International Journal of Multiphase Flow 37 (2011) 326–335

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International Journal of Multiphase Flow

journal homepage: www.elsevier .com/locate / i jmulflow

Modeling of oil–water flow using energy minimization concept

Anoop Sharma a, Abdelsalam Al-Sarkhi b,⇑, Cem Sarica a, Hong-Quan Zhang a

a Petroleum Engineering Department, The University of Tulsa, 800 South Tucker Drive, Tulsa, OK 74104, USAb Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 August 2010Received in revised form 6 October 2010Accepted 12 November 2010Available online 20 November 2010

Keywords:Total energy minimizationOil–water flow modelingFlow pattern transition

0301-9322/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijmultiphaseflow.2010.11.002

⇑ Corresponding author.E-mail address: alsarkhi@kfupm.edu.sa (A. Al-Sark

A new model is developed to predict flow behaviors including flow pattern, pressure gradient and holdupfor oil–water flow in horizontal and slightly inclined pipes. The model is based on the universal principlethat a system stabilizes to its minimum total energy. The structural configurations observed in two-phaseflow systems can be interpreted in terms of total energy minimization. Performance of the developedmodel is tested against several experimental data, and comparisons with existing models are presented.It is evident from the results and comparisons that the model estimates the pressure gradient and flowpattern very well. The model provides extensive information about oil–water flow characteristics.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The flow of two immiscible liquids is encountered in a diverserange of processes and equipments, particularly in the petroleumindustry, where mixtures of oil and water are often transportedin pipes over long distances. Accurate prediction of oil–water flowcharacteristics, such as flow pattern, water holdup and pressuregradient, is important in many engineering applications. However,despite their importance, liquid–liquid flow has not been exploredto the same extent as gas–liquid flow. The density difference be-tween the phases in a liquid–liquid system is relatively small,while the viscosity ratio encountered can extend over several or-ders of magnitude. Concepts of gas–liquid two-phase flow cannotbe readily applied to liquid–liquid systems.

Arirachakaran et al. (1989) collected extensive experimentaldata for oil–water flows in horizontal pipes for a wide range of vis-cosities of oil. A pressure gradient prediction model was developedfor both stratified and dispersed flows assuming smooth interfacefor stratified flow and full dispersion for dispersed flow.

Brauner and Moalem Maron (1989) developed a simple two-phase liquid–liquid stratified flow model. Brauner et al. (1998) im-proved the stratified flow model for the curved interface. Brauner(2001) suggested a unified approach for predicting the transitionto dispersed flow for liquid–liquid systems for all inclinations.Brauner (2002) conducted a theoretical study about two-phase li-quid–liquid flow modeling and control. She compiled severalworks related to the subject and made a general description of

ll rights reserved.

hi).

the flow patterns present in liquid–liquid flows with all the mod-eling approaches.

Trallero (1995) conducted extensive experimental and theoret-ical studies of flow patterns. A new classification of flow patternwas proposed containing segregated flow patterns and dispersedflow patterns. A total of six flow patterns are identified. Moreover,Trallero developed a modified mechanistic model for stratifiedflow and proposed a force balance between gravity and turbulentfluctuations normal to the axial flow direction to identify flow pat-tern transition from segregated to dispersed flow.

Angeli and Hewitt (2000a,b) conducted an experimental studyon drop size distributions in horizontal oil–water flow. They con-cluded that both the Rosin–Rammler and the upper limit Log-nor-mal distributions could satisfactorily predict their experimentaldata. Moreover, they proposed two different correlations to predictthe Sauter Mean Diameter (SMD, dSM) and the maximum diameter(dMAX) of droplets present in the dispersion, respectively:

dSMv1:8C ¼ 2� 10�8 � ð4f CÞ

�3:12; ð1Þ

dMAXv1:8C ¼ 4:2� 10�8 � ð4f CÞ

�3:13; ð2Þ

where vC and fC are continuous phase velocity and friction factor.The units in Eqs. (1) and (2) are m/s for the velocity and m for thediameter.

Alkaya (2000) conducted both theoretical and experimentalstudies on inclined oil–water pipe flows. Flow patterns, holdup,and pressure gradients were measured for near horizontal inclina-tions. Alkaya presented a new approach in which dispersion of oilin water and water flow pattern, and dispersion of water in oil andoil flow pattern are treated as segregated flow.

A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335 327

The flow structure in horizontal oil–water flow was investi-gated experimentally by Angeli and Hewitt (2000b) using two1-in. ID. pipes made from stainless steel and acrylic resin respec-tively. High speed video recording and measurement of the localphase fractions with a high frequency impedance probe were usedto identifying the flow pattern. The mixed flow pattern was foundin the steel pipe at lower mixture velocities than in the acrylic pipe,and oil was the continuous phase for a wider range of operatingconditions. The visual observations were consistent with the mea-surements of high frequency impedance probe. The distribution ofthe phases differed dramatically between the stainless steel andthe acrylic pipes for certain ranges of operational conditions.

Shi (2001) proposed a four layer model for near horizontal oil–water flow. The four layers consist of pure water, oil in water dis-persion (OW), water in oil dispersion (WO) and pure oil. They alsomodified the Brinkman viscosity correlation by incorporating We-ber number (We).

Zhang and Sarica (2006) presented a unified model for threephase gas–oil–water pipe flow. In their model, the phase distribu-tion and the hydrodynamics are described based on two criteria:gas–liquid flow pattern and oil water mixing status. Closure rela-tionships for mixing and inversion point are proposed for describ-ing the distribution between liquid phases.

Chakrabarti et al. (2005) performed an experimental and mod-eling study on pressure drop in liquid–liquid two-phase horizontalsegregated flow. They used kerosene and water for their study. Amodel was developed to predict pressure drop based on energyminimization and pressure equalization of both phases. The modelpredictions were validated with their experimental results. The en-ergy minimization approach seemed to be quite promising fromthe comparisons.

Guet et al. (2006) presented a new approach to handle viscosityof the mixture and suggested use of a hybrid water cut dependentmodel. For dispersion of oil in water, viscosity of mixture will beconstant and equal to the viscosity of water. For dispersion ofwater in oil, viscosity is calculated treating dispersion as emulsion.

Abduvayt et al. (2006) conducted an experimental study of oil–water flow in horizontal, hilly terrain and vertical pipes. They ac-quired flow pattern, holdup and pressure gradient data. The holdupdata was acquired by using a conductivity probe.

Vielma et al. (2008) acquired detailed experimental data includ-ing pressure gradient, holdup, phase distributions and droplet sizein horizontal and slightly inclined pipes (�1� to +1�). An empiricalcorrelation was developed to predict the SMD. Atmaca (2007) ex-tended Vielma’s work to higher inclinations (�5� to +5�).

Hadziabdic and Oliemans (2007) presented a parametric studyfor determining the liquid flow rates from the pressure drop andwater holdup in oil–water flows. A flow-pattern-dependent modelwas considered for calculation of the liquid production rates in oil–water horizontal flow. For stratified flow, the area-averaged stea-dy-state one-dimensional two-fluid model was used while thehomogeneous model was used for dispersed flow. It was concludedthat both interfacial waves and drop entrainment can be responsi-ble for the augmentation of the wall-shear stress in oil–water flow.In addition, consideration of the interfacial waves offers an expla-nation for some important phenomena of the oil–water flow, suchas the wall-shear stress reduction.

Piela et al. (2008) conducted an experimental study of the phaseinversion process in an oil–water flow through a pipe. The criticalconcentration of the dispersed phase at which phase inversion oc-curs and morphological structural changes during inversion wereinvestigated in this study. Pictures of the phase inversion processwere taken, and simultaneously the electrical conductivity of themixture was measured to identify the continuous and dispersedphases. The critical concentration was found to depend on theinjection phase volume fraction.

The present study focuses on facilitating gradual changes inflow structures using the energy minimization approach. There-fore, better predictions of the design parameters can be achieved.The objectives of this study are to develop a better model for oil–water flow capturing the physics of the process, especially for tran-sitions between flow patterns, and to validate the model usingavailable experimental data and models.

2. Modeling

For oil–water flow in horizontal and slightly inclined pipes themodel considers the Stratified Smooth (ST) flow pattern, the Dis-persion of Oil in Water and Oil layer (DOW/O) flow pattern, theDispersion of Water in Oil and Water layer (DWO/W) flow pattern,the Dual Dispersion (OW/WO) flow pattern, the Dispersion of Oil inWater (OW) flow pattern, and the Dispersion of Water in Oil (WO)flow pattern.

The model is based on the principle that a system stabilizes toits minimized total energy. Moreover, for the segregated flow pat-tern, the continuity and combined momentum balance equationsare solved. For full dispersion conditions, only total energy is min-imized, and the continuity equation is solved. Assumptions for themodel include: smooth interface and smooth pipe, negligible sur-face energy in between the pipe wall and fluids, homogeneous dis-persion, and steady-state flow.

The mixture properties for each continuous phase and its dis-persion are calculated as following:

q1 ¼ qoð1� HD1Þ þ qwHD1; ð3Þ

q2 ¼ qwð1� HD2Þ þ qoHD2; ð4Þ

l1 ¼ loð1� HD1Þ�2:5; ð5Þ

l2 ¼ lwð1� HD2Þ�2:5; ð6Þ

where subscripts 1 and 2 represent layer 1 (oil continuous phase)and layer 2 (water continuous phase), respectively; subscripts D, oand w represent dispersed phase, pure oil phase and pure waterphase, respectively; H, l and q represent the holdup, viscosityand density, respectively. Eqs. (5) and (6) use the Brinkman(1952) viscosity correlation to calculate the viscosity of dispersions.The quantities HD1 and HD2 vary from 0 to the phase inversion point.Therefore, HD1 is dispersed phase holdup in continuous oil phaseand HD2 is dispersed phase holdup in continuous water phase. Theinversion point in this model is calculated by using Eq. (7), as sug-gested by Zhang and Sarica (2006)

HO;INV ¼lOlW

� �0:4

1þ lOlW

� �0:4 ; ð7Þ

where HO,INV is the holdup of the dispersed phase (oil) in an oil–water dispersion at which inversion takes place. The velocity ofeach layer can be calculated by simultaneously solving continuityequations for both oil and water:

qO;T ¼ A1ð1� HD1Þv1 þ A2HD2v2; ð8Þ

qW ;T ¼ A2ð1� HD2Þv2 þ A1HD1v1; ð9Þ

where qO,T, qW,T, v1 and v2 are the total inlet flow rate of oil andwater and velocities of the oil continuous and the water continuousphases, respectively; A1 and A2 are the areas corresponding to oilcontinuous and water continuous phases as shown in Fig. 1,respectively.

Fig. 1. Cross-sectional view of pipe.Fig. 2. Side view of inclined pipe flow.

11

INVdrop1 HA

A≤⎟⎟⎠

⎞⎜⎜⎝

⎛

11 MAXMAX Ld ≤

328 A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335

Brauner and Moalem Maron (1989) then Trallero (1995) em-ployed two-fluid modeling approach to predict pressure drop intwo-phase segregated flow. The two-fluid model solves the com-bined momentum equation to predict the pressure drop. Assumingsmooth, equilibrium, stratified flow, the following momentum bal-ance equations can be derived for layers 1 and 2:

�A1dPdL

� �1� sW1S1 � sISI � q1A1g sin h ¼ 0; ð10Þ

�A2dPdL

� �2� sW2S2 þ sISI � q2A2g sin h ¼ 0: ð11Þ

Assuming equal pressure gradients for both layers, the followingcombined momentum equation can be obtained:

F ¼ � sW2S2

A2þ sW1S1

A1þ sISI

1A1þ 1

A2

� �þ ðq1 � q2Þg sin h ¼ 0: ð12Þ

In Eqs. (10) and (11), A, S and dPdL

� �denote area, perimeter and

pressure gradient, respectively; subscripts 1, 2 and I representthe oil continuous phase, water continuous phase and the inter-face, respectively; h is the inclination angle from horizontal; andsW1, sW2, sI are oil, water and interfacial shear stresses, respec-tively. The shear stresses can be expressed in terms of the corre-sponding fluid friction factors f1, f2 and fI.

The wall-shear stresses for the two layers are:

sW1 ¼f1q1v2

1

8; sW2 ¼

f2q2v22

8; ð13Þ

where the Moody friction factor is expressed (for any phase j,assuming a smooth pipe wall) as a function of Reynolds number,

fj ¼ Cdjv jqj

lj

!�n

: ð14Þ

Coefficient C and exponent n are 64 and 1 for laminar flow and0.184 and 0.2 for turbulent flow. Equivalent hydraulic diametersare determined on the basis of which phase is faster. In otherwords, closed and open channel flows are assumed for faster andslower moving phases, respectively. For equal velocities, bothphases are assumed to behave as open channel flow.

For v1 > v2:

d1 ¼4A1

S1 þ SI; d2 ¼

4A2

S2: ð15Þ

For v1 < v2,

d1 ¼4A1

S1; d2 ¼

4A2

S2 þ SI: ð16Þ

For v1 � v2,

d1 ¼4A1

S1; d2 ¼

4A2

S2: ð17Þ

The interfacial shear stress, sI, is calculated, as proposed byZhang et al. (2010):

sI ¼ ½qMIXCf ðsW1H1 þ sW2H2Þ�1=2ðv1 � v2Þ; ð18Þ

where

qMIX ¼ ðq1H1 þ q2H2Þ; Cf ¼f1H1 þ f2H2

8

� �;

H1 ¼A1

A1 þ A2

� �; H2 ¼

A2

A1 þ A2

� �:

The quantities ho and hw, shown in Fig. 1, are the centroids of the oilcontinuous phase and water continuous phase from the pipe base,respectively:

ho ¼d2

1þ 43

sin3 b2

� �ð2p� b� sinðbÞÞ

" #;

hw ¼d2

1� 43

sin3 b2

� �ðb� sinðbÞÞ

" #; ð19Þ

where b is a step parameter for the model. It is varied from 0� to360� with small increments. In a broader way, we are using b (anglesubtended by interface on the center of the pipe). Let us assumethat the height of lower layer normal to the pipe axis is x. The heightof the upper layer will be pipe diameter minus x. All the energiesand other parameters can be expressed and calculated using thevariable x. In our case, instead of x, we are using b to express andcalculate these. In 0� case, there will not be any lower layer, andsince there are two different fluids flowing, the only possibility isthat the lower phase fluid fully dispersed in upper phase fluid.The opposite is true for the case of 360�. In between cases, for eachb value, we will have two layers flowing. Possibility of existence ofdispersions is checked for each layer.

For an inclined pipe, the heights of the centroids of oil and watercontinuous phases from the bottom, as shown in Fig. 2, will be:

Fig. 3. Transition criteria for onset of entrainment.

Fig. 4. Model flow chart.

A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335 329

h1 ¼ ho cos h; h2 ¼ hw cos h: ð20Þ

The total potential energy per unit length of pipe can be calcu-lated as the summation of potential energies of both continuousphases:

PE ¼ A1q1gh1 þ A2q2gh2: ð21Þ

The total kinetic energy per unit length of pipe can be calculatedas the summation of kinetic energies of both continuous phases:

Fig. 5. Model predictions of pressure gradient comp

KE ¼ 12

A1q1v21 þ

12

A2q2v22: ð22Þ

The total surface energy per unit length can be calculated as thesummation of the surface energies of water droplets in oil phase,oil droplets in water phase and interface:

SE ¼ r d sinb2þ 6A1HD1

dSM1þ 6A2HD2

dSM2

� ; ð23Þ

where r, dSM1 and dSM2 are the interfacial tension between oil andwater, SMD of the water droplets and SMD of the oil droplets. SMD

ared with experimental data of Atmaca (2007).

Fig. 6. Model predictions of pressure gradient compared with experimental data of Alkaya (2000).

330 A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335

can be estimated by several different models and correlations. In thismodel, the correlations of Angeli and Hewitt (2000a,b) are used toestimate Maximum Diameter (dMAX) and SMD (dSM) (Eqs. (1) and (2)).

Finally, the total energy of the system can be represented as:

TE ¼ SEþ KEþ PE: ð24ÞThe pressure gradient is calculated by the two-fluid model for

all flow patterns in which two distinct phases are present in thepipe. For fully dispersed flow, the pressure gradient is calculatedfrom a homogeneous model as,

dPdL

� �¼ fMqMv2

M

2dþ qMg sin h; ð25Þ

where fM, qM and vM are the friction factor of the mixture, mixturedensity and mixture velocity, respectively.

The model inputs include fluid properties, superficial velocitiesof the oil and water. Changing b from 0� to 360� the model calcu-lates TE and F for the system. The solution is the condition wherethe total energy is minimum (TE ? minimum) and combined

Fig. 7. Model predictions of pressure gradient comp

momentum equation is near zero (F ? 0), simultaneously. For thetwo cases of full dispersions, i.e. b = 0� (dispersion of water inoil) and b = 360� (dispersion of oil in water), the model does notconsider the combined momentum equation (F = 0) and only con-siders the minimization of total energy. For Reynolds numbersmaller than 1000, it is assumed that the flow will be laminar,and the continuous phase will have a negligible amount of turbu-lent energy, if any, to cause any dispersion. Hence, the continuousphase remains in segregated form. To determine whether there isany dispersion for Reynolds number higher than 1000, dMAX is cal-culated and compared with the maximum distance (LMAX) betweenthe interface and pipe wall across the continuous phase. As shownin Fig. 3, dMAX must not be greater than LMAX,

dMAX 6 LMAX : ð26ÞMoreover, to avoid inversion, the maximum cross section area

swept by the biggest droplet should not cross the inversion pointof dispersed phase holdup in the continuous phase. Therefore, asshown in Fig. 3,

ared with experimental data of Trallero (1995).

Fig. 8. Model predictions of pressure gradient compared with experimental data of Abduvayt et al. (2006).

A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335 331

ðAdrop=AÞ 6 HINV ; ð27Þwhere Adrop, A and HINV represent the cross sectional area of thedrop, cross sectional area of the layer where the drop exists andthe holdup of the dispersed phase at the inversion point, respec-tively. These assumptions are reasonable, conservative and biasedtowards the dispersions. If either of the conditions is not satisfied,then it is not possible to have dispersions, and the continuous phasestill remains in segregated form. If the continuous phase canaccommodate dMAX, then the holdup of the dispersed phase HD isvaried from 0 to the phase inversion point, and for each case, bothTE and F are minimized. This is done simultaneously in both layers

A Visual Basic Application (VBA) code was written for the mod-el. A flow chart of the basic code algorithm is shown in Fig. 4.

3. Model validation

The model is validated against available experimental data.Model predictions are compared with oil–water experimental

Fig. 9. Model predictions of water holdup compar

results of Atmaca (2007), Alkaya (2000), Trallero (1995) and Abdu-vayt et al. (2006).

3.1. Pressure gradient

Fig. 5 shows the predicted pressure gradients compared withAtmaca’s (2007) experimental pressure gradients, for inclinationangles of 0�, 1�, 2�, �1�, �2�, and �5�. In Fig. 6, the predicted pres-sure gradients are compared with Alkaya’s (2000) data, for inclina-tion angles of 0�, 1�, �1�, 5�, and �5�. In Fig. 7, the predictedpressure gradients are compared with Trallero’s (1995) measure-ments, for horizontal oil–water flows. In Fig. 8, the predicted pres-sure gradients are compared with Abduvayt et al.’s (2006) data, forinclination angles of 0�, 0.5�, �0.5�, 3�, and �3�. It is seen in the fig-ures that the data points are quite close to the 45� line. This indi-cates that the current model performs very well in pressuregradient predictions.

ed with experimental data of Atmaca (2007).

Fig. 10. Model predictions of water holdup compared with experimental data of Alkaya (2000).

332 A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335

3.2. Holdup

Fig. 9 shows water holdup predictions of the current modelcompared with the Atmaca (2007) experimental data for inclinationangles of 1�, 2�, �1�, �2� and �05�. In Fig. 10, water holdup predic-tions of the current model are compared with the Alkaya (2000)measurements for inclination angles of 1�, 5�,�1�,�5� and horizon-tal. In Fig. 11, water holdup predictions of the current model arecompared with the Trallero (1995) experimental data for horizontalflow. It is seen that the current model predictions of water holdupsfor all flow patterns agree well with the experimental measurementsof Atmaca (2007), Alkaya (2000) and Trallero (1995).

In the water holdup predictions, some deviation is observed forstratified flow pattern. This may be due to inaccuracies in the esti-mation of interfacial shear stress in the current model. More workis needed to correctly estimate the interfacial shear stress. Somecluster of data points in fully dispersed flow patterns (OW andWO) are off the 45� line. This is because the experimentallyobserved flow patterns are not fully dispersed corresponding to

Fig. 11. Model predictions of water holdup compa

these data points. The model predicts fully dispersed flow patternfor these data points because the criteria for the onset of entrain-ment is conservative and biased towards dispersion. This leadsthe model to over predict dispersions in flow and hence causeerrors in water holdup prediction. A better criterion for the onsetof entrainment is required to more accurately predict both flowpattern and holdup.

3.3. Flow pattern

The current model flow pattern predictions are compared withexperimentally observed flow patterns. For comparison purpose,Fig. 12 shows the flow pattern map observed by Atmaca (2007)for horizontal flow, and Fig. 13 shows the flow pattern map pre-dicted by the current model for the same flow conditions. ST, STMI,DOW/W, DOW/O, O/W and W/O denote stratified, stratified mix-ing, dispersion of oil in water with water layer, dispersion of oilin water with oil layer, dispersion of oil in water and dispersionof water in oil, respectively.

red with experimental data of Trallero (1995).

Fig. 12. Flow pattern map observed by Atmaca (2007) for horizontal flow.

Table 1Statistical parameters for error.

Definitions

eri ¼ Predicted value�Measured valueMeasure value

� �� 100

ei = Predicted value �Measured value

e1 ¼ 1N

�PNi¼1eri

1 �PN

A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335 333

More flow pattern maps predicted by the current model arecompared with the Atmaca (2007) and Alkaya (2000) observationsfor different pipe inclinations Sharma (2009). The general agree-ment is good. The flow pattern map heavily depends on the dropletsize correlations and the definition of the criteria for onset ofentrainment. Over prediction of dispersed flow pattern is specu-lated to be due to utilization of a conservative criteria for the onsetof entrainment.

e2 ¼ N i¼1jerij

e3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1ðeri�e1Þ2

N�1

re4 ¼ 1

N

�PNi¼1ei

e5 ¼ 1N

�PNi¼1jeij

e3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1ðei�e4Þ2

N�1

r

3.4. Comparison with unified model

The current model predictions are also compared with the Zhangand Sarica (2006) TUFFP unified model predictions. The currentmodel uses a single criterion of minimization of energy to predict

Fig. 13. Flow pattern map predicted by present model for horizontal flow conditions of Atmaca (2007).

Table 3Statistical parameters for water holdup predictions by current model compared withexperimental studies.

e1 (%) e2 (%) e3 (%) e4 (–) e5 (–) e6 (–)

Atmaca (2007) 0.47 9.73 15.58 0.01 0.03 0.05Alkaya (2000) 16.45 27.79 51.81 0.02 0.05 0.07Trallero (1995) 13.06 15.74 58.43 0.03 0.04 0.05

Table 4Pressure gradient predictions by unified model and current model compared withAtmaca (2007) data.

e1 (%) e2 (%) e3 (%) e4 (Pa/m) e5 (Pa/m) e6 (Pa/m)

Unified model 2.5 20.6 40.7 41.5 63.6 99.4Current model �7.0 14.8 30.4 �0.4 49.4 86.9

Table 5Water holdup predictions by unified model and current model compared withAtmaca (2007) data.

e1 (%) e2 (%) e3 (%) e4 (–) e5 (–) e6 (–)

Unified model �2.94 8.93 13.21 �0.02 0.03 0.04Current model 0.47 9.73 15.58 0.01 0.03 0.05

Table 2Statistical parameters for pressure gradient predictions by current model compared with experimental measurements.

e1 (%) e2 (%) e3 (%) e4 (Pa/m) e5 (Pa/m) e6 (Pa/m)

Atmaca (2007) �6.79 14.65 30.16 �0.48 49.48 87.07Alkaya (2000) �1.85 16.33 33.39 �21.61 47.08 76.17Trallero (1995) �25.27 29.12 26.23 �33.83 55.54 91.37Abduvayt et al. (2006) 16.01 36.77 105.44 7.75 21.58 36.33

334 A. Sharma et al. / International Journal of Multiphase Flow 37 (2011) 326–335

flow pattern map, holdup as well as pressure gradient. The unifiedmodel first predict flow pattern and then predicts pressure gradientand holdup according to the respective flow pattern. The two modelsare compared against the same experimental data of Atmaca (2007)for pressure gradient and water holdup. The average absolute rela-tive error in predicting the pressure drop was 14.8% for the presentmodel compared to 20.6% for the unified model. For water holdupprediction the average absolute relative error was 8.93% for the uni-fied model and 9.73% for the current model.

The current model performs well in both pressure gradient predic-tions and holdup predictions. Its predictions are on par with the uni-fied model and in many occasions are better than the unified model.

3.5. Error analysis

For comparison purpose error estimation was performed byusing the six parameters listed in Table 1. These parameters arethe average relative error (e1), the average absolute relative error(e2), the standard deviation of relative errors (e3), the average actualerror (e4), the average absolute actual error (e5) and the standarddeviation of actual errors (e6), respectively. The relative error (eri)and actual error (ei) are also defined in the Table 1. Tables 2–5show the error analysis for each comparison explained in Sections3.1–3.4.

4. Concluding remarks

Energy minimization concept is used in this study to model thebehavior of oil and water flow in horizontal and near horizontalpipes. Unlike existing oil–water flow models, the current modeldoes not force the flow to conform to a particular flow pattern

and captures the gradual changes in flow configuration. Therefore,this results in a unified model for all flow configurations and betterpredictions of the design parameters.

The model provides extensive information about the flow, e.g.flow pattern (including dual dispersions, single layer dispersions),droplet size, holdup and dispersed phase holdup and pressure gra-dient. It can predict six different flow patterns namely: StratifiedSmooth (ST), Dispersion of Oil in Water and Oil layer (DOW/W),Dispersion of Water in Oil and Water layer (DWO/W), Dual Disper-sion (OW/WO), Dispersion of Oil in Water (OW) and Dispersion ofWater in Oil (WO). Unlike other previous models, it captures thegradual changes of the flow behaviors during the transition fromstratified to fully dispersed flow.

The model is validated against data from experimental studiesof Atmaca (2007), Alkaya (2000), Trallero (1995) and Abduvaytet al. (2006). It is evident from the comparisons that the modelestimates the pressure gradient, water holdup and flow patternwell. The model predictions are on par with the Zhang and Sarica(2006) unified model. In some cases the current model predictsmore accurately.

The model is sensitive to the closure relationships such as drop-let size, mixture viscosity, and onset of entrainment. The flow pat-tern map heavily depends on the droplet size correlations and thecriteria for onset of entrainment. The current model uses the corre-lation developed by Angeli and Hewitt (2000a,b) for estimation ofdrop size. The criteria used in current model are conservative andbiased towards formation of dispersions. The model performancecan further be improved by development of better closurerelationships.

Acknowledgment

The authors thank the member companies of the Tulsa Univer-sity Fluid Flow Projects for their support.

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