RESERVOIRVAPORLIQUID EQUILIBRIUM

INTRODUCTION Thedrivingforceforproduc0onofreservoiruids(oil,water,gas) androckmatrixwhicharecontainedunderhightemperatures pressures;theyarecompressedrela0vetotheirdensi0esat standardtemperaturesandpressure. Areservoirdrivemechanismisasourceofenergyfordrivingthe uidsoutthroughthewellbore. ItisnotnecessarilytheenergyliCingtheuidstothesurface, althoughinmanycases,thesameenergyiscapableofliCingthe uidstothesurface.

IDEALSOLUTIONSBehaviourofanidealsolu0on: nochemicalinterac0onoccurs nointermolecularforcesoccurwhenmixingcomponents resultinnohea0ngeectswhenidealsolu0onsaremixed thevolumeofthemixtureequalsthesumofthevolumesthepure componentswouldoccupyatthesamepressureandtemperature

RaoultsLaw Raoultsequa0onstatesthatthepar0alpressureofacomponentinthe gasisequaltotheproductofthemolefrac0on,xjintheliquid,mul0plied bythevapourpressureofthepurecomponentpvj. p j=xjpvj

where: pjisthepar0alpressureofcomponentjintheliquidwithacomposi0onxj pvjisthevapourpressureofthepurecomponentj.

DaltonsLaw LawDaltonslawstatesthatthepar0alpressureofacomponentpjis equaltothemolefrac0onofthatcomponentinthegas,yj0mesthe totalpressureofthesystemp. pj=yjp whereyjisthecomposi0onofthevapourandpisthepressureofthe system 2.3IdealEquilibriumRa0oBycombiningtheabovetwolaws,yjp=xjpvj (3)yxppjjvj=(4)

IdealEquilibriumRaDo BycombiningtheRaoultsandDaltonsLaws:

y jp = x jp vj yj p vj = = Kj xj p i.e.thera0oofthecomponentinthevapourandliquidphasesis givenbythera0oofthevapourpressureofthepurecomponentto thetotalpressureofthesystem.Thisra0oistermedtheEquilibrium raDo,Kj

Given: n=totalnumberofmolesofmixture zj=molefrac0onofcomponentjinthemixture ng=moleofgas nL=moleofliquid ng+nL=1 Then,

z jn = x jnL + y jn g p vj z jn = x jnL + x j ng p z jn xj = p vj nL + ng p

x j = 1.0 and y j = 1.0 z jn xj = = 1.0 p vj j=1 j=1 nL + ng p Similarly :c c

z jn yj = = 1.0 p j=1 j=1 n + nL g p vjc c

Ifabasisofonemoleofmixtureisusedi.e. n L + n g = 1.0

zj xj = = 1.0 p vj 1+ n g 1 p zj xj = = 1.0 p 1+ nL 1 p vj Usingtheseequa0onsinatrialanderrormethodthecomposi0ons ofvapourandliquidstreamsinaashsepara0oncanbe determined.

Theequilibriumra0oKjisdenedasthera0oofthecomposi0onofj inthevapourtoliquidphase,i.e.

yj Kj = xj ClearlyKjisdenedatapar0cularpressureandtemperature.

OthernamesforEquilibriumra0o,includeKfactors,Kvalues, equilibriumvapourliquiddistribu0onra0os.

Fugacity Theconceptoffugacityisintroduced,foruseinequilibriumcalcula0ons, toextrapolateorcorrectvapourpressures. Thisisrequiredsinceapurecomponentonlyhasavapourpressureupto itscri0calpoint. Thefugacityisathermodynamicquan0tydenedintermsofthechange infreeenergyinpassingfromonestatetoanother.Foranidealgas,the fugacityisequaltoitspressure,andthefugacityofeachcomponentis equaltoitspar0alpressure.Thera0ooffugacitytopressureistermed thefugacitycoecient,. Foramul0componetsystem,

fi = Pz i Allsystemsbehaveasidealgasesatverylowpressures,hence>1 whenP>0 Whenfugaci0esarenot1,thenthisgivesanindica0onofnonideality. Fugacityhasbeenimaginedasameasureoftheescapingtendencyof moleculesfromonephasetoanadjacentphase.

Fugacity Inmul0componentsystems,ifthefugacityofacomponentinadjacent phasesisthesame,thetwophaseswillbeinequilibriumwithnonet transferofmoleculesfromonephasetoanother.Atequilibriumtherefore

fg = fL Thefugacitycoecient,ofapurecomponentcanbecalculatedfrom thefollowinggeneralequa0on.

p z 1 1 v RT ln = Pdv dp = (z 1) lnz + RT v 0 P Thera0oofthefugacityatthestateofinteresttothatatareferencestate iscalledtheac0vity i = fi /fio Theac0vityisameasurethereforeofthefugacitycontribu0onor ac0venessofthecomponentinamixture fi = ifio Thera0oofac0vitytoconcentra0oniscalledtheac0vitycoecienti, where i i /x i =

Therefore : fi = ix ifio

NONIDEALSYSTEMS Idealsolu0onassump0onscannotbeappliedtothesystemsrelevantto mul0componenthydrocarbonuidsinreservoirow,transportand processingcondi0ons. Theidealassump0onsonlyapplytolowpressuresandmoderate temperatures,chemicallyandphysicallysimilarcomponentsand behaviourbelowthecri0calpoint. Dierentmethodshavebeendevelopedfortrea0ngvapourliquid equilibriumfornonidealsystems.ThepreviousKvalueisbasedon bothidealandidealsolu0onslaws. Toextendtheprincipleofequilibriumra0otomul0component hydrocarbonmixturestothepressuresandtemperaturesrelevantto petroleumengineers,methodsoftrea0ngnonidealsystemsneedtobe established. WeassumeinthatKvaluesareavailableeitherfromwhateversource, experimental,NGPSAdatacharts,orfromequa0onsofstateandother predic0vemethods.

EQUATIONSFORCALCULATING EQUILIBRIUMRELATIONSVapourLiquidCalculaDons Thecalcula0onsfordeterminingtheamountofliquidandvapour presentinamixturewhenthepressureandtemperatureisknownare obviouslyimportant,forexample,inop0misingtheperformanceofa separatorprocess. Thecalcula0onsfordeterminingthe amountofliquidandvapourpresentina mixturewhenthepressureand temperatureisknownareobviously important,forexample,inop0misingthe performanceofaseparatorprocess. Theequilibriumequa0onswhichare usedforaprocessseparatorarethe sameasthosewithinagridblockor elementofareservoirsimulator. Figurerepresentssuchasepara0on.

Given: F=totalmolesofsystembothliquidandgas L=totalmolesofmaterialwithinliquidphase V=totalmolesofmaterialwithinvapourphase zj=molefrac0onofjthcomponentinthemixture xj=molefrac0onofjthcomponentintheliquid yj=molefrac0onofjthcomponentinthevapour ItiscommontoexpressthefeedFas1.0or100molesandexpressLand Vasfrac0onsorpercentagesofF.

z j = x jL + x jK jV zj

F = L + V = 1.0 z jF = x jL + y jV z j = x jL + y jV yj Kj = xjm m j=1 j=1

z j = x j (L + K jV) xj =m

(L + K jV)m

y j = x jK jm

j=1 m

xj = yj = m

zj

j=1

(L + K jV)zj

= 1.0 = 1.0

x j = y j = z j = 1.0j=1

j=1

j=1

(V + L /K j )

m

j=1 m

zj

(L /V + K j )zj

=V

j=1 L VK + 1 j

=V

Theseequa0onsarethekeyequa0onsin vapourliquidequilibriumcalcula0ons Theiruseisthesamewhetherinthose calcula0onstodeterminephasebehaviourin aseparatororthosewhichtakeplacewithin thenumerousgridblocksofareservoir simulator.

ThemethodofcalculaDon: (1)SelectKjforeachcomponentatthetemperatureandpressureofthe system (2)Assumeavapourliquidspliti.e.V&LsuchthatV+L=1.0 (3)CalculateeitherV,L,xjoryjfromequa0ons (4)Either: (i)checkV&LcalculatedagainstassumedVorL (ii)determineifxjoryj=1.0 (5)Repeatthecalcula0onun0lassumedvalueiscalculatedvalueorun0l xjandyj=1.0.

DewPointCalculaDon Thedewpointiswhentherstdropofliquidbeginstocondense. Atthispointthecomposi0onoftheliquiddropishigherinheavier hydrocarbonswhereasthecomposi0onofthevapourisessen0allythe composi0onofthesystem. Atthedewpointtherefore:

z j = yj

z j = x jK j

Themixtureatthedewpointisthereforeinequilibriumwitha quan0tyofliquidhavingacomposi0ondenedbytheaboveequa0on. Clearly:

j=1

xj =

m

m

zj

j=1 K j

= 1.0

BubblePointCalculaDon Thebubblepointiswhentherstbubbleofgasappears. Atthispointthecomposi0onofthisbubbleofgasishigherinlighter hydrocarbonswhereasthecomposi0onoftheliquidisessen0allythe composi0onofthesystem. yj Atthebubblepointtherefore: z j = x j z j =

Kj

Themixtureatthebubblepointisthereforeinequilibriumwitha quan0tyofliquidhavingacomposi0ondenedbytheaboveequa0on.j=1

z y j = jK j = 1.0j=1

m

m

Thedewpointandbubblepointwheneithertemperatureorpressure areknownaredeterminedbytrialanderrortechniquesun0ltheabove rela0onshipsaresa0sed. Thedewpointpressureorbubblepointpressurearees0mated,K valuesobtainedandequa0onschecked. Ifthesumma0on1,dierentpressurevaluesaretriedun0l convergenceisreached.Whenconvergenceisreachedtherespec0ve dewpointorbubblepointpressurehasbeenobtained.

SEPARATORPROBLEMS Inaseparatorastreamofuidisbroughttoequilibriumatseparator temperatureandpressure.Vapourandliquidareseparatedwithinthe unitandcon0nueasseparatestreams. Severalseparatorscanbeoperatedinserieseachreceivingtheliquid phasefromtheseparatoropera0ngatthenexthigherpressure. Eachcondi0onofpressureandtemperatureatwhichvapourandliquid areseparatediscalledastagesepara0on.Henceaprocessusingone separatorandastocktankisatwostageprocessathreestageprocess hastwoseparatorsandonestocktank.(seenextFigure). Separatorcalcula0onsareperformedtodeterminethecomposi0onof products,theoilforma0onvolumefactorandthevolumeofgasreleased perbarrelofoilandtodeterminetheop0mumseparatorcondi0onsfor thepar0cularcondi0onsofuid. Usingequilibriumcalcula0onsalreadyderivedwecancalculatethe separa0onachievedateachstage,thecomposi0onofthephases separated,thegas/oilra0o,andtheoilforma0onvolumefactor.

StageSeparaDon

Gas/OilRaDo Gasisremovedfromeachstagesothatthesolu0onGORcanbe calculatedforeachstageorcombina0onofstages.

sum of gas volumes (scf) Total Gas /Oil Ratio = = RT volume of stock tank oil (bbl)(a) Calcula0onforStockTankOil,STO

Ifn1molesenter1ststage,molesofliquidentering2nd=n2=n1L1 whereL1=separa0oninstageonebasedonbasisofonemolefeed Numberofmolesenteringthirdstagen3=n2L2=L2L1n1 If3rdstageisthestocktankthen: nST=L3n3=L3L2L1n1m

nisthemolesofliquidinstocktankfornmolesintorstseparator:

nST = n1 L ii=1

m=numberofstages Li=molefrac0onofliquidoithstage n1=molesoffeedtorststage

Ifn1=1.0

nST = L ii=1 m

m

nST = L i = mole fraction of STO in the feedi=1

(b)Calcula0onofTotalGas

ngi=numberofmolesostagei ng1=V1n1 ng2=V2n2=V2L1n1 ng3=V3n3=V3L2L1n ingeneralfortotalgas:

n gT = n gi = n1 Vi L ji=1 i=1 j=1 m

m

m

i1

If n1 = 1 n gT = Vi L ji=1 j=1

i1

ngT=molefrac0onoftotalgasin thefeed Totalgasvolumepermoleoffeed =ngTVmcuCwhereVmisthe molarvolume

OilFormaDonVolume FactorVolumeofstocktankoilpermoleoffeed

(VST ) m =

n ST MST ST

MST=molecularweightofstocktankoil nST=molesofSTOpermoleoffeed ST=densityofSTOatstandardcondi0onslb/bbl Totalgastooilra0o,RT=

n gT Vm n gT VmST = (VST ) m nSTMST

Ifthefeedtotherststageisasinglephaseliquidintoitspointofentry intotheproduc0onstreamthenBocanbecalculated. res=densityoffeed(lb/bbl) Volumeofreservoiroilpermole=Vres=Mres/res

Vres M resST = Oilforma0onvolumefactor,Bo= VST resMST n STwhere: Ib.res Mres=molecularmolecularweightofreservoiruid=

Ib.mol

Ib.mol. stock tank fluid nST= Ib.mol.res.fluid

OpDmumPressureofSeparatorSystem Theopera0ngcondi0onsofpressureandtemperatureofaseparator inuencetheamountofgasandstocktankoilproduced.Changeinthese valueswillchangetheGORandtheBo. Inquo0ngthesevaluesthereforeitisimportanttokeepnoteofthe associatedsepara0oncondi0onsofpressureandtemperature.Anumber ofunitsinseriesalsoinuencetheseparameters. Itistheroleoftheprocessdesignerstoop0misetheopera0ng condi0onsofsuchlimitsandthenumberofunitsrequired.Itisthe equilibriumcharacteris0csoftheindividualcomponentsasafunc0onof temperature,pressureandcomposi0onwhichinuencethesetotal separa0oncharacteris0csforthemixturesateachsepara0onstage.

OpDmumPressureofSeparatorSystemFigurebelowillustratestheinuenceofachangeofpressureforatwostage separa0onprocessonGOR,Boandthedensityofstocktankoil.Equilibriumash calcula0ons,whichtheabovearecalled,areusedinmanyotherapplica0ons.In reservoirengineering,ashcalcula0onsareatthecoreofcomposi0onalsimula0on.

ExampleofSeparatorProblem ExamplefromMcCain(seehandout) IntegraDonoftheBlackOilandComposiDonalApproach Theexampleaboveillustratesthecombina0onofthecomposi0onal basedpredic0onofphasevolumesandassociatedproper0esandthat basedaroundtheblackoilmodel,centredaroundparametersofoil forma0onvolumefactorandgasoilra0o. Bysuchacombina0on,theweaknessesofthesimpletwocomponent blackoilmodelwhichisattheheartofdescribingoileldparameters, canbeovercomebyusingcomposi0onalderivedvaluesratherthan usingperhapsinappropriateempiricalcorrela0onsandcharts. InthenextchapterwewillexaminetheproceduresfordeterminingK.