Onset of water accumulation in oil/water pipe flow – experiments and modelling with LedaFlow

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Onset of water accumulation in oil/water pipe flow – experiments and modelling with LedaFlow

Jørn Kjølaas

^∗

, Marita Wolden

SINTEF, Norway

a rt i c l e i nf o

Article history:

Received 3 May 2020 Revised 14 August 2020 Accepted 21 September 2020 Available online 28 September 2020

a b s t r a c t

Inoil/waterflowswithverylowwaterrates,thesteady-statewaterfractionjumpsdiscontinuouslyfrom alowvaluetoahighvaluewhentheoilratefallsbelowthecriticalvalue.Thisjumpisunderstoodtobe connectedtotheexistenceofmultiplesolutions,andgenerallytakesplacewhentheoilratebecomestoo lowtosustainalowwaterfraction.Werefertothiscriticaloilflowrateastheonsetofwateraccumu- lation.Wateraccumulationinoiltransportlinesisundesirablebecauseitcanleadtocorrosionproblems thatcanthreatentheintegrityoftheinstallation,potentiallyleadingtooilleakingundetectedintothe environment,jeopardizingnearbywildlifeandecosystems.Itisthereforecriticaltomaintainaflowrate thatishighenoughtopreventwaterfromaccumulatinginoillines,andtheabilitytopredictthemin- imumallowableflowrateaccuratelyisthusofgreatimportance.To addressthischallenge,anewand uniquesetofexperimentswereconductedattheSINTEFMultiphaseLaboratory.Theexperimentswere speciallydesignedtomeasurethecriticalconditionsforwateraccumulationinoil/waterflowsandwere performedwithapipediameterof8inches(194mm)andapipeinclination of2.5degrees.Thefluid systemconsistedofExxsolD60astheoilphaseandregular tapwaterastheaqueousphase.Inthese experiments,themeasuredcriticalsuperficialwatervelocitieswereintherange0.1-2.6mm/s,whilethe criticalsuperficialoilvelocitieswereintherange0.3-0.5m/s.Wefoundthatthecustomaryapproachof modellingtheoil/waterinterfacialshearstressasasmoothwallwasinadequateforpredictingtheseex- periments,andthatinterfacialwavesmustbeconsidered.Thedataanalysisshowedthattheonsetofin- terfacialwavesiswellpredictedbyViscousKelvin-Helmholtztheory,andthatamodelfortheinterfacial shearstresscanbeconstructedwiththistheoryasastartingpoint.Anewmodelforoil/waterinterfacial shearstresswasdevelopedbasedonthisdataandtheassociateddataanalysis.Thenewmodelwasable tomatchtheexperimentaldatawellandaslightlymodifiedversionofitwasultimatelyimplementedin thecommercialmultiphaseflowsimulatorLedaFlow.

© 2020TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Becauseoftheincreasingawarenessandconcernabouttheim- pactoffossilfuels onclimatechange,aswell asthedepletion of currentoilreserves,oilproductionisexpectedtodeclinesubstan- tially in the coming years. Although this trend is desirable from an environmental perspective, itwill havenegativeconsequences foroiltransportlineswhichhavetypicallybeendesignedforhigh throughput.Oneoftheproblemsthatwilloccuristhatthedecline inthe flowvelocity allowssmallamounts ofwaterin thesystem oil (due to imperfect separation) to segregate out during trans- port andto accumulate in theinclined partsof the line.Specifi-

∗Corresponding author.

E-mail address: jorn.kjolaas@sintef.no (J. Kjølaas).

cally,ifthe flowvelocity becomessufficiently low,the watercan separateandaccumulateinuphill regions,creatingpoolsofwater that are essentially stagnant, see Fig. 1.This is a highlyundesir- able situationbecause free watercan lead tocorrosion problems ontheinsideofthepipewhich canultimatelylead tooilleaking undetectedintothesurroundings,contaminating theenvironment (Magill, 2012). Anexample wheresuch problemsare expectedto occuristheTransAlaskaPipelineSystem,whichtransportsoil800 miles from the North Slope oil fields south to Valdez on Prince WilliamSound(TransAlaskaPipelineSystem-Thefacts,2019).For thatsystemtheambienttemperatureistypicallysub-zero,andthe near-stagnant watercaneventually freeze, yielding aneven more hospitable environment forcorrosion. Inthat scenario, chunks of icecouldalsoperiodicallybepushedthroughthelineandpossibly causedamagetopumpstationequipment(Bluemink,2010).

https://doi.org/10.1016/j.ijmultiphaseflow.2020.103469

0301-9322/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Fig. 1. Illustration of water accumulation in uphill pipe sections.

Fig. 2. Example of the multiple solutions region in oil/water flows, where multi- ple water fractions are valid solutions to the steady-state equations for a range of superficial oil velocities USO .

Efficientmitigation proceduresrequireknowledgeaboutwhere andhow fastthewatercan accumulate,andaccurate modelsare then needed to make such assessments. Indeed, all preventive measures come at cost, andcould threaten the economic viabil- ity ofthe pipelineif they become toohigh. By havingpredictive models that canforecast theseproblems, one can obtaina more accuratepictureoftherequiredexpenditures.

Theprimary physicalmechanismthatdetermineswhetherwa- ter willaccumulate istheinterfacialshearstressbetweenoiland water.Whentheoilrateislow,thewaterwillsettleatthebottom ofthepipe butwillcontinuetoflow asa thinfilmiftheinterfa- cial shear stress is sufficientlyhigh. From a practical perspective, thissituationis arguablyacceptablebecause theexposureto cor- rosion problems are thenminimal. However, ifthe oilratedrops belowsome criticalvalue, thewaterwill almoststopmoving and it willinsteadaccumulate(see Fig.1).Thistransitioncanbevery distinct, andatsufficientlylowwaterrates,thetransitioniscom- pletely discontinuous.Werefertothisdiscontinuousjumpas"the onset of water accumulation",and in thispaper we will present experimentsthatexplorethisphenomenon.

From a modelling perspective, this phenomenon can be ex- plained by the notion that the steady-state flow equations have more than one solution in a certain range of oil flow rates (BarneaandTaitel,1992) (Landman,1991)(Ullmann etal.,2003), see Fig. 2. In the multiple solution region, there are three hy- pothetical solutions, and the system will then in most circum- stances"select"theonewiththelowestwaterholdup(Kjølaasand Holm, 2016). However, below a certain oil flow rate, the low- holdup-solutionceasestoexist,andthenthesystemhasnochoice thantoapproachthehigh-holdup-solution.Iftheoilratedropsbe- low thispoint, the waterwill accumulate until the high-holdup- solutionisobtained.Wewilldiscussthismatterinmoredetailin themodellingpartofthispaper.

Inthescientificliterature,wehavefoundnoarticlesaddressing thechallenge ofpredictingwateraccumulationinoil/waterflows.

Even the modelling of interfacial shear stress between oil/water in general seems to be universally neglected in the literature, as virtually allauthors(Brauneretal., 1998)(HallandHewitt,1993) (Valleand Kvandal, 1995) resortto assuming a smooth oil/water interface, disregarding the effectof interfacialwaves. Taitelet al.

(1995)andValle(2000),deviatedslightlyfromthisbasicapproach byintroducingalowerlimitof0.014ontheinterfacialfrictionfac- tor.Zaghlouletal.(2008)simplyelectedtouseaconstantvalueof 0.014.Ourdataanalysiswillhowevershowthatthissimplifiedap-

proachisinadequateforpredictingwateraccumulationininclined pipes,becausetheeffectofinterfacialwavesleadsto significantly elevatedvaluesofinterfacialshearstress.

InSection3ofthispaperwepresentasetofnewandunique large-scale oil/water experimentsthat were speciallydesigned to identifytheonsetofwateraccumulationfordifferentflowcondi- tions.Intheseexperimentsweuseaspecialtechniquethatises- sentiallythesameastheone usedpreviouslyforgas-liquidflows (Kjølaasetal.,2015).

InSection4ofthispaperweaddressthemodellingofthein- terfacialshearstress betweenoilandwater,focusing onthe pre- diction ofwateraccumulation inoil/water flowsasthe main ap- plication.It shouldhoweverbe emphasizedthat theoil/waterin- terfacialfrictionfactorisamodelthatcanbeusedinallscenarios whereoilandwaterflowasseparate phasesandcanthereforebe importantinmanyothercircumstances.

2. TheLedaFlow1Dmodel

Inthis paper,we have electedto usethe 1D multiphase flow simulator"LedaFlow"(“LedaFlow,” KongsbergDigitalAS,2020)as aframeworkforsimulatingthenewoil/waterexperiments,andfor addressingthemodellingofwateraccumulationinoil/waterflows.

Inthissectionweprovideabriefnon-exhaustivedescriptionofthe whatthissoftwaredoes,andwhatequationsitsolves.

2.1. Conservationequations

LedaFlow is a transient three-phase flow simulator designed tosimulatemultiphaseflowinpipes. Thetransport equationsare solvedinonedimensionalongtheflowdirection.Theflowisgen- erally represented by nine fields: continuous gas, oil and water, plusallpossibledispersedfields,seeFig.3.

The1Dmassconservationequationsaresolved foreachofthe ninefieldsk:

∂ (

^A

α

k

ρ

k

)

∂

^{t +}

∂

∂

^x

⁽

^A

α

k

ρ

ku_k

)

^=A

i=k

k,i+A

k,ext (1)

Here, A is the crosssection pipe area, and

α

k,

ρ

k and u_k are thevolumefraction,densityandvelocityoffieldk.k,irepresents mass transfer terms due to condensation/evaporation, and k,ext

representsexternalmasssources.

Each continuousfield combined withits two constituent dis- persedfieldsisdefinedasa"zone". Foreach zone,theassociated momentumequationissolved intheflow direction.Intheexper- imentspresentedinthispaper,theoilandwaterflowasseparate phaseswith negligible oil/water entrainment because ofthe low velocities. Thismakes themodelling more straightforward by al- lowingustoignorethedispersedfieldsinouranalysis.Indeed,for the special case of oil/water flow with no mass transfer and no dispersedfields,themomentumequationsreduceto:

∂∂^t

(

^A

α

^o

ρ

^ouo

)

⁺_∂^∂x

A

α

^o

ρ

^ou2o

=−A

α

^o∂_∂x^p

−A

α

o

ρ

ogx−A

α

ogy

( ρ

o−

ρ

g

)

^∂_∂^hx^o−So

τ

o−Sow

τ

ow

(2)

∂∂^t

(

^A

α

^w

ρ

^wuw

)

⁺_∂^∂x

A

α

^w

ρ

^wu2w

=−A

α

^w∂_∂^px

−A

α

w

ρ

wgx−A

α

wgy

( ρ

o−

ρ

g

)

^∂_∂^hx^o +

( ρ

w−

ρ

o

)

^∂_∂^hx^w

−Sw

τ

w+Sow

τ

ow

(3)

Here, the indiceso andw refer to "oil" and"water", pis the pressure,g_x istheaccelerationofgravityinthe flowdirection,g_y is the acceleration of gravity normalto the flow, ho and hw are the respective heights of the oil/water layers, So and Sw are the oil/waterwallperimeters,Sow istheoil/waterinterfacelength,

τ

o

and

τ

^{waretheoil/waterwallshearstresses,and}

τ

^{owistheinterfa-}

cialshearstressbetweentheoilandthewater.Fornear-horizontal

Fig. 3. Continuous and dispersed fields in LedaFlow.

flows,thegeometricalparametersSo,Swand,Sowarecalculatedas- sumingaflatoil/waterinterface.

Insteady-statefullydevelopedflow, themomentumequations can be simplified further by removing the temporal and spatial derivatives, exceptfor the pressure derivative (which for obvious reasonscannotbeneglected):

0=−A

α

^o

∂

^p

∂

^x−A

α

^o

ρ

^ogx−So

τ

^o−Sow

τ

^{ow (4)}

0=−A

α

^w

∂

^p

∂

^x−A

α

^w

ρ

^wgx−Sw

τ

^w+Sow

τ

^{ow (5)}

Theanalysespresentedinthispaperwillprimarilybebasedon Eqs.(4)and(5),astheprerequisitesforusingtheseequationsare generallyfulfilledinourexperiments.

2.2. Closurerelations

The shear stress terms

τ

^o,

τ

^{w and}

τ

^{ow are modelled using a}

certain set of closurerelations in LedaFlow, butwe will not de- scribethoseindetailhere.Instead,wehaveforthepurposeofthis analysisreplacedtheusual LedaFlowclosurelawswith"standard"

closurelawsfromtheliterature,andwe havethususedLedaFlow onlyasavehicleforsolvingthe1Dtransportequations.Wefound that ourresultsdidnotdependcriticallyonthe choiceofclosure laws,sowehavechosentousequitesimpleformulations:

ThewallshearstressforzonekisinLedaFlowexpressedas:

τ

k=1

2f_k

ρ

k

|

^uk

|

^{uk (6)}

Wehaveinthispaperelectedtocalculatethefrictionfactorf_k inEq.(6)as:

f_k= f_k^W_,lam·f_k¹_,⁻_turb^W (7) Here,f_k,lamisthefrictionfactorforlaminarflow:

f_k,lam= 16

Re_k (8)

and f_k,turb is the friction factor forturbulent flow, wherewe use theexpressionproposedbyHåland(1983):

1 fk,turb

=−3.6·log₁₀

_6.9

Re_k+

ε

3.7D_hk

¹.11

(9)

Here,Re_kistheReynoldsnumberwhichwedefineas:

Re_k=

ρ

ku_kD_hk

μ

k

(10)

Table 1 Fluid properties.

Property Value

Oil density [kg/m3] 795 Water density [kg/m3] 999

Oil viscosity [cP] 1.5

Water viscosity [cP] 1

Oil/water surface tension [mN/m] 19

where

μ

k isthephaseviscosityandD_hk isthehydraulicdiameter whichwedefineas:

D_hk=4

α

kA

S_k (11)

Finally,the laminar/turbulent weighting functionW isdefined as:

W = 1

1+

_Rek

2300

20 (12)

The starting point for this work is to model the interfacial shear stress as a smooth wall using the expression described by Eq.(20)laterinthispaper.Thepurposeofthisworkishoweverto findamoreappropriate expressionfortheinterfacialshearstress thanthesmoothwallassumption.

3. Experiments

Inthissectionwedescribetheflowloopsetupsandprocedures usedtoconducttheexperiments.

3.1. Experimentalsetup

The experimentswere conducted ina 94 mlong 8" pipe (in- ner diameter194 mm)with a 2.5° inclination using ExxsolD60 asthe oil phase and tapwater asthe aqueous phase. The setup isillustrated in Fig.4. The pipewasequipped withsixpressure- transmitters(labelledP)thatwerecoupledtoacommongas-filled reference line,six vertically mounted narrow-beam gamma den- sitometers to measure local water heights(labelled

γ

^{), and two}

traversing gamma densitometers (labelled T

γ

^{in Fig. 4). Finally,}

temperaturetransmitters(labelled T)were mountedatthebegin- ningandendofthetestsection.Thethermodynamicpropertiesof thefluidsarelistedinTable1.

3.2. Outlet

Theendofthe 8"inclinedpipeexpandsinto ashort12" pipe withthesameinclination,followedbyan80cmlong12"horizon-

Fig. 4. Schematic illustration of the test section used in the experiments.

talpipe,a90degreedownwardbendandavertical12"downward pipe.Initiallywehadsome concernsaboutthedesignofthisout- let, fearingthat watermightaccumulate inthe12" inclinedpipe, which couldperceivably influencetheflow inthe8" testsection.

Specifically,anywaterthataccumulatedtherecouldflowbackinto the8"pipeandincreasethewatercontent.

However, transient simulations of the experiments using LedaFlow (“LedaFlow,” Kongsberg Digital AS, 2020) showed that gas would always be present atthe top of the 12" pipe, so that theavailableareafortheliquidactuallydecreasedinthe12"pipe, yielding increased flow velocities instead of decreased velocities.

Based on this analysis, there was thus no real danger of water accumulating there, and the pipe expansion was ultimately not considered aproblem. Also,hadthisactuallybeena problem,we wouldhavereadilynoticeditduringtheexperiments.

3.3. Gammadensitometers

Allthe gammadensitometersconsistedofaCaesiumradiation sourceononesideofthepipeandaphotondetectorontheother side. Theattenuationofthephotonbeamdecreasesexponentially withthedensityofthemediumbetweenthesourceanddetector, allowing usto measuretheaverage densityalongthe rays’travel path.Byrecordingthephoton countrateswithpureoilandpure water inthepipe, themeasured photon ratesin oil/waterexper- iments could be converted to waterfractions, which is what we showinthispaper.

The gammadensitometerswere collimatedonboththesource side and the detectorside. With thistype of arrangement, scat- tered photons rarelyreach the detector, andthe prevailing accu- racy has been shownto be about 0.02foroil-water systems.All gammadensitometerswereloggedat50Hz,andthetraversingin- struments scanned the pipe from bottom to top witha constant velocityof0.4mm/s.

Example results from a traversing gamma densitometer (top graph) and a vertically mounted static gamma densitometer are showninFig.5(bottomgraph).Thetraversinggammadensitome- ter givesthetime-averagedwaterfractionprofile,whilethestatic gammadensitometergivestheinstantaneouswaterheight.Weob- servethat theoilandwaterflowasseparatephases,withsignifi- cantwavesonthesurface(thediffuseinterfaceobservedwiththe traversinggammaiscausedbywaves).Thetotalwaterfractionwas obtained by integrating the water fraction profiles over the pipe crosssection.

3.4. Steady-stateexperiments

Steady state experiments are experiments designed to obtain the phase fractions and pressure drop at steady (non-transient) conditions.Thesetypesofexperimentsarethemostcommonones forstudyingmultiphase flows.At thelowflow ratesexamined in thispaper,flow transientscanbe veryslow, anditwastherefore very important toallow the flowto stabilizefor sufficientlylong

Fig. 5. Example results from a traversing gamma densitometer (top graph) and a vertically mounted static gamma densitometer (bottom graph).

beforerecordingtheexperiments.Thestabilizationtimesbetween experimentsweretypicallyaround20-30minutes.

3.5. Screeningexperiments

Inpreviouspublications(KjølaasandHolm,2016)(Kjølaasetal., 2015), we have used the term "screening experiments" for ex- periments designed to findthe discontinuous transitionbetween high and low holdup in low liquid loading flows. Although this term is arguablynot a very descriptive one, we continue to use ithereforconsistency. Inthispaperwe usetheterminthecon- text of oil/water flows (as opposed to gas-liquid flows).In other words,theexperimentsdescribedhereareessentiallyequivalentto thosepresentedinKjølaasetal.(2015),exceptthatthegas/liquid systemhas beenreplaced by oil/water, andthe currentobjective was to find the discontinuous transition between high and low waterfractionforlowwaterrates.

Asmentionedinthe introduction,thediscontinuoustransition betweenlow/high waterholdup isclosely connectedto theexis- tence of multiple holdup solutions, i.e. that more than one flow configurationisinprinciple possibleforasetofboundarycondi- tions.WediscussthisinmoredetailinSection3.7.

Themainprincipleofthisexperimenttechniqueisthatweper- formmeasurements ina quasi-steady-statesituation,withahigh waterholdupinthefirstpartofthepipe,andalowholdupfurther downstream(seeFig.6).Hereweareinpracticeinatransientsit-

Fig. 6. Illustration of a screening experiment in two-phase oil/water flow.

uation, wheretheflow istransformingfroma highwaterholdup toalowwaterholdup,orviceversa,dependingontheoverallwa- ter mass balance in the pipe. The key aspect of this experiment is that we measurethe superficial watervelocity outof thepipe (USW_OUT).

Thereasoningbehindthisexperimenttechniqueisbasedonthe observationthatUSW_OUTisdecoupledfromUSW_INinthissituation (USW_INbeingthesuperficial watervelocityattheinlet).Thatim- pliesthatwecanadjustUSW_INtomatchUSW_OUT,sothatthewater front in thepipe becomesstationary. Starting fromthisscenario, any small increase in USW_IN will cause the liquid front to move towards theoutlet,eventually givingahighsolutionintheentire pipe. Conversely, asmall decreasein USW_IN would causetheliq- uid front tomove towards theinlet, givinga low solutioninthe entirepipe.Basedonthisreasoningwecanconcludethatthecrit- ical USO/USWpairgivenby USOandUSW_OUTdefinesthe "tipping point" wheretheflowgoesfromahighholdupsolutionto alow holdupsolution.

Building ontheseconsiderations, the followingprocedure was usedtomeasuretheonsetofthewateraccumulationintwo-phase flow:

• We started with an oil-filled pipe and set USO to a constant value.

• Weinjectedwateratarelativelyhighrate(USW≈0.02m/s),so thatthewaterbuiltupattheinlet,andawaterfront(hydraulic gradient)propagatedtowardstheoutlet.Whenthewaterfront wasaround themiddleofthe pipe,we stoppedthe waterin- jection.

• Wethenwaiteduntilthewaterrateoutofthepipewassteady, inordertoobtain asuitable timesequenceforcalculatingthe averagevalueofthenetwaterflowratein/outofthepipe.We willexplain howwe calculatedthe waterflow rateoutofthe pipeinSection3.6.

• Themeasured outletwaterflow rate(USW_OUT) combinedwith the current USO was subsequently interpreted as a "critical"

USO/USWpair,wheretheflowtransformsdiscontinuouslyfrom ahighholduptoalowholdup.

• Next,USOwasrampeduptoanewvalue,andanewvaluefor USW_OUTwasobtained,andsoon.

Iftheoilratehadbeenrampeduptothepointwhereonlyone solutionwaspossible(withalowwaterfraction),theexcesswater would presumablyhave been pushed out fromthe inlet. We did not trythisinthesetests, butearlierexperimentswithgas/liquid have shownthisto happen(Kjølaas etal., 2015). The reasonthis happensis that thehigh-holdupsolutionis nolonger avalidso- lutionatthoseconditions,anditisthusimpossibletomaintaina stablewaterfrontaswedointhesescreeningexperiments.

3.6. Calculationoftheoutletwaterrate

In thecurrentexperimentswe didnot havea designatedsys- tem for measuring the outlet water rate USW_OUT, so we had to come up with an alternative method for obtaining this parame- ter fromtheavailable instrumentation.We foundthat agoodap- proachwastousetheavailablepressuremeasurements:

Fig. 7. Pressure difference over the water front plotted versus time for a selected experiment.

We definethe watervolume V_w as thetotal amountofwater betweenthefirstandlastpressuretransmitteronthetestsection, andwedefinethedistancebetweenthesepressuretransmittersas

x.Theassociatedvolumebalancecanbewrittenas:

dVw

dt =A xd

α

^w

dt =A

(

^USWIN−USW_OUT

)

⁽¹³⁾

where A is the cross-sectional area of the pipe,

α

^{w is the}

average water fraction between the pressure transmitters, and USW_IN/USW_OUTarethesuperficialwatervelocitiesin/outofthevol- umeVw.Notethatintheexperimentsinquestion,thewaterfront illustratedinFig.6wasalwayssomewherebetweenthesepressure transmitters.RearrangingEq.(13),weobtain:

USWOUT=USWIN− xd

α

w

dt (14)

Thepressure difference P that wemeasurecan besafelyas- sumedtobe dominatedby gravitybecauseofthelow flowrates.

Thispressuredifferenceisthengivenby:

P≈[

α

^w

ρ

^w+

(

¹−

α

^w

) ρ

^o]gsin

φ

· x (15) Here,

ρ

^{o and}

ρ

^{w are the oil and water densities, g is the}

gravity acceleration, and

φ

^{is the} pipe angle. By differentiating Eq.(15)withrespecttotime,weget:

d P

dt ≈

ρ

^gsin

φ

^{· xd}

α

^w

dt (16)

AslightrearrangementofEq.(16)yields:

xd

α

^w

dt ≈ 1

ρ

^gsin

φ

^·

d P

dt (17)

Finally,wesubstituteEq.(17)intoEq.(14),yielding:

USWOUT=USWIN− 1

ρ

^gsin

φ

^{·ddtP (18)}

Inother words,wecanestimatethewaterrateoutofthepipe bymeasuringtherateatwhichthepressuredifferenceacrossthe water front varies intime. Fig. 7 showsan example of how the pressuredifference decreasesovertime. Theprocessis veryslow, so these measurements must be conducted over relatively long timescalestoobtainaccurateresults(typically1-2hours).

3.7. Interpretationofthescreeningexperiments

Wewill inthissection describe ourinterpretation andunder- standing of the screening experiments, and how they relate to steady-statepointmodelpredictions.

The bottom graph of Fig. 8 shows an example of a transient simulationofa screeningsimulationwithLedaFlow(“LedaFlow,”

Fig. 8. Example of LedaFlow simulation results. The top plot shows the water holdup (red line) plotted against USO (all solutions) for USW = 0.46 mm/s as pre- dicted by the LedaFlow point model. The bottom plot shows a time-averaged pic- ture of a screening simulation where USO = 0.37 m/s and USW = 0.46 mm/s.

KongsbergDigitalAS,2020),withUSOequalto0.37m/s.Thetran- sientsimulationisconductedinqualitativelythesamewayasthe experiment, whereawaterfronthasbeenestablishedinthepipe.

The plotshowsthepredictedtime-averagedwaterfractionprofile afterthewaterfronthasbeenestablished.

Inthissimulation,thewaterrateattheoutletwasfoundtobe USW_OUT=0.46mm/s.Whenadjustingtheinletwaterratetomatch USW_OUT, the oil/water front in the pipe was observed to remain stationary, asexpected. We maythus concludethat forthissim- ulated case, the critical USO (USO_C) is 0.37 m/swhen USW=0.46 mm/s.

The red curve in thetop graph showssteady-statesimulation results obtained by solving the steady-state flow equations de- scribedinSection2.1.Theconditionsassumedherewerethesame as inthe transientsimulation,andUSWwas setexactly equalto therecordedoutletrate(USW=0.46mm/s).

We observe here that the steady-state equations have three possible solutions in a certain range of USOs. The blue vertical line indicates the value ofUSO that was applied inthe transient simulation (0.37 m/s), which is the critical USO where the wa- ter holdupswitchesbetweenahighandlow value.Thisblueline intersects the red curve at the very start of the multiple solu- tionregion,indicatingthat thetransitionbetweenlow/highwater holdupoccursatthestartofthemultiplesolutionregion,i.e.that thelowsolutionis"preferred".

Thetwo blacklinesinthetopgraphofFig.8indicatethewa- terholdupsfoundbeforeandafterthewaterfrontinthetransient simulation (the location where these values were taken is indi- catedby theblacklinesinthebottomplot).Theseblacklinesin- tersect thered curveinthe sameplaceastheblueline,showing that thewaterholdupupstreamoftheliquidfrontcorrespondsto the highholdupsolution,whilethe waterholdupdownstream of the waterfront represents thelow solution (at theaccumulation point).

Fig. 9. Experimental and simulated outlet water rate plotted against USO for the screening experiments.

Theseresultsareconsistentwithourcurrentunderstandingof howmultiplesolutions work,i.e.that thetransitionbetweenlow andhighholdup occursatthe startof themultiple holdupsolu- tionregion.Manydynamicsimulations havebeencarriedoutus- ing LedaFlowtoconfirm thismatter, changingthe flowratesand examiningwhichholdup solutionprevails.Theresultsfromthese simulations haveturned out to always be the same,namely that thetransitionbetweenthehigh/lowholdupsolutionoccursatthat exactlocation.

Our understanding is that the reason for this "low-holdup- preference"hastodowiththeoutletboundarycondition.Specifi- cally,ifthegeometricalconfigurationoftheoutletallowsthewa- tertoflowout,yieldingalowwaterholdupattheoutlet,thelow- holdup-solutionwillprevail.Thishastodowiththeso-calledlevel gradientforces(the dh/dx-termsinEqs.(2)and(3)), whichtryto maketheinterfaceshorizontal,yieldingaforce onthewaterzone intheflowdirection.

Wehavealsosimulatedscenarioswherethetestsectionisfol- lowedbyapipe withan evenhigherinclination.Inthissituation wehavefoundthatthewaterholdupsolutioninthefirstpipede- pendsonthe waterholdup inthesecond pipe.Specifically,ifthe waterholdupinthesecondpipe ishigh,thewaterholdupinthe firstpipealsoremainshighifsuchasolutionispossible.Therea- son thishappens is presumably that the level gradient force on thewateratthejunctionnowpointsintheoppositedirection,and thatwaterisnotallowedtodrainnaturallyfromthefirstpipe.This

"low-holdup-preference"isthusnotageneraltraitofmultipleso- lutionscenarios,butratheraconsequenceoftheoutletboundary condition,whichinmostcircumstanceswillfacilitatealowwater holdup.

Wemayaddthatthelow-solutionpreferenceissupportedex- perimentallybyJohanssonetal.(2013),wheretheauthorswentto greatlengthstouncover hysteresisinthesecircumstances,butno hysteresiswaseverfound.

3.8. Experimentalresults

Fig.9showstheresultsobtainedintheexperiments,wherewe have plotted the outlet water rate versus the oil flow rate. The black markers are the experimental data, while the blue dashed linesarepredictionsmadeusingthesmoothinterface model.The interpretationoftheseresultsisthatwateraccumulatesinthere- giontotheleftofthesecurves,whilethelow-holdup-solutionpre- vailsontheright-handside.Itisquiteclearfromtheseresultsthat modellingtheinterfaceasasmoothwallyieldsoverly"pessimistic"

results.

Fig. 10. Experimental and simulated water fraction plotted against USO for the steady-state experiments.

Fig.10showstheresultsfromsteady-stateexperiments,where the superficial oil velocity USO was varies from 0.1 to 1.0 m/s, while superficial water velocity USW was fixed to 0.01 m/s. We observethatthe"smoothmodel"severelyover-predictsthewater holdup,indicatingthattheinterfacialshearstressistoosmall.

4. Dataanalysisandmodelling

In thissection we analysethe experimental datapresented in theprevioussectiontoobtainvaluesoftheinterfacialfrictionfac- tor forthedifferentcases, andwesubsequently usethose values toderiveanewmodelfortheoil/waterinterfacialshearstress.

4.1. Calculatingtheinterfacialfrictionfactorfromsteady-state experiments

Thestandardmethodforcalculatingtheinterfacialfrictionfac- torf_ifromsteady-statemultiphaseexperimentsistoinsertamix- ture of measured quantities and closure relationships into the steady-state momentum equations, and back out the interfacial shearstress.Thereareseveralpossiblewaysofdoingthis,butthe approachthatweusedinthispaperwastocombinetheoil/water momentumEqs.(4)and((5))andeliminatethepressuregradient.

Re-arranging the terms ofthe prevailing equation yields the fol- lowingexpression:

τ

ow=A

( ρ

w−

ρ

o

)

^gx+_α^Sw_w

τ

w−_α^So_o

τ

o Sow

α^w +^S_α^ow_o (19)

ByinsertingtheclosurelawssummarizedinSection2.2,thein- terfacialshearstress

τ

^{ow wasbackedout, andthe} interfacialfric- tionfactorcouldsubsequentlybecalculatedusingEq.(23).

Themainweaknessofthisapproachisthattheresultsdepend a greatdealonthevalidityoftheclosurelawsthatwehavesup- posed,aswellastheaccuracyofthephasefractionmeasurements.

Thelatterissueisespeciallyaproblemwhenoneofthephasefrac- tionsissmall,becausethemeasurementerrorforthesmallphase fractioncanthenyieldveryhighrelativeuncertaintiesinthefric- tionfactorestimates.

Becauseoftheseweaknesses,thisapproach isnot theprimary one usedin thispaper. Insteadwe mainlybasethemodellingon theapproachoutlinedinthenextsection.Also,itshouldbemen- tionedthatseveralofthesteady-stateexperimentswereexcluded fromthisanalysisonthegroundsthattheprevailinguncertainties inthefrictionfactorestimatesweresolargethatthefrictionfactor valueswerearguablynotmeaningful.

Fig. 11. Illustration of the procedure for calculating the interfacial friction factor from screening experiments.

4.2. Calculatingtheinterfacialfrictionfactorfromscreening experiments

The uncertainty problems listed in the previous section are largelymitigatedwiththeapproachthatweutilizehere,whichis identicaltothemethodusedinKjølaasandHolm(2016).Insimple terms,theprocedurethatweusetocalculatetheoil/waterinterfa- cialfrictionfactoristo"guess"thevalueoftheinterfacialfriction factor until the predictedwater accumulation point (the start of themultipleholdupsolutionregion)matchesthemeasured value.

Fig.11 illustrates how thisworks.Here we plottedwater holdup curveswithallsolutions foracertain waterflow rate(whichwas measured inthe experiment), usingdifferent valuesof theinter- facialfriction factor.The verticaldashed linerepresentsthemea- suredsuperficial oil velocity.The procedure issimply tofind the value off_ithat makes the modelmatchthemeasured accumula- tion point. In the example provided in Fig. 11, the black line (f_i

=0.01)givesthebestmatch.

It shouldbe noted that thisinterfacialfriction factoronly ap- pliesto the low-holdup solution,andnot to the high-holdupso- lution.The reason forthisis that the waterflow rateout of the pipe in the screeningexperiments are exclusively determined by theforce balanceontheliquidfilm downstreamofthegas/liquid front.Consequently,itistheoil’sabilitytopullthiswaterfilmthat wehavemeasured, andthe flowinthehigh-water-holdup region isdecoupledfromthis.

Wemustalsomentionthatinthisanalysis,wehaveinprinciple assumedthat theselectedclosurelawsforthewallshearstresses arecorrect. Thisishoweverofminorconcern,astheresultshave beenfoundtobeonlyweaklycoupledtothewallshearstresses.

Itisalsoworthnotingthatintheexperimentsaddressedinthis paper, there is essentially no droplet entrainment, meaning that theexperimental results can beassumed tobe a soleproduct of frictionandgravity forces.Ifthere wassignificant entrainmentin theseexperiments,theassociatedmomentumtransferwouldhave tobeaccountedforintheanalysis,whichwouldbeanexceedingly difficulttask.

4.3. Smoothoil/waterflow

Theunderlyingideabehindmodellinginterfacialshearstressis that theinterface can be viewedasa moving wall fromtheper- spective of the phaseson each side ofthe interface. Specifically, iftheinterfaceis smooth,i.e.withno interfacialwaves,itcan be modelledasasmoothwall.Ontheotherhand,iftherearewaves onthe interface,those waveswilltend toincrease theinterfacial

friction coefficient for the same reason as a rough wall will in- creasethewallfriction.InLedaFlow,weusethefollowingexpres- sionfortheinterfacialfrictioncoefficientf_i,smoothforsmoothinter- faces:

fi,smooth=

ρ

owf

(

^Reo

)

^f

(

^Rew

)

ρ

^of

(

^Reo

)

+

ρ

^wf

(

^Rew

)

2 (20) Here,theinterfacedensity

ρ

^{owisdefinedas:}

ρ

^ow=√

ρ

^o

ρ

^{w (21)}

Here, Reo and Rew are theReynolds numbers for oil andwa- ter,andf(Reo)andf(Rew)aresmoothwallfrictionfactorsobtained from the wall friction model described by Eq. (7), based on oil- andwaterzonepropertiesandtherespectivevelocitiesrelativeto the interface (uo-u_i anduw-u_i), where the interface velocity u_i is definedas:

ui=

ρ

of

(

^Reo

)

^uo+

ρ

wf

(

^Rew

)

^uw

ρ

of

(

^Reo

)

+

ρ

wf

(

^Rew

)

⁽²²⁾

Finally,theinterfacialshearstress

τ

iisexpressedas:

τ

i=1

2f_i

ρ

^ow

|

^uo−uw

| (

^uo−uw

)

⁽²³⁾

Inthecaseofasmoothinterface,theinterfacialfrictioncoeffi- cient f_itakesonthe valueoff_i,smooth,whileiftherearewaves on theinterface,ahighervalueisexpected.

Akey aspectofthisformulationisthat itissymmetricalwith respect tothe oil/waterphases, sothat interchanging theoil and waterpropertiesleadstothesameexpression.

4.4. Theonsetofoil/waterwaves

The primary challenge with modelling the interfacial shear stressistoincorporatetheeffectofinterfacialwavesontheinter- facialfrictionfactor.Here,thefirstquestionthatwemustaddress is:

Underwhat conditions doesthe interfacego fromsmooth to wavy?

To answerthis question,we turn to ViscousKelvin-Helmholtz (VKH)analyses,whichisthestudyofwavegrowth.Inthepresent analysis, we use the expression derived by Funada & Joseph (FunadaandJoseph,2001),whoincludedtheeffectofsurfaceten- sion and viscosityon the normalstress, while neglecting the ef- fectsoftheshearstresses.Using theseassumptions,they showed that the onset of waves with a dimensionless wave number k (madedimensionlessusingthepipediameterD)occurswhenthe slipvelocity uexceedsuc,givenby:

u²_c=

tanh(^k·α^o)+_μ^μ_w^otanh(^k·α^w)2

tanh(^k·α^o)+_μo

μw

2ρw

ρotanh(^k·α^w) 1 k

1+k² Eo

· ρ·g ρ^{o ·dd}α^hww

(24) Here,

μ

^{o and}

μ

^{w are theoil/water}viscosities,EoistheEötvös number (Eo=

ρ

^gD2/

σ

ow) and h_w is the water height. It should be noted that the original expression provided by Funada &

Josephwasderived forchannel flowandisslightlydifferentfrom Eq.(24)whichhasbeenadaptedtopipeflow.

Inorderto deducetheonsetofwavesfromEq.(24),we must selectacertain(dimensionless)wavenumberk.Thelogicalchoice hereistoselectthewave numberthatgivesthelowestvalue for uc,sincethe firstwavesthatwillgrow ontheinterface willhave thatwavenumber.Fig.12showsanexampleofhowthewaveon- setvelocityuctypicallydependsonthewavenumber.

We observe thatuc hasa minimum,andwe can say thatthe wave numberat that minimumrepresents the "mostdangerous"

Fig. 12. Example of how the wave onset velocity u cvaries with the dimensionless wave number k .

wave. Itis possibleto determinethis criticalwave numberkc by solving the equation duc/dk=0, but unfortunately, this equation doesnot havean analyticalsolution.We could alternativelyelect tosolvethisequationnumerically,butwehavefoundthatthecrit- icalwavenumbercanbeapproximatedwellbyfirstassumingthat kislargecomparedtothevolumefractions.Forlargek,thetanh- termsinEq.(24)canbereplacedbyunity,anditisthenastraight- forwardexercisetoshowthatthecriticalwavenumberk_cisgiven by:

k²_c≈Eo=

ρ

^gD2

σ

^{ow (25)}

Here,D is the pipediameter and

σ

^{ow isthe oil/watersurface}

tension.Thewave onset velocityuc can thenbe found bysubsti- tuting thedimensionlesswave numberk inEq.(24)by thewave numbergivenbyEq.(25):

u²_c =

tanh

(

^kc·

α

^o

)

⁺_μ^μw^o tanh

(

^kc·

α

^w

)

2

tanh

(

^kc·

α

^o

)

⁺

_μo

μ^w

2ρ^w

ρ^otanh

(

^kc·

α

^w

)

^· 2

kc ·

ρ

·g

ρ

o ·dhw

d

α

w

(26) 4.5. Modellingofwavyoil/waterflows

Wenowhaveanexpression forestimatingthewaveonsetve- locityu_c(Eq.(26)),andwemaypresumethatwaveswillappearas soonastheoil/waterslipvelocityexceeds thatvalue. Thismeans that the interfacial friction coefficient will at that point depart from the smooth model givenby Eq. (20)and become larger. A reasonableguessfor how theinterfacialfriction coefficient could beformulatedis:

fi=f_i,smooth

1+C·max

u−uc

uc ,0

(27)

Here, C could either be a constant or some function of the flow parameters.Thisparticular formulationprovides atransition fromsmoothtowavyflowthatisconsistentwiththeVKHanalysis above.Furthermore,thismodelassumesthattheinterfacialfriction coefficientincreaseswiththe"distance"fromtheonset,wherethe distanceisrepresentedbythedimensionlessgroup u/u_c.Were- fer to the second term inside the parenthesis in Eq. (27) as the

"wavefactor".Weusethistermbecauseitisafactorthatdescribes howthefrictionfactordepartsfromthesmoothmodelinthepres- enceofinterfacialwaves.

In Fig. 13 we have plotted the experimentally obtainedwave factoragainst u/uc-1.Notice thatthe parameterselectedforthe x-axisisthesameexpressionthat wehaveusedinthewavefac- torinEq.(27).Thewave factorslineupverywellforallthedata

Fig. 13. Wave factor plotted versus u/u c-1 . The black dashed line indicates the shape of the new model.

points calculated from the screening experiments, and the black dashed line that we have drawn through these points intersects theorigin.TheslopeofthislinecorrespondstothecoefficientCin Eq.(27),andthedatasuggeststhatavalueofC=4.0givesthebest fit. This resultis encouraging, becausethis means that the wave factorapproacheszerowhentheslipvelocityapproachesthewave onsetvelocityu_c,whichiswhatissupposedtohappen.Thislends credencetotheanalysisusedtodetermineuc,anditsuggeststhat thesimplificationsmadeinthatanalysisarereasonable.

ItisworthnotingthatwefirstattemptedusingInviscidKelvin- Helmholtz analysis instead of Viscous Kelvin-Helmholtz analysis, butthenthelinethroughthedatapointsdidnotintersecttheori- gin. Consequently,including theeffectofviscosity onthe normal stressesseemsimportantinthisanalysis.

Weobservethatdatapointsathigherslipvelocities(theblack circles, which are from the steady-state experiments with high holdup)donotcollapseonthesameline.Indeed,giventhescatter inthose pointsitisnot possibletodrawalinethat intersectsall ofthem,buttheaveragevalue forthisclusterofpointsisaround 7.0.Thereasonthatthesedatapointsdonotlineupsowellisnot clear, butthemostlikelyexplanation isthat theinterfacialshear stressisamorecomplexmatterathighwaterholdups.Indeed,the interfacialwavesathighwaterholdupshavelargeamplitudesand maydepend onother factorsthan whatwe haveassumedin our model. Also, aswe have pointed out earlier, the procedure used to calculatethose pointshas somesignificant weaknesses.In any case, itseemsthatareasonableextensionofthemodeldescribed byEq.(27)maybetointroduceaplateau,astheblackdashedline showninFig.13indicates.Thefinalversion ofthemodelisthus:

f_i= f_i,smooth

1+min

4.0·max

u−uc

uc ,0

,7.0

(28)

4.6. Modelverification

In this section we show the results obtained with the new modelproposedintheprevioussection(Eq.(28)).Thedatashown inthissectionisthesamedatathatweusedtoderivethemodel, so we perform thisexercise only to verifythat our analysis was carriedoutcorrectly.

InFig.14weshowtheoutletwaterrateplottedagainstUSOfor thescreeningexperiments.Theblackcirclesarethemeasuredval- ues,whilethebluetrianglesandredsquaresarepredictionsusing thesmoothinterfacemodelandEq.(28),respectively.Wefindthat thenewmodelmatchesthemeasuredvalueswell,certainlymuch betterthanthesmoothinterfacemodel.

Fig. 14. Experimental and simulated outlet water rate plotted against USO for the screening experiments. The red squares represent the results obtained with Eq. (28) .

Fig. 15. Experimental and simulated water fraction plotted against USO for the steady-state experiments. The red squares represent the results obtained with Eq. (28) .

In Fig. 15 we show the results from the steady-state experi- ments, where USO was varied from 0.1 to 1 m/s, using a fixed superficialwatervelocityofUSW=0.01m/s. Theblackcirclesrep- resent the measured values while the thin blue line and thick redlinearethepredictionsusingthesmooth interfacemodeland Eq. (28), respectively. Again, we see that the new model clearly outperforms the smooth interface model. The new model does howeverslightly under-predictthe water fractionforthe highest oil rates. The reason for this is unclear, but it could be due to weaknessesintheappliedwallfrictionmodel.

InFig.16 we havecombinedtheresults showninFig.14and Fig.15 to illustrate moreclearly what thescreening experiments actually represent. In each of the graphs in this figure we have plottedthewaterfractionversusthesuperficialoilvelocityfordif- ferent values ofUSW (which is indicated astext in each graph).

Mostofthedatapointsshowninthesegraphsare actuallytaken fromthesteady-stateexperimentsconductedwithUSW=0.01m/s, butwehavetakenthelibertytoincludetheminthesegraphsbe- cause the water holdup on the high-holdup branch is known to be virtually independent ofthe waterrateatsuch small superfi- cial watervelocities.This issomething that we haveobserved in theexperiments,andthe modelresultsthat we haveincludedin these graphs confirm this result. In other words, if we hadper- formedsteady-stateexperimentsatallthedesignatedwaterrates, the prevailing water fractions in the high-holdup-region would

Fig. 16. Water fraction plotted against USO for various USW -values. The thick red lines represent the results obtained with Eq. (28) .

presumablyhavebeen indistinguishablefromthe valuesobtained at USW=0.01 m/s. The physical rationale for this assumption is that atthese low waterrates(USW≤0.01 m/s)the watervelocity uw=USW/

α

^{w is}necessarilyverylowwhenthewaterholdup

α

^{w is}

high,andmaythusbereasonablyapproximatedbyavalueofzero in the momentum equations.The prevailing momentum balance, andby extension thewaterholdup, isthus virtuallyindependent ofthevalueofUSWinthissituation.

The last two (right-most) data points in each graph in Fig. 16 are taken from the screening experiments, where the highvalue is themeasured waterfractionupstream ofthe water front, andthe low values is the measured waterfraction down- stream of thewater front. Thesewater fractiondata pointsfrom the screening experiments were obtainedby converting the wa- ter heights measured by the static gamma densitometers into volume fractions by assuming a flat interface. As illustrated in Fig. 8, these two water fraction values represent the high/low solutions at the accumulation point. We have included no dat- apoints beyond the accumulation point because the data points obtained with USW=0.01 m/s would generally not be compara- ble to valuesobtained atlower USW on thelow-holdup-solution branch.

The thinbluelines are predictionsusingthe smooth interface model, while the thick red lines are predictions obtained using Eq. (28). We have elected to show all three water holdup solu- tions in thesegraphs, where the dashed part ofthe curvesrep- resentsthe solutions that arenot relevantforthese experiments.

As expected, we observe that the predictions obtained with the new model is in good agreement with the experimental values, whilethesmoothinterfacemodelisclearlyunsuitableforpredict- ingthesetypesofscenarios.

5. Conclusions

Inthispaperwe havedescribedasetofoil-waterexperiments conductedattheSINTEFMultiphaseLaboratoryina94meterlong 8"pipe. Theexperimentswere speciallydesignedtomeasure the criticalconditions forwater accumulationin oil/water flows. The experiments were performedata pipe inclination of 2.5degrees usingExxsol D60 asthe oilphase and tapwateras theaqueous phase.

Thedatashowedthatthecriticaloilvelocityforwateraccumu- lationincreaseswithincreasingwaterflowrate.Thedataanalysis also showed that a "smooth" interfacialfriction factor is unsuit- ableforpredictingtheonsetofwateraccumulation,andthat the effectofinterfacialwavesmustbeincorporatedtomodelsuchsce- narios.We foundthat the onsetofinterfacialwavesis accurately predictedbytheViscousKelvin-HelmholtztheorydescribedbyFu- nada&Joseph(FunadaandJoseph,2001).

Basedonthisdataanalysis,a newmodelforoil/waterinterfa- cialshear stresswasdeveloped,andthisnewmodel significantly improvestheagreementwiththemeasurements comparedtothe smooth friction factor. A slightly modifiedversion of this model hassincebeenimplementedinthetransientmultiphaseflowsim- ulatorLedaFlow(“LedaFlow,” KongsbergDigitalAS,2020).

DeclarationofCompetingInterest

Thereisnoconflictofinterestassociatedwiththismanuscript.

Acknowledgements

Theauthorswouldliketothankthetechnicalandscientificper- sonnelattheSINTEFMultiphase Laboratoryfortheirconsiderable efforts duringtheexperimental campaigns.Wewouldalsoliketo expressourappreciationtoLedaFlow TechnologiesDA(ownedby SINTEF, KongsbergDigital, TotalandConocoPhillips)andtheNor- wegianResearchcouncil(grantnumber281881),forfinancingthe modeldevelopmentdescribedhere,aswellasthispublication.

References

Barnea, D. , Taitel, Y. , 1992. Structural and interfacial stability of multiple solutions for stratified flow. Int J Multiphase Flow 18 (6), 821–830 .

Bluemink, E. “Fairbanks Daily News-Miner,” 3 Jan 2010. [On- line]. Available: http://www.newsminer.com/news/alaska _ news/

less- oil- may- spell- more- problems- for- trans- alaska- pipeline/article _ 3cecf9ac- 8076- 53c0- 8c4a- 9fc00acf2e61.html . [Accessed 24 June 2020].

Brauner, N. , Maron, D.M. , Rovinsky, J. , 1998. A two-fluid model for stratified flows with curved interfaces. International Journal of Multiphase Flow 24, 975–1004 . Funada, T. , Joseph, D. , 2001. Viscous potential flow analysis of Kelvin–Helmholtz in-

stability in a channel. Journal of Fluid Mechanics 445, 263–283 .

Johansson, P.S. , Pettersen, B.H. , Djoric, B. , 2013. Ramp-up and Rampdown of Low Liquid Loading Gas-Condensates Flowlines. International Conference on Multi- phase Flow .

Hall, A. , Hewitt, G. , 1993. Application of two-fluid analysis to laminar stratified oil-water flows. Int. J. Multiphase Flow 19, 711–717 .

Håland, S. , 1983. Simple and Explicit Formulas for the Friction Factor in Turbulent Flow. Journal of Fluids Engineering (ASME) 105 (1), 89–90 .

Kjølaas, J. , Holm, H. , 2016. Improvement of LedaFlow for low liquid loading conditions. 10th North American Conference on Multiphase Technology ISBN 9781855981553 .

Kjølaas, J. , Unander, T.E. , Wolden, M. , Johansson, P.S. , Holm, H. , 2015. Experi- ments for low liquid loading with liquid holdup discontinuities in two- and three-phase flows. 17th Multiphase Production Technology Conference, ISBN 9781855981478 .

Landman, M. , 1991. Non-unique holdup and pressure drop in simulations of two-phase stratified inclined pipe flow. Int J Multiphase Flow 17 (3), 377–394 .

“LedaFlow, ” Kongsberg Digital AS, [Online]. Available: http://www.ledaflow.com/ . [Accessed 24 June 2020].

Magill, B. “Popular mechanics,” 1 February 2012. [Online]. Avail- able: https://www.popularmechanics.com/science/energy/a7480/

how- much- life- is- left- in- the- trans- alaska- pipeline/ . [Accessed 24 June 2020].

Taitel, Y. , Barnea, D. , Brill, J. , 1995. Stratified three phase flow in pipes. International Journal of Multiphase Flow 21 (1), 53–60 .