An integrated simulation framework for the performance assessment of radioactive waste repositories

F.Cadinia,⇑,J.DeSanctisa,A.Cherubinia,E.Zioa,b,M.Rivac,A.GuadagninicaDipartimentodiEnergia–PolitecnicodiMilano,ViaPonzio34/3,I3Milan,ItalyEcoleCentraleParisandSupelec,Paris,FrancecDipartimentodiIngegneriaIdraulica,Ambientale,InfrastruttureViarie,Rilevamento–PolitecnicodiMilano,P.zzaLeonardodaVinci,32,20133Milano,Italybarticleinfoabstract

Wepresentanintegratedframeworkforaprocess-drivenperformanceassessmentofradioactivewasterepositories.Keyfeaturesoftheproposedmodelingstrategyinclude:(1)theuseofMonteCarlo-basedsimulationtomodelradionuclidesmigrationattherepositoryscale,whichallowssimplemanagementofrealisticscenariosand(2)theadoptionofanumericalcodetoproviderealisticdescriptionsofthedynamicsofradionuclidetransportinnaturalgroundwaterbodiesatthegeospherescale,fromthereleaselocationtopossiblehumanintakeoccurrence.Whilerepository-scalesimulationsareperformedbythein-housecodeMASCOT,thesubsequentgroundwaterflowandtransportfieldsaredepictedbymeansofthewidelyknownandextensivelyusednumericalcodesMODFLOWandMT3DMS.Anapplica-tiontoarealisticcasestudyispresentedtoshowthefeasibilityoftheapproach.Ó2011ElsevierLtd.Allrightsreserved.Articlehistory:Received26August2010Receivedinrevisedform31August2011Accepted2September2011Availableonline15October2011Keywords:RadioactivewasterepositoryPerformanceassessmentMonteCarloMODFLOWMT3DMS1.IntroductionAradioactivewasterepositoryisformedbybothartificialandnatural(geologic)barriers.Theseallowpreventingthereleaseofradionuclidesand/orretardingtheirmigrationtothegroundwater,andeventuallytothebiosphere(IAEA,1999a;Yimetal.,2000;NUREG-1573,2000).Aquantitativeperformanceassessment(PA)istypicallyper-formedtobuildtherequiredconfidenceinthesafecontainmentfunctionofaradioactivewasterepository.PAisperformedbytakingintoaccountaccidentalscenarioswhichmayleadtothereleaseofradioactivewastesfromtherepositoryandsubsequentintakebyatargetpopulation.Theassessmentusuallyentails:(i)theidentificationofthescenariosthatchallengetheintegrityoftherepositorybarriers;(ii)theestimationoftheirprobabilitiesofoccurrence;(iii)theestimationoftheconsequencesassociatedwiththereleaseofradionuclides,typicallyintermsoftheexpecteddosereceivedbyapre-definedcriticaltargetgroup;and(iv)theevalua-tionoftheuncertaintiesassociatedwiththeaforementionedestimates(HeltonandSallaberry,2009;Swiftetal.,2008).Inthisframework,afundamentalroleisplayedbytheconceptualandquantitativeanalysisoftheprocessesofradionuclidemigrationacrosstherepositorybarriersandofthedynamicsgoverningradionuclidesspreadingwithinthegeospheretoreachthemaintargetgroups.⇑Correspondingauthor.E-mailaddress:francesco.cadini@polimi.it(F.Cadini).0306-4549/$-seefrontmatterÓ2011ElsevierLtd.Allrightsreserved.doi:10.1016/j.anucene.2011.09.002Thelackofpreciseknowledgeonthephysicalprocessesinvolvedandthevaluesoftheassociatedmodelparametershamperacompletedepictionof(a)themechanismsofreleaseofradionuclidesfromthewasteforms,and(b)thetransportpro-cessesacrosstheengineeredandnaturalbarriers.Aprobabilisticapproachisasuitablewayoftreatingtheuncertaintyarisingintheanalysis.Thisrequiresaneffectivecouplingofprobabilisticcomputationalmodelsofradionuclidesreleasefromtherepositoryandsubsequentmigrationwithinthegeosphere.Inthispaper,anoriginalmodularmodelingframeworkispro-posedthatcouplesaMonteCarlosimulation-basedcompartmentmodelofradionuclidemigrationattherepositoryscale,calledMASCOT(MonteCarloAnalysisofSubsurfaceContaminantTrans-port)(Cadinietal.,2010)withnumericalmodelingoftheflowandtransportprocessesgoverningthemovementofradionuclidesinthesubsurfaceatthegeospherescale.ThelatterisperformedbymeansofthewidelyacceptedandusedcodesMODFLOWandMT3DMS(McDonaldandHarbaugh,1988;ZhengandWang,1999).Thekeyfeaturesoftheproposedframeworkare:(1)theuseofMonteCarlosimulationtomodelradionuclidesmigrationattherepositoryscale,whichallowssimplemanagementofmorerealisticfeatures,e.g.,complexgeometriesandheterogeneities;and(2)thepossibilityofadoptingawidespreadandtestednumer-icalcodetoproviderealisticdescriptionsofthedynamicsofradio-nuclidetransportinnaturalgroundwaterbodiesatthegeospherescale,fromthereleaselocationtopossiblehumanintakeoccur-rence.Moreover,thedetailedlevelofdescriptionoftheproblemofferedbybothmethodsattheexpenseofslightadditional2F.Cadinietal./AnnalsofNuclearEnergy39(2012)1–8modelingeffortsallowsusingmoredirectly-measuredphysicalparametersthanrelyingonconservativeassumptions;thiscanleadtomoreobjectiveanalysesaimedatidentifyingthemarginsofimprovementatthedesignstage,providedthatproperuncer-taintypropagationanalysesbeperformed.Thesefeaturesarehighlightedthroughapplicationtoarealistic,albeitsynthetic,casestudy.Thelatterisbuiltonareferencedesignofanearsurfacerepository(Marseguerraetal.,2001a,b;ENEA,1987)anddataavailableintheliterature(Zuloaga,2006;ENEA,1997).Thepaperisorganizedasfollows.InSection2,theMonteCarlosimulation-basedcompartmentmodelofradionuclidemigrationisillustratedwithreferencetothecasestudy(Cadinietal.,2010;Marseguerraetal.,2001a,b;ENEA,1987).Section3,describesthecouplingbetweentheMonteCarlo-basedcompartmentmodelandthegroundwaterflowandtransportanalysis.ConclusionsonthecapabilitiesoftheproposedprocedurearedrawninSection4.2.EstimationofradionuclidereleasefromanearsurfacerepositorybyMonteCarlosimulationInwhatfollows,aMonteCarlosimulation-basedcompartmentmodel(anAppendixAhasbeenaddedattheendofthepaper,forcompletion)(Cadinietal.,2010)ispresentedwithapplicationtotheestimationofthereleaseofradionuclidesfromanearsurfacerepositoryofadesignconceptstudiedbyENEA(Marseguerraetal.,2001a,b)ThedesignhassimilaritieswiththecurrentlyoperativedisposalfacilityofElCabril,Spain(Zuloaga,2006).Aone-dimensionalspatialrepresentationoftherepositoryalongtheverticaldirectionisconsidered.Fordemonstrationpurposes,groundwaterflowandtransportaredescribedwithinatwo-dimensionaldomaininthehorizontalplane.Themaincontainmentstructuresofthedisposalfacilityarethewastepackages,themodules,thecellsandthedisposalunits.Theseconstituteasetofmultiplebarriersagainstwaterflowandradionuclidemigration.TypicalwastepackagesaredepictedinFig.1.Theyconsistofsteeldrumscontainingtheradioactivewastewhichisimmobilizedinaconcretesolidifiedmatrix.Inourappli-cation,thediameterofawastepackageis0.791manditsheightis1.1m,foratotalvolumetriccapacityofaround400l.Themoduleisaconcretebox-shapedstructure,coveredandsealedwithacon-cretetopcover,whichcontainssixwastepackages.Fig.1depictsacross-sectionofamoduleadoptedinourtest-case.Theemptyspacesbetweenthepackagesarefilledbybentonite.Themoduleexternallengthis3.05m,withwidthandheightrespectivelyequalto2.09mand1.7m.Internaldimensionsare:length=2.75m;width=1.79m;andheight=1.37m.Themodulesarearrangedin5Â6Â8arrayswithinconcretestructurecellsbuiltbelowthenaturalgroundlevelandabovesomeaverageobservedwatertablelevel.Fig.2depictsthemodulesarrangementandtherepositoryplacementatadesignatedsite.Thedisposalunitisaconcretestructureembeddingarowof6–10cells,whereasthewholedisposalfacilityisingeneralmadeupofseveralunits,whicharetypicallyarrangedinparallelrows.Eachunitcanbeseenasrepresentativeofanindependentsystemwhichcanbebuiltandoperatedindividually,withoutinterferenceswiththeotherunits.Withoutanylossofgenerality,weadoptthefollowingsimplify-ingassumptions(Marseguerraetal.,2001a,b):(i)modulesareidentical;(ii)radionuclidemasstransportprevalentlyoccursalongtheverticaldirection;and(iii)lateraldiffusivespreadingissymet-rical.Undertheseconditions,theestimationoftheprobabilityofradionuclidemigrationthroughtherepositorystructuresandreleaseintothegroundwatersystembelowtherepositorycanbereducedtoaone-dimensionalproblemofestimatingthereleasefromacolumnoffiveidenticalverticallystackedmodules.Tothisaim,therepositoryverticalcolumnisrepresentedbyaone-dimensionalarrayoffourcompartments.Transitionsofradio-nuclidesacrosscompartmentsaredescribedstochastically.UndertheassumptionthatthemigrationprocessisMarkovian,theconstanttransitionratescanbeidentified,e.g.,bycomparisonofthegoverningequationswiththeclassicaladvection/dispersionmodel(MarseguerraandZio,2002).Asthetransportfeaturescanbenon-Fickian,otherdistributionscanbeusedtotakeintoaccountthecomplexityoftheprocessesthatcanbeobserved.Inthiscase,therelevanttransportparametersshouldbeestimatedbydetaileddataanalysisandmodelsatthetemporalandspatialscalesatwhichtheprocessesoccur(KawasakiandAhn,2006).Inthemodelingframeworkadopted,eachcompartmentischosensoastocorrespondtoamoduleofthecolumn,withtheexceptionofthemoduleatthetop,whichisnotcrossedbyradio-nuclidesreleasedbyothermodulesandservesonlyasaradionu-clidesourcetothedownstreamcompartment(Marseguerraetal.,2001a,b).Eachcompartmentisthenassignedaradionuclidesourceterm(S),whosetimedistributioncoincideswiththatoftheupstreammodulerelease.Thedistributionofthetransitiontimes(alsoindicatedascrossingtimes,CT)oftheseradionuclideparti-clestothenextcompartmentcanbereasonablyassumedtobeequaltothedistributionofthetimesthattheradionuclidestaketocrossamodule,fromtoptobottom.TheconceptualpictureatthebasisofthisschemeisreportedinFig.3.Themodulessourcetermandthetransition(crossing)timesdistributionsareestimatedbymeansofdetailednumericalsimu-lationsperformedwiththeMonteCarlosimulationcodeMASCOT(Marseguerraetal.,2003;KolmogorovandDmitriev,1945).Themigrationofalargenumberofparticlesof239Puthroughamoduleissimulatedonthebasisoftheprobabilitydistributionfunctionsdescribingthetransportprocessesandtakingintoaccountthephysical–chemicalpropertiesoftheradionuclideandthehostingsystem(MarseguerraandZio,2001).Thesimulationframeworkal-lowstakingintoaccountcomplexgeometriesandheterogeneitiesFig.1.Conceptualdesignofthewastepackage(left)andthemodule(right)(ENEA,1987).F.Cadinietal./AnnalsofNuclearEnergy39(2012)1–83Fig.2.Sketchofthe5Â6Â8arrayofmodulesinarepositorycell(Marseguerraetal.,2001a,b;ENEA,1987).Fig.4.One-dimensionalmodulediscretization.Fig.3.Conceptualschemeoftheproposedcompartmentmodelattherepositoryscale;acompartmentisassociatedtoamodule,excludingthefirstone.S=source;CT=crossingtime.Radionuclidemigrationprevalentlyoccursintheverticaldirection.inthephysical–chemicalpropertiesofthemediathroughwhichparticlesaredisplaced.Fig.4depictstheone-dimensionaldiscret-izationgrid(comprising3500cells)adoptedtocharacterizethemodulestructure.Theinitialdistributionoftheradionuclidesinthewastedrumregionisassumedtobeuniform.ThesourcetermandtransitiontimesdistributionestimatesarereportedinFig.5.Theseplotshavebeenobtainedbysimulatingthemigrationof5500particlestoensurestatisticalconvergenceoftheresults.ThesourcetermandtransitiontimesdistributionsdescribedabovedonotallowrelyingontheclassicalMarkovianrepresenta-tionoftheradionuclidemigrationthroughthecompartmentsoftherepositorycolumn.Thisrendersmandatorytheadoptionofnumericalapproximationschemesforestimatingtheprobabilities,Pn(t),thataradionuclidefallsincompartmentnattimet,n¼1;2;...;Nþ1(N+1indicatingthegroundwatercompart-ment).Inthiscontext,MonteCarlosimulationoffersaviablealter-nativeforestimatingtheprobabilitiesPn(t)and,consequently,theprobabilitydensityfunction,pdfout(t),oftheradionuclidereleaseintothegroundwatersystem.Thestochasticmigrationprocessofalargenumber,M,ofradionuclideparticlesinthefourcompart-mentdomainissimulatedbyrepeatedlysampling(i)thesourceprobabilitydensityofgenerationofradionuclidesineachcompart-ment,and(ii)theprobabilitydistributionoftheirtransitionsacrossthecompartments.Therandomwalkofanindividualradionuclideissimulatedeitheruntilitslifetimeexceedsthetimehorizon,T,oftheanalysis,oritexitsthedomaintoreachtheN+1compartment,i.e.,thegroundwatersystem.Thelatterisanabsorbingstatefromwhichtheparticlesarenotallowedtomovebacktotherepository.Thesimulationtimehorizon,T,isdiscretizedintoNtuniformtimesteps.Acounter,Count(n,k),isassociatedwiththenthcompartment(n¼1;2;...;Nþ1)andthekthtimeinterval(k¼1;2;...;Nt).Duringthesimulation,aunitvalueisaccumu-latedinthecounterCount(n,k)ifaradionuclideparticleresidesincompartmentnattimeintervalkduringitsrandomwalk.AttheendoftheMsimulatedrandomwalksoftheradionuclideparticles,thevaluesaccumulatedinthecountersallowestimatingthetime-dependentprobabilitiesofcompartmentoccupationPn(k)as:PnðkÞffiCountðn;kÞMð1Þ40.010.0090.0080.0070.0060.0050.0040.0030.0020.0010F.Cadinietal./AnnalsofNuclearEnergy39(2012)1–80.035Pu239 crossing time probability density [1/years]01002003004005006007008009001000Pu239 release probability density [1/years]0.030.0250.020.0150.010.005002004006008001000Time [years]Fig.5.239Time [years]239Pureleaseprobabilitydensityfunctionfromamodule(left)andPucrossingtimeprobabilitydensityfunctionofamodule(right).Analogously,thereleaseprobabilitydensityfunctioncanbeestimatedas:pdfoutðtÞffiCountðNþ1;kÞ;MÁDtkDt0andi;j¼1;2;...;Nþ1ð8ÞUnderthesehypotheses,itcanbeshown(Papoulis,2002)thatthestochastictimeTijthattheradionuclideresidesinstate(com-partment)ibeforemakingatransitiontostate(compartment)jisexponentiallydistributedwithparameterkij.Consequently,theprobabilityoftheprocessundergoingatransitionfromstate(com-partment)itostate(compartment)jinthetimeintervalmis(Papoulis,2002):pijðmÞ¼1ÀeÀkijmð9ÞConsideringatimeintervalm=DtsufficientlysmallthatonlyonetransitioncanoccurandapplyingtheTaylorexpansionofEq.(9),theone-steptransitionprobabilityfromcompartmentitocompartmentjcanbewrittenas:pijðDtÞ¼P½XðtþDtÞ¼jjXðtÞ¼i󰀇¼kijÁDtþOðDtÞð10Þwherekijistheconditionalprobabilitydensitydistributionofthetransitionfromstateitostatejconditionalonbeinginstatei,andlimOðDtÞfromDt¼0.Thus,theconditionalprobabilityofatransitionstate!0DtitostatejinthetimeintervalDtiskijÁDt,whereastheexpectednumberofradionuclidesmigratingfromcompartment8F.Cadinietal./AnnalsofNuclearEnergy39(2012)1–8itocompartmentjinthetimeintervalðt;tþDtÞisMÁPiðtÞÁkijÁDt,whereMisthetotalnumberofradionuclidesinthesystemandPiðtÞÁkijÁDtistheunconditionalprobabilityofatransitionfromitojinðt;tþDtÞ.Themigrationprocesscanthenbeprobabilisticallydescribedbythefollowingsystemofordinarydifferentialequations:dPdt¼PðtÞÁKð11Þwhere2NPþ1kÁÁÁk36Àk61j121Nþ17K¼6j¼2676N76Pþ1k2jÁÁÁk2Nþ177ð12Þ4k21Àj¼17j–25ÁÁÁÁÁÁÁÁÁÁÁÁisthetransitionratematrix.Inprinciple,Eq.(11)canbesolvedanalyticallytofindthestateprobabilitiesP(t),forgivenvaluesofthecompartmenttransitionrateskij(12).However,inrealisticcases,someoftheaboveassumptionsmustberelaxed,e.g.,toaccountfornon-homogeneitiesintimeandspace(Cadinietal.,2010;MarseguerraandZio,2001).Insuchcases,thetransitiontimesbetweenthecompartmentscannolong-erbedescribedbyexponentialdistributionsofconstantparame-terskijandanalyticsolutionsaredifficulttoobtain,ifnotimpossible;inthesecases,onemustresorttonumericalapproxi-mationschemes.AnavailablealternativeisgivenbyMonteCarlosimulation.ReferencesCadini,F.etal.,2010.MonteCarloestimationofradionuclidereleaseatarepositoryscale.Ann.Nucl.Energy37,861–866.ENEA,1987.Gestionedeirifiutiradioattivi.GuidaTecnican°26.ENEA,1997.InternalReport.ENEA,2000.Inventarionazionaledeirifiutiradioattivi.TaskForceperilsitonazionaledidepositodeimaterialiradioattivi,3aedizione(inItalian).Helton,J.C.,Sallaberry,J.C.,2009.Conceptualbasisforthedefinitionandcalculationofexpecteddoseinperformanceassessmentsfortheproposedhigh-levelradioactivewasterepositoryatYuccaMountain,Nevada.Reliab.Eng.Syst.Safe.94,677–698.IAEA,1999a.Safetyassessmentfornearsurfacedisposalofradioactivewaste.SafetyStandardsSeriesNo.WS-G-1.1,IAEAeds.,Austria.IAEA,1999b.Nearsurfacedisposalofradioactivewaste.IAEASafetyStandardsSeries–Requirements,WS-R-1,Vienna.Kawasaki,D.,Ahn,J.,2006.CompartmentModelforParticleMigrationwithResidenceTimeDistribution.IHLRWM2006,Apr.30-May4,LasVegas,NV.Kawasaki,D.,etal.,2005.MarkovChainModelParticleMigrationattheRepositoryScale.Global2005,October9–13,Tsukuba,Japan.Kolmogorov,A.N.,Dmitriev,N.A.,1945.C.r.Acad.Sci.,URSS56(1),7–10(inRussian).Lee,Y.M.,Lee,K.J.,1995.Nuclidetransportofdecaychaininthefracturedrockmedium:amodelusingcontinuoustimemarkovprocess.Ann.Nucl.Energy22(2),71–84.Marseguerra,M.,Zio,E.,2001.LookingatMonteCarlosimulationfordescribingnonlinearsorptioningroundwatercontaminanttransport.Math.Comput.Simul.55,167–176.Marseguerra,M.,Zio,E.,2002.BasicsoftheMonteCarloMethodwithApplicationtoSystemReliability.LiLoLe-VerlagGmbH,Hagen,Germany.Marseguerra,M.etal.,2001a.GroundwatercontaminanttransportinpresenceofcolloidsI.Astochasticnonlinearmodelandparameteridentification.Ann.Nucl.Energy28(8),777–803.Marseguerra,M.,etal.,2001b.SviluppodiunModelloStocasticoesuaImplementazioneinunCodiceMonteCarlo.InternalReport,PolitecnicodiMilano(inItalian).Marseguerra,M.etal.,2003.MonteCarlosimulationofcontaminantreleasefromaradioactivewastedeposit.Math.Comput.Simul.62,421–430.McDonald,M.G.,Harbaugh,A.W.,1988.Amodularthree-dimensionalfinite-differenceground-waterflowmodel,USGeologicalSurvey.Tech.Water-resour.InvestigationsBook6,586(ChapterA1).NUREG-1573,October2000.Aperformanceassessmentmethodologyforlow-levelradioactivewastedisposalfacility.Papoulis,A.P.,2002.Probability,RandomVariablesandStochasticProcesses.McGrawHill,NewYork.Swift,P.N.,etal.,2008.YuccaMountain2008performanceassessment:summary.In:Proceedingsofthe2008InternationalHighLevelRadioactiveWasteConference,LasVegas,NV,September7–11,2008,AmericanNuclearSociety,pp.575–581.Yim,Man-Sung,Simonson,S.A.,2000.Performanceassessmentmodelsforlowlevelradioactivewastedisposalfacilities:areview.Prog.Nucl.Energy36(1),1–38.Zheng,C.,Wang,P.P.,1999.MT3DMS:amodularthree-dimensionalmultispeciesmodelforsimulationofadvection,dispersionandchemicalreactionsofcontaminantsingroundwatersystems;DocumentationandUser’sGuide.ContractReportSERDP-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