Upward and downward two-phase flow of in a pipe: Comparison between experimental data and model predictions

(1)

International Journal of Multiphase Flow 138 (2021) 103590

ContentslistsavailableatScienceDirect

International Journal of Multiphase Flow

journalhomepage:www.elsevier.com/locate/ijmulflow

Upward and downward two-phase ﬂow of CO ₂ in a pipe: Comparison between experimental data and model predictions

Morten Hammer

^a

, Han Deng

^a

, Lan Liu

^b

, Morten Langsholt

^b

, Svend Tollak Munkejord

^a^,^∗

aSINTEF Energy Research, P.O. Box 4761 Torgarden, Trondheim NO-7465, Norway

bInstitute for Energy Technology, P.O. Box 40, Kjeller NO-2027, Norway

a rt i c l e i nf o

Article history:

Received 12 October 2020 Revised 28 January 2021 Accepted 8 February 2021 Available online 11 February 2021 Keywords:

Carbon dioxide CO 2injection Vertical ﬂow Friction Liquid holdup Fluid dynamics Thermodynamics

a b s t r a c t

Inorder todeployCO₂ capture and storage(CCS)systemstomitigate climatechange,it is crucialto developreliablemodelsfordesignandoperationalconsiderations.Akeyelementofthesystemisthein- terfacebetweentransportationandstorage,namelytheinjectionwell,wherevarioustransientscenarios involvingmultiphaseﬂowwilloccur.

Intheliteraturethere areveryfewdatarelevantforvalidation ofverticalmultiphaseflowmodelsfor CO₂.Hence inthiswork, wepresent measurements ofliquidholdup, pressuredrop and flow regime forupward anddownwardflowofCO₂ inapipeofinnerdiameter44mmatapressure of6.5MPa,a conditionrelevantforCO2-injectionwells.

Theexperimentalresultsindicatethattheﬂowisclosetono-slip.Wehavecomparedtheexperimental datatopredictionsbywell-knownmodels forphaseslipand frictionalpressure drop,and theresults showthatoverall,thebestmodelisthesimplestone– the fullyhomogeneousapproach, inwhichno slipisassumedandthefrictioniscalculatedsimplybyemployinggas-liquidmixturepropertiesinthe single-phasefrictionmodel.

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

1. Introduction

CO₂ capture and storage (CCS) is seen as one of the tech- nologies that are necessary to help mitigate climate change (Edenhofer et al., 2014). In order for CCS to attain the scale re- quired to do so, full-scale deployment must commence and be scaled up such that by the mid century, several gigatonnes of CO₂ are capturedeach year(IEA, 2017). This CO₂ must be trans- ported from the capture plants to the storage sites. In order to design and operatethe CO₂ transportation andinjectionsystems in a safeand efficient way, there is a need for flow models describing single- and multi-phase flow of CO₂ and CO₂-rich mixtures (Munkejord et al., 2016). CO₂ flows in pipes or tubes are alsorelevantinother applications,such asheat-pumpingsystems (Lorentzen, 1994; Pettersenet al., 2000), Brayton or Rankinecy- cles(Ayubetal.,2020),nuclearreactors(Eteretal.,2017)andheat storage(Ayachietal.,2016).

The injection well constitutes the interface between the CO₂ storage andthe transportation system. It is important tobe able

∗Corresponding author.

E-mail address: [email protected] (S.T. Munkejord).

topredicttheflowbehaviouroftheCO₂bothduringnormaloper- ation,andduringstart-up,shut-inorundesired eventslikeblow- outs. Duringnormaloperation, transientscanbe expecteddueto fluctuations in the CO₂ supply (Moe et al., 2020), dueto batch- wiseoffshoredeliveryfromships(Aursandetal.,2017;Munkejord et al., 2020), or during injectioninto depleted natural-gas reser- voirs(SacconiandMahgerefteh,2020).Amongotherthings,resulting temperature fluctuations could affectwell integrity (Aursand etal.,2017).

Depending on the maximum allowable pressure in the CO₂ reservoir and other operational conditions, the CO₂ could be in a two-phase state in part of the well (see e.g. Munkejord et al., 2013). This was also the case for the CO₂-production well stud- ied by Cronshaw et al. (1982). CO₂ has significantly different thermophysicalproperties compared tothose of e.g. oil andnat- ural gas. Therefore, existing models, validated for such fluids, may not be accurate for CO₂, andexperimental validation is re- quired. However, very few data are available in the literature forthe verticaltwo-phase flow ofCO₂ inrelevant configurations.

Cronshawetal.(1982)presented temperatureandpressuremea- suredatseverallocationsinaCO₂-productionwellforvaryingﬂow rates.Inthe upperpartofthewell, theCO₂-richmixture includ-

https://doi.org/10.1016/j.ijmultiphaseﬂow.2021.103590

(2)

M. Hammer, H. Deng, L. Liu et al. International Journal of Multiphase Flow 138 (2021) 103590

Nomenclature

C Dimensionlesspressuregradient,1 d Diameter,m

e Speciﬁcinternalenergy,Jkg⁻¹ ˆ

e Totalspeciﬁcenergy,Jkg⁻¹ f (Darcy)frictionfactor,1 Fr Froudenumber,1 F Frictionforce,Nm⁻³

gx Gravitationalaccelerationinaxialdirection,m⁻³ h Speciﬁcenthalpy,Jkg⁻¹

j Volumetricﬂux,ms⁻¹

˙

m Massﬂux,kgm⁻²s⁻¹

n Numberofexperimentalmeasurements,1 P Pressure,Pa

Q Heatﬂuxpervolume,Wm⁻³ Re Reynoldsnumber,1

t Time,s

u Velocity,ms⁻¹ x Axialcoordinate,m

x Gasmassfractionbasedonthemassﬂuxes,(5),kg kg⁻¹

y Elevation,m

α

^Volume^fraction,^m³^m⁻³

δ

^rms Root-mean-squaredeviation,(13)

σ

^Surface^tension,^N^m⁻¹

μ

^Dynamic^viscosity,^kg^m⁻¹^s⁻¹ ^Coeﬃcientⁱⁿ^(6),¹

ρ

^(Mass)^density,^kg^m⁻³

d Drift

f Friction

g Gas

k Phasek

Liquid

m Multiphasemixture CCS CO₂captureandstorage EOS Equationofstate

IFE InstituteforEnergyTechnology RMS Rootmeansquare

ingwaterwasinagas-liquidorgas-liquid-liquidmultiphasestate, although thegas fractionwasnot measured.Some ﬁeld datacan be foundforCO₂ wells(seeLu andConnell,2014;Lietal.,2017).

Thesedataare lessdetailedthandesirable forﬂow modelvalida- tion.

In principle,the complicatedtopology oftwo-phase ﬂows can be simulated in detail using front-capturing (Osher and Fedkiw, 2001; Sethian, 2001) or front-tracking (Tryggvason et al., 2001) methods.However,duetothecomputationalintensity,suchmeth- ods can onlybe used onrelatively smallcomputational domains.

Therefore oneresortsto consideringan average ofthetwo-phase ﬂow, not resolving the full details of the interfaces(Stewart and Wendroff,1984;DrewandPassman,1999).Eveninthiscase,when complicated equations of state are involved, three-dimensional simulationsarelimitedtosmalldomains(Gjennestadetal.,2017).

Asaresult,forengineeringpurposes,two-(ormulti-)phaseflows inpipesandwellsarecommonlydescribedusingone-dimensional models. The most generalapproach is usually referred to asthe two-fluid model(StewartandWendroff,1984). Herein,thediffer- ence between the gas and liquidvelocity is determined through inter-phasic friction models, the development of which involves extensive use of experimental data. For several flow regimes, it is possible to correlate the relative velocity between the phases, the slip velocity, as a function of the flow variables (Zuber and

Findlay, 1965; Ishii, 1977; Hibiki and Ishii, 2002). This a priori knowledge of the flow can be employed to reduce the number of transport equations to be solved, and the result is called the drift-flux model. In particular, drift-flux models have been developed for two- andthree-phase flows in wells(Shi etal., 2005b;

2005a).Inadditiontoslipmodels,modelsforthefrictionalpressure drop areneededinorder toperform simulations.Friction models fortwo-phaseﬂow existinvariousforms,rangingfromempirical (BeggsandBrill,1973)tophenomenologicalmodelsdescribingthe characteristicfeaturesofdifferentﬂow regimes(RELAP5Develop- mentTeam,1995).SeealsothereviewinDoraoetal.(2019).

Inthepresentwork,weaddressthelackofverticalexperimen- tal data for two-phase flow of CO2. We employ an experimental setup designed to generate liquid holdup and pressure-drop data,alongwithflow-regimeinformation,duringsteady-stateop- eration (Håvelsrud, 2012; Farokhpoor et al., 2020). A data series hasbeengeneratedforvaryinggasandliquidfluxesofpure CO₂, bothupwardsanddownwards,atapressureof6.5MPa.Thispres- surehasbeenchosensinceitisrelativelyclosetothecriticalpres- sure(7.38MPa),whileatthesametimegivingastatethatisclearly two-phase. As described inthe following, thishas allowed us to compareslipmodelsandfrictionalpressure-dropmodelsfromthe literature with experimental data, giving guidance to modellers wantingtodescribetheflowofCO₂ inwells.

2. Experimentalsetup

Vertical up-flow and down-flow experimental data have been acquiredfortwo-phase pureCO₂ saturatedat6.5MPainFALCON, IFE’sflowassuranceloopforCO₂ transport.Thecorrespondingsat- urationtemperatureis24.4^◦C.Themainpipeoftheflowloophas aninnerdiameterof44mm,alengthof13.7mandaneffectivesur- faceroughnessestimatedtobe 17m,givingarelativepipe roughness of3.9× 10⁻⁴ relevant forfriction calculations. Theexperi- mentalsetupisdescribedbyFarokhpooretal.(2020),whostudied horizontalandnearhorizontalflowofCO₂.Schematicdrawingsof thetestfacility’soveralldesignandtheinstrumentationofthetest section,forverticalpipeconfigurations,areshowninFig.1.

The temperature is controlled by a combined heating/cooling system where a coolant is circulated in copper-tubing-type heat exchangers ‘coiled’ on to the main separator and the test section. The coolant temperature is tuned so that the heat trans- fer to thesystem, via the heat exchangers, justbalance the heat addedby the pumpsandthe heatloss tothe ambient,justifying the assumption of an adiabatic system. In this waythe temperature/pressureiscontrolled inastableandaccurate way.Thenet heatloss,orgain,dependsonthepumps’rotationalspeed(i.e.,the ﬂow rates),the operatingtemperatureandthe ambienttempera- ture(heatloss/gain).The temperatureofthecoolantiscontrolled bycombinedheatingandcooling.Theeffectoftheelectricalheater andthecoolingplantenablesstableoperatingtemperaturesinthe range−10^◦C to 40^◦C ifthe ambient temperatureis around 10^◦C.

Allpipesandvesselsarewellinsulated.

Themaindifferencesbetweentheup-flowanddown-flowcon- figurationsarethepositionofthebroad-beamgammadensitome- ter andthe inlet andoutlet sections.A pre-separatoris included in the vertical-down setup and the inlet merger is Y-shaped. In thevertical-upsetup,thereisno outletpre-separator,onlytheY- split, andthe inletmerger is a jointof two half-circlesmade by steeltubes,seeFig.1foranoutline.Thegasandliquidphasesare drawnfrom,respectively,thetopandthebottomofthemainsep- aratorandconveyedassingle-phasefluids,inseparate feedlines, to the inlet merger of the test section. From a view cell on the liquid feed line, we can observe that no bubbles are present in theliquid,i.e.,noboilinghastakenplace.Fromtemperaturemea-

(3)

Fig. 1. Schematic of the FALCON test facility located at the Institute for Energy Technology (IFE).

surements just upstream of the merger, we can also verify that bothﬂuidphaseshavetemperaturesthatcloselycorrespondtothe vapour-liquidequilibriumline.Thismeansthatnoﬂashingorcon- densation should takeplace whenthegasandliquidstreams are merged.

The objective of the experimental campaign was threefold, namely, to measure the pressure drop, to measure the liquid holdup,andtodetecttheﬂowregimeatdifferentvolumetricphase ﬂuxes. The liquid holdup was measured using a broad-beam

γ

^-

densitometerandasinglecameraX-raysetup.Fromthemeasured holdup,thephase slipfactor(ug/u)can becalculated.Anarrow-

beam

γ

-densitometer is included in the flow loop setup, giving supplementary information on the liquid holdup, primarily used to evaluate the flow development. Using the X-ray results and images from a high-speed camera, visualizing the flow through a sight glass, the flow regimes were manually determined. The overallpressuredropwasdeterminedbyaveragingmeasurements fromsixpiezoresistivedifferential pressuresensors.Withthe liquid holdup measured by the X-ray system as input, the overall pressure dropwassplit ina hydrostaticandfriction contribution usingdensitiespredictedbytheSpanandWagner(1996)equation ofstate(EOS).

(4)

Fig. 2. Measured holdup plotted versus homogeneous holdup for the upward ﬂow experiments.

Table 1

Estimated measurement uncertainties in input and measured data.

Property Type Uncertainty

Pipe diameter Absolute ±0.1mm

Absolute pressure Relative ±1.5%

Delta pressure Relative ±7%

Temperature Absolute ±1.0 ^◦C

Liquid holdup – Broad-beam γ-meter Absolute ±0.02 Liquid holdup – Narrow-beam γ^-meter ^- ^±^0.035 Liquid holdup – X-ray system Absolute ±0.03

Liquid volumetric ﬂux Relative ±4%

Gas volumetric ﬂux Relative ±3%

In addition to the geometry (upward ordownward flow), the main experimental parameters are the volumetric gas flux, jg, and volumetric liquid flux, j. The test matrix consisted of all combinationsofj_g∈

{

⁰.2,0.4,0.6,1.3,2.0,3.0,4.0

}

^m^s⁻¹,andj∈

{

⁰.15,0.3,1.0,2.0,3.0

}

^m^s⁻¹^.^The^phase^mass^ﬂows^are^measured

using Coriolis ﬂow meters, and volumetric ﬂuxes are calculated fromdensityestimates.

The criticalpressure ofCO₂ is 73.8bar, andsince thepressure in the experiments is relatively close to that, the gas and liquid thermophysicalpropertiesaresimilar.Theliquid-to-gasdensityra- tio is

ρ

/

ρ

^g=2.83 while the viscosity ratio is about the same;

μ

/

μ

^g=2.75.Thesurfacetensionatthispressureisonlyapprox- imately 0.5mNm⁻¹.The phase slipfactorin theseexperimentsis thereforeexpectedtobeclosetoone,u_g/u≈1.

The main experimental uncertainties are listed in Table 1, and they have been estimated following the ISO Guide to the expression of uncertainty in measurement (Joint Committee for GuidesinMetrology,2008).Theuncertaintyinthemeasurements (flow stability, data acquisition, etc.) is handled as a Type A standard uncertainty with normal distribution of data, while in- strumentaccuracies(datasheets,previous experience,calibrations, inter-comparisons,etc.)arehandledasTypeBstandarduncertain- ties, withrectangulardistribution.Acoverage factorof2hasbeen used to get 95% confidence. For the flow rates, the contribution fromTypeAandTypeBtothe combineduncertaintyvarieswith

the magnitude of the ﬂow rates. For the pressure and differen- tial pressures, Type B dominates over Type A in the combined uncertainties. The holdup uncertaintiesare based on calibrations andlongtermexperience,whilethetemperatureuncertaintiesare based on the sensor accuracy and comparisons with redundant sensors.

3. Models

Since, in this work, we study vertical two-phase ﬂow where the two phaseshaverelatively similar thermophysicalproperties, thekeymodels forsimulationpurposes arethoseforfrictionand phase slip, in addition to the property models, which we brieﬂy discussinthefollowing.

One-dimensionalsingle-component two-phaseﬂow withequi- libriuminpressure,temperatureandchemicalpotentialcanbede- scribedbymassconservationandmomentumandenergybalance equationsasfollows.

∂ ∂

^t

k

α

k

ρ

k

+

∂

^x

k

α

k

ρ

kuk

=0, (1)

∂ ∂

^t

k

α

k

ρ

ku_k

+

∂

^x

k

α

k

ρ

ku²_k

+

∂

^P

∂

^x ⁼

ρ

^m^g^x⁻^F, ⁽²⁾

∂ ∂

^t

k

α

k

ρ

keˆk

+

∂

^x

k

α

k

ρ

kuk

(

^hk+1/2u²_k+gy

)

=Q. (3)

Herein,

α

k is the volume fraction of phase k and

ρ

^denotes

density, P denotes pressure andu is the velocity. The total spe- ciﬁc energy includes the internal, kinetic and potential energy;

ˆ

e_k=e_k+1/2u²_k+gy,wheregisthe gravitationalaccelerationand y is the elevation. Inthe momentum equation, gx isin the axial directionofthepipe.

Theenthalpyish_k=e_k+P/

ρ

k.Thesubscriptmdenotes(multiphase) mixture quantities. For example, the mixture density is

ρ

m=

k

α

k

ρ

k.Qistheheat fluxtransferred tothefluid through thepipewall andF isthewall friction.Inthiswork,wewill as- sumeadiabaticflow,Q=0.

(5)

Fig. 3. Measured holdup plotted versus homogeneous holdup for the downward ﬂow experiments.

Fig. 4. Flow regime observed in experiments with upward ﬂow.

In addition to the above equations, to close the system, one needs a sliprelation,i.e., amodelforthe differencebetweenthe phasicvelocities,andanequationofstate.

3.1. Frictionmodels

Forsingle-phaseﬂow, thewallfriction, F,iscommonlycalcu- latedas

F=f_km˙

|

^m^˙

|

2

ρ

kd, (4)

where f_kistheDarcyfrictionfactor,m˙ =

ρ

ûîs^the^mass^flux,ând

distheinnerpipediameter.

Two-phasefriction modelscanbe classiﬁedbasedonassump- tions andmodellingapproach (see Collier andThome,1994; He- witt, 2011). The simplest approach is that of the homogeneous model, where thephases are assumedto be well mixedso they canbetreatedasasinglephase,andthefrictioncanbedescribed

usinga friction factorobtainedfrom theReynolds numberbased onthegas-liquidmixtureproperties– essentiallyreplacingkbym in(4).Here,theReynoldsnumberiscalculatedusingamass-based harmonicaverageofthephaseviscosities,

1

μ

^m ⁼

x

μ

^g⁺

1−x

μ

, (5)

wherexdenotesthegasmassfractionbasedonthemassﬂuxes.

Several empirical modiﬁcations to obtain a two-phase friction factor have been suggested. One commonly used model is the Beggs and Brill (1973) correlation, which employs correction factors to the single-phase no-slip friction factor based on ﬂow regimeandinclination.

Anothermainapproachisthatofseparatedﬂow,i.e.,wherethe gasandliquidﬂowareaccountedforseparately,eachwithitsown velocity andarea fraction ofthe channel cross section. Here, the

(6)

Fig. 5. Flow regime observed in experiments with downward ﬂow.

wallfrictionisoftenmodelledas F=fm˙

|

^m^˙

|

2

ρ

d

, (6)

where denotes the liquid phase and îsâ ^two-phase ^friction multiplier. One commonly used separated-flow friction model is thatofFriedel(1979).

Inprinciple,moreaccuratefrictionpredictionscanbeachieved using phenomenological models, where the ﬂow regime is iden- tiﬁed, and separate adapted models are applied accordingly. The friction model employed in the RELAP5 model is one example (RELAP5 DevelopmentTeam, 1995), butwill not be further eval- uatedinthiswork.

All friction modelswill requirethe calculationof single-phase frictionfactorsbasedonReynoldsnumberandrelativepiperough- ness. In this work we will, as default, use the explicit formula of Haaland (1983) to calculate the Darcy friction factor, instead ofiteratively solvingthe moreaccurateColebrook-White equation (seee.g.White,1994).

3.2. Drift-ﬂuxmodels

The basic idea of drift-ﬂux modelling is that the gas velocity, ug,can be related tothe volumetric ﬂux, j=

α

^g^u^g+

α

u, of the mixtureanda driftvelocity, u_g_d,takinginto accountthe difference betweenthe mixture flux andthe gasvelocity, including thebuoyancyeffect.Thedrift-fluxconceptwasfirstintroducedby Zuber and Findlay (1965) for 1D flow, and due to its simplicity, many correlations have been developed for predictions of phase slipandholdupintwo-andthree-phaseflow.

The gasvelocity correlationin thedrift-ﬂux formalismis usu- allygivenas

ug=C0j+ugd, (7)

where the proﬁle parameter C₀ correlates the effect of cross- sectional velocity and holdup proﬁle information, and u_g_d is the drift velocity describing the local phase slip. According to Zuber andFindlay (1965), 1.0≤C₀≤1.5. In our simulation code, theu_g_d termisimplementedsuchthatitgivesapositivecontribu- tionagainstgravity.

Inthiswork,we evaluatethreedifferentslipmodels.First,we have implemented the Zuber and Findlay (1965) model for the churn-turbulentbubblyregime,

ug=1.18j+1.53

σ

^g

ρ ρ

²

¹/4

. (8)

Herein

σ

^is^the^surface^tension^and

ρ

=

ρ

−

ρ

^g^.

Second, we consider the model of Shi et al. (2005b), which wasdevelopedforverticaltonearhorizontalﬂowofoil/gas/water based onexperimental data fromlarge-diameter pipes. The third modelincludedistheoneofPanetal.(2011a,b),whichisanadap- tationoftheShietal.(2005b)modelforCO₂ ﬂowinwells.Here, welabelthismodelT2Well.

3.3. Dimensionlessparameters

Forvertical multiphase ﬂows, theFroude number, relatingin- ertiatogravity,isasigniﬁcantparameter.Severalformulationsare possible.Hereweuse

Frm= u²_m

gd, (9)

whereum=m˙/

ρ

^m îs^the mass-weighted mixture velocity.Inthis subsection, the mixture properties are calculated using volume fractions for homogeneous (no-slip) flow. This definition is em- ployedintheFriedel(1979)andBeggsandBrill(1973)correlations.

At timesa density-dependentprefactoris includedinthe Froude number for multiphase ﬂows, see e.g. Farokhpoor et al. (2020). Sincein thepresentexperimentsthe gasandliquiddensities are almostconstant,suchaprefactorisnotincludedhere.

AmultitudeofdifferentReynoldsnumbersareinuseformul- tiphase ﬂows. The Friedel (1979) correlation employs a gas-only anda liquid-onlyReynolds number,calculated assuming that the wholemassﬂow isgas,andliquid,respectively.IntheBeggsand Brill(1973) correlation andinthe homogeneous model,the two- phasemixtureReynoldsnumberiscalculatedas

Rem=

ρ

^m^u^m^d

μ

^m ^, ⁽¹⁰⁾

although with the difference that in the homogeneous model, we employ the relation (5) forthe two-phase mixture viscosity, whereasinBeggsandBrill(1973),avolumeaverageofthephasic viscositiesis used. It is also commonto calculate gas andliquid Reynoldsnumbersbasedonthevolumetricﬂuxes,

Re˜_k=

ρ

kjkd

μ

k . (11)

In the present experiments, we have Rem∈

{

¹.9×10⁵...3.6× 10⁶

}

,Re˜g∈

{

¹.0×10⁵...2.2×10⁶

}

,Re˜∈

{

⁸.4×10⁴...1.6×10⁶

}

^.

(7)

Fig. 6. Downward flow: Measured and calculated liquid holdup, α, as a function of gas volumetric flux, j g, for varying liquid volumetric flux, j .

Theexperimental pressuregradientdatacanbenormalizedby thedynamicpressurebasedonthemixturevelocityanddensityas follows.

C=

^P_x

d

1 2

ρ

^m^u²m

. (12)

ThisdeﬁnitionwillbeemployedinSection4.4.

3.4. Thermophysicalpropertymodels

In this work, the highly accurate Helmholtz-type equation of state(EOS)ofSpanandWagner(1996)forCO₂hasbeenused.The EOSisusedtocalculatewhatphasesarestable,andthedensities andenergiesoftheexistingphases.

The viscosity of pure CO₂ for conditions relevant for transport and capture is described using the correlation of Fenghour et al. (1998) to an accuracy below 2%. The thermal

(8)

Fig. 7. Upward flow: Measured and calculated liquid holdup, α, as a function of gas volumetric flux, j g, for varying liquid volumetric flux, j .

conductivity of pure CO₂ is correlated to a similar degree of accuracybyVesovicetal.(1990).Thegas-liquidinterfacialsurface tension is modelled using the correlation of Rathjen and Straub (1977).

For the ﬂash calculations, we utilize our framework for calculation of thermodynamic properties (Wilhelmsen et al., 2017;

Hammer et al., 2020). The framework interface the TREND

thermodynamics library (Span et al., 2016) for the Helmholtz EOS.

3.5. 1Dﬂuidﬂowsimulator

The non-linear system of governing equations for the ﬂow (1)–(3) are discretized on a regular forward-staggered grid us-

(9)

Fig. 8. Downward flow: Measured and calculated negative frictional pressure gradient as a function of gas volumetric flux, j g, for varying liquid volumetric flux, j .

ing aﬁrst-orderupwind-typeﬁnite-volumemethodsimilartothe one discussed by Zou et al. (2016). The resulting discrete equation system is solved by a Jacobian-free Newton–Krylov method as discussed by Knoll and Keyes (2004). Here we employ the PETSc library(Balayetal.,1997;2018) usingthe

SNESNEWTONLS

method,whichisaNewton-basednonlinearsolverthatusesaline search.Withinthismethod,theBiCGStab(stabilizedversion ofbi- conjugate gradient)method withSOR(successive over-relaxation) asapreconditionerisemployed.Furtherdetailsonthemodeland methodscanbefoundinMunkejordetal.(2020).

Toobtaintheresultspresented inthefollowing,weemployed agridof20cells,runningthesimulationsfor200stoarriveatthe steady-statesolution.

4. Resultsanddiscussion

Inthefollowingwewillpresentourexperimentaldatarelevant forCO₂wellﬂow,andcompareexperimentalresultstothemodels forfrictionandslippresentedinSection3.

(10)

Fig. 9. Upward flow: Measured and calculated negative frictional pressure gradient as a function of gas volumetric flux, j g, for varying liquid volumetric flux, j .

4.1. Experimentaldata

4.1.1. Liquidholdup

InFig.2,wehaveplottedthemeasuredholdup,basedontheX- ray system,againstthehomogeneousholdupfortheupwardflow geometry. The homogeneousholdup is simplythe fractionofthe liquid volumetric fluxto the total volumetricflux in each exper- iment. From this plot we can get qualitative information regard-

ingphaseslip.Ifthegasvelocityislargerthantheliquidvelocity, themeasured holdup willbe larger thanthe input homogeneous holdup,andtheexperimentaldatapointswilllietotheleftofthe dashed no-slipline. The ﬁgureshows that exceptfor theexperi- mentswithaninletholduplessthan20%,forwhichgasaccumu- lationandu>ug isregistered,theﬂowisessentiallyno-slip.The experimentswithlowinletliquidholdupwillhaveahighrelative uncertaintyintheholdupmeasurements,asthemeasurementun-

(11)

Fig. 10. Downward ﬂow: Comparison between measured and computed liquid holdup for different slip models.

certainty is absolute, see Table 1.The experimentswith an inlet holduplessthan20%,arethereforemostlikelyalsono-slip.

In Fig.3,we haveplottedthe measuredholdup, basedonthe X-raysystem,against thehomogeneousholdupforthedownward ﬂow geometry. The ﬁgure shows that exceptfor 2–3outliers, all experiments have liquid accumulation and ug>u. As the liquid densityisapproximately2.8timesthegasdensity,andthegravi- tationalforcewouldfavouru>u_g,thisisslightlysurprising.

4.1.2. Flowregime

The ﬂowregimes fortheseexperimentsaredominated bythe gas andliquidphasesbeingwell mixed. Thephasesareobserved to be segregated onlyto a very small extent.This is presumably because of the low density differencesand low surfacetensions.

The followingmeasurementsareusedto supporttheﬂowregime determination:

• holdupand

δ

^p/

δ

^x^time^series,^which^will^indicateintermittent ﬂowbehaviour

• theopticalvideos(veryshortrecordingtime)

• X-ray projections ofphase distribution (27 secondsside view projections)

Since there was very little intermittency in the flow andthe videos were tolimitedhelp, the flowregime findings aremainly basedontheX-rayprojections.Itmustbe admittedthattheflow regimes are encumbered with significant uncertainties and that theyinvolveguesswork.

Theflow regime mapidentified fortheupward flowisshown inFig. 4,and theflow regime map fordownward flow isshown inFig.5.Aqualitative descriptionoftheflow regimesisgivenin Table2.Inbothfigures,weseegas-continuousflowwithentrained liquiddroplets/dropsathighvolumetricgasflux.Forlowvolumet- ricgasflux,theflowismostlyliquidcontinuousorachaoticgas- liquidmixture.Some pointsalsoindicatesegregatedannularflow, butthereis noclearlydefinedannularregion intheflow-regime maps.

4.2. Comparisonofexperimentaldataandcalculatedresults

Toevaluateslipandfrictionmodels,ourdynamic1Dflowsim- ulator,Section3.5,wasconfiguredtomatchtheexperimentalflow geometriesandsimulatedtosteadystate. Asboundaryconditions forthesimulations,themassflowforeachphasewasspecifiedat theinletandthepressurewasspecifiedattheoutlet.Thepipewas initializedwithasaturatedstatedefinedbytheexitpressure,with a homogeneous flow based on the inlet condition. The pipe was consideredtobeadiabatic.

In thiswork, we havetested fourmodels for pipe friction,as tabulated in Table 3. The original Friedel (1979) correlation in- cludesanexplicitequation forthefrictionfactorthatdoesnotin- cludethe effectofpipe roughness. Asthe relative surfacerough- ness in the experimental setup is high, it was deemed relevant tousetheFriedelcorrelationwithboththeoriginalfrictionfactor modelandtheHaalandmodel.

(12)

Fig. 11. Upward ﬂow: Comparison between measured and computed liquid holdup for different slip models.

Table 2

Flow regime description.

Flow regime Description

Bubble Liquid-continuous ﬂow with entrained gas bubbles Droplet Gas-continuous ﬂow with entrained liquid droplets/drops

Annular Liquid-rich near the wall with a gas-rich core, not necessarily gas continuous Annular-bubble Little gas in the annular region near the wall, bubble flow in the centre Cap bubble Bubble flow where small bubbles have coalesced into larger cap bubbles Churn Bubble flow with larger, ‘chaotic’ gas structures

Table 3

Friction models considered.

Model Description

Homogeneous Friction calculated as for single-phase ﬂow, using gas-liquid mixture properties.

Friedel The Friedel (1979) friction model.

Friedel (Haaland) Friedel (1979) correlation with Haaland (1983) friction factor.

Beggs & Brill Two-phase friction factor correlation of Beggs and Brill (1973) .

Further, wehavetestedfourmodelsforgas-liquidslip, asdis- playedin Table4, whereonlythe T2Wellcorrelation is explicitly developedforCO₂ﬂow.

4.2.1. Liquidholdupvs.gasvolumetricﬂux

This section presents the calculation results for the different slip models presented in Table 4. The frictional pressure drop is calculatedbytheFriedel(Haaland)correlation.

Fig. 6 presents the downward-flow measured and calculated liquid holdup as a function of gas volumetric flux, j_g, with er- rorbandsindicatingtheestimatedexperimentaluncertainty.Each sub-figureisgeneratedforanapproximatelyconstantliquidvolu- metricflux, j.Weobservethatthereisquitegoodagreementbe- tweenmodelsandexperiments,butmostofthemodelspredicttoo lowliquidholdup,whichwouldcorrespondtoanunderprediction of thephase slip. The T2Well modelis seen to underpredict the holdupforlowvolumetricfluxes,i.e., for j at0.3ms⁻¹orbelow

(13)

M. Hammer, H. Deng, L. Liu et al. International Journal of Multiphase Flow 138 (2021) 103590 Table 4

Slip models considered.

Model Description

no-slip Homogeneous ﬂow ( u = u g)

Shi Drift-ﬂux correlation ﬂow of oil/gas/water based on large-pipe experimental data ( Shi et al., 2005b ).

T2Well Adaption of Shi model for CO 2well ﬂow ( Pan et al., 2011a,b ).

Zuber Equation (65) of Zuber and Findlay (1965) . See Equation (8) .

Fig. 12. Downward ﬂow: Comparison between measured and computed frictional pressure drop for different friction models.

and j_gbelowabout1.3ms⁻¹.Theno-slipmodel,theZuber-Findlay modelandtheShimodelhavesimilaroverallperformanceinthis case, althoughZuber-Findlay hasatendencyto overpredictionfor highgasvolumetricﬂuxesandtheother modelshavea tendency to underprediction.The resultisconsistentwiththefact thatthe measurementsshowsometendencytowards‘liquidaccumulation’, anditindicatesthattheestimateddriftvelocityislow.

Fig. 7 showsthe upward flow measured andcalculated liquid holdupasafunctionofgasvolumetricflux, jg.Weobservethatthe Zuber-Findlay modeloverpredictsthephase slip, consistentlygiv- ing a toolargeliquidholdup. Giventhequalitative resultsshown inFig.2,itisnotsurprisingthat theno-slipmodelgivesthebest fit withtheexperiments.The figurealsoshowsanoverprediction inholdupfromboththeShiandtheT2Wellmodel.Thisisaresult ofthosemodelspredictingalargerslip(ug−u) thanwhatisthe caseintheexperiment. Forthosemodels,the deviationislargest

forlowvolumetricﬂuxes, i.e., for j atorbelow0.3ms⁻¹ and j_g below0.6ms⁻¹.

4.2.2. Pressuredropvs.gasvolumetricﬂux

In addition to liquid holdup, the experimental results include the overall pressure change along the pipe. Given the measured holdup,thehydrostaticpressurecontributioniscalculated,andthe frictionalpressure dropcanbe determinedunderthe assumption thatonlyfrictionandgravitycontributetothepressuregradient.

Thissectionpresentsthecalculationresultsemployingthefric- tionmodels ofTable 3.Inthesecalculations,noslipbetweenthe phasesisassumed.

Fig.8showsthedownward-ﬂowmeasuredandcalculatedfric- tional pressure gradients as a function of gas volumetric ﬂux, jg, with error bands indicating the estimated experimental uncertainty. The Beggs & Brill model consistently overpredicts the

(14)

Fig. 13. Upward ﬂow: Comparison between measured and computed frictional pressure drop for different friction models.

friction. This is also the case for the homogeneous model, but the overpredictionissignificantly smaller.The FriedelandFriedel (Haaland) models underpredict the friction for liquid volumetric fluxesover1ms⁻¹,whilethereisquitegoodagreementbetween theFriedel(Haaland)modelandexperimentsforlowerliquidvol- umetric fluxes, and good agreementin generalfor low gas volumetricfluxes.

Weobservethattheexperimentaluncertaintyissignificant,es- pecially forlowfluxes.Onereasonforthisisthat theexperiment measures thetotalpressuredifference,andthefrictionalpressure dropiscalculatedbysubtractingthecontributionofgravity,which in this case isthe dominantpart. However, the good agreement betweenthecorrelationsandtheexperimentsatlowgasvolumet- ricfluxes,mayindicatethattheexperimentaluncertaintyissome- whatoverestimated.

Fig. 9 shows the upward-flow measured and calculated (negative) frictional pressuregradient asa function ofgas volumetric flux, jg.Thereisfairagreementbetweenmodelsandexperiments, with some exceptions: In this case, the Friedel model seems to generallyunderpredictthefrictionforthehigherliquidvolumetric fluxes, whereas the Beggs& Brill model overpredictsthe friction for thehighergas volumetricfluxes. Further,noneof themodels areabletopredictthefrictiontrendsforj_gbelow0.6ms⁻¹and j at1.0ms⁻¹andlower,wherethemeasuredfrictionismuchhigher

thanthemodelpredictions.Nevertheless,thedeviationsinthisre- gionaresmallerthantheexperimentaluncertainty.

4.3. Quantitativemodelperformance

In orderto quantify themodel performance, we calculate the root-mean-square(RMS)deviation(or2-norm)betweenthemodel prediction,y_calc,andtheexperimentalmeasurement,yexp:

δ

^rms⁼ ¹_nⁿ

i=1

y_calc_,i−yexp,i

2

. (13)

4.3.1. Liquidholdup

Table 5 gives the root-mean-square deviation between the holdup predictions and the experimental holdup from the X-ray measurements.Forupward flow,theno-slipmodelperforms best overall, whereas the no-slipmodel, theZuber-Findlay modeland the Shi model perform similarly for downward flow. The same can be seen from Figs. 10 and 11, where the performance of the different models are visualized by plotting predicted holdup againstmeasuredholdup,fordownwardandupwardflow,respectively.Thefiguresincludedashedlinesindicating ±30%deviation.

Fig.10(c)showsthatthe T2Wellmodelunderpredicts theholdup

(15)

M. Hammer, H. Deng, L. Liu et al. International Journal of Multiphase Flow 138 (2021) 103590 Table 5

RMS deviations between calculations and experiments for liquid holdup (–).

Data no-slip T2Well Zuber hi

downward, all data 0.025 0.059 0.028 0.027 downward, j = 0.15 m s ⁻¹ 0.036 0.072 0.035 0.026 downward, j = 0.30 m s ⁻¹ 0.017 0.082 0.034 0.026 downward, j = 1.00 m s ⁻¹ 0.021 0.053 0.025 0.030 downward, j = 2.00 m s ⁻¹ 0.022 0.038 0.021 0.028 downward, j = 2.77 m s ⁻¹ 0.022 0.033 0.020 0.027 upward, all data 0.015 0.061 0.075 0.036 upward, j = 0.15 m s ⁻¹ 0.020 0.101 0.098 0.059 upward, j = 0.30 m s ⁻¹ 0.017 0.086 0.097 0.050 upward, j = 1.00 m s ⁻¹ 0.011 0.028 0.063 0.011 upward, j = 2.00 m s ⁻¹ 0.016 0.019 0.053 0.016 upward, j = 2.77 m s ⁻¹ 0.011 0.016 0.048 0.012

Fig. 14. Dimensionless frictional pressure gradient versus mixture Froude number.

for downward flow, and from Fig. 11(c) we see that the same modeloverpredictstheholdupforupwardflow. Weobservefrom Figs. 10(d) and11(d) that the Zuber-Findlay modelslightly overpredicts theholdup for downwardflow, whereas the overpredic- tionislargeforupwardflowatlowvolumetricfluxes.Theno-slip model, on the other hand, predicts the upward flow rather well (Fig.11(a))whereasitunderpredictstheholdupfordownwardflow (Fig.10(a)).TheShimodelperformssimilarlytotheT2Wellmodel

for high measured holdups, but forlow holdups, it has a lower underpredictionfordownwardﬂow(Fig.10(b))andalower over- predictionforupwardﬂow(Fig.11(b)).

4.3.2. Frictionalpressuredrop

Table 6 displays the root-mean-square deviation between the calculated andmeasured frictional pressure gradient. The Friedel (Haaland) model andthe Beggs& Brill model performbetter for upward than for downward flow. The homogeneous model is by farthebestforthedownwardflow,whilethehomogeneousmodel, Friedel(Haaland)andBeggs&Brillhaveasimilarperformancefor upwardflow.

FromtheerrorplotsforthedownwardﬂowinFig.12,weob- servethat thehomogeneous model(Fig.12(a)) isthe onlymodel withthecorrectbehaviour athighvolumetric ﬂuxes,where both variantsof theFriedelmodel (Figs. 12(b)and12(c))underpredict thefriction.TheBeggs&Brillmodel(Fig.12(d))consistentlyover- predictsthepressuredrop.

From the error plots for the upward ﬂow in Fig. 13, we see that all modelsbehave reasonablywell, exceptthe Friedelmodel (Fig. 13(b)),which underpredictsthe frictiondueto thelack ofa termforthepiperoughness.

4.4. Differencesbetweenupwardanddownwardﬂow

AswasobservedinFigs.4–5,thedifferencesinﬂowregimebe- tweenupwardanddownwardﬂowarelimitedinthepresentcase.

Amainreasonforthisisthelowvaluesforthegas/liquidproperty ratios,

μ

/

μ

^g ^and

ρ

/

ρ

^g, bothapproximatelyequal to2.8. Never- theless,somedifferencesbetweenupwardanddownwardflowcan be seen. Toillustrate this, inFig. 14we have plottedthe dimensionless frictional pressure drop as a function of mixture Froude number,for both upward anddownwardflow. It is a cleartrend thatthefrictionalpressuredropishigherforupwardflow.Wealso observethat the data,particularly forupward flow, appear to be well correlated by theFroude number.Fordownward flow, there ismorescatter,whichmightberelatedtoflow-regime variations, ortheincreasedexperimentaluncertaintyduetothefactthatfric- tionandgravityhaveoppositeeffects.

A further illustration is given in Fig. 15, where upward and downward ﬂow data, forthe highest andlowest liquidvolumet- ricﬂux. Itcan beseen thatforthe liquidholdup(Fig.15(b)),the differencesare smalland mostlywithin the experimental uncertainty.ThisisconsistentwiththeobservationmadeforFigs.2–3. However,forthefrictionalpressuredrop(Fig.15(a)),thevaluesare

Fig. 15. Measured frictional pressure drop and liquid holdup: Comparison between upward and downward ﬂow.

(16)

M. Hammer, H. Deng, L. Liu et al. International Journal of Multiphase Flow 138 (2021) 103590 Table 6

RMS deviations between calculated and measured frictional pressure gradient (kPa m ⁻¹).

Data Homogeneous Friedel Friedel (Haaland) Beggs & Brill

downward, all data 0.123 0.642 0.398 0.644

downward, j = 0.15 m s ⁻¹ 0.138 0.090 0.129 0.378 downward, j = 0.30 m s ⁻¹ 0.082 0.157 0.075 0.320 downward, j = 1.00 m s ⁻¹ 0.138 0.332 0.159 0.533 downward, j = 2.00 m s ⁻¹ 0.091 0.824 0.527 0.739 downward, j = 2.77 m s ⁻¹ 0.152 1.113 0.682 1.000

upward, all data 0.293 0.640 0.250 0.357

upward, j = 0.15 m s ⁻¹ 0.128 0.125 0.146 0.158 upward, j = 0.30 m s ⁻¹ 0.155 0.156 0.197 0.218 upward, j = 1.00 m s ⁻¹ 0.210 0.295 0.215 0.335 upward, j = 2.00 m s ⁻¹ 0.334 0.726 0.233 0.414 upward, j = 2.77 m s ⁻¹ 0.484 1.181 0.389 0.529

higher forupwardflow. This tendencyismore significantforthe higherliquidvolumetricflux.ByinspectingFigs.8–9,weseethat theFriedelcorrelation(bothversions)capturesthistrend,atleast for highergas volumetricflux. The Beggs& Brillcorrelation cap- tures theincreasedfrictional pressuredrop forupward flowonly toa smallerextent.The homogeneousmodelonlycatersfortwo- phase flow via the liquidholdup andtherefore does not predict anydifferencebetweenupwardanddownwardflow.

5. Conclusion

Measurementsofliquidholdup,pressuredropandflowregime havebeenmadeforupwardanddownwardflowofCO₂inapipe ofinnerdiameter44mmatapressureof6.5MPa.Whilethispres- sure is relatively close to the critical pressure (7.38MPa), giving smalldifferencesinthe thermophysicalpropertiesofgasandliq- uid,weexpecttheflowtobegenuinelytwo-phase.Thiscondition is relevant for CO₂-injection wells, which may well be operated suchthatpartofthewellcontainsCO₂inatwo-phasestate.

The experimental resultsindicatethat theﬂow isclosetono- slip– withintheexperimentaluncertainty.Wehavecomparedthe experimental data to well-known models forphase slipandfric- tionalpressuredrop.Theresultsshowthatoverall,thebestmodel isthesimplestone– thefullyhomogeneousapproach,inwhichno slipisassumedandthefrictioniscalculatedsimplybyemploying gas-liquidmixturepropertiesinthesingle-phasefrictionmodel.

Inparticular, thehomogeneousmodelperforms bestforliquid holdupforupwardflowandforfrictionalpressuredropfordown- ward flow. For frictional pressure drop for upward flow, and for liquidholdupfordownwardflow, severalmodelsperformedsimi- larly.

The factthatthehomogeneous modelworkedbestoverall, in- dicatesthat the other modelstesteddo notcorrectlycapturethe ﬂow behaviourwhenthegasandliquidphasepropertiesbecome similarclosetothecriticalpoint.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

Morten Hammer: Methodology, Software, Investigation, Data curation, Writing - original draft, Writing - review & editing, Vi- sualization. Han Deng: Software, Formal analysis, Investigation, Data curation, Writing - review & editing,Visualization. Lan Liu:

Investigation, Data curation, Writing - review & editing. Morten Langsholt:Methodology,Investigation,Resources,Writing-review

& editing. SvendTollakMunkejord: Conceptualization, Writing - originaldraft,Writing-review&editing,Supervision,Fundingac- quisition.

Acknowledgements

ACT ELEGANCY,Project No 271498,has received fundingfrom DETEC (CH), BMWi (DE), RVO (NL), Gassnova (NO), BEIS (UK), Gassco,EquinorandTotal,andiscofundedbytheEuropeanCom- missionundertheHorizon2020programme,ACTGrantAgreement No691712.

AppendixA. Experimentaldata

The experimental data tables forupward and downward ﬂow areattachedassupplementaryﬁles.

Supplementarymaterial

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.ijmultiphaseﬂow.

2021.103590. References

Aursand, P., Hammer, M., Lavrov, A., Lund, H., Munkejord, S.T., Torsæter, M., 2017.

Well integrity for CO 2injection from ships: Simulation of the effect of ﬂow and material parameters on thermal stresses. Int. J. Greenh. Gas Con. 62, 130–141.

doi: 10.1016/j.ijggc.2017.04.007 . Jul.

Ayachi, F., Tauveron, N., Tartiére, T., Colasson, S., Nguyen, D., 2016. Thermo-electric energy storage involving CO 2transcritical cycles and ground heat storage. Appl.

Therm. Eng. 108, 1418–1428. doi: 10.1016/j.applthermaleng.2016.07.063 . Sep.

Ayub, A., Invernizzi, C.M., Di Marcoberardino, G., Iora, P., Manzolini, G., 2020. Car- bon dioxide mixtures as working ﬂuid for high-temperature heat recovery: a thermodynamic comparison with transcritical organic rankine cycles. Energies 13 (15), 4014. doi: 10.3390/en13154014 .

Balay, S., Abhyankar, S., Adams, M. F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., May, D. A., McInnes, L. C., Mills, R. T., Munson, T., Rupp, K., Sanan, P., Smith, B. F., Zampini, S., Zhang, H., Zhang, H., 2018. PETSc Web page. http://www.mcs.anl.gov/petsc . Balay, S. , Gropp, W.D. , McInnes, L.C. , Smith, B.F. , 1997. Eﬃcient management of

parallelism in object oriented numerical software Libraries. In: Arge, E., Bru- aset, A.M., Langtangen, H.P. (Eds.), Modern Software Tools in Scientiﬁc Comput- ing. Birkhäuser Press, pp. 163–202 .

Beggs, H.D. , Brill, J.P. , 1973. A study of two-phase ﬂow in inclined pipes. In: J. Petrol.

Technol., pp. 607–617 . Doi:10.2118/4007-PA

Collier, J.G. , Thome, J.R. , 1994. Convective Boiling and Condensation, third Oxford University Press, Oxford, UK . ISBN 0-19-856282-9

Cronshaw, M.B. , Bolling, J.D. , 1982. A numerical model of the non-isothermal ﬂow of carbon dioxide in wellbores. SPE California Regional Meeting. Society of Petroleum Engineers, San Francisco, California, USA . Doi:10.2118/10735-MS. Pa- per SPE-10735-MS.

Dorao, C.A., Ryu, J., Fernandino, M., 2019. Law of resistance in two-phase ﬂows in- side pipes. Appl. Phys. Lett. 114 (17), 173704. doi: 10.1063/1.5082808 .

Drew, D.A. , Passman, S.L. , 1999. Theory of Multicomponent Fluids, vol. 135 of Ap- plied Mathematical Sciences. Springer-Verlag, New York . ISBN 0-387-98380-5 Edenhofer, O., Pichs-Madruga, R., Sokona, Y., Farahani, E., Kadner, S., Seyboth, K.,

Adler, A., Baum, I., Brunner, S., Eickemeier, P., Kriemann, B., Savolainen, J.,