Study on the Pressure Drop Variation and Prediction Model

Research ArticleStudy on the Pressure Drop Variation and Prediction Model ofHeavy Oil Gas-Liquid Two-Phase Flow

Shanzhi Shi,1 Jie Li,1 Xinke Yang,1 Congping Liu,1 Ruiquan Liao ,2,3 Xingkai Zhang,2,3

and Jiadong Liao2,3

1Research Institute of Engineering Technology of Xinjiang Oilfield Company, Karamay, China2Petroleum Engineering College, Yangtze University, Wuhan, China3The Branch of Key Laboratory of CNPC for Oil and Gas Production, Yangtze University, Wuhan, China

Correspondence should be addressed to Ruiquan Liao; [email protected]

Received 8 August 2020; Revised 16 October 2020; Accepted 13 April 2021; Published 29 April 2021

Academic Editor: Reza Rezaee

Copyright © 2021 Shanzhi Shi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To explore the pressure drop variation with the viscosity of heavy oil gas-liquid two-phase flow, experiments with different viscositygas-liquid two-phase flows are carried out. The experimental results show that the total pressure drop increases with increasingliquid viscosity when the superficial gas and liquid flow rates are the same. The liquid superficial velocity is 0.52m/s, and thesuperficial gas velocity is 12m/s in the vertical and inclined pipes, as there is a negative friction pressure drop when thesuperficial gas and liquid velocities are small. Additionally, the increased range of the total pressure drop decreases withincreasing liquid viscosity. Considering the heavy oil gas-liquid two-phase flow, a prediction model of the pressure drop in high-viscosity liquid-gas two-phase flow is established. The new model is verified by experimental data and compared with existingmodels. The new model has the smallest error, basically within 15%. Based on the prediction of the wellbore pressuredistribution of four wells in the BeiA oilfield, the new model prediction results are closer to the measured results, and the erroris the smallest. The new model can be used to predict pressure drops in high-viscosity gas-liquid two-phase flow.

1. Introduction

As recoverable reserves of conventional crude oil are decreas-ing worldwide, heavy oil plays an important role in theenergy supply. During heavy oil development, the problemof high-viscosity fluid flowing in the wellbore is apparent.However, high viscosity poses great challenges for the pro-duction and transportation of heavy oil. In addition, gasand water are inevitably concurrently present with the oilin the pipeline flow process; thus, gas-liquid two-phase flowbehavior is more complex and difficult to predict [1].

However, most liquid holdup and pressure drop modelshave been developed based on low liquid viscosity [2, 3].Scholars have recently carried out theoretical and experimen-tal studies on high-viscosity gas-liquid two-phase flow.Schmidt et al. [4] conducted an experimental study on thephase and velocity distributions of the gas-liquid two-phaseflow of a high-viscosity fluid (viscosity up to 7000mPa∙s) in

a vertical upward pipe. The experimental results show thatthe existing void fraction model cannot agree with the voidfraction data under high-viscosity fluid. Zhang et al. [5] sum-marized the research progress of high-viscosity oil and com-pared it with the multiphase flow of low-viscosity oil(including the flow pattern, pressure gradient, and liquidholdup). The experimental results indicate that the flowbehaviors of high-viscosity oil and low-viscosity oil are quitedifferent. Jeyachandra et al. [6] studied the effects of high vis-cosity and pipe diameter on the drift velocity of horizontaland upward inclined pipes by conducting gas-liquid two-phase flow experiments and proposed a new high-viscositydrift velocity model suitable for horizontal to vertical pipes.Farsetti et al. [7] studied high-viscosity gas-liquid two-phase flow in horizontal and inclined pipes through experi-ments. The pressure gradient, bubble frequency, and lengthwere the main parameters measured in the experiment. Bycomparing the prediction results of existing low-viscosity

HindawiGeofluidsVolume 2021, Article ID 8813167, 20 pageshttps://doi.org/10.1155/2021/8813167

https://orcid.org/0000-0002-2109-1795

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https://doi.org/10.1155/2021/8813167

models, the existing models showed poor prediction abilityfor the flow behavior of high-viscosity fluids. Chung et al.[8] studied the effect of high-viscosity oil (122-560mPa∙s)on oil-gas flow behavior in vertical downward flow, mea-sured the pressure drop and liquid holdup data, and com-pared the experimental data of gas and water; they foundthat the viscosity has a significant impact on the flow behav-ior. Al-Ruhaimani et al. [9] studied high-viscosity oil-gastwo-phase flow in a vertical upward pipe and found thatthe friction pressure gradient of liquid increases with increas-ing viscosity. In addition, the negative friction pressure dropphenomenon of high-viscosity oil-gas multiphase pipe flowwas studied. Akhiyarov et al. [10] studied high-viscosity oil-gas two-phase flow in vertical pipes and found that a negativefriction pressure drop affected the prediction results. Liu [11]studied vertical gas-liquid two-phase pipe flow. The drop inthe liquid film in the Taylor bubble section resulted in a neg-ative friction pressure drop phenomenon, and the possibilityof a negative friction pressure drop phenomenon has beenproven by analysis of the quantity conservation equation.Al-Sarkhi et al. [12] studied the negative friction pressuredrop phenomenon of plug flow in high-viscosity oil-gastwo-phase flow in vertical pipes and qualitatively analyzedthe causes of the negative friction pressure drop phenome-non from the shear stress.

According to previous studies, most multiphase flowmodels are developed based on the experimental results oflow-viscosity fluids, although the flow behavior of high-viscosity fluids is significantly different from that of low-viscosity fluids. When these models are used to predict theflow behavior of high-viscosity fluids, they are quite differentfrom the measured data [13, 14]. In addition, research onhigh-viscosity liquid-liquid two-phase flow has recentlybegun, but due to the limitations of experimental conditions,the development of this research is relatively slow; therefore,there is no comprehensive model to predict the pressure dropunder different flow patterns of high-viscosity liquid.

Considering the existing problems, the gas-liquid two-phase flow in the wellbore of heavy oil production is takenas the research object in this paper. The change rule of thepressure drop in gas-liquid two-phase flow with the changein liquid viscosity is explored through an indoor physicalmodel experiment. By using the mechanical equation ofgas-liquid two-phase flow combined with the appropriateclosed relation formula, a prediction model of the pressuredrop in high-viscosity liquid-liquid two-phase flow isestablished, which provides theoretical support for accuratelypredicting the wellbore pressure distribution of gas-liquidtwo-phase flows of different viscosities.

2. Experiment of Gas-Liquid Two-PhaseFlow with Different Liquid Viscosities

2.1. Experimental Fluid and Equipment

2.1.1. Experimental Fluid. The gas phase of the gas-liquidtwo-phase flow experiment with different viscosities is air,and the liquid phase is a tackifying white oil. The viscositytest results of the tackifying white oil are shown in Figure 1.

By fitting the experimental data, the mathematical rela-tionship between the temperature and the viscosity of thetackifying white oil can be determined as follows:

μ1 = −0:0033T3 + 0:5599T2 − 33:597T + 765:36, ð1Þ

where μ1 is the viscosity of the tackifying white oil, mPa∙s;and T is the temperature, °C.

The gas-liquid two-phase flow experiment with differentoil viscosities can be carried out by changing the temperature.

2.1.2. Multiphase Flow Experiment Device and Process. Thisexperiment is carried out on a multiphase flow experimentalplatform. The main process of the experiment is described asfollows: the liquid in the mixing tank is pressurized by theliquid pump, stabilized, and measured, and the liquid is thenmixed with the measured compressed gas in the test pipe sec-tion. Finally, the gas is separated by the gas-liquid separator,and the liquid returns to the oil-water mixing tank. The gasfrom the test pipe section is directly discharged into theatmosphere. The experimental device and process are shownin Figure 2.

The pipe section selected in the experiment has an innerdiameter of 60mm and a total length of 11.5m. A pressuresensor, temperature sensor, differential pressure sensor,quick-closing valve, and other devices are installed in the pipesection (the measurement range and system error of thedevices are shown in Table 1). The distance between the twoquick-closing valves is 9.5m and includes a 7m plexiglass pipeand a 2.5m long stainless steel pipe. The drain valve is used toplace the liquid into the measuring cylinder after closing thequick-closing valve to measure the volume of the liquid phase.

2.1.3. Experimental White Oil Viscosity MeasuringEquipment. In this experiment, a Brookfield DV II rotaryviscosimeter and Thermosel heater are used to measure theviscosity of the experimental liquid phase fluid.

2.2. Experimental Principle

2.2.1. Viscosity. In the experiment, the same liquid samplesare obtained three times, and the measured liquid samplesare heated to the set temperature by the Thermosel heater.Then, the viscosities of the three liquid samples are measuredwith the Brookfield DV II rotary viscometer, and the data arerecorded. Finally, the average value is taken as the viscosity ofthe liquid at this temperature.

2.2.2. Flow Pattern. In the experiment, the flow pattern isdetermined by the combination of a high-speed camerainstalled on the test pipe section and experimental observa-tions and is recorded and photographed. The high-speedcamera (Canon Xtra NX4-S1) used to record the flow patternis shown in Figure 3. The main parameters of this camera areas follows: the pixel resolution is 1920 × 1080, and the maxi-mum frame rate is 50000 fps.

2.2.3. Differential Pressure. The differential pressure sensorinstalled on the test pipe section is used to measure the pres-sure difference when the flow is stable and to record and save

2 Geofluids

the value. To accurately measure the pressure drop in the gas-liquid two-phase flow, a Rosemount 3051S differential pres-sure sensor is installed in the laboratory. The high-pressureand low-pressure sides of the sensor are connected to the sen-sor by a pipeline filled with silicone oil, as the pressure istransmitted through the highly sensitive diaphragm vibration

at the high-pressure and low-pressure sides. An image of thedifferential pressure sensor is shown in Figure 4.

2.3. Experimental Scheme. Heavy oil refers to high-viscosityheavy crude oil with a viscosity greater than 50mPa∙s underformation conditions. This experiment is aimed at gas-liquidtwo-phase flow in highly inclined oil wells. Based on the vis-cosity characteristics of heavy oil and the characteristics ofthe experimental equipment, experiments on gas-liquidtwo-phase flow with different viscosities of 50mPa∙s,100mPa∙s, 290mPa∙s, and 480mPa∙s are carried out. Thespecific experimental scheme is shown in Table 2.

2.4. Experimental Procedure. The experiment is conducted asfollows:

(1) Check the state of the test device, including whetherthe valves are in the correct open or closed states

y = –0.0033x3 + 0.5599x2–33.597x + 765.36R² = 0.9999

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0 10 20 30 40 50 60 70

Visc

osity

(mPa

∙s)

Temperature (°C)

Figure 1: Viscosity vs. temperature curve of the tackifying white oil.

Pump

1. Water meter 2. Filter 3. Accumulator 4. Pressure sensor 5. Liquid flow meter 6. Quick-closevalve7. Differential pressure sensor 8. Temperature-pressure sensor 9. Gas-liquid mixture 10. Temperature sensor

Manual valve

Gas-powered valve

Test-pipe section

Water inlet

Gas outlet

Gas inlet

separator

Control center

WaterMixing

tanktank

1 3

24

59

10

8

76

tankOil

Gas-liquid

Figure 2: Device and process of the multiphase flow experiment.

Table 1: Measurement ranges and system errors of eachinstrument.

Device Measurement range System error

Pressure sensor 0-3.5MPa ±0.1%Temperature sensor 0-90°C ±0.5%Liquid mass flowmeter 0-20m3/h ±0.3%Gas mass flowmeter 0-2000m3/h ±1%Differential pressure sensor 0-0.25MPa ±0.25%

3Geofluids

and whether the pressure of the test pipe section isnormal

(2) Turn on the cooling water system and air compressor

(3) Adjust the experimental pipe section to the anglerequired by the experiment through the experimentalconsole

(4) Open the air inlet valve, adjust the air volume to alarger value to first dry the residual liquid in the pipe-line, and then close the air inlet valves and clear thedifferential pressure

(5) Open the agitator in the oil-water mixing tank, startthe electric heating, heat the liquid in the oil-watermixing tank to the predetermined temperature, openthe liquid inlet valve, adjust the liquid volume to therequired value, open the air inlet valve, adjust theair volume to the required value, adjust the openingof the outlet valve, and start the air heater to heatthe gas to the predetermined temperature. After thetemperature in the test tube reaches the experimentalset value, observe the gas-liquid two-phase flow sta-bility, start to record the data, acquire images withthe high-speed camera, and then observe and recordthe flow pattern

(6) After data recording, first, close the liquid inlet valveand open the air inlet valve. Check whether the pres-sure difference in the pipe is normal and repeat theexperiment outlined in step (5) after the liquid inthe pipe section is blown out

(7) Close the liquid inlet valve and air inlet valve after allexperiments at one inclination angle are completedand repeat the above experiments beginning withstep (3)

(8) After the completion of all experiments, close theplunger pump and liquid inlet valve, increase the airvolume to blow out the liquid in the test pipe section,and close the air inlet valve and air compressor. Then,restore the test pipe section to the horizontal positionand turn off the computer and power supply

2.5. Analysis of Experimental Results. The total pressure dropin gas-liquid two-phase flow is composed of a gravity pres-sure drop and friction pressure drop. The variation in thepressure drop with viscosity is discussed in terms of threeaspects: total pressure drop, gravity pressure drop, and fric-tion pressure drop.

2.5.1. Variation in the Gravity Pressure Drop with LiquidViscosity. The variation in the gravity pressure drop with liq-uid viscosity is shown in Figure 5. The figure shows that inthe vertical and inclined pipes, at the same apparent gasand liquid flow rates, the gravity pressure drop increases withincreasing viscosity. The main reason is that the increase inviscosity increases the viscous force between the liquid phaseand the pipe wall, causing more liquid to stay on the pipewall, which results in the increase in the liquid holdup.According to formulas (2) and (3), the gravity pressure dropis determined mainly by the liquid holdup. Therefore, thegravity pressure drop increases with increasing liquidviscosity.

2.5.2. Variation in the Friction Pressure Drop with LiquidViscosity. The variation in the friction pressure drop with liq-uid viscosity is shown in Figure 6. The figure shows that inthe vertical and inclined pipes, when the apparent liquid flowrate is greater than 0.16m/s and the apparent gas flow rate isgreater than 6m/s, the friction pressure drop increases withincreasing liquid viscosity at the same apparent gas and liq-uid velocity. When the apparent liquid flow rate is less than0.16m/s and the apparent gas flow rate is less than 6m/s,the friction pressure drop displays a negative value becauseunder the same apparent gas and liquid velocity, the negativevalue increases with increasing viscosity. The reason is thatthe shear force between the liquid phase and the pipe wallincreases with increasing viscosity at higher apparent gasand liquid flow rates; thus, the friction pressure dropincreases with increasing liquid viscosity. However, at lowerapparent gas and liquid flow rates, a negative wall shear stressmay exist due to the backflow of the liquid film (in the lami-nar state), which leads to a negative friction pressure drop.With increasing liquid viscosity, the liquid Reynolds number(Ref = ρlvfd/μl) in the liquid film decreases. According to thecorrelation formula (f l = C/Ref ) of the friction coefficient inlaminar flow, when the Reynolds number decreases, the cor-responding friction coefficient increases, which results in anincrease in the negative friction pressure drop.

Figure 3: High-speed camera.

Figure 4: Differential pressure sensor.

4 Geofluids

2.5.3. Variation in the Total Pressure Drop with LiquidViscosity. The variation in the total pressure drop with liquidviscosity is shown in Figure 7. The figure shows that in thevertical and inclined pipes, when the apparent gas and liquidflow rates are the same and the apparent liquid flow rate is0.52m/s and the apparent gas flow rate is 12m/s, the totalpressure drop increases with increasing liquid viscosity.

However, when the apparent gas and liquid flow rates aresmall and a negative friction pressure drop occurs, the nega-tive value of the negative friction pressure drop increaseswith increasing viscosity of the liquid because the gravitypressure drop increases with increasing viscosity and theincreased range of the total pressure drop decreases withincreasing liquid viscosity.

Table 2: Experimental scheme of gas-liquid two-phase flow under different viscosities.

Temperature(°C)

Oil viscosity(mPa∙s)

Inclination angle(°)

Pipe diameter(mm)

Apparent liquid flow rate(m/s)

Apparent gas flow rate(m/s)

10, 20, 40, 60 480, 290, 100, 50 90, 60 60 0.02, 0.08, 0.16, 0.52 0.2~23

0 100 200 300 400 5001.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

vsl = 0.08 m/s, vsg = 1.8 m/svsl = 0.16 m/s, vsg = 6 m/svsl = 0.52 m/s, vsg = 12 m/s

Gra

vity

pre

ssur

e dro

p (k

Pa/m

)

Oil viscosity (mPa·s)

(a) Vertical pipe


0 100 200 300 400 5001.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5


Gra

vity

pre

ssur

e dro

p (k

Pa/m

)

(b) Inclined pipe at 60 degrees

Figure 5: Variation in the gravity pressure drop with oil viscosity.

5Geofluids

3. Prediction Model of the Pressure Drop

3.1. Prediction Model of the Pressure Drop in Slug Flow

3.1.1. Gravity Pressure Drop in Slug Flow. The gravity pres-sure drop is mainly related to the density of the mixed fluidand the inclination of the pipeline, and the expression isdescribed as follows:

dPdL

� �h= ρmg sin θ, ð2Þ

ρm = ρlH l + ρg 1 −H lð Þ, ð3Þ

where ρm is the density of the mixture in the slug body,kg/m3; Hl is the liquid holding capacity of slug flow; ρl isthe density of the liquid phase, kg/m3; ρg is the density ofthe gas, kg/m3; and θ is the inclination angle, °.

The gravity pressure drop in slug flow is determinedmainly by the liquid holdup. To consider the influence ofviscosity, the liquid holdup calculation model of slug flowproposed by Liu et al. [1] is selected in this paper.

3.1.2. Friction Pressure Drop in Slug Flow

(1) Hydrodynamic Model of the Slug Flow Film Region. Taiteland Dukler [15] and Barnea [16] comprehensively analyzedslug flow and extended the research scope of slug flow to a

–2

–1

0

1

2

3

4

5

Fric

tion

pres

sure

dro

p (k

Pa/m

)

0 100 200 300 400 500



(a) Vertical pipe

–1

0

1

2

3

4

5

Fric

tion

pres

sure

dro

p (k

Pa/m

)

0 100 200 300 400 500




Figure 6: Variation in the friction pressure drop with oil viscosity.

6 Geofluids

unified model of horizontal flow, upward sloping flow, andupward vertical flow.

(a) Momentum Equation of the Liquid Film Region

In the translational velocity coordinate system, themomentum equations of the liquid film region and the airplug region are expressed as follows:

∂p∂z

=τFsFAF

−τlslAl

− ρlg sin θ, ð4Þ

∂p∂z

=τGsGAG

+τlslAl

− ρgg sin θ: ð5Þ

By combining the above two formulas, we obtain the fol-lowing expression:

τFsFAF

−τGsGAG

− τlsl1Al

+1AG

� �+ ρl − ρg

� �g sin θ, ð6Þ

where τF is the shear force in the liquid film region, N; τG isthe shear force in the air plug region, N; τl is the shear forcein the gas-liquid interface, N; sF is the perimeter of the liquidfilm region, m; sG is the perimeter of the gas plug region, m; slis the length of the gas-liquid interface, m; AF is the cross-

0

2

4

6

8

10

Tota

l pre

ssur

e dro

p (k

Pa/m

)

0 100 200 300 400 500



(a) Vertical pipe

0

2

4

6

8

10

Tota

l pre

ssur

e dro

p (k

Pa/m

)

0 100 200 300 400 500




Figure 7: Variation in the total pressure drop with oil viscosity.

7Geofluids

0 2 4 6 8 10

2

0

4

6

8

10

Experimental pressure drop measurement (kPa/m)

Calc

ulat

ed v

alue

of p

ress

ure d

rop

mod

el (k

Pa/m

)

Beggs-BrillAzizHasan

JPIKayaNew model

–20%

+20%

Oil viscosity 50 mPa·s

(a) 50 mPa∙s

–20%

+20%Oil viscosity 100 mPa·s

0 2 4 6 8 10Experimental pressure drop measurement (kPa/m)


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)(b) 100mPa∙s

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(c) 290mPa∙s

Figure 8: Continued.

8 Geofluids

sectional area of the liquid membrane, m2; AG is the cross-sectional area of the gas plug, m2; and Al is the cross-sectional area of the gas-liquid interface, m2.

τF = f Fρl vLTBj jvLTB

2, ð7Þ

τG = f Gρg vGTBj jvGTB

2, ð8Þ

τl = f lρg vGTB − vLTBj j vGTB − vLTBð Þ

2, ð9Þ

where f F is the friction coefficient of the liquid film and pipe-line wall; f G is the friction coefficient of the gas plug andpipeline wall; f l is the friction coefficient of the gas-liquidinterface; vLTB is the liquid flow velocity in the liquid filmregion, m/s; and vGTB is the gas flow velocity in the gas plugregion, m/s.

f F = CρldFvLTB

μl

� �−n

, ð10Þ

where hydraulic diameter dF = 4AF/SF.The friction coefficient of the gas phase can also be calcu-

lated by the same method, but dg = 4Ag/ðSg + SlÞ. For laminarflow, C = 16 and n = 1; for turbulent flow, C = 0:046 and n= 0:2.

It is complex to determine the friction coefficient of thegas-liquid interface. For low liquid and gas velocities, thesmooth interface friction coefficient can be applied, i.e., f l

= f g. For upward flow, the friction coefficient of the gas-liquid interface can be expressed as follows:

f l = 0:005 1 + 300hFD

� �, ð11Þ

where hF is the thickness of the liquid film, m.

(b) Determination of Flow Parameters in the Liquid FilmRegion

For the given flow conditions, including the apparent liq-uid velocity, the apparent gas velocity, the physical propertiesof the fluid, the diameter of the pipe, and the inclinationangle, the momentum equation in the liquid film regioncan be solved by combining the relevant closed relations.This equation is an implicit equation for the film thicknesshF. Therefore, an iterative calculation is needed in the processof solving the equation:

(1) First, based on the flow variables and the slug flowholdup calculation method, the slug average holdupHl and the related parameters vl, vb, and Hls arecalculated

(2) The liquid phase velocity vLLS in the liquid plug areais calculated, as is the closed relation equation (12)

(3) Assuming an appropriate film thickness hF, Hltb, AF,AG, sF, sG, and dF are calculated

(4) The velocity vLTB of the liquid phase in the liquidfilm region is calculated, as is the closed relationequation (13)

–20%

+20%




JPIKayaNew model

2

0

4

6

8

10

Calc

ulat

ed v

alue

of p

ress

ure d

rop

mod

el (k

Pa/m

)

(d) 480mPa∙s

Figure 8: Comparison between calculated values of different models and experimentally measured values in slug flow.

9Geofluids

0 2 4 6

–20%


+20%



JPIKayaNew model

2

0

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6

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ulat

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alue

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Pa/m

)

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–20%

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0 2 4 6



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10 Geofluids

(5) The friction coefficient of the liquid film and pipelinewall f F, the friction coefficient of the gas plug andpipeline wall f G, and the friction coefficient of thegas-liquid interface f l are calculated

(6) The shear force τF in the liquid film region, theshear force τG in the gas plug region, and theshear force τl at the gas-liquid interface arecalculated

(7) A step is performed to check whether the momentumequation converges. If it does not converge, steps (3)to (6) are repeated until it converges

(c) Correlation Closed Relation

The equation for calculating the liquid phase velocity vLLSin the liquid plug region is expressed as follows:

vLLS =vm − vGLS 1 −H lsð Þ½ �

H ls: ð12Þ

The equation for calculating vLTB of the liquid phasevelocity in the liquid film region is expressed as follows:

0 2 4Experimental pressure drop measurement (kPa/m)

–20%

+20%



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alue

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ress

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rop

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Pa/m

)

(d) 480mPa∙s

Figure 9: Comparison between calculated values of different models and experimentally measured values in agitation flow.

11Geofluids

–20%

+20%




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2

0

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(a) 50 mPa∙s

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12 Geofluids

vLTB = vt −vt − vLLSð ÞH ls

Hltb, ð13Þ

where H ltb is the liquid holdup of the liquid film region.The geometric relationship is expressed as follows:

H ltb =1π

π − cos−1 2hFd

− 1� �

+ 2hFd

− 1� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − 2hFd

− 1� �2

s24

35,

ð14Þ

sF = d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 2

hFd

− 1� �2

s, ð15Þ

sg = πd − sF, ð16Þ

AF = ApH ltb, ð17Þ

AG = Ap − AF: ð18Þ(2) Calculation of the Friction Pressure Drop in Slug Flow. Inslug flow, the liquid film in the liquid plug region flowsupward because the liquid phase in the liquid film region

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)

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Figure 10: Comparison between calculated values of different models and experimentally measured values in annular flow.

13Geofluids

flows downward. The friction pressure drop in the whole slugunit can be expressed as follows:

−dp =τsπdAp

Ls +τFSFAp

LF +τgSgAp

LF, ð19Þ

τs = f sρl vlsj jvls

2, ð20Þ

f s = Cρldvlsμl

� �−n

, ð21Þ

where vls is the average velocity of the liquid film along thepipe wall, m. In this paper, vls = vLLS/2.

The length of the liquid plug region is as follows:

Ls = 20d sin θ + 30d cos θ: ð22Þ

The length of the slug unit is as follows:

LU =Ls vLLSHls − vLTBHltbð Þ

vsl − vLTBH ltb: ð23Þ

The length of the liquid film region is as follows:

LF = LU − Ls: ð24Þ

3.2. Prediction Model of the Pressure Drop in Agitation Flow

3.2.1. Gravity Pressure Drop in Agitation Flow. The expres-sion for the gravity pressure drop is the same as equations(2) and (3). The characteristics of agitation flow oscillate upand down, and the research on its flow characteristics is inad-equate. For the convenience of calculation, the slug holdupmodel selected in this paper is used to calculate the holdupin agitation flow to calculate the gravity pressure drop in agi-tation flow.

Table 3: Error percentages of different pressure drop models.

Flow pattern Viscosity Error B-B Aziz Hasan JPI Kaya New model

Slug flow

50mPa∙sAAPE (%) 35.8 200.9 33.3 47.2 118.5 12.8

APE (%) 35.8 -167.2 33.3 47.2 -95.1 -6.5

100mPa∙sAAPE (%) 38.3 196.9 37.0 48.7 107.3 12.2

APE (%) 38.2 -163.5 37.0 48.7 -84.3 0.02

290mPa∙sAAPE (%) 51.5 154.0 46.9 57.9 251.3 12.3

APE (%) 51.5 -88.6 46.9 57.9 -241.3 -4.4

480mPa∙sAAPE (%) 48.1 143.3 40.1 55.4 166.8 11.9

APE (%) 48.0 -91.7 39.6 55.4 -163.0 -3.6

Agitation flow

50mPa∙sAAPE (%) 65.4 47.5 61.4 64.8 72.5 13.9

APE (%) 65.4 47.5 60.5 64.8 -19.0 -6.5

100mPa∙sAAPE (%) 73.0 56.3 78.0 69.3 34.9 15.3

APE (%) 73.0 55.7 78.0 69.3 34.9 -4.2

290mPa∙sAAPE (%) 84.3 62.6 89.6 75.4 404.2 19.4

APE (%) 84.3 62.6 89.6 75.4 -365.1 -19.4

480mPa∙sAAPE (%) 84.5 64.0 84.8 77.6 594.8 19.8

APE (%) 84.5 64.0 84.8 77.6 -594.8 -10.7

Annular flow

50mPa∙sAAPE (%) 66.5 43.7 69.3 60.3 115 14.6

APE (%) 66.5 40.2 0.1 60.3 110 -9.7

100mPa∙sAAPE (%) 71.7 50.6 63.3 63.9 35 13.8

APE (%) 71.7 47.2 6.1 63.9 -35 -0.7

290mPa∙sAAPE (%) 82.6 73.0 69.8 75.5 164 14.5

APE (%) 82.6 73.0 61.9 75.5 -148 10.6

480mPa∙sAAPE (%) 84.1 79.2 72.4 80.6 782 14.3

APE (%) 84.1 79.2 70.8 80.6 -774 1.6

Table 4: Viscosity data of dead oil at different temperatures.

Temperature (°C) 15.6 18.3 21.1 23.9 26.7 30.0 33.3 36.7 40.0 43.3 46.7

Viscosity (mPa∙s) 482.3 395.2 324.6 269.0 225.4 180.0 146.4 123.3 102.4 85.0 73.0

14 Geofluids

3.2.2. Friction Pressure Drop in Agitation Flow. The frictionpressure drop changes from a negative friction pressure dropto a positive friction pressure drop with increasing apparentgas velocity. Analysis of the characteristics of agitation flowindicates that the flow pattern is agitation flow when the fric-tion pressure drop is 0. The friction pressure drop in slugflow established in this paper can be used to predict the neg-ative and positive friction pressure drops. Therefore, the fric-tion pressure drop model of slug flow is used to calculate thefriction pressure drop in agitation flow: ðdP/dLÞf .3.3. Prediction Model of the Pressure Drop in Annular Flow

3.3.1. Gravity Pressure Drop in Annular Flow. Similarly, themethod for calculating the gravity pressure drop is the sameas that for slug flow and agitation flow. The difference is thatthe liquid holdup of annular flow is determined by the calcu-lation method given in this paper.

The model for calculating the liquid holdup of upwardannular flow can be divided into an empirical model and amechanistic model. Because the mechanistic model mustiteratively calculate the thickness of the liquid film aroundthe pipe wall, the equation has difficulty converging in theiterative calculation process, and the calculation results areoften quite different from the actual values. Therefore, themethod for calculating the liquid holdup in annular flow isobtained by fitting the experimental data based on the empir-ical formula of the liquid holdup.

The main purpose of this paper is to provide a model ofgas-liquid two-phase flow suitable for different viscosities.Therefore, the influence of viscosity should be consideredin the empirical model. The empirical formula given byMukherjee and Brill [17] includes dimensionless parametersthat consider viscosity. Therefore, we choose to modify theempirical constant in the relationship.

The relevant formula of Mukherjee and Brill is describedas follows:

H l = exp c1 + c2 sin θ + c3 sin2θ + c4N l2� �Nvg

c5

Nvlc6

, ð25Þ

Nvl = vslρlgσ

� �0:25, ð26Þ

Nvg = vsgρlgσ

� �0:25, ð27Þ

Nμl = μlg

ρlσ3

� �0:25: ð28Þ

Fitting the relationship between the liquid holdup H lmeasured in the experiment and the Nvl, Nvg, and Nμl ofthe correlation standard number given by Mukherjee andBrill, the new empirical coefficients c1-c6 are -0.131, 0.064,-0.0475, 0.0057, 0.6, and 0.099, respectively.

3.3.2. Friction Pressure Drop in Annular Flow

(1) Physical Model of Annular Flow. Alves et al. [18] estab-lished a vertical tube and a large inclined angle annular flowmodel. The annular liquid film flows upward at a certainspeed along the pipe wall, and the liquid drop in the middlemoves upward with the gas. The shear stress of the tube wallto the liquid film is opposite to the moving direction of theliquid film, so it is impossible for the annular flow to have anegative friction pressure drop.

(2) Annular Flow Friction Pressure Drop. According to theflow characteristics of annular flow, only the friction pressuredrop between the upward flowing liquid film and the wallsurface must be calculated, which is the friction pressuredrop in the annular flow. The friction pressure drop in theliquid film area can be expressed as follows:

∂p∂L

� �f= −τWL

sFAF

, ð29Þ

τWL = f lρlvF

2

2, ð30Þ

f l = CρldvFμl

� �−n

, ð31Þ

where τWL is the shear stress between the liquid phase andthe pipe wall; sF is the perimeter of the liquid film, m; AF isthe sectional area of the liquid film, m2; and vF is the velocityof the liquid film, m/s.

Table 5: Basic parameters of well XXXX-004.

Oil well typeMiddle depth of the

oil layer (m)Tubing size Casing size Bubble point pressure (MPa)

Straight well 3487 41/2″ + 31/2″ 95/8″ + 7″ 18.34

Water content (%) Gas oil ratio Relative density of water API gravity of crude oilRelative density of

natural gas

0 52.97 1.13 17.68 0.99

Production (m3/d) Wellhead oil pressure (MPa)Bottom hole flowpressure (MPa)

Wellhead temperature (°C) Bottom hole temperature (°C)

550.56 7.48 27.46 20 96

15Geofluids

–3500

–3000

–2500

–2000

–1500

–1000

–500

0

10 20 30Pressure (MPa)

Wel

l dep

th (m

)

Beggs-BrillAzizHasanJPI

KayaNew modelMeasured data

XXXX-004

(a) Well XXXX-004



–3000

–2500

–2000

–1500

–1000

–500

0

0 10 20 30Pressure (MPa)

Wel

l dep

th (m

)

XXXX-010

(b) Well XXXX-010


16 Geofluids

The speed of the liquid film can be determined by asimple mass balance calculation, expressed as follows:

vF = vsl1 − f Eð Þd24δl d − δlð Þ : ð32Þ

The gas and entrained droplets are assumed to flow uni-formly in the gas core without slippage, and the void ratio ofthe gas in the gas core is expressed as follows:

aC =vsg

vsg + vsl f E: ð33Þ

The relationship between the liquid holdup and the liq-uid film thickness is expressed as follows:

1 −H l = aC 1 − 2δld

� �2, ð34Þ

where δl is the thickness of the liquid film, m.

–3500

–3000

–2500

–2000

–1500

–1000

–500

0

0 10 20 30Pressure (MPa)

Wel

l dep

th (m

)

XXXX-012



(c) Well XXXX-012

–2500

–2000

–1500

–1000

–500

0

10 20 30Pressure (MPa)

Wel

l dep

th (m

)

XXXX-026



(d) Well XXXX-026

Figure 11: Predictions of wellbore pressure distributions by the different models.

17Geofluids

The equation for calculating the entrainment rate ofdroplets is expressed as follows:

f E = 1 − exp −0:125 φ − 1:5ð Þ½ �, ð35Þ

φ = 104vsgμgσ

ρgρl

� �0:5: ð36Þ

According to the geometric relationship,

sF = πd, ð37Þ

AC = πD2− δl

� �2, ð38Þ

AF = Ap − AC: ð39ÞThe method for calculating the annular flow friction

pressure drop is described as follows:

(1) According to the method for calculating the liquidholdup in annular flow, after calculating the liquidholdup, the liquid film thickness δl is calculatedaccording to the relationship between the liquidholdup and the liquid film thickness

(2) According to equation (35), the entrainment rate ofthe liquid drop f E is calculated

(3) According to equation (32), the liquid film velocity vFis calculated, and the relevant geometric parameterssF and AF are then calculated

(4) The friction coefficient f l is calculated according tothe friction coefficient calculation method, and theshear stress τWL between the liquid film and the wallis then calculated

(5) Finally, the friction pressure drop in annular flow iscalculated according to equation (29)

3.4. Verification of the Pressure Drop Prediction Model. Thepressure drop models of slug flow, agitation flow, and annu-lar flow are verified by experimental data, as the existingempirical model (Beggs-Brill [19]) and mechanistic models(Aziz [20], Hasan [21], JPI [22], and Kaya [23]) are usedfor comparison with the model in this paper. The calculated

pressure drop values of the six models with different viscosi-ties are compared with the experimental values, and theresults are shown in Figures 8–10 and Table 3.

The figure shows that compared with the experimentalmeasurement values, the error in the pressure drop in themodel established in this paper is largely within 20%, whichis smaller than the error of the existing five models. The cal-culation errors of most existing models are greater than 20%,with large errors. Comparison of the pressure drop calcula-tion errors of different models with different viscosities indi-cates that the errors of the existing models are relatively largeand increase as the viscosity increases. The average absoluteerror rate of the model established in this paper is less than15%, which is accurate for the prediction of experimentaldata and suitable for calculating the pressure drop for slugflow, agitation flow, and annular flow with differentviscosities.

4. Case Calculation and ComparativeVerification of WellborePressure Distribution

In this paper, the test data for four wells with low gas : oilratios (20~53m3/m3) and high production (227.2-656.3m3/d) in the BeiA oilfield of Iran are selected. Similarly,the wellbore pressure distribution is calculated by using theexisting empirical model (Beggs-Brill), existing mechanisticmodels (Aziz, Hasan, JPI, and Kaya), and multiphase flowpressure drop model established in this paper.

4.1. Viscosity of Degassed Crude Oil. The viscosity of thedegassed crude oil of the oilfield at different temperatures istested under laboratory conditions. The viscosity data areshown in Table 4. The data in the table show that the crudeoil of the oilfield is ordinary heavy oil, and the viscosity issimilar to that of the oil used in this experiment.

4.2. Example Well Calculation. The pressure test data for thefour selected oil wells are wellhead pressure and bottom holeflow pressure. The whole wellbore is divided into 14 sections,and different pressure drop models are used to calculate thewellbore pressure distribution from the wellhead to the bot-tom of the well. The distribution of wellbore pressure pre-dicted by the different models is given, and the calculationresults of the different models are compared. The basic

Table 6: Calculation errors of the different models.

Well Error B-B Aziz Hasan JPI Kaya New model

XXXX-004AAPE (%) 25.07 23.01 25.52 22.28 28.35 -1.23

APE (%) 25.07 23.01 25.52 22.28 28.35 12.61

XXXX-010AAPE (%) 27.93 28.25 28.59 28.03 31.25 13.7

APE (%) 27.93 28.25 28.59 28.03 30.82 7.1

XXXX-012AAPE (%) 34.28 19.32 19.59 19.28 36.4 16.3

APE (%) 34.28 17.68 18.1 17.61 36.4 7.04

XXXX-026AAPE (%) 35.45 36.65 36.09 35.25 38.15 13.33

APE (%) 35.45 36.65 36.09 35.25 38.14 8.24

18 Geofluids

parameters of well XXXX-004, as an example, are shown inTable 5.

4.2.1. Calculation of Wellbore Pressure Distribution byDifferent Models. The distribution of wellbore pressure calcu-lated by the six different models for four oil wells is shown inFigure 11. The figure shows that since the measured value hasonly two pressure points at the wellhead and bottom of thewell, the measured pressure distribution in the figure is rep-resented by a straight line, which is different from the actualwell distribution. The accuracy of the new model consideringthe effect of viscosity is higher, as the calculated value is nearthe measured wellbore pressure distribution curve and thewhole wellbore pressure distribution is closer to the mea-sured result. Therefore, the pressure drop model in this papercan be used to predict the pressure distribution of ordinaryheavy oil wells.

4.2.2. Calculation Errors of the Different Models. The resultsof the average relative error and average absolute error ratescalculated by the different models are shown in Table 6.The table shows that the error of the model established in thispaper is the smallest, as the average absolute error rate is basi-cally within 15%, and the calculation errors of most existingmethods are greater than 20%. Therefore, the predictionresults of the new model considering the influence of viscos-ity and predicting the negative friction pressure drop are bet-ter than those of the existing model, and the reliability of theprediction model given in this paper is verified from practicalengineering applications in the field.

5. Conclusions

To obtain an accurate method for calculating heavy oil-gastwo-phase pressure drops in different viscosities, experi-ments on gas-liquid two-phase flow with different viscosi-ties were carried out. Analysis of the experimental datashows that the total pressure drop increases with increasingliquid viscosity when the superficial gas and liquid flowrates are the same and that the liquid superficial velocityis 0.52m/s and the superficial gas velocity is 12m/s in thevertical and inclined pipes. The friction pressure drop isnegative when the superficial gas and liquid velocities aresmall, as the increased range of the total pressure dropdecreases with increasing liquid viscosity. A predictionmodel of the pressure drop in high-viscosity liquid-gastwo-phase flow is established. The new model is verifiedby experimental data and compared with existing models.The new model has the smallest error, which is basicallywithin 15%. The prediction results of the wellbore pressuredistribution of four wells in the BeiA oilfield show that thepredicted wellbore pressure distribution of the new modelis closer to the measured results, as the error of the newmodel is the smallest. The reliability of the new model con-sidering the influence of viscosity and predicting the nega-tive friction pressure drop is proven from practicalengineering applications in the field.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully express their thanks for the financialsupport for this study from the National Major Scientificand Technological Special Project of China(2016ZX05056004-002) and the National Natural ScienceFoundation of China (61572084).

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19Geofluids

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20 Geofluids

Documents

Study on the Pressure Drop Variation and Prediction Model