MATHEMATICAL MODELING OF HORIZONTAL TWO-PHASE FLOW THROUGH FULLY ECCENTRIC ANNULI

ÇİĞDEM ÖMÜRLÜ

MAY 2006

MATHEMATICAL MODELING OF HORIZONTAL TWO-PHASE FLOW THROUGH FULLY ECCENTRIC ANNULI

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

ÇİĞDEM ÖMÜRLÜ

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

PETROLEUM AND NATURAL GAS ENGINEERING

MAY 2006

Approval of the Graduate School of Natural and Applied Sciences

Prof. Dr. Canan Özgen

Director I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science.

Prof. Dr. Mahmut Parlaktuna

Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science

Assist. Prof. Dr. M. Evren Özbayoğlu Supervisor

Examining Committee Members Prof. Dr. Mahmut Parlaktuna (METU,PETE)

Assist. Prof. Dr. M. Evren Özbayoğlu (METU,PETE)

Prof. Dr. Tanju Mehmetoğlu (METU,PETE)

Assoc. Prof. Dr. Serhat Akin (METU,PETE)

Assoc. Prof. Dr. İ. Hakkı Gücüyener (TPAO)

iii

PLAGIARISM

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last name : ÇİĞDEM ÖMÜRLÜ

Signature:

iv

ABSTRACT

MATHEMATICAL MODELING OF HORIZONTAL TWO-PHASE FLOW

THROUGH FULLY ECCENTRIC ANNULI

Ömürlü, Çiğdem

M.Sc., Department of Petroleum and Natural Gas Engineering

Supervisor: Assist. Prof. Dr. M. Evren Özbayoğlu

May 2006, 107 pages

The primary objective of this study is to understand the mechanism, the

hydraulics and the characteristics, of the two-phase flow in horizontal annuli.

While achieving this goal, both theoretical and experimental works have been

conducted extensively. The METU-PETE-CTMFL (Middle East Technical

University, Petroleum and Natural Gas Engineering Department, Cuttings

Transport and Multiphase Flow Laboratory) multiphase flow loop consists of

4.84 m long eccentric horizontal acrylic pipes having 0.1143m inner diameter

(I.D) acrylic casing - 0.0571m outer diameter (O.D) drillpipe and 0.0932m I.D

acrylic casing - 0.0488m O.D drillipipe geometric configurations. During each

experiment, differential pressure loss data obtained from digital and analog

pressure transmitters at a given liquid and gas flow rate were recorded. The

v

flow patterns were identified visually. Meanwhile a mechanistic model has been

developed. The flow pattern identification criteria proposed originally for two-

phase flow through pipes by Taitel and Dukler1 has been inherited and modified

for the eccentric annular geometry. The complex geometry of eccentric annuli

has been represented by a new single diameter definition, namely

representative diameter dr. The representative diameter has been used while

calculating the pressure losses. A computer code based on the algorithm of the

proposed mechanistic model has been developed in Matlab 7.0.4. Both the flow

pattern prediction and the frictional pressure loss estimation are compared with

the gathered experimental data. Moreover, friction factor correlations have

been developed for each flow pattern using experimental data and statistical

methods. The performance of the proposed model and the friction factor

correlations has been evaluated from experimental data. The mechanistic

model developed in this study accurately predicts flow pattern transitions and

frictional pressure losses. The model’s pressure loss estimations are within

± 30% for two different annular flow geometries.

Keywords: Two-phase Flow, Frictional Pressure Loss, Friction Factor,

Mechanistic Model, Eccentric Annulus, Multiphase Experiment

vi

ÖZ

TAM EKSENTRİK HALKASAL ORTAMDA YATAY İKİ FAZLI AKIŞIN

MATEMATİKSEL MODELLENMESİ

Ömürlü, Çiğdem

Y.Lisans, Petrol ve Doğal Gaz Mühendisliği Bölümü

Tez Yöneticisi: Y.Doç. Dr. M. Evren Özbayoğlu

Mayıs 2006, 107 sayfa

Bu çalışmadaki başlıca hedef, iki fazlı akışın yatay halkasal ortamlardaki akış

mekanizmasını, hidroliğini ve karakteristiğini anlamaktır. Bu amaca ulaşırken

yoğun bir şekilde hem teorik hem de deneysel çalışmalar yapılmıştır. ODTÜ-

PETE-KTÇFAL (Orta Doğu Teknik Üniversitesi Petrol ve Doğal Gaz Mühendisliği

Bölümü Kesinti Taşıma ve Çok Fazlı Akış Laboratuvarı) iki fazlı akış deney

düzeneği 0.1143m iç çaplı akrilik muhafaza borusu - 0.0571m dış çaplı sondaj

borusu ve 0.0932m iç çaplı akrilik muhafaza borusu - 0.0488m dış çaplı sondaj

borusu geometrik özellikteki yatay halkasal ortamdan oluşmaktadır ve uzunluğu

4.84m’dir. Deneyler sırasında çeşitli gaz ve sıvı akış debilerinde oluşan basınç

kayıpları dijital ve analog olarak kaydedilmiştir. Akış biçimleri ise görsel olarak

tespit edilmiştir. Bir yandan da mekanistik model oluşturulmuştur. Borulardaki

vii

iki fazlı akış için Taitel ve Dukler1 tarafından geliştirilen akış biçimi tayini

kriterleri tercih edilmiş ve tam eksentrik halkasal ortama uyarlanmıştır. Eksentrik

halkasal ortamın karmaşık geometrisi tek bir yeni çap terimi, temsili çap dr ile

temsil edilmiştir. Temsili çap, basınç kayıpları hesaplamaları sırasında da

kullanılmıştır. Matlab 7.0.4 programı kullanılarak oluşturulan mekanistik modele

dayalı bilgisayar kodu yazılmıştır. Hem akış biçimi tayinleri hem de sürtünme

kaynaklı basınç kayıpları hesaplamaları deneysel verilerle karşılaştırılmıştır.

Ayrıca, her bir akış biçimi için sürtünme faktörü bağıntıları istatistiksel yöntem

kullanılarak oluşturulmuştur. Oluşturulan modelin ve sürtünme faktörü

bağıntılarının performansı deneysel veriler kullanılarak değerlendirilmiştir. Bu

çalışmada oluşturulan mekanistik model akış biçimleri geçişlerini ve basınç

kayıplarını doğru bir şekilde tayin edebilmektedir. Basınç kayıpları ± 30% hata

sınırları arasındadır.

Anahtar Kelimeler: İki Fazlı Akış, Sürtünme Basınç Kayıpları, Sürtünme Faktörü,

Mekanistik Model, Eksentrik Halkasal Ortam, Çok Fazlı Akış Deneyi

viii

To My Family

ix

ACKNOWLEDGMENTS

I would like to thank my advisor Assist.Prof.Dr. M. Evren Özbayoğlu for his

valuable support, guidance and encouragement during this study. I would like

to thank also to Naci Doğru and Ali Osman Atik for their help during the

construction of the experimental setup. Without all these assistances and helps,

this work would have not been accomplished. My thesis committee members

Prof. Dr. Tanju Mehmetoğlu, Prof. Dr. Mahmut Parlaktuna, Assoc. Prof. Dr. İ.

Hakkı Gücüyener, Assoc. Prof. Dr. Serhat Akın, and Assist. Prof Dr. M. Evren

Özbayoğlu are very appreciated for their comments and suggestions.

Finally my special and sincere thanks go to my whole family for their endless

love and support, to my grandmother who stands always by me with her great

understanding and prayers, to my father and mother for their care and

tolerance, to Tolga Metin who is always beside me with joy and love along with

kindness and never-ending support, and help, to my aunt for great

encouragement and guidance, to my sister İpek Ömürlü and my cousin Berk

Gercek, and to my friends İlkay Uzun and Sevtaç Bülbül for their supports,

encouragements and considerations. I would like to thank to Tolga Metin once

more for drawing the excellent figures in this thesis.

x

TABLE OF CONTENTS

PLAGIARISM .............................................................................................. III

ABSTRACT ..................................................................................................IV

ÖZ..............................................................................................................VI

ACKNOWLEDGMENTS..................................................................................IX

TABLE OF CONTENTS ................................................................................... X

NOMENCLATURE .......................................................................................... 1

CHAPTER .................................................................................................... 5

1.INTRODUCTION...................................................................................... 5

2.LITERATURE REVIEW.............................................................................. 8

2.1 Models for two-phase flow through pipe ............................................ 8

2.2 Models for two-phase flow through annuli ....................................... 13

3.STATEMENT of THE PROBLEM ............................................................... 18

4.SCOPE of THE STUDY............................................................................ 20

5.THEORY ............................................................................................... 21

xi

5.1 Flow Pattern Prediction................................................................... 21

5.1.1 Stratified Flow to Non-Stratified Flow Transition......................... 27

5.1.2 Intermittent Flow to Annular Flow Transition ............................. 29

5.1.3 Intermittent Flow to Dispersed Bubble Flow Transition ............... 31

5.1.4 Stratified Smooth Flow to Stratified Wavy Flow Transition .......... 33

5.2 Determination of Frictional Pressure Loss ........................................ 33

5.2.1 Stratified Flow ......................................................................... 33

5.2.2 Intermittent Flow ..................................................................... 36

5.2.3 Annular Flow ........................................................................... 42

5.2.4 Dispersed Bubble Flow ............................................................. 45

6.EXPERIMENTAL WORK .......................................................................... 47

6.1 Experimental Setup ........................................................................ 47

6.2 Test Section................................................................................... 51

6.3 Calibration Process......................................................................... 53

6.4 Experimental Test Procedure and Data Acquisition ........................... 55

7.COMPUTER WORK................................................................................. 57

8.RESULTS and DISCUSSION.................................................................... 59

xii

8.1. Validation of Flow Pattern Identification of Proposed Model with

Experimental Data ............................................................................... 59

8.2 Validation of Frictional Pressure Loss Estimations of Proposed Model

with Experimental Results .................................................................... 65

8.3 Empirical Friction Factor Correlations............................................... 73

9.CONCLUDING REMARKS ........................................................................ 80

RECOMMENDATIONS.................................................................................. 83

REFERENCES.............................................................................................. 85

APPENDIX.................................................................................................. 90

A.1 Modified Petalas and Aziz Model...................................................... 90

A.2 Modified Garcia et al Model............................................................. 95

A.3 Modified Beggs and Brill Model ....................................................... 96

A.4 Stratified-Intermittent Flow Transition ............................................. 97

A.5 Pictures Taken During Experiments ............................................... 101

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LIST OF FIGURES

Figure 5.1.1- Cross sections of pipe of representative diameter dr and annuli

with diameters of di and do ......................................................................... 23

Figure 5.1.2- Geometrical parameters for fully eccentric annuli...................... 24

Figure 5.1.1.1- Stratified Flow ..................................................................... 27

Figure 5.1.2.1- Intermittent Flow................................................................. 30

Figure 5.1.2.2- Annular Flow ....................................................................... 30

Figure 5.1.3.1- Dispersed Bubble Flow ......................................................... 31

Figure 6.1.1- Schematic view of the experimental setup................................ 48

Figure 6.1.2- Two-phase separator .............................................................. 49

Figure 6.1.3- Electropneumatic control valve ................................................ 50

Figure 6.2.1- Test section ........................................................................... 52

Figure 6.3.1- Frictional pressure loss gradient versus flow rate data of water

flowing through configuration 1................................................................... 54

Figure 6.3.2- Frictional pressure loss gradient versus flow rate data of water

flowing through configuration 2................................................................... 54

Figure 7.1- Flow chart of Matlab code for the flow pattern identification and

frictional pressure loss determination........................................................... 58

Figure 8.1.1- Comparison of flow pattern maps generated using dhyd and GDM

for configuration 1...................................................................................... 60

Figure 8.1.2- Comparison of flow pattern maps generated using dhyd and GDM

for configuration 2...................................................................................... 61

Figure 8.1.3- Comparison of flow pattern maps generated using dhyd and OOM

for configuration 1...................................................................................... 63

xiv

Figure 8.1.4- Comparison of flow pattern maps generated using dhyd and OOM

for configuration 2...................................................................................... 63

Figure 8.1.5- Validation of flow pattern maps generated using dr and modified

Beggs and Brill3 method with experimental data for configuration 1 ............... 64

Figure 8.1.6- Validation of flow pattern maps generated using dr and modified

Beggs and Brill3 method with experimental data for configuration 2 ............... 65

Figure 8.2.1- Comparison of frictional pressure loss estimations of the proposed

model with experimental data and mostly used models for stratified flow

through configuration 1 .............................................................................. 66

Figure 8.2.2- Comparison of frictional pressure loss estimations of the proposed

model with experimental data and mostly used models for stratified flow

through configuration 2 .............................................................................. 67

Figure 8.2.3- Comparison of frictional pressure loss estimations of EPDM with

experimental data and mostly used models for intermittent flow through

configuration 1 ........................................................................................... 69

Figure 8.2.4- Comparison of frictional pressure loss estimations of EPDM with

experimental data and mostly used models for intermittent flow through

configuration 2 ........................................................................................... 69

Figure 8.2.5- Comparison of frictional pressure loss estimations of OM with

experimental data and mostly used models for intermittent flow through

configuration 1 ........................................................................................... 71

Figure 8.2.6- Comparison of frictional pressure loss estimations of OM with

experimental data and mostly used models for intermittent flow through

configuration 2 ........................................................................................... 71

Figure 8.3.1- Friction factor and mixture Reynolds number relation of

experimental stratified flow data ................................................................. 75

xv

Figure 8.3.2- Friction factor and mixture Reynolds number relation of

experimental intermittent flow data ............................................................. 76

Figure 8.3.3- Friction factor and mixture Reynolds number relation of

experimental intermittent flow data for NRemixλ < 100000............................. 77

Figure 8.3.4- Friction factor and mixture Reynolds number relation of

experimental intermittent flow data for NRemixλ ≥ 100000 ............................ 78

Figure 8.3.5- Comparison of pressure losses determined by the empirical

correlations and experimental data for stratified flow.................................... 79

Figure 8.3.6- Comparison of pressure losses determined by the empirical

correlations and experimental data for intermittent flow ............................... 79

Figure A.1- The analysis of the forces during wave growth in the conduit ...... 98

Figure A.2- Stratified smooth flow through configuration 1.......................... 101

Figure A.3- Stratified wavy flow through configuration 1 ............................. 101

Figure A.4- Stratified wavy flow through configuration 1 ............................. 102

Figure A.5- Intermittent flow through configuration 1 ................................. 102

Figure A.6- Intermittent flow through configuration 1 ................................. 103

Figure A.7- Intermittent flow through configuration 1 at high liquid and gas flow

rates........................................................................................................ 103

Figure A.8- Intermittent flow through configuration 1 at high liquid and gas flow

rates........................................................................................................ 104

Figure A.9- Stratified smooth flow through configuration 2.......................... 104

Figure A.10- Stratified smooth flow through configuration 2 ........................ 105

Figure A.11- Stratified wavy flow through configuration 2 ........................... 105

Figure A.12- Stratified wavy flow through configuration 2 ........................... 106

Figure A.13- Intermittent flow through configuration 2 ............................... 106

Figure A.14- Intermittent flow through configuration 2 ............................... 107

xvi

LIST OF TABLES

Table 6.1.1- Geometrical configuration of annular section ............................. 48

Table 6.1.2- Capacity and brand name of experimental components.............. 51

Table 8.2.1- Error percentage for pressure loss estimation of mostly proposed

and modified models .................................................................................. 72

1

NOMENCLATURE

A area (L2)

AN Annular Flow

Bo Bond number

cg constant for gas phase

cl constant for liquid phase

DB Dispersed Bubble Flow

dC critical gas bubble diameter (L)

dCB critical bubble size below which bubbles can not migrate (L)

dCD critical bubble size above which the bubble is deformed (L)

dg hydraulic diameter for gas phase(L)

dhyd hydraulic diameter (L)

di outer diameter of inner pipe, drillpipe (L)

dl hydraulic diameter for liquid phase(L)

do inner diameter of outer pipe, casing (L)

dr representative diameter (L)

El liquid volume fraction in the slug unit

Els liquid volume fraction in the slug body

f friction factor

f friction factor

FE entrainment fraction

ff friction factor

g gravitational acceleration (L/T2)

g gravitational acceleration (L/T2)

2

GVF gas void fraction

Hl Liquid holdup

hl liquid level (L)

I Intermittent Flow

Lentrance length required from entrance for fully developed flow (L)

Lexit length required from exit for fully developed flow (L)

Lf length of gas zone (L)

Ls length of liquid slug body (L)

Lu length of a slug unit (L)

m constant for gas phase

n constant for liquid phase

NB dimensionless term

P pressure (M /(L T2))

Re Reynolds number

S contact perimeter (L)

s sheltering coefficient

SS Stratified Smooth Flow

SW Stratified Wavy Flow

v velocity (L/T)

z axial direction

l

l

dA

dh derivative of liquid area with respect to liquid level

LFrN Froude number

Re∞ high Reynolds number

P

L

∆

∆ pressure gradient (M/T2L2)

3

Subscripts

c core

d drift

d∞ drift at high Reynolds numbers

f film

ff formation fracture

fgp frictional pocket/liquid film

fsL frictional slug

g gas pocket

g related with the gas phase

i related with the interface

l related with the liquid phase

lf liquid film

ls liquid slug

m mixture

mixλ mixture based on liquid holdup

ml mixture related with liquid phase properties

r representative

sg superficial gas

sl superficial Liquid

t translational

wg wall gas

wl wall liquid

4

Greek

δ dimensionless liquid film thickness

Τµ weighting factor

δ liquid film thickness (L)

φ angle

η weighting factor

λl no-slip holdup

µ viscosity (M /(L T))

θ angle

ρ density (M/L3)

σ interfacial surface tension (M/T2)

τ shear stress (M /(L T2))

5

CHAPTER 1

INTRODUCTION

Two-phase flow is the flow phenomenon of two different fluid phases flowing

simultaneously through a conduit. Generally, liquid and gas phases are the

components of this commonly encountered flow type. Since 1950’s, the flow

mechanism of two-phase fluids has been the subject of research in many

different engineering practices. In petroleum industry, the applications of two-

phase flow start from drilling and continue till the refining process.

In depleted reservoirs, underbalanced drilling techniques are required. In this

integrated technology the drilling fluid pressure is less than the pore pressure in

the formation rock. Therefore, the balance between the borehole pressure and

formation pore fluid pressure is established. Air, gas, foam and aerated water

are the light fluids usually used during underbalanced drilling applications. The

number of wells drilled using this technology is increasing as a result of the

advantages of underbalanced drilling. Increased penetration rate, minimized

circulation loss especially in naturally fractured or pressure depleted reservoirs,

prolonged bit life, minimized differential sticking, improved formation

evaluation, reduced formation damage and environmental benefits are among

the main advantages of underbalanced drilling. This enhanced technology

6

diminishes the risks of contaminating the reservoir and eliminates the potential

pollution of drilling mud to environment2.

The hole cleaning efficiency is an important criterion that should be well

determined during drilling operations. The carrying capacity, the ability of

transporting the drilled particles to surface, is one of the major roles of drilling

fluids. Especially during underbalanced drilling operations in horizontal and

deviated wells, flow behavior of the two-phase fluid should be well determined

in order to improve the hole cleaning efficiency. Otherwise, an improper hole

cleaning may result in differential pipe sticking, increased torque and drag and

hence a severe economical loss. The efficiency of the hydraulic program during

drilling operations and the economical success of the operation are directly

related to the better understanding of cuttings transport phenomenon. The

minimum volumetric flow rates and the liquid and gas interface distributions are

the most important requirements for hole cleaning efficiency3. Therefore, an

accurate two-phase flow model is essential for an improved description of

transportation of cuttings from the wellbore to the surface.

Another important usage of two-phase flow takes place during the

transportation of the produced oil and gas via the pipelines. Since oil and gas

fields are mainly in remote onshore areas or in offshore, pipeline systems are of

great importance. As the demand for oil and gas in Asia and Europe is

increasing the Caspian region oil reserves are becoming more important.

Therefore pipelines are establishing, i.e., BTC (Baku-Tiflis-Ceyhan). Reliable

engineering calculations should be carried out as the overall distances of these

pipeline systems are considered. With the improving technology, innovative

7

methods provide more accurate results with better understanding of two-phase

flow systems.

When this wide range of application of two-phase flow in petroleum

engineering is considered, the importance of the appropriate determination of

flow parameters of two-phase fluid systems is remarkable. Several studies4-26

have been carried out for understanding the flow mechanism of two-phase fluid

systems through pipe. However limited researches3,27-35 are conducted for

annular two-phase flow.

This study attempts to propose a mechanistic mathematical model developed

for two-phase flow through horizontal fully eccentric annulus. Flow pattern

identification and frictional pressure loss determination procedures are

presented accordingly. The performance of the model is compared with the

experimental data collected from METU-PETE-CTMFL (Middle East Technical

University, Petroleum and Natural Gas Engineering Department, Cuttings

Transport and Multiphase Flow Laboratory) multiphase flow loop. Furthermore,

friction factor correlations are developed using the experimental data obtained

from same multiphase flow loop.

8

CHAPTER 2

LITERATURE REVIEW

Through the investigation of two-phase flow phenomenon, extensive theoretical

and experimental studies have been carried out. The early models developed

for two-phase fluid systems were flow pattern independent. These models

ignored the complex flow configurations, named as flow patterns, and treated

the two-phase flow as a single-phase fluid flow or as a flow of two separated

fluids. Wallis4, Lockhart and Martinelli5, and Duns and Ros6 models are among

the most important models that are the starting points through the progress of

modeling two-phase fluid flow. Recent studies focused on the determination of

flow patterns. The flow mechanism of two-phase fluid systems was examined

independently for each flow pattern. Then, governing flow equations were

proposed for a given flow pattern. These models were called mechanistic

models. As the knowledge of flow behavior of two-phase fluid systems has

improved, comprehensive and unified mechanistic models were developed.

2.1 Models for two-phase flow through pipe

In the “Homogeneous No-Slip Flow Model”, introduced by Wallis3, the two-

phase mixture wais treated as a pseudo single-phase fluid with average velocity

and physical properties. The physical properties of two-phase system were

9

determined from single-phase gas and liquid properties through the liquid

holdup.

The opposite approach was taken in the “Separated Flow Model” proposed by

Lockhart and Martineli5. In this model the gas phase and the liquid phase were

assumed to flow separately from each other. Thus, each of the phases was

analyzed utilizing single-phase flow methods such as friction factor concept.

Four flow mechanisms were established and transition criteria between these

flow mechanisms were suggested. Curves were presented for the prediction of

pressure drop and the liquid level in the pipe.

“Dimensional Analysis”, introduced by Duns and Ros6, was a powerful technique

to develop universal solutions from experimental data. This was achieved by

generating governing dimensionless groups that control a given flow system. It

has been applied successfully to various single-phase flow problems. However,

in two-phase flow, due to the large number of variables involved, dimensional

analysis could not be applied in a straightforward manner. Additional

assumptions were required to reduce the number of dimensionless groups.

“The Drift Flux Model”, developed by Wallis3, treated the two phases as a

homogeneous mixture. However, it allowed slippage between the gas and the

liquid phases. This was a significant improvement of the homogeneous model.

However, additional information was required about the relative movement of

the two phases and, this information was not always available.

10

Correlations were the most common procedure to predict the flow properties in

annulus, in the past. The predictions either have applied correlations developed

for flow in pipes by use of the hydraulic diameter concept or have applied

correlations developed from experimental data. Baxendell7, Gaither et al.8,

Angel and Welchon9, and Winkler10 presented empirical correlations for annular

two-phase flow.

Dukler et al.11 proposed a model based on dimensionless groups and developed

frictional pressure loss correlations using similarity analysis. This model

underestimated the liquid holdup. By using correlations proposed by Eaton and

Brown12, more accurate liquid holdup could be calculated. In the case of

inclined pipe flow, Flanigan 3 proposed a correlation for the determination of

the gravitational effects.

Beggs and Brill14 studied two-phase flow through pipes in an entire range of

inclination angles, i.e., vertical upward, horizontal and vertical downward. They

developed correlations for flow pattern determination by using Froude number

and no-slip holdup. The proposed model estimated the actual holdup and the

pressure loss for each flow pattern separately. Dukler et al11’s method was

referred to calculate the two-phase friction factor.

Taitel and Dukler1 analyzed the prediction of transition boundaries between the

flow patterns. The model begins with the equilibrium stratified flow assumption.

The equilibrium liquid holdup is determined by using the Lockhart and Martinelli

parameter. They modified the Kelvin-Helmholtz inviscid theory in order to

11

predict the initiation of slugs. The transition of intermittent to annular flow is

assumed to be dependent only on liquid level. Jeffrey’s theory for wave

initiation is used to determine the transition of stratified smooth to stratified

wavy flow pattern. Turbulent and buoyant forces acting on a gas pocket are

investigated for the boundary between dispersed bubble flow and intermittent

flow. The transition conditions were also expressed as dimensionless

parameters. This model, developed for Newtonian flow, was verified with the

experiments conducted in small diameter pipes under low-pressure conditions.

Dukler and Hubbard15 investigated the mechanisms and the hydraulic behavior

of slug flow. An idealized slug unit concept was introduced. Two main zones

that constituted a slug unit were defined as a liquid slug and stratified liquid

film/gas pocket. The proposed model estimated the liquid holdup, pressure loss

and velocity distributions within the slug unit.

Barnea16 studied the transition mechanisms for each individual boundary and

proposed a unified model. The flow chart of the model begins with the

dispersed bubble transition. The applicability of the developed mechanisms was

presented for the whole range of pipe inclinations. The results were compared

with the experimental data. The effects of flow rates, fluid properties, and pipe

size and inclination angle were incorporated in dimensionless maps.

Xiao et al17 developed a comprehensive mechanistic model for two-phase flow

in horizontal and near horizontal pipes. The flow pattern transitions were

inherited from Taitel and Dukler’s1, and Barnea et al’s18 models. Two different

methods were applied for calculating interfacial friction factor in stratified flow,

12

i.e., Baker et al19 and Andritsos and Hanratty20. Also the effect of pipe

roughness was taken into consideration during friction factor calculations.

Uniform liquid level in the film zone was assumed for intermittent flow.

Empirical correlations were used in order to predict the slug length and liquid

holdup of the slug body. Annular flow was treated as stratified flow with

different geometrical configuration. Liquid entrainment fraction (Oliemans et

al21) was also considered while calculating the liquid holdup in the gas core.

Gomez et al22 developed a unified mechanistic model for horizontal to vertical

upward flow of two-phase fluid systems. Unified transition flow pattern

prediction model and unified individual models for each flow type were

presented. Moreover, the proposed model implemented new criteria in order to

eliminate the discontinuity problems. The flow mechanisms in the flow pattern

boundaries were inherited from the models of Taitel and Dukler1, and Barnea16.

Empirical correlations were used for determination of several flow properties

such as, liquid/wall friction factor (Ouyang and Aziz23), entrainment fraction

(Wallis4) etc.

Petalas and Aziz24 proposed a mechanistic model applicable to a wide range of

pipe geometries and fluid properties. Empirical correlations were developed for

interfacial friction in stratified and annular-mist flows, for liquid entrainment

fraction and distribution coefficient in intermittent flow. A large amount of

experimental and field data were collected in order to develop these empirical

correlations. The transition mechanisms between the flow patterns were

presented in a similar way to Taitel and Dukler’s1, and Barnea’s16 models.

13

Garcia et al25 studied a large amount of two-phase flow data and developed

composite analytical expressions for friction factor covering both laminar and

turbulent flow regimes. Two different approaches were presented. The first

method is the universal composite correlation for friction factor estimation

regardless of the flow pattern. The second method represents the friction factor

correlations for a given flow pattern. This Fanning friction factor definition was

based on the mixture velocity and density.

Theofanous and Hanratty26 published a report summary of the study group on

flow regimes in multiphase fluid flow. In their work, the importance of

experiments, pattern-revealing diagnostics and the computer programs

developed for pattern identification was emphasized.

2.2 Models for two-phase flow through annuli

Sadatomi et al27 developed flow pattern maps for air-water flow through

vertical annuli. They determined the slug interval by considering the slug

frequency and the gas phase velocity. Then the flow pattern transitions were

estimated from the slug intervals. They developed flow pattern maps for

vertical air and water mixture through various noncircular conduits including

concentric annuli. From these flow pattern maps, they concluded that the

channel geometry has very little effect on the flow pattern transitions.

Hasan and Kabir28 conducted two-phase flow experiments in inclined annular

geometries. They studied air and water systems and developed a hydrodynamic

model in order to estimate the gas void fraction in slug flow and bubbly flow.

14

They concluded that for a small ratio of diameters of casing and tubing, the gas

void fraction in vertical annular conduit was similar to the gas void fraction in

circular pipes.

Caetano et al29 investigated the upward vertical flow of two-phase fluid systems

through an annulus. This experimental study covered the flow of water-air and

kerosene-air mixtures through concentric and eccentric annuli. The flow

patterns were identified visually. They proposed a mechanistic model for flow

pattern prediction by applying the Taitel and Dukler’s1 model to annular

concentric and eccentric geometries. Moreover, average liquid holdup and

pressure loss determination methods were developed. The effect of fluid

properties was observed as a result of the comparison of the developed flow

pattern maps for water-air and kerosene-air mixtures. Experiments were

conducted in a small scale experimental setup.

Salcudean et al30,31, investigated the effect of flow obstruction geometry on

pressure drops in horizontal two-phase flow. The effect of central and

peripheral obstructions on flow pattern transitions was also studied by

conducting experiments. The central obstruction corresponded to annular

geometry. They studied the effect of obstructions on void distribution of water

and air horizontal flow. They concluded that the central obstruction has the

strongest effect on stratified wavy-intermittent flow and stratified smooth-

stratified wavy flow transitions. This observation showed the importance of

developing a separate model for two-phase flow through annular conduit.

Moreover, they noted that the use of flow pattern determination models

developed for pipe flow may lead to inaccurate results.

15

Sunthankar32 conducted experiments in a field scale experimental setup for

water–air mixture through horizontal annuli. He modified the unified model

developed by Xiao et al.17 to predict the flow patterns for horizontal and nearly

horizontal annular flow. Hydraulic diameter concept was used while modifying

the model. The effects of eccentricity and inner pipe rotation were also

investigated. A simulator was developed based on the work of Gomez et al22 for

pressure loss estimations. The performance of the proposed model was

evaluated with the experimental data. It was concluded that the intermittent

flow was different than that was defined for pipe flow, i.e., the Taylor bubble

was distorted and the liquid slug was highly aerated. The developed flow

pattern maps showed shifts when compared with the flow pattern transition

boundaries of pipe flow.

Zhou33 studied cuttings transport with aerated mud in horizontal annulus under

elevated pressures and temperatures. Taitel and Dukler’s1 model was modified

for annular two-phase flow. A mechanistic model was developed to predict the

volumetric cuttings concentration in the annuli and the critical pressure gradient

for preventing cuttings from deposition. Experiments were also conducted

during this study in order to verify the accuracy of developed model. The

predictions of developed mechanistic model were in agreement with measured

data. It was concluded that liquid flow rate, gas liquid ratio and temperature

essentially affected the cuttings transport efficiency. Comparisons between

predictions and measurements for aerated mud flow showed an average error

of 12.2%.

16

Rodriguez3 carried out an experimental study in order to find the minimum air

and water flow rates that effectively transport cuttings through highly inclined

and horizontal wells. The experiments were carried out in a low pressure field

scale flow loop. The model proposed in the study of Sunthankar32 was inherited

for flow pattern identification and pressure loss determination. The model’s

results were compared with experimental data. It was concluded that the flow

patterns of cuttings are dependent on the total flow rate of the liquid and gas

phase. It was also concluded that in order to avoid the formation of a stationary

cuttings bed, an approximate boundary of minimum flow rate of each phase

can be determined. The minimum requirements for gas and liquid flow rates

were found to be always in the intermittent flow regime.

Gücüyener34 developed a multiphase hydrodynamic model for flow pattern

identification and pressure loss determination through drill string and annulus in

vertical and moderately deviated directional wells. The carrying capacity of the

aerated drilling fluid was evaluated by using two-phase flow properties and a

cuttings transport model. Moreover, a computer program was developed for the

prediction of flow patterns, circulating pressures, optimum two-phase flow

requirements, bit hydraulics and hole cleaning. It was concluded that dispersed

bubble flow did not develop in the drill string and the annulus, and that the

multiphase models calculated lower bottomhole pressures compared to

dispersed model.

Lage et al35 conducted an experimental and theoretical study on two-phase flow

in horizontal or slightly deviated fully eccentric annuli. Flow pattern prediction

and gas fraction and pressure drop calculation procedures were presented.

17

Flow pattern data and pressure drop measurements were compared with the

mechanistic models predictions. The results showed good agreement even

though the number of data points did not permit the development of a

complete and precise flow pattern map. The model performance was also

compared with Beggs and Brill14 correlation and modified Aziz et al36 method. It

was concluded that proposed model had better performance.

As it is remarked from literature review, few studies have been conducted on

two-phase flow in annular geometries especially for horizontal fully eccentric

annular conduit. In this study, a mechanistic model for accurate determination

of flow patterns and frictional pressure losses of two-phase systems through

fully eccentric horizontal annuli was proposed.

18

CHAPTER 3

STATEMENT of THE PROBLEM

Two-phase flow is a common aspect encountered in many major industrial

fields, i.e., aerospace, automotive, nuclear, and petroleum industries. The

better understanding of the flow mechanism of two-phase flow leads to more

accurate engineering solutions which can be sited as the design of steam

generators, internal combustion engines, cooling towers, and pipelines for

transport of gas and oil mixtures. As the emphasis is given to the petroleum

industry, numerous applications of two-phase flow come upon. During the

transportation of produced oil and gas, the two-phase flow occurs in horizontal,

inclined or vertical pipes. In offshore production, these lines can be of

substantial lengths before reaching separation facilities. Throughout drilling

stage of petroleum industry, the geometry of the flow conduit is no more

circular but concentric or eccentric annuli with inclinations from 00 to 900

(vertical to horizontal). Extensive researches have been conducted for

understanding the flow mechanism of two-phase flow through circular pipes.

Generally, hydraulic diameter is used to adapt these models to annular flow

geometries in order to explain the two-phase flow behavior in annulus.

However, the applicability of this method is questionable since many studies

showed mismatch between observed and calculated results when hydraulic

diameter is used. Therefore, experimental and theoretical studies are required

19

to comprehend the flow behavior of two-phase flow through horizontal annular

conduits using techniques different than hydraulic diameter.

20

CHAPTER 4

SCOPE of THE STUDY

The scope of this study is to develop analytical equations using fundamental

laws of physics and mathematics to predict the flow behavior, flow patterns and

their transition boundaries for two-phase flow through annular geometries.

Initially, an extensive literature review is conducted for understanding the two-

phase flow fundamentals. Then a new representative diameter approach is

introduced based on the equivalency of flow area, rather than applying

hydraulic diameter concept. With the purpose of determining the flow patterns

and the frictional pressure losses, a mechanistic model is developed using the

representative diameter concept. During the progress of the mathematical

model, experimental data acquired from METU-PETE-CTFL multiphase flow loop

is integrated. Flow pattern and pressure loss estimations are compared with the

experimental data and hence the model’s performance is evaluated. Geometry

independent model for transition of the flow pattern boundaries is obtained.

Moreover, empirical equations are proposed for friction factor determination

corresponding to each flow pattern individually and mutually as well.

21

CHAPTER 5

THEORY

In this chapter, the flow pattern determination and the pressure loss estimation

methods are presented in details. New approaches are developed hence

yielding in a mechanistic model for both flow pattern identification and frictional

pressure loss estimation of horizontal two-phase flow through eccentric annuli.

5.1 Flow Pattern Prediction

Accurate mapping of the flow patterns is the first step for determination of the

frictional pressure losses correctly. Major concern in mechanistic modeling is

the determination of flow patterns accurately. As discussed in previous section

most of the studies carried out inherited the flow pattern transition definitions

proposed by Taitel and Dukler1 and Barnea16. Since the flow area is fully

eccentric annuli; hydraulic diameter (Equation 1) approach yields to significant

errors during flow pattern determination.

hyd o id d d= − (1)

Therefore areal representative diameter approach is proposed. In this study,

representative diameter, dr is defined by

22

( )2 2

r o id d d= − (2)

The liquid holdup, Hl is determined from Lockhart and Martinelli5 parameter

using the superficial liquid (Equation 3) and superficial gas pressure gradients

(Equation 4) and the areal representative diameter, dr.

22

Re

sl

sl

l l

n

sl r

c vP

L d

ρ∆=

∆ (3)

22

Re

sg

sg

g g

m

sg r

c vP

L d

ρ∆=

∆ (4)

where 16l gc c= = and 1m n= = for superficial Reynolds numbers of liquid and

gas phases less than 2100. For greater values of superficial Reynolds numbers

the constants 0.046l gc c= = and 0.2m n= = . The superficial liquid and gas

Reynolds numbers are defined respectively as follows

Re r sl lsl

l

d v ρ

µ= (5)

Rer sg g

sg

g

d v ρ

µ= (6)

23

Then, after the determination of Hl from chart presented by Lockhart and

Martinelli5 parameter using the ratio of superficial liquid to superficial gas

pressure gradient, representative pipe liquid level hlr is calculated as

lr l rh H d= (7)

Equating the liquid pipe flow area to the liquid area in annular geometry (Alr=

Al), the liquid level hl in eccentric annular geometry (Figure 5.1.1) is calculated

using geometrical equations, which are functions of Al.

Figure 5.1.1- Cross sections of pipe of representative diameter dr and annuli with diameters of di and do

Equations 8-20 are given for determination of the liquid flow area Al and gas

flow area Ag,, as well as liquid contact perimeter Sl,, gas contact perimeter Sg

24

and interfacial length Si perimeters for a given liquid level in the annular

conduit. Figure 5.1.2 clearly represents the parameters used in these equations.

Figure 5.1.2- Geometrical parameters for fully eccentric annuli

The figures I and II in Figure 5.1.2 represent the case in Equation 8.

1 2cos if 2 2

ol

ol

o

dh

dh

dθ −

−

= <

(8)

Similarly, Equation 9 corresponds to the figure I in Figure 5.1.2

1 2cos if 2 2

il

il

i

dh

dh

dφ −

−

= <

(9)

Figure III in Figure 5.1.2 corresponds to Equations 10.

I II III

25

1 2cos if 2 2

ol

ol

o

dh

dh

dθ −

−

= − ≥

(10)

Equation 11 represents the case of figures II and III in Figure 5.1.2.

1 2cos if 2 2

il

il

i

dh

dh

dφ −

−

= − ≥

(11)

where the angles θ and φ are in radians. For a given liquid level in the annular

conduit, the geometrical parameters can be determined using appropriate

equations presented as follows;

if and 2 2

o il l

d dh h< <

22

sin sin 4 2 2 4 2 2

io o o i il l l

dd d d d dA h hθ θ φ φ

= − − − + −

(12)

l o iS d dθ φ= + (13)

2 2( )4

g o i lA d d Aπ

= − − (14)

26

sin sini o iS d dθ φ= − (15)

( )g o i lS d d Sπ= + − (16)

where the equations given for determination of Ag, Si and Sg are same for all

liquid level cases in the annular conduit.

if and 2 2

o il l

d dh h< ≥

22 2

sin sin -4 2 2 4 2 2 4

io o o i i il l l

dd d d d d dA h h

πθ θ φ φ

= − − + + −

(17)

( )l o iS d dθ π φ= + − (18)

if and 2 2

o il l

d dh h≥ ≥

22 2 2( )sin sin

4 2 2 4 2 2 4

io o o i i o il l l

dd d d d d d dA h h

πθ θ φ φ

− = − − − + + − +

(19)

( ) ( )l o iS d dπ θ π φ= − + − (20)

27

The liquid flow area Alr in the pipe of representative diameter dr is determined

using geometrical equations (Equations 8-20) and replacing do by dr and

equating di to 0.

Once the liquid level and the liquid level dependent parameters for annular

geometry are determined accurately, flow pattern transitions can be checked

accordingly.

5.1.1 Stratified Flow to Non-Stratified Flow Transition

The stratified flow (Figure 5.1.1.1) is assumed to take place, and the flow

variables are determined accordingly. Then the stability analysis is carried out.

The transition criterion of stratified flow to non-stratified flow based on the

modified Kelvin-Helmholtz stability analysis was originally proposed by Taitel

and Dukler1.

Figure 5.1.1.1- Stratified Flow

28

In this study this transition model is modified in two different methods. The first

approach is

( )

1

2

1l g gl

glo

g

l

g Ahv

dAd

dh

ρ ρ

ρ

− > −

(21)

Although this modified transition equation, i.e., geometry dependent model

(GDM), yields accurate flow pattern prediction for 0.1143m I.D - 0.05715m O.D

annular conduit, the results are not correct for 0.0932m I.D - 0.0488m O.D

annular two-phase flow, i.e., the experimental data are not within the

appropriate flow pattern boundaries for stratified flow. After the analysis of the

experimental data, it is noted that the geometric dependency of this transition

equation due to the term do should be eliminated. The representative diameter

dr is inserted instead of the inner diameter of the casing do. The resulting

geometry independent equation, i.e., Omurlu and Ozbayoglu method (OOM), is

as follows

( )

1

2

1l g gl

glr

g

l

g Ahv

dAd

dh

ρ ρ

ρ

− > −

(22)

where gas velocity vg is calculated from Equation 23.

29

g sg

g

Av v

A=

(23)

The term dA/dhl, can be calculated by taking the derivative of Al from

corresponding equations, i.e., Equations 12, 17 or 19 replacing do by dr and

equating di to 0. The stability analysis predicts whether an infinitesimal

disturbance on the surface will lead to a stable interface, a wavy interface or to

wave growth destroying the stratification between two layers. The derivation of

the transition equation is given in details in Appendix A.4.

5.1.2 Intermittent Flow to Annular Flow Transition

The liquid level in the conduit is low in case the liquid flow rate is low, i.e.,

superficial liquid velocity less than 0.1 m/s, but the gas flow rate is high, i.e.,

superficial gas velocity greater than 10 m/s. Due to the insufficient liquid supply

from the liquid film, waves formed on the interface (Figure 5.1.2.1) are

unstable. As a result the waves are swept up and an annulus of gas phase is

formed (Figure 5.1.2.2). The transition depends uniquely on the liquid level in

the annular geometry. The critical dimensionless liquid level was originally

suggested by Taitel and Dukler1 to be 0.5 for pipe flow. Later on, it has been

modified by Barnea et al18 as 0.35 (Equation 15) due to the presence of gas

void fraction in the liquid body in the pipe.

0.35lh

d< (24)

30

Figure 5.1.2.1- Intermittent Flow

Figure 5.1.2.2- Annular Flow

In this study, it is suggested that the critical liquid level in annular geometry is

the liquid level representing the half of the flow area as the geometrical

difference is involved. The transition criterion for intermittent flow to annular

flow is as follows

31

0.50lA

A< (25)

5.1.3 Intermittent Flow to Dispersed Bubble Flow Transition

The transition takes place when the turbulent fluctuations are strong enough to

overcome the buoyant forces, which keep the gas at the top of the annulus. At

sufficiently high liquid velocities, i.e., superficial liquid velocity greater than 10

m/s, gas pocket is broken into small dispersed bubbles mixing with the liquid

phase (Figure 5.1.3.1).

Figure 5.1.3.1- Dispersed Bubble Flow

This transition is originally proposed for pipe flow by Taitel and Dukler1.

Similarly, in this study the transition is given by;

32

1

24 ( )g l g

l

i l l

gAv

S f

ρ ρ

ρ

− ≥

(26)

where liquid velocity vl is as follows

ll sl

Av v

A= (27)

The friction factor and the Reynolds number of the liquid phase are defined by;

16

if Re 2100Re

l l

l

f = ≤ (28)

The Fanning friction factor for smooth pipe is:

1

6 3100.001375 1 if Re 2100

Rel l

l

f

= + >

(29)

The critical Reynolds number for each phase is taken as 3000 in order to

guarantee the turbulent flow. Reynolds number for liquid phase is as follows

( )Re

l l l l

l

l

A S v ρ

µ= (30)

33

5.1.4 Stratified Smooth Flow to Stratified Wavy Flow Transition

The mechanism of this transition is based on Jeffrey’s theory for wave initiation,

as suggested by Taitel and Dukler1. The pressure and shear forces exerted by

the gas phase overcomes the viscous dissipation force in the liquid phase, as a

result waves occur on the interface.

( )1

24 l l g

g

l g l

gv

s v

µ ρ ρ

ρ ρ

−>

(31)

Here, sheltering coefficient, s, is taken as 0.01 as suggested by Taitel and

Dukller1, since the liquid phase of the two-phase fluid, i.e, water, can be

considered as low viscous fluid.

5.2 Determination of Frictional Pressure Loss

After the identification of the flow patterns, the frictional pressure losses can be

estimated. The flow mechanism of each flow pattern is studied independently.

5.2.1 Stratified Flow

This flow pattern occurs at relatively low gas and liquid flow rates. The phases

are separated due to the gravitational forces as a result of density difference.

Interfacial shear stress occurs because of the variation in motion of each phase.

34

As the momentum balance is investigated (Figure 5.1.1.1) following equations

are obtained for liquid and gas phase respectively.

0l wl l i i

l

PA S S

zτ τ

∆ − − + =

∆ (32)

0g wg g i i

g

PA S S

zτ τ

∆ − − − =

∆ (33)

The wall shear stress of liquid phase wlτ , and the wall shear stress of gas

phase wgτ are calculated by;

2

=

2

l l lwl

f vρτ (34)

2

g

=2

g g

wg

f vρτ (35)

Similar to the liquid friction factor and Reynolds number, gas phase friction

factor and Reynolds number are calculated as

16

if Re 2100Re

g g

g

f N= ≤ (36)

35

The fanning friction factor for gas phase is given in Equation 37.

1

6 3100.001375 1 if Re 2100

Reg g

g

f N

= + >

(37)

and

( )( )Re

g g i g g

g

g

A S S v ρ

µ

+= (38)

An important parameter affecting the accuracy of the model is the

determination of interfacial friction factor, fi, and the interfacial shear stress iτ .

The interfacial friction factor fi, is assumed to be equal to gas friction factor fg

for stratified smooth and stratified wavy flow patterns. Then iτ is given by;

2( )=

2

g g g l

i

f v vρτ

− (39)

The frictional pressure loss of the two-phase flow system through the eccentric

annuli may be calculated using either Equation 32 or 33.

36

5.2.2 Intermittent Flow

Intermittent flow consists of two main zones, i.e., liquid slug body and the gas

pocket/liquid film region (Figure 5.1.2.1). In this alternate flow of gas pockets

and liquid slugs, gas phase in large bullet-shaped pockets flows in the upper

part of the pipe. The liquid film flows below the gas pocket. Dukler and

Hubbard15 investigated the flow mechanism of intermittent flow through pipes.

Due to the complexity of the phase distributions, numerical iterations are

required in order to determine the liquid holdup distributions within the slug

body and gas pocket/liquid film zone. Several different pressure loss calculation

methods were presented for intermittent flow afterward. Xiao et al17 assumed

the film thickness to be uniform along the gas pocket/film zone. This region

was treated to be analogous with stratified flow. Even with this simplifying

assumption, the iterative solution procedures are not reliable. Petalas and Aziz24

claimed that, this method contradicted with the experimental results and

developed empirical equations for pipe flow. In this study, two methods are

proposed. The first approach is similar to the one suggested by Petalas and

Aziz24. As the momentum balance over a slug unit is investigated Equation 40 is

obtained.

1 lf lf g gls rs f

u

S SdPL L

z L A A

τ ττ π + ∆ = +

∆ (40)

In this study, the proposed equation is:

37

fsl fgp fsl fgp

P P P P PT

z z z z zµ

∆ ∆ ∆ ∆ ∆ = + − −

∆ ∆ ∆ ∆ ∆ (41)

where, the weighting factor, Τµ is the function of dimensionless liquid holdup

and is presented in details in results and discussion chapter. This method will

be named as empirical pressure drop determination method (EPDM) throughout

this thesis. In this study Τµ is defined as;

= ( )lT f Hµ (42)

The translational velocity of liquid slug is calculated from equation (43) given by

Bendiksen37:

t o m dv C v v= + (43)

Co is taken as 1.2 as suggested by Niklin et al38. The drift velocity of gas pocket

is determined from Zukoski39 correlation:

d m dv f v ∞= (44)

where the correlation of vd∞ proposed by Bendiksen37 has only horizontal

velocity term and is determined by using Weber40 correlation:

38

0.56

( )1.760.54

r l g

d

l

gdv

Bo

ρ ρ

ρ∞

− = −

(45)

The Bond number is:

( )2( )l GrBo gd

ρ ρ

σ

−=

(46)

fm term in the Zukoski39 correlation is given as:

0.316 Re for 1m mf f∞= < (47)

Otherwise,

1mf = (48)

where

Re2

l d r

l

v dρ

µ∞

∞ = (49)

Once these parameters are determined, liquid volume fraction in the slug and

the average liquid volume fraction of the slug unit can be calculated from

39

Gregory et al41 correlation and mass balance equation of liquid phase over the

slug unit.

1.39

1

18.66

ls

m

Ev

=

+

(50)

(1 )l t g ls sgs

l

t

E v v E vE

v

+ − −= (51)

The entrainment fraction, FE, is estimated using Petalas and Aziz24 correlation:

0.2

0.0740.735

1

sg

B

sl

vFEN

FE v

=

− (52)

where

2 2

2

l sg g

B

l

vN

µ ρ

σ ρ= (53)

The dimensionless liquid film thickness is determined from equation (54)

40

( )11 (1 )

2

sl sg

l

sg

FE v vE

vδ

+ = − −

(54)

Mixture density and viscosity are defined in Equations 55 and 56 respectively as

mostly used in literature.

(1 )m l l l gE Eρ ρ ρ= + − (55)

(1 )m l l l gE Eµ µ µ= + − (56)

Once all the necessary parameters are defined, the frictional losses for slug

body can be determined using Equation 57.

22

sl

ml m m

f r

f vP

z d

ρ∆ =

∆ (57)

where fmL is determined from Reml ,

Re r l mml

l

d vρ

µ= (58)

For the gas pocket/liquid film zone,

41

4

gp

wl

f r

P

z d

τ∆ =

∆ (59)

If δ < 0.0001, then homogeneous two-phase model with slip proposed by

Petalas and Aziz24 can be used for pressure loss determination as given in

Equation 60.

2

2gp

f l f

f

f vP

z

ρ∆ =

∆ (60)

where liquid film velocity is determined from,

slf

l

vv

E=

(61)

From equation (41), total pressure loss is calculated. As the pressure loss

estimations of this method are compared with the experimental data, it is

observed that the method EPLDM gives more accurate results for 0.1143m -

0.05715m annular geometry than 0.0932m - 0.0488m annular conduit.

Therefore the second method, Omurlu Method (OM), is developed introducing

dimensionless group. The proposed equation in this study is as follows

42

2 2

1/4

1sl sg sl sgm m

l l

fsl fgpm m

i

o

v v v vP PE E

v l z v l zP

z dd

ρ ρ

ρ ρ

∆ ∆ + −

∆ ∆∆ = ∆

(62)

Pressure gradients in the liquid slug and the liquid film/gas pocket zone are

calculated as described in the previous method. Then the pressure gradients

are multiplied with a new dimensionless group and divided by ratio of the

casing diameter to drillpipe diameter.

5.2.3 Annular Flow

At very high gas flow rates, gas phase flows in a core of high velocity, which

contains entrained liquid droplets. The liquid phase flows as a thin film around

the pipe wall with a greater thickness at the bottom of the pipe than that at the

top (Figure 5.1.2.2). For practical purposes the film thickness δ, is assumed to

be uniform and equal to the average film thickness. When the momentum

balance equations of the liquid phase and the gas phase are examined, it is

observed that the mechanism of this flow pattern is analogous to stratified flow.

As the appropriate geometrical parameters are defined with empirical closure

equations, liquid film thickness can be determined from the combined

momentum equation by trial and error procedures. For liquid phase,

momentum equation is

43

0ff w f i i

PA S S

zτ τ

∆ − − + =

∆ (63)

For the gas phase in the gas core, momentum equation can be derived as

0c i i

PA S

zτ

∆ − − =

∆ (64)

Combining equations (63) and (64) yields

1 10

f

wf i i

f f c

SS

A A Aτ τ

− + =

(65)

It is assumed that the liquid droplets in the gas phase have the same velocity

with the gas core. Then, gas void fraction in the core, GVFc, is:

sg

c

sg sl

vGVF

v v FE=

+ (66)

Fluid density and viscosity inside the core are

(1 )c g c l cGVF GVFρ ρ ρ= + − (67)

44

(1 )c g c l cGVF GVFµ µ µ= + − (68)

respectively. With the geometrical configuration, liquid film velocity and core

velocity are defined as:

2(1 )

4 ( )

rf sl

r

FE dv v

dδ δ

−=

− (69)

2

2

( )

( 2 )

sg sl r

c

r

v v FE dv

d δ

+= −

(70)

Petalas and Aziz24 correlation is preferred for the interfacial friction factor, fi:

0.305

0.085

Re20.24

f

i

c c c c

fN

f v d

σ

ρ

=

(71)

The pressure loss can be estimated from either Equation 63 or 64, once the film

thickness is appropriately determined.

45

5.2.4 Dispersed Bubble Flow

This flow pattern occurs at very high liquid flow rates, i.e., superficial liquid

velocity higher than 10 m/s. The gas phase is dispersed as discrete gas bubbles

within the continuous liquid phase (Figure 5.1.3.1). A simple homogeneous flow

is assumed since the gas bubbles are moving at the same velocity as liquid

phase. Pressure loss can be calculated from equation (72).

22 m m m

r

f vP

z d

ρ∆ =

∆ (72)

where fm can be calculated using

Re r m mm

m

d vρ

µ= (73)

The mixture density, velocity and the mixture viscosity are calculated using the

no-slip liquid holdup λl term. The no-slip hold up is given by;

sll

sl sg

v

v vλ =

+ (74)

Where

sl sg mv v v+ = (75)

46

As mostly used in literature, mixture density and mixture viscosity are:

(1 )m l l l gρ λ ρ λ ρ= + − (76)

(1 )m l l l gµ λ µ λ µ= + − (77)

After the calculation of the necessary parameters, the frictional pressure loss of

the dispersed bubble flow can be calculated from Equation 72.

47

CHAPTER 6

EXPERIMENTAL WORK

The essential part of the experimental study is the experimental setup in

addition to the data acquisition system. The experimental work is presented in

four subtitles namely, experimental setup, test section, calibration process, test

procedure and data acquisition.

6.1 Experimental Setup

METU-PETE-CTMFL multiphase flow loop was constructed in order to perform

two-phase flow experiments in horizontal fully eccentric annuli. A schematic

view of the loop is presented in Figure 6.1.1.

48

Figure 6.1.1- Schematic view of the experimental setup

The experimental setup consisted of a 4.84 m transparent test section. Two

pairs of annular geometrical configuration are used as given in Table 6.1.1.

Table 6.1.1 Geometrical configuration of annular section

Configuration Casing Size (m) Drillpipe Size (m)

1 0.0932 0.0488

2 0.1143 0.0571

A thin plate is welded parallel to flow direction into the flange at the entrance of

the loop in order to contribute to the establishment of fully developed flow. An

air compressor and a centrifugal pump fed the two-phase air-water system. A

separator (Figure 6.1.2) is connected to the exit of the loop; hence water is

separated from two-phase mixture before being carried to the liquid tank. The

centrifugal pump is used with a magnetic flow meter and an electropneumatic

49

control valve (Figure 6.1.3) to measure and control the desired liquid flow rate.

Similarly, the air compressor is used with a volumetric flow meter and an

electropneumatic control valve to deliver required amount of gas into the loop.

The compressed air mixed with the water before entering to the annular

section. A pressure regulator is mounted before the gas flow meter as a safety

measure and to keep the air pressure controlled prior to entering to the test

section. The pressure of the gas phase is kept usually at 25 psi. The pressure of

the loop, frictional pressure losses, liquid and gas flow rates are measured

using the data acquisition system. Data logger and data acquisition software

are used to gather and store the experimental data digitally. The capacity and

brand name of each component in the experimental setup are presented in

Table 6.1.2.

Figure 6.1.2- Two-phase separator

50

Figure 6.1.3- Electropneumatic control valve

51

Table 6.1.2 Capacity and brand name of experimental components

Component Brand Name Capacity

Air Compressor TAMSAN 3000 l/min at 6 atm

Centrifugal Pump DOMAK 1.136 m3/min

Liquid Tank 2000 m3

Magnetic Liquid Flow

Meter

TOSHIBA 1.136 m3/min

Volumetric Gas Flow

Meter

COLE-PARMER INST. CO 0-1000 l/min at 25 psi

Electropneumatic Control

Valves

SAMSON

Digital Differential

Pressure Transducers

COLE-PARMER INST. CO 0-1 psi

Differential Pressure

Gauges

ASHCROFT 0-1” water

0-2” water

0-5” water

0-10” water

6.2 Test Section

The test section is 4.84 m. long and made of acrylic casing and steel drillpipe

(Figure 6.2.1). The geometrical configuration of the test section is given in

Table 6.1.1. The transparent casing allowed the observation and identification

of the flow patterns. Digital 1 psi differential pressure transducers and

differential pressure gauges of are used in order to measure frictional pressure

losses. Moreover, digital pressure transducers are mounted on the gas line at

52

the entrance of the gas flow meter and on the annular test section with the aim

of monitoring the system pressure at different locations. The average pressure

inside the loop was kept less than 20 psia. Table 6.1.2 clearly shows the

capacity and brand name of the pressure transmitters.

Figure 6.2.1- Test section

The determination of the locations of the pressure transmitters on the test

section was one of the important tasks during setup design and construction.

The data collected should be reliable since the mathematical model’s

performance would be evaluated using the experimental data. Therefore,

entrance and exit effects are calculated for each casing-drillpipe configuration

using Equations 69 and 70. The entrance length is calculated by;

53

50entrance hydL d= (78)

Equation 78 is porposed by Knudsen and Katz42 for pipe flow. In this study it

was modified by using hydraulic diameter.

The distance from the exit required to eliminate the chimney effect is given in

Equation 79 (reference 43). Similarly, the equation proposed for pipe flow was

modified for annular geometry inserting hydraulic diameter.

164.4 Reexit hydL N d= (79)

Thus, a fully developed region of 1.22 m for configuration1 and 0.61 m for

configuration 2 annular test sections are obtained.

6.3 Calibration Process

The most important components of the experimental study are the frictional

pressure loss measurement and flow pattern visualization. The verification of

fully developed flow in the test section is essential. To ensure the accuracy of

the experimental data, frictional pressure losses are recorded for single phase

water flow through the annular test section. The pressure and flow rate

readings are recorded and compared with theoretical estimations. Figure 6.3.1

and Figure 6.3.2 show the accordance between the calculated pressure losses

and the experimental data for configuration 1 and configuration 2 eccentric

annular flow.

54

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 20 40 60 80 100

Q (gpm)

DP

/DL

(in

ch

of

wate

r/in

ch

)

MEASURED DP/DL

CALCULATED DP/DL

Figure 6.3.1- Frictional pressure loss gradient versus flow rate data of water flowing through configuration 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 50 100 150 200 250

Q (gpm)

DP

/DL

(in

ch

of

wate

r/in

ch

))

MEASURED DP/DL

CALCULATED DP/DL

Figure 6.3.2- Frictional pressure loss gradient versus flow rate data of water flowing through configuration 2

55

The pressure losses are calculated for concentric annuli using narrow slot

approach. However, the experimental data were collected from fully eccentric

conduit. The slight difference between the experimental data and estimated

pressure losses may be due to this fact.

The calibration of liquid and gas flow meters are checked and the readings are

found to be within the 1% accuracy as given in the calibration data sheets. The

differential analogue pressure gauge is designed to measure the pressure

losses to an accuracy of 2% of full scale of reading. Similarly, the pressure

readings of digital differential pressure transducers and the pressure gauges are

within the 0.25% and 0.13% accuracy of the full scale, respectively.

6.4 Experimental Test Procedure and Data Acquisition

In the experimental work, air and water are used during two-phase flow tests.

Pressure losses and flow patterns are recorded at different gas and liquid flow

rates through horizontal fully eccentric annuli. The procedure of the two-phase

tests is as follows.

1) The water is pumped from the liquid collection tank to the loop using

centrifugal pump.

2) The water inside the water lines connecting the differential pressure

transducers’ low and high ends to the test section is flushed to prevent

the trapping of air bubbles in these lines.

56

3) Once the differential pressure transmitters are ready and the water flow

in the system is stabilized, control valve on the air line is opened and

adjusted to the desired flow rate.

4) After waiting for the stabilization of the flow rate of both phases, the

frictional pressure losses, system pressures and the flow rates are

recorded using data logger and data acquisition software. The record

time for each set of flow condition is one minute after steady state

condition is established. During the analysis of the experimental data the

average of this one minute recorded data is used.

5) Meanwhile, the observed flow pattern is recorded using high resolution

Sony Digital Video Camera Recorder.

A test matrix covering the minimum and maximum capacities of the system is

established to perform the experiments. While conducting the tests, air flow

rate is increased gradually keeping the water flow rate constant. Then the

water flow rate is increased to the next level and same procedure is followed.

57

CHAPTER 7

COMPUTER WORK

A computer program based on the proposed mathematical model in this study

is developed in order to facilitate the flow pattern determination and pressure

loss estimation using Matlab 7.0.4. The discussion of the model performance

and the analysis of the experimental data are given in details in the next

chapter. The flow chart of the computer program for the flow pattern

determination is given in Figure 7.1. The frictional pressure losses are

calculated accordingly once the flow pattern is identified.

58

START

Ql,Qg, di,

do,µl,µg,ρl,ρg

dr, hlr, hl, Al,Ag, Sl,

Sg, Si, vl, vg, vm,

µm,ρm

( )

1

2

1l g gl

glr

g

l

g Ahv

dAd

dh

ρ ρ

ρ

− ≤ −

Stratified Wavy( )

1

24 l l g

g

l g l

gv

s v

µ ρ ρ

ρ ρ

−>

Stratified Smooth

0.50lA

A<

Annular

1

24 ( )g l g

l

i l l

gAv

S f

ρ ρ

ρ

− ≥

Dispersed Bubble Intermittent

YES

NO

NO

NO

NO

YES

YES YES

Pressure Loss Calculation

END

Figure 7.1- Flow chart of Matlab code for the flow pattern identification and frictional pressure loss determination

59

CHAPTER 8

RESULTS and DISCUSSION

After the calibration of experimental setup is finished and the validation of the

data acquisition system and the experimental setup with single phase water

flow experiments is carried out, two-phase flow measurements are conducted.

Firstly, flow pattern maps are generated for 0.0932m I.D acrylic casing –

0.0488m O.D drillpipe (configuration 1) and 0.1143m I.D acrylic casing –

0.05715m O.D drillpipe (configuration 2) annular conduits. The validation of the

proposed model’s flow pattern prediction with the experimental data is given in

detail in the following section. The comparison with generated flow pattern

maps using hydraulic diameter is also discussed. The accuracy of the proposed

model’s frictional pressure loss estimations are explained using experimental

pressure loss measurements. Empirical friction factor correlations are presented

for each observed flow pattern. The results and the discussion are given in

details in the following sections.

8.1. Validation of Flow Pattern Identification of Proposed Model with

Experimental Data

The flow pattern map is generated using hydraulic diameter and the transition

criterions proposed by Taitel and Dukler1. Figures 8.1.1 and 8.1.2 represent the

60

comparison of flow pattern maps for hydraulic diameter, dhyd and representative

diameter, dr approaches (GDM) for two different annular geometries, i.e.,

configurations 1 and 2.

0.01

0.1

1

10

100

0.1 1 10 100

vsg (m/s)

vsl

(m/s

)

dr

SS

SW

I

dhyd

Figure 8.1.1- Comparison of flow pattern maps generated using dhyd and GDM for configuration 1

SS SW

AN

I

DB GDM

61

0.01

0.1

1

10

100

0.1 1 10 100

vsg (m/s)

vsl

(m/s

)

dr

SS

SW

I

dhyd

Figure 8.1.2- Comparison of flow pattern maps generated using dhyd and GDM for configuration 2

The flow patterns observed experimentally are stratified smooth (SS), stratified

wavy (SW) and intermittent (I) flow. Some of the flow pattern examples are

given as pictures taken during experiments in Appendix A.5. As compared with

the experimental data (Figures 8.1.1 and 8.1.2), it is observed that a significant

shift in flow pattern map generated using dhyd (blue dashed lines in Figures

8.1.1 and 8.1.2) occurred in stratified to non-stratified flow pattern boundaries

and stratified wavy to stratified smooth flow pattern transitions. Stratified

smooth and stratified wavy flow patterns are developed at considerably higher

liquid and gas flow rates than those estimated with hydraulic diameter

approach. It can be observed that the proposed first approach in this study

predicts accurately the flow pattern transitions for the conduit configuration 2.

The experimental data fits exactly within the predicted boundaries. However, as

the annular geometry changes, i.e., the flow area decreases, the accuracy of

the proposed GDM diminishes. From Figure 8.1.1, it is clearly observed that the

SS SW AN

DB

I

GDM

62

transition from intermittent flow to stratified smooth and stratified wavy flow

begins at higher liquid superficial flow rates than predicted velocities. Similarly,

annular flow pattern transition should shift towards right, to higher superficial

gas velocities. However, one should note that due to the experimental setup

limitations, i.e., capacity of experimental setup components given in Table

6.1.2, the majority of the data collected from configuration 1 annular conduit

consist of intermittent flow patterns. Therefore, more stratified flow data are

required in order to evaluate accurately the flow pattern transition models. The

liquid superficial velocity was greater than 0.1 m/s during the experiments. At

higher superficial liquid velocities, gas phase could not flow through the mixing

line since the pressure of the gas phase was kept less than 25 psi at the

entrance of flow meter. In case of hydraulic diameter used maps, the flow

pattern identifications are not correct for configurations 1 and 2 annular

conduits when compared with experimental results (blue dashed lines in Figures

8.1.1 and 8.1.2).

When the overall flow patterns in both annular geometries are observed it is

noted that the flow patterns are independent of the conduit dimensions.

However, the proposed first method (GDM) is geometry dependent because of

the presence of do in the transition equation, i.e., Equation 21. Therefore, the

proposed model is modified by replacing do with di (Equation 22). This second

method (OOM) becomes geometry independent. The comparison of the results

of OOM with the experimental data is presented for configuration 1 and

configuration 2 annular geometries in Figure 8.1.3 and Figure 8.1.4.

63

0.01

0.1

1

10

100

0.1 1 10 100

vsg (m/s)

vs

l (m

/s)

SS

SW

I

dhyd

dr, 2nd aproach

Figure 8.1.3- Comparison of flow pattern maps generated using dhyd and OOM for configuration 1

0.01

0.1

1

10

100

0.1 1 10 100

vsg (m/s)

vs

l (m

/s)

dr, 2nd approach

SS

SW

I

dhyd

Figure 8.1.4- Comparison of flow pattern maps generated using dhyd and OOM for configuration 2

As an expected result, the flow pattern predictions of OOM in this study are

highly accurate when compared with the experimental data. Similar to GDM,

SS

SW

AN

DB

I

OOM

SS

SW

AN

DB

I

OOM

64

the experimental results fit exactly within the appropriate flow pattern

boundaries configuration 2. In case of configuration 1, this geometry

independent method, OOM, gives more accurate flow pattern estimations than

GDM. Yet, more stratified flow data are needed in order to discuss and evaluate

the accuracy of OOM for configuration 1 annular geometry. The proposed

model still estimates the flow patterns more correctly than hydraulic diameter

approach.

The final part of the flow pattern comparison is the evaluation of Beggs and

Brill14 method (Appendix A.3) for horizontal fully eccentric annular two-phase

flow. Hydraulic diameter is used to represent the annular geometry. Figure

8.1.5 and Figure 8.1.6 show the comparison of the flow patterns generated

with the observed data.

0.01

0.1

1

10

100

0.1 1 10 100

vsg (m/s)

vs

l (m

/s)

SS

SW

I

modified

Beggs &Brill

Figure 8.1.5- Validation of flow pattern maps generated using dr and modified Beggs and Brill3 method with experimental data for

configuration 1

SEGREGATED

DISTRIBUTED

I

SEG-I TRANSITION

65

0.01

0.1

1

10

0.1 1 10 100

vsg (m/s)

vsl

(m/s

)SS

SW

I

modified

Beggs & Brill

Figure 8.1.6- Validation of flow pattern maps generated using dr and modified Beggs and Brill3 method with experimental data for

configuration 2

The empirical flow pattern prediction model of Beggs and Brill14 is modified

using dr. This widely used model in industry could not predict correctly the flow

patterns observed. Especially in annular conduit configuration 1, the results are

not accurate. In case of configuration 2, intermittent flow data are within the

appropriate boundaries but stratified flow data are also in this region. Stratified

flow takes place at higher liquid superficial velocities according to the

experimental observations.

8.2 Validation of Frictional Pressure Loss Estimations of Proposed

Model with Experimental Results

The flow patterns observed during experiments are only stratified (SS and SW)

flow and intermittent (I) flow. Although pressure loss determination equations

SEGREGATED

DISTRIBUTED

I

SEG-I TRANSITION

66

for annular (AN) and dispersed bubble (DB) flow are given in chapter 5, only

intermittent and stratified flow frictional pressure loss estimations are discussed

in this chapter.

The comparison of the experimental results and proposed model estimation for

pressure losses of stratified flow of water-air mixture flowing through fully

eccentric horizontal annulus of configurations 1 and 2 is presented in Figures

8.2.1 and 8.2.2.

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

measured DP/DL(Pa/m)

calc

ula

ted

DP

/DL

(Pa/m

)

modified Aziz et al.

modified Garcia et al.

Ti=f(fG),this study

modified Beggs & Brill

Figure 8.2.1- Comparison of frictional pressure loss estimations of the proposed model with experimental data and mostly used

models for stratified flow through configuration 1

+30%

-30%

67

0

50

100

150

200

250

0 50 100 150 200 250

measured DP/DL (Pa/m)

calc

ula

ted

DP

/DL

(P

a/m

)

modified Aziz et al.

modified Garcia et al.

Ti=f(fG),this study

modified Beggs & Brill

Figure 8.2.2- Comparison of frictional pressure loss estimations of the proposed model with experimental data and mostly used

models for stratified flow through configuration 2

The dashed lines in Figures 8.2.1 and 8.2.2 are ±30 % error margin, and the

solid line represents the perfect match between the experimental stratified flow

data and calculated results for configuration 1 and configuration 2 annular

conduits respectively. Mostly used models of Petalas and Aziz24, Garcia et al25

and Beggs and Brill14 for pipe flow are modified for annular geometry by using

dr instead of pipe diameter as shown in Appendix A.3. Modified Petalas and

Aziz24 and Beggs and Brill14 models give underestimated pressure loss results

when compared with the experimental results. The correct determination of

interfacial shear stress is the most important step during the pressure loss

calculations. Instead of using empirical correlation for interfacial friction factor

(Petalas and Aziz24), gas friction factor is preferred in this study. Although

modified Garcia et al’s model25 gives accurate results the proposed model’s

performance is agreeable when validated with experimental data. It can be

+30%

-30%

68

observed that the procedure presented in this study for stratified flow estimates

the frictional pressure losses with a reasonable accuracy for both annular

geometries. However, it should be remarked that the amount of data is not

sufficient to generalize the results due to the limitations of the experimental

setup, i.e. Table 6.1.2.

In chapter 5 two methods, i.e., EPDM and OM are presented for pressure loss

determination in case of intermittent flow. In this study, in EPDM an empirical

equation (Equation 80) is developed.

(1.6552 )=0.3805 lH

T eµ (80)

Figure 8.2.3 and Figure 8.2.4 represent the comparison of the experimental

results and EPDM’s estimations for pressure losses of intermittent flow. The

performances of the modified Petalas and Aziz24, modified Garcia et al.25 and

modified Beggs and Brill14 models are also evaluated with the experimental

data.

69

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

measured DP/DL (Pa/m)

calc

ula

ted

DP

/DL

(P

a/m

)

modified Aziz et al.

modified Garcia et al.

EPDM, this study

modified Beggs and Brill

Figure 8.2.3- Comparison of frictional pressure loss estimations of EPDM with experimental data and mostly used models for

intermittent flow through configuration 1

0

100

200

300

400

500

600

700

800

0 200 400 600 800

measured DP/DL(Pa/m)

calc

ula

ted

DP

/DL

(Pa/m

)

modified Aziz et a.

modified Garcia et al.

EPDM, this study

modified Beggs and Brill

Figure 8.2.4- Comparison of frictional pressure loss estimations of EPDM with experimental data and mostly used models for

intermittent flow through configuration 2

70

Similarly, in the figures above, the dashed lines are ±30 % error margin, and

the solid line represents the perfect match between the experimental and

calculated results. From Figures 8.2.3 and 8.2.4, it is observed that EPDM in

this study estimates accurately the frictional pressure losses for intermittent

flow. The results are more correct for configuration 2. As in case of

configuration 1, at relatively high measured pressure losses the model

overestimates the frictional pressure losses. However, the modified models, i.e.,

Garcia et al.25 and Beggs and Brill14, highly underestimate the pressure losses.

As the measured frictional pressure loss increases these modified models still

calculate very low pressure loss estimations. Modified Petalas and Aziz24 model

gives closer pressure results to the experimental data than other modified

models. However, EPDM proposed in this study is the most accurate model

among others.

The second method, OM developed in this study is more geometry independent

(Equation 62) than the first method EPDM (Equation 41). The higher accuracy

of OM is valid for configurations 1 and 2. Here, in the following figures, i.e.,

Figure 8.2.5 and Figure 8.2.6, the comparison of OM in this study is presented

along with the evaluation of the modified models with the experimental

intermittent flow data for configuration 1 and configuration 2 respectively.

71

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

measured DP/DL (Pa/m)

calc

ula

ted

DP

/DL

(P

a/m

)

modified Aziz et al.

modified Garcia et al.

OM, this study

modified Beggs and Brill

Figure 8.2.5- Comparison of frictional pressure loss estimations of OM with experimental data and mostly used models for

intermittent flow through configuration 1

0

100

200

300

400

500

600

700

800

0 200 400 600 800

measured DP/DL(Pa/m)

calc

ula

ted

DP

/DL

(Pa/m

)

modified Aziz et al.

modified Garcia et al.

OM, this study

modified Beggs and Brill

Figure 8.2.6- Comparison of frictional pressure loss estimations of OM with experimental data and mostly used models for

intermittent flow through configuration 2

72

In Figures 8.2.5 and 8.2.6, the dashed lines are ±30 % error margin, and the

solid line represents the perfect match between the experimental and

calculated results. It is remarked that OM proposed in this study estimates the

frictional pressure losses with a high accuracy for intermittent flow through

configurations 1 and 2 horizontal eccentric annular conduits. As discussed in the

validation of EPDM, the modified models, i.e., Garcia et al.25 and Beggs and

Brill14, highly underestimate the pressure losses, and modified Petalas and

Aziz24 model gives closer pressure results to the experimental data than other

modified models. However, OM proposed in this study is the most accurate

model among others, and most of the calculated pressure losses are within the

±30 % error margin. The reason for inaccurate pressure loss estimations of

modified models may be the fact that these models were originally developed

for two-phase pipe flow. Also, there are empirical equations used in these

models which are similarly proposed for circular pipes.

Table 8.2.1 Error percentage for pressure loss estimation of mostly

proposed and modified models

do-di (0.0932-0.0488m)

Mod. Beggs

and Brill model

Mod.Aziz

and

Petalas

Mod.

Garcia

First

Approach

(This Study)

Second

Approach

(This Study)

Mod.

Beggs

and Brill

model

Mod.Aziz

and

Petalas

Mod.

Garcia

This

study

Average Error % 86.6 42.2 93.4 35.1 30.5 70.7 78.2 36.7 51.1

Maximum Error % 93.0 56.8 95.4 102.6 170.0 90.8 126.7 89.4 166.4

Minimum Error % 53.0 3.8 82.9 0.1 0.1 48.8 52.9 7.1 0.5

do-di (0.1143-0.05715m)

Average Error % 90.5 54.2 95.0 31.0 15.1 93.5 113.9 57.0 25.3

Maximum Error % 95.9 68.6 96.7 84.4 39.4 95.5 131.1 77.4 58.3

Minimum Error % 74.9 21.3 90.5 0.2 0.0 40.7 2.1 2.3 1.4

Intermittent Flow Stratified Flow

73

Table 8.2.1 represents quantitatively the average, maximum and minimum

errors for pressure loss estimations. As remarked from this table, for

intermittent flow EPDM proposed in this study has an average error of 35.1%

and 31% for configurations 1 and 2. However, OM proposed in this study

determines the frictional pressure losses in both annular configurations more

accurately with an average error of 30.5% and 15.1%. Other compared models

have very low accuracy in pressure loss estimations. In stratified flow the

proposed procedure in this study calculates the pressure drop accurately with

an average error of 51.1% for configuration 1 and 25.3% for configuration 2.

In configuration 1 modified Garcia et al25 model estimates the pressure drop

more accurately than the proposed model with an average error of 36.7%. In

both annular geometries and for both flow patterns, proposed mechanistic

model has the least minimum errors. Moreover, as the amount of data is

considered, the proposed method in this study can be considered as accurate

and applicable. This proposed mechanistic model is highly useful for practical

purposes, since the calculation procedure is simple and accurate for both

annular geometries.

8.3 Empirical Friction Factor Correlations

In this study, although the simplicity and the accuracy of the model proposed

for stratified flow and OM for intermittent flow, for practical purposes empirical

friction factor correlations are developed. The frictional pressure losses are

estimated theoretically for each flow pattern independently using representative

diameter, dr. The correlations are developed based on data obtained from

METU-PETE-CTMFL multiphase flow loop using statistical methods. The data

74

obtained include only stratified and intermittent flow data due to the

geometrical and two-phase system restrictions, i.e., Table 6.1.2.

For stratified flow, the friction factor correlation is given in Equation 81.

( )1.1734

9880 Ref mf−

= (81)

Where Rem, the mixture Reynolds number can be calculated using Equation 73.

The friction factor versus mixture Reynolds number relation of the experimental

stratified flow data is presented in Figure 8.3.1. The friction factor is calculated

from Equation 72.

75

0.001

0.01

0.1

1

10000 100000 1000000

Rem

ff

Figure 8.3.1- Friction factor and mixture Reynolds number relation of experimental stratified flow data

Similarly, for intermittent flow, friction factor correlations are developed. It is

observed that mixture Reynolds number and the friction factor does not show

expected correlation (Figure 8.3.2).

76

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

100000 1000000

Rem

ff

Figure 8.3.2- Friction factor and mixture Reynolds number relation of experimental intermittent flow data

Therefore, a new mixture Reynolds number based on liquid holdup is proposed

as given in Equation 82.

Re Re (1 ) Remix l sl l sgN N Nλ λ λ= + − (82)

where liquid holdup, λl is calculated from Equation 74. As the results are

investigated, it is remarked that at NRemixλ = 100000, the slope of the curve

changes. Figures 8.3.3 and 8.3.4 show the relation between the friction factor

and the Reynolds number for each case.

77

0.001

0.01

0.1

1

10000 100000

NRemixλλλλ

ff

Figure 8.3.3- Friction factor and mixture Reynolds number relation of experimental intermittent flow data for NRemixλλλλ < 100000

Therefore, the friction factor correlations are developed accordingly. For

NRemixλ < 100000, the proposed equation is;

( )0.0594

0.0085 Ref mixf N λ

−= (83)

In case where NRemixλ ≥ 100000, friction factor can be calculated from

Equation 84.

( )0.4676

0.9435 Ref mixf N λ

−= (84

Figure 8.3.4 represents the correlation given in Equation 84.

78

0

0.001

0.002

0.003

0.004

0.005

0.006

100000 1000000

NRemixλλλλ

ff

Figure 8.3.4- Friction factor and mixture Reynolds number relation of experimental intermittent flow data for NRemixλλλλ ≥ 100000

Then, pressure loss can be determined using Equation 62. The performance of

the empirical correlations is compared with experimental data. Figures 8.3.5

and 8.3.6 represent the accuracy of the proposed correlations. Dashed lines

represent ±30% error margin. It can be noted that all of the estimated pressure

losses are within this error range. The empirical model can estimate the

frictional pressure losses accurately.

79

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

measured DP/DL (Pa/m)

calc

ula

ted

DP

/DL

(P

a/m

)

Figure 8.3.5- Comparison of pressure losses determined by the empirical correlations and experimental data for stratified

flow

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800 1000 1200 1400 1600

measured DP/DL (Pa/m)

calc

ula

ted

DP

/DL

(P

a/m

)

Figure 8.3.6- Comparison of pressure losses determined by the empirical correlations and experimental data for intermittent

flow

80

CHAPTER 9

CONCLUDING REMARKS

A mechanistic two-phase flow modeling study along with an experimental work

is carried out. The theoretical part of this study covers the development of the

mechanistic model for water-air horizontal two-phase flow through fully

eccentric annuli. Flow pattern identification, i.e., distribution of two phases in

the eccentric annuli, and frictional pressure loss calculation methods are

proposed. A new diameter term, i.e., representative diameter is introduced in

order to characterize the fully eccentric annuli. Empirical friction factor

correlations are developed for practical frictional pressure loss calculations. The

experimental study is carried out in METU-PETE-CTMFL multiphase flow loop.

Water and air are the fluids used during the experiments in two different

geometrical configurations of fully eccentric annular conduit. As the accuracy of

the proposed mechanistic model and empirical correlations are evaluated with

the experimental data, the following remarks are concluded.

• Hydraulic diameter can be used to represent the fully eccentric

horizontal annuli while determining the fully developed region.

• Omurlu and Ozabayoglu model (OOM) can estimate flow patterns

accurately for configurations 1 and 2 annular conduits when compared

81

with the experimental data. Geometry dependent model (GDM) model

identifies flow patterns correctly for configuration 2, but the

experimental stratified data of configuration 1 are not within the

appropriate flow pattern boundaries predicted by GDM.

• Hydraulic diameter approach is not applicable for two-phase fully

eccentric annular flow. Conduit geometry has a strong influence on flow

pattern transitions when compared with pipe flow. As the hydraulic

diameter approach is applied, noticeable shifts between the

experimental data occur at the flow pattern map.

• The flow pattern prediction and frictional pressure loss estimations of

modified models i.e., Petalas and Aziz24, Garcia et al25, and Beggs and

Brill14 are compared with experimental data collected at METU-PETE-

CTMFL multiphase flow loop. The modified models’ results are not

accurate and hence are not applicable for horizontal two-phase flow

through fully eccentric annuli.

• Interfacial shear stress should be well defined for stratified flow. Gas

friction factor is used as the interfacial friction factor. The model’s

pressure loss estimations are accurate when compared with

experimental results.

• Two different methods i.e., empirical pressure drop determination

model (EPDM) and Omurlu model (OM) are proposed for pressure loss

determination in intermittent flow based on weighting factor Τµ and

82

dimensionless group respectively. The pressure losses are estimated

accurately using EPDM for 0.1143m I.D – 0.05715m O.D horizontal

eccentric annuli when compared with experimental data. However, for

flow through 0.0932m I.D – 0.0488m O.D annular conduit, EPDM

overestimates the pressure losses at high liquid and gas flow rates, i.e.,

1 m/s superficial liquid velocity and 2.5 m/s superficial gas velocity. OM

determines the frictional pressure losses with a high accuracy for two-

phase flow through both annular geometries. OM is more accurate and

reliable as the amount of data compared is considered.

• For practical purposes, frictional pressure losses can be determined

using friction factor correlations proposed separately for each flow

pattern and flow properties of the mixture. A new mixture Reynolds

number NRemixl based on liquid holdup term is introduced for

intermittent flow. The friction factor equations are developed for NRemixl

< 100000 and NRemixl ≥ 100000 separately due to the friction factor and

new mixture Reynolds number relation.

83

RECOMMENDATIONS

This study is an important step for understanding the hydraulics of two-phase

flow through fully eccentric annuli. However, further studies are needed to

better comprehend the flow behavior of two-phase systems flowing through

horizontal fully eccentric annuli. Additional experimental and theoretical work

will allow analyzing the hydraulics of two-phase fluid and distinguishing

between the behaviors in circular pipe and annuli. The recommendations for

future works are as follows.

• Experiments should include more stratified flow, annular flow and

dispersed bubble flow data in order to complete the transition

boundaries of flow pattern map. The range for superficial liquid velocity

should be less than 0.1 m/s and greater than 2 m/s. The gas superficial

velocity should be between 0.1 m/s and 100 m/s. Then, the performance

of the proposed mechanistic model and the empirical correlations can be

evaluated and the accuracy may be generalized.

• Water and air was used during the experiments. Experiments with more

viscous liquid phase can be conducted in order to analyze the effect of

viscosity on flow pattern transition boundaries. Moreover, different fully

eccentric annular geometrical configurations are needed to generalize

the accuracy of the mechanistic model.

84

• The effect of eccentricity on flow pattern identification and pressure loss

determination may be analyzed using concentric and eccentric annular

conduits with different eccentricities.

• Flow patterns and flow pattern transitions may be recorded digitally by

high speed camera and may be analyzed to determine whether the flow

patterns in annular geometry are different than pipe flow.

• Experiments need to be conducted for better understanding the

determination of fully developed region in horizontal fully eccentric

annuli. Correlations different than pipe flow may be developed.

85

REFERENCES

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86

9. Angel, R.R. and Welchon, J.K.: “Low-Ratio Gas-Lift Correlation for

Casing-Tubing Annuli and Large-Diameter Tubing,” Drill.& Prod. Prac.

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10. Winkler, H.W.: “Single-and Two-Phase Vertical Flow Through 0.996 X

0.625-Inch Fully Eccentric Plain Annular Configuration,” PhD dissertation,

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Drop in Two-Phase Flow: An Approach Through Similarity Analysis”

AIChE J. (1964) 10, No. 1, 44.

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Holdup and Pressure Losses Occurring During Continuous Two-Phase

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14. Beggs, H.D., and Brill, J.P.: “A Study of Two-Phase Flow in Inclined

Pipes,” JPT (May 1973) 607; Trans., AIME, 255.

15. Dukler, A.E., and Hubbard, M.G.: “A Model For Gas-Liquid Slug Flow in

Horizontal and Near Horizontal Tubes,” Ind. Eng. Chem. Fundam. (1975)

14, 337.

16. Barnea, D.: “A Unified Model for Predicting Flow Pattern Transitions for

the Whole Range of Pipe Inclinations,” Int. J. Multiphase Flow (1987)

13, No. 1, 1.

17. Xiao, J.J., Shoham, O., and Brill, J.P.: “A Comprehensive Mechanistic

Model for Two-Phase Flow in Pipelines.” Paper SPE 20631 presented at

87

the 1990 SPE Annual Technical Conference and Exhibition, New Orleans,

23-26 September.

18. Barnea, D., Shoham, O., and Taitel, Y.: “Flow Patter Transition for

Vertical Downward Inclined Two-Phase Flow; Horizoontal to Vertical,”

Chem. Eng. Sc. 37, No. 5, 735-740 (1982).

19. Baker, A., Nielsen, K., and Gabb, A.: “Pressure loss, Liquid Holdup

Calculations developed,” Oil & Gas J., 55-59 (March 14, 1998).

20. Andritsos, N., and Hanratty, T. J.: “Influence of Interfacial Waves in

Stratified Gas-Liquid Flows,” AIChE J. 33, No. 3, 444-454 (1987).

21. Oliemans, R.V., A., Pots, B.F., and Trope, N.: “Modelling of Annular

Dispersed Two-Phase Flow in Vertical Pipes,” Int. J. Multiphase Flow 12,

No. 5, 711-732 (1986).

22. Gomez L.E., Shoham O., Schmidt Z., Chokshi R.N. and Northug T.:

“Unified Mechanistic Model for Steady-State Two-Phase Flow: Horizontal

to Vertical Upward Flow,” SPE Journal 5, No. 3, 339-350 (September

2000).

23. Ouyang, L.B., and Aziz, K.: “Development of New Wall Friction Factor

and Interfacial Friction Factor correlations for Gas/Liquid Stratified Flow

in Wells and Pipes,” paper SPE 35679 presented at the 1996 SPE

Western Regional Meeting, Anchorage, 22-24 May.

24. Petalas, N., and Aziz, K.: “A Mechanistic Model for Multiphase Flow in

Pipes,” JCPT 39, No. 6, 43-55 (2000).

25. Garcia, F., Garcia, R., Padrino, J.C., Mata, C., Trallero, J.L. and Joseph,

D.D.: “Power Law and Composite Power Law Friction Factor Correlations

for Laminar and Turbulent Gas-Liquid Flow in Horizontal Pipelines,” Int.

J. Multiphase Flow 29, 1605-1624 (2003).

88

26. Theofanous, T.G. and Hanratty, T.J.: “Appendix 1: Report of Study

Group on Flow Regimes in Multifluid Flow,” Int. J. Multiphase Flow 29,

1061-1068 (2003).

27. Sadatomi, M., Sato, Y., and Saruwatari, S.: “Two-Phase Flow in Vertical

Noncircular Channels,” Int. J. Multiphase Flow 8, 641-655 (1982).

28. Hasan, A.R., and Kabir, C.S.: “Two-Phase Flow in Vertical and Inclined

Annuli,” Int. J. Multiphase Flow 18, 279-293 (1992).

29. Caetano, E.F., Shoham, and O., Brill, J.P.: “Upward Vertical Two-Phase

Flow Through An Annulus, Part-I: Single Phase Friction Factor, Taylor

Bubble Velocity And Flow Pattern Prediction,” ASME J. Energy Resour.

Technol. 114, 1-13 (1992).

30. Salcudean, M., Chun, J.H., and Groeneveld, D.C.: “Effect of Flow

Obstructions on Void Distribution in Horizontal Air-Water Flow,” Int. J.

Multiphase Flow 9, 91-96 (1983).

31. Salcudean, M., Chun, J.H., and Groeneveld, D.C.: “Effect of Flow

Obstructions on the Flow Pattern Transitions in Horizontal Two-Phase

Flow,” Int. J. Multiphase Flow 9, 87-90 (1983).

32. Sunthankar, A.A.: “Study of the Flow of Aerated Drilling Fluids in Annulus

under Ambient Temperature and Pressure Conditions,” Ms. Thesis, The

University of Tulsa, 2000.

33. Zhou, L.: “Cuttings Transport with Aerated mud in Horizontal Annulus

under Elevated Pressure and Temperature Conditions,” PhD Dissertation,

The University of Tulsa, 2004.

34. Gucuyener, I.H.: “Design of Aerated Mud for Low Pressure Drilling,”

paperSPE 80491 presented at the SPE Asia Pacific Oil and Gas

Conference, Jakarta, 15-17 April, 2003.

89

35. Lage, A.C.V.M.; Rommetveit, R.; Time, R.W.: “An Experimental and

Theoretical Study of Two-Phase Flow in Horizontal or Slightly Deviated

Fully Eccentric Annuli,” SPE 62793, IADC/SPE Asia Pacific Drilling

Technology, Kuala Lumpur, Malaysia, 11–13 September 2000.

36. Aziz, K., Govier, G.W., and Fogarasi, M.: “Pressure Drop in Wells

Producing Oil and Gas,” J. Can. Petrol. Technol 11, 38 (1972).

37. Bendiksen, K.H.: “An Experimental Investigation of the Motion of Long

Bubbles in Inclined Tubes,” Int. J. Multiphase Flow 10, 467 (1984).

38. Niklin, D.J., Wilkes, J.O., and Davidson, J.F.: “Two Phase Flow in Vertical

Tubes,” Trans. Inst. Chem. Engr. 40, 61-68 (1962).

39. Zukoski, E. E.: “Influence of Viscosity, Surface Tension, and Inclination

Angle on Motion of Long Bubbles in Closed Tubes,” J. Fluid Mech. 25,

821-8337 (1966).

40. Weber, M.E.: “Drift in Intermittent Two-Phase Flow in Horizontal Pipes,”

Canadian J. Chem. Engg. 59, 398-399 (1981).

41. Gregory, G.A., Nicholson M.K., and Aziz K.: "Correlation of the Liquid

Volume Fraction in the Slug for Horizontal Gas-Liquid Slug Flow," Int. J.

Multiphase Flow 4, No. 1, 33-39 (1978).

42. Knudsen, J.G., and Katz, D.L.: “Fluid Dynamics and Heat Transfer,”

McGraw-Hill Book Co. Inc., New York City (1959)

43. ANSYS CFX version 8.0, Tutorial, Section Laminar to Turbulent Flow.

90

APPENDIX

A.1 Modified Petalas and Aziz Model

Pressure loss for intermittent flow can be determined using Equation A.1

( )1

sl gpf f

P P P

z z zη η

∆ ∆ ∆ − = + −

∆ ∆ ∆ (A.1)

where, the weighting factor, η, is the function slug length, Ls, to

slug unit length, Lu, ratio, s uL L . For 1.0η ≤

(0.75 )lE

lη λ−

= (A.2)

The translational velocity of liquid slug is calculated from equation

(A.3) given by Bendiksen37:

t o m dv C v v= + (A.3)

Co is taken as 1.2 as suggested by Niklin38. The drift velocity of

gas pocket is determined from Zukoski39 correlation:

d m dv f v ∞= (A.4)

91

where the correlation of vd∞ proposed by Bendiksen37 has only

horizontal velocity term and is determined by using Weber40

correlation:

0.56

( )1.760.54

r l g

d

l

gdv

Bo

ρ ρ

ρ∞

− = −

(A.5)

The Bond number is:

( )2( )l g

rBo gdρ ρ

σ

−= (A.6)

fm term in the Zukoski39 correlation is given as:

0.316 Remf ∞= (A.7)

for fm<1. Otherwise,

1mf = (A.8)

where

Re

2

l d r

l

v dρ

µ∞

∞ = (A.9)

92

Once these parameters are determined, liquid volume fraction in

the slug and the average liquid volume fraction of the slug unit

can be calculated from Gregory et al41 correlation and mass

balance equation of liquid phase over the slug unit.

1.39

1

18.66

ls

m

Ev

=

+

(A.10)

(1 )ls t g ls sg

l

t

E v v E vE

v

+ − −= (A.11)

The entrainment fraction, FE, is estimated from Petalas and Aziz24

correlation:

0.2

0.0740.735

1

sg

B

sl

vFEN

FE v

=

− (A.12)

where

2 2

2

l sg g

B

l

vN

µ ρ

σ ρ= (A.13)

93

The dimensionless liquid film thickness is determined from

equation (A.14)

( )11 (1 )

2

sl sg

l

sg

FE v vE

vδ

+ = − −

(A.14)

Once all the necessary parameters are defined, the frictional

losses for slug body can be determined

22

sl

ml m m

f r

f vP

z g d

ρ∆ =

∆ (A.15)

For the gas pocket/liquid film zone,

4

gp

wl

f r

P

z d

τ∆ =

∆ (A.16)

If δ < 0.0001, then

2

2gp

f L f

f

f vP

z

ρ∆ =

∆ (A.17)

From equation (A.1), total pressure loss is calculated.

94

For the stratified flow, pressure loss determination procedure is

similar the one presented in chapter 5. The differences are the

determination of liquid friction factor and interfacial friction factor

which are presented in Equations A.18 and A.19.

0.7310.452l slf f= (A.18)

1.3356

2(0.004 0.5 10 Re ) l r

i sl Frl

g g

d gf x N

v

ρ

ρ−

= +

(A.19)

The liquid diameter and gas diameter are as follows.

4 ll

l

Ad

S= (A.20)

4 g

g

g

Ad

S= (A.21)

Then, liquid and gas phase Reynolds number are calculated using

Equation A.22 and Equation A.23.

Re l l ll

l

d v ρ

µ= (A.22)

Reg g g

g

g

d v ρ

µ= (A.23)

95

The pressure loss for stratified flow can be calculated from

Equation 32 or Equation 33.

A.2 Modified Garcia et al Model

Pressure loss for intermittent flow can be determined using Equation A.24 and

A.25 for stratified flow.

0.9501 0.2

0.2534

0.1974.864

13.98 Re 0.0925 Re0.0925 Re

Re1

293

m mgar m

m

N Nff N

N

− −− −

= + +

(A.24)

2

2 mgar m

r

vPff

z dρ

∆=

∆ (A.25)

Similarly, the friction factor for intermittent flow is as follows;

0.9501 0.2629

0.2629

0.20293.577

13.98 Re 0.1067 Re0.01067 Re

Re1

293

m mgar m

m

N Nff N

N

− −− −

= + +

(A.26)

The frictional pressure loss for this flow pattern is calculated from Equation

A.25.

96

A.3 Modified Beggs and Brill Model

Flow pattern determination is as follows.

The flow regime is Segregated if λl<0.01 and NFrl<L1 or λl ≥0.01 and NFrl<L2

Where L1 and L2 are determined from Equations A.27 and A.28.

0.302

1 316 lL λ= (A.27)

1.4516

3 0.1 lL λ −= (A.28)

The transition zone between segregated and intermittent flow takes place if

λl ≥0.01 and L2 ≤ NFrl ≤L3. L2 can be determined from Equation A.29.

2.4684

2 0.0009232 lL λ −= (A.29)

The liquid holdup must be averaged as:

3 3( ) ( ) (int )

3 2 3 2

1Frl Frll transition l segregated l ermittent

L N L NH H H

L L L L

− −= + −

− − (A.30)

The intermittent flow takes place if 0.01 ≤ λl ≤0.4 and L3< NFrl ≤L1 or λl ≥0.4

and L3< NFrl ≤L4 where L4 is:

6.738

2 0.5 lL λ−= (A.31)

97

Similarly, the transition criterion for distributed flow is in case λl<0.4 and

NFrl≥L1 or λl ≥0.4 and NFrl ≥L4. Then dimensionless liquid holdup Hl(o) is

determined.

( )

b

ll o c

Frl

aH

N

λ= (A.32)

For all flow conditions Hl(o) ≥ λl should be satisfied. The friction factor is

determined from Dukler et al’15s method. Then the total pressure gradient is

calculated from Equation A.33.

(1 )

f

m m sg

P

zP

v vz

gP

ρ

∆

∆∆ =

∆−

(A.33)

This model is modified by using dr instead of pipe diameter.

A.4 Stratified-Intermittent Flow Transition

The condition for wave growth is as follows;

' ( ')( )g g l gP P h h gρ ρ− > − − (A.34)

98

Where hg and hg’ are the gas level in the conduit as shown in Figure A.1.

Figure A.1- The analysis of the forces during wave growth in the

conduit

From Bernoulli’s equation, Equation A.35 is obtained.

2 2'1' ( )

2 gg gP P v vρ− = − A.35

Substituting in Equation A.34,

2 2'( ) 2( ')( )g

l g

g g g

g

v v h h gρ ρ

ρ

−− > − A.36

Where from continuity equation gas velocity vg’ is:

2

2 2'

'

g

g g

g

hv v

h

=

A.37

P’ P

hg’ hg

hl’ hl’

vg vg’

vl

99

Equation A.37 can be expressed for round pipe in terms of Equation A.38.

2

2 2'

'

g

g g

g

Av v

A

=

A.38

and

''g g l lh h h h− = − A.39

Taylor expansion of Ag’ around Ag is:

'( )g

g g l l

l

dAA A h h

dh

= + −

A.40

Also writing Equation A.41, the derivative of gas flow area Ag to liquid flow area

Al can be taken.

g lA A A= − A.41

Ag’ is expressed in terms of hl and Al in Equation A.42.

' '( ) lg g l l

l

dAA A h h

dh

= − − −

A.42

100

Substituting the parameters calculated above in Equation A.36, Equation A.43

the criterion for transition to intermittent flow from stratified flow is obtained.

1 2

'

g l g

g g

g lg

l

Av gA

A dA

dh

ρ ρ

ρ

− >

A.43

Where

'

1g l

g r

A h

A d≅ − A.44

If the equilibrium level approaches to top of the conduit, the ratio of areas in

the left hand side of Equation A.44 goes to 0. Also, the right hand side of this

equation goes to 0. Therefore, the approximation presented in Equation A.44

can be used.

101

A.5 Pictures Taken During Experiments

Here is presented some pictures taken during experiments for configurations 1

and 2 annular geometries at different air and water flow rates.

Figure A.2- Stratified smooth flow through configuration 1

Figure A.3- Stratified wavy flow through configuration 1

102

Figure A.4- Stratified wavy flow through configuration 1

Figure A.5- Intermittent flow through configuration 1

103

Figure A.6- Intermittent flow through configuration 1

Figure A.7- Intermittent flow through configuration 1 at high liquid

and gas flow rates

104

Figure A.8- Intermittent flow through configuration 1 at high liquid

and gas flow rates

Figure A.9- Stratified smooth flow through configuration 2

105

Figure A.10- Stratified smooth flow through configuration 2

Figure A.11- Stratified wavy flow through configuration 2

106

Figure A.12- Stratified wavy flow through configuration 2

Figure A.13- Intermittent flow through configuration 2

107

Figure A.14- Intermittent flow through configuration 2