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©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.

MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016

PETROLEUM SCIENCE AND TECHNOLOGY

Vol. 21, Nos. 7 & 8, pp. 1101–1120, 2003

Prediction of the Occurrence of Oil–Water

Countercurrent Flow in Deviated Wells

Liang-Biao Ouyang*

ChevronTexaco E&P Technology Company,

Drilling Technology Center, Houston, Texas, USA

ABSTRACT

Water circulation and oil–water countercurrent flow in a deviated

well has been investigated in the present article. Flow patterns for

the countercurrent flow have been identified. Two practical criteria

based on the transport mechanisms of a liquid layer and a liquid

droplet have been developed for determining the occurrence of oil–

water countercurrent flow in deviated or multilateral wells. The

criteria can also be applied to determine the liquid loading-up

conditions of gas wells. Well deviation, well size, and oil density

are the three major factors that affect the occurrence of oil–water

countercurrent flow. The smaller the wellbore size, the larger the

well deviation, or the denser the oil phase, the lower the minimum

oil flow rate is required to avoid the occurrence of oil–water

*Correspondence: Liang-Biao Ouyang, Ph.D., ChevronTexaco Corporation,

2202 Oil Center Court, Rm A-162, Houston, TX, USA; E-mail:

[email protected].

1101

DOI: 10.1081/LFT-120017878 1091-6466 (Print); 1532-2459 (Online)

Copyright & 2003 by Marcel Dekker, Inc. www.dekker.com



countercurrent flow or water circulation. Excellent agreement has

been achieved between the model prediction and the field and labo-

ratory observations. A modified equation has also been proposed for

determining the water droplet size in oil–water wells.

1. INTRODUCTION

Oil–water, gas–oil, oil–gas–water flow in deviated wells is commonlyseen in the oil and gas industry. At moderate well deviation angles, watercirculation and countercurrent flow is anticipated under majority of fieldflow conditions, especially for wells producing at low to medium rates.Water circulation and countercurrent flow has become a key factor thatprevents production logging tools, including spinner flowmeter, densi-tometer and capacitance sensor, etc., to respond appropriately andmakes reasonable interpretation of production logs extremely difficult.Practical criteria to predict the occurrence of countercurrent flow, betterunderstanding of the flow phenomena and a good model to handle oiland water countercurrent flow would be very helpful for improvingproduction logging practice in deviated wells. Reliable interpretation ofproduction logs requires a good model for countercurrent flow. Therehave been some research reported on oil–water concurrent flow inhorizontal or inclined pipes, such as Trallero (1995), Hasan and Kabir(1998), and Ouyang (2000), etc. However, there is little investigationperformed in the area of countercurrent flow.

Significant efforts have been taken at ChevronTexaco Explorationand Production Technology Co. (EPTC) to investigate the oil–watercountercurrent flow in deviated wells. The mechanisms dominatingthe occurrence of water circulation and countercurrent flow havebeen identified. Two practical criteria based on the transport mech-anisms of a liquid layer and a liquid droplet have been developed fordetermining the occurrence of oil–water countercurrent flow indeviated or multilateral wells. Excellent agreement has beenachieved between the model prediction and the field and laboratoryobservations.

2. COUNTERCURRENT FLOW PATTERNS

Countercurrent flow, by its definition, is basically the flow formatwhere fluids are moving in opposite direction in a pipe or well as

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illustrated in Fig. 1. Different countercurrent flow patterns have beenobserved in the field and in the laboratory, which can be classified asthe following five basic categories according to Ouyang (2002).

2.1. Two-Layer Countercurrent Stratified Flow with

Clean Surface

As illustrated in Fig. 1, countercurrent stratified flow occurs when oiland water flow in the opposite direction, oil moving upwards while watermoving downwards. Because of gravity, oil is located in the upper part ofthe pipe, while the water in the lower part of the pipe. There is an inter-face between the two phases where little mixing and little dispersionoccurs and no oil or water droplets exist in the pipe.

2.2. Countercurrent Stratified Flow with Upward-Moving

Water Droplet

Again, oil and water flow in the opposite direction, oil movingupwards while majority of water moving downwards. There exists athin layer sandwiched by the upper oil layer and the lower waterlayer. Water droplets, oil droplets or mixing happens in this thinlayer where both oil and water move upwards. Water circulation isanticipated here.

Figure 1. Oil–water countercurrent stratified flow with clean interface.

Oil–Water Countercurrent Flow Prediction 1103



2.3. Stratified Flow with Upward-Moving Water Layer

Three fluid layers appear in this flow pattern, the upper upwards-moving oil layer, the bottom downwards-moving water layer and themiddle upwards-moving water layer. Water in the middle layer movesupwards, mainly due to the shear stress imposed by the upper oil layer.Mixing or dispersion may happen around the oil–water and water–waterinterfaces.

2.4. Intermittent Flow

In overall, oil and water flow in the opposite direction, oil, located inthe upper part of the pipe, moving upwards while majority of water in thebottom of the pipe moving downwards. Oil appears as elongated oilconglomerate while water appears as liquid slugs. Water circulationoccurs in the water slug region and the water thickness varies with pipelocation.

2.5. Dispersed Droplet Flow

Dispersed droplet flow is expected to occur at high oil and low waterflow rates. Water exists in discontinuous form and appears as droplets atdifferent sizes. Oil rate is high enough to transport small water dropletsbut is insufficient to lift large droplets. Consequently, large water dropletswill move downwards while their smaller counterparts move upwards,leading to countercurrent flow. Note that this flow pattern could also beidentified as water in oil emulsion since oil is the continuous phase whilewater is the discontinuous phase.

3. MECHANISMS FOR THE OCCURRENCE OF

COUNTERCURRENT FLOW

When oil and water flow upwards simultaneously in a pipe, the heavyphase, water, tends to flow downwards due to gravity. Depending on theflow conditions, water may exist as either droplet or bulk water (layer), orboth. The natural forces that may prevent water fallback are the inter-facial shear stress between oil and water and the pressure differenceimposed on the fluids. The higher the oil flow velocity, the larger thedifference in the oil and water velocities, the stronger the interfacial

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shear stress, and the more likely water will be pulled upwards. To makeoil–water co-current flow happen (Fig. 2), oil rate should be high enoughto transport both every single water droplet and bulk water layerupwards to the surface.

3.1. Mechanism No. 1: Bulk Water Lifting—The Layer Model

Consider the case where segregated oil and water flow happens in apipe. Oil flows upwards while the water could be moving upwards ordownwards (Fig. 3).

Figure 2. Schematic of cocurrent oil–water flow in an inclined pipe.

Figure 3. Segregated oil–water flow in a pipe.




As can be imagined, the pressure differential is the major resourcethat pushes oil and water upwards. Gravity, oil-wall shear stress, andinterfacial shear stress are the three forces that pulls the oil downwards.For water layer, in addition to pressure differential, the interfacial shearstress is also the force that tends to move the water upwards. If thepressure differential plus the interfacial shear stress are large enough toovercome the gravity and the water-wall shear stress, then water will alsomoves upwards, leading to concurrent flow. However, if the pressuredifferential plus the interfacial shear stress is not that high, water willfall back (downwards), and countercurrent flow occurs. The majordifference between concurrent and countercurrent flow reflects in thedirection of the water-wall shear stress, or �ww. Based on the definitiongiven in Fig. 3, the water-wall shear stress is positive for concurrent flow,and negative for countercurrent flow. Therefore, zero value for thewater-wall shear stress becomes the key criterion distinguishing the twotypes of flow formats.

The momentum balance equation for the water phase can be writtenas

�Awdp

dx¼ ��iSi þ �wwSw þ �wAwg sin � ð1Þ

where �i stands for the interfacial shear stress, Si is the perimeter ofthe interface, Sw is the perimeter for the water-wet surface, Aw is thewater-wet cross-sectional area.

Similarly, for the oil phase, the momentum balance follows

�Aodp

dx¼ �iSi þ �woSo þ �oAog sin � ð2Þ

Eliminating the pressure gradient term from Eqs. (1) and (2) leads to

�iSiðA�1w þ A�1

o Þ ¼ �wwSwA�1w � �woSoA

�1o þ ð�w � �oÞg sin � ð3Þ

In order to maintain the concurrent flow status, the water-wall shearstress must be greater than zero, or �ww > 0: Furthermore, the oil-wallshear stress is always positive. Therefore, the threshold interfacial shearstress may be estimated by

�i,TSi ¼ �w � �oð ÞAwAog sin �=A ð4Þ

As long as the Eq. (4) is satisfied, a nonnegative water-wall shearstress will be guaranteed, and hence concurrent flow sustains.

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Substituting the definition of the interfacial shear stress into Eq. (4)yields

Vo,T ¼2 �w � �oð ÞAwAog sin �

�o fiSiA

� �0:5ð5Þ

Therefore, the minimum oil flow rate required to maintain theconcurrent oil–water upward flow is given by

Qo,min ¼1

4�D2Eo

2 �w � �oð ÞAwAog sin �

�o fiSiA

� �0:5ð6Þ

The geometrical parameters in Eq. (6), including the oil hold up, theoil-wet cross-sectional area, the water-wet area, and the interfacial peri-meter are all merely the functions of the water height, hw. Replacing allthe geometrical parameters with the corresponding functions of the waterheight results in

Qo,min ¼1

8D2FðhwDÞ

�w � �oð ÞgD sin �

�o fi

� �0:5ð7Þ

where hwD is the dimensionless water height, and the function F is given by

FðhwDÞ ¼1ffiffiffi�

p arccosX � Xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� X2

ph i1:5X þ

�� arccosXffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� X2

p

� �0:5ð8Þ

where X¼ 2hwD� 1.The dependence of the function F upon the dimensionless water

height is displayed in Fig. 4.Equation (7) can also be written in the field units

Qo,min ¼ 21:87D2FðhwDÞð�w � �oÞD sin �

�o fi

� �0:5ð9Þ

where the pipe internal diameter is in inches, fluid densities in lbm/ft3,and flow rate in bbl/day.

Finally please note that the mechanism discussed in this sectionshould not be applied for vertical wells where no water layer isanticipated.

3.2. Mechanism No. 2: Water Droplet

Lifting—The Droplet Model

Consider a water droplet in an oil column as shown in Fig. 5. Thereare two forces acting on the droplet, the buoyancy force, FB, that tries to




move the droplet downwards, and the drag force, FD, that drags thedroplet to move along with the oil.

The buoyancy force is given by

FB ¼ ð�w � �oÞg1

6�d3 ð10Þ

Figure 4. Function F vs. the dimensionless water height.

Figure 5. Water droplet in an oil column.

1108 Ouyang



where d is the water droplet size. And the drag force, which is aligned tothe flow direction, is given by Levich (1962)

FD ¼1

8CD�oV

2�d2 ð11Þ

where CD is the drag coefficient, V the relative movement between theliquid droplet and the oil column.

Smooth, hard, spherical particles are presumed in developing both ofthe above equations. In reality, small bubbles of gas and droplets ofliquid often approach a spherical shape, but, because of fluid circulationwithin the bubble or droplet, their terminal velocities may differ some-what from those of solid particles. The circulation within a fluid dropletlessens the velocity gradients adjacent the drop, decreases the energydissipation, and results in a higher fall or rise velocity for a fluid ascompared with a solid particle.

For water droplets, based on the study by Levich (1962), we recom-mend to introduce ð2�o þ 3�wÞ=ð3�o þ 3�wÞ as a correction factor inEq. (11) to account for fluid circulation within a water droplet. Hence,

FD ¼1

24

2�o þ 3�w�o þ �w

CD�oðVo � VwÞ2�d2 ð12Þ

It is easy to imagine that if the drag force is greater than thebuoyancy force, or FD > FB sin �, then, the water droplet will moveupwards with the oil, concurrent flow ensures. On the contrast, if thedrag force is less than the buoyancy force, the oil flow is not fast enoughto lift the water droplet, consequently, countercurrent flow occurs.Mathematically, concurrent flow happens when

ðVo � VwÞ � 2�o þ �w

2�o þ 3�w

ð�w � �oÞgd sin �

�oCD

� �0:5ð13Þ

Hence,

Vo � 2�o þ �w

2�o þ 3�w

ð�w � �oÞgd sin �

�oCD

� �0:5ð14Þ

where D is the pipe internal diameter.In order to apply the Eq. (14), two parameters must be determined,

the water droplet size and the drag coefficient. Discussion on these twoparameters follows.




3.3. Droplet Size

The droplet size, d, is a function of many factors, including the fluidproperties, flow rates, pipe or well configuration, and geometry. Thedroplet size depends on the balance between surface tension forces thathold the drop together and the impact force of the oil that tends toshatter the drop. Hinze (1950) provided the maximum stable liquiddroplet size for gas-liquid pipe flow,

d ¼We,cr

�gV=2

ð15Þ

where We,cr is the critical Weber number and takes a value between 20and 30 for liquid droplets that are gradually accelerated, and V=2 repre-sents the turbulent fluctuation velocity that is related to the wall frictionand the energy dissipation rate per unit mass.

Turner et al. (1969) and Taitel et al. (1980) approximated theturbulent fluctuation velocity by the continuous phase velocity, whichis essentially the gas velocity for gas-liquid flow and the oil velocity foroil–water flow. For gas-liquid flow,

d ¼We,cr

�gV2g

ð16Þ

For oil–water flow,

d ¼We,cr

�oV2o

ð17Þ

Substituting Eq. (16) into Eq. (14) leads to

Vo � 1:414�o þ �w

2�o þ 3�w

ð�w � �oÞgWe,cr sin �

�2oCD

� �0:25ð18Þ

Correspondingly,

Qo � 1:112D2 �o þ �w2�o þ 3�w

ð�w � �oÞgWe,cr sin �

�2oCD

� �0:25ð19Þ

In field units,

Qo � 61:15D2 �o þ �w2�o þ 3�w

ð�w � �oÞWe,cr sin �

�2oCD

� �0:25ð20Þ

where the pipe diameter is in inches, density in lbm/ft3, surface tension indyne/cm, viscosity in cp, and flow rate in bbl/day.

1110 Ouyang



Note that Eq. (20) can also be used to determine the liquid-loadingcondition in a gas well where gas and liquid countercurrent flow mayoccur. For gas wells, Eq. (20) becomes

Qg � 0:343D2 �g þ �l2�g þ 3�l

ð�l � �gÞWe,cr sin �

�2gCD

" #0:25

ð21Þ

where again, field units should be used. In other words, pipe diameter isin inches, density in lbm/ft3, surface tension in dyne/cm, viscosity in cp,and flow rate in Mcf/day.

Finally it is worthwhile to discuss the prediction of oil and waterdroplet diameter by Eq. (16) or (17). Comparison with experimentalobservation by Zhu and Hill (1988) indicates that Eq. (16) significantlyoverestimates the droplet size for air–water flow (Table 1).

Equation (17) also significantly overestimates the water droplet sizefor oil–water flow by as many as 80 times, therefore, we introduce anadjustment factor, which is found to be around 0.015 for oil–water flowcase, into Eq. (17). So, for oil–water flow,

d ¼ 0:015We,cr

�oV2o

ð22Þ

The adjustment leads to significant improvement in the droplet sizeprediction as shown in Fig. 6.

It is strongly recommended that more experimental data need to besearched and used to validate Eqs. (16), (17), and (22).

Table 1. Comparison of predicted and observed water

droplet sizes.

Air rate

Predicted

droplet size

Observed

droplet size

Adjusted

predicted

droplet size

(ft3/day) (mm) (mm) (mm)

23000 21150 1–10 21.1503

47000 5065 1–6 5.0650

94000 1266 1–3 1.2662

140000 571 1–3 0.5708

190000 310 1.5–3 0.3099




3.4. Drag Coefficient

Generally speaking, the drag coefficient CD is dependent upon thedroplet Reynolds number, Re. The CD–Re relationship is flow regime-dependent as shown in Fig. 7. It may be determined via the followingequation (Govier and Aziz, 1972):

CD ¼

24R�1e laminar flow regime

30R�0:625e transition flow regime

0:44 turbulent flow regime

8<: ð23Þ

where the droplet Reynolds number is defined as

Re ¼�oVd

�oð24Þ

Under majority of field applications, the droplet Reynolds number isexpected to be greater than 1000, therefore, turbulent flow regimeis anticipated. That is why we recommend use a constant value of0.44 for the drag coefficient.

Figure 6. Droplet size for oil–water flow.

1112 Ouyang



4. IMPACT OF FLUID AND PIPE PARAMETERS ON

THE OCCURRENCE OF COUNTERCURRENT FLOW

The key factors that affect the occurrence of oil–water countercurrentflow, including well internal diameter, well deviation angle, oil densityand oil viscosity, have been investigated. The results will be brieflydiscussed in this section.

Figures 8–11 show the minimum oil flow rates required to prevent theoccurrence of the countercurrent flow under specified well and flow con-ditions as listed in Table 2. Except for the specific parameter investigated,the values shown in Table 2 are used in the calculation for all the otherparameters.

As can be found in Fig. 8, the pipe size has significant impact on theoccurrence of oil–water countercurrent flow. The minimum oil flow raterequired to prevent countercurrent flow increases rapidly with the pipesize. Under the specified flow conditions, the minimum oil rate is around1200 bbl/day for 4 inch well, while it approximately doubles for a 6 inchwell, and triples for a 7 inch well.

Figure 7. Influence of well diameter on occurrence of countercurrent

oil–water flow.




Figure 9. Influence of well deviation on occurrence of countercurrent

oil–water flow.

Figure 8. Influence of well diameter on occurrence of countercurrent

oil–water flow.

1114 Ouyang



Other important factors that affect the occurrence of oil–water coun-tercurrent flow are well deviation angle and oil density (Figs. 9 and 10).This is truly not a surprise considering the fact that gravity is thedominating force that controls countercurrent flow. The higher the well

Figure 10. Influence of oil density on occurrence of countercurrent

oil–water flow.

Figure 11. Influence of oil viscosity on occurrence of countercurrent

oil–water flow.




deviation, the less the gravitational effects, the less likely a countercurrentflow will occur, and thus the lower the minimum oil flow rate required tomaintain the concurrent flow (Fig. 9). On the other hand, the higher theoil density, the less difference between the oil and water density, the lesssevere the gravitational effect, the less likely the countercurrent flowhappens, hence, the lower the minimum oil flow rate needed to avoidthe countercurrent flow.

Compared with the three major factors discussed above, the oil vis-cosity only shows slight influence upon the occurrence of countercurrentflow. As can be seen in Fig. 11, the minimum oil flow rate only increasesfor about 140 bbls/day when the oil viscosity is increased by ten-fold from0.5 to 5.0 cp. Note that oil viscosity is mainly contributing to the stabi-lized water droplet sizes.

5. COMPARISON WITH LABORATORY AND

FIELD TEST RESULTS

The two criteria developed in this technical memorandum have beenapplied to predict the well loading-up conditions and compared with fieldand laboratory observation. There are few experimental data and fieldobservations available in the literature. Fortunately, we did find one inthe article by Zhu and Hill (1988) and some in the article by Turner et al.(1969).

Zhu and Hill (1988) ran a series experiments of oil and water flow ina perforated 7¼ inch pipe at 0, 5, and 15 degree deviations. The oil rateranges from 50 to 5000 bbl/day. And they found that at an oil rate of5000 bbl/day or above, all the water was lifted upwards through a vertical

Table 2. Parameters used for minimum oil

rate calculation.

Parameter Value

Pipe ID (inch) 6.0

Well deviation (deg) 45

Interfacial tension (dyne/cm) 30

Oil density (lbm/ft3) 50

Water density (lbm/ft3) 62.4

Oil viscosity (cp) 2.0

Water viscosity (cp) 1.0

Drag coefficient 0.44

1116 Ouyang



pipe. For oil rate lower than 5000 bbl/day, variable amount of watercould not be lifted. With the parameters given in the article, i.e., oildensity at 783 kg/m3 or 48.86 lbm/ft3, water density at 998.5 kg/m3 or62.4 lbm/ft3, oil viscosity at 2.46 cp, and water viscosity at 1 cp, the mini-mum oil rate required to lift all the water is calculated as 4976 bbl/day,less than 0.5% away from what Zhu and Hill (1988) observed in theirlaboratory tests.

Zhu and Hill (1988) also found that when the pipe was deviated fromvertical by 5 or 15 degrees, some water fell down through the oil stream,even at the highest oil rate and lowest water rate achieved in the experi-ment. Higher oil rates would be required in a deviated well to preventwater fallback than in a perfectly vertical well. This can be explained bythe two criteria described in Sec. 4 in this report. Under given flowconditions, the criterion No. 1—water layer lifting—tends to predict ahigher minimum oil/gas flow rates than those predicted via the criterionNo. 2—water droplet lifting. For vertical wells, the second criterion—water droplet lifting—is expected to be the only controlling water movingmechanisms. For deviated wells, however, both criteria play a role in thewater lifting, therefore, the minimum oil flow rate required for lifting thewater will be higher.

The wells reported in Turner et al. (1969) are mainly gas wells, withpotential production of gas condensate and/or water. Note that onlythose data in Turner et al. (1969) with tubing ID are considered. As wedemonstrated above in Sec. 5, well deviation is an important factor forthe minimum flow rate prediction. Nevertheless, no well deviation anglewas recorded, hence, all the wells are treated as vertical wells here whichwe believe should be a reasonable assumption for wells in the 1950s and1960s.

Three types of well conditions were observed and recorded in thefield, loaded-up, unloaded-up and near loaded-up. In total, there are 39unloaded wells, 5 near loaded-up wells, and 14 loaded-up wells. Most ofthe wells produced gas plus potential production of gas condensateand/or water. For wells with both water and gas condensate present,liquid properties are evaluated as an average weighted by condensatemake and water make.

Figure 12 shows the prediction results where loaded-up wells aredisplayed as squares, unloaded-up wells as triangles, and near loaded-up wells as circles. If the model prediction is 100% accurate, then, all thedata points for near loaded-up wells should be located around the diag-onal straight line; all the data points for unloaded wells should be locatedabove the diagonal straight line, whereas all the data points for loaded-upwells should be located below the diagonal line. It is clearly seen in Fig. 11




that the model does correctly predict all the 39 unloaded wells and does asatisfactory job for near loaded-up and loaded-up wells. All the datapoints for the 39 unloaded wells are indeed located above the diagonalline. The data points for all the 5 near loaded-up wells are located in theclose neighborhood of the diagonal line. And half of data points for the14 loaded-up wells are located below the diagonal line, and the remaininghalf are located around the diagonal line. Detailed statistics is provided inTable 3 below for reference.

In summary, the model makes accurate prediction of water/gas-condensate loading conditions for 51 wells out of a total of 58 (88%),and makes close prediction for the remaining 7 wells. Excellent agreementbetween the model prediction and field observation has been achieved.

Figure 12. Comparison of model prediction with field test data.

Table 3. Comparison between model prediction and field

observation.

Prediction

Field observation Unloaded-up Near loaded-up Loaded-up

Unloaded-up 39 0 0

Near Loaded-up 0 5 0

Loaded-up 0 7 7

1118 Ouyang



6. CONCLUDING REMARKS

Two practical criteria (Eqs. (9) and (20)) based on the transportmechanisms of a liquid layer and a liquid droplet have been developedfor predicting the occurrence of oil–water countercurrent flow in deviatedor multilateral wells. The criteria can also be applied to determine theliquid loading-up conditions of gas wells. Well deviation, well size, andthe oil density are the three major factors that affect the occurrence ofoil–water countercurrent flow. The smaller the wellbore size, the largerthe well deviation, or the denser the oil phase, the lower the minimum oilflow rate is required to avoid the occurrence of oil–water countercurrentflow or water circulation. Excellent agreement has been achieved betweenthe model prediction and the field and laboratory observations for bothoil and gas wells.

A modified equation has also been proposed for determining thewater droplet size in oil–water wells.

ACKNOWLEDGMENT

The author would like to take this opportunity to express our sincerethanks to ChevronTexaco EPTC management for permission to publishthis article.

REFERENCES

Govier, G. W., Aziz, K. (1977). The Flow of Complex Mixtures in Pipes.Van Nostrand, Reinhold, reprinted by Robert E. Kriger PublishingCo., Huntington, New York.

Hasan, N., Kabir, S. (1998). A simplified model for oil–water flow invertical and deviated wellbores. Paper SPE 49163, presented atthe 1998 SPE, Annual Technical Conference & Exhibition.New Orleans, LA, Sept 27–30.

Hinze, J. O. (1955). Fundamentals of the hydrodynamic mechanism ofsplitting in dispersion process. AIChE Journal 1:289–295.

Levich, V. G. (1962). Physicochemical Hydrodynamics. 1st ed. New Jersey:Prentice-Hall, Englewood Cliffs.

Ouyang, L. B. (2000). A mechanistic model based approach to evaluateoil/water slip at horizontal or highly-deviated wells. Paper SPE63262, presented at the 2000 SPE Annual Technical Conferenceand Exhibition, Dallas, TX, Oct 1–4.




Ouyang, L. B. (2002). Mechanistic and simplified models for oil–watercountercurrent flow in deviated and multilateral wells, Paper SPE77501, presented at the 2002 SPE Annual Technical Conference andExhibition, San Antonio, TX, Sept 29–Oct 2.

Taitel, Y., Barnea, D., Dukler, A. E. (1980). Modeling flow patterntransitions for steady upward gas–liquid flow in vertical tube.AIChE Journal 26(3):345–354.

Trallero, J. L. (1995). Oil–Water Flow Patterns in Horizontal Pipes. Ph. Dthesis, Department of Petroleum Engineering, University of Tulsa,Tulsa, OK.

Turner, R. G., Hubbard, M. G., Dukler, A. E. (1969). Analysis andprediction of minimum flow rate for the continuous removalof liquid from gas wells. Journal of Petroleum Technology21(11):1475–1482.

Zhu, D., Hill, A. D. (1988). The effect of flow from perforations on two-phase flow: implications for production logging. Paper SPE 18207,presented at the 1988 SPE Annual Technical Conference &Exhibition, Houston, TX, Oct 2–5.

Received March 25, 2002Accepted June 25, 2002

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