,6NoperturbationPerturbation

(C )("L+C")

0,97,975 0,98,985,99Degreeofsaturation,S- ] 0,995

Figure4 Pipelineupliftorceversusdifferentsaturationconditionsofseabedsediments.


13/14

SUBMARINEBURIEDPIPELINES147thus,Fig. howshegoverningproblemandFig. llustratesheesultsofcalculationsor ertainetfdatawhereheupliftorcesnfluencedydifferentvaluesofthedegreeofsaturation.

Itaneasilyecognizedhatheipelinepliftorceependserystronglyonhedegreeofsaturationandhasamaximumvalueorhedegreeofsaturationverycloseo1.00.

Changing valuefheegreeofsaturationwith stepf0.01i.e.,1%),forexample,hemostcriticalsituationcanbeeasilyomitted.Therefore,lookingforanabsolutemaximumvalueofthepipelineupliftforce,itisrequiredtoapplyevensmallerincrementofthedegreeofsaturationwhenperformingaparameterstudybymeansofnumericalcalculationsoobtainaprecisepictureofpossiblevariationsinhepipelineupliftorce.

Theelaboratedmethodseemstobeveryusefulinaoptimalizationdesignprocedureandgivesheresultwhichreflects,amongothers,hemostnconve-nientcaseforthepipelinestabilitywithrespectosaturationconditionsoftheseabedwhichare,ontheotherhand,extremelydifficultandalmostimpossibletodetermine'in-situ',usingengineeringmethodsoftesting,withheexactnesswhichiscomparabletohenecessarystepofcalculation.

Thecalculationprocedure,presentednhepaperandbasedonhead-vancedporewaterpressureheory,maketeasibleoncorporatemportantsoil/waterparametersntohepipelineupliftorceanalysis.btainedvaluesoftheupliftorceappeartobegreaterthanthesecomputedfromthepotentialtheory;hisindingsnaccordancewithomeobservationsromaboratorytestseportedinheliterature.REFERENCESDURSTHOFF ,W.,MAZURKIEWICZ,B.1985).

"Problemsrelatedosubmarinepipelines,"MitteilungendesFranzius-In-stitutsfurWasserbauundKiisteningenieurwesenderUniversitatHannover,Heft61,pp.176-238.LAI,N.W.,DOMINGUEZ,R.F.,DUNLAP,W.A.1974)."Numericalsolutionsfordeterminingwave-inducedpressuredistributionsaroundburiedpipelines,"TexasA&MUniversity,SeaGrantPub.No.TAMU-SG-75-205.

LENNON ,G.P.1985)."Wave-inducedorcesnuriedpipelines,"ournalfWaterway,ort,CoastalandOceanEngineering,Vol.Ill,No. ,pp.511-524.

MACPHERSON,H.1978)."Waveforcesonpipelineburiedinpermeableseabed,"JournalofWater-way,Port,CoastalandOceanDivision,Vol.104,No.WW4,pp.407-419.


14/14

3148OASTALENGINEERING992MADSEN ,O.S.1978).

"Wave-inducedporepressuresandeffectivestressesinaporousbed,"Geo-technique,Vol.8,No.4,pp.77-393.

MAGDA,W.1989)."Wave-inducedporewaterpressuregenerationn sanded,"InternalReportNo.,Sonderforschungsbereich-205"Kusteningenieurwesen,"TP A13,UniversitatHannover,June1989,pp.-94.

MAGDA ,W.1990)."Porewaterpressuregenerationinahighlysaturatedseabedforshallowwaterconditions,"Proceedingsofthes*nternationalSymposium"LIT-TORAL1990,"Marseille,July1990,pp.10-114.

MAGDA,W.,DAVIDOV,N.1990)."Ondeterminingofthedegreeofsaturationromin-situ'measurementsoftheseabedperformedonNorderneyIsland,"InternalReportNo.,Sonderforschungsbereich-205Kusteningenieurwesen,"PA13,Univer-sitatHannover,July1989,pp.-57.

MCDOUGAL,W.G.,DAVIDSON,S.H.,MONKMEYER,P.L.,SOLLITT,O.K.(1988)."Wave-inducedorcesnuriedpipelines,"ournalofWaterway,ort,Coastal,andOceanEngineering,Vol.14,No.,pp.220-236.

MEI,C.C.1989)."TheAppliedDynamicsofOceanurfaceWaves,"AdvancedSeriesnOceanEngineering-Vol.,WorldScientificPublishingCo.te.Ltd,pp.1-740.

MONKMEYER,P.L.,MANTOVANI,P.,VINCENT,H.1983)."Wave-inducedseepageeffectsonauriedpipeline,"ProceedingsofheCoastalStructures83Conference,pp.19-531.

MOON,P.,SPENCER,D.E.1971)."FieldTheoryHandbook;IncludingCoordinateSystems,DifferentialEqua-tionsandTheirSolutions,"Springer-Verlag,pp.1-236.

OKUSA,S.1985a)."Wave-inducedstressesinunsaturatedsubmarinesediments,"Geotechni-que,35,No.,pp.17-532.

PHILLIPS,B.A.,GHAZZALY,O.I.,KAIAJIAN,E.H.1979)."Stabilityofpipelinenandunderwavepressure,"roceedingsofheSpecialityConference"CivilEngineeringintheOceanIV,"SanFrancisco,pp.22-136.

Qiu,H.H.,SUN,Z.C.1987)."Wavenducedpressureson uriedpipeline,"roceedingsfhethInternationalSymposiumonOffshoreMechanicsandArcticEngineering(OMAE),Houston,Texas,March1-6,1987.

YAMAMOTO,T.,KONING,H.L.,SELLMEUER,H.,HIJUM,E.1978)."Onheresponseofaporo-elasticbedowaterwaves,"JournalofFluidMechanics,Vol.7,part ,pp.93-206.

Documents

Wave-Induced Pwp Acting on a Buried Pipeline