EFFECT OF PIPE ECCENTRICITY ON HOLE CLEANING AND

WELLBORE HYDRAULICS

A

THESIS

Presented to the Faculty of the

African University of Science and Technology

in Partial Fulfilment of the requirements for the

Degree of

MASTER OF SCIENCE

By

AKRONG, Joseph Annung

Abuja, Nigeria

December, 2010

EFFECT OF PIPE ECCENTRICITY ON HOLE CLEANING AND

WELLBORE HYDRAULICS

by

Akrong Joseph Annung

RECOMMENDED: ............................................................ Chair, Professor Godwin A. Chukwu

.......................................................... Professor Olurinde E. Lafe

........................................................... Professor Samuel Osisanya

APPROVED: .........................................................

Chief Academic Officer

......................................................... Date

i

ABSTRACT

The consideration of pipe eccentricity and its impact on hole cleaning has not been widely reported

in literature. Pipe eccentricity affects the average velocity of the drilling fluid in the annulus which

is one of the most important factors considered in hole cleaning.

In this study, developed equations of motion for a couette flow of non-Newtonian power-law fluid

through an eccentric annulus are used to determine the effect of pipe eccentricity on hole cleaning.

A graph for the velocity profile in the annulus is developed and the average velocity of the drilling

fluid for each eccentricity factor is used to analyse the parameters that indicate hole cleaning

efficiency. Some of the parameters include transport velocity, carrying capacity index, cuttings

concentration, and equivalent circulating density.

From the analysis, hole cleaning is said to be less effective when deviating from concentric annulus

to highly eccentric annulus. Secondly high eccentricity can cause high losses of the drilling fluid

after fracturing of the formation especially when the difference between the fracture pressure and

the pressure of the formation is very small.

ii

TABLE OF CONTENT

ABSTRACT i

TABLE OF CONTENT ii

LIST OF FIGURES v

LIST OF TABLES vii

ACKNOWLEDGEMENT viii

CHAPTER ONE 1

1.0 INTRODUCTION .............................................................................................................. 1

1.1 PROBLEM STATEMENT .................................................................................................. 3

1.2 OBJECTIVES .................................................................................................................... 3

1.3 APPROACH....................................................................................................................... 3

CHAPTER TWO 4

LITERATURE REVIEW 4

2.0 INTRODUCTION .............................................................................................................. 4

2.1 CUTTINGS REMOVAL ..................................................................................................... 4

2.1.1 Cuttings Transport in Wells ......................................................................................... 4

2.2 PARAMETERS AFFECTING HOLE CLEANING ............................................................. 7

2.2.1 Pipe Eccentricity ......................................................................................................... 8

2.2.2 Other Parameters ....................................................................................................... 9

2.3 COUTTE FLOW .............................................................................................................. 10

2.4 ECCENTRICITY EFFECTS ON ANNULUS PRESSURE DROP .................................... 11

2.5 HOLE CLEANING INDICATORS ................................................................................... 14

iii

2.5.1 Transport Ratio ......................................................................................................... 14

2.5.1 Carrying Capacity Index (CCI) ................................................................................. 15

CHAPTER THREE 16

DEVELOPMENT OF PERTINENT EQUATIONS 16

3.1 BASIC ASSUMPTIONS .................................................................................................. 16

3.2 GOVERNING EQUATIONS ............................................................................................ 16

3.3 GEOMETRIC ANALYSIS OF ECCENTRIC ANNULUS ................................................ 17

3.4 DIMENSIONLESS EXPRESSIONS ................................................................................ 19

3.5 LEADING-ORDER EQUATIONS AND SOLUTIONS .................................................... 20

3.6 FIRST-ORDER EQUATIONS AND SOLUTIONS ........................................................... 22

3.7 FLOW RATE .................................................................................................................... 23

3.8 MOORE’S CORRELATION FOR VS DETERMINATION ............................................... 25

CHAPTER FOUR 27

APPLICATION OF EQUATIONS 27

4.1 DIMENSIONLESS VELOCITY DETERMINATION ...................................................... 27

4.2 SLIP VELOCITY DETERMINATION USING MOORE’S CORRELATION ................ 36

4.3 AVAILABLE DATA (LIAO, 1993 AND CHUKWU, 2009)................................................... 37

CHAPTER FIVE 38

RESULTS AND DISCUSSION 38

5.1 VELOCITY PROFILE IN THE ANNULUS ................................................................... 38

5.2 TRANSPORT VELOCITY ............................................................................................ 38

5.3 CARRYING CAPACITY INDEX .................................................................................. 40

5.4 CUTTING CONCENTRATION .................................................................................... 41

5.5 EQUIVALENT CIRCULATING DENSITY................................................................... 42

5.5.1 Sample Problem ..................................................................................................... 43

iv

CHAPTER SIX 46

CONCLUSIONS AND RECOMMENDATIONS 46

6.1 CONCLUSIONS ........................................................................................................... 46

6.2 RECOMMENDATIONS ............................................................................................... 46

NOMENCLATURE 48

REFERENCES 51

APPENDIX 55

v

LIST OF FIGURES

Fig.2.1: Pipe eccentricity ................................................................................................................. 8

Fig.2.2. Effect of hole angle and weight on bit on eccentricity. ........................................................ 9

Fig 2.3: Effect of eccentricity on annulus pressure loss gradient for Newtonian fluid..................... 12

Fig 2.4: Relative pressure loss vs pipe eccentricity for five pipe rotation speeds; flow rate = 757

lt/min ............................................................................................................................................ 13

Fig 2.5: Relative pressure loss vs pipe eccentricity for five pipe rotation speeds; flow rate = 1514

lt/min ............................................................................................................................................ 13

Fig 2.6: Pipe rotation vs relative pressure loss for three different eccentricities; flow rate = 757

lt/min ............................................................................................................................................ 14

Fig 3.1 Geometric representation of an eccentric annulus .............................................................. 18

Fig.4.1. Dimensionless velocity against angle for dimensionless coordinate, x = 0.1 ..................... 29

Fig.4.2. Dimensionless velocity against angle for dimensionless coordinate, x = 0.2 ..................... 30

Fig.4.3. Dimensionless velocity against angle for dimensionless coordinate, x = 0.3 ..................... 31

Fig.4.4. Dimensionless velocity against angle for dimensionless coordinate, x = 0.4 ..................... 32

Fig.4.5. Dimensionless velocity against angle for dimensionless coordinate, x = 0.5 ..................... 33

Fig.4.6. Dimensionless velocity against dimensionless coordinate, x for angle, θ = 0.0o ................ 34

Fig.4.7: Dimensionless velocity against dimensionless coordinate, x for angle, θ = 15.0o .............. 35

Fig.5.1: Effect of eccentricity on transport velocity ....................................................................... 39

Fig.5.2: Effect of eccentricity on carrying capacity index .............................................................. 41

Fig.5.3: Effect of eccentricity on cuttings concentration ................................................................ 42

Fig.5.4: Effect of eccentricity on ECD ........................................................................................... 43

Fig.A-1: Dimensionless pressure versus pipe radii ratio for ε = 0.0................................................ 60

vi

Fig.A-2: Dimensionless pressure versus pipe radii ratio for ε = 0.1................................................ 61

Fig.A-3: Dimensionless pressure versus pipe radii ratio for ε = 0.2................................................ 62

Fig.A-4: Dimensionless pressure versus pipe radii ratio for ε = 0.3................................................ 63

Fig.A-5: Dimensionless pressure versus pipe radii ratio for ε = 0.4................................................ 64

Fig.A-6: Dimensionless pressure versus pipe radii ratio for ε = 0.5................................................ 65

Fig.A-7: Dimensionless pressure versus pipe radii ratio for ε = 0.6................................................ 66

Fig.A-8: Dimensionless pressure versus pipe radii ratio for ε = 0.7................................................ 67

Fig.A-9: Dimensionless pressure versus pipe radii ratio for ε = 0.8................................................ 68

Fig.A-10: Dimensionless pressure versus pipe radii ratio for ε = 0.9 .............................................. 69

vii

LIST OF TABLES

Table 4.1: Slip velocity determination ......................................................................................... 37

Table 5.1: ECD for different eccentricities .................................................................................. 45

Table 5.2: Hole Cleaning Indicators............................................................................................ 45

Table A.1: Velocity profile determination for concentric annulus (ε = 0) ..................................... 55

Table A.2: Velocity profile determination for eccentric annulus (ε = 0.50) .................................. 57

viii

ACKNOWLEDGEMENT

Pursuing masters degree is both a painful and an enjoyable experience. The eighteen months of

study in this institution has been accompanied with bitterness, hardships, frustration,

encouragement and trust and with so many people’s kind help. Though it will not be enough to

express my gratitude in words to all those people who helped me, I would still like to appreciate a

few whose contributions to successfully complete this study cannot be left out.

First of all, I’d like to give my sincere thanks to the almighty God. He gave me life, strength and

peace during my studies.

Special thanks go to my honorific supervisor, Prof. G.A. Chukwu, who accepted me as his MSc.

student without any hesitation when I expressed interest to work with him. He offered me advice,

patiently supervising me, and always guiding me in the right direction. I’ve learned a lot from him,

without his help I could not have finished my project successfully.

I also appreciate the input of the committee members, Prof. Olurinde Lafe, and Prof. Samuel

Osisanya for their critical comments, which enabled me to improve my work.

Last but not the least, I am greatly indebted to all the professors who taught me for the knowledge I

gained from them and to my family and friends especially Michael Akrong, Emelia Akrong, Dorcas

Karikari, Ebenezer Parker, Nii Odai and Mark Owusu, for their support through the period of my

study at AUST.

1

CHAPTER ONE

1.0 INTRODUCTION

Hole cleaning is the process of removing solid particles from the wellbore to the surface. These

solid particles are acted upon by four factors. They are Gravity, Viscous Drag, Impact and

Buoyancy. The transportation medium used to effect the removal of drilled solids is the drilling

fluid. Ability to lift particles of various sizes out of the hole is one of the most important functions

of drilling fluids. The factors which affect the carrying capacity of the fluid includes: fluid density

and rheology, annular velocity and flow regime, pipe rotation, cuttings density, size and shape of the

cutting, and annulus size and eccentricity. An optimum drilling fluid is expected to lift cuttings from

the wellbore and suspend them when circulation is stopped (Chukwu, 2009). The annulus size and

eccentricity is of great consideration in this study.

Effective hole cleaning is of great importance in oil well drilling operations, because inadequate

hole cleaning can lead to, but not limited to the following severe problems: fill, packing off, stuck

pipe and excessive hydrostatic pressure. Initially, it was considered that the primary purpose of the

mud was to remove the cuttings continuously (Chukwu, 2009).

Pipe eccentricity, usually expressed as a percentage, is a term that describes the deviation-from-

centre of a pipe within another pipe or open hole. A pipe is considered to be fully (100%) eccentric

if it lies against the inside diameter of the enclosing pipe or hole and concentric (0% eccentric) if it

is perfectly centred in the outer pipe or hole. One of the most important factors that affect the

carrying capacity of mud is the velocity profile in the annulus. The velocity profile, in turn, depends

on the annulus cross-sectional area, and the eccentricity of the inner core and its rotation. This work

involves the use of couette flow analysis to determine eccentricity. (Anon, 2010)

Couette flow is the phenomenon whereby the fluid is confined between two coaxial cylinders one of

which is stationary and the other is moving at a uniform velocity. This flow characteristic is

representative of flow in the wellbore annulus where the wall of the wellbore is represented by the

stationary cylinder and the drill string or casing pipe is represented by the moving cylinder. The

fluid average velocity is dependent on the velocity of the moving cylinder or pipe (Chukwu, 2009).

2

A real wellbore annulus is highly unlikely to be concentric and uniform through its entire length

especially in deviated wells in which the drill pipe lies along the “low side” of the wellbore annulus,

thereby creating extreme eccentricities. The eccentricity varies along the portions of the drill string

and it depends on the hole depth and its inclination angle with the vertical (Jawad and Akgun,

2002).

Hole cleaning is a more severe problem in high-angle holes than in vertical holes. It is not only

more difficult to carry the cuttings out of the hole, but they need to settle only to the low side of the

hole and causes problems like stuck pipe. Consequently, more attention should be paid to hole

cleaning requirements in directional holes (Annis and Smith, 1996).

From Zamora and Hanson (1991), four zones (0o-10o, 10o-30o, 30o-60o and 60o-90o) were

distinguished for critical hole cleaning and the zones where cleaning is most difficult are between

30o to 60o. This is due to slip of the cuttings bed and sagging. The experimental results confirmed

that probability of stuck pipe while drilling is higher when the angle is between 30o to 60o

(Saintpere and Marcillat, 2000). Tomren et al. (1983) also stated that increase in hole angle greatly

decreases the cuttings transport efficiency. He reported that 40o degree from vertical was found to

be the most difficult angle for hole cleaning (Zhou, 2006). Published experimental observations

(Jawad and Akgun, 2002) emphasized that during the transportation of cuttings in the annulus,

phase segregation occurs in the annulus. This causes cuttings accumulation on the low side of the

annulus, and leads to the cuttings bed formation in some drilling operations. Also, for the same

drilling conditions, the directional well requires higher flow rate than that of the vertical well to

obtain the same cleaning efficiency in both wells (Jawad and Akgun, 2002).

3

1.1 PROBLEM STATEMENT

In an eccentric annulus, fluid flows preferentially through the larger annulus which means that

varying the eccentricity will result in different rates at which the hole is cleaned. Increasing the

velocity of the fluid in the annulus is a way of trying to clean the hole efficiently. However, this

approach results in higher equivalent circulating density (ECD) which can lead to uncontrollable

losses after fracturing. There is therefore the need to investigate the effect of increasing or

decreasing eccentricity, on hole cleaning taking into consideration the actual mechanism of cutting

transport through an eccentric annulus.

1.2 OBJECTIVES

The objectives of this work are to:

1. Determine the velocity profile in the annulus for different eccentricities

2. Determine the effects of eccentricity on some hole cleaning indicators

3. Determine the effect of eccentricity on equivalent circulating density (ECD) and its impact

on hole cleaning.

1.3 APPROACH

The proposed approach consists of;

1. using the Couette flow analysis of Power-law fluids to determine the velocity for different

eccentricities

2. applying Moore’s correlation to determine slip velocity for different eccentricities

3. using graphical relationships to evaluate the impact of eccentricity on some hole cleaning

indicators

4

CHAPTER TWO

LITERATURE REVIEW

2.0 INTRODUCTION

The study of hole cleaning in deviated holes requires an understanding of the flow behaviour of

drilling fluids not only in a concentric annular geometry but also in an eccentric annular geometry

as well as the phenomena of transport of solids by fluids. Cuttings are mobilized and suspended

when the driving fluid forces acting on the solids are greater than the opposing gravitational and

frictional forces. These forces are dependent on the area open to flow and this area is a function of

eccentricity. Hence the study of hole cleaning in deviated wells requires an understanding of the

positioning of the inner pipe in the outer one (hole).

In this chapter, the existing literatures are reviewed. It should be noted that some important aspects

of hole cleaning like cuttings characteristics (density, shape and size) which are not within the scope

of this work are not discussed in details in this study.

2.1 CUTTINGS REMOVAL

2.1.1 Cuttings Transport in Wells

Early works to understand rheological effects on cutting transport in deviated wells were guided by

the successful approach employed in the cuttings transport study of vertical wells. In vertical wells,

the settling of the cuttings is opposed by the fluid drag force acting on it. This led to the

development of indices such as transport ratio and carrying capacity index, which in turn depend

on the knowledge of settling velocities of the cuttings through the drilling fluid, and the annular

velocity of the fluid through the bore hole. When the well is deviated from the vertical, it alters the

flow distribution around the annuli. This is due to the presence of settled cuttings and eccentricity of

the bore hole due to sagging of the drill string. (Vinod, 1994)

The Critical Transport Velocity (CTV) is defined as the minimum fluid velocity required to maintain

a continuous upward movement of the cuttings either by rolling or in full suspension. It is one of the

5

criteria used to evaluate efficiency of cuttings transport. Below this velocity, either a cuttings bed

will be formed or the cuttings will slide downwards towards the bottom of the hole. Cuttings

transport studies in deviated wells can be broadly classified as experimental, multi-layer modelling

and microscopic studies. The experimental studies focus on the simulation of the well bore

conditions in an experimental set up to understand the effect of the most relevant factors affecting

the carrying capacity of the fluid. These include fluid annular velocity, hole inclination, drilling

fluid rheology, rate of penetration of the drill bit, down hole geometry, particle size, density and

geometry and drill pipe rotary speed. In the multilayer modelling studies the problem is modelled as

a multiphase flow system, with the cuttings acting as the second phase. Microscopic studies involve

the estimation of the forces acting on an individual particle and evaluation of these forces as a

function of the microscopic variables (Vinod, 1994). A detailed description of the salient features of

these studies is attempted here.

Tomrem et al.(1983), from their experimental study have concluded that the major factors affecting

the carrying capacity of drilling fluids are fluid velocity, hole inclination and mud rheology. Actual

drilling fluids were used in their study and the cuttings concentration in the annulus was used as an

index for the evaluation of the carrying capacity of the fluid. Peden et al. (1990) and Ford et

al.(1990) from their experimental studies using clear model drilling fluids of the power law type

(Carboxyl Methyl Cellulose (CMC)/ Xantham Gum Bio-polymer (XC)) determined the minimum

transport velocities for various hole inclinations, inner pipe rotary speed and cuttings size. They

observed that the hole cleaning efficiency was dependent on the flow regime and the CTV appears

to pass through a maximum value for increasing viscosity. Hemphill and Larsen(1993), from their

experimental study of oil-based and water-based drilling fluids have concluded that fluid velocity

and the power law index of the fluid “n” are the significant factors affecting hole cleaning. The

study by Becker et al.(1991), which compared the effect of mud rheological properties on annular

drill cuttings build up in deviated wells, showed that the best correlations between mud rheological

parameters and cuttings transport performance were obtained with the low dial readings of the Fann

V-G meter. These results were consistent with the observations by Zamora and Hanson that drilling

fluids with higher low shear rate viscosities have better hole cleaning capabilities. (Zamora and

Hanson, 1991)

The multilayer models consider the shear stress at the solid-liquid interface as the driving force for

6

cuttings transport. The basis of the model is similar to the two phase liquid-liquid flow in pipes.

Wilson and Tse (1984) and Wilson (1976) have extended this approach to solid-liquid systems in

inclined pipes. It is assumed that the solids form a bed in the lower part of the annulus. The solids

are considered as one phase and the liquid as the other phase. The solids are assumed to move with

the fluid when the interfacial force at the solid-liquid interface and the pressure force at the end

overcome the friction force between the cuttings and the annulus wall. The solid phase was

modelled as two phases, the top layer moving in the direction of flow and the bottom layer near the

annular wall sliding down. Momentum equations for each phase (layer) are solved together with the

equations relating velocities, pressure drop and the thickness of each layer. Luo et al. (1990) using a

dimensionless group approach have shown that a modified version of Froude number and Reynolds

number based on local dynamic conditions near the cuttings bed can be used to correlate the flow

rate and cuttings concentration. The major drawback with these models is the evaluation of the

interfacial friction between cuttings and the liquid, and the cuttings and the annular wall. The

accuracy of these models depends on the accuracy of the equations used to describe these friction

coefficients. At present suitable values for these coefficients are chosen to fit the experimental data,

which imposes a restriction on extrapolating the information obtained from these models to other

experimental situations. (Vinod, 1994)

Rasi (1994) developed a hole cleaning criteria based on the free height above the cuttings. The

validity of the model was demonstrated with the help of experimental and field data. However it

should be kept in mind that the author assumes the existence of a cuttings bed and a fully offset

(100% eccentricity) drill pipe which may not be valid in non-horizontal wells. Luo et al. (1992)

developed simple charts to determine hole cleaning requirements in deviated wells. The model was

validated with field and experimental data, and involved a single parameter called the Transport

Index (TI). The individual factors which determine TI, namely, Rheology Factor (RF), Angle Factor

(AF) and Mud Weight (MW) were determined from charts. As expected, RF depends on the annular

geometry and new charts may be necessary for wells with long wash out sections. Iyoho and

Takahashi (1993) modelled the unstable cuttings transport regime to predict pressure fluctuations in

the bore hole from known drilling variables. The authors stated that, even though the present model

only gives trends at low solid concentrations, it can be used for the design and analysis of hole

cleaning parameters in horizontal wells. It should be noted that the model is based on Bernoulli and

continuity equations and does not take into account the non-Newtonian rheology effects.

The microscopic models take into account the forces acting on an individual cutting and assume

7

that drag and lift forces are the driving forces for cuttings transport. The forces acting on an

individual particle are gravity, buoyancy, drag and lift. In an inclined annuli the gravity force is

opposed by buoyancy together with the vertical component of lift and drag forces. If buoyancy and

the vertical component of drag and lift forces together are lower than gravitational forces, a bed will

be formed. If the drag forces together with the axial component of buoyancy is greater than the axial

component of gravity the particles will be transported up the annulus. The gravity forces and

buoyancy forces can be calculated easily but the calculation of drag and lift forces are not straight

forward and the correlations are dependent on the flow regime and the fluid rheology. Pilehvari et

al. (1999) used an empirical formula to calculate drag coefficient from the knowledge of Reynolds

number. A similar expression for lift forces was also assumed. A suitable hydraulic radius was

defined for the annuli for the computation of Reynolds number using a quasi-dimensional analysis

approach by Peden et al. (1990). Drag and lift forces were assumed to be functions of dimensionless

ratios and the empirical constants involved were evaluated from experimental data. (Peden et al,

1990)

Wicks (1971) assumed the presence of a particle bed and took into account the forces necessary for

a particle to roll over the particles on which it rests. His analysis is valid only for horizontal pipes

but the analysis can be extended to inclined annuli as well. He also assumed a laminar flow near the

particle even if the flow is turbulent in the bulk. This is probably a reasonable assumption near the

bed surface. The formation of a bed occurs only at high angles from the vertical and hence this

analysis might not be valid in low angle cases. Bed formation can alter the fluid flow over the bed

but this effect is not significant for open channels and can be easily taken into account for pipes.

However, for the case of eccentric annuli, this can be a difficult task even if we do not take into

account the difficulties due to non-Newtonian behaviour. Also as mentioned earlier these models

have inherent limitation of knowledge of the contribution from lift forces for which good empirical

correlations do not seem to be available even for relatively simple situations. (Vinod, 1994)

2.2 PARAMETERS AFFECTING HOLE CLEANING

8

2.2.1 Pipe Eccentricity

It is the term used to describe how off-centred a pipe is within another pipe or the open hole. It is

usually expressed as a percentage. A pipe would be considered to be fully (100%) eccentric if it

were lying against the inside diameter of the enclosing pipe or hole and concentric (0% eccentric) if

it were perfectly centred in the outer pipe or hole as shown in Fig.2.1 (Anon, 2010). Eccentricity

becomes important in field operations in estimating casing wear, wear and tear on the dill-string,

and the removal of cuttings from the low side of an inclined hole which is of importance to this

work. In an eccentric annulus, fluid flows preferentially through the wider annulus.

Fig.2.1: Pipe eccentricity. (Anon, 2010)

In the latter case, if the drill pipe lies on the low side of the hole (100% eccentric), the eccentricity

results in low-velocity fluid flow on the low side. Gravity pulls cuttings to the low side of the hole,

building a bed of small rock chips on the low side of the hole known as a cuttings bed. This cuttings

bed becomes difficult to clean out of the annulus and can lead to significant problems for the

drilling operation if the pipe becomes stuck in the cuttings bed, (Anon, 2010). Drill pipe

eccentricity was predicted as a function of hole inclination angle, weight on bit, and hole size. The

predicted values of eccentricity can be used to determine effective flow rate of the drilling fluid for

hole cleaning. It can also help in calculating the carrying capacity of the drilling fluid in the low

side of the annulus. Various hole angles were plotted versus eccentricities at different weights on bit

and for different hole sizes. It was concluded that as hole angle increases, eccentricity increases for

specific weights on bit. It was also observed that the drill pipe eccentricity is insignificant from

angles between 0o and 30o. However beyond this angle the drill pipe eccentricity is greatly affected

by the angle (Elsayed and Nasr El-Din, 2006). It was also shown that as the weight on bit increases

the drill pipe eccentricity decreases at any certain value of the hole inclination angle (Elsayed and

Nasr El-Din, 2006).This can be seen from the graph shown in fig. 2.2 below:

Partially Eccentric Fully Eccentric Concentric

9

Fig.2.2. Effect of hole angle and weight on bit on eccentricity. (Elsayed and Nasr El-Din, 2006)

2.2.2 Other Parameters

Fluid density and rheology are factors that affect the carrying capacity of the drilling fluid. The

density of the drilling fluid helps not only in carrying the cuttings, but also helps by providing the

pressure required in controlling the formation fluid. Fluid density is mostly altered using powdered

high density solids or dissolved salts to provide a hydrostatic pressure against the exposed

formations in excess of the pressure of the formation fluids. Fluid densities range from that of air

(1.0-2.0 ppg) to > 20.8 ppg. Most drilling fluids have densities greater than that of water. To

increase the fluid density, solid salts (barite) can be used in saturated salt brines. These salts are

mostly ground and packaged in several particle size grades (Colin Barker et al, 2007). According to

the experiment performed by Hopkin (1967), increasing the mud density from 8.8ppg to 15ppg

should reduce the slip velocity of the particle to about 50% of the slip velocity in water. However

this procedure was probably an impractical solution to improving carrying capacity unless mud

Hole Angle, degrees

Ecce

ntric

ity, %

10

weight is required to contain formation pressures.

Annular velocity and flow regime are other factors to be considered for carrying capacity or

cuttings removal. For cuttings to be removed efficiently, the annular flow velocity must be high

enough. The Critical Transport Velocity (CTV) is defined as the minimum fluid velocity required to

continuously maintain an upward movement of the cuttings either by rolling or in full suspension.

At velocities lower than the CTV, cuttings generated by the drill bit and other formation materials

will fall into the wellbore. The cuttings generated are transported up the annulus to the surface and

then separated. (Collin Barker et al., 2007)

Pipe rotation involves rotating the drill pipe during drilling. Limited laboratory test indicated that

thin, flat particles would turn on edge and result in a wide variation in their measured slip velocities.

From their results, rotating the drill pipe offset the torque effect and reoriented the particles to flat-

wise fall at rotary speeds as low as 35rpm (Hopkin, 1967).

Size and shape of the cuttings combined is one of the most important factors since it is the material

which is being carried from the bottom of the hole to the surface. The conclusions drawn from the

Wilson and Judge (1978) study for slurry flow in a single pipe are consistent with this study and

demonstrated that smaller particles are harder to clean out than larger ones when the particle size is

larger than 0.5mm but for the particles smaller than 0.5 mm, the smaller particles are easier to clean

out. The critical velocity required to transport different sizes of particles is also dependent on the

cuttings concentration and pipe eccentricity. (Walker and Li, 2000)

2.3 COUTTE FLOW

Fluid flow phenomenon whereby the fluid is confined between two coaxial cylinders one of which

is stationary and the other is moving at a uniform velocity is known as Couette flow. This flow

characteristic is representative of flow in the wellbore annulus where the wall of the wellbore is

represented by the stationary cylinder and the drill string or casing pipe is represented by the

moving cylinder. The fluid average velocity is dependent on the velocity of the moving cylinder or

pipe. The displacement of drilling fluid by the movement of the drill string or casing procedures

pressure variations in the borehole. The velocity of this displacement governs the magnitude of the

pressure change which can increase or decrease the hydrostatic pressure of the drilling fluid, to

11

produce surge or swab pressure, respectively. (Chukwu, 2009)

2.4 ECCENTRICITY EFFECTS ON ANNULUS PRESSURE DROP

The effects of eccentricity on the annular fluid flow are well studied From these studies, because no

exact solution is possible, approximated solutions are used (Luo and Pedan 1987; Escudier et al.

2001). However, numerical simulations have been performed to analyze the eccentric effects on the

annulus fluid flow (Ajay and Robello, 2009). Fig. 2.3 shows a variation of the annulus pressure loss

gradient with different eccentricity ε (in %). Eccentricity is defined by Eq. 2.1 and the pressure loss

gradient decreases with increase in eccentricity.

iRR

e=ε0

(2.1)

Where

e = offset of the inner pipe from the axis of the outer pipe and

Ro, Ri = inner radius of the outer pipe and outer radius of the inner pipe.

12

Fig 2.3: Effect of eccentricity on annulus pressure loss gradient for Newtonian fluid. (Ajay and

Robello, 2009)

From a paper by Hemphill and Ravi (2006), pressure drop results were calculated for five levels of

pump output (Q) as functions of drill string eccentricity and rotational speed. In their work, pressure

drops in the annulus were calculated for a fluid whose rheological properties are calculated by the

Herschel-Bulkley rheological model. This model is now the new API standard of oil well drilling

fluid rheology and hydraulic. The pressure drop results were presented as functions of pipe

eccentricity, axial flow rate, and pipe rotation speed. To present the results in a simple form, all

pressure drop results were scaled so that the base case of 100% represents the calculated pressure

drop for axial flow only at zero (0) eccentricity. All other calculated cases are then scaled relative to

this base case value and hence are called Relative dP/dL. (Hemphil and Ravi, 2006)

Results were shown for Q = 757 lt/min (200 gal/min) in Figure 2.4 and Q = 1514 lt/min (400

gal/min) in Figure 2.5. The results generally showed an increase in relative dP/dL with increasing

rotation speed across the range of pipe eccentricity. Between ε = 0 and ε = 0.25, there is little

change in pressure drop with rotation and eccentricity. However, pressure drop does begin to

increase with eccentricity after ε = 0.25 for all rotation speeds. This can be explained by the

13

narrowing annular gap under the rotating pipe. With the reduced gap width, the effect of the

rotational component of the coupled fluid velocity is greater. (Hemphil and Ravi, 2006)

Fig 2.4: Relative pressure loss vs pipe eccentricity for five pipe rotation speeds; flow rate = 757

lt/min. (Hemphil and Ravi, 2006)

Fig 2.5: Relative pressure loss vs pipe eccentricity for five pipe rotation speeds; flow rate = 1514

lt/min. (Hemphil and Ravi, 2006)

14

However, without rotation, it can be observed from their plot (Fig. 2.6) that relative pressure drop

decreases with increasing eccentricities. This also was true for very low rotational speeds up to

about 100 rpm. However, above this point, increasing the rotation speed increases the relative

pressure drop.

Fig 2.6: Pipe rotation vs relative pressure loss for three different eccentricities; flow rate = 757

lt/min. (Hemphil and Ravi, 2006)

2.5 HOLE CLEANING INDICATORS

2.5.1 Transport Ratio

Transport ratio is defined as the transport velocity (difference between the mean annular velocity

and the particle slip velocity) divided by the mean annular velocity. A positive value indicates that

some of the cuttings will be transported, and 100% indicates no cuttings remain in the hole(Bizanti

and Robinson, 1988). To optimize drill-cutting transport, the transport ratio should be maintained as

high as possible, though 100% in practice is not possible. To ensure adequate cutting transportation

to the surface, the drilling fluid velocity or the viscosity is usually increased depending on borehole

integrity.

15

Cuttings of different shapes move differently in the annulus. The fluid continuously move upwards

but the cuttings ride upward on different velocity profiles, rotate sideways, and fall downward. The

mathematical description of the process obviously is quite complicated. It involves the calculation

of slip velocity, models that describe transport constants, and descriptions of non-Newtonian flow in

annuli. (Robinson, 1993)

2.5.1 Carrying Capacity Index (CCI)

The three hole cleaning variables that can be controlled at the rig (mud weight, drilling fluid

viscosity, and annular velocity) improve hole cleaning when increased. Good hole cleaning is

indicated when the cuttings arrive at the surface with sharp edges. Rounded edges indicate tumbling

action in the annulus since the cuttings are not transported expediently to the surface (Robinson,

1993). In his studies he analyzed the cuttings removed by shale shakers for a period of 10 years and

came out with a dimensionless factor by multiplying the three drilling fluid parameters. For sharp

edge cuttings the factor was 400,000 and 200,000 for a round cuttings indicating that the particles

were grounded before they were transported to the surface. If the factor is about 100,000 or less it

gives an idea of small, almost grain sized particles. He used this concept to derive an equation for

the term carrying capacity index by dividing the product by 400000. This equation is shown below:

000,400

sVKMWCCI (2.2)

Good hole cleaning is expected when CCI equals 1 or above. The above equation is valid for

wellbore angle of deviation up to 350. Moreover, annular velocity for this equation should be the

lowest value encountered (e.g., for offshore operations, probably in the riser). The equation can be

rearranged if the value of CCI after the computation is low by assuming CCI to be one. Hence, the

value of K needed to transport cuttings to the surface can be predicted.

16

CHAPTER THREE

DEVELOPMENT OF PERTINENT EQUATIONS

3.1 BASIC ASSUMPTIONS

Some basic engineering assumptions are required to determine the flow of non-Newtonian Power-

law fluids through an eccentric annulus. The eccentric annular geometry is represented by two

cylinders positioned so that the inner cylinder moves with a uniform velocity, and the outer cylinder

is stationary. The average fluid velocity in the eccentric annulus is computed relative to the moving

inner pipe velocity. During drilling, this pipe velocity can be assumed to be the rate of penetration.

The following assumptions are made in the derivation of the equations applied in this study:

1) steady-state, single phase, incompressible fluid flow;

2) the flow is flow is isothermal with constant fluid properties;

3) slip effect is not considered;

4) gravitational effect is not considered; and

5) closed end pipe, i.e, no communication between the inside of the inner pipe and the annulus

3.2 GOVERNING EQUATIONS

From first principle, the equation of motion for a steady-state, incompressible fluid flowing through

an eccentric annulus can be expressed in cylindrical coordinates as

01)(1 ***

zP

rr

rrz

rz

(3.1)

Where P* is the dynamic pressure and z is measured axially in the direction of flow. The rheological

model that is used to represent power-law fluids in cylindrical coordinates can be written as

***rzrz (3.2)

17

***zz (3.3)

where the viscosity, µ* is defined by;

15.02*2**

n

rzrzK (3.4)

The shear tensor rate can be defined as

*

**

rVz

rz

(3.5)

*

** 1 zrz

Vr

(3.6)

Where Vz* is the local velocity of the fluid in the z-direction.

Considering a power-law fluid confined to the narrow space between two long cylindrical surfaces,

the inner one of which is eccentrically positioned and moving at a constant velocity, Vp, and the

outer cylinder (hole) is kept stationary. The equations below are the boundary conditions which

must be applied to solving these governing equations.

Piz VRV ),(* (3.7)

0)),(( ** RVz (3.8)

Where Ri is the outer radius of the inner or moving pipe and R*(θ) is the inside variable radius of

the outer or stationary pipe due to eccentricity.

3.3 GEOMETRIC ANALYSIS OF ECCENTRIC ANNULUS

The geometry of the eccentric annulus is shown in the Figure 3.1. Iyoho and Azar (1981) presented

the equation below to determine the local annular clearance h*(θ)

cossin)(*5.0222 eReRh io (3.9)

18

Where e is the offset between the centres of the two pipes, Ro is the inner radius of the outer or

stationary pipe and the angle θ is the eccentric angle.

The analysis which follows is based on the simplifying approximation that the annular clearance

between the two pipes is everywhere small compared with the mean radius. It is equivalent to

treating the annulus as if it were a slot of variable width. Under this assumption;

1

i

io

RRR

(3.10)

Fig 3.1 Geometric representation of an eccentric annulus. (Liao, 1993)

Expanding h*(θ) in power series of e/Ro, we obtain

422 sin

2cos1)(

o

io

o

io

RRR

RRRh (3.11)

Where ε is the fractional eccentricity and h (θ) is the dimensionless annular clearance defined by

io RR

hh

)(*)(

(3.12a)

io RRe (3.12b)

19

This value varies from 0 for a concentric annulus to 1 for a fully eccentric annulus. For a fully

eccentric operation (directional drilling),

io RR

e

(3.13)

Equation 3.11 may be rewritten in powers of δ, as

)()()()( 21 hhh o (3.14)

Where δ is the perturbation parameter defined by the equation

1

o

io

RRR

(3.15)

from equation 11, it can also be deduced that the dimensionless annular clearance terms are

cos1)( oh (3.16)

and

2

sin)(22

1

h (3.17)

3.4 DIMENSIONLESS EXPRESSIONS

The following dimensional scales are defined:

Length scale, ioo RRL (3.18.a)

Viscosity scale, KLV

n

o

P

1

(3.18.b)

Stress scale, KLV

n

o

P

(3.18.c)

Velocity scale, PP VV (3.18.d)

If we define a new dimensionless coordinate, x, then

20

xRr i 1* (3.19)

If equations 18.a-18.d are substituted into equation 1, then the equation of motion becomes

xzzxz

xx

1 (3.20)

Where τxz and τθz are dimensionless shear stresses and the dimensionless quantity Ω is given by

z

PV

L

P

o

*2

(3.21)

In terms of these new dimensionless scale units, the constitutive equations 3.2 and 3.3 become

xzxz (3.22)

zz (3.23)

And the dimensionless viscosity, µ is given by

15.022

n

zxz (3.24)

The dimensionless shear rate tensor can be expressed as

xV

xz

(3.25)

V

xz 1 (3.26)

Where V is the dimensionless velocity and the following defines the boundary conditions;

1),0( V (3.27.a)

0)),(( hV (3.27.b)

3.5 LEADING-ORDER EQUATIONS AND SOLUTIONS

21

For δ<<1, it is natural to employ the perturbation technique to seek a solution by expanding the

velocity term in powers of δ. Thus,

)( 21 VVV o (3.28)

It then follows that the shear stress components take the form

)( 210 xzxzxz (3.29.a)

)( 210 zzz (3.29.b)

Substituting equations 3.29.a and 3.29.b into equation 3.20, and substituting the expansion

coefficients of δ which is zero (zero order), the equation below is obtained:

xxz (3.30)

Similarly, the shear rate components are obtained as

)( 210 xzxzxz (3.31.a)

)( 210 zzz (3.31.b)

Where

11

11

00 ,, V

xV

xV

zxzxz (3.31.c)

Substituting equations 3.31.a and 3.31.b into the viscosity term of equation 3.24, and putting the

resulting expression for viscosity into equation 3.22 and 3.23, then equation 3.32 is obtained:

01

00 xzn

xzxz (3.32)

By applying the boundary conditions given in equations 3.7 and 3.8 to equations 3.30-3.32, the

leading order solution is obtained as follows:

)(;1)()(1

),(111

xxn

nxV nn

nn

no (3.33)

22

)(;)()()(1

),(111

xhxn

nxV nn

nn

no (3.34)

Where the implicit intermediary variable term, ξ(θ) is the value of x at which the value of τxz0 = 0.

At x = ξ(θ), equations 32 and 33 are equal and hence;

0))()(()(1)1()1(

n

nnn

hP (3.35)

Where

n

nnP

1

1

(3.36)

At assumed values of P and n for a given θ value, the corresponding values of ξ(θ) can be

determined from equation 3.35

3.6 FIRST-ORDER EQUATIONS AND SOLUTIONS

By substituting equations 3.28.a and 3.28.b into equation 3.20 and also substituting the expansion

coefficients of δ in the order of 1, the following equation is obtained;

001

xzxz

x

(3.37)

Similar to equation 3.31 for a leading order, equation 3.41 is obtained for first-order expression of

shear stress;

11

01 xzn

xzxz n (3.38)

Where γxz1 is defined in equation 3.31.c

By applying the boundary conditions to the leading-order solution, the velocity distributions given

by equations 3.39 and 3.40 are obtained

23

)(

;)()()()()12(2

1),(1112121

1

x

xFxn

xV nnnn

nn

n

(3.39)

)(;

)()(2

sin)()()(

)()()()12(2

1

),(1

0

221

0

1

12

0

12

1

1

xhhxF

hxn

xVnnn

nn

nn

n (3.40)

Where the intermediary variable term, F can be determined from equation 3.41

nn

nnn

nn

h

hhn

F11

0

1

0

221212

0

)()()(

)()(2

sin)()()()12(2

1

(3.41)

3.7 FLOW RATE

The fluid volumetric flow rate can be evaluated from equations 3.42;

)*(

**hR

Rz

i

i

ddrrVQ (3.42)

Let the dimensionless flow rate be defined by equation 3.43;

Pi VLR

QQ0

*

2 (3.43)

Equation 3.44 can be employed to expand the dimensionless flow rate termd in powers of δ, thus,

)( 21 QQQ o (3.44)

For a leading-order solution, the dimensionless flow rate is obtained from

0

)(

00

0

),(1 h

o ddxxVQQ (3.45)

24

Substituting the dimensionless velocity terms of equations 3.34 and 3.35 into equation 3.45, the

dimensionless flow rate given by equation 3.46 is obtained:

dhhPss

sQ s

0

1000 )()()(

21)(

211 (3.46)

Where

n

s 1 (3.47)

For the first-order solution, the dimensionless flow rate is expressed as

1QQQ o (3.48)

Where Q0 is obtained from equation 3.46 and Q1 is obtained from the equation below

0

)(

0101

0

),(1 h

ddxVxxVQ (3.49)

Substituting the velocity terms of equations 3.39 and 3.40 into equation 3.49 and evaluating the

resulting integrals, the following expressions are obtained:

0

432111 dIIIIQ (3.50)

Where

2

)()2)(3()1)(4()(

21 2

31

ssssI s (3.51)

shhsss

hPI

2

000

2 )()(2

)()()2)(3(

)()3()(

(3.52)

ss

ssssFPI 31

3 )()3(2

)1()( (3.53)

s

ss

hsss

hhsFPI

30

10

221

0

4

)()()3(2

)1(

)()(2

sin)()(

(3.54)

(Liao, 1993)

25

3.8 MOORE’S CORRELATION FOR VS DETERMINATION

Moore proposed that solid particles will slip through a mud system at a rate given by Eqn.3.55

fd

fsss C

dV

)(

4.92

(3.55)

The drag coefficient is obtained from a graph of particle Reynolds number versus drag coefficient (Chukwu,

2009). This particle Reynolds number is computed from

a

ssfp

dVR

47.15 (3.56)

For turbulent flow (Rp > 2000) and drag coefficient = 1.5. Hence using equation 3.57, Vs can be estimated.

fd

fsss C

dV

)(

4.92

(3.57)

For intermediate particle Reynolds number [(Rp > 3 < 300)] the drag coefficient is given by (transition from

laminar to turbulent).

p

d RC 22

(3.58)

Substituting equation 3.58 and 3.56 into equation 3.55 gives the equation below

333.0333.0

667.0)(175

fa

fsss

dV

(3.59)

For a particle Reynolds number of less than 3 (Rp ≤ 3) the flow is laminar, and the drag coefficient can be

determined from the equation:

p

d RC 40

(3.60)

26

Substituting equations 3.60 and 3.56 into equation 3.55 gives the equation below

a

fsss

dV

)(

49802

(3.61)

Where the apparent viscosity µa is given by

a

ph

ph

aa V

ddKn

ndd

V )(2003

124.22

(3.62)

(Chukwu, 2009)

27

CHAPTER FOUR

APPLICATION OF EQUATIONS

4.1 DIMENSIONLESS VELOCITY DETERMINATION

The effect of pipe eccentricity on hole cleaning can be best described by analysing the fluid

behaviour in the annulus. In this section dimensionless velocity developed in the previous chapter is

determined for each eccentricity factor. In order to obtain these dimensionless values, the following

procedure was followed:

1. From the available data, Ri/Ro is estimated.

2. Dimensionless pressure P is then estimated using fig. A-1 to A-10 in the appendix, and

assuming a value of n, for each eccentricity, ε. The parameters, δ and Ω are also determined.

3. The dimensionless leading order annular clearance term h0(θ) is then estimated from

equation 3.16 for different values of θ.

4. Similarly, the dimensionless annular clearance term h(θ) is also estimated from equation

3.11 for different values of θ.

5. The implicit intermediary variable term ξ(θ) is then determined numerically using wolfram

alpha software from equation 3.35 for different values of θ.

6. The intermediary variable parameter, F is calculated using equation 3.41

7. Knowing these parameters, the dimensionless leading order and first order velocities, V0 &

V1 respectively, are calculated for different θ and at different x using equations 3.33, 3.34,

3.39 & 3.40.

8. The dimensionless velocity, V is then calculated from the equation

28

1VVV o (4.1)

9. Steps 3 - 8 are then repeated for different eccentricities

Plots of dimensionless velocity (V) against angle (θ), and against dimensionless coordinate

(distance from moving pipe to stationary pipe or wall, x) for different eccentricities are

shown in figures 4.1 to 4.7 below;

29

Fig.4.1. Dimensionless velocity against angle for dimensionless coordinate, x = 0.1

-0.2000

-0.1000

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0 20 40 60 80 100 120 140 160 180

Dim

ensi

onle

ss V

eloc

ity

Angle,o

ε = 0 ε = 0.1 ε = 0.2 ε = 0.3 ε = 0.4

ε = 0.5 ε = 0.6 ε = 0.7 ε = 0.8 ε = 0.9

30

Fig.4.2. Dimensionless velocity against angle for dimensionless coordinate, x = 0.2

-1.2000

-1.0000

-0.8000

-0.6000

-0.4000

-0.2000

0.0000

0.2000

0.4000

0.6000

0 20 40 60 80 100 120 140 160 180

Dim

ensi

onle

ss V

eloc

ity

Angle,o

ε = 0 ε = 0.1 ε = 0.2 ε = 0.3 ε = 0.4

ε = 0.5 ε = 0.6 ε = 0.7 ε = 0.8 ε = 0.9

31

Fig.4.3. Dimensionless velocity against angle for dimensionless coordinate, x = 0.3

-2.0000

-1.5000

-1.0000

-0.5000

0.0000

0.5000

0 20 40 60 80 100 120 140 160 180

Dim

ensi

onle

ss V

eloc

ity

Angle,o

ε = 0 ε = 0.1 ε = 0.2 ε = 0.3 ε = 0.4

ε = 0.5 ε = 0.6 ε = 0.7 ε = 0.8 ε = 0.9

32

Fig.4.4. Dimensionless velocity against angle for dimensionless coordinate, x = 0.4

33

Fig.4.5. Dimensionless velocity against angle for dimensionless coordinate, x = 0.5

-2.5000

-2.0000

-1.5000

-1.0000

-0.5000

0.0000

0.5000

0 20 40 60 80 100 120 140 160 180

Dim

ensi

onle

ss V

eloc

ity

Angle,o

ε = 0 ε = 0.1 ε = 0.2 ε = 0.3 ε = 0.4

ε = 0.5 ε = 0.6 ε = 0.7 ε = 0.8 ε = 0.9

34

Fig.4.6. Dimensionless velocity against dimensionless coordinate, x for angle, θ = 0.0o

35

Fig.4.7: Dimensionless velocity against dimensionless coordinate, x for angle, θ = 15.0o

36

4.2 SLIP VELOCITY DETERMINATION USING MOORE’S CORRELATION

The apparent viscosity of a fluid can be defined as the noticeable internal resistance a fluid offers to

the flow. It is a rheological property calculated from rheometer readings on drilling fluids. In

determining the apparent viscosity, in order to use the correct equations for given Reynolds number

the following procedure is followed:

1. The apparent viscosity of the fluid is determined using equation 3.62

2. The slip velocity is also determined using any of equations 3.57, 3.59 or 3.61

3. The particle Reynolds number is determined using the slip velocity (Vs) from the step

above. Equation 3.56 is employed in this step.

4. The calculated Reynold’s number is compared to the range of the Rp in the particular Vs

(slip velocity equations used). If it falls within the Range of Rp, then the Vs calculated is

correct, otherwise use the Vs that matches the calculated Rp. Refer to table 4.1. below;

5. Hole cleaning indicators (transport velocity, carrying capacity index, etc) and equivalent

circulating density are then calculated for the different eccentricities.

The data above shows how the slip velocity was determined. Unlike the laminar and turbulent

column, Rp for intermediate corresponds to the initial guess and hence flow regime is intermediate.

37

Table 4.1: Slip velocity determination

Eccentricity (%)

Average Velocity (ft/min)

Apparent Viscosity

(cP)

Slip Velocity (ft/min)

RpSlip

Velocity (ft/min)

RpSlip

Velocity (ft/min)

Rp

0.00 188.99 13.52 400.20 1373.48 45.41 155.84 53.97 185.210.10 185.57 13.65 396.56 1348.63 45.41 154.42 53.80 182.970.20 179.74 13.87 390.28 1306.23 45.41 151.97 53.52 179.120.30 168.00 14.34 377.32 1220.95 45.41 146.93 52.92 171.240.40 157.17 14.83 364.95 1142.20 45.41 142.11 52.33 163.790.50 147.14 15.33 353.11 1069.30 45.41 137.50 51.76 156.750.60 136.73 15.90 340.40 993.67 45.41 132.55 51.13 149.270.70 124.13 16.69 324.33 902.10 45.41 126.29 50.32 139.950.80 109.91 17.73 305.19 798.74 45.41 118.84 49.31 129.050.90 96.68 18.91 286.23 702.59 45.41 111.46 48.27 118.48

Laminar Turbulent Intermediate

4.3 AVAILABLE DATA (Liao, 1993 and Chukwu, 2009)

Di = 7 inch

Do = 10 inch

n = 0.5

Vp = 150 fpm

ρf = 10 ppg

ρs = 22 ppg

ds = 0.3 inch

Cd = 1.5

db = 7 1/8 inch

D = 40 ft

K = 0.3 lb-secn/100ft2

38

CHAPTER FIVE

RESULTS AND DISCUSSION

From the appendices, the average annular velocity obtained from the excel spreadsheet for different

eccentricities was used to analyze the eccentricity effect on hole cleaning and some wellbore

hydraulics. This velocity was obtained by multiplying through the average dimensionless velocity

by the pipe velocity. The parameters analyzed include transport velocity, carrying capacity index,

cuttings concentration, and equivalent circulating density (ECD).

5.1 VELOCITY PROFILE IN THE ANNULUS

From figures 4.1 – 4.5, concentric annulus showed a uniform velocity profile from 0o-180o. For a

concentric annulus, the area open to flow is uniform throughout the annulus and hence velocity

distribution is constant.

For eccentricity of 0.1, velocity increases as the angle increases from 0o-180o. Initially, velocity for

eccentricity of 0.1 is smaller than that of concentric (0.0) which is due to the fact that area open to

flow is greater than that of concentric and hence lower velocity than that of concentric. However,

above 80o when the area open to flow for eccentricity of 0.1 has become less than that of the

concentric, the velocity becomes higher than that of concentric.

Similar results were obtained for the other eccentricity values. However, at higher eccentricities

(0.9), velocity becomes higher than that of concentric at about 40o. It can also be observed that at

higher eccentricities (above 0.7), there is a little decline between angles of 80o-120o. This could be

due to introduction of turbulence. From the results obtained for the velocities, Rp showed an

intermediate flow regime which explains this effect.

5.2 TRANSPORT VELOCITY

39

From table 4.1, the slip velocity and particle Reynolds Number (Rp) were calculated for the

different flow regimes. From the table, the flow regime that best describes the annulus average

velocity was that of the intermediate. This is because the particle Reynolds Number obtained for

that velocity coincided with that of intermediate flow regime, The Rp for both turbulent regime and

laminar regime calculated were not in the range established from literature. Hence, slip velocity

calculated for intermediate flow regime was subtracted from the annular velocity in order to obtain

the transport velocity. From fig 5.1, it can be clearly observed that the transport velocity decreases

with increase in eccentricity.

Fig.5.1: Effect of eccentricity on transport velocity

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tran

spor

t Vel

ocit

y (f

pm)

eccentricity (%)

40

This hole cleaning indicator implies that for a given depth, it will take less time for cuttings to reach

surface for a concentric annulus. As eccentricity increases, transport velocity decreases and hence,

cuttings take longer time to travel from bottom to surface. This can further be explained from

equation 5.1

timedistanceV (5.1)

5.3 CARRYING CAPACITY INDEX

One of the most important functions of a drilling fluid is the ability to transport the drilled particles

(cuttings) generated by the drill bit to surface through the wellbore annulus. This is commonly

referred to as the Carrying Capacity Index (CCI) of the drilling mud. This factor can be affected by

the velocity profile in the annulus.

From fig.5.2 and table 5.2, carrying capacity is shown to decrease as eccentricity increases. This

implies that the ability of the drilling fluid to lift cuttings from the bottom to surface is dependent

on the eccentricity. This eccentricity changes the fluid’s ability to carry the solids. The apparent

viscosity increases with increasing eccentricity as can be seen in table 4.1.

41

Fig.5.2: Effect of eccentricity on carrying capacity index

5.4 CUTTING CONCENTRATION

From fig. 5.3 and table 5.2, cutting concentration increased as eccentricity increases. For concentric,

cutting concentration is about 7% and increased to about 20% for eccentricity of 0.9. When cuttings

in the annulus are at an equilibrium concentration, the volume of cuttings flowing out of the annulus

equals the volume of cuttings being liberated at the bit. Depending on the velocity of cuttings,

annular area and cutting concentration, this volume of cuttings leaving the annulus per unit time can

be calculated.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

CCI

eccentricity (%)

42

Fig.5.3: Effect of eccentricity on cuttings concentration

From the results, it therefore implies that for a concentric annulus, the amount of cuttings left in the

annulus with respect to the amount of cuttings liberated at the bit is less than that of an eccentric

annulus. Similarly, that for the eccentric annulus increases as eccentricity increases. The equation

used to determine the cutting concentration is given by:

22

2

phsa

b

ddVVdROPCa

(5.2)

5.5 EQUIVALENT CIRCULATING DENSITY

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Ca

eccentricity (%)

43

Cuttings have a large effect on the effective circulating density and this is as a result of the excess

pressure due to the cuttings. From fig 5.4 and table 5.2, it can be observed that as the annular space

deviates from concentric, the pressure due to the cuttings increases which in effect causes an

increase in the equivalent circulating density.

For a depth of 40 ft and a fluid density of 10 ppg, the ECD recorded is about 10.8ppg for a

concentric annulus. This ECD increases to about 12.4 ppg for an eccentricity of 0.9. This implies

that, a formation with small mud weight window (offshore) can be fractured during drilling if

eccentricity effect is not considered.

Fig.5.4: Effect of eccentricity on ECD

5.5.1 Illustrative Example

44

At a depth of 6000ft, and with the available data given in chapter 4, determine the equivalent

circulating density at different eccentricities when drilling with a mud weight of 10ppg. If the

fracture gradient of the formation is 14.0ppg, what is the minimum eccentricity above which the

formation will be fractured? (Hydrostatic Pressure = Ph)

Solution

psiP

MWDP

h

h

3120106000052.0

052.0

Pressure drop due to cuttings (ΔPc) for concentric

psiP

P

CDP

c

c

afcc

37.267

071.0)00.1007.22(6000052.0

)(052.0

Equivalent circulating density (ECD) for concentric

ppgECD

ECD

DPMWECD c

85.10

6000052.037.2670.10

052.0

Below is the table that summarises the ECD calculations for eccentricity from 0.0-0.9

45

Table 5.1: ECD for different eccentricities

eccentricity (%) CaΔpcuttings

(psi) ECD (ppg)0 0.071 266.457 10.854

0.1 0.072 273.039 10.8750.2 0.076 285.044 10.9140.3 0.083 312.625 11.0020.4 0.091 343.198 11.1000.5 0.100 377.241 11.2090.6 0.112 420.337 11.3470.7 0.129 487.440 11.5620.8 0.158 593.719 11.9030.9 0.197 743.199 12.382

For a fracture gradient of 11.5ppg, it can be observed from the table above that at eccentricity of 0.7

and above, using a drilling fluid with mud weight of 10.0ppg will fracture the formation.

Table 5.2: Hole Cleaning Indicators

eccentricity (%)

Average Velocity

Slip Velocity

Transport Velocity

(fpm)Transport

Ratio CCI CaΔpcuttings

(psi) ECD (ppg)

0 188.992 53.966 135.026 0.714 1.285 0.071 1.776 10.8540.1 185.573 53.802 131.771 0.710 1.262 0.072 1.820 10.8750.2 179.738 53.517 126.221 0.702 1.222 0.076 1.900 10.9140.3 168.004 52.919 115.085 0.685 1.142 0.083 2.084 11.0020.4 157.168 52.335 104.833 0.667 1.069 0.091 2.288 11.1000.5 147.136 51.763 95.373 0.648 1.001 0.100 2.515 11.2090.6 136.729 51.135 85.595 0.626 0.930 0.112 2.802 11.3470.7 124.129 50.318 73.811 0.595 0.844 0.129 3.250 11.5620.8 109.908 49.309 60.599 0.551 0.747 0.158 3.958 11.9030.9 96.677 48.267 48.410 0.501 0.657 0.197 4.955 12.382

46

CHAPTER SIX

CONCLUSIONS AND RECOMMENDATIONS

6.1 CONCLUSIONS

From the analysis and discussion of results, the following conclusions are made in this study.

1. At constant flow rate, velocity increases with increase in angle (decrease in annular space)

for an eccentric annulus. This is similar for an eccentric annulus however, the average

velocity in the annulus decreases with increase in eccentricity (Table 4.1).

2. Increase in eccentricity reduces drilling fluid cleaning efficiency. Deviating from concentric

to nearly full eccentric annulus (0.0-0.9) reduces the transport velocity of the fluid and ther

carrying capacity of the fluid by about 64% & 49% respectively, whereas cuttings

concentration in the annulus is increased by about 64% for eccentricity of 0.9.

3. The effect of eccentricity goes a long way to affect the equivalent circulating density. As the

annulus deviates from concentric to eccentric and higher, cuttings concentration increases

creating excess pressure drop due to the cuttings. This increases the equivalent circulating

density as shown in the sample problem in chapter 5.

6.2 RECOMMENDATIONS

For a better understanding of the effect of pipe eccentricity on hole cleaning, the following

recommendations should be considered.

1. Development of relationships for the case of unsteady state flow of power-law fluids to

estimate eccentricity effect on hole cleaning.

2. The use of other rheological models can be applied since power-law fluid is not the only

model used in the industry.

47

48

NOMENCLATURE

CCI carrying capacity index

Cd drag coefficient

D depth, ft

dh inner diameter of outer pipe or hole, inch

dp outer diameter of inner or moving pipe, inch

ds particle diameter, inch

e offset of the inner pipe from the axis of the outer pipe

F intermediary variable

f function

h*(θ) annular clearance, inch

h(θ) dimensionless annular clearance

hmin minimum annular clearance, inch

h0(θ) leading order annular clearance, dimensionless

h1(θ) first order annular clearance, dimensionless

K power-law fluid consistency index, lbf-secn/ft2

n power-law fluid index, dimensionless

P dimensionless quality

P* surge or swab pressure, psi

P dimensionless pressure

Q dimensionless flow rate

Q* flow rate, ft3/sec

Q0 leading order flow rate, dimensionless

Q1 first order flow rate, dimensionless

R(θ) dimensionless radius

R*(θ) variable radius of the outer pipe due to eccentricity, inch

Rc radius of coupling or centralizer, inch

Ri outer radius of inner pipe, inch

Ro inner radius of outer pipe, inch

Rp particle Reynolds number, dimensionless

49

r* polar coordinate

r dimensionless coordinate

s reciprocal of n, dimensionless

V dimensionless velocity

V0 leading order velocity, dimensionless

V1 first order velocity, dimensionless

Va average velocity in annulus, ft/sec

Vp pipe velocity, ft/sec

Vs slip velocity, ft/sec

Vz* fluid flow velocity, ft/sec

x dimensionless coordinate

y intermediary variable

z coordinate along pipe axle

α pipe’s radii ratio, dimensionless

γrz* shear rate, 1/sec

γθz* shear rate, 1/sec

γrz dimensionless shear rate for exact annular model

γrz0 leading order dimensionless shear rate for exact annular model

γrz1 first order dimensionless shear rate for exact annular model

γxz dimensionless shear rate for narrow annular model

γxz0 leading order dimensionless shear rate for narrow annular model

γxz1 first order dimensionless shear rate for narrow annular model

γθz dimensionless shear rate

γθz0 leading order dimensionless shear rate

γθz1 first order dimensionless shear rate

δ perturbation parameter for narrow annular model, dimensionless

ε eccentricity, dimensionless

λ constant

µ dimensionless viscosity

µa apparent viscosity

dimensionless viscosity scale

50

ρf density of the fluid

ρs density of the solid particle

ξ intermediary variable

τrz dimensionless shear stress for exact annular model

τrz0 leading order dimensionless shear stress for exact annular model

τrz1 first order dimensionless shear stress for exact annular model

τxz dimensionless shear stress for narrow annular model

τxz0 leading order dimensionless shear stress for narrow annular model

τxz1 first order dimensionless shear stress for narrow annular model

τθz dimensionless shear stress

τθz0 leading order dimensionless shear stress

τθz1 first order dimensionless shear stress

Ω dimensionless quality

ω perturbation parameter for exact annular model, dimensionless

51

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55

θ h0 (θ) h (θ) ξ (θ) F 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

10.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

20.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

30.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

40.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

50.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

60.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

70.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

80.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

90.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

100.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

110.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

120.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

130.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

140.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

150.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

160.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

170.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

180.0 1.000 1.000 0.580 0.070 1.000 0.299 -0.162 -0.435 -0.568 -0.612 -0.616 -0.602 -0.528 -0.344 0.000 0.000 -0.188 -0.238 -0.214 -0.165 -0.126 0.114 0.091 0.045 0.001 0.000

V0, x = V1, x =

APPENDIX

Table A.1: Velocity profile determination for concentric annulus (ε = 0)

56

Table A.1: Velocity profile determination for concentric annulus (ε = 0) cont’d

θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

10.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

20.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

30.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

40.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

50.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

60.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

70.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

80.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

90.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

100.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

110.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

120.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

130.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

140.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

150.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

160.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

170.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

180.0 1.000 0.219 -0.264 -0.527 -0.639 -0.666 -0.567 -0.563 -0.509 -0.343 0.000

V, x =

57

Table A.2: Velocity profile determination for eccentric annulus (ε = 0.50)

θ h0 (θ) h (θ) ξ (θ) F 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0.0 1.500 1.500 0.801 0.144 1.000 0.017 -0.722 -1.251 -1.607 -1.823 -1.934 -1.975 -1.981 -1.975 -1.935 -1.826 -1.612 -1.260 -0.734 0.000

10.0 1.492 1.491 0.797 0.144 1.000 0.027 -0.703 -1.226 -1.575 -1.787 -1.894 -1.934 -1.939 -1.933 -1.891 -1.778 -1.560 -1.202 -0.669 0.000

20.0 1.470 1.465 0.786 0.145 1.000 0.056 -0.650 -1.152 -1.484 -1.682 -1.781 -1.814 -1.818 -1.810 -1.761 -1.639 -1.407 -1.031 -0.477 0.000

30.0 1.433 1.424 0.768 0.145 1.000 0.101 -0.566 -1.035 -1.342 -1.519 -1.604 -1.630 -1.631 -1.618 -1.560 -1.420 -1.165 -0.760 -0.170 0.000

40.0 1.383 1.368 0.745 0.145 1.000 0.158 -0.460 -0.888 -1.161 -1.313 -1.381 -1.398 -1.398 -1.377 -1.303 -1.140 -0.853 -0.408 0.231 0.000

50.0 1.321 1.299 0.718 0.144 1.000 0.223 -0.339 -0.720 -0.957 -1.083 -1.133 -1.143 -1.139 -1.107 -1.012 -0.818 -0.491 0.004 0.701 0.000

60.0 1.250 1.222 0.688 0.140 1.000 0.291 -0.213 -0.547 -0.747 -0.847 -0.881 -0.885 -0.877 -0.829 -0.708 -0.478 -0.104 0.448 1.213 0.000

70.0 1.171 1.138 0.657 0.134 1.000 0.357 -0.091 -0.381 -0.546 -0.623 -0.644 -0.645 -0.628 -0.562 -0.411 -0.141 0.284 0.897 1.734 0.000

80.0 1.087 1.050 0.628 0.125 1.000 0.417 0.018 -0.232 -0.368 -0.425 -0.437 -0.435 -0.408 -0.320 -0.139 0.173 0.648 1.322 2.231 0.000

90.0 1.000 0.963 0.603 0.113 1.000 0.467 0.109 -0.109 -0.222 -0.264 -0.270 -0.265 -0.226 -0.118 0.094 0.443 0.966 1.696 2.669 0.000

100.0 0.913 0.877 0.583 0.100 1.000 0.505 0.178 -0.017 -0.112 -0.144 -0.148 -0.138 -0.088 0.038 0.274 0.655 1.217 1.993 3.019 0.000

110.0 0.829 0.796 0.569 0.089 1.000 0.530 0.223 0.045 -0.040 -0.066 -0.068 -0.055 0.004 0.143 0.397 0.801 1.390 2.199 3.262 0.000

120.0 0.750 0.722 0.561 0.081 1.000 0.544 0.249 0.079 0.000 -0.023 -0.024 -0.009 0.055 0.202 0.467 0.884 1.489 2.316 3.401 0.000

130.0 0.679 0.657 0.558 0.078 1.000 0.550 0.260 0.094 0.017 -0.005 -0.005 0.011 0.077 0.227 0.497 0.920 1.532 2.367 3.461 0.000

140.0 0.617 0.601 0.557 0.077 1.000 0.552 0.263 0.098 0.022 0.001 0.000 0.017 0.083 0.234 0.505 0.930 1.544 2.382 3.478 0.000

150.0 0.567 0.558 0.557 0.077 1.000 0.552 0.263 0.098 0.022 0.001 0.000 0.017 0.084 0.235 0.506 0.931 1.545 2.383 3.480 0.000

160.0 0.530 0.526 0.557 0.077 1.000 0.552 0.263 0.098 0.022 0.001 0.001 0.017 0.084 0.235 0.506 0.931 1.545 2.384 3.480 0.000

170.0 0.508 0.506 0.556 0.077 1.000 0.552 0.263 0.099 0.023 0.002 0.001 0.018 0.085 0.236 0.507 0.932 1.547 2.385 3.482 0.000

180.0 0.500 0.500 0.556 0.077 1.000 0.552 0.264 0.099 0.023 0.002 0.002 0.018 0.085 0.236 0.507 0.933 1.547 2.386 3.483 0.000

V0, x =

58

Table A.2: Velocity profile determination for eccentric annulus (ε = 0.50) cont’d

θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0.0 0.000 -0.364 -0.521 -0.539 -0.475 -0.376 -0.279 -0.210 -0.185 0.161 0.092 -0.004 -0.104 -0.169 -0.154 0.000

10.0 0.000 -0.355 -0.505 -0.518 -0.451 -0.351 -0.255 -0.187 0.164 0.138 0.068 -0.029 -0.128 -0.192 -0.173 0.000

20.0 0.000 -0.327 -0.458 -0.458 -0.385 -0.284 -0.189 -0.125 0.106 0.075 0.001 -0.099 -0.197 -0.254 -0.224 0.000

30.0 0.000 -0.286 -0.388 -0.371 -0.289 -0.185 -0.094 -0.038 0.024 -0.016 -0.097 -0.199 -0.293 -0.340 -0.289 0.000

40.0 0.000 -0.237 -0.306 -0.269 -0.179 -0.076 0.009 0.055 -0.067 -0.118 -0.205 -0.308 -0.395 -0.424 -0.341 0.000

50.0 0.000 -0.187 -0.224 -0.170 -0.075 0.025 0.099 0.132 -0.150 -0.211 -0.305 -0.405 -0.479 -0.480 -0.353 0.000

60.0 0.000 -0.143 -0.153 -0.088 0.009 0.100 0.162 -0.181 -0.210 -0.281 -0.377 -0.469 -0.520 -0.483 -0.297 0.000

70.0 0.000 -0.109 -0.102 -0.031 0.060 0.141 0.188 -0.200 -0.241 -0.318 -0.409 -0.485 -0.504 -0.415 -0.156 0.000

80.0 0.000 -0.088 -0.072 -0.004 0.078 0.145 0.178 -0.191 -0.240 -0.316 -0.397 -0.447 -0.424 -0.273 0.071 0.000

90.0 0.000 -0.079 -0.064 -0.003 0.067 0.121 0.141 -0.159 -0.211 -0.281 -0.343 -0.362 -0.290 -0.070 0.365 0.000

100.0 0.000 -0.079 -0.071 -0.020 0.038 0.080 -0.092 -0.115 -0.164 -0.223 -0.264 -0.247 -0.125 0.161 0.683 0.000

110.0 0.000 -0.085 -0.085 -0.044 0.005 0.038 -0.047 -0.070 -0.115 -0.163 -0.182 -0.136 0.029 0.371 0.962 0.000

120.0 0.000 -0.090 -0.097 -0.063 -0.020 0.008 -0.016 -0.039 -0.080 -0.118 -0.124 -0.056 0.135 0.513 1.148 0.000

130.0 0.000 -0.093 -0.102 -0.071 -0.031 -0.005 -0.002 -0.025 -0.064 -0.099 -0.098 -0.021 0.183 0.576 1.231 0.000

140.0 0.000 -0.092 -0.102 -0.071 -0.032 -0.006 -0.001 -0.024 -0.063 -0.096 -0.094 -0.016 0.191 0.588 1.247 0.000

150.0 0.000 -0.092 -0.101 -0.070 -0.031 -0.004 -0.002 -0.026 -0.064 -0.098 -0.097 -0.018 0.188 0.584 1.243 0.000

160.0 0.000 -0.092 -0.101 -0.070 -0.031 -0.005 -0.002 -0.025 -0.064 -0.097 -0.096 -0.017 0.190 0.586 1.246 0.000

170.0 0.000 -0.093 -0.103 -0.072 -0.033 -0.007 0.000 -0.023 -0.061 -0.094 -0.092 -0.012 0.195 0.593 1.254 0.000

180.0 0.000 -0.093 -0.103 -0.073 -0.034 -0.008 0.002 -0.021 -0.060 -0.093 -0.090 -0.010 0.198 0.597 1.259 0.000

V1, x =

59

Table A.2: Velocity profile determination for eccentric annulus (ε = 0.50) cont’d

θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0.0 1.000 -0.139 -0.945 -1.482 -1.810 -1.984 -2.053 -2.065 -2.060 -1.906 -1.896 -1.828 -1.657 -1.333 -0.800 0.000

10.0 1.000 -0.125 -0.920 -1.448 -1.769 -1.937 -2.004 -2.014 -1.869 -1.874 -1.861 -1.791 -1.615 -1.284 -0.743 0.000

20.0 1.000 -0.085 -0.846 -1.348 -1.649 -1.804 -1.862 -1.868 -1.773 -1.777 -1.761 -1.681 -1.491 -1.140 -0.572 0.000

30.0 1.000 -0.022 -0.732 -1.194 -1.465 -1.599 -1.644 -1.646 -1.621 -1.625 -1.601 -1.505 -1.291 -0.906 -0.294 0.000

40.0 1.000 0.057 -0.591 -1.003 -1.237 -1.346 -1.377 -1.375 -1.427 -1.428 -1.391 -1.272 -1.022 -0.589 0.084 0.000

50.0 1.000 0.143 -0.435 -0.793 -0.989 -1.072 -1.091 -1.086 -1.204 -1.198 -1.142 -0.992 -0.696 -0.202 0.550 0.000

60.0 1.000 0.230 -0.279 -0.585 -0.743 -0.804 -0.812 -0.962 -0.967 -0.950 -0.869 -0.679 -0.327 0.241 1.086 0.000

70.0 1.000 0.311 -0.135 -0.394 -0.521 -0.562 -0.563 -0.730 -0.731 -0.698 -0.586 -0.349 0.067 0.719 1.667 0.000

80.0 1.000 0.380 -0.013 -0.234 -0.335 -0.363 -0.361 -0.517 -0.510 -0.456 -0.309 -0.019 0.466 1.205 2.261 0.000

90.0 1.000 0.433 0.081 -0.110 -0.193 -0.212 -0.210 -0.333 -0.316 -0.238 -0.054 0.288 0.842 1.666 2.826 0.000

100.0 1.000 0.471 0.147 -0.025 -0.096 -0.110 -0.187 -0.187 -0.158 -0.058 0.161 0.549 1.163 2.062 3.311 0.000

110.0 1.000 0.494 0.187 0.026 -0.038 -0.050 -0.088 -0.085 -0.046 0.073 0.319 0.743 1.402 2.358 3.674 0.000

120.0 1.000 0.506 0.207 0.052 -0.009 -0.019 -0.030 -0.025 0.021 0.151 0.413 0.860 1.547 2.536 3.893 0.000

130.0 1.000 0.511 0.216 0.063 0.004 -0.006 -0.006 0.000 0.049 0.185 0.455 0.911 1.610 2.614 3.989 0.000

140.0 1.000 0.512 0.219 0.067 0.008 -0.002 0.000 0.006 0.056 0.193 0.464 0.923 1.626 2.633 4.012 0.000

150.0 1.000 0.513 0.220 0.068 0.009 -0.001 -0.001 0.006 0.056 0.193 0.464 0.923 1.625 2.633 4.013 0.000

160.0 1.000 0.513 0.220 0.068 0.009 -0.001 0.000 0.007 0.057 0.193 0.465 0.924 1.627 2.635 4.014 0.000

170.0 1.000 0.513 0.219 0.068 0.009 -0.001 0.001 0.008 0.058 0.196 0.468 0.927 1.630 2.639 4.020 0.000

180.0 1.000 0.512 0.219 0.067 0.009 -0.002 0.002 0.009 0.059 0.197 0.469 0.929 1.632 2.642 4.023 0.000

V, x =

60

Fig.A-1: Dimensionless pressure versus pipe radii ratio for ε = 0.0, (Liao, 1993)

61

Fig.A-2: Dimensionless pressure versus pipe radii ratio for ε = 0.1, (Liao, 1993)

62

Fig.A-3: Dimensionless pressure versus pipe radii ratio for ε = 0.2, (Liao, 1993)

63

Fig.A-4: Dimensionless pressure versus pipe radii ratio for ε = 0.3, (Liao, 1993)

64

Fig.A-5: Dimensionless pressure versus pipe radii ratio for ε = 0.4, (Liao, 1993)

65

Fig.A-6: Dimensionless pressure versus pipe radii ratio for ε = 0.5, (Liao, 1993)

66

Fig.A-7: Dimensionless pressure versus pipe radii ratio for ε = 0.6, (Liao, 1993)

67

Fig.A-8: Dimensionless pressure versus pipe radii ratio for ε = 0.7, (Liao, 1993)

68

Fig.A-9: Dimensionless pressure versus pipe radii ratio for ε = 0.8, (Liao, 1993)

69

Fig.A-10: Dimensionless pressure versus pipe radii ratio for ε = 0.9, (Liao, 1993)