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LostCirculation MechanismsandSolutions AlexandreLavrov
SINTEFPetroleumResearch,Trondheim,Norway
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Preface Lostcirculationisoneofthemosttroublesomedrillingproblems.Itseconomic costisduetothelossofexpensivedrillingfluidintotheformationanddueto nonproductivetimespentonregainingcirculation.Ifuntreated,lossesmaylead towellcontrolissues,poorholecleaning,packoffs,andstuckpipe.Choosing preventivemeasuresatthewellplanningstagecanmitigatelost-circulation problemssignificantly.Iflostcirculationcouldnotbeavoidedthroughpreventivemeasures,onehastolivewithlossesortrytocurethem.Bothpreventionand curingrequireagoodunderstandingofwhatcauseslosses,wheretheyoccur, whereandhowpreventivemeasuresandcuresmaywork,andwhereandwhy theywillnotwork.
Despitetheimportanceofthesubject,veryfewbookshavebeendedicated specificallytolostcirculationsincethepublicationofJosephU.Messenger’s LostCirculation in1981.Thereisaneedforanup-to-dateintroductorytextthat summarizesthecurrentlyavailableknowledgeandtheremainingknowledge gaps.Thepresentbookisastepinthatdirection.Theauthor’sambitionwasto provideabalancedoverviewofthecurrentstateoftheartwithinthetopicoflost circulation,takingintoaccountitsmultidisciplinarynatureandcurrentindustrial practices.Thereiscurrentlynocure-allagainstlostcirculation.Therefore,the bookdoesnotadvocateanyspecificcommercialproduct,orproductsofany specificvendor.Therearenocommercialnamesinthistext.Hundredsoflost circulationproductshavebeenintroducedonthemarketinthelastdecades. Afterreadingthistext,thereaderwillbecomebetterpreparedforchoosing effectivetreatments.
Withtheincreasingshareofdifficultwells(deepwater,deviated,highpressure,hightemperature,etc.)inthedrillingportfolio,lostcirculationincidentsare likelytooccurmoreofteninthefuture.Thebookisintendedtopreparethe readerforlostcirculationchallengesoftomorrow.Thefirstfivechaptersshould equipthereaderwithasolidunderstandingofthephysicsandmechanicsoflost circulation.Thesubsequenttwochaptersprovideabroadoverviewofavailable curesandpreventivetechniques.
Followingtheintroductorychapter1,“ TheChallengeofLostCirculation,” chapters2and3,“StressesinRocks”and“NaturalFracturesinRocks,” introducestressesandfracturesinrocks,thetwomajorfactorsthatcontrollost circulationandareessentialforunderstandinglossescausedbynaturaland inducedfractures.
Preface
Chapter4,“DrillingFluid,”offersabriefintroductiontodrillingfluids,with specialemphasisontheirpropertiesandbehaviorrelevanttolostcirculation. Understandinghowadrillingfluidworksisessentialforunderstandingtreatmentsdiscussedinlaterchapters.
Chapter5,“MechanismsandDiagnosticsofLostCirculation,”openswith foursectionsonthemechanismsoflostcirculation,namelylossesintothe porousmatrix,vugs(cavities),andnaturalanddrilling-inducedfractures.Losses indifferenttypesofwellsandformationsarediscussed(deepwaterwells, deviatedwells,depletedreservoirs,etc.).Somecurrentlyavailabletheoretical modelsarediscussedinchapter5too.
Mechanismsoflostcirculationdetailedinchapter5provideabackgroundfor thediscussionoflosspreventiontechniquesinchapter6,“PreventingLost Circulation.”Inparticular,geologicalcharacterizationprovidesimportantclues abouthowtomitigatethelostcirculationriskbybetterplanningthewelltrajectory.Chapter6alsocoverswellborestrengtheningandmanagedpressure drilling.
Chapter7,“CuringtheLosses,”coversavailabletreatmentsoflostcirculationonceithasoccurred.Itexplainshowlostcirculationmaterialsworkwhen theyenterthethiefzone.Anoverviewoflostcirculationmaterialsandsettable materialsisgiven.Commonlyacceptedand,preferably,standardizedtestingand evaluationtechniquesareessentialforprogressinanybranchofengineering,and lostcirculationisnoexception.Chapter7concludeswithanoverviewoflaboratorytestingtechniquescurrentlyemployedintheindustrytoevaluatethe effectivenessofdifferenttreatmentsandthepropertiesoflostcirculation materials.
Despiteconsiderableprogressmadeintherealmoflostcirculationinthelast 25years,therearestillmanyunresolvedissuesandoutstandingchallenges requiringresearchanddevelopment.Thesechallengesarereviewedinchapter8, “KnowledgeGapsandOutstandingIssues.”
ThepreferredsystemofunitsinthisbookisSI.Otherunitsareusedoccasionally,e.g.,whenquotingliteraturesourceswherethoseunitswereused. EquationsarebasedonSI.
LISTOFACRONYMS APL Annularpressureloss
BHP Bottomholepressure
CBP Constantbottomholepressure
CCS Continuouscirculationsystem
CWD Casingwhiledrilling
ECD Equivalentcirculatingdensity
FBP Formationbreakdownpressure
FCP Fractureclosurepressure
FIP Fractureinitiationpressure
FIT Formationintegritytest
FMCD Floatingmudcapdrilling
FPP Fracturepropagationpressure
FRP Fracturereopeningpressure
HPHT Highpressure,hightemperature
ISIP Instantaneousshut-inpressure
KGD Khristianovitch Geertsma deKlerk
LCM Lostcirculationmaterial
LOT Leakofftest
LPM Losspreventionmaterial
MD Measureddepth
MPD Managedpressuredrilling
MWD Measurementwhiledrilling
NADF Nonaqueousdrillingfluids
OBM Oil-basemud
PDC Polycrystallinediamondcompact
PKN Perkins Kern Nordgren
PMCD Pressurizedmudcapdrilling
PSD Particlesizedistribution
RCD Rotatingcontroldevice
SBM Synthetic-basemud
TVD Trueverticaldepth
UBD Underbalanceddrilling
UCS Unconfinedcompressivestrength
WBM Water-basemud
WSM Wellborestrengtheningmaterial
XLOT Extendedleakofftest
Chapter1 TheChallengeofLost Circulation Drillingawellisthemostcommonwaytoaccessoilandgasresources andgeothermalreservoirs.Duringdr illing,afluidiscirculatedinthewell.
Thisfluid(the drillingfluid)coolsthedrillstring,transportsrockcuttings outofthewell,andpreventsthesurroundingformationfromcollapse. Thebottomholepressureofthedrillingfluidiskeptwithinacertain “window.”Thelowerboundofthewellborepressureisusuallydictatedby theformationporepressureorthem inimumpressureobtainedfromthe boreholestabilityanalysis,whicheveris greater.Ifthebo ttomholepressure dropsbelowtheporepressure,aninfluxofformationfluidsintothewellmay occur.Ifthebottomholepressuredropsbelowtheminimumvalueobtained fromboreholestabilityanalysis,theformationmaycavein.
Theupperoperationalboundofthebottomholepressureischosensoasto avoidlostcirculation. Lostcirculation isasituationwherelessfluidis returnedfromthewellborethanispumpedintoit.Whenlostcirculation occurs,somedrillingfluidislostintotheformation.Lostcirculationgives risetononproductivetimespentonregainingcirculation.Accordingto Ref. [1],lostcirculationwasresponsibleformorethan10%ofnonproductive timespentwhendrillingintheGulfofMexicobetween1993and2003. Theinabilitytocurelossesandresumedrillingmay,intheworstcase, necessitatesidetrackingorabandoningthewell.
Theeconomicimpactoflostcirculationincludes,inaddition,thecostsofthe lostdrillingfluidandofthetreatmentusedtocuretheproblem.Accordingtoone estimate,thecostofdrillingfluidsamountsto25% 40%oftotaldrillingcosts [2] Giventhatbothregulardrillingfluidsandlostcirculationmaterialsareoften quiteexpensive,thedirecteconomicimpactoflosingthesesubstancesintothe formationmaybesubstantial.Thecostissueisespeciallyrelevantforoil-based mudsthatareusuallymorecostlythanwater-basedfluids.
Inadditiontothedirecteconomicimpact(costofexpensivedrillingfluid andnonproductivetime),lostcirculationmaycauseadditionaldrillingproblems.Inparticular,thereducedrateofreturnsmayimpaircuttingstransport outofthewell.Thisleadstopoorholecleaning,especiallyindeviatedand horizontalwells [3].Poorholecleaningmayeventuallyresultinpack-offsand stuckpipe.
LostCirculation. http://dx.doi.org/10.1016/B978-0-12-803916-8.00001-7 Copyright © 2016ElsevierInc.Allrightsreserved.
Losingdrillingfluidintotheformationinthepayzoneincreasesformation damageasporesandfracturesinthereservoirrockbecomeplugged withparticlespresentinthedrillingmud(barite,bentonite,cuttings,solids usedaslostcirculationmaterial,etc.).Theformationdamagecreatedbylost circulationneedstoberemovedbeforeproductioncanstart,whichleadsto additionalcosts.
Severecasesoflostcirculationmayleadtowellcontrolproblems. Inparticular,themudcolumndisappearingintotheformationmayreducethe fluidpressureinthewell,whichwillenabletheinfluxofformationfluids,in particulargas,intothewell.Thismayeventuallyleadtoakickorborehole collapse.Lostcirculationintopholesectionsmayleadtoshallowwaterflow events.
Giventhescopeofitsnegativeconsequences,lostcirculationhasbeen identifiedas“oneofthedrillingindustry’smostsingularproblems” [4]. Accordingtosomeestimates,theannualcostoflostcirculationproblems, includingthecostofmaterialsandtherigtime,isaroundonebilliondollars globally [5,6]
Lostcirculationintheoverburdencanbeequallyasbadasinthereservoir, eventhoughformationdamageisofnoconcernthere.Iflossesarenottreated properlyanddrillingproceedswithoutfirstsealingthethiefzone,subsequent cementjobscanbecompromised.Thequalityofwellcementingdepends cruciallyonplacingthecementcolumnallthewayuptothetargetheight. Ifanunpluggedthiefzoneexistsagainsttheannulustobecemented,cement slurrymayescapeintothiszoneduringthecementjob,andthecemented lengthoftheannuluswillbeshorterthanplanned.Remedialcementingcanbe employedtocuretheproblem,butthiswillincreasenonproductivetimeand incurextracosts.
Lostcirculationiscommoningeothermaldrilling(Boxes1.1and1.2) [7,8].Largefractureapertures(ontheorderofcm)oftencausesevereortotal losseswhiledrillingtheoverburdenorthereservoir.AccordingtoRef. [9],lost circulationproblemsareresponsiblefor10%ofwellcostsinmature geothermalfieldsandoftenmorethan20%ofwellcostsinexplorationwells intheUnitedStates.InIceland,ananalysisrevealedthatlostcirculationor holecollapsewastheprimarycauseofdrillingtroublesin18outof24wellsin theHengillGeothermalArea [10].Theseproblemsmayfurtherleadtocement lossesintotheformationduringsubsequentwellcementing.
Wellsdrilledinfieldswithelevatedgeothermalgradientareoftenproneto lossescausedbycooling.Whentherelativelycolddrillingfluidcomingfrom thesurfacecontactsthemuchhotterformation,therockcontractsandthehoop stressaroundtheholebecomessmaller ie,lesscompressive.Therockisthen easiertofracturebecauseofthiseffect.Ballooningandlossesobservedin someGulfofMexicowellsareattributedtothiseffect [11]
Lostcirculationiscommoninnaturallyfracturedformations.Severe ortotallossesarecommonincarbonaterocksintheMiddleEast [12]
BOX1.1LostCirculationinaGeothermalWellinIceland Lostcirculationisacommonproblemingeothermaldrilling,whereitisexacerbatedbyhightemperaturesandhardrocks.Asequenceoflostcirculationevents whiledrillingageothermalwellintheKraflafieldinIcelandin2008 09was describedbyPa ´ lssonetal.intheirpaper“DrillingofthewellIDDP-1” (Geothermics,2014,49,23 30).
Thewell,IDDP-1,waspartoftheIcelandDeepDrillingProjectandwas originallydesignedtoreacha4500-mdepth.Severeproblemswereencountered duringdrilling,andthewellhadtobesidetrackedeverytimeitapproached magmaat2100m.Mudlosseswereexperiencedoften,becomingworseasthe depthincreased.Also,thewellbecameincreasinglymoreunstableandwashed outwithdepth,whichaffectedtheholecleaningandreducedtherateof penetrationthatalreadywaslowduetothehardrock.
Minormudlossesoccurredinthefirst1000mwhiledrillingfortheintermediate18-5/800 casing.Thelosseswerecuredwithlostcirculationmaterial.
Lossesof20L/swerethenexperiencedat1432m.Theproblemwascuredby cementingthethiefzone.
Lossesinexcessof60L/swereexperiencedat2043m.Thelossescouldnotbe cured.Theweighteddrillingfluidwasreplacedwithwater,anddrillingcontinued. At2101m,thebottomholeassemblybroke.Afterunsuccessfulfishing,a cementplugwasplaced,andthewellwassidetracked.
Uponthesidetrack,totallossesoccurredat2054m.Acementplugwasplaced inthewell,andthedrillingcontinuedfrom2060m.Lossesresumedat2067m, andtotallossofcirculationoccurredat2076m.
Continuedproblemswithstuckpipeandunsuccessfulfishingattemptsat 2103mforcedasecondsidetrack.Mudlossescontinuedafterthesidetrack,and totallossofcirculationoccurredat2071m.Drillingwasterminatedupon reachingmagmaat2100m.Thewellwasthentestedandcompleted.
InanaturallyfracturedcarbonatefieldinIran,mudlosseswerereportedin 35%ofdrilledwells [3].InSaudiArabia,32%ofwellsinthenaturally fracturedcarbonateKhuffFormationexperienceballooning,while10% experiencelostcirculation [13]
Thebestwaytodealwithlostcirculationistopreventitfromhappening altogetherinthefirstplace.Inpractice,however,thismaybedifficultto achieve.Nevertheless,technologicalimprovementsinformationcharacterizationanddrillingfluiddesignenablethepreventionoflossesinmanywells. Preventinglostcirculationrequiresthatthemechanicsandphysicsofthis drillingproblemarefullyunderstood.
Themostobviouswaytopreventlostcirculationistokeepthedownhole pressuresufficientlylow ie,belowtheupperoperationalpressurebound. Inpractice,however,itisnotquiteobvioushowthisupperboundshouldbe chosen.Incompetentintactformations,theupperboundoftheoperational
BOX1.2LossesinGeothermalWellWK204inNewZealand Adramaticsequenceoflostcirculationevents,culminatinginablowout,occurred duringdrillingofaninvestigationwellatWairakeiGeothermalFieldinNew Zealandin1960.ThecasehistorywasdescribedbyBoltonetal.intheirpaper “DramaticincidentsduringdrillingatWairakeiGeothermalField,NewZealand” (Geothermics,2009,38,40 47).
Thewellwasdrillednearafault.Theinitialdesignwasasfollows:
l 406-mmcasingto27m(surfacecasing);
l 298-mmcasingto122m(anchorcasing);
l 219-mmcasingto305mordeeper,iftheconditionsallow.
Majorlosseswereexperiencedwhiledrillingforthe298-mmcasing. Cementingthecasingrequiredsixtimestheannularvolume,indicatingthat cementwentintothiefzones.
Whiledrillingforthe219-mmcasing,athiefzonewasencounteredat134m. Thezonewassealed,anddrillingcontinuedtothetargetdepthof305mwithfull returns.Thedecisionwasmadetocontinuedrillingdeeperthan305m.
Majorlossesstartedat350m.Attemptstocurethelosseswithincreasingly coarserlostcirculationmaterialswereunsuccessful.Thedrillbitdroppedby1.5m at373m.Furtherdevelopmentseventuallyresultedinstoppingthepumps.
Intheaftermathoftheevents,thelostcirculationexperiencedat350 373m wasattributedtothewellpenetratingahigh-pressurehigh-temperature(HPHT) zonenearthefault.Buildupoftemperatureandpressureintheholeafterthe pumpswerestoppedeventuallyledtobreakdownofthesealsetat134m,anda blowout.
bottomholepressureisoftensetequaltotheminimuminsitustress(minus somesafetymargin).Theupperpressureboundisoftencalled fracture pressure (Weshallprefertheterm“fracturingpressure”ratherthan“fracture pressure”inthisbooktoavoidpossibleconfusionwiththefluidpressure insideafracture.Fracturegradient,routinelyusedindrillingpractice,isthe fracturingpressuredividedbytheheightofthemudcolumn(psi/ftorkPa/m). Thefracturegradient,ingeneral,increaseswithdepthsincethebulkdensityof rocks,ingeneral,increaseswithdepth.Deviationsfromthistrend,however, arepossible.Inparticular,depletedformationsmayexhibitsignificantlylower porepressuregradientandfracturegradient.)inthiscase,sinceaninduced fracturewillnotpropagateifthewellborepressurestaysbelowtheminimum insitustress.However,asweshallsee,mudcanbelostnotonlyintoinduced fractures,butalsointohigh-permeabilityzones(gravel,unconsolidatedsand, etc.),largecavities,andnaturalfractures.Theminimuminsitustressplays onlyaminorroleinthosescenarios.Also,inducedfracturesdonotalways causelostcirculation.Aslongastheinducedfractureisshortandnarrow, lossesmightbeacceptableornotnoticeableatall.
Therefore,itwouldbemoreappropriatetocalltheupperoperational pressurebound“lost-circulationpressure”ratherthan“fracturingpressure.” Lost-circulationpressuremeanssimplythebottomholepressureabove whichlostcirculationwilloccur,withoutreferencetoanyspecific(andoften unknown)mechanism.
Thelost-circulationpressureisamajoruncertainty,evenincompetent rocks.Thisuncertaintyisincreasedindepletedorcomplexreservoirswhere porepressureandstressdistributionsarerarelyknown.Innaturallyfractured rocks,thelost-circulationpressuredependsonboththeorientationandthe apertureofnaturalfractures.Aperturesofnaturalfracturesmayindeedbeso smallthatthedrillingfluidwillnotbeabletoenterthem.Differentfracture orientationsmeanthatdifferentfractureswillopenandcauselossesat differentwellborepressures.Sincethereisusuallyagreatvarietyinboth aperturesandorientationsofnaturalfractures,itmakessensetoconsidera spectrum oflost-circulationpressures,ratherthanasinglevalue,insuch formations.Thisshiftofparadigmmayhelpinsituationswheretheupper pressureboundestimatedfromformationintegrityandleakofftestsperformed onashortopenholesectionbelowthecasingshoeislaterfoundtobe misguiding.Indeed,ashortopenholepressurizedinsuchtestsprovidesonlya sampleofthenaturalfracturesthatmaybeencounteredbythedrillbitduring subsequentdrilling.Theresultsofthetestsarethereforenotalways representativeofwhatliesahead.
Theprofilesofporepressureandfracturingpressure(or,alternatively,pore pressuregradientandfracturegradient)versusdepthdeterminethemaximum lengthoftheintervalthatcanbedrilledwiththesamemudweight.Thus,they determinethelocationofcasingpointsalongthewell.Thisisillustratedfora fictitiousverticalonshorewellin Fig.1.1 byplottingtheupperandlower operationalpressureprofiles.Thestaticbottomholepressureisshownwiththe dottedline.Inclinedpartsofthe dottedline mustpassthroughzeropressureat thesurfacesincetheannularpressureiszeroatthesurface(Thisistruefor conventionaldrillingwherethecirculationsystemisopenedtotheatmosphere.Inmanagedpressuredrilling,abackpressuremaybeapplied.).Jumps inthe dottedline signifychangesinthemudweight.Thelargestpossible lengthsoftheintervalsareshownbyasterisks.Indeed,movingthecasing pointlocatedat D1 deeperalongtheholewouldbringthestaticbottomhole pressurebelowtheporepressureinthelowerpartoftheinterval.Increasing themudweighttoremedythisproblemwouldincreasetheslopeofthe dotted line,whichwouldviolatetheupperpressureboundintheupperpartofthe interval.Thisexampleshowsthatthelost-circulationpressureandthepore pressure(ortheboreholestabilitylimit)playcrucialrolesinsettingupthe casingprogram.
Analternativerepresentationoftheproblemispossibleintermsofthe pore pressuregradient and lost-circulationpressuregradient.Thisisillustratedfor
FIGURE1.1 Casingpointsdeterminedfromthelower(porepressureorboreholestabilitylimit) andupper(lost-circulationpressure)operationalpressureboundsalongthewell.Thestaticbottomholepressureisshownwitha dottedline. Asterisks denotecasingpoints.
afictitiousverticalonshorewellin Fig.1.2.Theresult ie,thelocationsofthe casingpoints is,ofcourse,thesameasin Fig.1.1
Theabilitytopredictthelost-circulationpressureisthereforecrucialfor optimizingthecasingprogram. Figs.1.1and1.2 alsosuggestthatifthe lost-circulationpressurecouldbeincreased,longerintervalscouldbedrilled withthesamemudweight.Itwouldreducethenumberofcasingpointsand increasethewelldiameteratthetargetdepth.Thisisthemotivationfor applyingspecialtreatmentstoincreasethelost-circulationpressureintheopen hole(“wellborestrengthening”),discussedinchapter“PreventingLost Circulation.”
Evenifthebottomholepressurestaysbelowthelost-circulationpressure duringnormaldrilling,pressuresurgesduringtripsandconnectionsmay exceedthisupperboundandcauselostcirculation.Whenaconnectionis made,circulationissuspended.Duringconnection,thedrillingfluidstarts developinggelstrength.Whencirculationisresumedafterconnection,the pressureneedstobeincreasedsufficientlytobreakthegel.Thismayleadto
FIGURE1.2 Casingpointsdeterminedfromthegradientprofilesalongthewell.Thestaticmud weightisshownwitha dottedline Asterisks denotecasingpoints.
substantialpressurechangesduringconnections.Iftheoperationalpressure windowisnarrow eg,inadeepwaterwell thereductionofpressurebefore connectionmayleadtoaformationfluidinflux,andthepressuresurgeafter connectionmayleadtolostcirculation.
Tripsmaycauseformationfluidinfluxwhenpullingoutofthehole,and lostcirculationwhenrunninginhole.Preventinginfluxesandlostcirculation duringtripscanbeachieved,forexample,byoptimizingthedrillingfluid rheology.
Drillingthroughdepletedformationsisoftenrequiredinordertoaccess deeperreservoirs.Depletedformationsarepronetomudlosses.Insomewells drilledindepletedformations,lossesontheorderofthousandsofbarrelshave beenreported [14].Theminimumhorizontalstressisusuallyreducedin depletedreservoirs(chapter:StressesinRocks).Thisreductionaffectsthe operationalpressurewindowbyreducingthefracturingpressureandthereby increasingtheriskofmudlosses.
Deviatedorhorizontalwellsarepronetolostcirculation.Theoperational pressurewindowisnarrowinsuchwells.Insomecases,thewindowmayclose altogetherastheinclinationincreases.Inextendedreachwells,theproblemis additionallyaggravatedbyincreasingannularpressurelossesalongthehorizontalsection.Sincethefracturingpressureremainsapproximatelythesame atthesamedepth,thebottomholepressurewilleventuallyexceedit.
Aconsiderableshareofmudlossesoccurwhenrunningcasingorpumping cement.Runningthecasingpipegeneratesanexcessivebottomholepressure thatcanleadtoformationbreakdown.Duringcementing,highdensityand rheologyofcementresultinanelevatedbottomholepressure,likelytobethe highestpressuretheformationiseverexposedtoduringwellconstruction. Thismayleadtolostcirculationduringacementjob.
Anotherexampleofformationwherelossesarecommonissubsaltrubble zones [15].Suchformationsareoftenrepresentedbyrelativelyweakand/or fracturedshale.Ithasbeenarguedthatfracturesintheseshalesarecaused bydeformationintheadjacentsaltthroughoutgeologicalhistory [15].Pore pressureinthesubsaltshalecanbeeitherlowerorhigherthanthepore pressureinthesalt.Theformerscenarioisthecase,forexample,intheHassi Messaoudfield,whereseverelosseswereexperienced [16];thelatterscenario isthecase,forexample,intheGulfofMexico,withporepressurevs.depth schematicallyshownin Fig.1.3.Highporepressureinshalebelowthesaltis
FIGURE1.3 Porepressureprofile(solidline)aboveandbelowasaltbodytypicalofsomeGulf ofMexicowells.Thesaltbodyisshownwitha dottedline.Overpressureinthesubsaltshaleis causedbythesalt“trapping”theporepressure. BasedonSweatmanR,FaulR,BallewC.New solutionsforsubsalt-welllostcirculationandoptimizedprimarycementing.SPEpaper56499 presentedatthe1999SPEannualtechnicalconferenceandexhibitionheldinHouston,Texas; 3 6October1999.
causedbythesalt“trapping”theporepressure.Itresultsinanarrowmargin betweentheporepressureandthelost-circulationpressureinshale.Aswith otherfracturedrocks,fillingfracturesinthesubsaltshalewithcementorlost circulationmaterialisnotaneasytask.Accordingtoonereportfrom1999, “thecostofdrillingformationsapproximately1500ftabovethesaltto approximately1500ftbelowthesalthavereachedseveralmilliondollars” [15].Thenonproductivetimeassociatedwithsuchintervalscanbeweeks. Asanexample,morethanhalfofthewellsdrilledintheHassiMessaoud fieldexperiencedtotallossesinsubsaltshale [16].
Lostcirculationisexacerbatedindeepwaterdrilling.Thefracturegradient isoftenquitelowindeepwaterwells [17] (Fig.1.4).Thisresultsinanarrow operationalpressurewindowinsuchwells.Exceedingthefracturegradient canleadtomudlosses.
Lostcirculationisamultidisciplinarychallenge.Combattinglostcirculationrequiresacomplexapproachthatincludesrockmechanicalanalysis, carefulwelltrajectoryplanning,optimizationofdrillingfluidrheologyand composition,optimizationoflosspreventionandlostcirculationmaterials, andoptimizationofdrillinghydraulics [18].
Considerableadvancesinpetroleum-relatedrockmechanics,including hydraulicfracturemechanics,overthepastdecadeshaveimprovedour understandingoflostcirculation.Bettermethodsofmudlosspredictionand moreeffectivetreatmentsmakeitpossibletodrillwellsthatwouldbe impossibletodrillafewdecadesago.Atthesametime,increasinglymore difficultdrillingconditionsareencounteredaseverdeeperreservesare involvedinexplorationandproduction.Thisresultsinpersistenceofthe lost-circulationchallengetothisday.
FIGURE1.4 Fracturegradientasafunctionofverticaldepthindeepwaterdrilling:waterdepth incaseAisgreaterthanincaseB.
Lost-circulationincidentshappenregularlyallovertheworld.Inthefuture, theincidenceoflostcirculationislikelytoincrease.Fivetypesofchallenging wellsarebecomingincreasinglycommoninoilandgasindustryandwill continuetobesoinfuture:
l deepwaterwells;
l wellsindepletedreservoirs;
l deviated,horizontal,andextended-reachwells;
l HPHTwells;
l combinationsoftheabove.
Aswewillseeinsubsequentchapters,theriskoflostcirculationis increasedinallfiveofthesetypesofwells.
Theaimofthisbookistopreparethereaderforthelost-circulation challengesoftomorrow.Thisisdonebyprovidinganup-to-dateexplanation oflost-circulationmechanismsandofcurrentindustrialpractices.Westartour journeyintotherealmoflostcirculationbyreviewingsomebasicconceptsof rockmechanicsinthenextchapter.
REFERENCES [1]RehmB,SchubertJ,HaghshenasA,PaknejadAS,HughesJ,editors.Managedpressure drilling.Houston(TX):GulfPublishingCompany;2008.
[2]Le ´ colierE,HerzhaftB,Ne ´ auL,QuillienB,KiefferJ.Developmentofananocompositegel forlostcirculationtreatment.SPEpaper94686presentedattheSPEEuropeanformation damageconferenceheldinScheveningen,TheNetherlands;25 27May2005.
[3]AbdollahiJ,CarlsenIM,MjaalandS,SkalleP,RafieiA,ZareiS.Underbalanceddrillingas atoolforoptimizeddrillingandcompletioncontingencyinfracturedcarbonatereservoirs. SPE/IADCpaper91579presentedatthe2004SPE/IADCunderbalancedtechnologyconferenceandexhibitionheldinHouston,Texas,USA;11 12October2004.
[4]SandersMW,ScorsoneJT,FriedheimJE.High-fluid-loss,high-strengthlostcirculation treatments.SPEpaper135472presentedattheSPEdeepwaterdrillingandcompletions conferenceheldinGalveston,Texas,USA;5 6October2010.
[5]AlMaskaryS,AbdulHalimA,AlMenhaliS.Curinglosseswhiledrilling&cementing. SPEpaper171910presentedattheAbuDhabiinternationalpetroleumexhibitionand conferenceheldinAbuDhabi,UAE;10 13November2014.
[6]DrogerN,EliseevaK,ToddL,EllisC,SalihO,SilkoN,etal.Degradablefiberpillfor lostcirculationinfracturedreservoirsections.IADC/SPEpaper168024presentedatthe 2014IADC/SPEdrillingconferenceandexhibitionheldinFortWorth,Texas,USA;4 6 March2014.
[7]Pa ´ lssonB,Ho ´ lmgeirssonS,GuðmundssonA ´ ,BoassonHA ´ ,IngasonK,SverrissonH,etal. DrillingofthewellIDDP-1.Geothermics2014;49:23 30.
[8]BoltonRS,HuntTM,KingTR,ThompsonGEK.Dramaticincidentsduringdrillingat wairakeigeothermalfield,NewZealand.Geothermics2009;38:40 7.
[9]FingerJ,BlankenshipD.Handbookofbestpracticesforgeothermaldrilling.SandiaNationalLaboratories;2010.ContractNo.:SAND2010 6048.
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[12]SavariS,WhitfillDL.Managinglossesinnaturallyfracturedformations:sometimesnanois toosmall.SPE/IADCpaper173062presentedattheSPE/IADCdrillingconferenceand exhibitionheldinLondon,UnitedKingdom;17 19March2015.
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[15]SweatmanR,FaulR,BallewC.Newsolutionsforsubsalt-welllostcirculationandoptimizedprimarycementing.SPEpaper56499presentedatthe1999SPEannualtechnical conferenceandexhibitionheldinHouston,Texas;3 6October,1999.
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Chapter2 StressesinRocks Thestressstateinthereservoirandaroundtheborehole,androckfailure causedbymechanicalorthermalstresses,playcentralrolesinlostcirculation. Theseconceptsarereviewedinthischapter.
2.1STRESS,STRENGTH,ANDFAILURE Whenmechanicalloadsareappliedtotherock,orwhentherockisheatedor cooled,stressesareinduced. Stress isdefinedastheforceactingonasurfaceof unitarea.Ifwecutoutanimaginarycubeinsidetherockwiththefacesnormal tothecoordinateaxes,thestressoneachfacecanbedecomposedintothe normalstressandtheshearstress.Thelattercanbedecomposedonceagain intotwocomponentsactingalongtwocoordinatedirectionsparalleltotheface asshownin Fig.2.1.Normalstressesaredenotedby s,andshearstresses aredenotedby s,in Fig.2.1.Thefirstindexintheshearstress eg,“x”in
FIGURE2.1 Stresscomponents.
http://dx.doi.org/10.1016/B978-0-12-803916-8.00002-9
sxy denotesthefaceonwhichthestressisacting.Thesecondindexdenotes thedirectionofthestress.Thedirectionsofnormalstressesin Fig.2.1 correspond tocompressivestresses.Normalstressesinducelongitudinaldeformations. Compressivenormalstressesinducecontraction,whereastensilenormalstresses induceextension.Asignconventionisneededfornormalstresses.Inrockmechanics,compressivestressesareusuallyassumedpositive,andtensilestresses negative.
Equilibriumconditionsrequirethattheshearstresseswiththesamebut permutedindexesbeequal,forinstance sxy ¼ syx,andsimilarlyforothershear stresscomponents.
Theninestresscomponentsmakeupasecond-order stresstensor:
Duetothesymmetrypropertiesoftheshearstresses,thestresstensoris symmetricandthushassixindependentcomponents.
Rotatingthecubein Fig.2.1 would,ingeneral,changethevaluesof normalandshearstresses(exceptinthecaseofauniform,so-called hydrostatic compressionortensioninwhichcasetherearenoshearstresses,andall normalstresseshavethesamemagnitude).Thereexistsonespecialorientation ofthecubesuchthattherearenoshearstressesonitsfaces(Fig.2.2).Normal stressesactingonthecubefacesinthiscasearecalled principalstresses.They aredenotedby s1, s2,and s3,with s1 beingthegreatestofthethree(themost compressiveone)and s3 beingthesmallestofthethree(thetensileorleast compressiveone).Thedirectionsofthesestressesarecalled principalaxes
FIGURE2.2 Principalstresses.
Inthecoordinatesystemcoincidentwiththeprincipalaxes,thestresstensor hasonlydiagonalentries:
Influid-saturatedporousmedia,stressesarecarriedpartiallybytherock grainsandpartiallybythesaturatingfluid.Thepartofthestresscarriedbythe rockgrainsiscalledthe effectivestress.Sincetypicalsaturatingfluids,suchas oilorgas,cannotcarryshearstresses,theconceptoftheeffectivestressonly makessensefornormalstresses.Aneffectivenormalstress, s0 ,isrelatedto thetotalnormalstress, s,andtheporepressure, Pp,asfollows:
where a isadimensionlesscoefficientcalledthe effectivestresscoefficient (orthe Bioteffectivestresscoefficient).Insoilsandunconsolidatedorweak rocks,thevalueof a iscloseto1.Forothertypesofrocks,suchassandstone orshale,thevalueof a istypicallysmallerthan1.
Theeffectivestressesareresponsibleforthedeformationoftheporous media.Forporousmedia,theconstitutiveequationsoflinearporoelasticityare givenby
where εx ; εy ; εz arenormalstrainsinthe x , y,and z directions,respectively; gxy ; gyz ; gzx areshearstrains; Ed and n d aretheYoung’smodulusandthe Poisson’sratiointhedrainedregime,respectively(alsoknownasthe Young’smodulusandthePoisson’sratiooftherockframework);and G is theshearmodulus. Eq.[2.4] areananalogueofHooke’slawofclassical linearelasticity.
Inadditiontocontrollingthedeformationofporousmedia,theeffective stressesalsoenterthefailurecriteria.Foragivenrock,thevalueof a usedin thefailurecriteriacanbedifferentfromthatusedinporoelasticityequations.
Arockcanonlytoleratestressesuptoacertainlimit.Whenthatlimitis exceeded,therockfails.Rockfailurecanbeeitherintensionorinshear.
Tensilefailurehappenswhentheeffectivetensilestressactingonsomeplane exceedsthe tensilestrength.Sincethelowest(ie,themosttensile)principal stressis s3,thetensilefailurecriterionisgivenby:
where T0 isthetensilestrengthoftherock.Theeffectivestresscoefficient, a, usedtoevaluatetheeffectivestressin Eq.[2.5] isoftensetequalto1.0for bothunconsolidatedandconsolidatedrocks [1]
Thetensilestrengthcanbedeterminedinlaboratorybyadirecttensiontest. Inthistest,acylindricalspecimenoftherockissubjecttouniaxialtensionalong itsaxis.Dividingthemaximumloadthatthespecimencansustainbytheareaof thecrosssectionnormaltoitsaxisyields T0.Inpractice,thedirecttensiontestis difficulttoperform.Instead,anindirecttensiontestisused.Oneofthemost popularindirecttensiontestsistheso-called Braziliantest,inwhichadiskor cylinderofrockissubjectedtocompressionalongoneofthediametersshown asadashedlinein Fig.2.3.Inthecentralareaofthedisk,tensilestressis producedinthedirectionnormaltotheloadingdiameter.Whenthisstressexceedsthetensilestrength,thedisksplitsalongtheloadingdiameter.
Apartfromtensilefailure,arockcanfailbyshearfailure.Thishappens whentheshearstressexceedsacertainlimitoncertainfavorablyoriented planes(failureplanes).Thislimit, smax,dependsontheeffectivenormalstress s0 n actingonthefailureplane,theconceptknownasMohr’shypothesis [1]:
Inthe s0 s coordinates,theMohr’shypothesiscanberepresentedusing thefailureenvelopeshownasthesolidlinein Fig.2.4.Thestressstateslocated betweenthefailureenvelopeandthe s0 -axisareadmissible.Thestressstates locatedaboveortotheleftofthefailureenvelopecannotbeattainedwithout breakingtherock.Therockfailswhenapointinthestressspacereachesthe failureenvelopeduringloading.Itisconvenienttoillustratethisbymeansofa Mohr’scircle whichisshownasadashedlinein Fig.2.4.TheMohr’scirclein Fig.2.4 issupportedbytheminimumandmaximumeffectivestressesand showsthestressstateonplaneswithdifferentorientations.Whilethefailure envelopedescribesthepropertiesoftherockitself,theMohr’scircledescribes theactualstateofstressinducedintherockbyloading.Inthesituationshown in Fig.2.4,theMohr’scircledoesnotcrossthefailureenvelope.Thus,no failureoccurs(notethattheminimumprincipalstress, s0 3 ,iscompressivein Fig.2.4,andthusnotensilefailureoccursatthatpoint,either).If,asaresultof loading,theMohr’scircleisexpandedbyincreasingthemaximumeffective stressordecreasingtheminimumeffectivestress,theMohr’scirclemay eventuallytouchthefailureenvelope,resultinginshearfailure.Shearfailure canbeinducedalsobymovingtheMohr’scircleasawholeleftwardby increasingtheporepressureandthusreducingtheeffectivestresses.Eventually,thecirclewilltouchtheenvelopeinthiscase,againresultinginfailure.
Inpracticalrockmechanicsanalysis,thecurvedfailureenvelopeof Fig.2.4 isoftenapproximatedwithastraightline,asshownin Fig.2.5.The resultingfailurecriterionisknownasthe Mohr Coulombfailurecriterion. Theinterceptofthefailureenvelopewiththe s-axisisknownasthe cohesion, S0.Theslopeofthefailureenvelopeischaracterizedbythe coefficientof internalfriction, m.Thus,theMohr Coulombcriterionhasthreeparameters: S0, m,and T0.
Ifacylindricalrockspecimenissubjectedtocompressiveloadingalong thecylinder’saxiswhileitssidesurfaceremainsstress-free,failureoccurs
FIGURE2.4 FailureenvelopeandMohr’scircle. StressesinRocks Chapter j 2 17
whenthecompressivestressreachesthe unconfinedcompressivestrength (UCS).UCSisanimportantrockmechanicalpropertyoftherockandislinked tothecohesionandthecoefficientofinternalfrictionasfollows:
wheretan 4 ¼ m (theangle 4 isknownasthe angleofinternalfriction).In ordertoobtainbothparameters S0 and m,auniaxialcompressivetestisnot enough,sinceitonlyprovidesonepointontheMohr Coulombfailureenvelope.Atleastonetriaxialtestisneededtoobtainasecondpointonthe failureenvelopeinordertocalculateboth S0 and m.Atriaxialtestwith s1 > s2 ¼ s3 > 0isusuallyusedtothisend.
Fig.2.4 illustratesanimportantaspectoftheMohr’stheory:failureisnot affectedbytheintermediateprincipalstress, s2.Thisassumptionisnotstrictly correct,andmoreelaboratefailurecriteriahavebeenproposedtoaccountfor theeffectoftheintermediateprincipalstress;forexample,seeRef. [1] for moredetails.
2.2INSITUSTRESSES Rockstressisoneoftheinsitufactorscontrollinglostcirculation.Insitu stressescanchangeasaresultofproductionfromorinjectionintothe reservoir.Production-andinjection-inducedchangesofinsitustresseswillbe discussedin Sections2.4and2.5.Inthissection,wefocusontheoriginal insitustateofstress.
Todefinetheinsitustressesmeanstodefineallthreeprincipalstresses, withtheirmagnitudesanddirections.Sincedeformationandfailureofrocks
FIGURE2.5 Mohr Coulombfailurecriterion.
dependontheeffectiveratherthanthetotalstresses,theinsituporepressure playsacrucialrolehere.
Inanidealcaseofsedimentsdepositedashorizontallayers,andhorizontal Earth’ssurface,thetotalverticalstressatagivendepth, D,isoneofthe principalstressesandcanbeestimatedfromtheweightoftheoverlayingrocks asfollows:
where rb isthebulkdensityoftherockatdepth z; g istheaccelerationof gravity.Assumingfullsaturation,thebulkdensityisgivenby: rb ¼ frf þ (1 f)rs,where f, rf,and rs aretherockporosity,theporefluid density,andthedensityofthemineralgrains,respectively [2].
Assessinghorizontalinsitustressesisconsiderablymoredifficult,evenin thissimplestcaseofhorizontallayers.Intheabsenceoftectonicforces,the horizontalstressesaretheresultoflateralconfinementonly.Inthiscase,and assumingtherockbehaveslinearelastically,thehorizontalinsitustressescan beevaluatedbysetting
Eq.[2.4], whichyields
withthemaximum, s0 H ,andminimum, s0 h ,horizontaleffectivestressesbeing equalinthiscase.In Eq.[2.9], nd isthedrainedPoisson’sratiooftherock.The effectiveverticalstressisgivenby
where Pp istheporepressureand sv isgivenby Eq.[2.8].
Eq.[2.9] showsthattheverticalstressisthelargestcompressivestressin thissimplemodelsince0 < nd < 0.5.Thisstressregimeisknownas extensional (sv > sH).Iffaultsarepresent,thisstressregimewouldpromote normalfaulting.
Tectonicforcesmayproduceaninsitustressstatewiththelargeststress beinghorizontal.Iftheverticalstressisthesmallestprincipalstress,thestress regimeiscalled compressional (sH sh > sv).Itresultsin reversefaulting (thrustfaulting).Ifboththelargestandthesmallestinsitustressesarehorizontal(sH > sv > sh),thestress-regimeis strike-slipfaulting.
Inadditiontothebasictectonicregimes ie,normal,reverse,and strike-slip combinationsarepossible,includingnormalfaultingwithsome strike-slipcomponent,strike-slipfaultingwithsomenormalorreverse component,orreversefaultingwithsomestrike-slipcomponent. StressesinRocks Chapter j 2 19
Apartfromtectonicforces,elevatedhorizontalstressescanbecausedby geologicalhistory,inparticulariftheformationunderwentupliftcausedby erosionoftheoverburden.Thehorizontalstressesactingbeforetheuplift remainlockedintherockunlesstheyaregraduallydissipatedbystress relaxationcausedbyviscoelasticbehavioroftherock.
Therefore,inreality,thecoefficientinfrontof s0 v in Eq.[2.9] isnot necessarilyequalto nd =ð1 nd Þ,andtherelationshipbetweenthehorizontal andtheverticaleffectivestresscouldmoreaccuratelybewrittenasfollows:
Thecoefficient K0 typicallyvariesbetween1and10atshallowdepth,and between0.2and1.5atgreatdepth [1]
Crudeestimatesofhorizontalstressescanbeobtainedbyusingempirical relations,suchastheBreckels vanEekelenequationsforthetotalminimum horizontalstressattheUSGulfCoast [1]: sh ¼ 0:0053D1 145 þ 0:46ðPp Ppn Þ at
where D isthedepth; Pp istheporepressure;and Ppn isthe“normalpore pressure”correspondingtotheporepressuregradientof10.5MPa/km.
Contrarytothesimplifiedmodelgivenby Eq.[2.11],thehorizontal stressesareoftennotequal: sH s sh.
Sofar,wehaveassumedthatoneoftheinsituprincipalstressesisvertical. Eventhoughthisistrueinmanycases,thedirectionsofprincipalstressesmay begravelyaffectedbysurfacetopography.Inparticular,nearaslope,the verticaldirectionmightnotcoincidewithanyoftheprincipalstressdirection (eg,seeRef. [3]).Inthevalley,thehorizontalstresscanbelargerthanthe verticalstressevenintheabsenceoftectonicforces.
Fig.2.6 illustratestheimpactofstructuralfeaturesontheinsitustresses onshore.Indeepwaterdrilling,horizontalstressanomaliessimilartothose
FIGURE2.6 Effectofsurfacetopographyonprincipalstressdirectionsandmagnitudes.
Chapter j 2 21
shownin Fig.2.6 areobservednearsurfacesofescarpments,asopposedto intrabasinareas.Nottakingthereducedhorizontalstressesintoaccountwhen designingawellmayresultinlostcirculationincidentswhendrillingthe top-holesection [4].Directionsofinsitustressesinthereservoircanbe affectedbynearbyfaults,domes,andothergeologicalfeatures.
Toillustratecomplexitiesofinsitustressstates,letusconsiderstress distributionsaroundsaltbodies.Drillingthroughsaltisoftenaccompanied withboreholeinstabilitiesandonexitingthesalt,lostcirculation.Anumerical geomechanicalanalysisperformedinRef. [5] providesavaluableinsightinto theproblemsexperiencedwhiledrillingsalt.Saltcannotsustainshearstresses overlongperiods.Stressrelaxationleadstogradualreductionofshearstresses insalt,sothatthestateofstressgraduallyapproacheshydrostaticcompression,with sv z sH z sh.Theinabilitytosustainshearstressesforlong periodsmakessaltsimilartoaveryviscousfluidwhich,asanyfluid,would eventuallyapproachhydrostaticequilibrium.Theinsitustressesinsurroundingrocks(eg,shale)mayontheotherhandbefarfromhydrostatic.In theGulfofMexico,thetectonicregimeisnormalfaulting,andthefar-field stressesinrocksaroundthesaltaresuchthat sv > sH z sh.Thediscrepancybetweenthestressfieldsinsidethesaltbodyandoutsideitleadsto significantcomplexityofthestressfieldaroundtheboundaryofthesalt. Aroundasaltbodythatisclosetosphericalinshape,thestressesarealtered comparedwiththeirfar-fieldvalues,asshownin Fig.2.7.Horizontalstresses arereducedaboveandbelowthesalt,whichexplainsthereducedfracturing pressureexperiencedonexitingthesaltwhiledrilling.Inthesideburden ie, intherockslaterallyadjacenttothesalt thestresschangesaremore
FIGURE2.7 Alterationofinsitustressesaroundanidealized,spheroidalsaltbody(shownin gray). BasedontheresultsofFredrichJT,CoblentzD,FossumAF,ThorneBJ.Stressperturbations adjacenttosaltbodiesinthedeepwaterGulfofMexico.SPEpaper84554presentedattheSPE AnnualTechnicalConferenceandExhibitionheldinDenver,Colorado,USA,October5 8,2003.
complex:horizontalstressesinthoseareasbecomeanisotropic,evenifthe far-fieldhorizontalstressesareisotropic.Inaddition,principalstressesmaybe rotatedinthoseareas,withtheverticaldirectionnotbeingaprincipaldirection anymore [5].Thesestresschangeshaveadetrimentaleffectonborehole stability.
Reducedverticalstressesabovesaltbodiesindicatethatusingdensity profilestoevaluatetheverticalinsitustress, Eq.[2.8],wouldoverestimatethe insituverticalstressintheseareas.Thereductionintheverticalstressabove thesaltundernaturalconditionsisanexampleofthe archingeffect,similarto theoneobservedduringdepletion(Section2.4).Note,however,thatthe archingeffectin Fig.2.7 iscausedsolelybynaturalstressredistributiondueto stressrelaxationbeforeanyreservoirdepletion.Inparticular,sincesaltis unabletosustainshearstresses,theverticalstressneedstoberedistributedso thatsomepartoftheverticalloadthatwouldotherwisebebornebythesaltis carriedbythesurroundingrock.
Thesphericalgeometryofthesaltbodyin Fig.2.7 isastrongidealization. Numericalanalyseswithmorerealisticgeometriesconfirmthatreducedhorizontalinsitustressespersistrightaboveandrightbelowthesalt [6].Thisis consistentwithlostcirculationincidentsexperiencedwhenenteringorexiting thesalt.Areductionofthefracturegradientby3ppg(0.36g/cm3)underneath thesaltwasreportedforawellintheGulfofMexico [6].
Iftheshapeofthesaltbodyisnotspheroidal(eg,inthecaseofasalt sheet),stressperturbationsaroundthesaltarelesssevere.Accordingtothe numericalresultsofFredrichetal. [5],thehydrostaticstressmagnitudeina spheroidalsaltbodyliesbetweenthefar-fieldvaluesof sv and sH z sh.Ina saltbodywithlateralextensionlargerthanitsheightbyafactorof4ormore, thehydrostaticstressinthesaltisclosetothefar-fieldverticalstress.
Inadditiontoreducedhorizontalstresses,therockbelowthesaltisoften damagedorbroken.These rubblezones furthercontributetomudlosseson exitingthesalt.
Theuncertaintyaboutthevaluesof sH, sh,andtheirdirectionscallfor experimentalmethodsthatwouldenablemeasurementoftheseparametersin thefield.Someofthemethodscurrentlyusedintheindustrywillbediscussed in Section2.7.
Numericalmodelingisanotherwaytoobtainestimatesofbothverticaland horizontalstressesinandaroundthereservoir.Theuseofnumericalmodeling requiresthattherocks(reservoir,overburden,andsideburden)areproperly characterizedintermsofrockmechanicalproperties,oratleastthevaluesof theseparametersareconstrainedtowithinsufficientlynarrowmargins.Dueto scarcityofcorematerial,particularlyfromthecaprock,suchconstrainingis oftendifficult.Inspiteofthisdifficulty,examplesofnumericalevaluationof rockstressesareavailable.Forinstance,finite-elementstressanalysiswas usedtoevaluatestressesinandaroundasaltdiapirinthedeepwaterGulfof Mexico [6]
