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Model for Interpretation of Pipeline Survey Data


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MODELFORINTERPRETATIONOFPIPELINESURVEYDATAByCHENCHENQIUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA 2003

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ACKNOWLEDGMENTSIwouldliketoexpressmysincerestgratitudeandappreciationtoDr.MarkE.Orazemforhissupport,inspirationandguidanceindirectingthisstudy.Hishardworkandattentiontodetailwereanexcellentexampleduringmystudies.Healwayshelpedmetolearnmoreandtaughtmetheimportanceofthinkingcreatively.Manyoftheadvancesachievedduringmygraduatestudywouldnothavebeenpossiblewithouthissupport.Iwouldalsoliketogratefullyacknowl-edgeDr.LocVuQuocforteachingmetheboundaryelementmethodattheearlystageofthiswork.Moreover,IwishtothankDr.AnthonyJ.Ladd,Dr.OscarD.Crisalle,andDr.DarrylButtfortheirtime,usefuldiscussionsandguidanceasmembersofthesupervisorycommittee.Iwouldliketoacknowledgethe-nancialsupportofthePipelineResearchCouncil,InternationalandGasResearchInstitute.Mycolleagues,DouglasP.Riemer,KerryAllahar,NelliannPerez-Garcia,PavanShukla,whohavecontributedtothisresearchbytheirvaluablediscussionsandfriendship,arealsogratefullyacknowledged.Finally,Iwouldliketothankmyhusband,KuideQin,forhisunselshsupportandunderstandingthroughoutmydoctoratework,whichwouldnothavebeenpossiblewithouthim.Iwouldalsoliketothankmyparentsandmysisterfortheirencouragementandinspira-tion.Eventhoughtheywerehalfaplanetawayfromme,theyhavealwaysbeenwithmeinmyheart. ii

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TABLEOFCONTENTSpage ACKNOWLEDGMENTS ::::::::::::::::::::::::::::ii LISTOFTABLES ::::::::::::::::::::::::::::::::vii LISTOFFIGURES :::::::::::::::::::::::::::::::viii ABSTRACT :::::::::::::::::::::::::::::::::::xii NOTATION :::::::::::::::::::::::::::::::::::xvCHAPTER 1INTRODUCTION :::::::::::::::::::::::::::::1 2PIPELINECORROSION,PROTECTIONANDMEASUREMENTS :::4 2.1Introduction ................................ 4 2.2BasicConceptsofCorrosion ....................... 5 2.3CorrosionProtection ........................... 7 2.3.1Coating .............................. 7 2.3.2CathodicProtection ....................... 7 2.4CathodicProtectionCriteria ....................... 10 2.4.1)]TJ/F19 11.955 Tf 9.672 0 Td[(850mVPotentialCriterion ................... 10 2.4.2100mVPotentialCriterion .................... 11 2.5PipelineMeasurements .......................... 11 2.5.1PotentialSurvey ......................... 11 2.5.2LineCurrentSurvey ....................... 13 2.5.3OtherSurveyTechniques .................... 14 2.6Conclusions ................................ 15 3APPLICATIONOFTWO-DIMENSIONALFORWARDANDINVERSEMODELSTOCORROSION ::::::::::::::::::::::::16 3.1Introduction ................................ 16 3.2ThinPlateMethod ............................ 17 3.2.1Introduction ............................ 17 iii

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3.2.2EvaluationProcess ........................ 18 3.2.3Summary ............................. 20 3.3Two-DimensionalBoundaryElementMethod ............. 20 3.3.1Introduction ............................ 20 3.3.2EvaluationProcess ........................ 22 3.3.3Two-DimensionalProblem ................... 22 3.4InverseAnalysisinTwoDimensions .................. 24 3.4.1ObjectiveFunction ........................ 24 3.4.2RegressionMethodAnalysis .................. 25 3.4.3DownhillSimplexMethod ................... 25 3.4.4AccuracyoftheParameters ................... 26 3.4.5RegressionResults ........................ 28 3.5Conclusions ................................ 29 4DEVELOPMENTOFTHREE-DIMENSIONALFORWARDANDINVERSEMODELSFORPIPELINEWITHPOTENTIALSURVEYDATAONLY :30 4.1Introduction ................................ 30 4.2ConstructionoftheForwardModel .................. 33 4.2.1CathodicProtectionSystem ................... 33 4.2.2GoverningEquationandBoundaryConditions ....... 34 4.2.3Theoreticaldevelopment .................... 35 4.3Three-DimensionalBoundaryElementMethod ............ 38 4.3.1InnityDomain .......................... 38 4.3.2Half-InnityDomain ....................... 41 4.3.3BoundaryDiscretization ..................... 44 4.3.4CoordinatesDenitionandTransformation .......... 45 4.3.5DiscretizationofBoundaryElementMethod ......... 46 4.3.6RowSumElimination ...................... 49 4.3.7Self-Equilibrium ......................... 50 4.4ForwardModel .............................. 52 4.4.1ConstantSteelPotentialAssumption .............. 52 4.4.2SimulationResults ........................ 54 4.5InverseModel ............................... 54 4.5.1ObjectiveFunction ........................ 55 4.5.2AnalysisofRegressionMethods ................ 55 4.5.3SimulatedAnnealingMethod ................. 56 4.5.4SimulationResultsandDiscussions .............. 57 4.5.5InverseStrategies ......................... 62 4.6Conclusions ................................ 70 5DEVELOPMENTOFTHREE-DIMENSIONALFORWARDANDINVERSEMODELSFORPIPELINEWITHBOTHPOTENTIALANDCURRENTDENSITYSURVEYDATA :::::::::::::::::::::::::72 5.1ForwardModel .............................. 72 5.1.1Introduction ............................ 72 5.1.2PipelinewithVaryingSteelPotential .............. 72 iv

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5.1.3SimulationResults ........................ 82 5.1.4AnalysisoftheSimulationResults ............... 86 5.1.5ValidationwithCP3D ...................... 87 5.2InverseModel ............................... 91 5.2.1Introduction ............................ 91 5.2.2RegressiontoNoise-FreeData ................. 93 5.2.3RegressiontoNoisyData .................... 99 5.3Conclusions ................................ 102 6APPLICATIONOFOFF-POTENTIALDATATOTHEINVERSEMODEL 104 6.1ForwardModel-CalculationoftheSoilSurfaceOff-Potential .... 104 6.1.1PhysicalProcess .......................... 104 6.1.2Assumptions ........................... 105 6.1.3ModelImplementation ...................... 105 6.1.4SimulationResultsandAnalysis ................ 106 6.2InverseModel-UsingThreeKindsofHeterogeneousDataSets ... 107 6.2.1DatawithoutNoise ........................ 107 6.2.2DatawithAddedNoise ..................... 109 6.3Conclusions ................................ 112 7SUMMARYANDFUTUREWORK ::::::::::::::::::::113 7.1Summary .................................. 113 7.2FutureWork ................................ 114 APPENDIX ABOUNDARYINTEGRAL :::::::::::::::::::::::::115 A.1IntegralalongOneObject ........................ 115 A.1.1IntegraloverCylindricalElements ............... 115 A.1.2IntegraloverCircleElements .................. 116 A.2IntegralbetweenDifferentObjects ................... 117 A.2.1IntegraloverCylindricalElements ............... 118 A.2.2IntegraloverCircleElements .................. 119 BCODESTRUCTURE ::::::::::::::::::::::::::::121 B.1CodeStructureoftheForwardModel ................. 121 B.2CodeStructureoftheInverseModel .................. 121 CINTERFACE ::::::::::::::::::::::::::::::::124 C.1InputWindows .............................. 124 C.2OutputWindows ............................. 124 REFERENCES ::::::::::::::::::::::::::::::::::127 v

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BIOGRAPHICALSKETCH :::::::::::::::::::::::::::134 vi

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LISTOFTABLESTable page 3.1Regressionresultfortwo-dimensionalinversemodeldescribedbyAokietal.Theunderlinedsymbolsrepresenttheparametersesti-matedbyregression. ........................... 28 4.1Parametervaluesobtainedusingthethree-dimensionalinversemodeldevelopedinthepresentworkfora10mpipesegmentwithtwocoatingdefects. .............................. 60 4.2Testcaseparameterswithvecoatingdefectsonthepipeusedtodemonstratethemethodfordeterminationofthenumberofstatis-ticallysignicantparametersseeFigure 4-15 .Theintactcoatingresistivityhadavalueof5:0107/Wm. ............... 62 4.3Regressionresultsfromthethree-dimensionalinversemodelfora100mpipesegmentwiththreecoatingdefects.Thesequentialre-gressionprocedurewasusedtoidentifythenumberofdefectsthatwerestatisticallysignicant.Theintactcoatingresistivityhadavalueof5:0107/Wm. ......................... 64 5.1Regressionresultsusingeitherequation 5-19 or 5-18 forhomoge-neoussyntheticdatawithoutaddednoiseorequation 5-20 forhet-erogeneoussyntheticdatawithoutaddednoise.Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92m. 95 5.2Regressionresultsusingequation 5-24 forheterogeneoussyntheticdatawithaddednoise.Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92m. ............. 101 6.1Regressionresultsfromthedatawithoutaddednoiseobtainedbyusingthegeneralformoftheobjectivefunction 5-25 5-26 and 5-27 .Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92m. ......................... 108 6.2Regressionresultsfromthenoise-addeddatabyusinggeneralformoftheobjectivefunction 5-25 5-26 and 5-27 .Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92m. 111 vii

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LISTOFFIGURESFigure page 2-1Polarizationcurveofthecathodicprotection. ............. 8 2-2SacricialelectrodeCP. .......................... 9 2-3ImpressedcurrentCP. .......................... 9 2-4Potentialsurveymethod. ......................... 12 2-5Pipe-to-soilpotentialsurveymethodandimportanceofthehalf-celllocation. .................................. 13 2-6Linecurrentsurvey. ............................ 14 3-1Finitedifferencemethod. ......................... 19 3-2ComparisonsoftheforwardandinverseresultsofTPAmethod. ... 20 3-3Schematicrepresentationofthetwo-dimensionalproblem. ..... 22 3-4Reproducingresultsforthetwo-dimensionalforwardmodel. .... 23 3-5Schematicrepresentationofthetwo-dimensionalinversemodel. .. 24 3-6Schematicrepresentationofthedownhillsimplexmethod. ..... 26 4-1Schematicillustrationofthepipesegmentandanodeusedtotesttheinversemodel. ............................. 33 4-2Linearrelationshipbetweenpotentialdropandcurrentdensityoverthepipecoating. .............................. 36 4-3Schematicillustrationofthepipesurfaceresistivitymodel. ..... 38 4-4Interiorproblem. ............................. 39 4-5Multiconnectedregionofinteriorproblem. .............. 40 viii

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4-6Schematicillustrationofmirrorreectiontechnique. ......... 43 4-7TheFundamentalsolutiontothehalf-innitydomainsatisfyingtheNeumanb.c's. ............................... 43 4-8Pipediscretisionandcollocationpoints. ................ 44 4-9Coordinaterotation. ........................... 46 4-10Falsecolorimageoftheon-potentialonthesoilsurfacethatwasgeneratedbytheforwardmodelcorrespondingtoFigure 4-1 .... 55 4-11Flowchartfortheinversemodelcalculations. ............. 58 4-12Gridshowingthelocationof303surfaceon-potentialscalculatedusingthethree-dimensionalforwardmodeldevelopedinthepresentwork.Thegridshownisfora10mpipesegment.Ascaledversionofthegridwasusedfora100mpipesegment. ............ 59 4-13Theregressionobjectivefunctionasafunctionofthenumberofevaluationsforapipecoatingwithonedefectregion.Thesimu-latedannealingmethodwasusedforthisregression. ........ 60 4-14Comparisonofthesetandttedresultsforpipecoatingwithtwocoatingdefects. .............................. 61 4-15Theregressionstatisticasafunctionofthenumberofcoatingde-fectsassumedforthemodel.Theminimuminthisvalueisusedtoidentifythemaximumnumberofcoatingdefectsthatcanbejusti-edonstatisticalgrounds. ........................ 63 4-16Comparisonbetweentheinputvaluesandregressionresultsfornoise=0:1mV. .............................. 66 4-17Comparisonbetweentheinputvaluesandregressionresultsfornoise=1:0mV. .............................. 67 4-18Comparisonbetweentheinputvaluesandregressionresultsfornoise=2:0mV. .............................. 69 5-1Cathodicprotectionsystemwithvariantpotentialalongthepipeandanode. ................................. 73 5-2RelationofpotentialVandcurrentdensity. .............. 74 5-3Axialdirectioncurrentowalongthepipeline. ............ 75 ix

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5-4Axialdirectioncurrentowalongtheanode. ............. 76 5-5Potentialandcurrentdensityalongtheanode. ............ 83 5-6Simulatedaxialdistributionsalongthepipeline. ........... 85 5-7PotentialofpipecalculatedbyusingCP3D. .............. 88 5-8PotentialofpipesteelcalculatedbyusingCP3D. ........... 89 5-9AxialdirectioncurrentdensityofpipecalculatedbyusingCP3D. 90 5-10RadialdirectioncurrentdensityofanodecalculatedbyusingCP3D. 90 5-11Schematicillustrationofthecouplingofasoil-surface-levelpoten-tialsurveymethodtoanaerialmagnetometercurrentsurveytoas-sesstheconditionofaburiedpipeline. ................. 92 5-12SyntheticpipecurrentandsurfacepotentialdatacorrespondingtoTable 5.1 .Dashedlinesindicatethelocationofcoatingdefects. ... 93 5-13The2=statisticcorrespondingtoTable 5.1 asafunctionofthenumberofcoatingdefectsassumedintheregression. ........ 96 5-14The2=statisticcorrespondingtoregressionofequation 5-21 seeTable 5.1 asafunctionofthenumberofcoatingdefectsas-sumedintheregression. ......................... 98 5-15Syntheticdatawithaddednoise.Thelinerepresentsthenoise-freevaluesandthesymbolsrepresentsyntheticdatawithaddednoise.asurfaceon-potentialvaluewith=0:4mV;baxialcurrentden-sityvaluewithproportionalnoisecorrespondingto2percentofthevalue. ................................... 100 6-1SwitchoffCPpowersourcetodetermineinstantoff-potential. ... 105 6-2On-potentialandoff-potentialonsoilsurface. ............. 106 6-3Typicalclose-intervalpotentialgraph. ................. 107 6-4Decidethenumberofdefectsusingnoisefreesoilsurfaceonandoff-potentialandcurrentdensitydata. ................. 109 6-5Setvalueandnoise-addedoff-potentialonsoilsurfacedata. .... 110 x

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6-6Decidethenumberofdefectsusingnoise-addedonandoff-potentialandcurrentdensitydata. ......................... 111 A-1Integralbetweendifferentobjects. ................... 118 B-1Codestructureoftheforwardmodel. .................. 122 B-2Codestructureoftheinversemodel. .................. 123 C-1Windowforpipeproperties. ....................... 125 C-2Windowforanodeproperties. ...................... 125 C-3Windowforcoatingproperties. ..................... 126 C-4Windowforgures. ............................ 126 xi

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophy MODELFORINTERPRETATIONOFPIPELINESURVEYDATAByChenchenQiuDecember2003 Chair:MarkE.OrazemMajorDepartment:ChemicalEngineeringPipelinecorrosionannuallycoststheU.S.sevenbillionsofdollars.Ninetypercentofthecostisthecapital,operationandmaintenance.Itisdesirabletooptimizethisexpenditurewhileensuringthattheintegrityofthepipeline,whichrepresentsavaluableasset,ismaintained.Regularinspectionofpipelinesusingavarietyofsurveytechniquesisroutinelyconducted,butinterpretationoftheresultsisconfoundedbystochasticandsystematicerrors.Theobjectiveofthepresentworkwastoinvestigateuseofinversemodelthatcouldinterpretsurveydatainthecontextofthephysicsofthesystem.Aninverseanalysismodelwasdevelopedwhichprovidesamathematicalframeworkforinterpretationofsurveydatainthepresenceofrandomnoise.Aboundary-elementforwardmodelwascoupledwithaweightednonlinearregres-sionalgorithmtoobtainpipesurfacepropertiesfromtwotypesofsurveydata:soil-surfacepotentialsandlocalvaluesofcurrentowingthroughthepipe.Theforwardmodelaccountedforthepassageofcurrentthroughathree-dimensional xii

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homogeneousmediumandyieldedsoilsurfacepotentialsforgivenpipe/anodecongurationsandpipecoatingproperties.Thenumberofregressedparameterswasreducedbyusingafunctionforcoatingresistivitythatallowedspecicationofcoatingdefects.Aweightedsimulate-annealingnonlinearregressionalgorithmfacilitatedanalysisofnoisydata.Amethodtodeterminetheappropriatenumberofttedparameterswasdeveloped.Themodelwasdemonstratedforsyntheticdatageneratedforasectionofacoatedundergroundpipelineelectricallyconnectedtoaverticalsacricialanode.Thesuccessoftheregressionwassensitivetotherelativeweightingappliedintheobjectivefunctiontotherespectivetypesofdata.Ageneralizedweightedandscaledobjectivefunctionwasproposed. xiii

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NOTATION Potentialandpotentialdistribution,V Potentialandpotentialdistributionrefertothereferenceelectrode,denedasV)]TJ/F47 12.457 Tf 12.116 0 Td[(,V k Resistancereductionofthecoatingdefectk,Wm k Thehalfwidthofthecoatingdefectk,m ~n Normalvectorofboundary cj Concentrationofspeciesj,mol=cm3 Dj Diffusioncoefcientofspeciesj,cm2=s F Faraday'sconstant,96487C=equiv i Currentdensity,A=m2 uj Mobilityofspeciesj,cm2mol=Js V Voltagedistributionofpipesteeloranodemetal,V v Bulkvelocity,cm=s xk Centerpointofthecoatingdefectsk,m zj Chargenumberofspeciesj Resistivityofthepipecoating,Wm xv

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CHAPTER1INTRODUCTIONIntheU.S.,thereare779,000km,000miofgasandliquidtransmissionpipelines.Sevenbillionofdollarsisspentduetopipelinecorrosioneveryyear.90%ofthecostisthecapital,operationandmaintenance.Itisdesirabletoop-timizethisexpenditurewhileensuringthattheintegrityofthepipeline,whichrepresentsavaluableasset,ismaintained.Regularinspectionofpipelinesusingavarietyofsurveytechniquesisroutinelyconducted,butinterpretationoftheresultsisconfoundedbystochasticandsystematicerrors.Thegoalofthisworkwastodevelopacomputerprogramthatcanbeusedtoextracttheconditionofthepipetakingintoaccountawidevarietyofelddata,suchaspotentialandlinecurrentsurveydata.Inaddition,theintentionofthispreliminaryeffortwastoestablishaproofofconceptwhichcanmotivatefuturedevelopment.Thepipelineprotectiontechniquesthatareusedmostoftenarecathodicpro-tectionandcoating.Thenormaluseofcomputerprogramsformodelingofca-thodicprotectionistoassesswhetherapipelinewithanassumedcoatingcondi-tioncanbeprotectedbyagivencathodicprotectiondesign.Inthisstrategy,calledaforwardsolution,allphysicalpropertiesofthepipe,anodes,groundbeds,andthesoilareassumed,andthecorrespondingdistributionofcurrentandpotentialonthepipeiscalculated.Over-protectedandunder-protectedsectionsofthepipelinecanbeidentiedinthisway.Suchprogramscanalsobeusedtocalculatethepo-tentialandthecurrentdensityatotherlocationssuchasatthesoilsurfaceorat 1

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2 thespeciedlocationswherecouponsmaybeburied.Thisstrategylendsitselfwelltoansweringwhatifquestions.Forexample,whatwillhappenifdiscretecoatingholidaysexposevepercentofthepipesurface?,orwhatwillhappenifanewpipeisconstructedintheright-of-way?Theforwardstrategyisnotappropriate,however,fortheinterpretationofelddatatoassesswhetheragivenpipeisprotected.Theconceptbehindtheapproachdevelopedinthisworkistosolvetheinverseprobleminwhichthepropertiesofthepipecoatingareinferredfrommeasurementsofthecurrentandthepotentialdistributions.Thisapproachwouldallowinterpretationofelddatainamannerthatwouldtakeintoaccountthephysicallawsthatconstraintheowofelectricalcurrentfromanodetopipe.Inaddition,thedevelopmentoftheinversemodelcannotproceedalone.Onlywhentheforwardmodeliswelldevelopedcantheinversemodelbeabletointer-pretmultiplekindsofsurveydata.Thebasicconceptsofthecorrosion,corrosionprotectionmethods,andpipelinesurveymethodsarepresentedinChapter 2 .Vericationofpublishedmethodsformodeltwo-dimensionalsystemsarepre-sentedinChapter 3 .Thischapterincludesthethin-platemethodandthebound-aryelementmethodtosolvethetwo-dimensionalLaplace'sequation.Thetwo--dimensionalinversemodelisalsoestablishedinthischapter.Thisworkinvolvedtheconstructionoftheobjectivefunction,theformalismofregressionstrategies,andregressionmethodsanalysis.Thedevelopmentoftheforwardandtheinversemodelsfortwo-dimensionalsystemsprovidedinsightsintothemethodsneededtoaddresstheinverseprob-lemforpipelines.However,thetwo-dimensionalforwardmodelcannotdescribepreciselythegeometryofthepipelinecathodicprotectionsystem.Inaddition,the

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3 moresophisticatedthree-dimensionalforwardmodelssuchasCP3D, 1 PROCAT, 2 andOKAPPI 3 arenotappropriate,sincetheyrequiretoomuchcomputationworkfortheinversemodel.AsimpliedversionofaforwardispresentedinChapter 4 .Boththeforwardmodelandinversemodelareappliedtosoilsurfacepotentialsur-veydata.Fortheforwardpart,athree-dimensionalboundaryelementmethodwithcylindricalelementswasdevelopedinthiswork.Thesimulatedannealingmethodwasshowntobeideallysuitableforthethree-dimensionalinversemodel.Inaddition,severalstrategieswereexploredtoassessthecondenceleveloftheinversemodelresults.TheforwardmodelpresentedinChapter 4 providedthefoundationforanex-tendedmodel,presentedinChapter 5 ,whichaccountedforthepotentialdropalongthepipesteel.Boththepotentialonsoilsurfaceandthecurrentdensityalongthepipelinecouldbeobtainedfromtheforwardmodel.Aninversemodelwascreatedtoapplythesedata.Anewgeneralformoftheobjectivefunctionwasestablishedbyaddingdifferentweightingfactorsforeachdatapointandbyincludingthenumberofthedatapoints.Chapter 6 veriesthegeneralformoftheobjectivefunctionbuiltinChapter 5 .Off-potentialonthesoilsurfacewassimulatedintheforwardmodel.Therefore,threekindsofdatasets,i.e.,on-potential,off-potentialandcurrentdensitydata,wereusedfortheinversemodel.SummaryofthestudyandfutureworkarepresentedinChapter 7

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CHAPTER2PIPELINECORROSION,PROTECTIONANDMEASUREMENTS2.1IntroductionThecorrosionofmetallicstructureshassignicantimpactontheU.S.economyinmanyelds.In1975,Battelle-NBS'sbenchmarkstudyestimatedthatthecostofcorrosionwas$70billionperyear,4.2percentofthenation'sgrossnationalproductGNP.Iftheeffectiveandpresentlyavailablecorrosiontechnologycanbeapplied,$10billionofthiscostcouldbeavoided.From1999to2001,theFederalHighwayAdministrationFHWAindicatedthatthetotaldirectcostofcorrosionwasabout$276billionperyear,3.1percentoftheU.S.grossdomesticproductGDP.Theindirectcostsisunpredictable. 4 Pipeline-relatedcorrosioncostsapproximately$7:0billionannually.Thereare528,000km,000miofnaturalgastransmissionpipelines,119,000km,000miofcrudeoiltransmissionpipelinesand132,000km,000miofhazardousliquidtransmissionpipelinesintheU.S. 4 Inthepastfewyears,anumberofgasandliquidpipelinefailureshavedrawnpeople'sattentiontothepipelinesafety.Inordertopreservetheassetofthepipelineandtoavoidthosefailureswhichmayjeopardizepublicsafety,resultinproductloss,orcausepropertyandenvironmentaldamage,somenewregula-tionswereestablished.Regularpipelineinspections,suchashydrostatictesting,directassessment,andin-lineinspectionILI,arerequired.Furthermore,corro-sionpredictionmodelsneedtobedevelopedinordertodetermineandprioritizethemosteffectivecorrosionpreventivestrategies. 4

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5 Developmentofanunderstandingofcathodicprotectionofburiedstructuressuchaspipelinesrequiresanunderstandingoftheelectrochemicalreactionsasso-ciatedwithcorrosion.Thissectionprovidesanoverviewoftheprinciplesofcorro-sion,methodsforcathodicprotection,andthecriteriausedtoassesstheeffective-nessofcorrosion-preventionstrategies.Moreover,thepipelinesurveymethodsareincluded.2.2BasicConceptsofCorrosionCorrosiontakesplaceinresponsetothetendencytoreducetheoverallfreeen-ergyofasystem.Forexample,whenmetalisinadiluteaeratedneutralelectrolyteatmosphere,moisturelms,whichcontainoxygenandwater,usuallycoverthemetalsurface.Someatomsinthemetalsurfacetendtogiveoutelectronsandbe-comeionsinthemoisturelayer.Inthisway,theyareinalowerenergystateinsolutionthanwhentheyareinthelatticeofthesolidmetal.Atthesametimethemetalsurfacebuildsupahugeexcessnegativecharge.TheseelectronsmoveoutfromthemetalandattachthemselvestoprotonsH+ionsandmoleculesoxygenO2.Theprocessrepeatsitselfonvariouspartsofthesurface.Thenthemetaldissolvesawayasions. 5 Theprimaryconstituentofpipeline-gradesteelsisiron.Thereforetheoverallcorrosionreactioncanbewrittenintermsofdissolutionofiron,e.g.,2Fe+O2+2H2O!2FeOH2-1Reaction 2-1 canbeconsideredastheresultoftwohalf-cellreactions.OneisFe!Fe2++2e)]TJ/F19 11.955 Tf 154.478 -4.937 Td[(-2Thisisanoxidationreactionwithanincreaseofoxidationstateforironfrom0to2+.Itiscalledananodicreaction.Theotheris2H2O+O2+4e)]TJ/F49 12.457 Tf 9.934 -4.937 Td[(!4OH)]TJ/F19 11.955 Tf 127.181 -5.4 Td[(-3

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6 ThisreactionisareductionreactionwithadecreaseofoxidationstateforO2toOH)]TJ/F19 11.955 Tf 6.946 -5.4 Td[(.Itisdenedtobeacathodicreaction.Thechangeinfreeenergyassociatedwithreaction 2-1 canbeexpressedintermsofacellpotentialasDG=)]TJ/F48 11.955 Tf 9.791 0 Td[(nFDE-4wherenisthenumberofelectronsexchangedinthereaction,FisFaraday'scon-stant,96487coulombs/equivalent, 6 andEistheelectrochemicalpotential.Ac-cordingtotheNernstequation,obtainedbyneglectingbothactivitycoefcientsandliquid-junctionpotential, 7 8 DE=DE0)]TJ/F48 11.955 Tf 13.789 8.093 Td[(RT nFlnasii=DE0)]TJ/F48 11.955 Tf 13.789 8.093 Td[(RT nFlncsii-5whereE0istheequilibriumpotential,andaiistheactivityofthespeciesi,siisthestoichiometriccoefcientofspeciesi,andciistheconcentrationofthespeciesi.Forreaction 2-2 ,thepotentialinNernstequationformisEa=E0Fe2+=Fe)]TJ/F48 11.955 Tf 13.789 8.093 Td[(RT 2Fln\002Fe2+-6Likewise,forreaction 2-3 ,thepotentialinNernstequationformisEc=E0O2=OH)]TJ/F49 12.457 Tf 8.157 1.056 Td[()]TJ/F48 11.955 Tf 13.79 8.094 Td[(RT 4FlnOH)]TJ/F67 11.955 Tf 6.946 5.35 Td[(4 PO2!-7whereameansanodicreactionandcreferstothecathodicreaction.Tothewholereaction,wherena=2andnc=4,DG=2DGa+DGc=)]TJ/F19 11.955 Tf 9.672 0 Td[(2naFEa)]TJ/F48 11.955 Tf 12.116 0 Td[(ncFEc=)]TJ/F19 11.955 Tf 9.671 0 Td[(4FEa+Ec-8Forthetwohalf-cellreactions,EaandEcarepositive.Hence,thefreeenergyoftheoverallreactionmustbenegative,whichmeansthecorrosionreactionoccursspontaneously.

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7 2.3CorrosionProtectionTocontrolthecorrosionsituations,severalmethodshavebeenprovedeffec-tive,whichincludecoatingandcathodicprotection.2.3.1CoatingElectricallyinsulatingmaterialscanbeusetocoverthepipesurfaceandtherebyisolatethepipemetalfromcontactwiththesurroundingelectrolyte.Inthisway,ahighelectricalresistanceintheanode-cathodecircuitisadded.Theoretically,sincenosignicantcorrosioncurrentowsfromtheanodetothecathode,nocor-rosionwouldoccur.Manykindsofmaterialshavebeenappliedascoatings,suchasenamels,tapes,andplasticcoating.Recentimprovementsinpipelinecoatingmaterialswillalsoreducetheriskofacorrosion-relatedfailure.However,arecentsurveyofmajorpipelinecompaniesindicatedthatabout30%oftheprimarylossofthepipelineprotectionwasduetocoatingdeterioration. 4 Therefore,coatingalonecannotprovidefullprotectionforthepipeline.Practically,effectivepipelinecorrosioncontrolcomprisesuseofgoodcoatingsalongwithcathodicprotectionasasecondarydefense.2.3.2CathodicProtectionIn1824,SirHumphreyDavysuccessfullyprotectedcopperagainstcorrosionfromseawaterbyusingironanodes.Itwastherstapplicationofcathodicpro-tectionCPand,atthattime,ithadnotheoreticalfoundation.Fromthatbegin-ning,CPhasbeenappliedtomarineandundergroundstructures,waterstoragetanks,pipelines,oilplatformsupports,reinforcingsteelandmanyotherfacilitiesexposedtoacorrosiveenvironment.Bynow,thetheoryofCPiswellestablished.NaturalgasandoilcompanieshavealreadybeenusingCPforeconomic,aswellassafetyreasons,sincethe1930's.ThePipelineSafetyActof1972madetheappli-

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8 Figure2-1:Polarizationcurveofthecathodicprotection. cationofCPonpipelinestransportinghazardousmaterialmandatoryforsafetyconcerns.Cathodicprotectionresultsfromcathodicpolarizationofacorrodingmetalsurfacetoreducethecorrosionrate.Inthissystem,thecorrosionpotentialreducestherateofthehalf-cellreaction 2-2 withanexcessofelectrons,whichdrivestheequilibriumfromrighttoleft.TheexcessofelectronsalsoincreasestherateofoxygenreductionandOH)]TJ/F19 11.955 Tf 11.179 -5.399 Td[(productionbyreaction 2-3 inasimilarmannerduringcathodicpolarization.Cathodicprotectionreducesthecorrosionrateofametallicstructurebyreducingitscorrosionpotential,bringingthemetalclosertoanimmunestate,whichisrepresentedbytheblackcurveinFigure 2-1 .Twomethodsareusuallyusedtoachievethisgoal. 6 Thereincludesacricialanodeandimpressedcurrent.SacricialAnode.SacricialAnode.SacricialAnode.OnemethodtoprovideCPistoconnectasacricialmetalwithahighernaturalelectromotiveforcethroughametallicconductororawiretothestructureintendedtobeprotectedasFigure 2-2 .Magnesiumisacommonsacricialorgalvanicanode.Thistypeofgalvaniccathodicprotectionreliesonthe

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9 Figure2-2:SacricialelectrodeCP. Figure2-3:ImpressedcurrentCP. naturalelectricalpotentialbetweenthetwometalstocausethecathodicprotectioncurrenttoow.Sincethedrivingvoltageislimitedtotheverysmallpotentialdifferenceexistingbetweenthemetals,andthecurrentoutputisrelativelylow,thistypeofcathodicprotectionisnormallyassociatedwithverysmallorverywellcoatedstructures.Impressed-Current.Impressed-Current.Impressed-Current.ThesecondcommonmethodtoprovideCPinvolvesuseofimpressed-current.ItreliesonanexternaldirectcurrentsourcesuchasarectierorbatteryasFigure 2-3 .Ananodematerial,placedintheelectrolytewiththeprotectedstructure,ismademorepositivethanthestructurebyconnectingboththeanodeandthestructuretothedirectcurrentsupply.Anyconductivematerialscanbeutilizedasanimpressedcurrentanode,butsincecorrosiontakesplaceattheanode,materialswithverylowconsumptionratesaremostdesirable.

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10 2.4CathodicProtectionCriteriaAsthecathodicprotectioniswidelyused,thereliablecriterianeedtobedevel-opedtodirectthecathodicprotectionsystemstoanoptimumlevel.Thecriteriaarebasedonthepotentialdifferencesbetweenthepipelineanditsenvironment.Twocriteriawillbeintroducedinthissection,)]TJ/F19 11.955 Tf 9.672 0 Td[(850mVand100mVpotentialcrete-ria.2.4.1)]TJ/F19 11.955 Tf 9.671 0 Td[(850mVPotentialCriterionThecriteriaforprotectingsteelpipe,RP-10-69,wasestablishedbyNACEin1969.Itwasrevisedlaterin1972,1983andagainin1996.Itdenesanegativepo-tentialatleast850mVwithrespecttoasaturatedcopper/coppersulfatereferenceelectrodeCSEwhenCPisapplied.Thiscriterionismostlyusedsincethepoten-tialmeasurementswithcurrentappliedrequiresminimumequipment,personnelandtakeslesstimetoobtainintheeldsurvey.Normally,thedesiredpotentialrangeofthecathodicprotectionforapipelineisbetween)]TJ/F19 11.955 Tf 9.672 0 Td[(850mVand)]TJ/F19 11.955 Tf 9.672 0 Td[(1200mVmeasuredwithrespecttoaCu=CuSO4refer-enceelectrodelocatedatthesurfaceofthesteelorcoatingdefects. 9 Ifexcessiveamountsofcathodicprotectionareapplied,thedirectreductionofwaterbecomesthermodynamicallypossible,e.g.,2H2O+2e)]TJ/F49 12.457 Tf 9.935 -4.938 Td[(!H2+2OH)]TJ/F19 11.955 Tf 126.906 -5.4 Td[(-9Hydrogenisevolveduponthemetal,and,atlargecathodicover-potentials,asmallfractionofhydrogencanenterthemetal,whichmakesthemetalbrittle.Thisundesirableprocessiscalledhydrogenembrittlement.Ifthepipelinehasacoating,thedevelopinghydrogengasbubblecanexerttremendouspressure.Whenthepressureiscreatedunderthetorncoating,astrippingactionexposesthemetalandresultsinrapiddeterioration.Obviously,ifsmalleramountsofCP

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11 isappliedresultinginpotentiallessnegativethan)]TJ/F19 11.955 Tf 9.671 0 Td[(850mV,thestructurewillnotbefullyprotected.Inaddition,the)]TJ/F19 11.955 Tf 9.672 0 Td[(850mVcriteriondoesnotdirectlyaddressthepolarization.ItisvalidonlywhentheeffectofIRdrophasbeenconsideredandeliminated.2.4.2100mVPotentialCriterionThe100mVpolarizationcriterionwasrstproposedbyS.P.Ewingin1951. 10 Thecriterionmeansthatthechangeinpotentialpolarizationtoprotectpipelineisalwayslessthan100mV.Toutilizethiscriterion,theIRdropeffectatmeasurementareaneedstobeestimated.Normally,couponsareusedtogettheIRdropvaluesbymeasuringthepotentialswhenthecouponisonandoff,andcomparingthetwopotentialscurvestoestablishthepotentialdifferencesbetweenthetwocurves,whichistheIRdropvalue.RecentresearchintheUKhasshownthattheappliedcurrentdensitycouldbereducedby25%afterthreemonthofusingthe100mVcriterionandby65%afteroneyear,withoutlosinganyprotection. 11 2.5PipelineMeasurementsCathodicprotectiondesignshouldbebasedonactualdata.Monitoringmeth-odsarenecessarytoinvestigatethecorrosiveconditionsandtoevaluatethede-greeoftheappliedCP.Thetechniquesusedincludemeasurementofpotentialalongthegroundabovethepipe,measurementofcurrentowingthroughthepipes,anduseofauxiliaryelectrodes,suchasareferenceelectrodeandcoupon.2.5.1PotentialSurveyConstructionofapotentialsurveyinvolvesmeasuringthepotentialdifferencebetweenthepipeandareferenceelectrodeatthegroundsurfacelevel.Therefer-enceelectrodeismovedtodifferentlocationstosamplemanypositionsatgroundlevelnearthepipe.Inthisway,amappingofpotentialiscreatedwhichcanbe

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12 Figure2-4:Potentialsurveymethod. usedtoidentifyregionswherethepipeisgrosslyunder-protected.ThesurveyinvolvesuseofavoltmeterandareferenceelectrodewhichistypicallybasedontheCu=CuSO4reaction.Aschematicrepresentationofthesurveymethodispre-sentedinFigure 2-4 12 Potentialsurveymeasuresthepotentialsbetweenburiedpipelineandenvironment. 9 Thissurveyneedsanappropriatevoltmeterwithitsnegativeterminalconnectedtothepipelineandthepositiveterminalconnectedtoareferenceelectrode.Theprocedureinvolvesmovingthereferenceelectrodeat3to10ftintervalsdownthefulllengthofthepipeandusingthetrailingcabletodeterminedistanceandtoconnecttothepipeline.Avastquantityofdataisgatheredbyadatalogger.Fromthesurveyreadings,theworstcorrosiontakesplacewherethereadingsarethehighestlessnegativevalueandlittleornocorrosionshouldtakeplacewherethepotentialismorenegative.Forthismeasurement,thelocationofthereferenceelectrodehalf-cellisim-portant. 13 Themostdesirableplaceisdirectlyabovethepipeline.Ifthereferencehalf-cellisplaced23meterstothesideofthepipeline,themeasuredpotentialwillbeagroupofscattereddataasthesecondcurveshowninFigure 2-5 .Insuchacase,itisdifculttomakeajudgement.Toavoidthiscase,measurementsonbothsidesofthepipemayberequiredtomakesurethattheoverthepipepotentials

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13 Figure2-5:Pipe-to-soilpotentialsurveymethodandimportanceofthehalf-celllocation. aretakenasclosetothepipesaspossible.Aninstantoff-potentialIR-freesurvey 9 14 isusedtoeliminatethepotentialdropthroughthesoilinthepipetosoilpotentialmeasurementbyinterruptingtheowofprotectivecurrenttothepipeline.Thistechniquecanbeusedtodeterminetheeffectivenessofthecathodicprotectionsystem.ItisbasedontheprinciplethattheIReffectinthepotentialmeasurementdecaysalmostinstantaneously,whilethepipe-to-soilinterfacepolarizationdecaysrelativelyslowly,thusallowingthecorrectpipe-to-soilpotentialtobemeasuredfreeoftheIRerror.2.5.2LineCurrentSurveyThelinecurrentsurveytechniqueisusedtomeasuretheelectricalcurrentow-ingonthepipe.Ifcorrosionistakingplaceonapipeline,currentwillowtothelineatsomepointsandowoutofthelineatothers.Forsmalllocalcells,thecurrentpathmaybetooshorttodetect.Forlargecells,thecurrentmayfollowthepipeforhundredsorthousandsoffeet.Itistheselonglinecurrentsthatcanbedetectedinalinecurrentsurvey.

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14 Figure2-6:Linecurrentsurvey. Becausethepipeitselfhassomeresistancetotheowofelectriccurrent,therewillbeavoltagedropinthepipeifcurrentisowingthroughthepipesteel.Normally,thevoltagedropsareverysmall.Thusthespanofthetwotestpointsisoftenxedat100feettoincreasethespanresistance.Knowingthespanresistanceofthepipebeingsurveyed,thevoltagedropsmaybeconvertedtoequivalentcurrentbytheapplicationofOhm'sLaw:Icurrent=Epotential Rresistance-10Thevaluesofcurrenttogetherwiththedirectionofowthenmaybeplottedasfunctionoflinelength. 9 FromFigure 2-6 ,itisnotedthatinoneareacurrentowsfrombothdirectionstowardaparticularpointontheline.Thispointmustbeaplaceofcurrentdischarge.2.5.3OtherSurveyTechniquesTheairbornecathodicmonitoringsystemsACMS 15 detectupsetconditionsonthepipelinesprotectedbyimpressedcurrent.Sensitiveandlteredmagneticeldcoilsareinstalledonahelicopter,whichcontinuouslymeasurethemagneticeldgeneratedbytheripplefromanalternatingcurrentsource.TheprincipleisbasedonB=0 2I R-11whereBisthemagneticeldgeneratedbytheimpressedcurrentI,Risthedis-tancefromthecenterofthepipetothesensorinstalledonthehelicopter,andthe

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15 constant0=2isthepermeabilityofthemedium.Thus,ifthemagneticeldcanbemeasuredalongwiththedistance,thenthecurrentcanbecalculated.Inaddition,useofaninsulatedcouponisavariationoftheinstant-offpotentialtosoilmethod.Thecouponusuallyisburiedclosetotheprotectedstructurestohavethesamecharacteristicofthechemicalenvironmentandelectricaleldasthoseatthestructuresurface.Theexposedcouponsurfacesimulatesacoatingdefect.Cathodicprotectioncurrenttothecouponcanbeinterruptedwithoutanyeffectontheprotectedstructures.Inthisway,thedifcultiesoftheinstant-offpotential-to-soilmethodcanbeeliminated.2.6ConclusionsAbriefintroductionofthecorrosionsituationswasgiveninthischapter.Thebasicconceptsofcorrosionwereexplained.Thepipelineprotectiontechniques,es-peciallythecathodicprotectiontechnique,havebeenintroduced.Inaddition,thecommonlyusedpipelinesurveymethods,suchaspotentialsurvey,linecurrentsurvey,airbornecathodicmonitoringsystemsandcoupons,arealsosummarized.

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CHAPTER3APPLICATIONOFTWO-DIMENSIONALFORWARDANDINVERSEMODELSTOCORROSION3.1IntroductionGiventhegoverningequationandtheboundaryconditionstoestimatethepo-tentialandcurrentdensitydistributiononthesoilsurfacebysolvingthegovern-ingequationdirectlyisknownastheforwardmodel.Incontrast,inaninverseprob-lem,usuallyamodeloragoverningequationandmeasurementsofsomevari-ablesaregiven,suchasthepotentialandcurrentdensitymeasurements.Bound-aryconditionsandtherestofthevariablesmaynotbeknownexplicitly.Thepurposeoftheinversemodelistoidentifytheunknownparametersusingthein-formationfromthemeasurements.Itisatypeofill-conditionedprobleminwhichthesolutionisextremelysensitivetothemeasureddata.Inthischapter,theforwardandinversemodelsappliedintheliteraturearere-viewedandreproduced.Twomethodswerestudied,thethinplateapproximationmethodandtheboundaryelementmethod.Theexperiencesofdevelopingandevaluatingthesemodelsaresummarized. 16

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17 3.2ThinPlateMethod3.2.1IntroductionInglese 16 17 solvedtheinverseproblemfortwoelectrodeseparatedbyasmallgap,a,asfollowing:8>>>>>>>><>>>>>>>>:52=0inW;x=0xx2[0;1];y=0;nx+xx=fxx2[0;1];y=a;n;y=n;y=0y2[0;a]-1whereisthecorrosioncoefcient,whichrepresentsthecorrosionrate,isthepotential,andnistheoutwardnormaluxandaisthethicknessofthedo-main.Thegoalwastondfromthepotential0ontheaccessibleboundaryx2[0;1];y=0.TheauthorintroducedthethinplateapproximationTPAmethod.TheideaistoperformanexpansionofthesolutionintermsofpowersofthethicknessofthedomainW.Owingtotheassumptionthatthethicknessaismuchsmallerthan1a1,higherordertermsoftheexpansioncanbeignored.Inglesededucedaformulaofasfunctionofxxx;0,whichisthesecondderivativeofwithrespecttox.TPAx=xxx;0+0x x;0-2Inordertoprovetheaccuracyofequation 3-2 ,thedirectproblemwassolvedforaprescribedcorrosionrategivebyx=exp)]TJ/F19 11.955 Tf 9.671 0 Td[(100x)]TJ/F19 11.955 Tf 11.996 0 Td[(0:252-3Firstofall,fortheforwardpart,x;0iscalculatedaccordingtothegivenx.Afterthat,xx,0canbeobtainedwithoutproblems.Fortheinversepart,the

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18 corrosionratewasobtainedfromequation 3-2 anditcanbecomparedwiththevalueofxfromequation 3-3 .3.2.2EvaluationProcessThedomainofWinxdirectionis[0;1]andinydirectionis[0;a].Tosimplifytheproblem,theydirectionisnormalizedto[0;1].Thus,thedecomposedformofequations 3-1 is:8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:a2xx+yy=0inW;x;y=0;x;y=0;yx;0=)]TJ/F48 11.955 Tf 9.755 0 Td[(a2;yx;1+a2exx;1=0;aex=exp[)]TJ/F19 11.955 Tf 9.671 0 Td[(100x)]TJ/F19 11.955 Tf 11.997 0 Td[(0:252]:-4Tosolveequation 3-4 ,theLaplace'sequationwithRobinboundarycondi-tions,twomethodsweretested.Onewastheseparationofvariablesmethodandtheotherwasthenitedifferencemethod.However,theseparationvariablemethodfailedforthiskindofproblembecauseacontradictionoccursbetweentheresultandboundaryconditionyx;0=)]TJ/F48 11.955 Tf 9.755 0 Td[(a26=0.ToapplythenitedifferencemethodasshowninFigure 3-1 ,weset8>>>>>>>><>>>>>>>>:i;j=xi;yjxi=ihyj=jhDx=Dy-5

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19 Figure3-1:Finitedifferencemethod. Therefore,8>>>>>>>><>>>>>>>>:x;x=1 Dx2i+1;j)]TJ/F19 11.955 Tf 11.996 0 Td[(2i;j+i)]TJ/F19 7.97 Tf 6.448 0 Td[(1;jy;y=1 Dy2i;j)]TJ/F19 7.97 Tf 6.447 0 Td[(1)]TJ/F19 11.955 Tf 11.996 0 Td[(2i;j+i;j+1x;y=)]TJ/F19 7.97 Tf 6.448 0 Td[(1;j)]TJ/F47 12.457 Tf 12.115 0 Td[(1;jyx;0=i;1)]TJ/F47 12.457 Tf 12.116 0 Td[(i;)]TJ/F19 7.97 Tf 6.448 0 Td[(1-6Equation 3-4 becomes:8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:i;j)]TJ/F19 7.97 Tf 6.448 0 Td[(1+a2i+1;j)]TJ/F19 11.955 Tf 11.996 0 Td[(2a2+1i;j+a2i)]TJ/F19 7.97 Tf 6.448 0 Td[(1;j+i;j+1=0;)]TJ/F19 7.97 Tf 6.448 0 Td[(1;j)]TJ/F47 12.457 Tf 12.116 0 Td[(1;j=0j=0;1;N;N+1;j)]TJ/F47 12.457 Tf 12.116 0 Td[(N)]TJ/F19 7.97 Tf 6.448 0 Td[(1;j=0j=0;1;N;i;1)]TJ/F47 12.457 Tf 12.116 0 Td[(i;)]TJ/F19 7.97 Tf 6.447 0 Td[(1=)]TJ/F19 11.955 Tf 9.671 0 Td[(2ha2ei=0;1;N;1 2h[)]TJ/F47 12.457 Tf 9.791 0 Td[(i;N)]TJ/F19 7.97 Tf 6.447 0 Td[(1+i;N+1]=)]TJ/F48 11.955 Tf 9.755 0 Td[(a2eihi;N=)]TJ/F48 11.955 Tf 9.756 0 Td[(aihi;Ni=0;1;N;aex=exp[)]TJ/F19 11.955 Tf 9.672 0 Td[(100x)]TJ/F19 11.955 Tf 11.996 0 Td[(0:252]:-7thesolutiontoequation 3-7 yieldsx;xx;0,xandx;0.Duetothenormal-izationinydirection,correspondingly,equation 3-2 willbe,=axxx;0+x x;0-8TheresultsareshowninFigure 3-2 .Wherethesolidcurverepresentsx=

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20 Figure3-2:ComparisonsoftheforwardandinverseresultsofTPAmethod. exp[)]TJ/F19 11.955 Tf 9.672 0 Td[(100x)]TJ/F19 11.955 Tf 12.327 0 Td[(0:252]andthepointsrepresentxcalculatedfromtheanditsderivativesaccordingtoequation 3-8 .3.2.3SummaryThismethodisonlyusefulfortwo-dimensionproblemofathinstripandforthree-dimensionproblemofathinplatesincetheinstabilityincreasesdramaticallywiththicknessa.Therefore,thismethodisinapplicabletotheburiedundergroundpipelineundercathodicprotectionsystem.Inaddition,thecorrosionparameterprovidesaninadequatedescriptionofthecorrosionproblem.TheselimitationsshowthatTPAmethodisnotsuitableforthisapplication.3.3Two-DimensionalBoundaryElementMethod3.3.1IntroductionAokietal., 18 19 solvedatwo-dimensionalinverseproblem.Thenonlinearrela-tionshipbetweenthepotentialandcurrentdensitywasassumedtobeknownontheboundaryofcathodeasequation 3-9 andanodeasequation 3-10 .q=q0cfexpc)]TJ/F47 12.457 Tf 12.116 0 Td[( c)]TJ/F19 11.955 Tf 11.996 0 Td[(expc)]TJ/F47 12.457 Tf 12.116 0 Td[( )]TJ/F47 12.457 Tf 9.791 0 Td[(cg-9

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21 q=q0afexpa)]TJ/F47 12.457 Tf 12.115 0 Td[( a)]TJ/F19 11.955 Tf 11.996 0 Td[(expa)]TJ/F47 12.457 Tf 12.115 0 Td[( )]TJ/F47 12.457 Tf 9.791 0 Td[(ag-10Boththe-potentialandq-currentdensitywereunknown.AokiappliedaNewton-RaphsoniterativeprocedureandusedsingularvaluedecompositionSVDmethodtogetandq.Whereq0a,q0c,c,a,a,c,aandcareparameters.Exceptfortheparametertorepresentstheconnectingpositionoftwoelectrodes,fourparam-etersareusedforeachelectrode.Thus,theseparametersneedtobettedintheinverseprocess.KenjiandAoki 20 21 simpliedequations 3-9 and 3-10 byusinganinversehyperbolicsinefunction.Thepolarizationcurvesforanodeandcathodeturnouttobe,fq=1sinh)]TJ/F19 7.97 Tf 6.447 0 Td[(1q+1-11fq=2sinh)]TJ/F19 7.97 Tf 6.447 0 Td[(1q+2-12where1,1,2,and2aretheparameters.Thenumberofparameterswasre-ducedafactorof2.Theauthorsusedaconjugategradientmethodtominimizetheobjectivefunction 3-13 gi;i;x0=kj=1j)]TJ/F47 12.457 Tf 12.116 0 Td[(j2i=1;2-13wherejandjrepresentmeasuredpotentialandcalculatedpotentialrespec-tively.x0istheconnectingpointoftwoelectrodes.However,theresultsofthet-tedparametersandtheirstandarddeviationswerenotgiven.Inordertoimprovetheaccuracyoftheresults,theauthorscheckedtheeigenvaluesoftheHessianmatrixoftheobjectivefunctionattheconvergencepointandfoundthatthreeoftheveeigenvalueswereapproachingzero.Inotherwords,theveparametersoftheobjectivefunctionactuallyoccupiedaboutatwo-dimensionalsub-domain

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22 Figure3-3:Schematicrepresentationofthetwo-dimensionalproblem. insteadofave-dimensionaldomain.Thisisthereasonthattheresultscannotbeclosetotheexactsolutionsandwhyitisdifculttogivetheaccuracyoftheparam-eters.Inaddition,anaprioritechniquewasusedintheirwork.Apriorimeanstheprobablevaluesofsomeparametersthatneedtobeestimatedareknown.Insuchcases,onemaywanttoperformatthattakesthisadvanceinformationproperlyintoaccount,neithercompletelyfreezingaparameteratapredeterminedvaluenorcompletelyleavingittobedeterminedbythedataset.3.3.2EvaluationProcessTheprocedureofevaluatingtheinversemodeldescribedbytheliteraturetookplaceintwostages.Attherststage,aforwardmodelforthetwo-dimensionalsystemwascreated,whichwassimilartotheworkofAokietal.'s. 18 20 Basedontheforwardmodel,thesecondstageinvolvedtestingtheregressionapproachesandcomparingthem.3.3.3Two-DimensionalProblemThetwo-dimensionalproblemstudiedhereisinarectangulardomainasshowninFigure 3-3 ,whichisbydescribedAokietal.. 18 20 Thelowerhorizontalbound-aryconsistsofacathodeandananode.Thecathodemaybeconsideredtoplaytheroleofthepipeline,andtheanodeplaystheroleofasacricialanode.Theupper

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23 a bFigure3-4:Reproducingresultsforthetwo-dimensionalforwardmodel:aPoten-tialsattheboundaryofdomain;bCurrentdensityalongtheelectrodesurface. horizontalboundarycanbeconsideredasthesoilsurface.Neitherthepotentialnorcurrentdensityqareknownontheelectrodeboundary,butthenonlinearre-lationshipbetweenthemisknown.Thenonlinearrelationshipbetweenpotentialandcurrentdensityqisthepolarizationcurveoftheelectrodes,whicharegivenasequations 3-9 and 3-10 .Theparametersareq0c=q0a=1Am)]TJ/F19 7.97 Tf 6.448 0 Td[(2,c=0:845V,a=)]TJ/F19 11.955 Tf 9.671 0 Td[(0:985V,c=0:001V,a=0:025V,c=0:05Vanda=0:05V.Theparame-ters,andq0arepolarizationparameterstobeobtainedbytheinversetechniquedescribedinsection 3.4 .Thesubscriptscandarepresentthecathodeandanode,respectively.PotentialwithintheinteriorofthesystemisgovernedbyLaplace'sequation.WefollowedAoki'smethod,applyingaNewton-Raphsoniterativeprocedurefortheforwardproblem.Thepotentialandcurrentdensitydistributionsfromthefor-wardmodelareshowninFigure 3-4 .AsseeninFigure 3.4a ,Aokietal.showedthatthevariationofpotentialontheuppersurfaceismuchlessthanthatseenontheelectrodeboundary. 18 20 Itisevidentthatalargedistancebetweentheelec-trodesandthesoilsurfacewillblurthemeasurementdataonthesoilsurface.

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24 Figure3-5:Schematicrepresentationofthetwo-dimensionalinversemodel. 3.4InverseAnalysisinTwoDimensionsThepurposeofthisportionofeffortwastoapplyaninverseboundaryele-mentmethodtodevelopanefcientapproachforidenticationofthepolariza-tioncurveinacathodicprotectionsystem.Differentregressionmethodswillbeexploredinthissection.3.4.1ObjectiveFunctionAnobjectivefunction,whichdescribesthedifferencebetweenthemeasuredpotentialandcalculatedpotentialonthesoilsurface,wasconstructedasfollowinggi;i;x0=kj=1j)]TJ/F47 12.457 Tf 12.116 0 Td[(j2i=1;2-14wherejandjrepresentmeasuredandcalculatedsoilsurfacepotentials,respec-tively.Forthetwo-dimensionalinverseanalysis,Aokietal. 18 20 describeditinFigure 3-5 .Theparametersofthepolarizationcurveswerenotknown.Whatwasknownisthemeasuredpotentialonthesoilsurface.Thegoaloftheinverseanaly-sisistoidentifytheparametersofthepolarizationcurvesthatwouldprovidetheminimumvaluefortheobjectivefunction.

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25 3.4.2RegressionMethodAnalysisWetestedseveralregressionmethods,includingtheLevenberg-Marquart,conju-gategradientandquasi-Newtonmethods.Thesemethods,distinguishedbytheneedtoevaluateboththefunctionvalueandthefunctiongradient,wererejectedforthepipelineinversemodel.Forthepresentproblem,evaluationofthefunctiongradi-entcanonlyusenite-differencemethod,whichrequiresextracalculationsoftheforwardmodel.Itinevitablyaddssignicantworktothecomputation.Besides,themostseriousdeciencywasthatthesemethodswerefoundtobeextremelysensitivetothevalueofaninitialguessandtohaveatendencytondlocalratherthanglobalminima.Thedownhillsimplexmethodrequiresevaluationonlyoftheobjectivefunctionitself,i.e.,thederivativeoftheobjectivefunctionisnotneeded.Astheresultsofthiswork,itisshownthatthedownhillsimplexmethodisthemostrobustone,independentofthemeasurementpositionandtheinitialguesses.Itwassuccessfulforourtwo-dimensionalinverseanalysisandwillbefurtherde-velopedforthethree-dimensionalproblems.3.4.3DownhillSimplexMethodThedownhillsimplexmethodwasrstcreatedbyNelderandMead. 22 Onlytheobjectivefunctionneedstobeevaluated,nosuchneedtoitsderivative.Anobjectivefunction,whichhasNnumberofadjustableparameters,isevaluatedforN+1points.TheN+1pointsaregeneratedbydeningastartingpointP0,thenotherpointsareobtainedbyusingPi=P0+iei-15whereeiareNunitvectors,iareconstantssetaccordingtotheproblem'scharac-teristiclengthscaleatdifferentdirections.ThegeometricalgureincludingN+1vertices,alltheinterconnectinglinesegmentsandpolygonalfacesiscalleda`sim-

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26 Figure3-6:Schematicrepresentationofthedownhillsimplexmethod. plex'.AccordingtothefunctionvaluesonN+1points,byusingsteps,suchasre-ection,reectionandexpansion,contractionandmultiplecontraction,asshowninFigure 3-6 ,onecanndanewpointtocontinuethefunctionevaluationuntiltheobjectivefunctionencountersaminimum.3.4.4AccuracyoftheParametersOneoftheimportantissuesoftheregressionprocessishowtodeterminetheaccuracyoftheparameters.Fora2meritfunction2ak=Ni=1yi)]TJ/F48 11.955 Tf 12.642 0 Td[(yi i2-16akrepresentstheparameters.yiandyiarethemodelandthemeasurementvaluesrespectively.iisthestandarddeviationforthemeasurementdatapointyi.Inthissection,wearegoingtodiscusstheuncertaintiesoftheparametersak.

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27 Thevarianceandthecovarianceofparameterakaregivenas2ak;aj=Ni2i@aj @yi@ak @yi;j;k=1;2;:::;M-17Ifj=k,2aj=Ni2i@aj @yi2-182ajakisaMMmatrix.Wecanset2ajak=[C]-19Thegradientof2withrespecttotheparametersakwillbezeroatthe2mini-mum.Thus,fromequation 3-16 ,weget@2 @ak=)]TJ/F19 11.955 Tf 9.672 0 Td[(2Ni=1yi)]TJ/F48 11.955 Tf 12.642 0 Td[(yi 2i@yi @akk=1;2;:::;M:-20Takinganadditionalpartialderivativegives@22 @aj@ak=2Ni=1@yi @aj@yi @ak)]TJ/F19 11.955 Tf 11.996 0 Td[(yi)]TJ/F48 11.955 Tf 12.642 0 Td[(yi@2yi @aj@ak1 2i-21Notethatthecomponentsofequation 3-21 dependbothontherstderivativesandonthesecondderivativesofthebasisfunctionswithrespecttotheirparam-eters.Inclusionofthesecondderivativetermcaninfactbedestabilizingifthemodeltsbadly.Fromthispointon,wealwaysdenejkasjk=1 2@22 @aj@ak=Ni=1@yi @aj@yi ak1 2i-22Comparingequation 3-19 withequation 3-22 ,weget[C]=[])]TJ/F19 7.97 Tf 6.448 0 Td[(1-23Thediagonalelementsofthematrix[])]TJ/F19 7.97 Tf 6.447 0 Td[(1aretheuncertaintiesoftheparameters.Thesemethodswillbeusedtoestimatetheuncertaintiesoftheparametersinthenextsection.

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28 Table3.1:Regressionresultfortwo-dimensionalinversemodeldescribedbyAokietal.Theunderlinedsymbolsrepresenttheparametersestimatedbyregression. InitialParameters FinalParameters g StandardDeviations 1=)]TJ/F19 11.955 Tf 9.672 0 Td[(0:2 1=)]TJ/F19 11.955 Tf 9.672 0 Td[(0:1 1:3260310)]TJ/F19 7.97 Tf 6.447 0 Td[(12 1=1:59410)]TJ/F19 7.97 Tf 6.447 0 Td[(61=0:7, 1=0:7, 2=)]TJ/F19 11.955 Tf 9.672 0 Td[(1:0, 2=)]TJ/F19 11.955 Tf 9.672 0 Td[(1:0, 2=0:0, 2=0:0, xconnection=0:5. xconnection=0:5. 1=)]TJ/F19 11.955 Tf 9.672 0 Td[(0:1, 1=)]TJ/F19 11.955 Tf 9.672 0 Td[(0:1, 1:3260310)]TJ/F19 7.97 Tf 6.447 0 Td[(12 1=1:4610)]TJ/F19 7.97 Tf 6.448 0 Td[(61=0:1 1=0:7 2=)]TJ/F19 11.955 Tf 9.672 0 Td[(1:0, 2=)]TJ/F19 11.955 Tf 9.672 0 Td[(1:0, 2=0:0, 2=0:0, xconnection=0:5. xconnection=0:5. 1=)]TJ/F19 11.955 Tf 9.672 0 Td[(0:2 1=)]TJ/F19 11.955 Tf 9.672 0 Td[(0:1576 7:2515710)]TJ/F19 7.97 Tf 6.448 0 Td[(4 1=3:1610+11=0:7, 1=0:7, 2=)]TJ/F19 11.955 Tf 9.672 0 Td[(1:3 2=)]TJ/F19 11.955 Tf 9.672 0 Td[(1:2112 2=1:1310+32=0:0, 2=0:0, xconnection=0:5. xconnection=0:5. 1=)]TJ/F19 11.955 Tf 9.672 0 Td[(0:2 1=)]TJ/F19 11.955 Tf 9.671 0 Td[(0:164 7:3543910)]TJ/F19 7.97 Tf 6.448 0 Td[(3 1=1:1810+21=0:5 1=0:6632 1=3:1610+12=)]TJ/F19 11.955 Tf 9.672 0 Td[(1:3 2=)]TJ/F19 11.955 Tf 9.671 0 Td[(1:0708 2=1:2610+32=0:1 2=0:497 2=1:010+2xconnection=0:5 xconnection=0:5 xconnection=4:3510+1 3.4.5RegressionResultsTheresultsoftheregressionforthetwo-dimensionalproblemarepresentedinTable 3.1 .Theunderlinedsymbolsrepresenttheparametersestimatedbyregres-sion.Itisclearthatthenumberofindependentparametersthatcanbeobtainedinastatisticallysignicantmannerislimited.Whenfourparametersarexed,thefthcouldbeobtainedwithhighcondence.Whentwoandmoreparametersweredeterminedfromtheregression,thecondenceintervalbecametoolarge.Thisanalysisprovidedtwoimportantinsights.Therstwasthatthenumberofstatisticallysignicantparametersthatcouldbeobtainedbyregressionwaslimitedbytheamountandqualityofthedata.Thesecondwasthattheweightedregressionprocedureproposedinthisworkcouldprovideanindicationofwhenthenumberofstatisticallysignicantparameterswasexceeded.

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29 3.5ConclusionsDifferentregressiontechniques,suchasconjugategradientmethod,quesi-Newtonmethodanddownhillsimplexmethod,werecompared.Astheresultsofthestud-iesinthischapter,thedownhillsimplexapproachwasshowntobethemostrobustmethodanditisindependentoftheinitialguesses.Itwillbeappliedfurtherforthethree-dimensionproblem.However,theresultsfromalltheregressionprocedureswereinsufcientlyre-liable.Onepossiblereasonmaybetheinsufciencyofdata,sinceonlyoneseriesofpotentialdatahasbeenused.Futureworkwillincorporatenotonlytheon-potentialdata,butalsotheoff-potentialandthecurrentsurveydata.Thecomplexobjectivefunctiontobeminimizedwillbe:2a1;:::;am="n1i=1yi)]TJ/F48 11.955 Tf 12.642 0 Td[(yi2 2i#on)]TJ/F48 7.97 Tf 6.966 0 Td[(potential+"n2i=1yi)]TJ/F48 11.955 Tf 12.642 0 Td[(yi2 2i#off)]TJ/F48 7.97 Tf 6.966 0 Td[(potential+"n3i=1yi)]TJ/F48 11.955 Tf 12.642 0 Td[(yi2 2i#current-24wherea1,...,amaretheparameterstobetted,iisthestandarddeviationofameasuredvalue,andn1,n2,n3arethenumberofdatapoints.

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CHAPTER4DEVELOPMENTOFTHREE-DIMENSIONALFORWARDANDINVERSEMODELSFORPIPELINEWITHPOTENTIALSURVEYDATAONLY4.1IntroductionFromthepreviouschapter,itisknownthatpipelinesurveysareusuallyusedtodeterminetheadequacyofcathodicprotectionCPfortheundergroundpipelines.However,thesesurveydatamaycontainsignicantscatterornoise.Thelimita-tionsofmeasurementtechniques,instruments,andanalysismethodsmakethedatainterpretationdifcult.Therefore,theobjectiveofthischapteristodevelopamathematicalframeworkforinterpretationofsurveydatainthepresenceofran-domnoise.Therearetwotypesofapproachesusedinmodelingcathodicprotection.Aforwardmodelyieldsthedistributionofcurrentandpotentialforagivensystemgeometryandforgivenphysicalpropertiesofpipes,anodes,groundbeds,andsoil.Aninversemodelyieldssystempropertiessuchaspipecoatingresistivitygivenvaluesforcurrentand/orpotentialdistributions.Thehistoryofthedevelopmentofanalyticandnumericalforwardmodelsforcathodicprotectionofpipelinesencompassesmorethanftyyears.Waberetal. 23 derivedananalyticmodelintheformofaFourierseries,whichcanonlybeusedforsimplegeometriesandboundaryconditions.Piersonetal. 24 developedase-riesofanalyticequationstoaccountforattenuationforcoatedpipelineswhichextendtheusualresistanceformulassuchasDwight'sequation. 9 )]TJET1 0 0 1 6.448 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(26 Doigetal. 27 utilizedthenitedifferencemethodtosimulatethegalvaniccorrosionwithcom30

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31 plexpolarization.Inthe1980's,Brebbia 28 developedtherevolutionaryboundaryelementmethodBEM.TheBEMhassincebeenappliedinmanyengineeringeldsbecauseitisaccurate,effectiveforbothinniteandsemi-innitedomains,andcomputationallyefcient. 28 29 Aokietal. 18 20 30 appliedtheBEMtobothtwo--dimensionalandthree-dimensionalsystems.Tellesetal. 31 improvedtheBEMforCPsimulationbyintroducingthecurrentdensityself-equilibriumlimitation,whicheliminatedtheneedtodiscretizeaboundarylocatedatinnity.BrichauandDeconinck 3 32 coupledtheinternalandexternalLaplaceequationswhichweresolvedbycoupledBEMandniteelementmethodsFEM.Kennelleyetal. 33 34 usedFEMtomodeltheinuenceonCPprotectionofdiscretecoatingholidaysthatexposedbaresteel.TheconceptofdiscreteholidayswasextendedtothreedimensionsusingBEMbyOrazemetal. 35 )]TJET1 0 0 1 6.448 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(37 RiemerandOrazem 1 38 )]TJET1 0 0 1 6.447 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(40 combinedtheadvantagesofthepreviousworkintheirdevelopmentoftheCP3Dmodel.Adistinguishingpointofthissoftwareisthatitcanaccountforlocalizeddefectsandyetissuitableforlongpipelinesandpipenetworks.SomecommercialprogramsareavailablesuchasPROCAT 2 andOKAPPI. 3 AllthesimulationsdescribedabovesolveLaplace'sequationcoupledwithlinearornonlinearboundaryconditionstoobtainthepotentialdistributionandcurrentdistribution.Theanalyticmodelscanonlybeusedfortwo-dimensionaldomainswithsimplegeometries.Thenumer-icalmodelscanbeappliedtoalmostanycomplexdomain;thus,theyarewidelyused.WhilemanynumericalmodelsarebasedontheFEM,thenewergenerationofmodelsuseeitherBEMoracombinationofBEMandFEM.Theuseofcomputerprogramstointerpretpipesurveydataintermsoftheun-knownconditionofthepiperequiressolutionoftheinverseproblem,inwhichthepropertiesofthepipecoatingareinferredfrommeasurementsofcurrentandpo-tentialdistributions.Thisapproachallowsinterpretationofelddatainamanner

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32 thatwouldtakeintoaccountthephysicallawsthatconstraintheowofelec-tricalcurrentfromanodetopipe.Aokietal.hasmadesignicantcontributionstotheapplicationofinversemodeltoCP.Theirstudiesincludedsimplifyingtheunknownparametersinthepolarizationcurves,changingtheformofobjectivefunction,andtryingdifferentkindsofminimizationmethods.Tominimizetheob-jectivefunction,theyhaveemployedavarietyoftechniquesconjugategradient,fuzzyapriori, 20 andgeneticalgorithmmethods. 41 42 Theirworkinvolvesmod-elingprotectionofpipes,ships,andreinforcingsteelinconcretestructures. 41 )]TJET1 0 0 1 6.448 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(42 WrobelandMiltiadoudescribetheapplicationofgeneticalgorithmstoinverseproblems,includingidenticationofcoatingholidays. 44 45 Aokietal.usedanin-verseBEManalysistoeliminateOhmicerrorfrommeasurementofpolarizationcurves. 46 Theconceptbehinduseoftheinversemodelforcathodicprotectionisrelatedtotheapproachusedtoreconstructthedistributionofelectricalresistanceformed-ical,chemicalprocess,andgeologicalapplications.Formedicalapplications,thereadingsfromanarrayofsensingelectrodesareusedtoconstructanimageas-sociatedwithconductivityvariationswithinabody. 47 )]TJET1 0 0 1 6.447 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(50 Thetechniqueiscalledelectrical-impedancetomography. 51 52 Electrical-impedancetomographyhasalsobeenusedtodeterminetheinterfacialareafortwo-phasegas-liquidandparticu-lateowsinchemicalprocesses 53 )]TJET1 0 0 1 6.448 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(56 and,throughidentifyingthedistributionofelectricalresistivity,toidentifycompositiondistributionsinlaboratoryandplant-scaleprocessequipment. 57 Electrical-resistivitytomographyisusedtointerpretcross-boreholeresistivitymeasurementstoobtainelectricalresistivitydistribu-tionsofgeologicalformations. 58 )]TJET1 0 0 1 6.448 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(64 Electricalresistivityisanimportantpetroleumreservoirparameterbecauseitissensitivetoporosity,typeofporeuid,andde-greeofsaturation.Whileuseofneuralnetworkshavebeensuggested, 65 66 most

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33 Figure4-1:Schematicillustrationofthepipesegmentandanodeusedtotesttheinversemodel. approachesarebasedonnon-linearregression.Inthischapter,theconstructionoftheforwardmodel,inversemodelforCPsystem,theoreticalbasisanddevelopmentofthethree-dimensionalBEMwillbeintroduced.4.2ConstructionoftheForwardModel4.2.1CathodicProtectionSystemForthethree-dimensionalproblem,apipewasconsideredtobeburiedhori-zontallyunderthegroundsurface.Acylindricalverticalanodewasplacedadis-tanceawayfromthepipe.Theundergroundregionwasconsideredtobeboundedbythesoilsurfaceandtoextendinnitelyintheotherdirectionsinthesoil.Theanodewasconnectedtothepipewithawire,asshowninFigure 4-1 .Thepipewasplacedhorizontally1.0m.75feetbelowthesoilsurfaceandhadadiame-terof0.457m.5feet.Theanodewasplacedinaverticalpositionandadistanceawayfromthecenterofthepipe.Theanodediameterwas0.152mandthelengthwas1.0m.28feet.Thesoilresistivitywas100Wm.A0.5mm-thickcoatingwasassumedtocoverthesideareaofthepipe,withtheexceptionthatthetwoendswereassumedtobeinsulated.Thepotentialofthepipesteelwasassumedto

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34 beuniform.Thisassumption,validfortheshortpipesegmentsconsideredinthispreliminarywork,willneedtobeextendedinfuturework.TheforwardmodelwasusedtocalculatethepotentialthatonemightmeasurewithavoltmeterconnectedtothepipelineandaCu/CuSO4referenceelectrodeCSE.4.2.2GoverningEquationandBoundaryConditionsThecurrentdensityinadiluteelectrolytesolutioncanbedescribedas 8 i=)]TJ/F48 11.955 Tf 10.209 0 Td[(F2rjz2jujcj)]TJ/F48 11.955 Tf 12.534 0 Td[(FjzjDjrcj+Fvjzjcj-1whereiisthecurrentdensityvector,visthebulkvelocity,FisFaraday'sconstant,zjisthechargeforspeciesj,ujisthemobilityforspeciesj,cjistheconcentrationforspeciesj,andDjisthediffusioncoefcientforspeciesj.Inthisstudy,onlytheelectrodesurfaces,suchaspipewallsareconsidered.Sincethevelocityvissmallinthesoil,theconvectioncanbeneglected.Inaddition,concentrationgradientsonlyhaveasignicanteffectinthediffusionlayer,whichisclosetothestruc-tureandisrelativelythincomparingwiththecharacteristiclengthofthesystem.Thus,concentrationgradientsgenerallyareneglectedinlarge-scalesimulations.Therefore,thecurrentdensityinthesoilisi=)]TJ/F48 11.955 Tf 10.21 0 Td[(F2rjz2jujcj-2Theconductivityoftheelectrolyteisdenedtobe=F2rjz2jujcj-3Thecurrentdensityequationcanbereducedtoi=)]TJ/F47 12.457 Tf 9.708 0 Td[(r-4whichiscalledOhmsLaw.Theconservationofchargeinthebulkyieldsri=0-5

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35 Substitutingequation 4-4 intoequation 4-5 r)]TJ/F47 12.457 Tf 9.707 0 Td[(r=0-6Thusr2=0-7Equation 4-7 isknownasLaplace'sequationfortheelectrochemicalpotential.TheboundaryconditionsareshowninFigure 4-1 .Thecorrosionpotentialofpipesteelis)]TJ/F19 11.955 Tf 9.671 0 Td[(0:6V.Thesacricialanodehasagivenpotentialof)]TJ/F19 11.955 Tf 9.672 0 Td[(1:2V.4.2.3TheoreticaldevelopmentAsmentionedpreviously,theforwardmodelhastobeefcientintermsofcom-putercalculationsduetoitscouplingwiththeinversemodel.Toachievethisgoal,alinearrelation,consideringthecoatingcoveredpipewithvariousresistivity,isproposedinthissection.LinearBoundaryConditionsThepolarizationcurve,whichdescribestherelationshipbetweenthepotentialandthecurrentdensity,indicatesthecorrosionconditiononthepipesurfaceandisnormallyusedastheboundarycondition.Itisnotaneasytasktodeterminethepolarizationcurvesinceitstronglydependsonanumberofphenomena.Furthermore,thepolarizationcurvecanalsobeafunctionoftimeandhistory.Thepolar-izationcurveusedbyRiemerandOrazem 1 39 hadeightparameters.AmayaandAokietal. 20 21 simpliedthepolarizationcurvefromtheButler-Volmerequationandreducedtheparameterstofourforonecathodeandoneanodesystem.How-ever,ifthecorrosionconditionalongthepipeisnotuniform,morepolarizationcurvesareneeded.Inpresentstudy,thepolarizationcurvewassubstitutedbyalinearrelationshipbetweenthepotentialdropandthecurrentdensityoverthepipecoatingforthree-dimensionalsimulation,showninFigure 4-2

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36 Figure4-2:Linearrelationshipbetweenpotentialdropandcurrentdensityoverthepipecoating. i=)]TJ/F19 11.955 Tf 16.272 8.093 Td[(1 c[V)]TJ/F47 12.457 Tf 12.116 0 Td[(outer)]TJ/F19 11.955 Tf 11.996 0 Td[(V)]TJ/F47 12.457 Tf 12.116 0 Td[(inner]-8whereiisthecurrentdensityalongthepiperadicaldirection,cistheunitresis-tanceofcoatingmaterial,isthecoatingthickness,Visthepotentialofthepipesteel,assumedtobeconstant,isthepotentialintheelectrolyte,V)]TJ/F47 12.457 Tf 12.198 0 Td[(outeristhepotentialofthepipereferredtoareferenceelectrodeplacedattheoutsidecoatingsurface,andV)]TJ/F47 12.457 Tf 12.191 0 Td[(inneristhepotentialofthepipewithrespecttoareferenceelec-trodeplacedattheinsidecoatingsurface.Undertheassumptionthatthecurrentdensitiesthroughthecoatingandthoseatthepipesteelsurfaceareequal,i=)]TJ/F19 11.955 Tf 10.867 8.093 Td[([V)]TJ/F47 12.457 Tf 12.115 0 Td[(outer)]TJ/F19 11.955 Tf 11.997 0 Td[(V)]TJ/F47 12.457 Tf 12.116 0 Td[(inner] Rc=)]TJ/F19 11.955 Tf 10.868 8.093 Td[([V)]TJ/F47 12.457 Tf 12.116 0 Td[(inner)]TJ/F19 11.955 Tf 11.997 0 Td[(V)]TJ/F47 12.457 Tf 12.116 0 Td[(corr] Rkinetic-9whereRcandRkineticrepresentthecoatingandpolarizationresistance,respec-tively,andV)]TJ/F47 12.457 Tf 12.116 0 Td[(corristhecorrosionpotential.AsRcRkinetic,V)]TJ/F47 12.457 Tf 12.116 0 Td[(inner)]TJ/F19 11.955 Tf 11.996 0 Td[(V)]TJ/F47 12.457 Tf 12.116 0 Td[(corr]0-10Therefore,i=)]TJ/F19 11.955 Tf 16.273 8.093 Td[(1 c[V)]TJ/F47 12.457 Tf 12.116 0 Td[(outer)]TJ/F19 11.955 Tf 11.996 0 Td[(V)]TJ/F47 12.457 Tf 12.116 0 Td[(corr]-11Equation 4-11 providesalinearrelationshipbetweenthepotentialandthecur-rentdensity.Thislinearboundarycondition,replacesthenonlinearpolarization

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37 curveandgreatlysimpliesthecalculationsofforwardmodel.DescriptionofthePipeSurfaceAccordingtothecoating-coveredspecialfeatureofpipe,acoatingresistanceapproximationwasmade.Itallowsthecoatingresistancetovaryalongthelengthofthepipeaccordingto:=0+kke)]TJ/F19 7.97 Tf 6.448 0 Td[(x)]TJ/F48 7.97 Tf 6.663 0 Td[(xk2=22k-12wherein0referstotheaveragegoodcoatingresistanceofthepipe,krepresentstheresistancereductionassociatedwithcoatingdefectk,xkisthecenterpointofthecoatingdefect,andkisthehalfwidthofthedefectregion.Equation 4-12 showsthatifthecoatingpositionisclosetoadefectcenterxk,i.e.,fallingintothedefectregion2k,thecoatingresistancedecayisevident;whereas,ifthecoatingpositionisfarawayfromthedefectcenter,thecoatingresistancedecayisinsignif-icant.Thepresentapproachhastwosignicantadvantagesoverassigningaresis-tivitytoeachcylindricalelement.Forinstance,useofequation 4-12 torepresenttwocoatingdefectsona150feetpipelineisshowninFigure 4-3 .Ifthepipelineisdiscretizedwitha10footspaceinterval,onlysevenparametersarerequired0,1,x1,1,2,x2and2fortheequation 4-12 ,ascomparedto15parametersareneededforanelement-by-elementcoatingresistivityestimation.Ifthepipelineisdiscretizedwitha5footspacing,stillthesesevenparametersarerequired,whileitisnowthirtyparametersfortheregularmethod.Likewise,ifthespaceintervalisdecreasedtobesosmallthatitmakestheresistivityvaluecontinuous,nomorethanthesesevenparametersareenoughtodescriberesistivityalongthewholepipeline.Thedegreeoffreedomfortheproblemisincreaseddramatically,andcorrespondingly,therequiredcomputationaltimeisreduced.

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38 Figure4-3:Schematicillustrationofthepipesurfaceresistivitymodel. 4.3Three-DimensionalBoundaryElementMethodThetheoryforapplicationoftheboundaryelementmethodtothethree-dimensionalhalf-innitymulti-connecteddomainisintroducedinthissection.Thelinearcylin-dricalelementsareadoptedtodiscretizethepipe/anodesurfaceinpresentstudy.4.3.1InnityDomainInthetwo-dimensionalsystem,aclosedrectangulardomainwasinvestigated.Thiskindofsystem,calledaninteriorproblem,isshowninFigure 4-4 andwhereSistheboundaryofthevolumeV,theshadedareaistheinteriordomain.~nisthenormalvectoroftheboundarySatthatpoint.AccordingtoBrebbiaetal., 29 thebasicequationofboundaryelementmethodcorrespondingtotheLaplace'sequationiscxsuxs=Zs[qyuxs;y)]TJ/F48 11.955 Tf 12.115 0 Td[(qxs;yuy]dSy-13Equation 4-13 iscalledboundaryintegralequation.Whereinxsisthesourcepoint,yistheeldpoint.uxs;yandqxs;yarethefundamentalsolutionsofGreen'sfunctionforpotentialandux.uandqarethegeneralsolutionofLaplace'sfunctionforpotentialandux.cxsdependsonthepositionofthe

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39 Figure4-4:Interiorproblem. sourcepointxs,asshowninequation 4-14 ,cxs=8>>>><>>>>:xs=2V;cxs=0xs2V;cxs=1xs2S;cxs=! 4-14!isthesolidanglesubtendedbyVatxs,whichisontheboundaryS.Forthethree-dimensionalsystem,thestudieddomainisoutsidethepipelineandthean-odesurface,andisbetweenthesoilatinnity.Itisamulti-connectedregionofaninteriorproblem,asshowninFigure 4-5 .TheboundaryScanbeconsideredtobetheboundaryofthepipelineortheanode.TheboundarySRcanbetheboundaryatinnity.WhenR!1,equation 4-13 isalsovalidforthiscase.Forxs2S,cxsuxs=Zs[qyuxs;y)]TJ/F48 11.955 Tf 12.115 0 Td[(qxs;yuy]dSy+limR!1ZSR[qyuxs;y)]TJ/F48 11.955 Tf 12.115 0 Td[(qxs;yuy]dSy -15 ThefundamentalsolutionstotheGreen'sfunctionforthree-dimensionalproblemattheboundarySRareuxs;y=1 4Rqxs;y=@uxs;y @n=)]TJ/F19 11.955 Tf 21.334 8.094 Td[(1 4R2 -16

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40 Figure4-5:Multiconnectedregionofinteriorproblem. AndthegeneralsolutiontotheproblemovertheinnityboundarySRareuy=Kuxs;y+u1O1 Rqy=)]TJ/F48 11.955 Tf 10.162 0 Td[(K 4R2O1 R2 -17 whereKandu1aretwoundeterminedconstantsattheinnityboundary.Kisthesumofsourceterme.g.,electricchargedistributedoverS,K=ZSqydSyO-18Equation 4-16 and 4-17 areintroducedintoequation 4-15 .SincelimR!1RSR[qyuxs;y)]TJ/F48 11.955 Tf 12.116 0 Td[(qxs;yuy]dSy=limR!1RSR[qyuxs;y)]TJ/F48 11.955 Tf 12.115 0 Td[(qxs;yKuxs;y+u1]dSy=limR!1RSRfuxs;y[qy)]TJ/F48 11.955 Tf 12.487 0 Td[(Kqxs;y])]TJ/F48 11.955 Tf 12.115 0 Td[(qxs;yu1gdSy -19 wheretherstterminsidethebracketislimR!1RSRuxs;y[qy)]TJ/F48 11.955 Tf 12.486 0 Td[(Kqxs;y]dSy=limR!1RSR1 4R[O1 R2)]TJ/F19 11.955 Tf 11.997 0 Td[()]TJ/F48 7.97 Tf 17.488 4.711 Td[(K 4R2]dSylimR!1[O1 R34R2]!0 -20

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41 andthesecondtermislimR!1ZSRqxs;yu1dSy=limR!1u1)]TJ/F19 11.955 Tf 21.334 8.093 Td[(1 4R24R2=)]TJ/F48 11.955 Tf 9.791 0 Td[(u1 -21 Thenequation 4-15 becomes,cxsuxs+Zs[)]TJ/F48 11.955 Tf 9.791 0 Td[(qyuxs;y+qxs;yuy]dSy=u1-22Equation 4-22 indicatesthatthevalueofKhasnoimportanteffectonthebound-aryS.TosatisfythepipelineandtheanodeCPsystem,theunitoutwardnormalnonSinFigure 4-5 needstochangeitsdirection.Correspondingly,equation 4-22 willbecxsuxs+Zsf)]TJ/F48 11.955 Tf 16.02 0 Td[(qyuxs;y+[)]TJ/F48 11.955 Tf 9.792 0 Td[(qxs;yuy]g+dSy=u1-23Inaddition,accordingtothephysicalcondition,thereisnocurrentowoutofthesoilsurface,therefore,K=0isgivenasboundaryconditionandu1isunknown.4.3.2Half-InnityDomainThehalf-innitydomainisobtainedbyusingaplanetosplitaninnitedo-main.Twohalfspaceslieoneachsideoftheplane.EithertheDirichletorNeu-mannboundaryconditionissatisedattheplane.Fortheburiedpipelinessys-tem,theundergroundsoildomainisthehalf-spacebeingstudied.Sincethereisnonetcurrentowingoutofthesoilintotheair,theNeumannboundarycondi-tionvanishesatthesoilsurface.TheGreen'sfunctioncansatisfythatconditionbyusingthereectionproperties. 29 )]TJET1 0 0 1 6.448 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(70 IfwesetW)]TJ/F19 11.955 Tf 9.619 -4.34 Td[(tobetheregionx0,P2W)]TJ/F19 11.955 Tf 9.618 -4.34 Td[(andletP0)]TJ/F48 11.955 Tf 9.995 0 Td[(x;y;zbethemirrorimageofagivensourcepointPx;y;zwithrespecttotheplanex=0or@W,P0becomesthereectedsourcepoint,andr=jQ)]TJ/F48 11.955 Tf 12.618 0 Td[(Pj,r0=jQ)]TJ/F48 11.955 Tf 12.776 0 Td[(P0jseeFigure 4-6 .Qistheeldpoint.Consequently,half-spacefun-damentalsolutionssatisfyingahomogeneousNeumannconditionHNP;Q=0

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42 on@Wcanbededucedbyadequatesuperpositionoffree-spacefundamentalsolu-tionsforsourcesatPandP0,thatis@GNP;Q @QnQ=HNP;Q=0 -24 Here,thesuperscriptNisfortheNewmannboundaryconditions;nQisthenormalvectoratpointQ.Assumingtwofullfree-spaceproblemswith8Q2W)]TJ/F19 11.955 Tf 6.946 -4.34 Td[(,equation 4-24 canbewrittenas@G)]TJ/F19 11.955 Tf 6.946 -4.339 Td[(P;Q @QnQr=H)]TJ/F19 11.955 Tf 6.946 -4.937 Td[(P;Q=)]TJ/F19 11.955 Tf 19.373 8.094 Td[(1 4r2nQr -25 @G+P0;Q @QnQr0=H+P0;Q=)]TJ/F19 11.955 Tf 20.592 8.094 Td[(1 4r02nQr0 -26 FromFigure 4-7 ,thenormalvectorsnQrandnQr0of~rand~r0atpointQarenQr=1 -27 nQr0=)]TJ/F19 11.955 Tf 9.672 0 Td[(1 -28 Meanwhile,accordingtotheNeumannboundaryconditiongiveninequation 4-24 HNP;Q=H)]TJ/F19 11.955 Tf 6.946 -4.937 Td[(P;Q+H+P0;Q=0 -29 WhenQ2@W,r=r0,thefundamentalsolutionintwofullfree-spacehavetobeG)]TJ/F19 11.955 Tf 6.946 -4.938 Td[(P;Q=1 4r -30 G+P0;Q=1 4r0 -31 tosatisfyequation 4-24 ,indicatingthatthetwosourceintensitiesPandP0areequalandhavethesamesign.Therefore,thenalformoftheGreen'sfunctionisGNP;Q=G)]TJ/F19 11.955 Tf 6.946 -4.937 Td[(P;Q+G+P0;Q=1 41 r+1 r0-32

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43 Figure4-6:Schematicillustrationofmirrorreectiontechnique. Figure4-7:TheFundamentalsolutiontothehalf-innitydomainsatisfyingtheNeumanb.c's.

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44 Figure4-8:Pipediscretisionandcollocationpoints. 4.3.3BoundaryDiscretizationRectangularandtriangularelementsarecommonlyusedintheliteratureforBEMmeshesonthesurfaceofathree-dimensionalobject.Aokietal. 71 used352constanttriangularelementstodiscretizethesideareaofa160mmlong,155mm-diametercylinder.RiemerandOrazem'smodelCP3Dmayinvolvethousandsofelementstodiscretizeapipelinewithdiscretecoatingholidays.Thislevelofde-tailisappropriateforaforwardmodel,butitisinappropriateforaninversemodelwheretheamountofdatamaybeinsufcienttoextractahighlevelofdetailscon-cerningthepipecondition.Besides,themodelneedstobecalculatednotonce,butthousandsoftimes.Todecreasethenumberofelements,aseriesoflinearcylindricalelementswereusedtotakeadvantageofthecylindricalshapeofthepipeandtheanode.Thesourcepointsalsocalledcollocationpointswerelocal-izedonthecylindricalsidearea.Atthetwoendsoftheobjectpipe/anode,thetwodiscsweresetastwoconstantcircleelements.ThecenterofeachcirclewasacollocationpointseeFigure 4-8 .Forthecylindricalelements,therewasnovaria-tionalongthecircumference.Thevariationwasonlyalongtheaxialdirection 72 )]TJET1 0 0 1 6.448 0 cm0 g 0 G1 0 0 1 0.498 0 cm0 g 0 GBT/F19 7.97 Tf 0 0 Td[(75 suchthatthecollocationpointscouldbechosenatthereferenceline,whichwasonthetopofthecylinder.TheelementsonthesideareaofthecylinderwerefromS0toSnel)]TJ/F19 7.97 Tf 6.448 0 Td[(1,and,atthetwoendsofthepipe,werethetwocircleareasSnelandSnel+1.Thecollocationpointsonthesideareaofcylinderwerefrom0ton)]TJ/F19 11.955 Tf 12.126 0 Td[(1,

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45 whiletwocollocationpointsatthecenterofcirclesarecountedasnandn+1.Therelationbetweenthenumberofelementsandthenumberofnodesdependsontheorderoftheinterpolationfunctions,whichcanbeconstant,rst-orderorevenhigherorder.Theidenticalmeshwasusedfortheelementseparationbecauseinthefollow-inginversemodel,thepositionofthecoatingdefectswasdistributedrandomlyalongthepipeline.Ifthepipelinemeshingdependedonthepositionofcoatingdefects,everytimewhenthenewcoatingdefectpositionswouldbeassumed,thesystemwouldhavetoberemeshed.Tosavecomputationaltime,byusingtheidenticalmesh,thematricesGandHfortheBEMneedtobecalculatedonlyonce.4.3.4CoordinatesDenitionandTransformationInthisprogram,twokindsofobjectsweredened,oneispipe,andtheotherisanode.Theuserneedstoinputthebeginningpointofthepipe/anode'saxletotheprogram.Boththeglobalcoordinateandthelocalcoordinatewereusedinthisstudy.Theglobalcoordinateswassetas:zaxialparalleltothesoilsurfaceanddirectedtonorthdirection;xaxialperpendiculartothesoilsurfaceandpointedtowardthesky.Theyaxialandzaxialbuildthesoilsurfaceplane.Theoriginisonthesoilsurface.Theanglebetweenthepipe/anode'saxleandsoilsurfaceisdenedastheelevationanglee.e=0meansthattheobjectisparalleltothesoilsurface.e=90ore=)]TJ/F19 11.955 Tf 9.672 0 Td[(90meansthattheobjectisperpendiculartothesoilsurface.Thedirectionangledreferstheanglebetweenthepipe/anode'saxleandzaxialpositivedirection.Thelocalcoordinateisusedwhenthecollocationpointandtheeldpointareonthesameobjecti.e.,thepipe/anode,whiletheglobalcoordinateisusedwhenthecollocationpointandeldpointareonthedifferentobjectstoclarifytherelativepositionsoftheseobjects.Transformationbetweentwodifferentcoordinates,suchastheglobalandthe

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46 Figure4-9:Coordinaterotation. localcoordinatesortwodifferentlocalcoordinates,usuallyconsistsoftwosteps.Oneisorigintransformation,whichistomovefromoneoriginalpointtoaneworigin.Theotherisrotation,whichistorotatepreviousaxialsaroundtheneworigin.Coordinaterotationactuallyincludestworotations.Figure 4-9 showsthepreceduretorotateacoordinatearoundapointandoneaxis.Therstrotationistorotatethexaxisandthezaxisabouttheyaxiswithelevationangleetogetnewx0axisandz0axis.Thesecondrotationistoxx0,andtorotateyandz0axeswithdirectionangledtogetanewaxesy00andz00.Finally,coordinatesx;y;zisconvertedtox0;y00;z00.4.3.5DiscretizationofBoundaryElementMethodRewritingequation 4-23 usingPassourcepoint,Qaseldpointand@Wasboundary,wegetcuP+Z@WuQqPQd@W=Z@WqQuPQd@W+u1 -33

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47 whereuandqarethefundamentalsolutionoftheGreenfunction,andtheyhavetherelationqPQ=@uPQ @nQ -34 Discretizingequation 4-33 ,ciuiP+nelj=1ZSjujQ@uPQ @nQdSj=nelj=1ZSjqjQ@uPQdSj+u1-35wherenelmeansthenumberofelement.Sinceuandqofeachlinearcylindricalelementvarylinearlyalongthezaxialdirection,thenuz=u1l)]TJ/F48 11.955 Tf 12.331 0 Td[(z+u2zqz=q1l)]TJ/F48 11.955 Tf 12.331 0 Td[(z+q2z -36 wherelisthelengthoftheelement.Lett=z l,thenuandq,thefunctionsofz,canbetransferredtobethefunctionsoft,ut=l[u1l)]TJ/F48 11.955 Tf 12.116 0 Td[(t+u2t]=l[u1N1t+u2N2t]qt=l[q1l)]TJ/F48 11.955 Tf 12.116 0 Td[(t+q2t]=l[q1N1t+q2N2t] -37 Thebasisfunctionscanbeextractedfromequation 4-37 ,N1t=1)]TJ/F48 11.955 Tf 12.116 0 Td[(t;N2t=t: -38 Uponsubstitutingequation 4-37 intoequation 4-35 ,thediscretizedboundaryintegralequationbecomesciuiP+nelj=1ZSjlj[uj;1N1t+uj;2N2t]@uij @njdSj=nelj=1ZSjlj[qj;1N1t+qj;2N2t]uijdSj+u1 -39

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48 Sincethefundamentalsolutionsareuij=1 4rijqij=@uij @nj=1 4@1 rij @nj -40 Accordingtoequation 4-39 ,wedeneGij=ZSj2k=1ljqjkNktuijdSj -41 Hij=ZSj2k=1ljujkNktqijdSj -42 IftheeldpointQisonthecylindricalelementsidearea,dSj=Rddt,Ristheradiumofcylinder,thenGijbecomesGij=Z10Z202k=1ljqjkNjtuijRddt=ljR 4Z102k=1qjkNktdtZ201 rijd -43 andHijbecomesHij=Z10Z202k=1ljujkNktqijRddt=ljR 4Z102k=1ujkNktdtZ20@1 rij @njd -44 IftheeldpointQisonthecircleelements,dSj=rdrd,0rR,GijchangesitsformtoGij=qj1 4ZR0Z201 rijrdrd -45 andHijisinformHij=uj1 4ZR0Z20@1 rij @njrdrd -46 TheboundaryintegralforGijandHijwiththedifferentpositionsofthesourcepointPandtheeldpointQisshowninAppendix A

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49 Inaddition,addingtheunknownpotentialatinnityu1inequation 4-39 toHij,HmatrixbecomesH=264Hp;pHp;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1Ha;pHa;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1375np+nanp+na+1-47where-1columncorrespondstou1.AndtheGmatrixisG=264Gp;pGp;aGa;pGa;a375np+nanp+na-48Thesubscriptpandaindicatethepipeandtheanoderespectively.Thesubscriptp;panda;aindicatethatthecollocationpointsandtheeldpointsareonthesameobject.Whilep;ameansthatthecollocationpointsareonthepipeandtheeldpointsareontheanode.Likewisea;pmeansthatthecollocationpointsareontheanodeandtheeldpointsareonthepipe.Thenumberofnodeonthepipeandtheanodearerepresentedbynpandna.4.3.6RowSumEliminationThediagonalelementsofthematrixHareusuallydifculttocalculatsincethelinearorhigherorderelementshavebeenusedandtheconstantciinequation 4-14 involvesthecalculationofthesolidanglesubtendedbytheregionVatxsonS.Inordertoovercomethisdifculty,Gibbs'stheoremcanbeappliedtondthevaluesofthediagonal. 76 ForadomaingovernedbyLaplace'sequation,ifthepotentialisuniformthroughoutthedomain,anditsgradientofthepotentialiszeroatinnityifitexists,thegradientofthepotentialwillbeequaltozerowithinthedomain.Forthethree-dimensionalpotentialproblem,uy1isdenedintheinteriorregion.Sinceqy=@uy @n=0ontheboundary,qy=0inthedomain.Thusequation 4-13 becomescxs=)]TJ/F68 10.76 Tf 11.332 14.938 Td[(Zsqxs;ydSy-49

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50 Itmeansthat,i;jHi;j=0 -50 wherethediagonaltermsareunknown.Thediagonaltermsarenowspeciedbythenegativesumoftheoff-diagonaltermsofeachrowHi;i=)]TJ/F17 17.215 Tf 11.332 -2.553 Td[(j6=iHi;j-51wherejgoesfromonetothenumberoftermsinarow.Thesametechniqueusedfortheinteriordomainisappliedtopresentwork.Assuminguy1,sinceqy=@uy @n=0ontheboundaryandinthedomain,equation 4-15 becomescxs=)]TJ/F68 10.76 Tf 11.333 14.938 Td[(Zsqxs;ydSy)]TJ/F19 11.955 Tf 14.442 0 Td[(limR!1ZSRqxs;ydSy -52 Additionally,sincelimR!1ZSRqxs;ydSy=)]TJ/F19 11.955 Tf 21.334 8.093 Td[(1 4R2R2=)]TJ/F19 11.955 Tf 9.671 0 Td[(1 -53 cxsinequation 4-52 becomescxs=)]TJ/F68 10.76 Tf 11.332 14.938 Td[(Zsqxs;ydSy+1 -54 whichmeansthat,Hi;i=1)]TJ/F17 17.215 Tf 11.996 -2.553 Td[(j6=iHi;j-55Therefore,byusingGibbs'stheorem,thediagonalelementsofthematrixHcanbeobtainedfromtheoff-diagonalelementsandthecalculationdifcultiesareavoided.4.3.7Self-EquilibriumCathodicprotectionsystemsdonotloseorgaincurrentfromtheirsurround-ings.Inotherwords,thecurrentisconserved.Therefore,iftheuxbetweenthe

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51 anodeandthepipeisself-equilibrium,theuxatinnitywillbezero.Toimple-mentthiscondition,anextraequation,equation 4-56 isaddedtothesystem. 31 77 ZSqdS=0-56ItindicatestheintensityofanequivalentsourcedistributedoverS.Italsocanbewrittenasi1=kip;kAp;k+kia;kAa;k=0-57whereAk=ZSkdSk;k2[0;np]or[0;na]-58Addingequation 4-57 toequation 4-47 and 4-48 ,anddeningthepotentialvectoruas=266664pa1377775np+na1-59theuxvectorqasnr=264nrpnra375np+na1-60weget266664Hp;pHp;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1Ha;pHa;a)]TJ/F19 11.955 Tf 9.672 0 Td[(10003777758>>>><>>>>:pa19>>>>=>>>>;=266664Gp;pGp;aGa;pGa;aApAa3777758><>:)]TJ/F48 11.955 Tf 9.791 0 Td[(n5p)]TJ/F48 11.955 Tf 9.791 0 Td[(n5a9>=>;-61HereHmatrixissingularbecausethereisstillarowofzerosinit.However,itwillonlybethecasewhenNeumanntypeboundaryconditionsarespeciedeverywhere.TheNeumannproblemresultsinaninnitenumberofsolutions.

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52 Therefore,atleastoneelementinthesystemmusthaveaDirichletboundarycon-ditiontomaketheHmatrixnonsingularandresultinauniquesolution.Inaddi-tion)]TJ/F48 11.955 Tf 9.792 0 Td[(n5hasaminussignbecausethenormalvectordirectionisoutwardofboundaryS.4.4ForwardModel4.4.1ConstantSteelPotentialAssumptionSincetheaxialvariationofpotentialinpipesteelistrivialforashortsectionorlowresistancepipelines,thepotentialofpipesteelisassumedtobeaconstantinthischapter.Rewritingequation 4-61 bymakingavariablesubstitution=V)]TJ/F47 12.457 Tf 11.914 0 Td[(,weget266664Hp;pHp;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1Ha;pHa;a)]TJ/F19 11.955 Tf 9.672 0 Td[(10003777758>>>><>>>>:V)]TJ/F47 12.457 Tf 12.116 0 Td[(pV)]TJ/F47 12.457 Tf 12.116 0 Td[(aV)]TJ/F47 12.457 Tf 12.116 0 Td[(19>>>>=>>>>;=266664Gp;pGp;aGa;pGa;aApAa3777758><>:)]TJ/F48 11.955 Tf 9.791 0 Td[(n5V)]TJ/F47 12.457 Tf 12.116 0 Td[(p)]TJ/F48 11.955 Tf 9.791 0 Td[(n5V)]TJ/F47 12.457 Tf 12.115 0 Td[(a9>=>;-62whereVisthepotentialofpipesteel.DecomposingthevectorV)]TJ/F47 12.457 Tf 12.526 0 Td[(and)]TJ/F48 11.955 Tf 9.791 0 Td[(n5V)]TJ/F47 12.457 Tf 12.116 0 Td[(,equation 4-62 becomes266664Hp;pHp;a)]TJ/F19 11.955 Tf 9.671 0 Td[(1Ha;pHa;a)]TJ/F19 11.955 Tf 9.671 0 Td[(10003777758>>>><>>>>:VVV9>>>>=>>>>;)]TJ/F67 11.955 Tf 11.996 38.376 Td[(266664Hp;pHp;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1Ha;pHa;a)]TJ/F19 11.955 Tf 9.672 0 Td[(10003777758>>>><>>>>:pa19>>>>=>>>>;=266664Gp;pGp;aGa;pGa;aApAa3777758><>:n5pn5a9>=>; -63 SincethepotentialofsteelVisassumedtobeaconstant,)]TJ/F48 11.955 Tf 9.791 0 Td[(n5V=0.Moreover,

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53 accordingtoequation 4-55 Hp;p+Hp;a=1 -64 Ha;p+Ha;a=1 -65 Thus,thersttermonthelefthandsideofequation 4-63 isvanished,andwecanreadilyobtain266664Hp;pHp;a)]TJ/F19 11.955 Tf 9.671 0 Td[(1Ha;pHa;a)]TJ/F19 11.955 Tf 9.671 0 Td[(10003777758>>>><>>>>:pa19>>>>=>>>>;=)]TJ/F67 11.955 Tf 11.332 38.376 Td[(266664Gp;pGp;aGa;pGa;aApAa3777758><>:n5pn5a9>=>;-66Equation 4-11 canberewriteasn5p=soil coatp)]TJ/F47 12.457 Tf 12.116 0 Td[(corr-67Substitutingequation 4-67 intoequation 4-66 ,wecanget266664Hp;pHp;a)]TJ/F19 11.955 Tf 9.671 0 Td[(1Ha;pHa;a)]TJ/F19 11.955 Tf 9.672 0 Td[(10003777758>>>><>>>>:pa19>>>>=>>>>;=)]TJ/F67 11.955 Tf 11.332 38.376 Td[(266664Gp;pGp;aGa;pGa;aApAa3777758><>:soil coatp)]TJ/F47 12.457 Tf 12.116 0 Td[(corrn5a9>=>;-68Equation 4-68 willbeappliedinthetwocasesasbelow.SacricialanodecaseForthesacricialanodeprotectionmethod,wesetthepotentialoftheanodeasaconstantandthecurrentdensityoftheanodeisunknownexceptthatitiszeroatthetwoendsoftheanode.Thepotentialofthepipeisunknown.Thecurrentdensityofpipeisalsounknownexceptthezerovalueisgivenattheendsofpipe.Inaddition,thepotentialandthecurrentdensityofthepipesatisfytherelationofequation 4-67

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54 Rearrangingtheequation 4-68 ,266664Hp;p+Gp;psoil coatGp;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1Ha;p+Ga;psoil coatGa;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1+Apsoil coatAa03777758>>>><>>>>:pn5a19>>>>=>>>>;=266664Gp;psoil coat)]TJ/F48 11.955 Tf 10.245 0 Td[(Hp;aGa;psoil coat)]TJ/F48 11.955 Tf 10.246 0 Td[(Ha;aApsoil coat03777758><>:corra9>=>;-69itcanbeusedtomodelsacricialanodeCPsystem.ImpressedcurrentcaseTosimulatetheimpressedcurrentprotectionmethod,wesetthecurrentden-sityoftheanodeasaconstantvalue,butthepotentialoftheanodeisunknown.Theconditionsofcurrentdensityandpotentialofpipearesameasthoseofthesacricialanodecase.Likewise,thefollowingequationcanbeobtained.266664Hp;p+Gp;psoil coatHp;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1Ha;p+Ga;psoil coatHa;a)]TJ/F19 11.955 Tf 9.672 0 Td[(1+Apsoil coat003777758>>>><>>>>:pa19>>>>=>>>>;=266664Gp;psoil coat)]TJ/F48 11.955 Tf 9.911 0 Td[(Gp;aGa;psoil coat)]TJ/F48 11.955 Tf 9.91 0 Td[(Ga;aApsoil coat)]TJ/F48 11.955 Tf 10.389 0 Td[(Aa3777758><>:corrn5a9>=>;-704.4.2SimulationResultsTypicalsimulationresultsforpotentialsonsoilsurfacedirectlyabovea10mpipelinewithonecoatingdefectareshowninFigure 4-10 .Thepotentialchangeatthepositioncorrespondingtothecoatingdefectsisevident,butthemagnitudeofthechangeissmall,inagreementwithpreviouscalculations. 35 38 4.5InverseModelThepurposeoftheinversemodelistoconstructanobjectivefunctionandtocomparedifferenttypesofregressionmethodsinordertominimizetheobjectivefunction.Thesimulatedannealingmethodwasfoundappropriateinthepresentstudysinceitcanescapefromthelocalminimaandtherebyavoidtheinuenceoftheinitialguess.

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55 Figure4-10:Falsecolorimageoftheon-potentialonthesoilsurfacethatwasgen-eratedbytheforwardmodelcorrespondingtoFigure 4-1 4.5.1ObjectiveFunctionAnobjectivefunction,whichdescribesthedifferencebetweenthemeasuredpotentialandcalculatedpotentialonthesoilsurface,isgivenasgx;;=Nj=1j)]TJ/F67 11.955 Tf 13.066 3.503 Td[(bj2-71wherejandbjrepresentmeasuredandcalculatedsoilsurfacepotentials,re-spectively.Thecalculatedsoilsurfacepotentialdependsonthepipecoatingcon-ditions,suchasposition,resistivityandwidthofcoatingdefect.Thus,oncetheminimumoftheobjectivefunctionisreached,thettedcoatingparameterswillreecttherealphysicalconditionsofpipeline.4.5.2AnalysisofRegressionMethodsSeveralregressionmethodsweretested,includingtheconjugategradient,Levenberg--Marquart,andquasi-Newtonmethods.AsummaryofthesemethodsisprovidedbyPressetal. 78 Thesemethods,distinguishedbytheneedtoevaluateboththefunctionvalueandthefunctiongradient,wererejectedforthepipelineinversemodel.Theextracalculationsrequiredforevaluationofthefunctiongradient

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56 addedsignicantlytothecomputationalrequirement.Inaddition,thesemethodswereextremelysensitivetothevalueofaninitialguessandhadatendencytondlocalratherthanglobalminima.Thedownhillsimplexmethod, 78 whichrequiresevaluationofonlytheobjec-tivefunctionandnotitsgradient,wasusedsuccessfullyforpreliminarytwo--dimensionalinverseanalysis.Thedownhillsimplexmethoddidnotworkverywellforthree-dimensionalproblemsbecauseitwasstronglyaffectedbytheinitialguess.Geneticminimizationalgorithms,intendedtomimicthecourseofnaturalse-lection,havebeenappliedtoinverseproblemsforcathodicprotection. 42 44 45 Pa-rametersarenormallycodedasbinarystringstoreducethesearchingpopulation.Proceduresofselection,matingandmutationareusedrepeatedlytocreatethenewgenerationuntilthespeciedstopcriterionissatised.Thismethodisnotverysensitivetothevaluesofinitialguessesandcanescapefromthelocalmin-ima.However,ithasdifcultyselectingbetweenclosefunctionvalues. 79 Aftertryingthealternatives,thesimulatedannealingoptimizationapproachwasselectedforthepresentwork.Thismethodisattractivebecauseitissuit-ableforlarge-scaleproblemsandcansearchforaglobalminimum,whichmaybehiddenamongmanylocalminima.Simulatedannealingwasbetterthandown-hillsimplexbecausethesimplexmethodacceptsonlydownhillstepsduringitssearching;whereas,simulatedannealingcanacceptboththedownhillanduphillsteps.Inthisway,simulatedannealingmethodcanstepoutofthelocaloptimaandsuccessfullylocatetheglobalminimum.4.5.3SimulatedAnnealingMethodThetermsimulatedannealingcomesfromaphysicalprocessanalogy. 78 Whenamaterialisheatedandthenisslowlycooleddown,astrongcrystallinestructure

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57 willbeobtained.Thiscrystalistheminimumenergystateofthesystem.Inasim-ulationprocess,aminimumoftheobjectivefunctioncorrespondstothegroundstateofthesubstance.TheBolzmannprobabilitydistribution,whichdescribesthedifferentenergystateinathermalequilibriumundergiventemperature,canalsobeusedtomimicthedifferentobjectivefunctionvalueatcertainsearchingregion.Thesimulatedannealingmethodstartswithabriefviewofthesearchingdomainbymakinglargemovesandthenitfocusesonthemostpossibleregion.Theinher-entrandomuctuationpermitstheannealingprocedureavoidthelocalminima.TheMetropolisrulep=e)]TJ/F19 7.97 Tf 6.447 0 Td[(g2)]TJ/F48 7.97 Tf 6.703 0 Td[(g1=kT-72isusedtocontrolwhethertheuphillstepisaccepted.Inequation 4-72 ,preferstotheprobability,giistheobjectivefunctionvalueandTisanimportantparam-eterinthesimulatedannealingmethod,whichresemblesthetemperatureinthethermalsystem.Forg2g1,theprobabilitypiscomparedwithauniformlydistributedrandomnumberfrom[0;1]todecidewhetheruphillstepsareaccepted.Sinceeachparameterhasitsownlimit,oncetheparameterisoutofitslimit,thefollowingequationcanbeusedtocorrecttheparameter.x=xL+xU)]TJ/F48 11.955 Tf 12.319 0 Td[(xL-73whereisauniformlydistributedrandomnumberamong[0;1],xListhelowerlimitofxandxUistheupperlimitofx.Theparameterxcanthusbeguaranteedtoliewithinitsbounds.4.5.4SimulationResultsandDiscussionsExamplesoftheapplicationoftheproposedforwardandinversemodelsaregiveninthissection.Inaddition,severalinversestrategiesareintroduced.There-

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58 Figure4-11:Flowchartfortheinversemodelcalculations. gressionalgorithmemployedbytheinversemodelisshownintheFigure 4-11 .Thesetparameterswereusedasinputstotheforwardmodeltogetthepotentialonsoilsurface.Theseresultsweretreatedasthemeasureddata.Thesemeasuredpo-tentialsbelongtoeachpointofa3x101-gridarea,asshowninFigure 4-12 .Therearethreelinesinthegridarea:themiddlelineisrightabovethepipecenterline,andtwootherlinesofcalculatedvalueswereplaced1mfromthecenterlineofthepipe.Thegrid,therefore,comprised303datapoints.Initializetheparameters,thenumberofdefects,eachdefect'scenterposition,resistivityreductionandthewidthofthedefects,andinputthemintotheinversemodel.Amongallthesepa-rameters,thenumberofdefectshasthemostsignicantimpactontheregressionresults.Inthisstudy,wepurposeamethodtodecidethenumberofdefectswhichcanbeobtainedfromthemeasurements.Theinverseanalysisresultbyusingsim-ulatedannealingmethodisshowninTable 4.1 fora10-meterlongpipesegment

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59 Figure4-12:Gridshowingthelocationof303surfaceon-potentialscalculatedus-ingthethree-dimensionalforwardmodeldevelopedinthepresentwork.Thegridshownisfora10mpipesegment.Ascaledversionofthegridwasusedfora100mpipesegment. withtwocoatingdefects.Thecalculateddatawerefordefectslocatedatthe3:2and6:0mpositionswithcharacteristicdimensionsof0:05and0:5m,respectively.Theintactcoatingresistivityhadavalueof5:0107/Wm,whichwasxedfortheregressionprocedure.Theinputvaluesforsurfacepotentialswerecalculatedatthelimitsoftheprecisionoftheforwardmodel,i.e.,nonoisewasadded.Theregressionyieldedthecorrectlocations,characteristicdefectdimensions,andre-sistivitychangesassociatedwiththecoatingdefects.TheprocessofndingtheminimumoftheobjectfunctionisshowninFigure 4-13 .Forthistestproblem,withtheforwardmodelandnonoiseaddedtothesyn-theticdata,theminimumvalueofthecostfunctionwas10)]TJ/F19 7.97 Tf 6.448 0 Td[(15.Ofthetechniquesused,onlythesimulatedannealingmethodcouldndthisglobalminimum.Thebestoftheothertechniqueswereabletoidentifylocalminimawithvaluesontheorderof10)]TJ/F19 7.97 Tf 6.448 0 Td[(4to10)]TJ/F19 7.97 Tf 6.448 0 Td[(8precedingtheplateau.Acomparisonofthesetandttedresultsforpipecoatingresistivity,poten-tialandcurrentdensity,respectively,isshowninFigures 4-14 .Correspondingtothecoatingresistivitydecay,thepotentialandcurrentdensitydistributionshavesignicantchanges.Agoodagreementbetweenthesyntheticdataandregressionresultscanbeobserved.

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60 Table4.1:Parametervaluesobtainedusingthethree-dimensionalinversemodeldevelopedinthepresentworkfora10mpipesegmentwithtwocoatingdefects. Coating Position Resistivity Characteristicdimension Defect xk/m k/Wm k/m SetValues 1 3.2 )]TJ/F19 11.955 Tf 9.671 0 Td[(4:0107 0.22 2 6.0 )]TJ/F19 11.955 Tf 9.671 0 Td[(3:5107 0.71 InitialValues 1 2.5 )]TJ/F19 11.955 Tf 9.671 0 Td[(3:0107 0.1 2 5.0 )]TJ/F19 11.955 Tf 9.671 0 Td[(3:0107 0.1 RegressionResult 1 3.206 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:81107 0.26 2 6.004 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:51107 0.70 Figure4-13:Theregressionobjectivefunctionasafunctionofthenumberofevalu-ationsforapipecoatingwithonedefectregion.Thesimulatedannealingmethodwasusedforthisregression.

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61 a b cFigure4-14:Comparisonofthesetandttedresultsforpipecoatingwithtwocoatingdefects:acoatingresistivity;bpotentials;andccurrentdensity.

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62 Table4.2:Testcaseparameterswithvecoatingdefectsonthepipeusedtodemonstratethemethodfordeterminationofthenumberofstatisticallysignif-icantparametersseeFigure 4-15 .Theintactcoatingresistivityhadavalueof5:0107/Wm. CoatingDefect Position Resistivity Characteristicdimension xk/m k/Wm k/m SetValues 1 10 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107 0.45 2 30 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:5107 0.32 3 40 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:0106 0.55 4 70 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:0107 0.45 5 85 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:0107 0.63 4.5.5InverseStrategiesThestrategies,includingthedeterminationofthenumberofsignicantpa-rameters,theproceduresusedtoreducetheinuenceofinitialguessandtheeval-uationoftheroleofnoiseinthemeasureddata,wereexploredtoassessthecon-denceleveloftheinversemodelresults.DeterminationoftheNumberofStatisticallySignicantParametersWhenthecollecteddataarescattered,itisdifculttodecidehowmanypos-sibledefectsorcoatinganomaliesshouldbeincludedintheregression.Theap-proachtakentoaddressthisissuewastoincreasesequentiallythenumberofde-fects,usingtheregressionstatisticstodeterminewhenthenumberofstatisticallysignicantparameterswasexceeded.Thisapproachissimilartothatusedtoas-sessthenumberofstatisticallysignicantmeasurementmodelparameterscanbeobtainedbyregressiontoimpedancespectroscopydata. 80 81 Anexamplewithvedefects,showninTable 4.2 ,wasusedtoillustratetheapproachtakentoassessthecorrectnumberofstatisticallysignicantparameters.Therelationbetweentheregressionstatisticlog2=andthenumberofcoatingdefectsassumedintheregressionispresentedinFigure 4-15 .Here,=N)]TJ/F48 11.955 Tf 12.964 0 Td[(Mrepresentsthedegreeoffreedomoftheproblemwhichisreducedasthenumberof

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63 Figure4-15:Theregressionstatisticasafunctionofthenumberofcoatingde-fectsassumedforthemodel.Theminimuminthisvalueisusedtoidentifythemaximumnumberofcoatingdefectsthatcanbejustiedonstatisticalgrounds. parametersisincreased.Nrepresentsthenumberofdatapoint,andMrepresentsthenumberofparameters.2istheweightedobjectivefunction.Thelowestpointofthecurvedenotesthenumberofstatisticallysignicantcoatingdefectsobtain-ablebyregression.Here,itisfour.Thefthcoatinganomaly,identiedinTable 4.2 asdefect3,couldnotbeidentiedbytheregressionprocedure.Thenumberofcoatinganomaliesidentiedbythisprocedurewilldependontheamountandqualityofdataandonthesensitivityofthedatatocoatingcondition.TestingtheRobustnessoftheInverseModelToassesstheinuenceofuncertaintyinthemeasureddata,normallydis-tributedstochasticerrorswereaddedtothesoilsurfacepotentialgeneratedbytheforwardmodel.Thenoiseaddedhadstandarddeviationsnoiseof0.1,1.0,and2.0mVrespectively.Thesetvalues,initialguesses,andregressionresultsarepresentedinTable 4.3 .Threecoatingdefectsplacedat30,40,and70mpositionswereusedtogeneratesyntheticdata.Forthelowestnoiselevel,noise=0:1mV,thesequentialproceduredescribedintheprevioussectionallowedonlytwostatisticallysignicantdefects.

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64 Table4.3:Regressionresultsfromthethree-dimensionalinversemodelfora100mpipesegmentwiththreecoatingdefects.Thesequentialregressionprocedurewasusedtoidentifythenumberofdefectsthatwerestatisticallysignicant.Theintactcoatingresistivityhadavalueof5:0107/Wm. Coating Position Resistivity Characteristicdimension Defect xk/m k/Wm k/m SetValues 1 30 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:5107 0.32 2 40 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:0106 0.55 3 70 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:0107 0.45 InitialGuess 1 50 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107 0.92 2 50 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:5106 0.92 3 50 )]TJ/F19 11.955 Tf 9.671 0 Td[(3:5106 0.92 RegressionResults noise=0:1mV 1 29.85 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:25107 0.66 2 69.83 )]TJ/F19 11.955 Tf 9.672 0 Td[(7:57106 1.30 noise=1:0mV 1 32.01 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:67107 0.087 noise=1:0mV 1 32.01 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:68107 0.057 2 90.29 )]TJ/F19 11.955 Tf 9.672 0 Td[(1:22106 0.86 noise=2:0mV 1 29.95 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:33107 1.50 Theinitialguessesplacedthedefectsatthemidpointofthepipesegment.There-gressionresultssuggestedthatthedefectsexistnearthe30and70mlocations.Themissingdefectistheonewiththesmallestdeviationincoatingresistivity.Thus,theregressionprocedureidentiedthecorrectlocationofthetwomostsignicantreductionsincoatingresistivity.Fornoise=1:0mV,thesequentialprocedureusingtheminimizationofthe2=criterionallowedtwocoatingdefects.Thedefectlocatedat32.01mwasconsistentwiththemostsignicantdefectlocatedat30m,butthedefectidentiedat90mdidnotconformtotheinputdata.Inaddition,theregressionfailedtoidentifythesecondmostsignicantreductionincoatingresistivityat70m.Theproblemheremaybeaninadequatesensitivityofthe2=statisticforidentifyingoverttingofdata.Othercriteria,suchastheAkaikiinformationcri-teria, 82 83 84 provideadditionalpenaltiesforaddingparameterstoamodel.For

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65 noise=1:0mV,theAkaikiperformanceindexAPI=21+np=nob 1)]TJ/F48 11.955 Tf 12.116 0 Td[(np=nob-74suggestedthatonlyonedefectcouldbeidentied;whereas,theAkaikiinforma-tioncriterionAIC=log)]TJ/F17 17.215 Tf 5.476 -12.242 Td[(2)]TJ/F19 11.955 Tf 5.476 -9.69 Td[(1+2np=nob-75suggeststhattwodefectscouldbeidentied.Regressionforasinglecoatingdefectidentiedadefectintheclosevicinityofthemostsignicantcoatingreduction.Byanyofthestatisticalmeasurestested,onlythelargestdefectcouldbeidentiedfornoise=2:0mV.Thelocationofthedefectwascorrectlyidentied;however,thewidthofthedefectwasincorrectlydetermined.Theseresultscanbeexplainedbyexaminationofthesyntheticsurfacepoten-tialdatausedfortheinversemodelanalysis.Thecorrespondingresultsoftheinverseanalysisfornoise=0:1mVareshowninFigure 4-16 AsisshowninFig-ure 4.16a ,thelevelofaddednoisedidnotobscurethesurface-potentialfeaturesintroducedbythepresenceofthemajorcoatingdefects.Theregressedandnoise-freetargetvaluesforpotentialcannotbedistinguished;thus,theabsenceoftheminorcoatingdefectdidnotinuencethetofthemodeltothesyntheticdata.AsseeninFigure 4.16b ,theregressionprocedureidentiedthethetwomostsignicantreductionsincoatingresistivityat30and70mlocations.Incontrast,therandomnoiseaddedwithnoise=1:0mV,showninFigure 4.17a ,almostcompletelyobscurestheinuenceofthecoatingdefects.Neverthe-less,amajordefectcanberesolvedbytheregressionprocedureinthevicinityofthelargestdefect,asshowninFigure 4.17b .Theanomalousdefectintroducedbytheregressionprocedureat90mhasassociatedwithitasmallreductiononcoat-ingresistivity.Thereisaquestionastowhetherthisdefectcanbeconsideredto

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66 a bFigure4-16:Comparisonbetweentheinputvaluesandregressionresultsfornoise=0:1mV:asoil-surfacepotentialatthecenterlineofthepipe;bcoatingresistivity.

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67 a bFigure4-17:Comparisonbetweentheinputvaluesandregressionresultsfornoise=1:0mV:asoil-surfacepotentialatthecenterlineofthepipe;bcoatingresistivity.

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68 bestatisticallysignicant,but,nevertheless,theregressionprocedurewouldhavegivenadequateguidanceforexcavationofthepipe.Byvisualinspection,therandomnoiseaddedwithnoise=2:0mV,showninFigure 4.18a ,wouldappeartoobscuretheinuenceofeventhemajorcoatingdefect.Theregressionprocedureidentiedasinglestatisticallysignicantcoatingdefectlocatednearthe30mdefect.Theagreementbetweentheregressedpoten-tialproleandthenoise-freetargetvalueswassurprisinglygood.AsshowninFigure 4.17b ,thecorrectlocationfortheprincipaldefectwascorrectlyidentied,eventhoughthebreadthofthedefectwasnotcorrectlydetermined.Theresultssuggestthat,whileitisdifculttoextractcoatingconditionsfromthedatawhentheinuenceofthecoatingdefectonthesurfacepotentialiscompa-rablewithnoise,theregressionprocedureshowedasurprisingabilitytoidentifythelocationofthemostsignicantcoatingdefectsfromnoisydata.Thisresultsuggeststhataninversemodelisfeasible,inparticularwhenothertypesofdataareincluded.SensitivitytoInitialGuessThevaluesusedforaninitialguesshaveasignicantinuenceonmostoftheconventionalnonlinearregressionmethods.Toreducethiseffectandtotesttherobustnessofthepresentinversemodel,identicalparametersforeachdefectwereused.Forexample,astheinitialguessinTable 4.3 ,thepositionsofthedefectswereallsettothemiddleofthepipeline.Thiskindofinitialguesscreatedsignicantdifcultiesforconventionalregressionmethods,butposednoproblemsforthesimulatedannealingmethodusedinthepresentinversemodel.

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69 a bFigure4-18:Comparisonbetweentheinputvaluesandregressionresultsfornoise=2:0mV:asoil-surfacepotentialatthecenterlineofthepipe;bcoatingresistivity.

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70 4.6ConclusionsInthischapter,boththeforwardmodelandtheinversemodelforthethree--dimensionalCPsystemwasintroduced.Thesimpliedforwardmodelhasthefollowingadvantages. 1. TheLaplace'sequationwiththesimpliedlinearboundaryconditionwasapplied.Thisapproachnotonlysimpliestheforwardcalculation,butalsoreducesthecomputationalefforttotheinversecalculation. 2. Thelineshapeapproximationforthepipecoatingresistivitymakesitpossi-bletolimitthenumberofparametersneededtodescribethecoatingcondi-tionsalongthewholepipe.Itdramaticallyincreasesthedegreeoffreedomoftheproblemascomparedtothemethodthatrepresentsthecoatingcondi-tionselementbyelement. 3. TheBEMmethodwithcylindricalelementwasdeveloped.ThebasictheoryofBEM,suchasuseofthemirrorreectiontechniqueforthehalf-innitydomain,rowsumelimination,andself-equilibrium,wereintroduced.Inad-dition,theimplementationtechniques,forexample,thecoordinatestrans-formation,elementdiscritization,werealsoincluded. 4. Thesimpliedforwardmodelwasusedtoobtainthepotentialsofthepipeonthesoilsurface.Thepotentialandthecurrentdensityofthepipeandtheanodecouldalsobecalculated.Theinversepartofworkrepresentsanambitiousresearcheffortwithapoten-tiallylargepayofffortheoilandgastransmissionindustry.Thecentralquestioniswhetheraregressionapproachcouldbeusedtoassesspipelinecoatingconditionsfromelddatainawaythatisconsistentwiththelawsofphysics.Therationale,

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71 methodology,andresultsarepresented,whichdemonstratethefeasibilityoftheinversemodelforpipelines. 1. Thisworkhasdemonstratedthatitispossibletocoupleaboundaryele-mentforwardmodelwithanonlinearregressionalgorithmtoobtainpipesur-facepropertiesfrommeasuredsoil-surfacepotentials.Theresultingmodeliscalledaninversemodel. 2. Thetechniqueidentiesthelocationofcoatinganomaliesaswellasthebreadthoftheanomalyandtheamountthatthelocalresistivityhaschanged. 3. Theperformanceoftheinversemodelissensitivetotheregressionprocedureused.Thesimulatedannealingalgorithmprovedtobethemostrobustandhadgreatestcapabilitytoseektheglobalminimumforthisproblem. 4. Analgorithmwasdevelopedthatcouldbeusedtoidentifythemaximumnumberofcoatinganomaliesthatcanbedetected.Thisnumberissensitivetothequalityofdataaswellastotheactualcoatingcondition. 5. Ifthenumberofcoatinganomaliesdetectedissmallerthantheactualnum-berofcoatingdefects,thetechniquewillidentifythemostseriousanomalies.Theresultsofthiseffort,limitedtoasinglepipeinaright-of-way,demonstratethefeasibilityofaprogramtointerpretsurveydataintermsofthestateofprotectionofthepipe.

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CHAPTER5DEVELOPMENTOFTHREE-DIMENSIONALFORWARDANDINVERSEMODELSFORPIPELINEWITHBOTHPOTENTIALANDCURRENTDENSITYSURVEYDATA5.1ForwardModel5.1.1IntroductionInRiemer'swork,thereweretwoseparatedomainsfortheowofcurrent.Therstwasthesoildomainuptothesurfacesofthepipesandanodes.Theboundaryconditionsforthesoildomainwerethekineticsofthecorrosionreactions.Theseconddomainwastheinternalpipemetal,anodemetalandconnectingwiresforthereturnpathofthecathodicprotectioncurrent. 38 TheBEMmethodwasusedfortherstdomain.TheFEMmethodwasappliedtotheseconddomain.BrichaualsocoupledtheBEMandFEMtosolvethetwodomains. 72 Inthischapter,tosimplifythecalculations,notonlythepotentialbutalsothecurrentdensityalongthepipesteelwillbeobtainedbyusingtheBEMmethod.5.1.2PipelinewithVaryingSteelPotentialInchapter 4 ,thepotentialVonthepipesteelwasassumedtobeaconstant.Actually,thepotentialonthepipesteelvariesalongthepipesincethepipelinehasitsowninternalresistance.Forashortsectionofpipe,sincethepotentialdropduetothepiperesistanceisverysmall,itisreasonabletoassumedittobeconstant.However,foralongpipelineorahighresistancepipe,thepotentialdropalongthepipesteelissignicantandcannotbeignored.Themodelinchapter 4 wasextendedtoaccountfortheelectricalpotentialofthepipelinesteel. 72

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73 Figure5-1:Cathodicprotectionsystemwithvariantpotentialalongthepipeandanode. ThemethodofcalculationisillustratedschematicallyinFigure 5-1 ,whereahorizontallyplacedpipelineisconnectedbyawiretoaverticallyplacedsacriceanode.Theconnectionpointsarec1onthepipeandc2ontheanode.InFigure 5-1 ,irepresentsthecurrentdensityalongtheaxialdirectionofpipelineoranode,andsubscriptspandadesignatepipelineandanode,respectively.Thecurrentdensityenteringthepipeattheendsisdesignatedbyi0pandinp,andthecurrentdensityenteringtheanodeattheendsisdesignatedbyi0aandina.Thecurrentdensityenteringthepipecoatingintheradialdirectionisgivenbyq.AsshowninFigure 5-1 ,theprotectingcurrentowsawayfromtheanodetothesoil,thenowstothepipeline.Onthepipeline,thecurrentsowfromthetwooppositedirectionstotheconnectionpointc1andtheyareconcurrentascurrentI.ThecurrentIowsbacktotheanodethroughthewire.Thecurrentdensityinthepipelinesteelalongaxialdirectionisgivenbyi=1 steel)]TJ/F48 11.955 Tf 9.791 0 Td[(nrV=)]TJ/F19 11.955 Tf 19.066 8.094 Td[(1 steeldV dz-1whereVrepresentsthepotentialofthesteelinthepipe.Equation 5-1 canbe

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74 Figure5-2:RelationofpotentialVandcurrentdensity. integratedintheaxialdirectionalongasegmentwithlengthdz,showninFigure 5-2 ,suchthatZVkVk)]TJ/F19 5.978 Tf 4.836 0 Td[(1dV=)]TJ/F47 12.457 Tf 9.791 0 Td[(steelZzkzk)]TJ/F19 5.978 Tf 4.836 0 Td[(1idz-2ThepotentialchangeacrossthesegmentisgivenbyVk)]TJ/F48 11.955 Tf 12.236 0 Td[(Vk)]TJ/F19 7.97 Tf 6.448 0 Td[(1=)]TJ/F47 12.457 Tf 9.791 0 Td[(steelZzkzk)]TJ/F19 5.978 Tf 4.836 0 Td[(1z)]TJ/F48 11.955 Tf 12.33 0 Td[(zk)]TJ/F19 7.97 Tf 6.448 0 Td[(1 zk)]TJ/F48 11.955 Tf 12.331 0 Td[(zk)]TJ/F19 7.97 Tf 6.448 0 Td[(1ik)]TJ/F19 7.97 Tf 6.448 0 Td[(1+zk)]TJ/F48 11.955 Tf 12.331 0 Td[(z zk)]TJ/F48 11.955 Tf 12.331 0 Td[(zk)]TJ/F19 7.97 Tf 6.447 0 Td[(1ikdz-3orVk)]TJ/F48 11.955 Tf 12.235 0 Td[(Vk)]TJ/F19 7.97 Tf 6.447 0 Td[(1=steelzk)]TJ/F48 11.955 Tf 12.331 0 Td[(zk)]TJ/F19 7.97 Tf 6.448 0 Td[(1ik)]TJ/F19 7.97 Tf 6.448 0 Td[(1+ik 2-4wherethepotentialdifferencebetweentwopointsisfoundastheproductoftheaveragecurrentdensityik)]TJ/F19 7.97 Tf 6.448 0 Td[(1+ik=2andtheresistancesteelzk)]TJ/F48 11.955 Tf 12.331 0 Td[(zk)]TJ/F19 7.97 Tf 6.447 0 Td[(1.Conservationofchargerequiresthatthecurrentowingintothepipesegmentisequaltothecurrentowingout.Thus,ik)]TJ/F19 7.97 Tf 6.448 0 Td[(1=ik+As Acq-5whereqistheaveragecurrentdensityenteringthecoatedsurfaceoftheelementintheradialdirection.AcandAsrepresenttheareasofthesteelcross-sectionandsidewalls,respectively.Forthecathodicprotectedsystem,theprotectingcurrentowawayfromtheanodetothesoil,andthenowtothepipeline.Onthepipeline,thecurrentsfromthetwosidesoftheconnectionpointc1owhead

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75 Figure5-3:Axialdirectioncurrentowalongthepipeline. toheadandareconcurrentascurrentIowsbacktotheanodethroughthewirewhichisconnectingthepipeandtheanode.Therefore,foreachsegmentofthepipebeforeconnectionpointc1,thepotentialdifferenceis,V0)]TJ/F48 11.955 Tf 12.236 0 Td[(V1=i0+i1R=[2i0)]TJ/F67 11.955 Tf 11.996 16.863 Td[(Aps Apcq0+q1 2]RV1)]TJ/F48 11.955 Tf 12.236 0 Td[(V2=i1+i2R=[2i0)]TJ/F67 11.955 Tf 11.996 16.863 Td[(Aps Apcq0+2q1+q2 2]R...Vc1)]TJ/F19 7.97 Tf 6.448 0 Td[(1)]TJ/F48 11.955 Tf 12.236 0 Td[(Vc1=ic1+ic1)]TJ/F19 7.97 Tf 6.448 0 Td[(1R=[2i0)]TJ/F67 11.955 Tf 11.996 16.863 Td[(Aps Apcq0+2q1++2qc1)]TJ/F19 7.97 Tf 6.448 0 Td[(1+qc1 2]RForeachsegmentofthepipeaftertheconnectionpointc1)]TJ/F48 11.955 Tf 12.235 0 Td[(Vc1+1+Vc1=ic1+1+ic1R=[2in+Aps Apcqn+2qn)]TJ/F19 7.97 Tf 6.448 0 Td[(1++2qc1+1+qc1 2]R...)]TJ/F48 11.955 Tf 9.911 0 Td[(Vn+Vn)]TJ/F19 7.97 Tf 6.448 0 Td[(1=in+in)]TJ/F19 7.97 Tf 6.448 0 Td[(1R=[2in+Aps Apcqn+qn)]TJ/F19 7.97 Tf 6.448 0 Td[(1 2]RTherearenequationsifthepipelineisdiscritizedasnsegments,whiletherearen+1theunknownvariablesV0;V1;Vn.Onemoreequationisneededinordertogetauniquesolution.ThepotentialV=0canbesetatanypositionalongthepipe.Inthisstudy,zeropotentialwassetatthepipeandanodeconnectionpointc1,thatis,Vc1=0.Thecurrentowintheanodethroughthewirewillowtothedifferentdirectionalongtheanode.Likewise,foreachsegmentoftheanode

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76 Figure5-4:Axialdirectioncurrentowalongtheanode. beforetheconnectionpointc2,V0)]TJ/F48 11.955 Tf 12.236 0 Td[(V1=[2i0)]TJ/F67 11.955 Tf 11.996 16.863 Td[(Aas Aacq0+q1 2]RV1)]TJ/F48 11.955 Tf 12.236 0 Td[(V2=[2i0)]TJ/F67 11.955 Tf 11.996 16.863 Td[(Aas Aacq0+2q1+q2 2]R...Vc2)]TJ/F19 7.97 Tf 6.448 0 Td[(1)]TJ/F48 11.955 Tf 12.236 0 Td[(Vc2=[2i0)]TJ/F67 11.955 Tf 11.996 16.862 Td[(Aas Aacq0+2q1++2qc2)]TJ/F19 7.97 Tf 6.448 0 Td[(1+qc2 2]RForeachsegmentoftheanodeafterconnectionpointc2)]TJ/F48 11.955 Tf 12.235 0 Td[(Vc2+1+Vc2=[2im+Aas Aacqm+2qm)]TJ/F19 7.97 Tf 6.448 0 Td[(1++2qc2+1+qc2 2]R...)]TJ/F48 11.955 Tf 9.911 0 Td[(Vm+Vm)]TJ/F19 7.97 Tf 6.448 0 Td[(1=[2im+Aas Aacqm+qm)]TJ/F19 7.97 Tf 6.447 0 Td[(1 2]RCorrespondingtothemsegmentsoftheanode,therewillbem+1unknownvari-ablesV0;V1;Vm.Anotherequationisneeded.IfthewireconnectingthepipeandtheanodehasresistanceRwire,andthecurrentowthroughthewireisI,thenVc1=Vc2)]TJ/F48 11.955 Tf 12.714 0 Td[(IRwire=0-6Inordertosatisfytheself-equilibriumlimitation,thecurrentIneedstoobeytherelationsasbelow.

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77 Forthepipeline,asshowninFigure 5-3 ,ic1F0=i0)]TJ/F67 11.955 Tf 11.996 16.863 Td[(Aps Apcq0+2q1++2qc1)]TJ/F19 7.97 Tf 6.448 0 Td[(1+qc1 2 -7 ic1Fn=in+Aps Apcqc1+2qc1+1++2qn)]TJ/F19 7.97 Tf 6.447 0 Td[(1+qn 2 -8 whereic1F0isthecurrentdensitycomingfromleftendofpipetothepointc1,andic1Fnisthecurrentdensitycomingfromrightendofpipetothepointc1.Bycombiningthesetwoequations,ic1F0)]TJ/F48 11.955 Tf 12.032 0 Td[(ic1FnApc=I-9Wehavei0)]TJ/F48 11.955 Tf 12.032 0 Td[(inApc)]TJ/F19 11.955 Tf 13.192 8.093 Td[(q0+2q1++2qn)]TJ/F19 7.97 Tf 6.448 0 Td[(1+qn 2Aps=I-10Fortheanode,asshowninFigure 5-4 ,ic2T0=i0)]TJ/F67 11.955 Tf 11.996 16.862 Td[(Aas Aacq0+2q1++2qc2)]TJ/F19 7.97 Tf 6.448 0 Td[(1+qc2 2 -11 ic2Tm=im+Aas Aacqc2+2qc2+1++2qm)]TJ/F19 7.97 Tf 6.448 0 Td[(1+qm 2 -12 whereic2T0isthecurrentdensityenteratc2andowtotheupperendofanode,andic2Tmisthecurrentdensityenteratc2andowtothelowerendofanode.Again,bycombiningthesetwoequations,ic2T0)]TJ/F48 11.955 Tf 12.033 0 Td[(ic2TmAac=I-13Wehave)]TJ/F48 11.955 Tf 9.708 0 Td[(i0+imAac)]TJ/F19 11.955 Tf 13.192 8.094 Td[(q0+2q1++2qm)]TJ/F19 7.97 Tf 6.448 0 Td[(1+qm 2Aas=I-14Summarizingthen+1equationsforpipelineinmatrixform,anddeningKp,Fp,

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78 Vp,qpandiendpasKp=2666666666666666666666641)]TJ/F19 11.955 Tf 9.671 0 Td[(10001)]TJ/F19 11.955 Tf 9.672 0 Td[(100001)]TJ/F19 11.955 Tf 9.671 0 Td[(100001)]TJ/F19 11.955 Tf 9.671 0 Td[(100001)]TJ/F19 11.955 Tf 9.672 0 Td[(100100377777777777777777777775Tp=2R2666666666666666666666666664101010010101003777777777777777777777777775

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79 Fp=)]TJ/F67 11.955 Tf 11.332 16.863 Td[(Aps ApcR 226666666666666666666666411001210012210000)]TJ/F19 11.955 Tf 9.672 0 Td[(1)]TJ/F19 11.955 Tf 9.672 0 Td[(2)]TJ/F19 11.955 Tf 60.217 0 Td[(2)]TJ/F19 11.955 Tf 9.672 0 Td[(100)]TJ/F19 11.955 Tf 9.672 0 Td[(1)]TJ/F19 11.955 Tf 9.672 0 Td[(100377777777777777777777775Vp=266666666666666666666664V0V1...Vc1Vc1+1...Vn)]TJ/F19 7.97 Tf 6.448 0 Td[(1Vn377777777777777777777775;qp=26666666666666666664q0q1...qc1qc1+1...qn37777777777777777775;iendsp=264i0in375WegetKpVp=Tpiendsp+Fpqp-15

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80 Likewise,deneKa,Fa,Va,qaandiendaasKa=2666666666666666666666641)]TJ/F19 11.955 Tf 9.672 0 Td[(10001)]TJ/F19 11.955 Tf 9.672 0 Td[(100001)]TJ/F19 11.955 Tf 9.672 0 Td[(100001)]TJ/F19 11.955 Tf 9.672 0 Td[(100001)]TJ/F19 11.955 Tf 9.672 0 Td[(100100377777777777777777777775Ta=2R2666666666666666666666666664101010010101)]TJ/F48 7.97 Tf 11.553 4.967 Td[(Ac 2RRwire)]TJ/F48 7.97 Tf 11.553 4.967 Td[(Ac 2RRwire3777777777777777777777777775

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81 Fa=)]TJ/F67 11.955 Tf 11.332 16.863 Td[(Aas AacR 226666666666666666666666411001210012210000)]TJ/F19 11.955 Tf 9.672 0 Td[(1)]TJ/F19 11.955 Tf 9.672 0 Td[(2)]TJ/F19 11.955 Tf 60.217 0 Td[(2)]TJ/F19 11.955 Tf 9.672 0 Td[(100)]TJ/F19 11.955 Tf 9.672 0 Td[(1)]TJ/F19 11.955 Tf 9.672 0 Td[(1)]TJ/F48 7.97 Tf 11.346 4.968 Td[(As RRwire)]TJ/F48 7.97 Tf 239.817 4.968 Td[(As RRwire377777777777777777777775Va=266666666666666666666664V0V1...Vc2Vc2+1...Vm)]TJ/F19 7.97 Tf 6.448 0 Td[(1Vm377777777777777777777775;qa=26666666666666666664q0q1...qc2qc2+1...qm37777777777777777775;iendsa=264i0im375Them+1equationsoftheanodeinmatrixformare,KaVa=Taiendsa+Faqa-16Introducingtheequations 5-15 5-16 intoequation 4-61 inthepreviouschap-

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82 ter,weobtain2666666666664Kp00000Ka00000HppHpa)]TJ/F19 11.955 Tf 9.672 0 Td[(100HapHaa)]TJ/F19 11.955 Tf 9.672 0 Td[(10000037777777777758>>>>>>>>>>><>>>>>>>>>>>:VpVapa19>>>>>>>>>>>=>>>>>>>>>>>;=2666666666664FpTp0000FaTaGpp0Gpa0Gap0Gaa0ApAa37777777777758>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:)]TJ/F48 11.955 Tf 9.791 0 Td[(nrpi0pinp)]TJ/F48 11.955 Tf 9.791 0 Td[(nrai0aima9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;-17Theforwardmodel,equation 5-17 ,canbeusedtocalculateboththepotentialonthesoilsurface,andpipesteelpotentials.Furthermore,pipesteelpotentialsallowscalculationofthecurrentowinginthepipe,whichcanbecomparedwiththeeldmeasurements.5.1.3SimulationResultsInordertoshowthesimulationresultswithvariantpipesteelpotentials,atestcaseofCPsystemwasstudied.A500mlongpipeisassumedtobeburied1.45m.75feetbelowthesoilsurface.Itsdiameteris0.457m.5feet.Aanodeisplaced50mawayfromthecenterofthepipe.Thediameteroftheanodeis0.152manditslengthis1.22mfeet.Thesoilresistivityis100Wm.TheboundaryconditionsareshowninFigure 4-1 .A0.5mm-thickcoatingisassumedtocoverthesideareaofthepipe,withtheexceptionatthetwoend,whichareassumedtobeinsulated.Awireconnectedthepipelinetotheanodeat50mpositionand0.5mposition,respectively.Thepotentialofthepipesteelvariesalongthepipe.Acoatingdefectissetonthemiddleofthelengthofpipeline.ThesimulationresultsofanodeandpipelineareshowninFigure 5-5 andFig-ure 5-6 respectively.

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83 a b c dFigure5-5:Potentialandcurrentdensityalongtheanode:apotentialYalongtheanodegivenvalue;bpotentialValongtheanode;cradicaldirectionofcurrentdensityalongtheanode;daxialdirectionofcurrentdensityalongtheanode.

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84 a b c

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85 d eFigure5-6:Simulatedaxialdistributionsalongthepipeline:ainputvalueforcoatingresistivity;bradialcomponentofcurrentdensity;ccalculatedvalueforpotential;dcalculatedvalueforsteelpotentialV;andeaxialcomponentofcurrentdensityinthepipe.

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86 5.1.4AnalysisoftheSimulationResultsFigure 5.5a showsthata=)]TJ/F19 11.955 Tf 9.671 0 Td[(1:1V,becausetheanodeisasacricialanodeanditspotentialisagivenboundarycondition.Figure 5.5b indicatesthepoten-tialsalongtheanodesteelarevariant.Atthepositionwherethepotentialequalszero,awireconnectstheanodetothepipeline.Inordertoprotectthepipeline,thecurrentoftheanodeowsawayfromtheanode,andowsbacktotheanodethroughthewire.Therefore,thefartherthepositionisfromthewireconnectedposition,thelowertheanodesteelpotentialthanthatofthepotentialatthewireconnectionpoint.Figure 5.5c exhibitstheradialdirectionofcurrentdensity.Fig-ure 5.5d showsthattheaxialdirectionofcurrentdensityisequaltozeroatthetwoendsoftheanode.Thisisagivenboundarycondition.Thecurrentdensitiesatthetwosidesofthewireconnectionpointhavedifferentsigns,whichmeansthatthecurrentowsawayfromthewireconnectingpositiontotwooppositedirections.ThecoatingresistivityofthepipelineisshowninFigure 5.6a wherethenom-inalresistivityofdefect-freecoatingisseentobe5:0107Wm.Theresistivitywasassumedtodecreaseabruptlyatapositionof250m,correspondingtotheposi-tionofasignicantcoatingdefect.Thecathodicradialcurrentdensityincreaseddramaticallyatthepositionofthecoatingdefect,asshowninFigure 5.6b .Figure 5.6c showsthatthepotentialislessnegativeatthecoatingdefectpositionthanotherplaces.Itmeansthatpipeislessprotectedatthecoatingdefect.ThevariationofpotentialwithinthepipesteelispresentedinFigure 5.6d .Sincethelengthofthe500mpipewasnotlarge,thepotentialdropoverthepipewasverysmall,ontheorderof10)]TJ/F19 7.97 Tf 6.448 0 Td[(310)]TJ/F19 7.97 Tf 6.447 0 Td[(2V.Thecalculationofsteelpotentialwasprimarilyusefulforallowingcalculationofthecurrentowinthepipe.Thesteelpotentialwassettoavalueofzeroatthe50mpositionwheretheanodeand

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87 pipewereconnected.Thisrepresentstheminimumvalueforsteelpotential.Thecurrentcomingfromsoilentersthepipelineandowsalongthepipefromtwooppositedirectionstothewireconnectingpointonthepipe.Hence,thepotentialsoneithersideoftheconnectionpositionhavehighervalues.Asharpchangeintheslopeofthepotentialisseenatthepositionofthecoatingdefect.TheaxialdirectioncurrentdistributionshowninFigure 5.6e revealsthatthecurrentdensityatthetwoendsofpipewassettozerovalues.Thecurrentdensitychangedsignatthebondlocationwherethewirewasconnectedtothepipe.Asignicantstepincreaseincurrentisobservedatthecoatingdefectduetothecontributionoftheenhancedaxialdirectioncurrentdensity.5.1.5ValidationwithCP3DTheforwardmodelusedinthepresentworkisasimplicationascomparedwithCP3D.Therearesomecommonfeaturesbetweenthetwomodels,suchas Thesamegoverningequation,i.e.,Laplace'sequation,wasappliedtothethree-dimensionalcathodicprotectionpipelinesystem. Thesystemgeometryandpositioncanbethesame.However,therearealsosomemajordifferencesbetweenthetwomodels: Coatingconditionsforthepipeweredifferent.CP3Dallowedmodelingacoating,thatexposesbaresteel;whereas,usingthepresentmodel,defectivecoatingwasmodeledasasectionoflowcoatingresistivity. Theboundaryconditionsforthepipeweredifferentastheconsequenceofthedifferentcoatingconditions.CP3Dappliedanonlinearpolarizationcurve,whereasthepresentworkusedlinearboundaryconditions.

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88 Figure5-7:PotentialofpipecalculatedbyusingCP3D. MoreelementsofhigherorderwereusedinCP3D.Thepresentmodelap-pliedlinearcylindricalelementtoreducethenumberofsegmentsandcom-putationaltask. CP3DcombinedtheBEMandFEMtosolvethesystem.ThecurrentworkappliedonlytheBEMtosimplifythecalculation.Tocomparetheresultsobtainfromthesetwomodels,thesameconditions,suchassoilresistivity,thedimensionsofthepipeandanodesystem,theappliedpotentialandthedefect-freecoatingresistivity,asthoseusedintheexampleofsection 5.1.3 ,wereappliedtotheCP3D.TheresultsareshowninFigures 5-7 ,whichcouldbecomparedwiththeFigure 5.6c .Atthepositionofthedefect,250m,thepotentialhasapeakvalue.Thepeakvalueinthetwoguresarebotharound)]TJ/F19 11.955 Tf 9.672 0 Td[(0:78V.Attheotherpositiononthepipe,thecurvetendstobehorizontal.Thevaluefortheformergureisabout)]TJ/F19 11.955 Tf 9.672 0 Td[(0:88V,whichforthelatteroneisabout)]TJ/F19 11.955 Tf 9.672 0 Td[(0:82V.Inaddition,inFigure 5.6c ,thepotentialsatthetwoendsofthepipedroppedalittle.ThisphenomenaislesspronouncedinFigure 5-7 .Figure 5-8 andFigure 5.6d hassimilartendency,thatis,atthe50mposition,wherethewireconnectsthepipeandtheanode,thepotentialofthepipe

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89 Figure5-8:PotentialofpipesteelcalculatedbyusingCP3D. steeliszero.Moreover,atthe250mposition,whichisthepositionofthecoatingdefect,thepotentialofthepipesteelhasaslopechange.Thedifferenceofthesetwoguresisthatthepotentialvalueisaround6:010)]TJ/F19 7.97 Tf 6.448 0 Td[(6Vvs.CSEreferenceelectrodefortheformer,anditisaround1:610)]TJ/F19 7.97 Tf 6.448 0 Td[(8Vforthelatter.CP3Ddoesnotprovidethecurrentdensityofthepipeattheaxialdirection,butitcanbecalculatedfromthedataofthepotentialofthepipesteelandthepiperesistivity.Figure 5-9 wasobtainedusingthismethod.ThecurrentalongthepipeseeFigure 5-9 andFigure 5.6e showsqualitativelythesamebehavior,whichgivessupporttotheideathatthesimpliedmodelmaybesufcienttoidentifyregionsofdefectivecoating.ThecomparisonbetweenFigure 5-10 andFigure 5.5c showsthatboththecurveshapeandtheorderofvalueareapproximatelythesame.However,thegurefromCP3Dhassmoothercurvethanthatfromcurrentworkatthetwoendsoftheanode.ThereasonisthatCP3Dusedtwohalfspheresastheendandtheproleoftheanodeiscontinuous.Onthecontrast,thecirclesareappliedasthetwoendsinpresentstudy.Itclearlyshowsthat,fromallthecomparisons

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90 Figure5-9:AxialdirectioncurrentdensityofpipecalculatedbyusingCP3D. Figure5-10:RadialdirectioncurrentdensityofanodecalculatedbyusingCP3D.

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91 above,thetwomodelscandescribethesimilartendency.Butthedifferentcoatingconditionsandcorrespondinglydifferentboundaryconditionsmayresultinthevaluedifferences.Inaddition,thetotalcurrentowstothepipeinCP3Dmodelisabout0:056618Amps,whileitisabout0:05489Ampsinpresentmodelduetothedifferenceofcoatingdefect.ThedefectsetinCP3Disasectionofbarepipesteel.Itneededmorecurrenttoprotectthepipelinethanasectionoflowresistivitycoating.Moreover,thepresentmodeldoesnotaccountthedepositionofscalesonthepipesteel,whichmayreducethecurrentnecessaryforcathodicprotection.5.2InverseModel5.2.1IntroductionAlltheinversemodelsdevelopedtodaterelyonasingletypeofdata,i.e.,po-tentialreadingsfromsoilsurfacesurveysorfromindividualsensorelectrodes.AsindicatedbyWrobelandMiltiadou, 45 forsuccessfulidenticationofcoatingde-fects,potentialreadingsshouldbeincloseproximitytothecoatingdefects.Evenonemeteroftopcovermaybesufcienttoobscuretheeffectsoflocalizedunder-protectedregions. 35 38 Thus,insomecases,aseriouscoatingdefectmayleadtoinsignicantchangesinpotentialatthesoilsurface.Thelowsensitivityofpoten-tialmeasurementstocoatingfailurescanbemitigatedbyincludingothertypesofdata,forexample,currentdensitiesinthepipelinemetal.Aschematicrepresen-tationoftheincorporationofpotentialsurveyandcurrentdataisgivenasFigure 5-11 .MasilelaandPereiradescribetheuseofsoilsurfacepotentialgradientsur-veystoassesstheconditionofpipelinecoatings. 85 Fischeretal.discussthedif-cultiesofobtainingIR-freemeasurementsofsoilpotential. 86 GummowandEngdescribeuseoffour-electrodemethodstomeasurecurrentsinpipelines. 87 Aerialsurveyswithmagnetometerscanbeusedtoassesstheowofcurrentinpipelines.

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92 Figure5-11:Schematicillustrationofthecouplingofasoil-surface-levelpotentialsurveymethodtoanaerialmagnetometercurrentsurveytoassesstheconditionofaburiedpipeline. Conversely,thepresenceofpipelinesundercathodicprotectionconfoundsinter-pretationofhigh-resolutionaeromagneticdata. 88 CampbellandZimmermande-scribetheuseofremotesensingofcurrentintheTrans-AlaskaPipeline. 89 Mur-phyetal.coupledimpedancespectroscopywithuseofSQUIDmagnetometerforremotesensingofcurrenttoassesslocalizedcorrosionratesonburiedmetallicstructures. 90 Tocreatetheinversemodel,theBEMcodedescribedintheprevioussectionwasusedaspartoftheobjectivefunctioninanonlinearregressionalgorithm.Thesimulatedannealingoptimizationapproachwasselectedforthepresentworkaccordingtochapter 4 .Theinversemodelactstominimizethevalueofanobjectivefunctionwhichrepresentsthedifferencebetweenmeasuredandcalculatedvalues.Theparametersetwhichresultsinthesmallestvalueoftheobjectivefunctioncanbeassumedtoreecttheconditionofthecoatedpipe.Theinverseresultsarestronglydependent

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93 Figure5-12:SyntheticpipecurrentandsurfacepotentialdatacorrespondingtoTable 5.1 .Dashedlinesindicatethelocationofcoatingdefects. ontheformoftheobjectivefunctionchosen.5.2.2RegressiontoNoise-FreeDataPreliminaryregressionresultswereobtainedusingsyntheticdatageneratedwithmachineprecision.Thephysicalsituationconsistedofasinglepipecon-nectedtoasingleanode,aspresentedschematicallyinFigure 4-1 .Thepipelinewas100mlong.ThreecoatingdefectswereassumedtoexistonthepipecoatingwithparameterspresentedinTable 5.1 .Theforwardmodelwasusedtoobtainsyn-theticsurfacepotentialandcurrentdensitydatawhichwerethenusedastheelddatafortheinversemodel.ThesyntheticdataareshowninFigure 5-12 .Dashedlinesindicatethelocationofcoatingdefects.Thepresenceofdefectsat30and70mpositionscanbeinferredfromsubtlechangesinthecurrentandsurfacepo-tentialdistributionsshowninFigure 5-12 ;however,itisdifculttodiscernthedefectlocatedat40m.Thepotentialdatasetcomprised303valuescorrespondingtothreelinesof101pointseachlocateddirectlyabovethepipesegmentandonemetertoeitherside.Therewere99currentdensitydatapoints.Regressionwasperformedtohomoge-

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94 neousdatasetswhichcomprisedeitherthecurrentorpotentialdata.Regressionwasalsoperformedtoheterogeneousdatasetswhichincludedbothcurrentandpotentialdata.RegressiontoHomogeneousDataSetsAnobjectivefunction,whichdescribesthedifferencebetweenthemeasuredpotentialandthecalculatedpotentialonthesoilsurface,ordifferencebetweenthemeasuredcurrentdensityalongthepipeandthecalculatedcurrentdensitydata,isgivenasgx;;=Nck=1)]TJ/F48 11.955 Tf 5.511 -9.69 Td[(ik)]TJ/F67 11.955 Tf 10.46 0.634 Td[(bk2-18forcurrentsurveydataandgx;;=Npj=1j)]TJ/F67 11.955 Tf 13.066 3.503 Td[(bj2-19forpotentialsurveydata.Theobjectivefunctionvaluegisafunctionofthecoatingdefectparametervectorsx,and,showninequation 4-12 .Inequations 5-18 and 5-19 ,ikandjrepresentthemeasuredcurrentdensityalongthepipelineandsoilsurfacepotentials,andbkandbjrepresentthecorrespondingvaluesobtainedbythemathematicalmodel.ThenumberofcurrentdensitydatapointsisgivenbyNc,andthenumberofpotentialdatapointsisgivenbyNp.Bothtypesofdataareinuencedbythepipecoatingcondition;thereforetheminimumoftheobjectivefunctionisobtainedwhenthecoatingparameterspacereectstheconditionofthepipelinecoating.AnexplorationoftheregressionstrategyispresentedinTable 5.1 regressionusingeitherequation 5-19 or 5-18 forhomogeneoussyntheticdatawithoutaddednoise.Theinuenceofinitialguessesforthenonlinearregressionwasminimizedbyassigninginitialvaluesofxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92mforeachdefect.

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95 Table5.1:Regressionresultsusingeitherequation 5-19 or 5-18 forhomoge-neoussyntheticdatawithoutaddednoiseorequation 5-20 forheterogeneoussyntheticdatawithoutaddednoise.Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92m. Data Equation Coating Position Resistivity Dimension Type Defect xk/m k/Wm k/m 2= SetDefectValues 1 30 )]TJ/F19 11.955 Tf 9.671 0 Td[(4:5107 0.316 2 40 )]TJ/F19 11.955 Tf 9.671 0 Td[(3:0106 0.548 3 70 )]TJ/F19 11.955 Tf 9.671 0 Td[(2:0107 0.447 RegressionResult Current 5-18 1 29.91 )]TJ/F19 11.955 Tf 9.671 0 Td[(4:94107 0.217 6:710)]TJ/F19 7.97 Tf 6.448 0 Td[(11 2 70.00 )]TJ/F19 11.955 Tf 9.671 0 Td[(1:95107 0.460 Potential 5-19 1 30.02 )]TJ/F19 11.955 Tf 9.671 0 Td[(3:90107 0.999 9:710)]TJ/F19 7.97 Tf 6.448 0 Td[(8 Potential 5-19 1 30.03 )]TJ/F19 11.955 Tf 9.671 0 Td[(2:62107 2.37 1:210)]TJ/F19 7.97 Tf 6.448 0 Td[(7 2 70.06 )]TJ/F19 11.955 Tf 9.671 0 Td[(3:04107 1.07 Potential 5-19 1 29.95 )]TJ/F19 11.955 Tf 9.671 0 Td[(1:60107 2.10 8:010)]TJ/F19 7.97 Tf 6.448 0 Td[(8 2 30.01 )]TJ/F19 11.955 Tf 9.671 0 Td[(2:66107 0.31 3 43.03 )]TJ/F19 11.955 Tf 9.672 0 Td[(0:17107 2.32 Both 5-20 1 30.03 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:12107 1.54 3:710)]TJ/F19 7.97 Tf 6.448 0 Td[(8 2 70.00 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:99107 0.012 Both 5-21 1 29.98 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:51107 0.30 3:010)]TJ/F19 7.97 Tf 6.448 0 Td[(4 2 38.96 )]TJ/F19 11.955 Tf 9.672 0 Td[(0:14107 1.41 3 69.75 )]TJ/F19 11.955 Tf 9.672 0 Td[(0:50107 2.46

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96 Figure5-13:The2=statisticcorrespondingtoTable 5.1 asafunctionofthenum-berofcoatingdefectsassumedintheregression. The2=statisticispresentedinFigure 5-13 asafunctionofthenumberofcoatingdefectsassumedintheregression.ThemethodproposedbyQiuandOrazem 91 fordeterminationofthemaximumnumberofdefectsrevealedaclearminimumat2coatingdefectswhenregressionwasmadetocurrentdataalone.Onedefectwaslocatedatthe30mpositionandtheotherwasatthe70mpo-sition.Sincetheresistivityreductionofthesetdefectatthe40mpositionwasrelativelysmall,itwasdifculttondthisdefectbyregression.Themethodforselectingthestatisticallysignicantcoatingdefectsyieldedmoreequivocalresultswhenregressionwasmadetosurfacepotentialdataonly.Noclearminimumcouldbefound.Whenregressionwasmadetoamodelwithonecoatingdefect,thedefectfoundwasthatatthe30mposition.Asmallim-provementinthe2=statisticwasfoundbyregressingtotwocoatingdefects.Thesecondcoatingdiscoveredwaslocatedatthe43mposition,whichwasclosetothedefectsetatthe40mposition.Thesmallestvalueofthe2=statisticwas

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97 foundforthreecoatingdefects,buttheresults,showingtwodefectslocatedatthesameposition,werenotstatisticallysignicant.UseoftheAkaikeinformationcri-teria 82 83 84 didnotyieldaclearerdenitionofthemaximumnumberofresolvableparameters.Thedifcultyofobtainingthestatisticallysignicantnumbercoatingdefectsforregressiontopotentialdataindicatesthatworkisneededtoestablishrenedmethodsforevaluatingtheregressionresults.Betterinverseresultswereobtainedbyusingcurrentdensitydata,eventhoughtherewerefewercurrentdatathanpotentialdata.UnweightedRegressiontoHeterogeneousDataSetsInordertotakeadvantagesoftheinformationfromboththepotentialandthecurrentdensitydata,anobjectivefunctionwasobtainedasthesummationoftheleastsquaredifferencesofthetwosetsofdata,i.e.,gx;;=Npj=1j)]TJ/F67 11.955 Tf 13.067 3.503 Td[(bj2+Nck=1ik)]TJ/F67 11.955 Tf 12.578 0.634 Td[(bk2 -20 Theinverseresultsofapplyingpotentialandcurrentdensitydatatoequation 5-20 areshowninTable 5.1 .AsshowninFigure 5-13 ,the2=statisticindicatedthattwodefectscouldbeidentied,andthesewerelocatednearthe30and70mposi-tions.Theresultsobtainedusingequation 5-20 arecomparabletothoseobtainedusingthecurrentdataalone.Useofequation 5-20 forregressiontoheteroge-neousdatadidnotimprovetheresolutionofpipecoatingcondition.WeightedRegressiontoHeterogeneousDataSetsTheregressionstrategyusedforequation 5-20 failedbecausethemagnitudeofthetwotypesofdatawasvastlydifferent.Aweightingstrategywasemployedinwhichtheregressionwasscaledbythemagnitudeofthedata,i.e.,gx;;=Npj=1j)]TJ/F67 11.955 Tf 13.066 3.503 Td[(bj2 2j+Nck=1ik)]TJ/F67 11.955 Tf 12.577 0.633 Td[(bk2 2k-21

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98 Figure5-14:The2=statisticcorrespondingtoregressionofequation 5-21 seeTable 5.1 asafunctionofthenumberofcoatingdefectsassumedintheregression. Undertheassumptionthattheweightingstrategyshouldbebasedonthevari-anceofthedata, 92 93 useofequation 5-21 isconsistentwiththeassumptionofaproportionalerrorstructure.The2=statisticispresentedinFigure 5-14 asafunctionofthenumberofcoatingdefectsassumedintheregression.Eachofthethreecoatingdefectscouldbefound.Thedefect1approximatelyhasthesamecoatingparametersasthoseofthedefectatthe30mposition.Thepositionofdefect2isclosetothedefectatthe40mposition,andtherelativeerrorforthepositionisabout3percent.Thedefect3referstothedefectatthe70mposition.Forthedefects2and3,thecoatingresistivityreductionandthedefectwidthdifferfromthesetvaluesduetothecorrelationamongdefectparametersevidentinequation 4-12 .Theimportanceofweightingforregressionstrategiesisparticularlyevidentforheterogeneousdatasets.Proportionalweightingisgenerallyrecommendedforsyntheticdataforwhichtheuncertaintyisgovernedbytheaccuracyofthe

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99 calculation.Forexperimentaldata,aweightingstrategyshouldbeguidedbytheperceivedormeasuredvarianceoftheexperimentaldata.5.2.3RegressiontoNoisyDataForthepurposeoftestingtherobustnessoftheinversemodel,noisewasaddedtothedataset.Noisewasaddedtothepotentialdataaccordingto=+P;1-22whereP;1representsanormallydistributedrandomnumberwithmeanvalueequaltozeroandstandarddeviationequaltounity.Therandomlygeneratednumberwasscaledbyastandarddeviation,whichwasassignedavalue=0:4mV.Thenoiseinthecurrentsignalwasassumedtobeproportionaltothecurrentsuchthati=i1+iP;1-23Thevalueofiwas0:02;thus,thenoiselevelforthecurrentmeasurementrepre-sented2percentofthemeasuredvalue.Thecomparisonbetweenthebasevalueandthenoise-addedvalueforthepotentialdataandthecurrentdensitydataisshowninFigures 5.15a and 5.15b ,respectively.Whiletheobjectivefunction 5-21 wasverywellsuitedforregressiontonoise-freesyntheticdata,onlythelargestcoatingdefectatthe30mpositioncouldbefoundwhenitwasusedtoregresstothesyntheticdatawithaddednoise.Thesituationwasnotimprovedbyreplacingtheproportionalweightingshowninequation 5-21 withweightingbasedontheassumederrorstructure.WeightedandScaledRegressiontoHeterogeneousDataSetsThereasonforthepoorresultsobtainedbyregressionofequation 5-21 couldberelatedtothelargenumberofpotentialmeasurementswhicharerelativelyin-

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100 a bFigure5-15:Syntheticdatawithaddednoise.Thelinerepresentsthenoise-freevaluesandthesymbolsrepresentsyntheticdatawithaddednoise.asurfaceon-potentialvaluewith=0:4mV;baxialcurrentdensityvaluewithproportionalnoisecorrespondingto2percentofthevalue.

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101 Table5.2:Regressionresultsusingequation 5-24 forheterogeneoussyntheticdatawithaddednoise.Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92m. Data Equation Coating Position Resistivity Dimension Type Defect xk/m k/Wm k/m 2= SetDefectValues 1 30 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:5107 0.316 2 40 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:0106 0.548 3 70 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:0107 0.447 RegressionResult Both 5-24 1 30.1 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:59107 0.50 3.02 Both 5-24 1 30.1 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:99107 0.31 2.71 2 69.4 )]TJ/F19 11.955 Tf 9.672 0 Td[(0:77107 1.27 Both 5-24 1 38.8 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:05107 0.88 10.8 2 46.4 )]TJ/F19 11.955 Tf 9.672 0 Td[(0:66107 2.07 3 46.8 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:13107 1.50 sensitivetocoatingcondition.Are-scaledobjectivefunctionwasproposedasgx;;=1 NpNpj=1j)]TJ/F67 11.955 Tf 13.066 3.503 Td[(bj2 2+1 NcNck=1ik)]TJ/F67 11.955 Tf 12.578 0.633 Td[(bk2 i2-24where2andi2=iik2provideaweightingbasedontheerrorstructureofthedata.Equation 5-24 providesanobjectionfunctioninwhichthecontributionsofpotentialandcurrentdataarescaledsuchthattheyprovideequalweighttotheregression.TheregressionresultsareshowninTable 5.2 .Theweighted2=statisticreachedaminimumvaluefortwocoatingdefects,indicatingthattwodefectscouldbeidentied.Thescaledobjectivefunction 5-24 hastheadvantagesthat,foreachdatatype,theweightingstrategyisbasedontheerrorstructureofthatdataandthatitdecreasestheinuenceofsuperuousinputinformation.GeneralizationoftheObjectiveFunctionTheproperscalingoftheobjectivefunctionwillbedeterminedbytherelativesensitivityofaparticulartypeofdatatothecoatingcondition.Aformofequation

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102 5-24 thatisreadilygeneralizabletomultipletypesofdatacanbeexpressedasgx;;=Mk=1kgk-25wheregkistheobjectivefunctionforagiventypeofelddatascaledbythenum-berofdata,i.e.,gkx;;=1 NkNkj=1xk;j)]TJ/F67 11.955 Tf 12.785 0.634 Td[(bxk;j2 k;j2-26andkisascalingfactorthataccountsfortherelativesensitivityofthedatatypetocoatingcondition.Accordingtotheerrorstructureofthedata,2k;jcanbeavalueorequaltokxk;j2.Tofacilitateinterpretationofthe2=statistic,itisappropriatetoconstrainksuchthatMk=1k=1 -27 Inequation 5-26 ,xk;jandbxk;jrepresentthemeasuredandcalculatedvaluesre-spectivelyfordatatypek,andk;jrepresentsthecorrespondingstandarddevia-tionforthemeasuredvalues.Thedatasetsmaycomprise,amongothers,surfaceon-potential,surfaceoff-potential,currentdensitydata,orreadingsfromburiedreferenceelectrodesorcoupons.5.3ConclusionsInthischapter,thepotentialofthepipesteelwasassumedtovaryalongthepipe.Thecurrentdensityofthepipeinaxialdirectionwascalculatedinthefor-wardmodelwithoutusingtheniteelementmethodFEM,whichsimpliedthecalculationprocess.Inaddition,asthecurrentdensitydatawereavailablefromtheforwardmodel,thereweretwokindsofdatasets,potentialdataandcurrentdensitydata,thatcouldbeusedasmeasureddatafortheinversemodel.

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103 Astrategywasdevelopedthatcanbeusedtoidentifythemaximumnumberofcoatingdefectsthatcanbedetected.Thisnumberissensitivetothequalityofdataaswellastotheactualcoatingcondition.Ifthenumberofcoatingdefectsdetectedissmallerthantheactualnumberofcoatingdefects,thetechniquewillidentifythemostseriousones.Duringtheproceduretobuildthefourobjectivefunctions,addingdifferentweightforeachdatapointandincludingthenumberofthedatapointshavebeenprovedeffectivetoformtheobjectivefunction.Thepresentworkprovesthefeasibilityofcouplingaboundaryelementfor-wardmodelwithanonlinearregressionalgorithmtoobtainpipesurfaceproper-tiesfrompipelinesurveydata.

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CHAPTER6APPLICATIONOFOFF-POTENTIALDATATOTHEINVERSEMODELInChapter 4 ,thepotentialcalculatedintheforwardmodelistheon-potential,orthepotentialmeasuredwhentheanodeisconnectedtothepipeline.Inthischapter,theforwardmodelisextendedtocalculatethepotentialonthesoilsur-faceimmediatelyaftertheanodeisdisconnectedwiththepipeline.Thiskindofpotentialiscalledoff-potential.Threekindsofdatasets,i.e.,on-potential,off-potentialandcurrentdensitydatacouldbeobtainedfromtheforwardmodel.TheywerethenappliedtotheinversemodeltoprovethevalidityofthegeneralformoftheobjectivefunctionbuiltinChapter 5 .6.1ForwardModel-CalculationoftheSoilSurfaceOff-Potential6.1.1PhysicalProcessWhentheanodeisdisconnectedwiththepipe,nocurrentwillowtothepipe.Likewisenocurrentwillowbacktotheanodethroughthewire.Thepoten-tialmeasuredonthesoilsurfaceatthismomentiscalledtheoff-potential.ThetechniqueisestablishedontheprinciplethattheIRcomponentinthepotentialmeasurementdecaysalmostinstantaneously,whilethepipe-to-soilinterfacepo-larizationdecaysrelativelyslowly.Thusthecorrectpipe-to-soilpotentialfreeoftheIRdropcanbemeasured.Thereisnogeneralstandardtoidentifythetimerequiredtomeasurethepotentialafterswitchofftheprotectioncurrent.Thetimecanrangefrom100microsecondstoseconds,orevendays. 13 104

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105 Figure6-1:SwitchoffCPpowersourcetodetermineinstantoff-potential. 6.1.2AssumptionsAccordingtoCox'sstudy, 94 asharppeakisobservedinthepipe-to-soilpoten-tialasthepotentialisinterruptedFigure 6-1 .Thespikelastsabout10microsecondsto200microseconds,thendepolariza-tionwilltakeplace.Inpresentstudy,afterthecurrentisshutoff,theconditionrightafterthedevelopmentofthespikeandjustbeforethedepolarizationoccur-renceisstudied.Atthatmoment,thepotentialinthesoilwillnotchangesincethedepolarizationhasnotyetbegun.However,thepotentialofthepipesteelVwillberedistributedtokeepthepotentialuniformalongthepipe,whichmakescurrentowalongthepipefromregionofhighpotentialtoregionoflowpotential.Theresultisthatthepotentialofthepipesteelbecomeszeroalongthepipeline.SincethepotentialofthevoltmeterreadingdependsonthepotentialsandVas=V)]TJ/F47 12.457 Tf 12.116 0 Td[(,willbechangedafterthesacricialcurrentisswitchedoff.6.1.3ModelImplementationThesystemsatisestheLaplace'sequation.Thepipelineistheonlyobjectofthesystem.ApplyingthesametechniquesmentionedinChapter 4 ,suchasthe

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106 Figure6-2:On-potentialandoff-potentialonsoilsurface. BEMforhalf-innitydomain,andtheself-equilibriumlimitation,wecanobtain264)]TJ/F48 11.955 Tf 9.911 0 Td[(Gpp)]TJ/F19 11.955 Tf 9.672 0 Td[(1)]TJ/F48 11.955 Tf 10.389 0 Td[(Ap03758><>:nrp19>=>;=264)]TJ/F48 11.955 Tf 10.245 0 Td[(Hpp0375p-1Thelastequationintheabovematrixformmeansthatthecurrentowingawayfrompipe,forexample,atthecoatingdefectposition,willnotbelostinthesoildo-main.Thecurrentwillowbacktothepipe.Therefore,nocurrentwillbegainedorlostattheinnityboundary.Becausewillremainthesameasthatwhentheanodeisconnectedtothepipe,itisaknownboundaryconditioninequation 6-1 .Theunknownsarethecurrentdensityofthepipelineandthepotentialatinnity.6.1.4SimulationResultsandAnalysisThetestexamplewith500mlongpipeshowninChapter 5 issolvedtogettheoff-potentialonthesoilsurface.Thecomparisonoftheon-potentialandoff-potentialisshowninFigure 6-2 ,whichagreeswellwiththeexpectedtrendFigure 6-3 95 Theon-potentialismorenegativethantheoff-potential.Inaddition,theon-potentialonthesoilsurfaceshowsobservabledifferencesrightabovethepipeandatcertaindistancesawayfromthecenterlineofthepipeasFigure 4-10 .However,thevariationfortheoff-potentialonthesoilsurfacesurroundingthe

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107 Figure6-3:Typicalclose-intervalpotentialgraph. pipeareaissmall.Therefore,thenumberoftheoff-potentialdatacanbereduced.6.2InverseModel-UsingThreeKindsofHeterogeneousDataSetsTheheterogeneousdatasets,suchasthesoilsurfaceon-potential,off-potentialandthecurrentdensityintheaxialdirectionalongthepipe,wasusedintheinversemodeltotesttheeffectivenessofthegeneralformofobjectivefunction,equations 5-25 5-26 and 5-27 .Thetestingprocessinvolvedtwodifferentcases,i.e.,thedatawithoutaddednoiseanddatawithaddednoise.6.2.1DatawithoutNoiseThreesetsofdatawithoutnoiseareappliedtothegeneralformofobjectivefunction.Sincethedatawereindependentofthenoise,thek;j=xk;j.There-gressionresultsobtainedbysettingtheinitialguessasone,twoandthreedefectswiththesamecoatingparametersareshowninTable 6.1 .Whentherewasonlyonedefect,themostseriousdefectat30mpositionwasfound.Assumingtwodefectsatthebeginning,thetwomostseriousdefectswerefound.Thesearethedefectsatthe30mandthe70mpositions.Allthethreeparametersforholiday1wereclosetothesetvalues.Whenthreedefectswereassumed,thepositions

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108 Table6.1:Regressionresultsfromthedatawithoutaddednoiseobtainedbyus-ingthegeneralformoftheobjectivefunction 5-25 5-26 and 5-27 .Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.671 0 Td[(3:5107Wm,andk=0:92m. DataType Coating Position Resistivity Dimension Defect xk/m k/Wm k/m 2= SetDefectValues 1 30 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:5107 0.316 2 40 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:0106 0.548 3 70 )]TJ/F19 11.955 Tf 9.671 0 Td[(2:0107 0.447 RegressionResult Three 1 30.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:53107 0.09 4:3510)]TJ/F19 7.97 Tf 6.447 0 Td[(5 Three 1 30.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:50107 0.34 2:2410)]TJ/F19 7.97 Tf 6.447 0 Td[(6 2 69.6 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:04107 3.14 Three 1 30.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:50107 0.316 2:2310)]TJ/F19 7.97 Tf 6.447 0 Td[(8 2 40.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(1:06106 1.70 3 70.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(1:66107 0.58 Three 1 30.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:18107 0.24 5:9710)]TJ/F19 7.97 Tf 6.447 0 Td[(7 2 30.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:31107 0.45 3 40.5 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:63105 3.02 4 69.9 )]TJ/F19 11.955 Tf 9.672 0 Td[(8:06106 1.43 ofallthethreepresetcoatingdefectscouldbefound.Thetwoparameters,i.e.,theresistivityreductionandthewidth,ofdefect1wereapproximatelysameasthoseofsetvalues.However,thoseparametersfordefect2anddefect3weredif-ferentfromsetvalues,indicatingfurtherthattheexistenceofthecouplingeffectbetweentheparametersasinequation 4-12 .Whenfourdefectswereassumed,bothdefect1and2referredtothedefectat30mposition.Thedefectsat40mand70mpositionscouldalsobefound.Thestrategyusedinchapter 5 wasappliedtodeterminedtheappropriatenumberofdefects.2=asafunctionofthenumberofdefectsisshowninFigure 6-4 .Theappropriatenumberofdefectsisthreeinthecurrentexample.ComparingthelasttworowsofTable 5.1 andTable 6.1 ,itisfoundthattheoptimalnumberofdetectabledefectsistwofortheformercase,whileitisthreeforthelattercase.Itcanalsobeconcludedthataddingmoredatasets,e.g.,addingtheoff-potentialdataasinputinformation,yieldedregressionre-

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109 Figure6-4:Decidethenumberofdefectsusingnoisefreesoilsurfaceonandoff-potentialandcurrentdensitydata. sultsthatwereclosertothesetvalues.Thegeneralformoftheobjectivefunctioniseffectivefordatasetswithoutaddednoise.6.2.2DatawithAddedNoiseThedatasetswithaddednoisewereusedtotestthecorrectnessoftheequa-tions 5-25 5-26 and 5-27 .Thenoiseon-potentialandcurrentdensitydatawerethesameasshowninFigures 5.15a and 5.15b .Thenoisewith=0:2mVstandarddeviationwasaddedtotheoff-potentialonthesoilsurface,asshowninequation 5-22 .Theresultingsoilsurfaceoff-potentialwereshowninFigure 6-5 .Thethreesetsofnoisydatawereusedasthemeasureddata.Thus,on=0:4mVisfortheon-potentialdataset,off=0:2mVisfortheoff-potentialdatasetandi=2%ikisforthecurrentdensitydataset.Alltheinitialguessesofthedefectparameterswerethesameasthepreviouscase.Iftherewasonlyonedefect,thelargestcoatingdefectatthe30mpositionwasfound.Assumingtwodefects,the

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110 Figure6-5:Setvalueandnoise-addedoff-potentialonsoilsurfacedata. defectsatthe30mandthe40mpositionscouldbefound.Whenthreedefectswereassumedinitially,nodefectscouldbefound.TheresultsareshowninTable 6.2 .Figure 6-6 shows2=asafunctionofthenumberofdefects.Itindicatesthattheappropriatenumberofdefectsforthecurrentexampleistwo.Withthenoisedata,itisdifculttocomparetheregressionresultTable 6.2 withTable 5.2 .Ap-parently,theproblembecomesmoredifcultinestimatingtheeffectofthenoiseaddedoff-potentialdata.Thecommonobservationisthatonlytwodefectscanbefoundinbothcases.Themostseriousdefectatthe30mpositioncanalwaysbefound.Buttheresultonotherdefectsvarywiththenoise-addeddata,indicatingthedisturbanceofthenoise.Morerenedoff-potentialdatamaybenecessarytothemodel.Sincetheoff-potentialdataarefreeofIRdrop,itwillbemorereliabletoassesspipeconditionbyusingoff-potentialdataifthepresentmodelcanaccountfortheeffectofdeposits.Currently,thismodelcannotsimulatetheinuenceofscaledeposits,butitwillbeasubjectforfutureresearch.

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111 Table6.2:Regressionresultsfromthenoise-addeddatabyusinggeneralformoftheobjectivefunction 5-25 5-26 and 5-27 .Theinitialvaluesforeachdefectwasxk=50m,k=)]TJ/F19 11.955 Tf 9.672 0 Td[(3:5107Wm,andk=0:92m. DataType Coating Position Resistivity Dimension Defect xk/m k/Wm k/m 2= SetDefectValues 1 30 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:5107 0.316 2 40 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:0106 0.548 3 70 )]TJ/F19 11.955 Tf 9.672 0 Td[(2:0107 0.447 RegressionResult Three 1 30.2 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:36107 0.67 2.94 Three 1 30.1 )]TJ/F19 11.955 Tf 9.672 0 Td[(4:50107 0.46 2:77 2 42.8 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:47107 0.29 Three 1 49.8 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:57107 1.20 7:81 2 50.0 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:56107 0.92 3 53.1 )]TJ/F19 11.955 Tf 9.672 0 Td[(3:46107 0.10 Figure6-6:Decidethenumberofdefectsusingnoise-addedonandoff-potentialandcurrentdensitydata.

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112 6.3ConclusionsInthischapter,theeffectivenessofthegeneralformoftheobjectivefunctionwasstudiedbyusingthreekindsofsurveydata.Itisconcludedthat 1. Sincetheoff-potentialonsoilsurfacedoesnotvarymuchontheneighboringsoilsurfacearearightabovethecenterlineofpipe,fewdatapointsneedtobemeasured. 2. ComparingFigure 6-4 withFigure 6-6 ,itcanbendthatmorecoatingde-fectscanbefoundbyusingnoisefreedatathanthatwithaddednoise.Thisresultagreeswellwiththendingsfromotherresearch,i.e.,theimpedanceanalysis. 3. Withthegeneralformoftheobjectivefunction,themorekindsofdatasetsapplied,themoreprecisetheregressionresultsareforthenoisefreedatasets. 4. Sincetheoff-potentialdataarefreeofIRdrop,itwillbemorereliabletoassesspipeconditionbyusingoff-potentialdataifthepresentmodelcanaccountfortheeffectofdeposits.

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CHAPTER7SUMMARYANDFUTUREWORK7.1SummaryPipelinecorrosioncoststheU.S.sevenbillionsofdollars.Ninetypercentofthecostisthecapital,operationandmaintenance.Topreservetheassetofthepipelineandtoensuresafeoperations,regularpipelineinspectionsneedtobedonetoavoidpossiblefailures.Inaddition,methodstoanalyzeandinterpretpipelinesurveydatatomaximizeinformationbecomesmoreimportantthanever.Inthisstudy,acomputerprogram,whichcanbeusedtoextracttheconditionsofthepipetakingintoaccountawidevarietyofeldsurveydata,hasbeendevel-oped.Theprogramconsistsoftwoparts,theforwardmodelandtheinversemodel.Theintegrationofthesetwomodelsmakestheprogrameffective.Besides,thiseffortestablishedaproofofconceptwhichcanmotivatefuturedevelopments.Theforwardmodelwasdevelopedtoaddresshowthecathodicprotectionpipelinesystemworks.Theresistivityofpipecoatingwasapproximatedbytheparametersofthecoatingdefects.Thelinearboundaryconditionswereappliedtosimplifythecalculation.Theboundaryelementmethodwithcylindricalelementswasdevel-opedtoreducethecomputationtask.Withthevaryingpotentialsonthepipesteel,theon-potentialonthesoilsurfaceandthecurrentdensityofthepipeintheaxialdirectioncouldbeobtained.Whentheanodeisdisconnectedwiththepipeline,theoff-potentialonthesoilsurfacecouldalsobecalculated.Theinversemodelfocusedonconstructionofaneffectiveobjectivefunction.Thescaledobjectivefunctionpossessestheadvantagesthattheweightingstrategy 113

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isbasedontheerrorstructureofeachtypeofdataandthattheinuenceofsuper-uousinputinformationcanbereduced.Byminimizingtheobjectionfunction,theparameterstodescribethecoatingconditionswereestimated.Severalkindsofheterogeneousdatasets,i.e.,theon-potential,off-potentialandthecurrentden-sitydata,wereappliedtotheobjectivefunctiontoprovideasmoreinformationaspossible.Sincetheforwardmodelwasappliedfrequentlyintheinversecalculation,itneededtobepreciseandfast.Thepresentstudyshowsthatthesimpliedversionoftheforwardmodelcancouplewellwiththeinversemodelininterpretingthepipelinesurveydata.7.2FutureWorkCurrently,themodelwasappliedtothesyntheticdatageneratedforasectionofacoatedundergroundpipelinewhichiselectricallyconnectedtoaverticalsac-ricialanode.Forthenextstep,thesyntheticdatageneratedbytheCP3Dmodelcanbeusedasthepipelinesurveydata.ComparingtheresultsofthecoatingconditionsobtainedfromthisinversemodelwiththesetconditionfortheCP3Dmodel,therobustnessoftheinversemodeldevelopedinpresentstudywillbefur-therveried.Inaddition,anundergroundpipelinewithmulti-anodecanbesimulatedbythepresentmodel.Otherapplicationsforcylindricalshapedobject,suchastank,willbeanotherdirectiontoextendthemodel'sapplications.Eventually,auserfriendlyinterfaceneedstobecreatedforpracticalapplica-tions. 114

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APPENDIXABOUNDARYINTEGRALA.1IntegralalongOneObjectWhentheintegraliscalculatedononeobject,itisappropriatetouselocalco-ordinate.Theintegralcanbeovercylindricalelementsorcircleelements.A.1.1IntegraloverCylindricalElementsTherearetwokindsofintegralovercylindricalelements,dependingonthepositionofthecollocationpoints.Whenthecollocationpointsareonthecylinderreferenceline,thelocalcoordi-nateofthecollocationpointPisR;0;zPandtheintegrationpointQisRcos;Rsin;zQ.jrPQj=rPQ=q Rcos)]TJ/F48 11.955 Tf 12.594 0 Td[(R2+Rsin2+zQ)]TJ/F48 11.955 Tf 12.33 0 Td[(zP2A-1Substituteequation A-1 intoequation 4-43 togetGijGij=ljR 4Z102k=1qjkNktdtZ201 p Rcos)]TJ/F48 11.955 Tf 12.594 0 Td[(R2+Rsin2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2dA-2Sincenj=cos~i+sin~jatthecylindricalsurfacebySubstitutingequation A-1 intoequation 4-44 ,HijbecomesHij=ljR 4Z102k=1ujkNktdtZ20@1 rij @eRd=ljR 4Z10Z20)]TJ/F48 11.955 Tf 10.269 0 Td[(R)]TJ/F19 11.955 Tf 11.997 0 Td[(cos2k=1ujkNktdtd [p 2R2)]TJ/F19 11.955 Tf 11.996 0 Td[(2R2cos+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2]3 A-3 Wherezjisafunctionoft.Itcanbedescribedaszj=lp+ljtA-4 115

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116 wherelprepresentsthelengthbeforeelementj.lpisaconstantiftheelementjisxed.Whenthecollocationpointsareonthecenterofthecircles,whicharethetwoendsofcylinder,thelocalcoordinatesofP,Qpointsare;0;zPandRcos;Rsin;zQrespectively.jrPQj=rPQ=q Rcos)]TJ/F19 11.955 Tf 11.996 0 Td[(02+Rsin)]TJ/F19 11.955 Tf 11.996 0 Td[(02+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2=q R2+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2 A-5 Substitutingequation A-5 intoequation 4-43 ,GijbecomesGij=ljR 4Z102k=1qjkNktdtZ201 p R2+zJ)]TJ/F48 11.955 Tf 12.331 0 Td[(zI2d=ljR 42 p R2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2Z102k=1qjkNktdt=[ljR 2p R2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2]Z102k=1qjkNktdt A-6 AndSubstitutingequation A-5 intoequation 4-44 ,HijbecomesHij=ljR 4Z102k=1ujkNktdtZ20@1 p R2+zj)]TJ/F48 7.97 Tf 6.67 0 Td[(zi2 @eRd=)]TJ/F48 11.955 Tf 9.791 0 Td[(ljR2 2p R2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi23Z102k=1ujkNktdt A-7 A.1.2IntegraloverCircleElementsThecollocationpointscanbeonthecylindersideareaorthecenterofcircleelementastheintegraliscalculatedovercircleelements.Whenthecollocationpointsareonthecylinderside,P,QareR;0;zPandrcos;rsin;zQinlocalcoordinates.Here,risdistancefromcenterofcircleO1orO2toanypointonthecircle,asshowninFigure 4-8 .jrPQj=rPQ=q rcos)]TJ/F48 11.955 Tf 12.594 0 Td[(R2+rsin2+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2=q R2+r2)]TJ/F19 11.955 Tf 11.996 0 Td[(2Rrcos+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2 A-8

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117 Substitutingequation A-8 intoequation 4-45 ,wegetGijGij=qj1 4ZR0Z20rdrd p R2+r2)]TJ/F19 11.955 Tf 11.996 0 Td[(2Rrcos+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2 A-9 Atthecirclesurfacenj=ez,where+ezreferstothenormalvectoroftheendingcirclesurface,while)]TJ/F64 11.955 Tf 9.672 0 Td[(ezreferstothenormalvectorofthebeginningcirclesurface.Substitutingequation A-8 intoequation 4-46 ,wegetHijHij=uj1 4ZR0Z20@1 p R2+r2)]TJ/F19 7.97 Tf 6.448 0 Td[(2Rrcos+zj)]TJ/F48 7.97 Tf 6.671 0 Td[(zi2 @ezrdrd=uj[zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi 4]ZR0Z20rdrd [p R2+r2)]TJ/F19 11.955 Tf 11.997 0 Td[(2Rrcos+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2]3 A-10 Whenthecollocationpointsarethecenterofthecircleelements,thelocalco-ordinatesofP,Qare;0;zPandrcos;rsin;zQ.jrPQj=rPQ=q rcos2+rsin2+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2=q r2+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2 A-11 Substitutingequation A-11 intoequation 4-45 ,GijbecomesGij=qj1 4ZR0Z20rdrd p r2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2 A-12 Substitutingequation A-11 intoequation 4-46 ,HijbecomesHij=uj1 4ZR0Z20@1 p r2+zj)]TJ/F48 7.97 Tf 6.671 0 Td[(zi2 @ezrdrd=uJ[zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi 4]ZR0Z20rdrd [p r2+zj)]TJ/F48 11.955 Tf 12.33 0 Td[(zi2]3 A-13 A.2IntegralbetweenDifferentObjectsIntegralbetweenanobjectanditsmirrorreectionwithrespecttothesoilsur-faceandintegralbetweenpipeandanodeallbelongtotheintegralbetweendif-ferentobjects.Inordertosimplifythecalculation,thecoordinatetransformation

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118 FigureA-1:Integralbetweendifferentobjects. betweentwodifferentlocalcoordinatesneedstobedone.Forexample,tointe-gratefromcollocationpointP,whichisnotbelongtoobjectA,toanypointontheobjectA,P'scoordinatehastobechangedtoobjectA'slocalcoordinateSeeFigure A-1 .Thus,P'scoordinatebecomesxAP;yAP;zAPinA'scoordinate.TheequationstoobtainmatricesGandHareinducedbelow.A.2.1IntegraloverCylindricalElementsWhenthecollocationpointsareonthecylindersidearea,thecoordinateofPaboutobjectA'slocalcoordinatecanbesimplywrittenasx;y;zP.TheintegrationpointQisRcos;Rsin;zQ,whichisonthesurfaceofobjectA.jrPQj=rPQ=q Rcos)]TJ/F48 11.955 Tf 12.318 0 Td[(x2+Rsin)]TJ/F48 11.955 Tf 12.641 0 Td[(y2+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2=q )]TJ/F19 11.955 Tf 9.671 0 Td[(2Rxcos+ysin+[x2+y2+R2+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2] A-14

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119 Substitutingequation A-14 intoequation 4-43 ,GijbecomesGij=ljR 4Z102k=1qjkNktdtZ201 p )]TJ/F19 11.955 Tf 9.672 0 Td[(2Rxcos+ysin+[x2+y2+R2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2]d A-15 Equation A-15 canbearrangedasGij=ljR Kk q p x2+y2+R2+zJ)]TJ/F48 11.955 Tf 12.33 0 Td[(zI2Z102K=1qJKNKtdtA-16WhereKkistherstkindofcompleteellipticintegral.FormatrixH,substitutingequation A-14 intoequation 4-44 ,wegetHij=ljR 4Z102k=1ujkNktdtZ20@1 rij @eRdA-17Equation A-17 canbearrangedasHij=ljR f[Kk)]TJ/F17 11.955 Tf 11.997 0 Td[(Pk] )]TJ/F19 11.955 Tf 9.672 0 Td[(2Rq p x2+y2+R2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2)]TJ/F17 11.955 Tf 53.49 8.47 Td[(Pkp x2+y2+R [q p x2+y2+R2+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2]3gZ102k=1ujkNktdt A-18 WherePkisthethirdkindofcompleteellipticintegral.A.2.2IntegraloverCircleElementsTointegrateovercircleelements,P'scoordinatewithrespecttoobjectAisstillx;y;zP.ThecoordinateofcollocationpointQisrcos;rsin;zQ.jrPQj=rPQ=q rcos)]TJ/F48 11.955 Tf 12.319 0 Td[(x2+rsin)]TJ/F48 11.955 Tf 12.642 0 Td[(y2+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2=q x2+y2+r2)]TJ/F19 11.955 Tf 11.996 0 Td[(2rxcos+ysin+zQ)]TJ/F48 11.955 Tf 12.331 0 Td[(zP2 A-19 Substitutingequation A-19 intoequation 4-45 ,GijbecomesGij=qj1 4ZR0Z20rdrd p x2+y2+r2)]TJ/F19 11.955 Tf 11.997 0 Td[(2rxcos+ysin+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2 A-20

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120 Substitutingequation A-19 intoequation 4-46 ,HijbecomesHij=uj1 4ZR0Z20@1 p x2+y2+r2)]TJ/F19 7.97 Tf 6.448 0 Td[(2rxcos+ysin+zj)]TJ/F48 7.97 Tf 6.671 0 Td[(zi2 @ezrdrd=uj[zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi 4]ZR0Z20rdrd [p x2+y2+r2)]TJ/F19 11.955 Tf 11.996 0 Td[(2rxcos+ysin+zj)]TJ/F48 11.955 Tf 12.331 0 Td[(zi2]3 A-21

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APPENDIXBCODESTRUCTUREThecodeinpresentworkwerecreatedandcompiledbyusingVisualC++6.0.Objectorientationprogrammingmethodwasused.ThecodestructurefortheforwardmodelandtheinversemodelareshowninFigure B-1 and B-2 respectively.B.1CodeStructureoftheForwardModelB.2CodeStructureoftheInverseModel 121

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122 FigureB-1:Codestructureoftheforwardmodel.

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123 FigureB-2:Codestructureoftheinversemodel.

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APPENDIXCINTERFACECurrently,theinterfacewasmadefortheforwardmodel.Twokindsofwin-dowswerecreated.Onewasfortheinputtingconditions,theotherisforout-puttingresults.C.1InputWindowsThepropertiesofpipeline,anodeandcoatingconditionsareinputthroughthethreewindows,asshowninFigures C-1 C-2 and C-3 .C.2OutputWindowsIntheoutputwindow,theguresofpotential,coatingresistivity,andcurrentdensitycanallbeshownbypressingthecorrespondingbuttons.SeeFigure C-4 124

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125 FigureC-1:Windowforpipeproperties. FigureC-2:Windowforanodeproperties.

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126 FigureC-3:Windowforcoatingproperties. FigureC-4:Windowforgures.

PAGE 141

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BIOGRAPHICALSKETCHChenchenQiuwasbornonDecember20,1971,inTianjin,China.In1994,shegraduatedwithaBachelorofEngineeringdegreeinchemicalengineeringfromTianjinUniversity,Tianjin,China.FromSeptember1994toMarch1997,shewaswiththeChemicalEngineeringResearchCenteratTianjinUniversity,wheresheperformedresearchonbio-materialseparationandreceivedherMasterofEngi-neeringdegree.ShethenjoinedtheBeijingPetro-chemicalEngineeringCorpora-tioninBeijing,China,wheresheworkedasaprocessengineerforoneandahalfyears.Inpursuitofahighqualityeducation,shecametotheUniversityofFloridainMay1999andjoinedtheDepartmentofChemicalEngineeringwithProfessorMarkE.Orazemasheradvisor.Sheworkedonthestudyoftheinterpretationofsurveydataofthecathodicprotectedburiedundergroundpipeline. 134


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MODEL FOR INTERPRETATION OF PIPELINE SURVEY DATA


By

CHENCHEN QIU















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2003














ACKNOWLEDGMENTS

I would like to express my sincerest gratitude and appreciation to Dr. Mark

E. Orazem for his support, inspiration and guidance in directing this study. His

hard work and attention to detail were an excellent example during my studies.

He always helped me to learn more and taught me the importance of thinking

creatively. Many of the advances achieved during my graduate study would not

have been possible without his support. I would also like to gratefully acknowl-

edge Dr. Loc Vu Quoc for teaching me the boundary element method at the early

stage of this work. Moreover, I wish to thank Dr. Anthony J. Ladd, Dr. Oscar

D. Crisalle, and Dr. Darryl Butt for their time, useful discussions and guidance

as members of the supervisory committee. I would like to acknowledge the fi-

nancial support of the Pipeline Research Council, International and Gas Research

Institute. My colleagues, Douglas P. Riemer, Kerry Allahar, Nelliann Perez-Garcia,

Pavan Shukla, who have contributed to this research by their valuable discussions

and friendship, are also gratefully acknowledged. Finally, I would like to thank

my husband, Kuide Qin, for his unselfish support and understanding throughout

my doctorate work, which would not have been possible without him. I would

also like to thank my parents and my sister for their encouragement and inspira-

tion. Even though they were half a planet away from me, they have always been

with me in my heart.
















TABLE OF CONTENTS


page


ACKNOWLEDGMENTS ............................


LIST OF TABLES ................................


LIST OF FIGURES ...............................


ABSTRACT ..................


N O TATIO N . . . . . . . . .


CHAPTER

1 INTRODUCTION .............................


2 PIPELINE CORROSION, PROTECTION AND MEASUREMENTS .

2.1 Introduction . . . . . . . .
2.2 Basic Concepts of Corrosion .......................
2.3 Corrosion Protection ...........................
2.3.1 C oating . . . . . . . .
2.3.2 Cathodic Protection .......................
2.4 Cathodic Protection Criteria .......................
2.4.1 -850mV Potential Criterion ...................
2.4.2 100mV Potential Criterion ...................
2.5 Pipeline Measurements ..........................
2.5.1 Potential Survey .........................
2.5.2 Line Current Survey .......................
2.5.3 Other Survey Techniques ...................
2.6 Conclusions . . . . . . . .


3 APPLICATION OF TWO-DIMENSIONAL FORWARD AND
MODELS TO CORROSION ...................


3.1 Introduction . ..
3.2 Thin Plate Method . .
3.2.1 Introduction . .


INVERSE
. . 16

. . 16
. . 17
. . 17


. . . .
. . . .
. . . .









3.2.2 Evaluation Process .................. ....... 18
3.2.3 Summary ....... ...... ... ....... 20
3.3 Two-Dimensional Boundary Element Method ............. 20
3.3.1 Introduction .......... ......... ...... 20
3.3.2 Evaluation Process .................. ....... 22
3.3.3 Two-Dimensional Problem .... . . . 22
3.4 Inverse Analysis in Two Dimensions . . . 24
3.4.1 Objective Function .................. ....... 24
3.4.2 Regression Method Analysis . . . 25
3.4.3 Downhill Simplex Method .................. ..25
3.4.4 Accuracy of the Parameters ........ ......... 26
3.4.5 Regression Results .................. ..... 28
3.5 Conclusions .. ........................... 29

4 DEVELOPMENT OF THREE-DIMENSIONAL FORWARD AND INVERSE
MODELS FOR PIPELINE WITH POTENTIAL SURVEY DATA ONLY .30

4.1 Introduction .............. ............... 30
4.2 Construction of the Forward Model .............. 33
4.2.1 Cathodic Protection System .......... ....... 33
4.2.2 Governing Equation and Boundary Conditions ...... 34
4.2.3 Theoretical development ......... ....... .. 35
4.3 Three-Dimensional Boundary Element Method ............ 38
4.3.1 Infinity Domain ........... .............. 38
4.3.2 Half-Infinity Domain .......... ....... ...... 41
4.3.3 Boundary Discretization ....... .... .......... 44
4.3.4 Coordinates Definition and Transformation ......... 45
4.3.5 Discretization of Boundary Element Method ........ 46
4.3.6 Row Sum Elimination ........... ...... ...... 49
4.3.7 Self-Equilibrium ........... ............. 50
4.4 Forward Model ................... ........ 52
4.4.1 Constant Steel Potential Assumption .............. 52
4.4.2 Simulation Results ........... ...... ....... 54
4.5 Inverse Model ............................... 54
4.5.1 Objective Function ........... ...... ....... 55
4.5.2 Analysis of Regression Methods ........ ...... 55
4.5.3 Simulated Annealing Method ....... ........ 56
4.5.4 Simulation Results and Discussions .............. 57
4.5.5 Inverse Strategies ........... ............. 62
4.6 Conclusions .............................. 70

5 DEVELOPMENT OF THREE-DIMENSIONAL FORWARD AND INVERSE
MODELS FOR PIPELINE WITH BOTH POTENTIAL AND CURRENT
DENSITY SURVEY DATA ......................... 72

5.1 Forward Model .............................. 72
5.1.1 Introduction ........ ......... ......... 72
5.1.2 Pipeline with Varying Steel Potential .............. 72









5.1.3 Simulation Results ................... ....... .. 82
5.1.4 Analysis of the Simulation Results . . ..... 86
5.1.5 Validation with CP3D ............... . 87
5.2 Inverse Model ........... .... ......... 91
5.2.1 Introduction ........... .... ....... 91
5.2.2 Regression to Noise-Free Data . . . 93
5.2.3 Regression to Noisy Data ... . . . 99
5.3 Conclusions ............. . . ... 102

6 APPLICATION OF OFF-POTENTIAL DATA TO THE INVERSE MODEL 104

6.1 Forward Model-Calculation of the Soil Surface Off-Potential . 104
6.1.1 Physical Process .......... ....... ....... 104
6.1.2 Assumptions ........... ............... 105
6.1.3 Model Implementation ... . . .105
6.1.4 Simulation Results and Analysis . . . ... 106
6.2 Inverse Model-Using Three Kinds of Heterogeneous Data Sets 107
6.2.1 Data without Noise .......... .......... ... 107
6.2.2 Data with Added Noise ................ 109
6.3 Conclusions .......... .......... .......... .. 112

7 SUMMARY AND FUTURE WORK . . . .. ... 113

7.1 Summary ........ ........... ...... ........ 113
7.2 Future Work ................................ 114

APPENDIX

A BOUNDARY INTEGRAL ......................... 115

A.1 Integral along One Object .......... ....... ........ 115
A.1.1 Integral over Cylindrical Elements ......... ...... 115
A.1.2 Integral over Circle Elements ............. 116
A.2 Integral between Different Objects ............... 117
A.2.1 Integral over Cylindrical Elements ......... ...... 118
A.2.2 Integral over Circle Elements ............. 119

B CODESTRUCTURE ................... ....... 121

B.1 Code Structure of the Forward Model ............... 121
B.2 Code Structure of the Inverse Model ................. 121

C INTERFACE ... ............... .. ......... 124

C.1 Input Windows .............................. 124
C.2 Output Windows ................... ......... 124
REFERENCES .................................. 127











BIOGRAPHICAL SKETCH ........................... 134















LIST OF TABLES


Table page

3.1 Regression result for two-dimensional inverse model described by
Aoki et al. The underlined symbols represent the parameters esti-
mated by regression. .................. ....... .. 28

4.1 Parameter values obtained using the three-dimensional inverse model
developed in the present work for a 10m pipe segment with two
coating defects. .................. ........... .. 60

4.2 Test case parameters with five coating defects on the pipe used to
demonstrate the method for determination of the number of statis-
tically significant parameters (see Figure 4-15). The intact coating
resistivity had a value of 5.0 x 107 / m. . . . 62

4.3 Regression results from the three-dimensional inverse model for a
100m pipe segment with three coating defects. The sequential re-
gression procedure was used to identify the number of defects that
were statistically significant. The intact coating resistivity had a
value of 5.0 x 107 / Qm. ................. ..... .. 64

5.1 Regression results using either equation (5-19) or (5-18) for homoge-
neous synthetic data without added noise or equation (5-20) for het-
erogeneous synthetic data without added noise. The initial values
for each defect was Xk = 50 m, pk = -3.5 x 107 Qm, and ok = 0.92 m. 95

5.2 Regression results using equation (5-24) for heterogeneous synthetic
data with added noise. The initial values for each defect was Xk =
50 m, pk = -3.5 x 107 Qm, and ak = 0.92 m. . . .... 101

6.1 Regression results from the data without added noise obtained by
using the general form of the objective function (5-25), (5-26) and
(5-27). The initial values for each defect was Xk = 50 m, pk = -3.5 x
107 Qm, and ok = 0.92 m. ................ ..... .. 108

6.2 Regression results from the noise-added data by using general form
of the objective function (5-25), (5-26) and (5-27). The initial values
for each defect was Xk = 50 m, pk = -3.5 x 107 Qm, and ok = 0.92 m. 111















LIST OF FIGURES


Figure page

2-1 Polarization curve of the cathodic protection. .............. 8

2-2 Sacrificial electrode CP ............... ........ 9

2-3 Impressed current CP ............... ........ 9

2-4 Potential survey method. ............. .. ..... 12

2-5 Pipe-to-soil potential survey method and importance of the half-cell
location. ................. ...... ......... 13

2-6 Line current survey. .............. . . .... 14

3-1 Finite difference method. ................ ........ 19

3-2 Comparisons of the forward and inverse results of TPA method. .. 20

3-3 Schematic representation of the two-dimensional problem. . 22

3-4 Reproducing results for the two-dimensional forward model. . 23

3-5 Schematic representation of the two-dimensional inverse model. .. 24

3-6 Schematic representation of the downhill simplex method. . 26

4-1 Schematic illustration of the pipe segment and anode used to test
the inverse model. ............. . . ...... 33

4-2 Linear relationship between potential drop and current density over
the pipe coating .................. ............ 36

4-3 Schematic illustration of the pipe surface resistivity model. . 38

4-4 Interior problem. .................. .......... .. 39

4-5 Multi connected region of interior problem. . . ... 40









4-6 Schematic illustration of mirror reflection technique. . ... 43

4-7 The Fundamental solution to the half-infinity domain satisfying the
Neuman b.c's. ............... . . ..... 43

4-8 Pipe discretision and collocation points. . . . 44

4-9 Coordinate rotation .... ........... ....... .. 46

4-10 False color image of the on-potential on the soil surface that was
generated by the forward model corresponding to Figure 4-1. . 55

4-11 Flow chart for the inverse model calculations. . . .... 58

4-12 Grid showing the location of 303 surface on-potentials calculated
using the three-dimensionalforward model developed in the present
work. The grid shown is for a 10m pipe segment. A scaled version
of the grid was used for a 100m pipe segment. . . ... 59

4-13 The regression objective function as a function of the number of
evaluations for a pipe coating with one defect region. The simu-
lated annealing method was used for this regression. . ... 60

4-14 Comparison of the set and fitted results for pipe coating with two
coating defects. .................. ........... .. 61

4-15 The regression statistic as a function of the number of coating de-
fects assumed for the model. The minimum in this value is used to
identify the maximum number of coating defects that can be justi-
fied on statistical grounds. ................... ...... 63

4-16 Comparison between the input values and regression results for
noise = 0.1 mV. ................ ..... .. .. .. .. 66

4-17 Comparison between the input values and regression results for
noise = 1.0 mV. ........... . . ...... 67

4-18 Comparison between the input values and regression results for
noise = 2.0 mV. ........... . . ....... 69

5-1 Cathodic protection system with variant potential along the pipe
and anode. ................. . . .... 73

5-2 Relation of potential V and current density. . . ... 74

5-3 Axial direction current flow along the pipeline. . . .. 75









5-4 Axial direction current flow along the anode. . . .... 76

5-5 Potential and current density along the anode. . . ... 83

5-6 Simulated axial distributions along the pipeline. . . ... 85

5-7 Potential of pipe calculated by using CP3D .. . . ... 88

5-8 Potential of pipe steel calculated by using CP3D . . .... 89

5-9 Axial direction current density of pipe calculated by using CP3D 90

5-10 Radial direction current density of anode calculated by using CP3D 90

5-11 Schematic illustration of the coupling of a soil-surface-level poten-
tial survey method to an aerial magnetometer current survey to as-
sess the condition of a buried pipeline. . . . ..... 92

5-12 Synthetic pipe current and surface potential data corresponding to
Table 5.1. Dashed lines indicate the location of coating defects. 93

5-13 The X2/ statistic corresponding to Table 5.1 as a function of the
number of coating defects assumed in the regression. . ... 96

5-14 The X2/ statistic corresponding to regression of equation (5-21)
(see Table 5.1) as a function of the number of coating defects as-
sumed in the regression. ................. ..... .. 98

5-15 Synthetic data with added noise. The line represents the noise-free
values and the symbols represent synthetic data with added noise.
a) surface on-potential value with = 0.4 mV; b) axial current den-
sity value with proportional noise corresponding to 2 percent of the
value ................. ........ .......... 100

6-1 Switch off CP power source to determine instant off-potential. 105

6-2 On-potential and off-potential on soil surface. . . ... 106

6-3 Typical close-interval potential graph. . . . 107

6-4 Decide the number of defects using noise free soil surface on and
off-potential and current density data. . . . 109

6-5 Set value and noise-added off-potential on soil surface data. . 110









6-6 Decide the number of defects using noise-added on and off-potential
and current density data. .................. ....... 111

A-1 Integral between different objects. .. . . . 118

B-1 Code structure of the forward model. ..... . . . 122

B-2 Code structure of the inverse model. .... . . . 123

C-1 Window for pipe properties. ................... .. 125

C-2 Window for anode properties ............. . 125

C-3 Window for coating properties ...... . . . 126

C-4 Window for figures .................. . .. 126














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


MODEL FOR INTERPRETATION OF PIPELINE SURVEY DATA
By

Chenchen Qiu

December 2003


Chair: Mark E. Orazem
Major Department: Chemical Engineering


Pipeline corrosion annually costs the U.S. seven billions of dollars. Ninety

percent of the cost is the capital, operation and maintenance. It is desirable to

optimize this expenditure while ensuring that the integrity of the pipeline, which

represents a valuable asset, is maintained. Regular inspection of pipelines using

a variety of survey techniques is routinely conducted, but interpretation of the

results is confounded by stochastic and systematic errors. The objective of the

present work was to investigate use of inverse model that could interpret survey

data in the context of the physics of the system.

An inverse analysis model was developed which provides a mathematical

framework for interpretation of survey data in the presence of random noise. A

boundary-element forward model was coupled with a weighted nonlinear regres-

sion algorithm to obtain pipe surface properties from two types of survey data:

soil-surface potentials and local values of current flowing through the pipe. The

forward model accounted for the passage of current through a three-dimensional









homogeneous medium and yielded soil surface potentials for given pipe/anode

configurations and pipe coating properties. The number of regressed parameters

was reduced by using a function for coating resistivity that allowed specification

of coating defects. A weighted simulate-annealing nonlinear regression algorithm

facilitated analysis of noisy data. A method to determine the appropriate number

of fitted parameters was developed.

The model was demonstrated for synthetic data generated for a section of a

coated underground pipeline electrically connected to a vertical sacrificial anode.

The success of the regression was sensitive to the relative weighting applied in

the objective function to the respective types of data. A generalized weighted and

scaled objective function was proposed.














NOTATION

Potential and potential distribution, V

4 Potential and potential distribution refer to the reference electrode, defined

as V V


Pk Resistance reduction of the coating defect k, Q.m

Jk The half width of the coating defect k, m

if Normal vector of boundary

Cj Concentration of species j, mol/cm3

Di Diffusion coefficient of species j, cm2/s

F Faraday's constant, 96487C/equiv

i Current density, A/m2

Uj Mobility of species j, cm2 mol/J s

V Voltage distribution of pipe steel or anode metal, V

v Bulk velocity, cm/s

Xk Center point of the coating defects k, m

zi Charge number of species j


p Resistivity of the pipe coating, Q.m














CHAPTER 1
INTRODUCTION

In the U.S., there are 779,000 km (484,000 mi) of gas and liquid transmission

pipelines. Seven billion of dollars is spent due to pipeline corrosion every year.

90% of the cost is the capital, operation and maintenance. It is desirable to op-

timize this expenditure while ensuring that the integrity of the pipeline, which

represents a valuable asset, is maintained. Regular inspection of pipelines using

a variety of survey techniques is routinely conducted, but interpretation of the

results is confounded by stochastic and systematic errors.

The goal of this work was to develop a computer program that can be used to

extract the condition of the pipe taking into account a wide variety of field data,

such as potential and line current survey data. In addition, the intention of this

preliminary effort was to establish a proof of concept which can motivate future

development.

The pipeline protection techniques that are used most often are cathodic pro-

tection and coating. The normal use of computer programs for modeling of ca-

thodic protection is to assess whether a pipeline with an assumed coating condi-

tion can be protected by a given cathodic protection design. In this strategy, called

a forward solution, all physical properties of the pipe, anodes, ground beds, and the

soil are assumed, and the corresponding distribution of current and potential on

the pipe is calculated. Over-protected and under-protected sections of the pipeline

can be identified in this way. Such programs can also be used to calculate the po-

tential and the current density at other locations such as at the soil surface or at






2

the specified locations where coupons may be buried. This strategy lends itself

well to answering "what if" questions. For example, "what will happen if discrete

coating holidays expose five percent of the pipe surface?", or "what will happen

if a new pipe is constructed in the right-of-way?"

The forward strategy is not appropriate, however, for the interpretation of field

data to assess whether a given pipe is protected. The concept behind the approach

developed in this work is to solve the inverse problem in which the properties of

the pipe coating are inferred from measurements of the current and the potential

distributions. This approach would allow interpretation of field data in a manner

that would take into account the physical laws that constrain the flow of electrical

current from anode to pipe.

In addition, the development of the inverse model can not proceed alone. Only

when the forward model is well developed can the inverse model be able to inter-

pret multiple kinds of survey data.

The basic concepts of the corrosion, corrosion protection methods, and pipeline

survey methods are presented in Chapter 2.

Verification of published methods for model two-dimensional systems are pre-

sented in Chapter 3. This chapter includes the thin-plate method and the bound-

ary element method to solve the two-dimensional Laplace's equation. The two-

-dimensional inverse model is also established in this chapter. This work involved

the construction of the objective function, the formalism of regression strategies,

and regression methods analysis.

The development of the forward and the inverse models for two-dimensional

systems provided insights into the methods needed to address the inverse prob-

lem for pipelines. However, the two-dimensional forward model cannot describe

precisely the geometry of the pipeline cathodic protection system. In addition, the






3

more sophisticated three-dimensional forward models such as CP3D,1 PROCAT,2

and OKAPPI3 are not appropriate, since they require too much computation work

for the inverse model. A simplified version of a forward is presented in Chapter 4.

Both the forward model and inverse model are applied to soil surface potential sur-

vey data. For the forward part, a three-dimensional boundary element method

with cylindrical elements was developed in this work. The simulated annealing

method was shown to be ideally suitable for the three-dimensional inverse model.

In addition, several strategies were explored to assess the confidence level of the

inverse model results.

The forward model presented in Chapter 4 provided the foundation for an ex-

tended model, presented in Chapter 5, which accounted for the potential drop

along the pipe steel. Both the potential on soil surface and the current density

along the pipeline could be obtained from the forward model. An inverse model

was created to apply these data. A new general form of the objective function

was established by adding different weighting factors for each data point and by

including the number of the data points.

Chapter 6 verifies the general form of the objective function built in Chapter

5. Off-potential on the soil surface was simulated in the forward model. Therefore,

three kinds of data sets, i.e., on-potential, off-potential and current density data,

were used for the inverse model.

Summary of the study and future work are presented in Chapter 7.














CHAPTER 2
PIPELINE CORROSION, PROTECTION AND MEASUREMENTS

2.1 Introduction

The corrosion of metallic structures has significant impact on the U.S. economy

in many fields. In 1975, Battelle-NBS's benchmark study estimated that the cost

of corrosion was $70 billion per year, 4.2 percent of the nation's gross national

product (GNP). If the effective and presently available corrosion technology can

be applied, $10 billion of this cost could be avoided. From 1999 to 2001, the Federal

Highway Administration (FHWA) indicated that the total direct cost of corrosion

was about $276 billion per year, 3.1 percent of the U.S. gross domestic product

(GDP). The indirect costs is unpredictable .4

Pipeline-related corrosion costs approximately $7.0 billion annually. There are

528,000 km (328,000 mi) of natural gas transmission pipelines, 119,000 km (74,000

mi) of crude oil transmission pipelines and 132,000 km (82,000 mi) of hazardous

liquid transmission pipelines in the U.S.4

In the past few years, a number of gas and liquid pipeline failures have drawn

people's attention to the pipeline safety. In order to preserve the asset of the

pipeline and to avoid those failures which may jeopardize public safety, result

in product loss, or cause property and environmental damage, some new regula-

tions were established. Regular pipeline inspections, such as hydrostatic testing,

direct assessment, and in-line inspection (ILI), are required. Furthermore, corro-

sion prediction models need to be developed in order to determine and prioritize

the most effective corrosion preventive strategies.









Development of an understanding of cathodic protection of buried structures

such as pipelines requires an understanding of the electrochemical reactions asso-

ciated with corrosion. This section provides an overview of the principles of corro-

sion, methods for cathodic protection, and the criteria used to assess the effective-

ness of corrosion-prevention strategies. Moreover, the pipeline survey methods

are included.

2.2 Basic Concepts of Corrosion

Corrosion takes place in response to the tendency to reduce the overall free en-

ergy of a system. For example, when metal is in a dilute aerated neutral electrolyte

atmosphere, moisture films, which contain oxygen and water, usually cover the

metal surface. Some atoms in the metal surface tend to give out electrons and be-

come ions in the moisture layer. In this way, they are in a lower energy state in

solution than when they are in the lattice of the solid metal. At the same time the

metal surface builds up a huge excess negative charge. These electrons move out

from the metal and attach themselves to protons (H+ ions) and molecules oxygen

(02). The process repeats itself on various parts of the surface. Then the metal

dissolves away as ions.5 The primary constituent of pipeline-grade steels is iron.

Therefore the overall corrosion reaction can be written in terms of dissolution of

iron, e.g.,

2Fe + 02 + 2H20 -- 2Fe(OH)2 (2-1)

Reaction (2-1) can be considered as the result of two half-cell reactions. One is

Fe Fe2+ + 2e (2-2)

This is an oxidation reaction with an increase of oxidation state for iron from 0 to

2+. It is called an anodic reaction. The other is


2H20 + 02 + 4e 40H


(2-3)









This reaction is a reduction reaction with a decrease of oxidation state for 02 to

OH It is defined to be a cathodic reaction.

The change in free energy associated with reaction (2-1) can be expressed in

terms of a cell potential as

AG = -nFAE (2-4)

where n is the number of electrons exchanged in the reaction, F is Faraday's con-

stant, 96487 coulombs/equivalent,6 and E is the electrochemical potential. Ac-

cording to the Nernst equation, obtained by neglecting both activity coefficients

and liquid-junction potential,7'8

RT RT
AE = AE ln(H(a~')) = AEo ln(H(c~')) (2-5)
nEE

where EO is the equilibrium potential, and ai is the activity of the species i, si is the

stoichiometric coefficient of species i, and ci is the concentration of the species i.

For reaction (2-2), the potential in Nernst equation form is

RT
E, = Eoe2+Fe 2 n ([Fe2+]) (2-6)

Likewise, for reaction (2-3), the potential in Nernst equation form is

RT In [OH ]4 (2-7)
Ec = E2/OH- n P2

where a means anodic reaction and c refers to the cathodic reaction. To the whole

reaction, where n, = 2 and nc = 4,


AG = 2AGa + AGc = -2naFEa ncFEc = -4F(E, + Ec) (2-8)


For the two half-cell reactions, E, and Ec are positive. Hence, the free energy of

the overall reaction must be negative, which means the corrosion reaction occurs

spontaneously.









2.3 Corrosion Protection

To control the corrosion situations, several methods have been proved effec-

tive, which include coating and cathodic protection.

2.3.1 Coating

Electrically insulating materials can be use to cover the pipe surface and thereby

isolate the pipe metal from contact with the surrounding electrolyte. In this way,

a high electrical resistance in the anode-cathode circuit is added. Theoretically,

since no significant corrosion current flows from the anode to the cathode, no cor-

rosion would occur. Many kinds of materials have been applied as coatings, such

as enamels, tapes, and plastic coating. Recent improvements in pipeline coating

materials will also reduce the risk of a corrosion-related failure. However, a recent

survey of major pipeline companies indicated that about 30% of the primary loss

of the pipeline protection was due to coating deterioration.4 Therefore, coating

alone can not provide full protection for the pipeline. Practically, effective pipeline

corrosion control comprises use of good coatings along with cathodic protection

as a secondary defense.

2.3.2 Cathodic Protection

In 1824, Sir Humphrey Davy successfully protected copper against corrosion

from seawater by using iron anodes. It was the first application of cathodic pro-

tection (CP) and, at that time, it had no theoretical foundation. From that begin-

ning, CP has been applied to marine and underground structures, water storage

tanks, pipelines, oil platform supports, reinforcing steel and many other facilities

exposed to a corrosive environment. By now, the theory of CP is well established.

Natural gas and oil companies have already been using CP for economic, as well

as safety reasons, since the 1930's. The Pipeline Safety Act of 1972 made the appli-










-0.4
Fe Fe2+ + 2e-
-0.6


-0.8

H20 +2e- H2+20H-
S-1.0
{.. O_02+H20+4e ->40H
-1.2
0.01 0.1 1 10 100
Current Density, mA/ft2

Figure 2-1: Polarization curve of the cathodic protection.

cation of CP on pipelines transporting hazardous material mandatory for safety

concerns.

Cathodic protection results from cathodic polarization of a corroding metal

surface to reduce the corrosion rate. In this system, the corrosion potential reduces

the rate of the half-cell reaction (2-2) with an excess of electrons, which drives

the equilibrium from right to left. The excess of electrons also increases the rate

of oxygen reduction and OH production by reaction (2-3) in a similar manner

during cathodic polarization. Cathodic protection reduces the corrosion rate of a

metallic structure by reducing its corrosion potential, bringing the metal closer to

an immune state, which is represented by the black curve in Figure 2-1.

Two methods are usually used to achieve this goal.6 There include sacrificial

anode and impressed current.

Sacrificial Anode. One method to provide CP is to connect a sacrificial metal

(with a higher natural electromotive force) through a metallic conductor or a wire

to the structure intended to be protected (as Figure 2-2). Magnesium is a common

sacrificial or galvanic anode. This type of galvanic cathodic protection relies on the









Ground level

Earth
Current environment

*Pipe 4-- Mg
--anode


Figure 2-2: Sacrificial electrode CP.
i -------------
rrenti r Current
F-- f ------


SCurrent I I




Figure 2-3: Impressed current CP.

natural electrical potential between the two metals to cause the cathodic protection

current to flow. Since the driving voltage is limited to the very small potential

difference existing between the metals, and the current output is relatively low,

this type of cathodic protection is normally associated with very small or very

well coated structures.

Impressed-Current. The second common method to provide CP involves use of

impressed-current. It relies on an external direct current source such as a rectifier

or battery (as Figure 2-3). An anode material, placed in the electrolyte with the

protected structure, is made more positive than the structure by connecting both

the anode and the structure to the direct current supply. Any conductive materials

can be utilized as an impressed current anode, but since corrosion takes place at

the anode, materials with very low consumption rates are most desirable.









2.4 Cathodic Protection Criteria

As the cathodic protection is widely used, the reliable criteria need to be devel-

oped to direct the cathodic protection systems to an optimum level. The criteria

are based on the potential differences between the pipeline and its environment.

Two criteria will be introduced in this section, -850mV and 100mV potential crete-

ria.

2.4.1 -850mV Potential Criterion

The criteria for protecting steel pipe, RP-10-69, was established by NACE in

1969. It was revised later in 1972, 1983 and again in 1996. It defines a negative po-

tential at least 850mV with respect to a saturated copper/copper sulfate reference

electrode(CSE) when CP is applied. This criterion is mostly used since the poten-

tial measurements with current applied requires minimum equipment, personnel

and takes less time to obtain in the field survey.

Normally, the desired potential range of the cathodic protection for a pipeline

is between -850mV and -1200mV measured with respect to a Cu/CuSO4 refer-

ence electrode located at the surface of the steel or coating defects.9 If excessive

amounts of cathodic protection are applied, the direct reduction of water becomes

thermodynamically possible, e.g.,


2H20 + 2e -, H2 + 20H (2-9)


Hydrogen is evolved upon the metal, and, at large cathodic over-potentials, a

small fraction of hydrogen can enter the metal, which makes the metal brittle.

This undesirable process is called "hydrogen embrittlement". If the pipeline has

a coating, the developing hydrogen gas bubble can exert tremendous pressure.

When the pressure is created under the torn coating, a stripping action exposes

the metal and results in rapid deterioration. Obviously, if smaller amounts of CP









is applied (resulting in potential less negative than -850mV), the structure will not

be fully protected. In addition, the -850mV criterion does not directly address the

polarization. It is valid only when the effect of IR drop has been considered and

eliminated.

2.4.2 100mV Potential Criterion

The 100mV polarization criterion was first proposed by S.P. Ewing in 1951.10

The criterion means that the change in potential polarization to protect pipeline is

always less than 100mV. To utilize this criterion, the IR drop effect at measurement

area needs to be estimated. Normally, coupons are used to get the IR drop values

by measuring the potentials when the coupon is "on" and "off", and comparing

the two potentials curves to establish the potential differences between the two

curves, which is the IR drop value. Recent research in the UK has shown that the

applied current density could be reduced by 25% after three month of using the

100mV criterion and by 65% after one year, without losing any protection.11

2.5 Pipeline Measurements

Cathodic protection design should be based on actual data. Monitoring meth-

ods are necessary to investigate the corrosive conditions and to evaluate the de-

gree of the applied CP. The techniques used include measurement of potential

along the ground above the pipe, measurement of current flowing through the

pipes, and use of auxiliary electrodes, such as a reference electrode and coupon.

2.5.1 Potential Survey

Construction of a potential survey involves measuring the potential difference

between the pipe and a reference electrode at the ground surface level. The refer-

ence electrode is moved to different locations to sample many positions at ground

level near the pipe. In this way, a mapping of potential is created which can be









Voltmeter


CuS04
Electrode






Figure 2-4: Potential survey method.

used to identify regions where the pipe is grossly under-protected. The survey

involves use of a voltmeter and a reference electrode which is typically based on

the Cu/CuSO4 reaction. A schematic representation of the survey method is pre-

sented in Figure 2-4.12 Potential survey measures the potentials between buried

pipeline and environment.9 This survey needs an appropriate voltmeter with its

negative terminal connected to the pipeline and the positive terminal connected

to a reference electrode.

The procedure involves moving the reference electrode at 3 to 10 ft intervals

down the full length of the pipe and using the trailing cable to determine distance

and to connect to the pipeline. A vast quantity of data is gathered by a data logger.

From the survey readings, the worst corrosion takes place where the readings are

the highest (less negative value) and little or no corrosion should take place where

the potential is more negative.

For this measurement, the location of the reference electrode (half-cell) is im-

portant.13 The most desirable place is directly above the pipeline. If the reference

half-cell is placed 2 3 meters to the side of the pipeline, the measured potential

will be a group of scattered data as the second curve shown in Figure 2-5. In such a

case, it is difficult to make a judgement. To avoid this case, measurements on both

sides of the pipe may be required to make sure that the over the pipe potentials


















I:1


I5 Reference electrode placed 2~3 meter to side
of pipeline
+ ----------------------------------
Figure 2-5: Pipe-to-soil potential survey method and importance of the half-cell
location.
are taken as close to the pipes as possible.
An instant off-potential (IR-free) survey9'14 is used to eliminate the potential

drop through the soil in the pipe to soil potential measurement by interrupting the
flow of protective current to the pipeline. This technique can be used to determine
the effectiveness of the cathodic protection system. It is based on the principle that
the IR effect in the potential measurement decays almost instantaneously, while
the pipe-to-soil interface polarization decays relatively slowly, thus allowing the
correct pipe-to-soil potential to be measured free of the IR error.

2.5.2 Line Current Survey

The line current survey technique is used to measure the electrical current flow-
ing on the pipe. If corrosion is taking place on a pipeline, current will flow to the
line at some points and flow out of the line at others. For small local cells, the
current path may be too short to detect. For large cells, the current may follow the
pipe for hundreds or thousands of feet. It is these long line currents that can be
detected in a line current survey.










350
300 -
250
200
150
100
50
IT 0
0 1000 2000 3000 4000
Pipeline Length /feet

Figure 2-6: Line current survey.

Because the pipe itself has some resistance to the flow of electric current, there

will be a voltage drop in the pipe if current is flowing through the pipe steel.

Normally, the voltage drops are very small. Thus the span of the two test points is

often fixed at 100 feet to increase the span resistance.

Knowing the span resistance of the pipe being surveyed, the voltage drops

may be converted to equivalent current by the application of Ohm's Law:

E(potential)
I(current) = (l (2-10)
R(resistance)

The values of current together with the direction of flow then may be plotted

as function of line length.9 From Figure 2-6, it is noted that in one area current

flows from both directions toward a particular point on the line. This point must

be a place of current discharge.

2.5.3 Other Survey Techniques

The airborne cathodic monitoring systems (ACMS)15 detect upset conditions

on the pipelines protected by impressed current. Sensitive and filtered magnetic

field coils are installed on a helicopter, which continuously measure the magnetic

field generated by the ripple from an alternating current source. The principle is

based on

B = ( (2-11)
2- R

where B is the magnetic field generated by the impressed current I, R is the dis-

tance from the center of the pipe to the sensor installed on the helicopter, and the









constant 0o/27 is the permeability of the medium. Thus, if the magnetic field can

be measured along with the distance, then the current can be calculated.

In addition, use of an insulated coupon is a variation of the instant-off potential

to soil method. The coupon usually is buried close to the protected structures to

have the same characteristic of the chemical environment and electrical field as

those at the structure surface. The exposed coupon surface simulates a coating

defect. Cathodic protection current to the coupon can be interrupted without any

effect on the protected structures. In this way, the difficulties of the instant-off

potential-to-soil method can be eliminated.

2.6 Conclusions

A brief introduction of the corrosion situations was given in this chapter. The

basic concepts of corrosion were explained. The pipeline protection techniques, es-

pecially the cathodic protection technique, have been introduced. In addition, the

commonly used pipeline survey methods, such as potential survey, line current

survey, airborne cathodic monitoring systems and coupons, are also summarized.















CHAPTER 3
APPLICATION OF TWO-DIMENSIONAL FORWARD AND INVERSE MODELS

TO CORROSION

3.1 Introduction

Given the governing equation and the boundary conditions to estimate the po-

tential and current density distribution on the soil surface by solving the govern-

ing equation directly is known as the forward model. In contrast, in an inverse prob-

lem, usually a model or a governing equation and measurements of some vari-

ables are given, such as the potential and current density measurements. Bound-

ary conditions and the rest of the variables may not be known explicitly. The

purpose of the inverse model is to identify the unknown parameters using the in-

formation from the measurements. It is a type of ill-conditioned problem in which

the solution is extremely sensitive to the measured data.

In this chapter, the forward and inverse models applied in the literature are re-

viewed and reproduced. Two methods were studied, the thin plate approximation

method and the boundary element method. The experiences of developing and

evaluating these models are summarized.









3.2 Thin Plate Method

3.2.1 Introduction

Inglese16,17 solved the inverse problem for two electrode separated by a small

gap, a, as following:

V2 =0 in 0Q;

(x) = 00(x) x [0, 1], y = 0;
(3-1)
On(x) + 7(x)9(x) = f(x) x e [0,1], y = a;

n (0, y) = O(l, y)= 0 y [0,a]

where 7 is the corrosion coefficient, which represents the corrosion rate, 0 is the

potential, and ,n is the outward normal flux and a is the thickness of the do-

main. The goal was to find 7 from the potential 00 on the accessible boundary

x [0, 1],y = 0.

The author introduced the thin plate approximation (TPA) method. The idea

is to perform an expansion of the solution 7 in terms of powers of the thickness of

the domain Q. Owing to the assumption that the thickness a is much smaller than

1 (a < 1), higher order terms of the expansion can be ignored. Inglese deduced

a formula of 7 as function of ,xx(x, 0), which is the second derivative of 0 with

respect to x.
#xx(x, 0)+ So(x)
rTPA(x) = (X 0) + 0 (3-2)
((x,0)
In order to prove the accuracy of equation (3-2), the direct problem was solved for

a prescribed corrosion rate give by


7(x) = exp(-100(x 0.25)2) (3-3)

First of all, for the forward part, O(x, 0) is calculated according to the given 7(x).

After that, ,xx, Oo can be obtained without problems. For the inverse part, the









corrosion rate 7 was obtained from equation (3-2) and it can be compared with the

value of 7(x) from equation (3-3).

3.2.2 Evaluation Process

The domain of Q in x direction is [0, 1] and in y direction is [0, a]. To simplify

the problem, the y direction is normalized to [0, 1]. Thus, the decomposed form of

equations (3-1) is:

a29xx + yy = 0 in Q;

x(O, Y)= 0

x(,y) = 0 (3-4)

Oy(X, O) = -a2

Oy(x, 1) + a27(x)(x, 1) = 0

ay(x) = exp[-100(x 0.25)2]

To solve equation (3-4), the Laplace's equation with Robin boundary condi-

tions, two methods were tested. One was the separation of variables method

and the other was the finite difference method. However, the separation variable

method failed for this kind of problem because a contradiction occurs between the

result and boundary condition Oy(x, 0) = -a2 0.

To apply the finite difference method as shown in Figure 3-1, we set

o,,j (xi, yj)

xi = ih
(3-5)
yj = jh
Ax = Ay















i-1 ,j ij i+1 ,j



i,j-1


Figure 3-1: Finite difference method.


Therefore,


Oy,y

Ox(1, y)

Oy(x, 0)


' (i+l,j -


0-1,j- 1,;- j

i,i i,- 1


20ij, + i- i,j)

20i,i + -i,j+1)


Equation (3-4) becomes:

i,j-1 + a20i+1i, 2(a2 + 1)oi,j + a20i-1, + -i,j+l = 0

0-1,] 01,i = 0

N+ ,] PN-1, = 0

i,i 0i,-1 = -2ha2 0


2[- i,N-1 + Oi,N 1] = -a2/(ih)O(i, N) = -a-(ih)Oi,N i = 0, 1, ... N;

I a(x) = exp[-100(x 0.25)2]

the solution to equation (3-7) yields Ox,x(x, 0), O(x) and O(x, 0). Due to the norma

ization in y direction, correspondingly, equation (3-2) will be,

aoxx(x, 0) + O(x)
--- ( -
S (x,0)

The results are shown in Figure 3-2. Where the solid curve represents 7(x)


(3-6)


(3-7)


8)


0,1,..- N;

0, 1, ...N;

0, 1, -.. N;


al-












08
calculated-gamma
-exact-gamma
06

04

02

0 .' ... .. .' .
00 01 02 03 04 05 06 07 08 09 10


Figure 3-2: Comparisons of the forward and inverse results of TPA method.

exp[-100(x 0.25)2] and the points represent 7(x) calculated from the 0 and its

derivatives according to equation (3-8).

3.2.3 Summary

This method is only useful for two-dimension problem of a thin strip and for

three-dimension problem of a thin plate since the instability increases dramatically

with thickness a. Therefore, this method is inapplicable to the buried underground

pipeline under cathodic protection system. In addition, the corrosion parameter

7 provides an inadequate description of the corrosion problem. These limitations

show that TPA method is not suitable for this application.

3.3 Two-Dimensional Boundary Element Method

3.3.1 Introduction

Aoki et al.,18s19 solved a two-dimensional inverse problem. The nonlinear rela-

tionship between the potential and current density was assumed to be known on

the boundary of cathode as equation (3-9) and anode as equation (3-10).


q =qo{exp( exp (0C } (3-9)
3PC ( (-c ) I











q = qoa{exp (O -) exp } (3-10)

Both the 0-potential and q-current density were unknown. Aoki applied a Newton-

Raphson iterative procedure and used singular value decomposition (SVD) method

to get 0 and q. Where qoa, qoc, P,, aL, pa ca and ac are parameters. Except for

the parameter to represents the connecting position of two electrodes, four param-

eters are used for each electrode. Thus, these parameters need to be fitted in the

inverse process.

Kenji and Aoki20'21 simplified equations (3-9) and (3-10) by using an inverse

hyperbolic sine function. The polarization curves for anode and cathode turn out

to be,

f(q) = a1 sinhl q + P/ (3-11)



f(q) = a2 sinh1 q + 2 (3-12)

where a,, pi, a2, and 32 are the parameters. The number of parameters was re-

duced a factor of 2. The authors used a conjugate gradient method to minimize

the objective function (3-13)
k
g(ai, ;i, xo) = (j j)2 (i = 1, 2) (3-13)
j-1

where (j and )j represent measured potential and calculated potential respec-

tively. xo is the connecting point of two electrodes. However, the results of the fit-

ted parameters and their standard deviations were not given. In order to improve

the accuracy of the results, the authors checked the eigen values of the Hessian

matrix of the objective function at the convergence point and found that three of

the five eigen values were approaching zero. In other words, the five parameters

of the objective function actually occupied about a two-dimensional sub-domain









q=O or 4 =const


q=O q=O
Cathode Anode

q=f(f) q=fa(4)
) guess ) guess

Figure 3-3: Schematic representation of the two-dimensional problem.

instead of a five-dimensional domain. This is the reason that the results cannot be

close to the exact solutions and why it is difficult to give the accuracy of the param-

eters. In addition, an a priori technique was used in their work. A priori means the

probable values of some parameters that need to be estimated are known. In such

cases, one may want to perform a fit that takes this advance information properly

into account, neither completely freezing a parameter at a predetermined value

nor completely leaving it to be determined by the data set.

3.3.2 Evaluation Process

The procedure of evaluating the inverse model described by the literature took

place in two stages. At the first stage, a forward model for the two-dimensional

system was created, which was similar to the work of Aoki et al. 's.l1820 Based on

the forward model, the second stage involved testing the regression approaches

and comparing them.

3.3.3 Two-Dimensional Problem

The two-dimensional problem studied here is in a rectangular domain as shown

in Figure 3-3, which is by described Aoki et al. .18,20 The lower horizontal bound-

ary consists of a cathode and an anode. The cathode may be considered to play the

role of the pipeline, and the anode plays the role of a sacrificial anode. The upper











-084 *--* --* Electrode 2 -IElectrode
Potential on Boundary
Electrode Surface E Boundary
> -088 .. I
-! 1,- t
Potential at c 0 -
C -092 Ground Surface "
0 96

S-*2. -2
-1 00 I I I
00 02 04 06 08 10 00 02 04 06 08 10
Position / m Position Im


(a) (b)

Figure 3-4: Reproducing results for the two-dimensional forward model: a) Poten-
tials at the boundary of domain; b) Current density along the electrode surface.

horizontal boundary can be considered as the soil surface. Neither the potential

nor current density q are known on the electrode boundary, but the nonlinear re-

lationship between them is known. The nonlinear relationship between potential

Sand current density q is the polarization curve of the electrodes, which are given

as equations (3-9) and (3-10). The parameters are qoc = qoa = 1Am 2, 0c = 0.845V,

a = -0.985V, ac = 0.001V, a = 0.025V, )c = 0.05V and a = 0.05V. The parame-

ters a, 3 and qo are polarization parameters to be obtained by the inverse technique

described in section 3.4. The subscripts c and a represent the cathode and anode,

respectively.

Potential within the interior of the system is governed by Laplace's equation.

We followed Aoki's method, applying a Newton-Raphson iterative procedure for

the forward problem. The potential and current density distributions from thefor-

ward model are shown in Figure 3-4. As seen in Figure 3.4(a), Aokiet al. showed

that the variation of potential on the upper surface is much less than that seen on

the electrode boundary.1820 It is evident that a large distance between the elec-

trodes and the soil surface will blur the measurement data on the soil surface.









X measured potential j



i=O in=O
Cathode Anode


in=fc() in=fa)
Polarization curve is unknown

Figure 3-5: Schematic representation of the two-dimensional inverse model.

3.4 Inverse Analysis in Two Dimensions

The purpose of this portion of effort was to apply an inverse boundary ele-

ment method to develop an efficient approach for identification of the polariza-

tion curve in a cathodic protection system. Different regression methods will be

explored in this section.

3.4.1 Objective Function

An objective function, which describes the difference between the measured

potential and calculated potential on the soil surface, was constructed as following


k
g(ai, A;, xo)= (j __ j)2 (i = 1, 2) (3-14)
j-1
where (j and Oj represent measured and calculated soil surface potentials, respec-

tively. For the two-dimensional inverse analysis, Aoki et al.18'20 described it in

Figure 3-5. The parameters of the polarization curves were not known. What was

known is the measured potential on the soil surface. The goal of the inverse analy-

sis is to identify the parameters of the polarization curves that would provide the

minimum value for the objective function.









3.4.2 Regression Method Analysis

We tested several regression methods, including the Le';ilderg-MA lirquart, conju-

gate gradient and quasi-Newton methods. These methods, distinguished by the need

to evaluate both the function value and the function gradient, were rejected for the

pipeline inverse model. For the present problem, evaluation of the function gradi-

ent can only use finite-difference method, which requires extra calculations of the

forward model. It inevitably adds significant work to the computation. Besides,

the most serious deficiency was that these methods were found to be extremely

sensitive to the value of an initial guess and to have a tendency to find local rather

than global minima. The downhill simplex method requires evaluation only of the

objective function itself, i.e., the derivative of the objective function is not needed.

As the results of this work, it is shown that the downhill simplex method is the

most robust one, independent of the measurement position and the initial guesses.

It was successful for our two-dimensional inverse analysis and will be further de-

veloped for the three-dimensional problems.

3.4.3 Downhill Simplex Method

The downhill simplex method was first created by Nelder and Mead.22 Only

the objective function needs to be evaluated, no such need to its derivative. An

objective function, which has N number of adjustable parameters, is evaluated for

N + 1 points. The N + 1 points are generated by defining a starting point Po, then

other points are obtained by using


Pi = Po + Aiei (3-15)


where ei are N unit vectors, Ai are constants set according to the problem's charac-

teristic length scale at different directions. The geometrical figure including N + 1

vertices, all the interconnecting line segments and polygonal faces is called a 'sim-










Reflection


high val reflection and
expansion




high value


contraction 4


high value



/ multiple
contraction


high value


Figure 3-6: Schematic representation of the downhill simplex method.

plex'. According to the function values on N + 1 points, by using steps, such as re-

flection, reflection and expansion, contraction and multiple contraction, as shown

in Figure 3-6, one can find a new point to continue the function evaluation until

the objective function encounters a minimum.

3.4.4 Accuracy of the Parameters

One of the important issues of the regression process is how to determine the

accuracy of the parameters. For a x2 merit function

N 9i iYi2
x2(ak) = Y (3-16)
i=1 i

ak represents the parameters. gi and yi are the model and the measurement values

respectively. ,6 is the standard deviation for the measurement data point yi. In this

section, we are going to discuss the uncertainties of the parameters ak.









The variance and the covariance of parameter ak are given as

i Dai dak
0a2 ( ak j,k = 12,... M (3-17)
ak,aj J o i )i

If = k,
Oa
or 2 a j)2 (3-18)
aj I Oyi

2ak is a M x M matrix. We can set

2ak = [C] (3-19)


The gradient of X2 with respect to the parameters ak will be zero at the X2 mini-

mum. Thus, from equation (3-16), we get

OX2- N [i- y k = 1,2,..., M. (3-20)
Dak i=1 I \i OakJ

Taking an additional partial derivative gives
02 X2 N 09i i d. yi1 (3-21)
=2 (9i Yi) (3-21)
aijOak -i=1 aj Oak (ajak 2

Note that the components of equation (3-21) depend both on the first derivatives

and on the second derivatives of the basis functions with respect to their param-

eters. Inclusion of the second derivative term can in fact be destabilizing if the

model fits badly. From this point on, we always define ajk as

1 02 2 i y 1 (3-22)
2 OaiOak i=1 aj k _

Comparing equation (3-19) with equation (3-22), we get


[C] = [a1 (3-23)

The diagonal elements of the matrix [a] 1 are the uncertainties of the parameters.

These methods will be used to estimate the uncertainties of the parameters in the

next section.









Table 3.1: Regression result for two-dimensional inverse model described by Aoki
et al. The underlined symbols represent the parameters estimated by regression.
Initial Parameters Final Parameters g Standard Deviations
al = -0.2, al = -0.1, 1.32603 x 10 12 a = 1.594 x 10 6
1 = 0.7, 1 = 0.7,
a2 = -1.0, a2 = -1.0,
32 =0.0, 32 =0.0,
Connection = 0.5. Connection = 0.5.
al = -0.1, al = -0.1, 1.32603 x 10-12 aU, = 1.46 x 10 6
A3 = 0.1, 1 = 0.7,
a2 = -1.0, a2 = -1.0,
32 =0.0, 32 =0.0,
Connection = 0.5. Connection = 0.5.
al = -0.2, al = -0.1576, 7.25157 x 10-4 oa = 3.16 x 10+
1 = 0.7, i = 0.7,
a2 = -1.3, a2 = -1.2112, 02 = 1.13 x 10+3
/32 =0.0, /32 = 0.0,
Connection = 0.5. Connection = 0.5.
al = -0.2, al = -0.164, 7.35439 x 103 aa = 1.18 x 10+2
1 = 0.5, /1 = 0.6632, p, = 3.16 x 10+1
a2 = -1.3, a2 = -1.0708, 02 = 1.26 x 10+3
/32 = 0.1, /32 = 0.497, 32 = 1.0 x 10+2
Connection = 0.5. Connection = 0.5. Uxo,,,,cion = 4.35 x 101


3.4.5 Regression Results

The results of the regression for the two-dimensional problem are presented in

Table 3.1. The underlined symbols represent the parameters estimated by regres-

sion. It is clear that the number of independent parameters that can be obtained

in a statistically significant manner is limited. When four parameters are fixed,

the fifth could be obtained with high confidence. When two and more parameters

were determined from the regression, the confidence interval became too large.

This analysis provided two important insights. The first was that the number

of statistically significant parameters that could be obtained by regression was

limited by the amount and quality of the data. The second was that the weighted

regression procedure proposed in this work could provide an indication of when

the number of statistically significant parameters was exceeded.









3.5 Conclusions

Different regression techniques, such as conjugate gradient method, quesi-Newton

method and downhill simplex method, were compared. As the results of the stud-

ies in this chapter, the downhill simplex approach was shown to be the most robust

method and it is independent of the initial guesses. It will be applied further for

the three-dimension problem.

However, the results from all the regression procedures were insufficiently re-

liable. One possible reason may be the insufficiency of data, since only one series

of potential data has been used. Future work will incorporate not only the on-

potential data, but also the off-potential and the current survey data. The complex

objective function to be minimized will be:

2(a am) (i yi)2' + n (- yi)2

L = i on-potential i i off potential


S (y[ i- yi)2] (3-24)
-=1 current
where al,...,am are the parameters to be fitted, ci is the standard deviation of a

measured value, and ni, n2, n3 are the number of data points.














CHAPTER 4
DEVELOPMENT OF THREE-DIMENSIONAL FORWARD AND INVERSE

MODELS FOR PIPELINE WITH POTENTIAL SURVEY DATA ONLY

4.1 Introduction

From the previous chapter, it is known that pipeline surveys are usually used to

determine the adequacy of cathodic protection (CP) for the underground pipelines.

However, these survey data may contain significant scatter or noise. The limita-

tions of measurement techniques, instruments, and analysis methods make the

data interpretation difficult. Therefore, the objective of this chapter is to develop a

mathematical framework for interpretation of survey data in the presence of ran-

dom noise.

There are two types of approaches used in modeling cathodic protection. A

forward model yields the distribution of current and potential for a given system

geometry and for given physical properties of pipes, anodes, ground beds, and

soil. An inverse model yields system properties such as pipe coating resistivity

given values for current and/or potential distributions.

The history of the development of analytic and numerical forward models for

cathodic protection of pipelines encompasses more than fifty years. Waber et al.23

derived an analytic model in the form of a Fourier series, which can only be used

for simple geometries and boundary conditions. Pierson et al.24 developed a se-

ries of analytic equations to account for attenuation for coated pipelines which

extend the usual resistance formulas such as Dwight's equation.9 26 Doig et al.27

utilized the finite difference method to simulate the galvanic corrosion with com-









plex polarization. In the 1980's, Brebbia28 developed the revolutionary boundary

element method (BEM). The BEM has since been applied in many engineering

fields because it is accurate, effective for both infinite and semi-infinite domains,

and computationally efficient.28'29 Aoki et al.ls820'30 applied the BEM to both two-

-dimensional and three-dimensional systems. Telles et al.31 improved the BEM

for CP simulation by introducing the current density self-equilibrium limitation,

which eliminated the need to discretize a boundary located at infinity. Brichau

and Deconinck3'32 coupled the internal and external Laplace equations which were

solved by coupled BEM and finite element methods (FEM). Kennelley et al.33'34

used FEM to model the influence on CP protection of discrete coating holidays

that exposed bare steel. The concept of discrete holidays was extended to three

dimensions using BEM by Orazem et al.35 37 Riemer and Orazem1'38 40 combined

the advantages of the previous work in their development of the CP3D model. A

distinguishing point of this software is that it can account for localized defects and

yet is suitable for long pipelines and pipe networks. Some commercial programs

are available such as PROCAT2 and OKAPPI.3 All the simulations described above

solve Laplace's equation coupled with linear or nonlinear boundary conditions to

obtain the potential distribution and current distribution. The analytic models can

only be used for two-dimensional domains with simple geometries. The numer-

ical models can be applied to almost any complex domain; thus, they are widely

used. While many numerical models are based on the FEM, the newer generation

of models use either BEM or a combination of BEM and FEM.

The use of computer programs to interpret pipe survey data in terms of the un-

known condition of the pipe requires solution of the inverse problem, in which the

properties of the pipe coating are inferred from measurements of current and po-

tential distributions. This approach allows interpretation of field data in a manner









that would take into account the physical laws that constrain the flow of elec-

trical current from anode to pipe. Aoki et al. has made significant contributions

to the application of inverse model to CP. Their studies included simplifying the

unknown parameters in the polarization curves, changing the form of objective

function, and trying different kinds of minimization methods. To minimize the ob-

jective function, they have employed a variety of techniques conjugate gradient,

fuzzy a priori,20 and genetic algorithm methods.41'42 Their work involves mod-

eling protection of pipes, ships, and reinforcing steel in concrete structures.41 42

Wrobel and Miltiadou describe the application of genetic algorithms to inverse

problems, including identification of coating holidays.44,45 Aoki et al. used an in-

verse BEM analysis to eliminate Ohmic error from measurement of polarization

curves.46

The concept behind use of the inverse model for cathodic protection is related to

the approach used to reconstruct the distribution of electrical resistance for med-

ical, chemical process, and geological applications. For medical applications, the

readings from an array of sensing electrodes are used to construct an image as-

sociated with conductivity variations within a body.47 50 The technique is called

electrical-impedance tomography.51'52 Electrical-impedance tomography has also

been used to determine the interfacial area for two-phase gas-liquid and particu-

late flows in chemical processes53 56 and, through identifying the distribution of

electrical resistivity, to identify composition distributions in laboratory and plant-

scale process equipment.57 Electrical-resistivity tomography is used to interpret

cross-borehole resistivity measurements to obtain electrical resistivity distribu-

tions of geological formations.58 64Electrical resistivity is an important petroleum

reservoir parameter because it is sensitive to porosity, type of pore fluid, and de-

gree of saturation. While use of neural networks have been suggested,65,66 most























Figure 4-1: Schematic illustration of the pipe segment and anode used to test the
inverse model.
approaches are based on non-linear regression.

In this chapter, the construction of the forward model, inverse model for CP

system, theoretical basis and development of the three-dimensional BEM will be

introduced.

4.2 Construction of the Forward Model

4.2.1 Cathodic Protection System

For the three-dimensional problem, a pipe was considered to be buried hori-

zontally under the ground surface. A cylindrical vertical anode was placed a dis-

tance away from the pipe. The underground region was considered to be bounded

by the soil surface and to extend infinitely in the other directions in the soil. The

anode was connected to the pipe with a wire, as shown in Figure 4-1. The pipe

was placed horizontally 1.0 m (4.75 feet) below the soil surface and had a diame-

ter of 0.457 m (1.5 feet). The anode was placed in a vertical position and a distance

away from the center of the pipe. The anode diameter was 0.152 m and the length

was 1.0 m (3.28 feet). The soil resistivity was 100 Qm. A 0.5 mm-thick coating was

assumed to cover the side area of the pipe, with the exception that the two ends

were assumed to be insulated. The potential of the pipe steel was assumed to









be uniform. This assumption, valid for the short pipe segments considered in this

preliminary work, will need to be extended in future work. The forward model was

used to calculate the potential that one might measure with a voltmeter connected

to the pipeline and a Cu/CuSO4 reference electrode (CSE).

4.2.2 Governing Equation and Boundary Conditions

The current density in a dilute electrolyte solution can be described as8

i= -F2V z z ujcj F zjDjVcj + Fvz zjci (4-1)

where i is the current density vector, v is the bulk velocity, F is Faraday's constant,

zi is the charge for species j, uj is the mobility for species j, cj is the concentration

for species j, and Dj is the diffusion coefficient for species j. In this study, only the

electrode surfaces, such as pipe walls are considered. Since the velocity v is small

in the soil, the convection can be neglected. In addition, concentration gradients

only have a significant effect in the diffusion layer, which is close to the struc-

ture and is relatively thin comparing with the characteristic length of the system.

Thus, concentration gradients generally are neglected in large-scale simulations.

Therefore, the current density in the soil is

i= -F2V zYujcj (4-2)

The conductivity of the electrolyte is defined to be

S= F2V Z UjCj (4-3)

The current density equation can be reduced to

i = Vo (4-4)

which is called Ohms Law. The conservation of charge in the bulk yields

V i = 0 (4-5)









Substituting equation(4-4) into equation (4-5)

V (-KVO)= 0 (4-6)

Thus

KV20 = 0 (4-7)

Equation (4-7) is known as Laplace's equation for the electrochemical potential 0.

The boundary conditions are shown in Figure 4-1. The corrosion potential of pipe

steel is -0.6V. The sacrificial anode has a given potential of -1.2V.

4.2.3 Theoretical development

As mentioned previously, the forward model has to be efficient in terms of com-

puter calculations due to its coupling with the inverse model. To achieve this goal,

a linear relation, considering the coating covered pipe with various resistivity, is

proposed in this section.

Linear Boundary Conditions

The polarization curve, which describes the relationship between the potential

and the current density, indicates the corrosion condition on the pipe surface and

is normally used as the boundary condition. It is not an easy task to determine the

polarization curve since it strongly depends on a number of phenomena. Further

more, the polarization curve can also be a function of time and history. The polar-

ization curve used by Riemer and Orazem1'39 had eight parameters. Amaya and

Aoki et al.20'21 simplified the polarization curve from the Butler-Volmer equation

and reduced the parameters to four for one cathode and one anode system. How-

ever, if the corrosion condition along the pipe is not uniform, more polarization

curves are needed. In present study, the polarization curve was substituted by

a linear relationship between the potential drop and the current density over the

pipe coating for three-dimensional simulation, shown in Figure 4-2.









Coating
V-outer Resistance



SPipe Steel



V- Inner
i= -(1/ p5)[(V-0'outer)-(V-, )inner)]

Figure 4-2: Linear relationship between potential drop and current density over
the pipe coating.


1
i = [(V Oouter) (V inner)] (4-8)
PcO
where i is the current density along the pipe radical direction, pc is the unit resis-

tance of coating material, 6 is the coating thickness, V is the potential of the pipe

steel, assumed to be constant, ( is the potential in the electrolyte, V outer is the

potential of the pipe referred to a reference electrode placed at the outside coating

surface, and V inner is the potential of the pipe with respect to a reference elec-

trode placed at the inside coating surface. Under the assumption that the current

densities through the coating and those at the pipe steel surface are equal,

[(V Oouter) (V Oinner)] [(V inner) (V Ocorr)] (
Rc Rkinetic

where Rc and Rkinetic represent the coating and polarization resistance, respec-

tively, and V Ocorr is the corrosion potential. As Rc > Rkinetic,


(V- Oinner) (V corr)] 0 (4-10)


Therefore,
1
i = (V- Oouter) (V Ocorr)] (4-11)
PcO
Equation (4-11) provides a linear relationship between the potential and the cur-

rent density. This linear boundary condition, replaces the nonlinear polarization









curve and greatly simplifies the calculations of forward model.

Description of the Pipe Surface

According to the coating-covered special feature of pipe, a coating resistance

approximation was made. It allows the coating resistance to vary along the length

of the pipe according to:


P = po + pke (x-xk)2/22 (4-12)
k

wherein po refers to the average good coating resistance of the pipe, pk represents

the resistance reduction associated with coating defect k, xk is the center point of

the coating defect, and Ok is the half width of the defect region. Equation (4-12)

shows that if the coating position is close to a defect center xk, i.e., falling into the

defect region 2ak, the coating resistance decay is evident; whereas, if the coating

position is far away from the defect center, the coating resistance decay is insignif-

icant. The present approach has two significant advantages over assigning a resis-

tivity to each cylindrical element. For instance, use of equation (4-12) to represent

two coating defects on a 150 feet pipeline is shown in Figure 4-3. If the pipeline

is discretized with a 10 foot space interval, only seven parameters are required

Po,P1,x1,a1,p2,x2 and -2 for the equation (4-12), as compared to 15 parameters are

needed for an element-by-element coating resistivity estimation. If the pipeline is

discretized with a 5 foot spacing, still these seven parameters are required, while

it is now thirty parameters for the regular method. Likewise, if the space interval

is decreased to be so small that it makes the resistivity value continuous, no more

than these seven parameters are enough to describe resistivity along the whole

pipeline. The degree of freedom for the problem is increased dramatically, and

correspondingly, the required computational time is reduced.









1200 . ..

1000

800

600 -Model
60 -10-foot elements
400 -5-foot elements

200


0 50 100 150
x,ft

Figure 4-3: Schematic illustration of the pipe surface resistivity model.

4.3 Three-Dimensional Boundary Element Method

The theory for application of the boundary element method to the three-dimensional

half-infinity multi-connected domain is introduced in this section. The linear cylin-

drical elements are adopted to discretize the pipe/anode surface in present study.

4.3.1 Infinity Domain

In the two-dimensional system, a closed rectangular domain was investigated.

This kind of system, called an interior problem, is shown in Figure 4-4 and where

S is the boundary of the volume V, the shaded area is the interior domain. f is

the normal vector of the boundary S at that point. According to Brebbia et al. ,29

the basic equation of boundary element method corresponding to the Laplace's

equation is

c(xs)u(xs) = [q(y)u*(xs, y) q*(xs, y)u(y)]dS(y) (4-13)

Equation (4-13) is called boundary integral equation. Wherein xs is the source

point, y is the field point. u*(xs, y) and q*(xs, y) are the fundamental solutions

of Green's function for potential and flux. u and q are the general solution of

Laplace's function for potential and flux. c(xs) depends on the position of the










n











Figure 4-4: Interior problem.

source point xs, as shown in equation (4-14),

Xs X V, c(xs) = 0

c(x) = xs V, c(x) = 1 (4-14)

Xs e S, c(xs) = 4

w is the solid angle subtended by V at xs, which is on the boundary S. For the

three-dimensional system, the studied domain is outside the pipeline and the an-

ode surface, and is between the soil at infinity. It is a multi-connected region of an

interior problem, as shown in Figure 4-5. The boundary S can be considered to be

the boundary of the pipeline or the anode. The boundary SR can be the boundary

at infinity. When R oc, equation (4-13) is also valid for this case. For xs C S,

c(xs)u(xs) = [q(y)u*(xs, y) q(xs, y)u(y)]dS(y)

+ lim [q(y)u*(xs, y) q(x, y)u(y)]dS(y) (4-15)
R-oo JSR

The fundamental solutions to the Green's function for three-dimensional problem

at the boundary SR are

1
u*(x, y) =
47 R
Ou*(xs, y) 1
q*(xs, y) = (4-16)
On 47R2












-0.
SR n






n





Figure 4-5: Multi connected region of interior problem.
And the general solution to the problem over the infinity boundary SR are

1
u(y) = Ku*(xs, y) + u, O( )
K 1
q(y) = 4 0( ) (4-17)
47R2 R2

where K and u, are two undetermined constants at the infinity boundary. K is the

sum of source term (e.g., electric charge) distributed over S,

K =j q(y)dS(y) 0(1) (4-18)
Js

Equation (4-16) and (4-17) are introduced into equation (4-15). Since

limRo JSR [q(y)u*(xs, y) q*(xs, y)u(y)]dS(y)

= limR fS J[q(y)u*(xs, y) q*(xs, y)(Ku*(xs, y) + uo)]dS(y)

= limR,, Is {u*(xs, y)[q(y) Kq*(xs, y)] q*(xs, y)u,}dS(y) (4-19)

where the first term inside the bracket is

limRoo sR u*(xs, y)[q(y) Kq*(xs, y)]dS(y)

= lim s s, 4 [O() (- )]dS(y)

limn ,[O(T)) 47rR2] -- 0 (4-20)









and the second term is

f 1
lim q*(xs, y)udS(y) = lim u,(- 47R 2)
R4oo JSR Roo 47rR2
= -u_ (4-21)

Then equation (4-15) becomes,

c(xs)u(xs) + [-q(y)u*(xs y) + q*(xs, y)u(y)]dS(y) = u. (4-22)

Equation (4-22) indicates that the value of K has no important effect on the bound-

ary S. To satisfy the pipeline and the anode CP system, the unit outward normal n

on S in Figure 4-5 needs to change its direction. Correspondingly, equation (4-22)

will be

c(xs)u(xs) + {-q(y)u*(x, y) + [-q*(xs, y)u(y)]} + dS(y) = u (4-23)

In addition, according to the physical condition, there is no current flow out of the

soil surface, therefore, K = 0 is given as boundary condition and u. is unknown.

4.3.2 Half-Infinity Domain

The half-infinity domain is obtained by using a plane to split an infinite do-

main. Two half spaces lie on each side of the plane. Either the Dirichlet or Neu-

mann boundary condition is satisfied at the plane. For the buried pipelines sys-

tem, the underground soil domain is the half-space being studied. Since there is

no net current flowing out of the soil into the air, the Neumann boundary condi-

tion vanishes at the soil surface. The Green's function can satisfy that condition by

using the reflection properties.29 70 If we set Q- to be the region x < 0, P E Q- and

let P'(-x, y, z) be the mirror image of a given source point P(x, y, z) with respect

to the plane x = 0 (or 9OD), P' becomes the reflected source point, and r = | Q P|,

r' = | Q P' ( see Figure 4-6). Q is the field point. Consequently, half-space fun-

damental solutions satisfying a homogeneous Neumann condition HN(P, Q) = 0








on O9 can be deduced by adequate superposition of free-space fundamental solu-

tions for sources at P and P', that is

aGN(P, Q)
G n(Q) = HN(, Q) = 0 (4-24)
aQ

Here, the superscript N is for the Newmann boundary conditions; n(Q) is the

normal vector at point Q. Assuming two full free-space problems with VQ C Q ,

equation (4-24) can be written as

GC (P, Q) 1
( Q) n(Qr) = H(P, Q) = )n(Qr) (4-25)
OQ 47r2
G+(P" Q) n(Qr,) = H(P', Q)= (- ) n(Qr') (4-26)
OQ 4ir'2

From Figure 4-7, the normal vectors n(Qr) and n(Qr') of Fand r' at point Q are

n(Qr) = 1 (4-27)

n(Qr') = -1 (4-28)

Meanwhile, according to the Neumann boundary condition given in equation (4-

24)

HN(, Q) = H(P, Q) + H(P', Q) = 0 (4-29)

When Q e O9d, r = r', the fundamental solution in two full free-space have to be

1
G-(P, Q) = (4-30)
47rr
1
G+(P', Q) = 1 (4-31)
47r'

to satisfy equation (4-24), indicating that the two source intensities P and P' are

equal and have the same sign. Therefore, the final form of the Green's function is

GN(P, Q) = G-(P, Q) + G (P', Q) = 1 + (4-32)
47 r r'




















-,--t-
1~----------------



S\r


0\


R=Radium



x

Figure 4-6: Schematic illustration of mirror reflection technique.


Figure 4-7: The
Neuman b.c's.


Fundamental solution to the half-infinity domain satisfying the









x
eo eR
0 1 2 1 3 3..... n-2 n-1
n n+1
-------- .----- -- -- --- ----'--, ,'----- -- .---- -- ------
Sne- i i nel+

S0 S2 S3 Snel-2 Snel-1

Figure 4-8: Pipe discretision and collocation points.

4.3.3 Boundary Discretization

Rectangular and triangular elements are commonly used in the literature for

BEM meshes on the surface of a three-dimensional object. Aoki et al.71 used 352

constant triangular elements to discretize the side area of a 160mm long, 155mm-

diameter cylinder. Riemer and Orazem's model CP3D may involve thousands of

elements to discretize a pipeline with discrete coating holidays. This level of de-

tail is appropriate for a forward model, but it is inappropriate for an inverse model

where the amount of data may be insufficient to extract a high level of details con-

cerning the pipe condition. Besides, the model needs to be calculated not once,

but thousands of times. To decrease the number of elements, a series of linear

cylindrical elements were used to take advantage of the cylindrical shape of the

pipe and the anode. The source points (also called collocation points) were local-

ized on the cylindrical side area. At the two ends of the object-pipe/anode, the

two discs were set as two constant circle elements. The center of each circle was a

collocation point (see Figure 4-8). For the cylindrical elements, there was no varia-

tion along the circumference. The variation was only along the axial direction72 75

such that the collocation points could be chosen at the reference line, which was

on the top of the cylinder. The elements on the side area of the cylinder were from

So to (Sne1-1), and, at the two ends of the pipe, were the two circle areas Snei and

S(ne, 1). The collocation points on the side area of cylinder were from 0 to (n 1),









while two collocation points at the center of circles are counted as n and (n + 1).

The relation between the number of elements and the number of nodes depends

on the order of the interpolation functions, which can be constant, first-order or

even higher order.

The identical mesh was used for the element separation because in the follow-

ing inverse model, the position of the coating defects was distributed randomly

along the pipeline. If the pipeline meshing depended on the position of coating

defects, every time when the new coating defect positions would be assumed, the

system would have to be remeshed. To save computational time, by using the

identical mesh, the matrices G and H for the BEM need to be calculated only once.

4.3.4 Coordinates Definition and Transformation

In this program, two kinds of objects were defined, one is pipe, and the other

is anode. The user needs to input the beginning point of the pipe/anode's axle to

the program. Both the global coordinate and the local coordinate were used in this

study. The global coordinates was set as: z axial parallel to the soil surface and

directed to north direction; x axial perpendicular to the soil surface and pointed

toward the sky. The y axial and z axial build the soil surface plane. The origin is

on the soil surface. The angle between the pipe/anode's axle and soil surface is

defined as the elevation angle ,O. e, = 0 means that the object is parallel to the soil

surface. 8O = 90 (or e, = -90) means that the object is perpendicular to the soil

surface. The direction angle Od refers the angle between the pipe/anode's axle and

z axial positive direction. The local coordinate is used when the collocation point

and the field point are on the same object (i.e., the pipe/anode), while the global

coordinate is used when the collocation point and field point are on the different

objects to clarify the relative positions of these objects.

Transformation between two different coordinates, such as the global and the
























y

Figure 4-9: Coordinate rotation.

local coordinates or two different local coordinates, usually consists of two steps.

One is origin transformation, which is to move from one original point to a new

origin. The other is rotation, which is to rotate previous axials around the new

origin. Coordinate rotation actually includes two rotations. Figure 4-9 shows the

procedure to rotate a coordinate around a point and one axis. The first rotation is

to rotate the x axis and the z axis about the y axis with elevation angle 0e to get

new x' axis and z' axis. The second rotation is to fix x', and to rotate y and z' axes

with direction angle Od to get a new axes y" and z". Finally, coordinates (x, y, z) is

converted to (x', y", z").

4.3.5 Discretization of Boundary Element Method

Rewriting equation (4-23) using P as source point, Q as field point and O0Q as

boundary, we get

cu(P) + u(Q)qod(O0) = q(Q)upod(O) + u (4-33)
I P-









where u* and q* are the fundamental solution of the Green function, and they have

the relation
OU*po
q (4-34)

Discretizing equation (4-33),
nel f u* nel
ciui(P) + uj(Q) dS, = I qj(Q)Ou*odSj + u, (4-35)
jl Jsj Ono j-1 S,
where nel means the number of element. Since u and q of each linear cylindrical

element vary linearly along the z axial direction, then

U(z) = i(l z) +2Z

q(z) = q(1 -z)+q2z (4-36)

where 1 is the length of the element. Let t = ', then u and q, the functions of z, can

be transferred to be the functions of t,

u(t) = l[u(l- t) u2t]

= l[lNi(t) + u2N2(t)

q(t) = l[q(1 t) + q2t]

= l[qiNi(t) + q2N2(t)] (4-37)

The basis functions can be extracted from equation (4-37),

Ni(t) = t;

N2(t) = t. (4-38)

Upon substituting equation (4-37) into equation (4-35), the discretized boundary

integral equation becomes
nel &u*
ciui(P) + l lj[uj,Nil(t) + uj,2N2(t)] 1jdSj
j1 Sj Onj
nel
= lj[qj,lNl(t) + qj,2N2(t)]uidSj + um (4-39)
j=1 j









Since the fundamental solutions are

1
S= 47rij

qi* O = 4 j (4-40)
zy 9ni 47r 9O{ni

According to equation (4-39), we define

Gij = lqjkN dS (4-41)
Jik=l
Hij = lukNktdS (4-42)
Jik=l
If the field point Q is on the cylindrical element side area, dSj = RdOdt, R is the

radium of cylinder, then Gij becomes
j l J27 2 2
Gi = 1 lqjkNj(t)u Rd0dt
k=1
1R 1 2 22
k 1
= li ) (lkNk(t))dt ii dO (4-43)

and Hij becomes
H11 l j*27 2T
Hii = ljujkNk(t) qiRdOdt
k=1
1 2 27 0(1)
= l )(4) (I ukNk(t))dt j dO (4-44)
47 Jo k-1 JO Onj

If the field point Q is on the circle elements, dSj = rdrdO, 0 < r < R, Gij changes its

form to

1 jR j27,
Gij = qj(4i) rdrdO (4-45)
47 JO Jo rij

and Hii is in form

1 R 2 (1)
Hij = uij() j / rdrdO (4-46)
47r Jo Jo 9n

The boundary integral for Gij and Hii with the different positions of the source

point P and the field point Q is shown in Appendix A.









In addition, adding the unknown potential at infinity u, in equation (4-39) to

Hij, H matrix becomes


H= p H, 1 (4-47)
Ha,p Ha,a,, -1
[ H (np+na)x(np+n,+l1)
where -1 column corresponds to u,. And the G matrix is


G= GP Gy,a (4-48)
Ga, p Ga,a
(np+na)x(np+na)
The subscript p and a indicate the pipe and the anode respectively. The subscript

(p, p) and (a, a) indicate that the collocation points and the field points are on the

same object. While (p, a) means that the collocation points are on the pipe and the

field points are on the anode. Likewise (a, p) means that the collocation points are

on the anode and the field points are on the pipe. The number of node on the pipe

and the anode are represented by n, and na.

4.3.6 Row Sum Elimination

The diagonal elements of the matrix H are usually difficult to calculat since the

linear or higher order elements have been used and the constant ci in equation

(4-14) involves the calculation of the solid angle subtended by the region V at xs

on S. In order to overcome this difficulty, Gibbs's theorem can be applied to find

the values of the diagonal.76 For a domain governed by Laplace's equation, if the

potential is uniform throughout the domain, and its gradient of the potential is

zero at infinity if it exists, the gradient of the potential will be equal to zero within

the domain. For the three-dimensional potential problem, u(y) 1 is defined in

the interior region. Since q(y) = aY = 0 on the boundary, q(y) = 0 in the domain.

Thus equation (4-13) becomes

c(s)= q*(x, y)dS(y) (4-49)
JIs









It means that,


YHi,i = 0 (4-50)
ij
where the diagonal terms are unknown. The diagonal terms are now specified by

the negative sum of the off-diagonal terms of each row


H,i = Hi,i (4-51)
j7i
where j goes from one to the number of terms in a row.

The same technique used for the interior domain is applied to present work.

Assuming u(y) 1, since q(y) = (y) = 0 on the boundary and in the domain,
On
equation (4-15) becomes

c(x) = *(xs, y)dS(y) lim q(xs, y)dS(y) (4-52)
Js Rao JSR

Additionally, since

lim q*(xs,y)dS(y)= (- 1)(4R2) = -1 (4-53)
R-oo JSR 47r2i

c(xs) in equation (4-52) becomes

c(xs) = q(xs, y)dS(y) + 1 (4-54)

which means that,

H, = 1 Hi, (4-55)
ii
Therefore, by using Gibbs's theorem, the diagonal elements of the matrix H can

be obtained from the off-diagonal elements and the calculation difficulties are

avoided.

4.3.7 Self-Equilibrium

Cathodic protection systems do not lose or gain current from their surround-

ings. In other words, the current is conserved. Therefore, if the flux between the









anode and the pipe is self-equilibrium, the flux at infinity will be zero. To imple-

ment this condition, an extra equation, equation (4-56) is added to the system.31'77



i qdS =0 (4-56)

It indicates the intensity of an equivalent source distributed over S. It also can be

written as

i' = ip,kAp,k + Y ia,kAa,k = 0 (4-57)
k k
where

Ak= dSk, k c [0, np] or [0, n] (4-58)

Adding equation (4-57) to equation (4-47) and (4-48), and defining the potential

vector u as
Op

= (4-59)

0.- (np+na)xl
the flux vector q as

n Vp = n (pn(4-60)
S(np+na)xl
we get


Hp,p Hp,, -1 p GP,p Gya

Ha,p Ha,a -1 = Ga,p Ga,a (4-61)
-n V^a
0 0 0 0, Ap Aa V

Here H matrix is singular because there is still a row of zeros in it. However,

it will only be the case when Neumann type boundary conditions are specified

everywhere. The Neumann problem results in an infinite number of solutions.









Therefore, at least one element in the system must have a Dirichlet boundary con-

dition to make the H matrix nonsingular and result in a unique solution. In addi-

tion -n VO has a minus sign because the normal vector direction is outward of

boundary S.

4.4 Forward Model

4.4.1 Constant Steel Potential Assumption

Since the axial variation of potential in pipe steel is trivial for a short section or

low resistance pipelines, the potential of pipe steel is assumed to be a constant in

this chapter.

Rewriting equation (4-61) by making a variable substitution = V 9, we get



H,,, H,,, -1 V- Gpp Gp,a G V

H,,p H,,, -1 V -,', = G, Ga, -n (4-62)
-n (V -'.,)
0 0 0 V o A, Aa

where V is the potential of pipe steel. Decomposing the vector V 0 and -n *

V(V ), equation (4-62) becomes


H,,,p H,,a -1 V H,,,p H,,,a -1
H,,, H,, -1 V H,,, H,,, -1

0 00 V 0 0 0

Gpp GPa

= G,,p Ga, n (4-63)
n 7 ...
AP A

Since the potential of steel V is assumed to be a constant, -n VV = 0. Moreover,









according to equation (4-55)


Thus, the first term

can readily obtain


on the left hand side of equation (4-63) is vanished, and we


Hp,p Hp,a -1

Ha,p Ha,a -1 ',

0 0 0

Equation (4-11) can be rewrite as


Gp,p Gp,a

Ga,p Ga,a

Ap Aa


n Psoil cor
V ',' o (',= '1'- corr)
PcoatO

Substituting equation(4-67) into equation (4-66), we can get


Hp,p Hp,a -1 GY Gp,a

Ha,p Ha,a -1 a Ga, p Ga,a I

0 0 0 Aoo Ap Aa

Equation (4-68) will be applied in the two cases as below.

Sacrificial anode case


(4-66)


For the sacrificial anode protection method, we set the potential of the anode

as a constant and the current density of the anode is unknown except that it is zero

at the two ends of the anode. The potential of the pipe is unknown. The current

density of pipe is also unknown except the zero value is given at the ends of pipe.

In addition, the potential and the current density of the pipe satisfy the relation of

equation (4-67).


H,p + Hp,a

Ha,p + Ha,a,,


(4-64)

(4-65)


(4-67)


,''. Vcorr) (4-68)
.V,7









Rearranging the equation (4-68),
HP'P + PPpco Gp,a -11 "1. PPpcoilS Hpa

H,p+ G,psoi Ga,a -1 n V' PsoiG -Ha,a
S~ap + Pcoat6 P1 n a Pcoat6
+A Psoi Aa 0 J A Pil 0
P coatO PPcoat
(4-69)

it can be used to model sacrificial anode CP system.

Impressed current case

To simulate the impressed current protection method, we set the current den-

sity of the anode as a constant value, but the potential of the anode is unknown.

The conditions of current density and potential of pipe are same as those of the

sacrificial anode case. Likewise, the following equation can be obtained.

Hp,p Gpp Psoi Hp,a -1 G'1' PPpcoS Gp,a
H, + G Psoil H, -1 { G= Psoil -Ga,a

+AP 0 0 Ap so -Aa
P PcoatO PcoatO
(4-70)

4.4.2 Simulation Results

Typical simulation results for potentials on soil surface directly above a 10m

pipeline with one coating defect are shown in Figure 4-10. The potential change at

the position corresponding to the coating defects is evident, but the magnitude of

the change is small, in agreement with previous calculations.35'38

4.5 Inverse Model

The purpose of the inverse model is to construct an objective function and to

compare different types of regression methods in order to minimize the objective

function. The simulated annealing method was found appropriate in the present

study since it can escape from the local minima and thereby avoid the influence of

the initial guess.










Potential on the Soil Surface
-1.109

-1.11




-1.112
o_. 1
0 2 4 6 8 10
Axial Position/m 1 11

-1.114

-1.115

Figure 4-10: False color image of the on-potential on the soil surface that was gen-
erated by the forward model corresponding to Figure 4-1.

4.5.1 Objective Function

An objective function, which describes the difference between the measured

potential and calculated potential on the soil surface, is given as

N
g(x,p, ) = j(Vj- j)2 (4-71)
j-1

where 4j and 4j represent measured and calculated soil surface potentials, re-

spectively. The calculated soil surface potential depends on the pipe coating con-

ditions, such as position, resistivity and width of coating defect. Thus, once the

minimum of the objective function is reached, the fitted coating parameters will

reflect the real physical conditions of pipeline.

4.5.2 Analysis of Regression Methods

Several regression methods were tested, including the conjugate gradient, Levenberg-

-Marquart, and quasi-Newton methods. A summary of these methods is provided

by Press et al.78 These methods, distinguished by the need to evaluate both the

function value and the function gradient, were rejected for the pipeline inverse

model. The extra calculations required for evaluation of the function gradient









added significantly to the computational requirement. In addition, these methods

were extremely sensitive to the value of an initial guess and had a tendency to find

local rather than global minima.

The downhill simplex method,78 which requires evaluation of only the objec-

tive function and not its gradient, was used successfully for preliminary two-

-dimensional inverse analysis. The downhill simplex method did not work very

well for three-dimensional problems because it was strongly affected by the initial

guess.

Genetic minimization algorithms, intended to mimic the course of natural se-

lection, have been applied to inverse problems for cathodic protection.42'44,45 Pa-

rameters are normally coded as binary strings to reduce the searching population.

Procedures of selection, mating and mutation are used repeatedly to create the

new generation until the specified stop criterion is satisfied. This method is not

very sensitive to the values of initial guesses and can escape from the local min-

ima. However, it has difficulty selecting between close function values.79

After trying the alternatives, the simulated annealing optimization approach

was selected for the present work. This method is attractive because it is suit-

able for large-scale problems and can search for a global minimum, which may be

hidden among many local minima. Simulated annealing was better than down-

hill simplex because the simplex method accepts only downhill steps during its

searching; whereas, simulated annealing can accept both the downhill and uphill

steps. In this way, simulated annealing method can step out of the local optima

and successfully locate the global minimum.

4.5.3 Simulated Annealing Method

The term "simulated annealing" comes from a physical process analogy.78 When

a material is heated and then is slowly cooled down, a strong crystalline structure









will be obtained. This crystal is the minimum energy state of the system. In a sim-

ulation process, a minimum of the objective function corresponds to the ground

state of the substance. The Bolzmann probability distribution, which describes the

different energy state in a thermal equilibrium under given temperature, can also

be used to mimic the different objective function value at certain searching region.

The simulated annealing method starts with a brief view of the searching domain

by making large moves and then it focuses on the most possible region. The inher-

ent random fluctuation permits the annealing procedure avoid the local minima.

The Metropolis rule

p = e (g2-)/kT (4-72)

is used to control whether the uphill step is accepted. In equation (4-72), p refers

to the probability, gi is the objective function value and T is an important param-

eter in the simulated annealing method, which resembles the temperature in the

thermal system. For g2 < gi, the probability p is greater than 1, which means that

the downhill steps are always accepted. For g2 > gi, the probability p is compared

with a uniformly distributed random number from [0, 1] to decide whether uphill

steps are accepted. Since each parameter has its own limit, once the parameter is

out of its limit, the following equation can be used to correct the parameter.

x = XL + a(xu XL) (4-73)

where a is a uniformly distributed random number among [0, 1], XL is the lower

limit of x and xu is the upper limit of x. The parameter x can thus be guaranteed

to lie within its bounds.

4.5.4 Simulation Results and Discussions

Examples of the application of the proposed forward and inverse models are

given in this section. In addition, several inverse strategies are introduced. The re-
































Figure 4-11: Flow chart for the inverse model calculations.

gression algorithm employed by the inverse model is shown in the Figure 4-11. The

set parameters were used as inputs to the forward model to get the potential on soil

surface. These results were treated as the measured data. These "measured" po-

tentials belong to each point of a 3 x 101-grid area, as shown in Figure 4-12. There

are three lines in the grid area: the middle line is right above the pipe centerline,

and two other lines of calculated values were placed +1 m from the centerline of

the pipe. The grid, therefore, comprised 303 data points. Initialize the parameters,

the number of defects, each defect's center position, resistivity reduction and the

width of the defects, and input them into the inverse model. Among all these pa-

rameters, the number of defects has the most significant impact on the regression

results. In this study, we purpose a method to decide the number of defects which

can be obtained from the measurements. The inverse analysis result by using sim-

ulated annealing method is shown in Table 4.1 for a 10-meter long pipe segment









x=O,y=1,Z=0~-10




I x=0,y=-1,Z=O0lO
--- -- -- -- -- ---------------------- --------- ----- ----- ----- ----- --- -
S10m
0 100
Figure 4-12: Grid showing the location of 303 surface on-potentials calculated us-
ing the three-dimensional forward model developed in the present work. The grid
shown is for a 10m pipe segment. A scaled version of the grid was used for a 100m
pipe segment.

with two coating defects. The calculated data were for defects located at the 3.2

and 6.0 m positions with characteristic dimensions of 0.05 and 0.5 m, respectively.

The intact coating resistivity had a value of 5.0 x 107 / Qm, which was fixed for

the regression procedure. The input values for surface potentials were calculated

at the limits of the precision of the forward model, i.e., no noise was added. The

regression yielded the correct locations, characteristic defect dimensions, and re-

sistivity changes associated with the coating defects.

The process of finding the minimum of the object function is shown in Figure

4-13. For this test problem, with the forward model and no noise added to the syn-

thetic data, the minimum value of the cost function was 10 15. Of the techniques

used, only the simulated annealing method could find this global minimum. The

best of the other techniques were able to identify local minima with values on the

order of 10-4 to 10 8 preceding the plateau.

A comparison of the set and fitted results for pipe coating resistivity, poten-

tial and current density, respectively, is shown in Figures 4-14. Corresponding to

the coating resistivity decay, the potential and current density distributions have

significant changes. A good agreement between the synthetic data and regression

results can be observed.

















Table 4.1: Parameter values obtained using the three-dimensional inverse model
developed in the present work for a 10m pipe segment with two coating defects.

Coating Position Resistivity Characteristic dimension
Defect Xk / m pk / Q m 0k / m
Set Values 1 3.2 -4.0 x 107 0.22
2 6.0 -3.5 x 107 0.71
Initial Values 1 2.5 -3.0 x 107 0.1
2 5.0 -3.0 x 107 0.1
Regression Result 1 3.206 -3.81 x 107 0.26
2 6.004 -3.51 x 107 0.70


10-2
0



> 10-6
U-


0)
1o'
0 10-14


10-18


0 50 100 150 200 250 300 3
# Evaluations


Figure 4-13: The regression objective function as a function of the number of evalu-
ations for a pipe coating with one defect region. The simulated annealing method
was used for this regression.


I I I I I


' I I
Initial Guess


\- Evaluation of Local Minima






Best Minimum
Identified
I I I I I



























5x109 cm


- - - - -





--Set data
0 Regression result


0 2 4 6 8 10
Position /m


1 148


- Set data
o Regression result


-1 152


-1 156


-1 160


I I I


4 6
Position /m


8 10


0

E
<
-lx10l

(3)
C
c

S-2X10l


3x1
-3x104


Set data
O Regression result


S 2 4 6 8 10


Position Im



(c)

Figure 4-14: Comparison of the set and fitted results for pipe coating with two
coating defects: a) coating resistivity; b) potentials; and c) current density.


E
a 50

S40

D 30
0)
S20

8 10
10
0


. .


1




:il


1









Table 4.2: Test case parameters with five coating defects on the pipe used to
demonstrate the method for determination of the number of statistically signif-
icant parameters (see Figure 4-15). The intact coating resistivity had a value of
5.0 x 107 / Qm.
Coating Defect Position Resistivity Characteristic dimension
Xk / m pk / Qm Ok / m
Set Values 1 10 -3.5 x 107 0.45
2 30 -4.5 x 107 0.32
3 40 -3.0 x 106 0.55
4 70 -2.0 x 107 0.45
5 85 -4.0 x 107 0.63


4.5.5 Inverse Strategies

The strategies, including the determination of the number of significant pa-

rameters, the procedures used to reduce the influence of initial guess and the eval-

uation of the role of noise in the measured data, were explored to assess the confi-

dence level of the inverse model results.

Determination of the Number of Statistically Significant Parameters

When the collected data are scattered, it is difficult to decide how many pos-

sible defects or coating anomalies should be included in the regression. The ap-

proach taken to address this issue was to increase sequentially the number of de-

fects, using the regression statistics to determine when the number of statistically

significant parameters was exceeded. This approach is similar to that used to as-

sess the number of statistically significant measurement model parameters can be

obtained by regression to impedance spectroscopy data.808,s

An example with five defects, shown in Table 4.2, was used to illustrate the

approach taken to assess the correct number of statistically significant parameters.

The relation between the regression statistic log(x2/v) and the number of coating

defects assumed in the regression is presented in Figure 4-15. Here, v = N M

represents the degree of freedom of the problem which is reduced as the number of










I '' I I I
-7.2 o


S-7.6
_--o-
0 \ /0---
oT /
0 -8.0
\ /
\ /
-8.4 o

0 2 4 6 8 10
# defects

Figure 4-15: The regression statistic as a function of the number of coating de-
fects assumed for the model. The minimum in this value is used to identify the
maximum number of coating defects that can be justified on statistical grounds.

parameters is increased. N represents the number of data point, and M represents

the number of parameters. x2 is the weighted objective function. The lowest point

of the curve denotes the number of statistically significant coating defects obtain-

able by regression. Here, it is four. The fifth coating anomaly, identified in Table

4.2 as defect 3, could not be identified by the regression procedure. The number

of coating anomalies identified by this procedure will depend on the amount and

quality of data and on the sensitivity of the data to coating condition.

Testing the Robustness of the Inverse Model

To assess the influence of uncertainty in the measured data, normally dis-

tributed stochastic errors were added to the soil surface potential generated by

the forward model. The noise added had standard deviations Onoise of 0.1, 1.0, and

2.0 mV respectively.

The set values, initial guesses, and regression results are presented in Table 4.3.

Three coating defects placed at 30, 40, and 70 m positions were used to generate

synthetic data. For the lowest noise level, noise = 0.1 mV, the sequential procedure

described in the previous section allowed only two statistically significant defects.






64

Table 4.3: Regression results from the three-dimensional inverse model for a 100m
pipe segment with three coating defects. The sequential regression procedure was
used to identify the number of defects that were statistically significant. The intact
coating resistivity had a value of 5.0 x 107 / Qm.
Coating Position Resistivity Characteristic dimension
Defect Xk / m pk / Q m 0k / m
Set Values 1 30 -4.5 x 107 0.32
2 40 -3.0 x 106 0.55
3 70 -2.0 x 107 0.45
Initial Guess 1 50 -3.5 x 107 0.92
2 50 -3.5 x 106 0.92
3 50 -3.5 x 106 0.92
Regression Results
noise = 0.1 mV 1 29.85 -4.25 x 107 0.66
2 69.83 -7.57 x 106 1.30
Noise = 1.0 mV 1 32.01 -4.67 x 107 0.087
Noise = 1.0 mV 1 32.01 -4.68 x 107 0.057
2 90.29 -1.22 x 106 0.86
Noise = 2.0 mV 1 29.95 -3.33 x 107 1.50

The initial guesses placed the defects at the midpoint of the pipe segment. The re-

gression results suggested that the defects exist near the 30 and 70 m locations. The

missing defect is the one with the smallest deviation in coating resistivity. Thus,

the regression procedure identified the correct location of the two most significant

reductions in coating resistivity.

For -noise = 1.0 mV, the sequential procedure using the minimization of the

Sx2/V criterion allowed two coating defects. The defect located at 32.01 m was

consistent with the most significant defect located at 30 m, but the defect identified

at 90 m did not conform to the input data. In addition, the regression failed to

identify the second most significant reduction in coating resistivity at 70 m.

The problem here may be an inadequate sensitivity of the X2/V statistic for

identifying overfitting of data. Other criteria, such as the Akaiki information cri-

teria,82'83,84 provide additional penalties for adding parameters to a model. For









noise = 1.0 mV, the Akaiki performance index

Ap = X21 ob (4-74)
1 np/nob

suggested that only one defect could be identified; whereas, the Akaiki informa-

tion criterion

Aic = log (I X2 (1 + 2np/nob)) (4-75)

suggests that two defects could be identified. Regression for a single coating defect

identified a defect in the close vicinity of the most significant coating reduction. By

any of the statistical measures tested, only the largest defect could be identified for

noise = 2.0 mV. The location of the defect was correctly identified; however, the

width of the defect was incorrectly determined.

These results can be explained by examination of the synthetic surface poten-

tial data used for the inverse model analysis. The corresponding results of the

inverse analysis for noise = 0.1 mV are shown in Figure 4-16 As is shown in Fig-

ure 4.16(a), the level of added noise did not obscure the surface-potential features

introduced by the presence of the major coating defects. The regressed and noise-

free target values for potential cannot be distinguished; thus, the absence of the

minor coating defect did not influence the fit of the model to the synthetic data.

As seen in Figure 4.16(b), the regression procedure identified the the two most

significant reductions in coating resistivity at 30 and 70 m locations.

In contrast, the random noise added with -noise = 1.0 mV, shown in Figure

4.17(a), almost completely obscures the influence of the coating defects. Neverthe-

less, a major defect can be resolved by the regression procedure in the vicinity of

the largest defect, as shown in Figure 4.17(b). The anomalous defect introduced by

the regression procedure at 90 m has associated with it a small reduction on coat-

ing resistivity. There is a question as to whether this defect can be considered to








66












-0.988



U -0.990


0
-0.992



-0.994 .
0 20 40 60 80 100

Position /m



(a)



60
E
C 50

40

3 30

c n
S20

10

I I I lIl
0 20 40 60 80 100
Position /m



(b)

Figure 4-16: Comparison between the input values and regression results for
noise =0.1 mV: a) soil-surface potential (at the centerline of the pipe); b) coating
resistivity.




















-0.986

0 '
0 00 0
> -0.988 Q 0 0 00 000

o 00o o000
C -0.990 -0 OO 0 o 0

0Q
-0.992


-0.994 I , I , I ,
0 20 40 60
Position /m


0 20 40 60

Position /m


80 100


80 100


Figure 4-17: Comparison between the input values and regression results for
noise =1.0 mV: a) soil-surface potential (at the centerline of the pipe); b) coating
resistivity.









be statistically significant, but, nevertheless, the regression procedure would have

given adequate guidance for excavation of the pipe.

By visual inspection, the random noise added with noise = 2.0 mV, shown in

Figure 4.18(a), would appear to obscure the influence of even the major coating

defect. The regression procedure identified a single statistically significant coating

defect located near the 30 m defect. The agreement between the regressed poten-

tial profile and the noise-free target values was surprisingly good. As shown in

Figure 4.17(b), the correct location for the principal defect was correctly identified,

even though the breadth of the defect was not correctly determined.

The results suggest that, while it is difficult to extract coating conditions from

the data when the influence of the coating defect on the surface potential is compa-

rable with noise, the regression procedure showed a surprising ability to identify

the location of the most significant coating defects from noisy data. This result

suggests that an inverse model is feasible, in particular when other types of data

are included.

Sensitivity to Initial Guess

The values used for an initial guess have a significant influence on most of the

conventional nonlinear regression methods. To reduce this effect and to test the

robustness of the present inverse model, identical parameters for each defect were

used. For example, as the initial guess in Table 4.3, the positions of the defects were

all set to the middle of the pipeline. This kind of initial guess created significant

difficulties for conventional regression methods, but posed no problems for the

simulated annealing method used in the present inverse model.








69












0-.984 c-- i --p -- --0-- i-0000 -- | -- i --
-0.984 '
0 0
of 0 00
> 0 00 0 0
-0.988 -qp 0 Oo 0O00 n ^ -


t3 O OO, O O OOOOO
0 0

0 O0


-0.996 -

0 20 40 60 80 100
Position /m



(a)



60 -

E 50o

S40

> 30 -

S20 -
10
10 -


0 20 40 60 80 100
Position /m



(b)

Figure 4-18: Comparison between the input values and regression results for
Noise =2.0 mV: a) soil-surface potential (at the centerline of the pipe); b) coating
resistivity.









4.6 Conclusions

In this chapter, both the forward model and the inverse model for the three-

-dimensional CP system was introduced. The simplified forward model has the

following advantages.

1. The Laplace's equation with the simplified linear boundary condition was

applied. This approach not only simplifies the forward calculation, but also

reduces the computational effort to the inverse calculation.

2. The line shape approximation for the pipe coating resistivity makes it possi-

ble to limit the number of parameters needed to describe the coating condi-

tions along the whole pipe. It dramatically increases the degree of freedom

of the problem as compared to the method that represents the coating condi-

tions element by element.

3. The BEM method with cylindrical element was developed. The basic theory

of BEM, such as use of the mirror reflection technique for the half-infinity

domain, row sum elimination, and self-equilibrium, were introduced. In ad-

dition, the implementation techniques, for example, the coordinates trans-

formation, element discritization, were also included.

4. The simplified forward model was used to obtain the potentials of the pipe

on the soil surface. The potential and the current density of the pipe and the

anode could also be calculated.

The inverse part of work represents an ambitious research effort with a poten-

tially large payoff for the oil and gas transmission industry. The central question is

whether a regression approach could be used to assess pipeline coating conditions

from field data in a way that is consistent with the laws of physics. The rationale,









methodology, and results are presented, which demonstrate the feasibility of the

inverse model for pipelines.

1. This work has demonstrated that it is possible to couple a boundary ele-

ment forward model with a nonlinear regression algorithm to obtain pipe sur-

face properties from measured soil-surface potentials. The resulting model

is called an inverse model.

2. The technique identifies the location of coating anomalies as well as the

breadth of the anomaly and the amount that the local resistivity has changed.

3. The performance of the inverse model is sensitive to the regression procedure

used. The simulated annealing algorithm proved to be the most robust and

had greatest capability to seek the global minimum for this problem.

4. An algorithm was developed that could be used to identify the maximum

number of coating anomalies that can be detected. This number is sensitive

to the quality of data as well as to the actual coating condition.

5. If the number of coating anomalies detected is smaller than the actual num-

ber of coating defects, the technique will identify the most serious anomalies.

The results of this effort, limited to a single pipe in a right-of-way, demonstrate the

feasibility of a program to interpret survey data in terms of the state of protection

of the pipe.















CHAPTER 5
DEVELOPMENT OF THREE-DIMENSIONAL FORWARD AND INVERSE

MODELS FOR PIPELINE WITH BOTH POTENTIAL AND CURRENT DENSITY

SURVEY DATA

5.1 Forward Model

5.1.1 Introduction

In Riemer's work, there were two separate domains for the flow of current. The

first was the soil domain up to the surfaces of the pipes and anodes. The boundary

conditions for the soil domain were the kinetics of the corrosion reactions. The

second domain was the internal pipe metal, anode metal and connecting wires for

the return path of the cathodic protection current.38 The BEM method was used

for the first domain. The FEM method was applied to the second domain. Brichau

also coupled the BEM and FEM to solve the two domains.72 In this chapter, to

simplify the calculations, not only the potential but also the current density along

the pipe steel will be obtained by using the BEM method.

5.1.2 Pipeline with Varying Steel Potential

In chapter 4, the potential V on the pipe steel was assumed to be a constant.

Actually, the potential on the pipe steel varies along the pipe since the pipeline has

its own internal resistance. For a short section of pipe, since the potential drop due

to the pipe resistance is very small, it is reasonable to assumed it to be constant.

However, for a long pipeline or a high resistance pipe, the potential drop along

the pipe steel is significant and cannot be ignored. The model in chapter 4 was

extended to account for the electrical potential of the pipeline steel.














i0 in
I q
--- __ -y--

Pipeline


+iqa
Lc2
L2 r:
Anode
rim
a
Figure 5-1: Cathodic protection system with variant potential along the pipe and
anode.

The method of calculation is illustrated schematically in Figure 5-1, where a

horizontally placed pipeline is connected by a wire to a vertically placed sacrifice

anode. The connection points are cl on the pipe and c2 on the anode. In Figure 5-1,

i represents the current density along the axial direction of pipeline or anode, and

subscripts p and a designate pipeline and anode, respectively. The current density

entering the pipe at the ends is designated by i and i;, and the current density

entering the anode at the ends is designated by i, and i. The current density

entering the pipe coating in the radial direction is given by q. As shown in Figure

5-1, the protecting current flows away from the anode to the soil, then flows to the

pipeline. On the pipeline, the currents flow from the two opposite directions to

the connection point cl and they are concurrent as current I. The current I flows

back to the anode through the wire.

The current density in the pipeline steel along axial direction is given by

1 1 dV
i = (- n VV) 1 dV (5-1)
Psteel Psteel dz

where V represents the potential of the steel in the pipe. Equation (5-1) can be













Vk-1 Vk


As


Figure 5-2: Relation of potential V and current density.

integrated in the axial direction along a segment with length dz, shown in Figure

5-2, such that

V1 dV= -psteelZk idz (5-2)

The potential change across the segment is given by


Vk Vk- = -Psteel k ( zk k-1 1 Zk )dz (5-3)
Jk-l Zk z-1 Z Z-k 1

or
(ik-1 + k)
Vk Vk-1 = Psteel(Zk Zk-1) k (5-4)

where the potential difference between two points is found as the product of the

average current density (ik1 + ik)/2 and the resistance psteel(Zk Zk-).

Conservation of charge requires that the current flowing into the pipe segment

is equal to the current flowing out. Thus,


ik = k + (5-5)


where q is the average current density entering the coated surface of the element

in the radial direction. Ac and As represent the areas of the steel cross-section

and side walls, respectively. For the cathodic protected system, the protecting

current flow away from the anode to the soil, and then flow to the pipeline. On

the pipeline, the currents from the two sides of the connection point cl flow head
















Figure 5-3: Axial direction current flow along the pipeline.
to head and are concurrent as current I flows back to the anode through the wire
which is connecting the pipe and the anode. Therefore, for each segment of the

pipe before connection point cl, the potential difference is,

Vo V = (io + i)R = [2io (A-'. (q0 + q ) ]R

V1 V2 = (i 2)R = [2i-(A (qo + 2ql + 2)
,A) 2


V, V, = ( = (ic'+ic) = [2io- (A (qo + 2q + -+ 2q1 + ql)]R

For each segment of the pipe after the connection point cl

SA (q'+2q" P^1+...2qcl q1+ql)
-Vc+l+V, = (ic+l +ic)R=[2in+(Ac. +2q" +( +2qc + qc]



-Vn+Vn 1 = (in+i- 1)R=[2in+ (A (q + -)]R

There are n equations if the pipeline is discritized as n segments, while there are
n + 1 the unknown variables Vo, V1, Vn. One more equation is needed in order
to get a unique solution. The potential V = 0 can be set at any position along the
pipe. In this study, zero potential was set at the pipe and anode connection point
cl, that is, Vc, = 0. The current flow in the anode through the wire will flow to
the different direction along the anode. Likewise, for each segment of the anode








Pipeline




I


Anode


Figure 5-4: Axial direction
before the connection point c2,

Vo V = [2io
V V [2 () A


current flow along the anode.


S(q+0 qi)
2
(q0 + 2q + q2)
2


S[2i (Aj+2 (o+2q1+ +2q2+ qC)
( C)

For each segment of the anode after connection point C2

S(Aa (qm + 2qm1 + .. .+2qC2+1 + qC2
-VC+2i+V = [2im+ A 2


-Vm + Vm-1 = [2im+ A (m + q2- ]

Corresponding to the m segments of the anode, there will be m + 1 unknown vari-
ables Vo, V1, Vm. Another equation is needed. If the wire connecting the pipe
and the anode has resistance Rwire, and the current flow through the wire is I, then


V, = Vc2 IRire = 0


(5-6)


In order to satisfy the self-equilibrium limitation, the current I needs to obey the
relations as below.









For the pipeline, as shown in Figure 5-3,

ic = io _A\ (q + 2q1 +.+ 2qc-l + qc) (5-7)
FO ~AP 2
ic A A, (qCl +2qcl+ +.. +2q +q) (5
Fn AP 2 (5-8)

where ifO is the current density coming from left end of pipe to the point cl, and ic

is the current density coming from right end of pipe to the point cl. By combining

these two equations,

(io ic )AP = I (5-9)

We have

(io i")A ( 2 2q ) AP = I (5-10)

For the anode, as shown in Figure 5-4,

C2 = i A(As' (qO + 2q + .+ 2qC21 + qC2)
iO = 2 (5-11)
ro\ A 2

S= m A (C2 + 2qC21 + ... + 2qm- + qm) (5-12)
Tm + kA" 2

where iC0 is the current density enter at C2 and flow to the upper end of anode, and

iT 2 is the current density enter at c2 and flow to the lower end of anode. Again, by
combining these two equations,


(iT -0 iT2)A = I (5-13)

We have

(- i im)A- (q 2' (5-14)

Summarizing the n + 1 equations for pipeline in matrix form, and defining K,, F,,










Vp, qp and iend as


-1 0

1 -1


0 1

0


-1 0

1 -1


... ... ... ... 0 1
.. 1 0 ......
010**


S0


. 0

0


0

0



0

0










2R

















A, 2(
IA?2 )


2

0


0

0


1 0

-1 -2


q0






qCl 1 -






S= Fpq


KV =Ti ends + Fpqp
p P p


Vo

V,



VI,

Vc +1



Vn-1

Vn


We get


(5-15)










Likewise, define Ka, Fa, V,, q, and i"d as


Ka




















Ta = 2R


1 -1 0


S. 0


0 1 -1 0


0 1 -1 0


0


0 1 -1 0 *** 0



0 1 -1
. .... ... .. 1 -


0 1 0


AcR AcR
2R wire 2R Rwire

















As
Aaj 2
c


Va =









The m + 1


1

1



1

0



0

A Rwire


Vo

V1



C2 e mindss = 0
,qa qC2 ia
VC2+ qC2



Vm-
t qM
Vn

equations of the anode in matrix form are,


KaVa = Tai d + Faaq


Introducing the equations (5-15), (5-16) into equation (4-61) in the previous chap-


0

0


I U



2

0


1 0

-1 -2


0

-1



-1

A Rwire


(5-16)









ter, we obtain

(-n V7),
K, 0 0 0 0 Vp Fp Tp 0 0

0 Ka 0 0 0 Va 0 0 F, T,
in
0 0 Hy, Hpa -1 < p = Gpp 0 Gpa 0
(-n V )a
0 0 Hap Haa -1 Oa Gap 0 Gaa 0
1a
0 0 0 0 0 Ap Aa
i"
(5-17)

The forward model, equation (5-17), can be used to calculate both the potential

on the soil surface, and pipe steel potentials. Furthermore, pipe steel potentials

allows calculation of the current flowing in the pipe, which can be compared with

the field measurements.

5.1.3 Simulation Results

In order to show the simulation results with variant pipe steel potentials, a test

case of CP system was studied. A 500 m long pipe is assumed to be buried 1.45 m

(4.75 feet) below the soil surface. Its diameter is 0.457 m (1.5 feet). A anode is

placed 50 m away from the center of the pipe. The diameter of the anode is 0.152 m

and its length is 1.22 m (4 feet). The soil resistivity is 100 1Qm. The boundary

conditions are shown in Figure 4-1. A 0.5 mm-thick coating is assumed to cover

the side area of the pipe, with the exception at the two end, which are assumed

to be insulated. A wire connected the pipeline to the anode at 50 m position and

0.5 m position, respectively. The potential of the pipe steel varies along the pipe.

A coating defect is set on the middle of the length of pipeline.

The simulation results of anode and pipeline are shown in Figure 5-5 and Fig-

ure 5-6 respectively.






























I I I I I I I I '
-24 -22 -20 -1 8 -1 6 -1 4
Position /m


-24 -20
Position /m


-1.0x107


75
-2.0x107

-3.0x10'


-2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.2
Position /m


012


-24 -22 -20 -1 8 -1 6
Position /m


-14 -1 2


Figure 5-5: Potential and current density along the anode: a) potential Y along the
anode (given value); b) potential V along the anode; c) radical direction of current
density along the anode; d) axial direction of current density along the anode.


0020

E
0016
.--i

- 0012

0
0008


I I,


























E 60
C
r 50

S40

"3 30

S20

co
o 10
0


0 100 200 300
Position / m




(a)


0

E


-20
-40
0)

" -40


O


-60


500


0 100 200 300
Position / m


I I I I I














0 100 200 300 400 50

Position / m


I I I I


I I I I




















0.020

- 0.016

c 0.012
()
0
a_ 0.008
()
(D
g 0.004

0


0 100 200 300
Position / m


c, U.4 I I
E |location
Q-
E 0 ------ -
location of bond
'c -0.4
(D
0
C

S-1.2-


-1.6 I
0 100 200 300

Position / m


400 500


400 500


Figure 5-6: Simulated axial distributions along the pipeline: a) input value for
coating resistivity; b) radial component of current density; c) calculated value for
potential 4; d) calculated value for steel potential V; and e) axial component of
current density in the pipe.









5.1.4 Analysis of the Simulation Results

Figure 5.5(a) shows that 4a = -1.1V, because the anode is a sacrificial anode

and its potential is a given boundary condition. Figure 5.5(b) indicates the poten-

tials along the anode steel are variant. At the position where the potential equals

zero, a wire connects the anode to the pipeline. In order to protect the pipeline,

the current of the anode flows away from the anode, and flows back to the anode

through the wire. Therefore, the farther the position is from the wire connected

position, the lower the anode steel potential than that of the potential at the wire

connection point. Figure 5.5(c) exhibits the radial direction of current density. Fig-

ure 5.5(d) shows that the axial direction of current density is equal to zero at the

two ends of the anode. This is a given boundary condition. The current densities

at the two sides of the wire connection point have different signs, which means

that the current flows away from the wire connecting position to two opposite

directions.

The coating resistivity of the pipeline is shown in Figure 5.6(a) where the nom-

inal resistivity of defect-free coating is seen to be 5.0 x 107'2m. The resistivity was

assumed to decrease abruptly at a position of 250 m, corresponding to the posi-

tion of a significant coating defect. The cathodic radial current density increased

dramatically at the position of the coating defect, as shown in Figure 5.6(b). Figure

5.6(c) shows that the potential is less negative at the coating defect position than

other places. It means that pipe is less protected at the coating defect.

The variation of potential within the pipe steel is presented in Figure 5.6(d).

Since the length of the 500 m pipe was not large, the potential drop over the pipe

was very small, on the order of 10 3 10 2 /V. The calculation of steel potential

was primarily useful for allowing calculation of the current flow in the pipe. The

steel potential was set to a value of zero at the 50 m position where the anode and