++ drift-velocity closure relationships for slug two-phase high-viscosity oil flow in pipes

June 2012 SPE Journal 593

Drift-Velocity Closure Relationships for Slug Two-Phase High-Viscosity Oil Flow in Pipes B.C. Jeyachandra and B. Gokcal, University of Tulsa; A. Al-Sarkhi, King Fahd University of Petroleum & Minerals;

and C. Sarica and A.K. Sharma, University of Tulsa

SummaryThe drift velocity of a gas bubble penetrating into a stagnant liquid is investigated experimentally in this paper. It is part of the translational slug velocity. The existing equations for the drift velocity are either developed by using the results of Benjamin (1968) analysis assuming inviscid fluid flow or correlated using air/water data. Effects of surface tension and viscosity usually are neglected. However, the drift velocity is expected to be affected by high oil viscosity. In this study, the work of Gokcal et al. (2009) has been extended for different pipe diameters and viscosity range. The effects of high oil viscosity and pipe diameter on drift veloc-ity for horizontal and upward-inclined pipes are investigated. The experiments are performed on a flow loop with a test section with 50.8-, 76.2-, and 152.4-mm inside diameter (ID) for inclination angles of 0 to 90°. Water and viscous oil are used as test fluids. New correlation for drift velocity in horizontal pipes of different diameters and liquid viscosities is developed on the basis of experi-mental data. A new drift-velocity model/approach are proposed for high oil viscosity, valid for inclined pipes inclined from horizontal to vertical. The proposed comprehensive closure relationships are expected to improve the performance of two-phase-flow models for high-viscosity oils in the slug flow regime.

IntroductionThe translational velocity (velocity of slug units), is one of the key closure relationships in two-phase-flow modeling. It is described as the summation of the maximum mixture velocity in the slug body and the drift velocity. The drift velocity and translational velocity are affected by high oil viscosity. High-viscosity oils are being produced from many oil fields around the world. Oil-production systems are currently flowing oils with viscosities as high as 10 Pa·s. Current multiphase-flow models and correlations are largely based on experimental data with low-viscosity liquids. The labora-tory liquids commonly used have viscosities less than 0.02 Pa·s. Multiphase flows are expected to exhibit significantly different behavior for higher-viscosity oils.

Gokcal et al. (2008) experimentally observed slug flow to be the dominant flow pattern for high-viscosity oil and gas flows. Accurate predictions of slug-flow characteristics are crucial in the design of pipelines and process equipment. In order to improve the accuracy of slug characteristics for high-viscosity oils, new and improved models for flow characteristics such as drift velocity and translational velocity are required.

Slug translational velocity is the sum of the bubble velocity in stagnant liquid (i.e., the drift velocity vd) and the maximum veloc-ity in the slug body. Research efforts have typically been focused on the drift velocity in horizontal and upward-inclined pipes.

Nicklin et al. (1962) proposed an equation for translational velocity as

v C v vt s s d= + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

The parameter Cs is approximately the ratio of the maximum to the mean velocity of a fully developed velocity profile. Cs is

approximately 1.2 for turbulent flow and 2.0 for laminar flow. vs

is the mixture velocity, which is the sum of the superficial liquid and gas velocities. The drift velocity contributes to the translational velocity of the slug unit for all pipe inclination. The value of the drift velocity will be the translational velocity at zero mixture velocity (the intersection point of the vertical axis on the vt-vs.-vs curve). For horizontal pipes, the drift velocity is acting in the same direction as the mixture velocity so it contributes to the magnitude of the slug translational velocity.

Vertical Flow. Dumitrescu (1943) and Davies and Taylor (1950) performed a potential fl ow analysis to fi nd the drift velocity for vertical fl ow. Both derived the same dimensionless group (Froude number) and found that Froude number has a constant value. Davies and Taylor (1950) estimated the constant value as 0.328. Dumitrescu (1943) made more-accurate calculations and theoreti-cally determined this value as 0.351, which agreed well with the air/water experimental data of Nicklin et al. (1962).

v gDd = 0 351. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

For vertical flow, Joseph (2003) proposed a model for the bub-ble-rise velocity in vertical flow, taking viscosity, surface-tension, and shape of the bubble-nose effects into consideration. From the experimental results, it is observed that the bubble nose is almost spherical. When the bubble nose is spherical (axisymmetric cap), the effect of the surface tension vanishes and the equation becomes a function of only the fluid viscosity and the radius of the spheri-cal-cap bubble, as shown in Eq. 3:

vr

grrd = − + +4

3

4

9

16

9

2

2

�

�

�

�( ), . . . . . . . . . . . . . . . . . . . . . (3)

where r is the radius of cap and � and � are the density and viscos-ity of the liquid, respectively. It was shown that the experimental data of Bhaga and Weber (1981) and the model predictions were in good agreement.

Inclined Flow. For the inclined case, Zukoski (1966), Bendiksen (1984), Weber et al. (1986), Hasan and Kabir (1986), and Carew et al. (1995) experimentally studied drift velocity and found that the drift velocity increases with inclination angle and then decreases to its lowest value for vertical fl ow, reaching a maximum value at an intermediate angle of inclination approximately 40 to 60° from the horizontal. This fact was explained qualitatively by Bonnecaze et al. (1971). They discussed that the gravitational potential fi rst increases and then decreases as the inclination angle changes from the vertical to the horizontal position.

Weber et al. (1986) experimentally studied bubble-rise velocity (in relatively small pipe diameters from 0.6 to 3.7 cm) for high-viscosity Newtonian liquids. Froude number Fr was correlated as a function of Eotvos number Eo, the Morton number (M=g�4/ρ� 3), and the inclination angle �.

Bendiksen (1984) performed an experimental study for veloci-ties of single elongated bubbles in flowing liquids at different inclination angles. The measured velocities were plotted against the liquid velocity for each inclination angle. Then, drift velocities were found by the extrapolation of the data to zero liquid velocity. He correlated the drift velocity for inclined flow by using the drift velocities for horizontal and vertical flow:

Copyright © 2012 Society of Petroleum Engineers

This paper (SPE 151616) was accepted for presentation at the SPE Annual Technical Conference and Exhibition, Tuscany, Italy, 20–22 September 2010, and revised for publication. Original manuscript received 11 November 2010. Revised manuscript received 14 July 2011. Paper peer approved 21 July 2011.

594 June 2012 SPE Journal

v v vd dh

dv= +cos sin� � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

Hasan and Kabir (1986) performed an experimental study in the range of 90° > � > 30° and proposed the relation

v vd dv= +sin ( cos ) .� �1 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

Carew et al. (1995) studied the motion of long bubbles in inclined pipes experimentally with viscous Newtonian and non-Newtonian liquids. They proposed an empirical correlation for the drift velocity of an elongated bubble in inclined pipes. The correlation depends on inclination angle and surface tension and is valid at Re > 200 and Eo > 60.

Shosho and Ryan (2001) experimentally investigated the effects of tube size and fluid type (including Newtonian and non-Newtonian fluids) on drift velocity for vertical and inclined tubes. The drift velocity in terms of Froude number was correlated with Eotvos and Morton numbers. Froude number increased and decreased as the angle of inclination increased for both Newtonian and non-Newtonian fluids. For non-Newtonian fluids with high Morton number, Froude number was affected by both viscous forces and tube size.

Alves et al. (1993) proposed a model for the drift velocity including surface-tension effect in inclined flow using inviscid flow theory. The model was compared against their experimental data and Zukoski (1966) data. Gokcal (2008) used the Alves et al. (1993) results and proved that the effect of surface tension on drift velocity is negligible when the ID is ≥ 50.8 mm.

Van Hout et al. (2002) used an approach similar to that of this study to identify inclination effects on drift velocity using optical sensors. Two different pipe diameters (0.024 and 0.054 m) were studied. The fluids were air and water. They found that the Ben-diksen (1984) correlation predicted well for smaller diameter. But as the pipe diameter increased, the correlation was valid only for slightly inclined case. Substantial discrepancy was found when the correlation was applied to higher inclination angles.

Horizontal Flow. Zukoski (1966) experimentally investigated the effects of liquid viscosity, surface tension, and pipe inclination on the motion of single elongated bubbles in stagnant liquid for different pipe diameters. He also found that the effect of viscosity is negligible on the drift velocity for Re = vd �D/� > 200. Wallis (1969) and Dukler and Hubbard (1975) claimed that there is no drift velocity for horizontal fl ow because gravity cannot act in the horizontal direction. However, Nicholson et al. (1978), Weber (1981), and Bendiksen (1984) showed that drift velocity exists for the horizontal case and the value of drift velocity can exceed the vertical fl ow value. The drift velocity is a result of hydrostatic pres-sure difference between the top and bottom of the bubble nose.

Benjamin (1968) proposed the following relationship for the drift velocity in horizontal pipes:

v gDd = 0 542. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

Benjamin calculated the value of the drift-velocity coefficient by using inviscid potential flow theory that inherently neglects surface tension and viscosity. The drift velocity in horizontal slug flow is the same as the velocity of the penetration of a bubble when liquid is drained out of a horizontal pipe. Bendiksen (1984) and Zukoski (1966) supported the study of Benjamin (1968) experimentally.

Weber (1981) developed a correlation for drift velocity in horizontal pipes on the basis of the experimental data of Zukoski (1966) for liquids of low viscosities, as shown in Eq. 7:

v gD Eod / . ..

= −−

0 54 1 760 56

, . . . . . . . . . . . . . . . . . . . . . . . (7)

where Eotvos number is defined as Eo D g= � �2 / .A dimensional analysis for the drift velocity in horizontal pipes

was presented by Ben-Mansour et al. (2010). The step-by-step

method was used, and the following dimensionless numbers have been found:

Frv

gDd= , N

D g�

�

�= 3 2 1 2 , Eo

D g= �

�

2

. . . . . . . . . . . . . (8)

The first dimensionless group is the Froude number Fr, the second is the viscosity number N�, and the third is Eotvos number Eo. It was concluded that the drift velocity in a horizontal pipe can be modeled using those three dimensionless groups.

From the literature review related to drift velocity for horizon-tal, inclined, and vertical pipes, it is apparent that detailed research has been conducted on the effects of surface tension and pipe diameter on drift velocity at different inclination angles. However, for the effect of high viscosity on drift velocity, experimental and theoretical studies are scarce and have been conducted for rela-tively small pipe diameter.

Shi et al. (2005a, b) recently conducted experimental and modeling studies to determine the drift-flux model parameters of water/gas, oil/water, and oil/water/gas flow in a 15-cm-diameter pipe at different angles ranging from vertical to slightly down-ward. Experiments were performed with kerosene, tap water, and nitrogen. The viscosity of the oil was 1.5 cp. A correlation for drift velocity similar to that of Hasan and Kabir (1999) is used in their drift-flux modeling of oil/water flow and similar to Wallis (1969) for gas/liquid flows.

The approach in this study differs from that of the previous studies in that single-viscosity oil was used to conduct the experi-ments at different viscosities by controlling the temperature of the oil. The oil-viscosity range varied from 0.155 to 0.574 Pa·s. The other salient feature is that the large diameters that are prevalent in the field were selected. Experiments were conducted on 50.8-, 76.2-, and 152.4-mm-diameter acrylic pipe for inclination angles from 0 to 90°. The purpose of the present paper is to develop a uni-fied drift-velocity closure relationship based on viscosity and pipe diameter and compare it with experimental results for horizontal and upward inclined pipes.

Experimental Setup and Procedure The experimental facility consists of an oil-storage tank, a 20-hp screw pump, a 3.05-m-long acrylic pipe, heating and cooling loops, and transfer hoses and instrumentation. Details of the experimen-tal setup are given in Gokcal et al. (2008) and shown in Fig. 1. Experiments were conducted on 50.8-, 76.2-, and 152.4-mm-ID pipes. The acrylic pipe is located close to the storage tank. The inclination of the pipe can be varied using a pulley arrangement. The pipe inclination can be changed from 0 to 90°.

The heating and cooling loops are used to maintain the desired temperature and thereby control the viscosity of the oil. The oil pump supplies the pipe with oil. Then, the main inlet valve and the auxiliary inlet valve are closed. The drainage valve is opened to drain the residual oil captured and thereby create a gas pocket. Next, the drainage valve is closed and the main inlet valve is opened to release the gas bubble into the stagnant oil column. The drift velocity is measured by two lasers (for 50.8- and 76.2-mm-ID pipe) or optical sensors (for 152.4-mm-ID pipe) separated by a distance of 0.9144 m. The optical sensors work by the principle that the light intensity changes when it reflects from/refracts through the oil or the gas phase. This is stored as voltage readings in a data-acquisition system with a frequency of 500 readings/sec. The data are used to calculate the drift velocity by dividing the distance between the two sensors by time difference between the two voltage peaks.

The facility was modified for the horizontal case by replacing the end plate of the pipe with a plug. This facilitated proper drain-ing of oil as the gas bubble penetrated into the liquid.

Water and viscous oil were used as test fluids. The properties of the oil are given in Table 1. The most important characteristic of the oil is its large range of viscosity owing to strong temperature dependence. The oil sample was tested before experiments, and the surface tension remained constant at 29 to 30 dynes/cm for the temperature change used in the present experiments. Oil-viscosity


and density variation with temperature are shown in Figs. 2 and 3,respectively.

Results and DiscussionsInitially, an experiment is conducted with water for horizontal pipe to prove that the system is working properly. The results for water are compared with the Benjamin (1968) model prediction. The pre-dictions of drift velocity of the water from Benjamin’s model show excellent agreement with the data. The rest of the experiments are

conducted at different oil temperatures corresponding to a range of viscosities from 0.154 to 0.574 Pa·s.

Drift Velocity in Horizontal Pipes. Fig. 4 shows the experimental results for drift velocity vs. viscosity for horizontal pipes of dif-ferent diameters. The fi rst point for all plots (at lowest viscosity =0.001 Pa·s) is for the water case, and the other points are for oil. Drift velocity decreases with increasing viscosity. As the pipe diameter increases, the drift velocity increases. The decrease in

Fig. 1—Schematic of indoor high-viscosity test facility.

TABLE 1–PROPERTIES OF OIL

Gravity at 15.6°C (°API )

Density at 15.6°C (kg/m3)

Flash Point (°C)

Pour Point (°C)

Viscosity at 40°C (Pa·s)

27.6 889 250 –12.2 0.22

Temperature (°C)

Visc

osity

(Pa·

s)

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

010 20 30 40 50 60 70

Fig. 2—Oil viscosity vs. temperature.


the drift velocity with viscosity is steeper in small pipes than in large pipes. The plots tend to have an asymptotic level at very high viscosity, which leads to a small variation in drift-velocity variation with viscosity.

As shown by Ben-Mansour et al. (2010), the drift velocity in a horizontal pipe can be correlated using Froude number, viscosity number, and Eotvos number. Fig. 5 shows the correlation for drift velocity in horizontal pipes at different viscosities, diameters, and surface tensions. It is worth noting that data include oil at different viscosities and water in different pipe diameters. The experimental data for drift velocity in horizontal pipes of different diameters with liquid of various viscosities are well correlated by Eq. 9:

Fr e N Eo= − −

0 53 13 7 0 46 0 1

. . . .� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

As mentioned in the Horizontal Flow subsection, the effect of surface tension is negligible for a pipe diameter larger than 50 mm. However, for horizontal flow, the effects of surface tension on drift velocity were considered in the equation by using Eotvos number (Eotvos number in Eq. 9 is defined as it was defined in Eq. 8).

Evaluation of Drift-Velocity Correlation in Horizontal Pipes. Fig. 6 shows the comparison between the measured data and the predicted result using Eq. 8. On the same plot, the prediction by Weber (1981) for low viscosities is shown. It can be seen clearly that most of the data collapse in a range of +8% and –13%. The Weber (1981) correlation matches only for points at high values of Froude number, which is expected because the correlation was developed on the basis of the Zukoski (1966) data in a water/air system. The data points where Fr is approximately 0.5 are for water experiments.

Temperature (°C)

Den

sity

(kg

/m3 )

895

890

885

880

875

870

865

860

8550 10 20 30 40 50 60 70

Fig. 3—Oil density vs. temperature.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Dri

ft V

elo

city

(m

/s)

Viscosity (Pa·s)

D=50.8 mm

D=76.2 mm

D= 152.4 mm

Fig. 4—Drift-velocity variation with viscosity in horizontal pipes.


Drift Velocity in Vertical Pipes. Several correlations are available in literature for rise velocity in stagnant fl uid (Davies and Taylor 1950; White and Beardmore 1962; Joseph 2003). Fig. 7 shows the comparison of the model predictions of Joseph (2003) with experimental data from Weber et al. (1986), Shosho and Ryan (2001), and this study. The bubble radius and liquid viscosity must be known to calculate the drift velocity from Eq. 3. It is experi-mentally observed that the radius of a bubble is approximately 0.55 to 0.6 times the radius of the pipe. This value is used for the remaining calculations to compare model predictions with experi-mental results. Weber et al. (1986) performed their experiments in 37.3-mm-ID pipe (relatively small) for viscosities between 0.051 and 0.183 Pa·s. The Shosho and Ryan (2001) experiments were for same diameter for viscosities between 0.003 and 0.883 Pa·s. They are the only available data set for a higher-viscosity range with comparable pipe diameter in the literature. The Joseph (2003) model correlated the experimental data well, within ±20%. The present work’s experimental data agreed very well with this correlation. A better match can be obtained if the curvature of the bubble cap is included.

As mentioned in the Horizontal Flow subsection, the effect of viscosity is negligible for pipe diameters larger than 50 mm. For vertical flow, in the Joseph (2003) model, viscosity and surface

tension and shape of bubble-nose effects on drift velocity were considered in his model in Eq. 3.

Drift Velocity in Upward-Inclined Pipes. Water/Air Case. The drift velocity for water/air experiments in large diameter will be a function of only pipe diameter and inclination angle (surface-tension effect is negligible). Fig. 8 shows the measured Froude number in different pipe diameters for air/water system. Froude-number plots vs. pipe inclination follow similar behavior for all pipe diameters. The new closure relationship correlated with Froude number at any inclination angle and pipe diameter is pro-posed in Eq. 10. Evaluation of Eq. 10 is shown in Fig. 9. All the data are well predicted within ±5%.

Fr = − + +0 248 0 299 0 4972. . .� � . . . . . . . . . . . . . . . . . . . . . (10)

Oil/Air Case. Effect of pipe diameter and inclination angle at certain oil viscosity is presented in Figs. 10 through 13. Effect of vis-cosity at certain pipe diameters is shown in Figs. 14 through 16.

From Figs. 10 through 16, it is observed that there is a clear effect of diameter on drift velocity. For horizontal flow, when diameter increases, the gravitational potential increases. This leads to a stronger drive for the gas bubble to penetrate into the stagnant liquid column, hence the higher drift velocity. As the inclination gradually increases, so does the drift velocity, and it peaks at an inclination of approximately 30 to 50°. The gravitational potential is at its maximum at this inclination angle. As the inclination is increased further, the effect of drainage area comes into consid-eration. Even though the gravitational potential is high, the area available for oil to drain is low. This, in turn, creates a resistance for the air bubble to penetrate into the oil, effectively reducing the drift velocity. With increasing pipe inclination, the extended air bubble location, which is in contact with the upper part of the pipe (at low inclination angle), moves toward the centerline of the pipe. At a right angle, the extended air bubble location is in the center of the pipe.

From Figs. 14 through 16, it is observed that as the viscos-ity increases, drift velocity decreases. As viscosity increases, the resistance for the gas bubbles to intrude into the stagnant col-umn increases. Thereby, the drift velocity is reduced. The same parabolic trend of drift velocity increasing as the inclination angle increases, reaching a maximum at 30 to 50° and then decreasing, is observed.

Drift-Velocity Correlation in Upward-Inclined Pipes. Fig. 17shows the comparison between the experimental data and the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.0001 0.001 0.01 0.1 1

Fro

ud

e N

um

ber

(F

r)

Nµµ0.46 / Eo0.1

Data

Equ. (8)

Fig. 5—Horizontal Froude-number correlation vs. experimental data.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

Fro

ud

e N

um

ber

-Pre

dic

ted

Froude Number -Measured

Present

Weber (1981)

+8%

-13%

Fig. 6—Comparison of correlation predictions with measured Froude number for horizontal flow.


0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Dri

ft V

elo

city

- P

red

icte

d (

m/s

)

Drift Velocity - Measured (m/s)

Present Data

Weber et al.

Shosho and Ryan +20%

-20%

Fig. 7—Comparison of Joseph (2003) model predictions with measured drift velocities for vertical case.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90

Inclination Angle (°)

Present, 50.8-mm

Present, 76.2-mm

Present, 152.4-mm

Alves (1993)

Zukoski (1966)

Fro

ud

e N

um

ber

(F

r)

Fig. 8—Measured Froude number vs. inclination angle for water.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fro

ud

e N

um

ber

-P

red

icte

d

Froude Number - Measured

PresentZukoski (1966)Alves (1993)

+5%

-5%

Fig. 9—Comparison of measured Froude number with Eq. 9 predictions.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90

Dri

ft V

elo

city

(m

/s)


152.4 -mm

76.2-mm

50.8-mm

Fig. 10—Effect of pipe diameter on drift velocity for 0.574-Pa·s oil.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90

152.4-mm

76.2-mm

50.8-mm


Dri

ft V

elo

city

(m

/s)



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90

152.4-mm

76.2-mm

50.8-mm

Dri

ft v

elo

city

(m

/s)



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90

152.4-mm

76.2-mm

50.8-mm

Dri

ft V

elo

city

(m

/s)



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80 90

0.574

0.378

0.256

0.154

Dri

ft V

elo

city

(m

/s)


Fig. 14—Viscosity effect on drift velocity for 152.4.-mm-diam-eter pipe (viscosities in Pa·s).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 10 20 30 40 50 60 70 80 90

0.574

0.378

0.256

0.154

Dri

ft V

elo

city

(m

/s)


Fig. 15—Viscosity effect on drift velocity for 76.2-mm-diameter pipe.

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 10 20 30 40 50 60 70 80 90

Dri

ft V

elo

city

(m

/s)


0.645 Pa.s 0.412 Pa.s 0.296 Pa.s 0.185 Pa.s 0.104 Pa.s 0.001 Pa.s

Fig. 16—Viscosity effect on drift velocity for 50.8-mm-diameter pipe (Gokcal et al. 2008).


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fro

ud

e N

um

ber

-M

easu

red

Froude Number - Predicted

Oil Data

water Data

+12%

-12%

Fig. 17—Comparison of the correlation predictions with meas-ured Froude number for inclined flow.

predicted result using an approach similar to that of Bendiksen (Eq. 4). The modified Bendiksen correlation shown in Eq. 11 for all pipe inclination simply uses the Froude number instead of the drift velocity. The horizontal and vertical components of the Froude number in the equation are obtained from the experimental values for 0 and 90°, respectively, which accounts for the effects of viscosity.

Fr Fr Frh v� � �= +cos sin , . . . . . . . . . . . . . . . . . . . . . . . . . (11)

where Fr� is the Froude number at angle of inclination, � is the angle of inclination, and h and v are for the horizontal and vertical cases, respectively.

The predicted values in Fig. 17 are calculated by substituting Eq. 3 in terms of Froude number for Frv and Eq. 8 for Fr h.

It can be observed that most of the data points for high-viscosity oil fall within the ± 12% limit. It can also be observed that all data points for water lie very close to the 45° line. These observations imply that there is good match between the predicted value and the experimental data, provided that viscosity effects are considered.

Conclusions Drift-velocity experiments were conducted on high-viscosity oil for varying viscosities, pipe diameters, and inclination angles. It was observed that viscosity has a profound effect on drift velocity. Drift velocity increases with increasing pipe diameter and decreases with increasing liquid viscosity. For horizontal flow, a dimension-less analysis was performed, and a new correlation for horizontal drift velocity was developed and tested with the available data set. For the inclined case, the effect of pipe inclination on drift velocity was explained. As the pipe inclination increases, the drift veloc-ity increases, and it peaks at an inclination near 30 to 50°, then it decreases again. For the inclined case of an air/water system, a new correlation for different pipe diameter was developed. For the vertical case of an oil/air system, the correlation developed by Joseph (2003) was tested with the data set, and a very good match was obtained. For inclined flow, it was observed that the correlation developed by Bendiksen (1984) modified by using Froude number instead of drift velocity worked well, provided that the viscosity effects are considered for the horizontal and vertical components. Therefore, the correlation developed for horizontal flow and the Joseph (2003) correlation for vertical flow can be used in conjunction with the modified Bendiksen equation to provide accurate values for drift velocity in the horizontal and the upward-inclined case.

AcknowledgmentThe authors would like to thank the member companies of the TUFFP for their support.

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Benin Jeyachandra is a petroleum engineer with Schlumberger Information Solutions in Houston, Texas, USA. He holds an MS degree in petroleum engineering from The University of Tulsa.

Bahadir Gokcal is a senior flow assurance engineer with ConocoPhillips in Houston, Texas, USA. His research interests are multiphase flow in pipes, CFD modeling, and flow assurance. Gokcal holds a BS degree in petroleum and natural gas engineer-ing from Middle East Technical University in Turkey, and MS and PhD degrees in petroleum engineering from The University of Tulsa.

Abdelsalam Al-Sarkhi is an associate professor at King Fahd University of Petroleum and Minerals, Saudi Arabia. His research interests are experimentation and modeling of multiphase flow, thermodynamics, and heat transfer. He conducted sev-eral researches on the effect of drag-reducing polymers on multiphase-flow behavior. He holds BS and MS degrees in mechanical engineering from Jordan University of Science and Technology and a PhD degree in mechanical engineer-ing from Oklahoma State University, Stillwater, Oklahoma, USA.

Cem Sarica is a professor of petroleum engineering and the director of two industry supported consortia at the University of Tulsa (TU): Tulsa University Fluid Flow Projects (TUFFP) and Tulsa University Paraffin Deposition Projects (TUPDP). He is also serving as coprincipal investigator of Tulsa University High Viscosity Oil Projects (TUHOP). He was as an associate professor of petro-leum and natural gas engineering at The Pennsylvania State University and an assistant professor of petroleum and natural gas engineering at Istanbul Technical University (ITU) prior to joining TU. His research interests are production engineering, multiphase flow in pipes, flow assurance and horizontal wells. He holds BS and MS degrees in petroleum engineering from ITU and PhD degree in petroleum engineering from TU. He currently serves as a member of SPE Projects, Facilities and Construction Advisory Committee. He has previously served as a member of SPE Production Operations and Books Committees and was a member of SPE Journal Editorial Board between 1999 and 2007. He is the recipient of 2010 SPE International Production and Operations Award. He has more than 100 publications mostly in SPE Journals and Proceedings.

++ drift-velocity closure relationships for slug two-phase high-viscosity oil flow in pipes

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