Experimental investigation on flow patterns and pressure gradient through two pipe diameters in horizontal oil–water flows

Experimental investigation on flow patterns and pressure gradientthrough two pipe diameters in horizontal oil–water flows

T. Al-Wahaibi a,n, Y. Al-Wahaibi a, A. Al-Ajmi a, R. Al-Hajri a, N. Yusuf b, A.S. Olawale b,I.A. Mohammed b

a Department of Petroleum and Chemical Engineering, Sultan Qaboos University, P.O. Box 33, Al-Khoud, P.C. 123, Omanb Department of Chemical Engineering, Ahmadu Bello University, Zaria, Nigeria

a r t i c l e i n f o

Article history:Received 19 July 2013Accepted 14 July 2014Available online 1 August 2014

Keywords:Oil–water flowPipe diameter effectFlow pattern mapFlow pattern transitionPressure gradient

a b s t r a c t

The flow patterns and pressure gradient of immiscible liquids are still subject of immense researchinterest. This is partly because fluids with different properties exhibit different flow behaviors indifferent pipe's configurations under different operating conditions. Recently, Yusuf et al. (2012)investigated experimentally the flow patterns and pressure gradient of horizontal oil–water flow in25.4 mm acrylic pipe. This paper describes similar works in 19 mm ID pipe to examine how significant isthe effect of a small decrease in pipe diameter on flow patterns and pressure gradient. The results reveala remarkable influence of pipe diameter on flow patterns and pressure gradient. The region of dualcontinuous and dispersed oil in water flows are enlarged as the pipe diameter increases from 19 to25.4 mmwhile the extent of stratified, bubble and annular flow regions are found to decrease as the pipediameter increases.

The pressure gradient values obtained in the 19 mm pipe are greater than those measured in the25.4 mm pipe at similar superficial oil and water velocities. The differences in pressure gradient resultsbecome bigger with higher oil and water velocities. The experimental pressure gradient results werecompared with the two-fluid, homogenous and drift-flux models. The drift-flux model showed a goodprediction to the experimental results while the two-fluid and the homogenous models were found tohighly overpredict the experimental results especially for the smaller pipe diameter.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

The simultaneous flow of two immiscible fluids (e.g. oil andwater) in pipes is a common phenomenon in oil, chemical andpetrochemical industries. It is known that the fundamental differ-ence between single phase flow and two-phase flow is theexistence of flow patterns or flow regimes in two-phase flow.When a mixture of two fluids flows simultaneously in a channel,the two phases can distribute themselves in several configurationswhich are largely dependent on the physical properties of thefluids, like density and viscosity, the operational variables such asflow-rate and volume fraction of each phase, and the geometry ofthe channel (pipe material, diameter and inclination, etc.).

The understanding of oil–water flow behavior is of significantimportance in many applications in the petroleum industry. Forexample, the performance of separation facilities and multiphasepumps is a strong function of the flow pattern. Understanding theflow structure in liquid–liquid flow in horizontal pipes will go long

way in developing predictive models that could aid in the design andconstruction of flow equipments.

Different researchers have investigated flow pattern and pres-sure drop in horizontal oil–water flow. Russell and Charles (1959)observed annular flow when they introduced water into viscouscrude oil to reduce the pressure gradient along the pipeline.Charles et al. (1961) used equal density oil–water mixtures asworking fluids and observed oil-slugs-in-water. Both Angeli andHewitt (2000) and Raj et al. (2005) used 25.4 mm ID acrylic pipesin their work. Amongst the flow pattern observed by the latter wasbubbly flow pattern which was not reported by the former.Al-Wahaibi et al. (2012) attributed this finding to the lowerinterfacial tension in Angeli and Hewitt (2000) system comparedto that of Raj et al. (2005).

Lovick and Angeli (2004) used a stainless steel pipe(ID¼38 mm) and mineral oil as working fluids. They classifiedthe flow pattern into stratified, dual continuous, dispersion ofwater in oil and dispersion of oil in water. Al-Wahaibi et al. (2007a)used acrylic pipe with ID¼14 mm and oil with similar physicalproperties to that of Lovick and Angeli (2004). Al-Wahaibi et al.(2007a) observed bubbly, slug and annular flows in their studywhich were not observed by Lovick and Angeli (2004). Al-Wahaibi

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Journal of Petroleum Science and Engineering

http://dx.doi.org/10.1016/j.petrol.2014.07.0190920-4105/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: [email protected] (T. Al-Wahaibi).

Journal of Petroleum Science and Engineering 122 (2014) 266–273

et al. (2012) attributed this to the smaller pipe diameter used byAl-Wahaibi et al. (2007a) compared to that of Lovick and Angeli(2004).

Pressure drop in concurrent flow of oil and water is of interestsince it was discovered that introduction of water into crude oilpipeline under certain conditions reduces pressure drop andimproves transportation of the oil (Russell and Charles, 1959).Recently, Angeli and Hewitt (1998), Lovick and Angeli (2004),Chakrabarti et al. (2005) and Rodriguez and Oliemans (2006)employed the two-fluid and the homogeneous models to predictthe pressure gradient in oil–water flow. There were considerablediscrepancies between the measured and predicted values, espe-cially in dual continuous flow. For example, Angeli and Hewitt(1998) compared their experimental pressure gradient data withboth models and found that none of them were able to predict thedata adequately. They attributed this poor agreement to thewetting phenomena and drag reduction effects. Lovick andAngeli (2004) concluded that the two-fluid model was not ableto predict the pressure gradient during dual continuous flow.Rodriguez and Oliemans (2006) predicted the pressure gradientfor horizontal oil–water flow with accuracies of 35% with bothmodels.

Despite the large number of studies currently available in theliterature of flow pattern and pressure drop in horizontal oil–water flow, very few works have tried to understand the effect ofphysical properties or geometry of the channel on flow patternsand pressure drop (Angeli and Hewitt, 1998, 2000; Mandal et al.,2007; Sotgia et al., 2008; Yusuf et al., 2012). Investigating theeffect of fluid properties, pipe geometries and materials underdifferent operating conditions will help us to obtain a clear pictureand understanding on liquid–liquid behavior. Such studies can belinked to obtain a generalized flow pattern map and would go along way in developing more robust predictive models for liquid–liquid flow.

This paper focuses on quantifying the significant effect of asmall change in pipe diameter on flow patterns and pressuregradient in horizontal oil–water flow. This is achieved by combin-ing the results obtained in this study using 19 mm ID acrylic pipewith those previously presented by Yusuf et al. (2012) with25.4 mm ID acrylic pipe using the same experimental facility.

2. Experimental set-up

The experimental studies on flow patterns and pressure gra-dient were carried out in the oil–water flow experimental facilityshown in Fig. 1. Oil and water were used as test fluids withproperties given in Table 1. Each fluid was transferred from theirstorage tank with a pump to the test section made up of 19 mmacrylic pipe that consists of two 8 m long sections joined by aU-bend. The two fluids entered the test section from two pipes viaa Y like-junction. The water phase was allowed to enter from thebottom while the oil joined from the top to reduce the effect ofmixing. A complete description of the experimental set-up and theprocedure was given by Yusuf et al. (2012).

High-speed camera that can record up to 1000 fps (Fastec –

Troubleshooter) and visual observation were used to identify thedifferent flow patterns. The camera was located 6.5 m from thefirst eight-meter part of the test section. In this work, 250 fps wasselected and the images were processed using MiDAS 4.0 expresssoftware.

Pressure gradient experiment was conducted in the test sectionby measuring the pressure drop between two points 1 m apartalong the flow line 6.5 m from the entry point. The pressure dropwas measured with a Dywer 490 digital differential manometer.It has a full scale accuracy of 70.5%. The manometer has two ports

(negative and positive); the negative port was connected to thenegative pressure side of the flow, while the positive side wasconnected to the positive pressure side with the aid of flexibletubes. The manometer displays the pressure drop between the twopoints when oil and water flows through the line.

The experiments were conducted using oil superficial velocitiesranging between 0.10 and 2.0 m/s and water superficial velocitiesfrom 0.10 to 2.6 m/s. In all experimental runs, the pipe wasprewetted with oil before the two phase mixture was introducedand measurements were taken. Angeli and Hewitt (1998) showedthat that prewetting can affect two-phase flow. Therefore, it is vitalto follow the above procedure for consistency in the results.

3. Results and discussion

3.1. Flow patterns

Refined version of the flow pattern map from the sameexperimental facility previously presented by Yusuf et al. (2012)using 25.4 mm ID acrylic pipe is shown in Fig. 2. Six flow patternswere identified for the range of superficial oil and water velocitiesinvestigated. These are stratified (ST), bubble (Bb), dual continuous(DC), annular, (AN), dispersed oil in water (Do/w) and dispersedwater in oil (Dw/o). They are defined as follows:

Stratified (stratified smooth, SS, and stratified wavy, SW):where the two fluids flow as separated layers with lighterliquid on the top of the denser one forming an interfacebetween them.Dual continuous (DC): where both oil and water form con-tinuous layers at the top and bottom of the pipe respectivelybut drops of one phase appear in the continuum of theother phase.Annular (AN): where water forms an annular film at the walland oil flows in the pipe core.Bubble (Bb): where the oil appears in the form of elongateddrops (slightly longer than the pipe diameter) within watercontinuum.Dispersed oil in water (Do/w): where the pipe cross-sectionalarea is occupied by water containing dispersed oil droplets.Dispersed water in oil (Dw/o): where oil is the continuousphase and water is present as droplets across the pipe cross-sectional area.

In this study, similar experiments to those conducted by Yusufet al. (2012) in 25.4 mm ID pipe were performed in the 19 mm IDacrylic test section. The resultant flow pattern map is shown inFig. 3 together with the pattern boundaries observed in the25.4 mm pipe. In general, an increase in pipe diameter from 19to 25.4 mm enlarges the region of DC and Do/w flows while itshrinks the region where ST, Bb and AN flows occurred.

3.1.1. Stratified flow (ST)In both pipes, stratified flow was observed to transform into

bubbly flow at Usoo0.1 m/s and to dual continuous flow atUso40.1 m/s. However, the boundary between stratified (ST) anddual continuous (DC) flow shifted slightly to the right as the pipediameter decreased (Fig. 3). To explain such behavior, it isimportant to understand the forces responsible of destabilizingthe interface and those tend to stabilize it. Two forces are knownto destabilize the interface and cause it to grow in amplitude.These are the relative movement and viscosity difference betweenthe two fluids (see Al-Wahaibi and Angeli (2011) and Al-Wahaibiet al. (2007b, 2012)). The relative movement between the twophases will create a suction pressure force that acts below and

T. Al-Wahaibi et al. / Journal of Petroleum Science and Engineering 122 (2014) 266–273 267

above the interface and makes waves to grow in amplitude (seeAl-Wahaibi and Angeli (2007) and Al-Wahaibi et al. (2012)). Thehigher the viscosity difference between the two fluids enhancesthe instability of oil–water interface and grows of waves ampli-tude. The viscosity difference is eliminated from the comparisonsince the same oil is used. On the other hand interfacial tensionand gravity forces tend to stabilize the interface (see Al-Wahaibiet al. (2007b, 2012)). When the destabilizing forces are greaterthan the stabilizing ones, waves become unstable and eventuallydrops will form.

Thus, to check which force has a dominant effect in the twopipes, Eőtvős number (Eo) as defined by Brauner and MoalemMaron (1992) was calculated for the two pipes where Eo41indicates that surface tension forces dominate.

EO ¼ 4π2σ

ðρW �ρOÞgD2 ð1Þ

The Eőtvős number (Eo) for the 25.4 and 19.0 mm pipes werecalculated to be 1.04 and 1.87 respectively. Based on Brauner andMoalem Maron (1992) model, interfacial forces will have moredominant effect in the smaller pipe.

Therefore, early drops were observed in the larger pipe becausethe interfacial forces were less than that in the smaller pipediameter (the Eőtvős number for the 25.4 mm pipe is lower thanthat for the 19.0 mm pipe). In the smaller pipe diameter, theincrease in the two layers relative velocities was opposed by thehigher interfacial forces which tend to stabilize the growth ofinterfacial waves, hence, delay drop formation. Fig. 4 showspicture of the flow at superficial oil and water velocities of 0.38and 0.21 m/s, respectively in the two pipes. It is clear that the flow

Fig. 1. Schematic diagram of the oil–water experimental flow facility.

Table 1Properties of oil and water used in this study.

Parameters Mineral oil Water

Density (g/cm3) 0.875 0.998Viscosity (cP) 12 1Interfacial tension (mN/m) 20.1

0.1

1

11.010.0

Usw

, m/s

Uso, m/s

ST Bb AN DC Do/w Dw/o

Fig. 2. Flow pattern map constructed using a 25.4 mm ID pipe, m¼12 cP, ρ¼0.87g/cm3, σ¼20.1 mN/m (after Yusuf et al. (2012)).

0.1

1

11.010.0

Usw

, m/s

Uso, m/sST Bb AN DC Do/w Dw/o

BbDC

AN

Do/w

Dw/oST

Fig. 3. Comparison of flow patterns in the 19 mm ID pipe with flow patternboundaries in the 25.4 mm ID pipe.

T. Al-Wahaibi et al. / Journal of Petroleum Science and Engineering 122 (2014) 266–273268

is still stratified in the smaller pipe while it is already dualcontinuous in the larger pipe.

3.1.2. Bubble flow (Bb)Bubbly flow pattern appeared in both pipes at low superficial

oil velocities (Uso¼0.06–0.1 m/s) and moderate superficial watervelocities. However, the bubble flow region extended to highersuperficial water velocities as the pipe diameter decreased (Fig. 3).This is attributed to the relatively higher interfacial forces (asindicated by the Eőtvős number) in the 19 mm pipe in comparisonto the 25.4 mm pipe. This also agrees with the speculation madeby Poesio (2008). Visual observation from both pipes revealed thatas superficial water velocity increased, the bubbles lengths alsodecreased. A comparison between the bubble flow observed in thetwo pipes at Uso¼0.10 m/s is given in Fig. 5.

3.1.3. Dual continuous flow (DC)The extent of dual continuous regime was found to increase as

the pipe diameter is increased from 19 to 25.4 mm. For example, atUso¼0.60 m/s, the dual continuous flow regime extended up toUsw¼0.55 m/s and 1.0 m/s in the 19 and 25.4 mm pipe respec-tively. This is because dual continuous flow transformed intoannular flow. From the flow pattern map (Fig. 3), it is obviousthat annular flow tends to form easier in the smaller pipe.

On the contrary the continuity of the oil layer depends on theturbulence intensity of the two layers. Oil layer in the 19 mm pipebreaks into drops at relatively higher values of Usw comparing tothe 25.4 mm pipe. This is attributed to the higher turbulenceintensity in the larger pipe for the same values of superficial oil

and water velocities. Fig. 6 shows a comparison between the DCpattern in the 19 and 25.4 mm pipe.

3.1.4. Annular flow (AN)Comparing 19 and 25.4 mm data, it is clear that annular flow

appeared at lower superficial water velocities and over a widerrange of conditions in the 19 mm pipe than in the 25.4 mm pipe(Fig. 3). This is because in liquid–liquid system, wall wetting(curved interface formed rather than a plane) was found toincrease as the interfacial forces increases (see Ng et al. (2001,2004)). The higher the value of the Eőtvős number, the morecurved is the interface. At certain oil velocity, the oil layer isbecoming thinner as the water velocity increases while the surfacetension pulls the curved interface up to the top of the pipe andforms annular flow. Since the interface curvature between oil andwater is larger in the smaller pipe diameter as indicated from thevalue of the Eőtvős number of the two pipes, annular flow formedat lower values of superficial water velocity in the smaller pipe.Fig. 7 shows the effect of increasing oil velocity on annular flow inboth pipes.

3.1.5. Dispersed oil in water (Do/w)The range of dispersed oil in water (Do/w) regime increases as

the pipe diameter is increased from 19 to 25.4 mm. In the 25.4 mmpipe, the dispersed oil in water (Do/w) regime appeared at lowersuperficial water velocity compared to the 19 mm pipe. This isbecause the turbulence is higher for larger pipe diameter and it isknown that dispersion of oil in water is a function of the flowturbulence.

Fig. 4. Flow pattern at Uso¼0.38 m/s and Usw¼0.21 m/s in (a) 25.4 mm pipe and (b) 19.0 mm pipe.

Fig. 5. Effect of increasing Usw on bubble flow pattern at Uso¼0.11 m/s in (a) 25.4 mm pipe and (b) 19.0 mm pipe.

T. Al-Wahaibi et al. / Journal of Petroleum Science and Engineering 122 (2014) 266–273 269

3.1.6. Dispersed water in oil (Dw/o)Pipe diameter seems to have a small effect on dispersed water

in oil. The transition between dual continuous (DC) and (Dw/o)flow appears at relatively higher oil velocity in the smaller pipe(Fig. 3). For example, at Usw¼0.1 m/s, Dw/o started at Uso¼0.63and 0.98 m/s in the 25.4 and 19.0 mm pipes respectively.

3.2. Pressure gradient

Pressure gradient due to friction was measured in this studyusing two pipe diameters over a broad range of superficial oil andwater velocities ranging from 0.1 to 1.25 m/s and 0.1 to 2.25 m/srespectively. The results averaged over at least three measure-ments are presented in Figs. 8 and 9 against superficial watervelocity. It is clear that pressure gradient increased as superficialoil and water velocities increased.

3.2.1. Effect of pipe diameter on pressure gradientThe effect of pipe diameter on pressure gradient is shown in

Figs. 8 and 9. The data obtained from the two pipe diameters isplotted in terms of pressure gradient against superficial watervelocity for some Uso. The pressure gradient values obtained in the

19 mm pipe are greater than those measured in the 25.4 mm pipeat similar superficial oil and water velocities. This is because as thepipe diameter decreases, the friction inside the pipe increases.From the figures, it is clear that the differences in pressuregradients become bigger with higher oil and water velocities.

3.2.2. Comparison with modelsPressure gradient values calculated using the differential pres-

sure measurements were compared with the two-fluid, homo-genous and Shi et al. (2005) drift-flux models (see Appendices Aand B) as shown in Figs. 10, 11 and 12 respectively. Stratified anddual continuous flows are classified as separated flow. Hence, thetwo-fluid model was employed to predict the pressure gradient ofthis regime. For the dispersed and bubble flows, the homogenousand drift-flux models were used.

In contrast to the drift-flux model which showed a goodprediction to the experimental results, the two-fluid and thehomogenous models were found to highly overpredict the experi-mental results especially for the smaller pipe diameter. This isattributed to the increase in the effect of interfacial tension as pipediameter decreased (as indicated from the Eőtvős number of thetwo pipes) which is not taken into account by the two models.

Fig. 6. Dual continuous flow at Uso¼0.21 m/s and Usw¼0.48 m/s in (a) 25.4 mm pipe and (b) 19.0 mm pipe.

Fig. 7. Effect of increase in superficial oil velocity on annular flow at Usw¼0.80 m/s in (a) 25.4 mm pipe and (b) 19.0 mm pipe.

T. Al-Wahaibi et al. / Journal of Petroleum Science and Engineering 122 (2014) 266–273270

The accuracy of the predictions was measured by calculatingthe average percent error (APE), average absolute percent error(AAPE) and standard deviation (SD) of each data source (seeTable 2).

The average percent error is defined as

APE¼ 1n

∑n

k ¼ 1

ðdp=dxÞpred�ðdp=dxÞexpðdp=dxÞexp

" #� 100 ð2Þ

where subscripts “pred” and “exp” represent the predicted andexperimental values, respectively and n is the number of datapoints.

The average percent error (Eq. (2)) is used to quantify thedegree of overprediction or underprediction of the experimentaldata. Positive values indicate overprediction while negative valuesindicate underprediction.

The average absolute percent error (AAPE) is calculated toevaluate the prediction capability of the correlation. Unlike theaverage percent error (APE), the absolute errors are considered sothat the positive and negative errors are not canceled out. Theequation is given by

AAPE¼ 1n

∑n

k ¼ 1

ðdp=dxÞpred�ðdp=dxÞexpðdp=dxÞexp

����������

" #� 100 ð3Þ

The standard deviation (SD) of the predicted values from theexperimental data is used to measure how close the predictions tothe experimental data. The equation which is also known as the

root mean square percent error can be expressed as

SD¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n�1∑n

k ¼ 1

ðdp=dxÞpred�ðdp=dxÞexpðdp=dxÞexp

!2vuut264

375� 100 ð4Þ

The APE, AAPE and SD for the two-fluid model, the homo-genous model and the drift-flux model were calculated and arepresented in Table 2. The two-fluid model and the homogenous

0

1000

2000

3000

4000

0 1000 2000 3000 4000

(dp/

dx) p

red,

Pa/m

(dp/dx) exp, Pa/m

19mm ID pipe 25.4mm ID pipe

Fig. 10. Comparison between the experimental pressure gradient results obtainedin separated flow (stratified and dual continuous flows) of the two different pipediameters and the two-fluid model.

0

2000

4000

6000

8000

10000

0 2000 4000 6000 8000 10000

(dp/

dx) p

red,

Pa/m

(dp/dx) exp, Pa/m

19mm ID pipe 25.4mm ID pipe

Fig. 11. Comparison between the experimental pressure gradient results obtainedin dispersed flow of the two different pipe diameters and the homogenous model.

0

2000

4000

6000

8000

10000

0 2000 4000 6000 8000 10000

(dp/

dx) p

red,

Pa/m

(dp/dx) exp, Pa/m

19mm ID pipe 25.4mm ID pipe

Fig. 12. Comparison between the experimental pressure gradient results obtainedin dispersed flow of the two different pipe diameters and Shi et al. (2005) drift-flux model.

0

2000

4000

6000

8000

0 0.4 0.8 1.2 1.6 2 2.4

dp/d

x, P

a/m

Usw, m/s

Uso = 0.92 m/s (19mm ID) Uso = 0.9 m/s (25.4mm ID)Uso = 1.25 m/s (19mm ID) Uso = 1.21 m/s (25.4mm ID)

Fig. 9. Pressure gradients versus superficial water velocities in the 19 and 25.4 mmID pipes at certain superficial oil velocities.

0

1000

2000

3000

4000

5000

6000

0 0.4 0.8 1.2 1.6 2 2.4

dp/d

x, P

a/m

Usw, m/s

Uso = 0.31 m/s (19mm ID) Uso = 0.33 m/s (25.4mm ID)Uso = 0.72 m/s (19mm ID) Uso = 0.7 m/s (25.4mm ID)

Fig. 8. Pressure gradients versus superficial water velocities in the 19 and 25.4 mmID pipes at certain superficial oil velocities.

T. Al-Wahaibi et al. / Journal of Petroleum Science and Engineering 122 (2014) 266–273 271

models were found to better predict the experimental data of the25.4 mm pipe as indicated from the average percentage error (APEof 17.6% and 13.9% of two-fluid and homogeneous models respec-tively) while the two-fluid and the homogeneous models werehighly overpredicted the 19 mm ID pipe data (APE¼47.8 and 48.4respectively). The comparison also revealed that the two-fluid andthe homogeneous models have better capability to predict thelarger pipe diameter experimental pressure gradient data asshown from the average absolute percent error values (51.5% and48.4% for the 19 mm pipe and 21.6% and 15.9% for the 25.4 mmpipe). On the contrary, although the drift-flux model prediction isin general better than that of the two-fluid and homogenousmodels, it better predicts the smaller pipe diameter pressuregradient data as shown from the APE and AAPE values (12.1%and 14.5% for the 25.4 mm pipe and 0.3% and 12.6% for the19 mm pipe).

4. Conclusions

The effect of decreasing the pipe diameter from 25.4 mm to19 mm pipe during horizontal oil–water flow on flow pattern andpressure gradient was investigated experimentally. Six flow pat-terns were identified in the two pipes. These are stratified(stratified smooth, SS, and stratified wavy, SW), bubble (Bb), dualcontinuous (DC), annular, (AN), dispersed oil in water (Do/w) anddispersed water in oil (Dw/o). The dual continuous and dispersedoil in water regimes appeared for a smaller range of conditions inthe 19 mm pipe than that in the 25.4 mm pipe. On the contrary,the region of stratified, bubbly and annular flows extended towider range of conditions as the pipe diameter decreased. This isattributed to the competition between destabilizing forces (turbu-lence force and the relative movement between the two phases)and stabilizing ones (interfacial tension and gravity forces). Theincrease in destabilizing forces causes waves to grow in amplitudeand hence increases the instability.

The pressure gradient was found to increase as the pipediameter decreased. The effect of pipe diameter on the accuracyof the two-fluid, the homogenous and the drift-flux models wasexamined. The two-fluid and the homogenous models were foundto better predict the experimental results obtained from the largerpipe diameter (25.4 mm).

Appendix A. Calculation of the pressure gradient using thetwo-fluid model

The two-fluid model is derived by considering smooth equili-brium horizontal stratified flow and taking the momentum bal-ance on each phase. The wall and interfacial shear forces balancethe pressure gradient as shown in Eqs. (A.1) and (A.2)

water phase : �Awdpdz

� ��τwSw7τiSi ¼ 0 ðA:1Þ

oil phase : �Aodpdz

� ��τoSo7τiSi ¼ 0 ðA:2Þ

where τw, τo, τi are the water, oil and interfacial shear stressesrespectively; and Si, Sw, So, are the interfacial length and the wallwetted perimeter of the oil and water phases respectively; Aw andAo are the cross sectional areas of the water and oil respectively;dp/dz is the pressure gradient.

The wall shear stresses τw and τo are expressed in terms ofcorresponding fluid friction factors, fw and fo. The wall shear stressacting on each phase and friction factors are written as

τw ¼ f wρwU

2w

8;

f w ¼

64Rew

for laminar flow

�1:8 log6:9Rew

þ ϵ

3:7Dw

� �1:1" # !�2

for turbulent flow

8>>>><>>>>:

ðA:3Þ

τo ¼ f oρoU

2o

8;

f o ¼

64Reo

for laminar flow

�1:8 log6:9Reo

þ ϵ

3:7Do

� �1:1" # !�2

for turbulent flow

8>>>><>>>>:

ðA:4Þwhere Rew and Reo are the water and oil Reynolds numberrespectively.

Dw and Do are the hydraulic diameters of the water and oilphases respectively. Therefore, they can be defined as follows:

Dw ¼ 4Aw

ðSwþSiÞ; Do ¼

4Ao

ðSoÞfor Uw4Uo ðA:5Þ

Do ¼ 4Ao

ðSoþSiÞ; Dw ¼ 4Aw

ðSwÞfor UwoUo ðA:6Þ

Do ¼4Ao

ðSoÞ; Dw ¼ 4Aw

ðSwÞfor Uw �Uo ðA:7Þ

Table A.1Geometric parameters used in the two-fluid model.

Parameter

Interfacial length (Si)D� 1� 2

hw

D�1

� �2 !0:5

Wall perimeter of oil phase (So) D� cos �1 2hw

D�1

� �Wall perimeter of water phase (Sw) πD�SoCross sectional area of the pipe π

4D2

Area oil phase (Ao) D4

� So�Si � 2hw

D�1

� �� �Area water phase (Aw) Aw ¼ A�Ao

Oil hold-up (Ho) Ao=AWater hold-up (Hw) Aw=AIn-situ oil velocity (Uo) Uso=Ho

In-situ water velocity (Uw) Usw=Hw

Table 2Comparison of the effect of pipe diameter on the accuracy of pressure gradientmodels.

Pipe diameter Two-fluid model Homogeneous model Drift-flux model

APE AAPE SD APE AAPE SD APE AAPE SD

19 mm ID 47.8 51.5 60.6 48.4 48.4 55.0 0.3 12.6 17.625.4 mm ID 17.6 21.1 31.8 13.9 15.9 23.3 12.1 14.5 21.6

T. Al-Wahaibi et al. / Journal of Petroleum Science and Engineering 122 (2014) 266–273272

where the parameters Si, So, Sw, Ao and Aw are defined in Table A.1.In this study, the interfacial shear stress is given by

τi ¼ f iρf ðUw�UoÞ2

8; f w

f i ¼ f w and ρf ¼ ρw for Uw4Uo

f i ¼ f o and ρf ¼ ρo for UwoUo

(ðA:8Þ

where fi is the interfacial friction factor, ρo is the density of oil, ρw isthe density of water and ρf is the density of the faster phase.

When the ratio of the two phase velocities is between 0.98 and1.05 (Brauner and Moalem Maron, 1989) then there is no inter-facial shear stress and both phases are assumed to flow as in anopen channel. In this case the hydraulic diameters are calculatedby Eq. (A.7).

Appendix B. Calculation of the pressure gradient using thehomogeneous model

The single phase equation for pressure drop prediction is usedwith the mixture treated as a pseudo-fluid. The average propertiesof the two fluids are used as the mixture properties. The equationis expressed as follows:

dpdz

¼ � f mρmU2m

2DðB:1Þ

Different models have been proposed for the determination ofaverage viscosity of the mixture since the viscosity can haveanomalous behavior during liquid–liquid flow. Dukler et al.(1964) proposed for gas–liquid flow average viscosity in terms offlow volume fraction. Applying the same principle for liquid–liquidflow, the average viscosity is given as

μm ¼HwμwþHoμo ðB:2Þwhere Hw and Ho are the water and oil hold-up respectively.

The friction coefficient can be determined by inserting themixture Reynolds number which is given by Eq. (B.3) into thefriction factor equation

Rem ¼ ρmUmdμm

: ðB:3Þ

References

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