# Torque and drag calculations in three-dimensional wellbores

TABLE OF CONTENTS

Page ABSTRACT ..................................................................................................................... iii

ACKNOWLEDGMENTS .................................................................................................. v

TABLE OF CONTENTS ................................................................................................ vii

LIST OF FIGURES ...........................................................................................................xi

LIST OF TABLES ........................................................................................................ xvii

CHAPTER

I INTRODUCTION ............................................................................................ 1

1.1 Literature Review ....................................................................................... 2 1.1.1 T&D equations for three-dimensional wellbore ........................ 2 1.1.2 Tortuosity effects ....................................................................... 3 1.1.3 Stress concentration factor ......................................................... 4 1.1.4 Buckling ..................................................................................... 4 1.2 Objectives and Organization ...................................................................... 5

II TORQUE AND DRAG CALCULATIONS IN THREE- DIMENSIONAL WELLBORE METHODOLOGY ....................................... 7

2.1 Introduction ................................................................................................ 7 2.2 Original Concept for Calculating Normal Contact Force .......................... 7 2.3 Soft-string Model for Three-dimensional T&D Calculation .................... 11 2.4 Lowering the Pipe into the Hole .............................................................. 13 2.4.1 Lowering the pipe into the hole in the build section................ 13 2.4.2 Lowering the pipe into the hole in the hold section ................. 15 2.4.3 Lowering the pipe into the hole in the drop section ................ 16 2.4.4 Lowering the pipe into the hole while the wellbore turns ......................................................................................... 19 2.5 Pulling the Pipe out of the Hole ............................................................... 21 2.5.1 Pulling the pipe out of the hole in the build section ................ 21 2.5.2 Pulling the pipe out of the hole in the hold section ................. 24 2.5.3 Pulling the pipe out of the hole in the drop section ................. 25 2.5.4 Pulling the pipe out of the hole while the wellbore turns ......................................................................................... 28 2.6 Conclusion ................................................................................................ 30

viii

CHAPTER Page

III WELL PLANNING IN THREE-DIMENSIONAL WELLBORE ................ 31

3.1 Introduction .............................................................................................. 31 3.2 System Modeling ...................................................................................... 32 3.2.1 Well-planning and math modeling between surveys ............... 32 3.2.1.1 Finding minimum curvature (DLS) to intersect target ........................................................ 33 3.2.1.2 Example tie-on surveys and target directions ................................................................ 36 3.2.2 Well-planning and math modeling for build type .................... 38 3.2.3 Well-planning and math modeling for build and hold type........................................................................................... 39 3.2.4 Well-planning and math modeling for build hold and drop type .................................................................................. 39 3.2.5 Well-planning and math modeling for horizontal well design type ............................................................................... 40

IV TORTUOSITY IN THREE-DIMENSIONAL WELLBORES AND THE EFFECT OF TORTUOSITIES ON TORQUE CALCULATION ........................................................................................... 42

4.1 Introduction .............................................................................................. 42 4.2 Oscillation in the Wellbore....................................................................... 42 4.3 Borehole Oscillations ............................................................................... 44 4.4 Model of Borehole Oscillations ............................................................... 48 4.5 Mathematical Model for Torque Calculations ......................................... 50

V STRESS CONCENTRATION WITHIN TOOL JOINT ............................... 52

5.1 Introduction .............................................................................................. 52 5.2 Stress Concentration ................................................................................. 52 5.3 Mathematical Model for the Stress Concentration Factor........................ 55

VI BUCKLING ................................................................................................... 59

6.1 Introduction .............................................................................................. 59 6.2 System Modeling for a Deviation Wellbore............................................. 60 6.3 Buckling in a Vertical Well ...................................................................... 61

ix

CHAPTER Page

VII NUMERICAL METHOD SOLVING T&D CALCULATION ..................... 63

7.1 Introduction .............................................................................................. 63 7.2 Euler’s Theory .......................................................................................... 63 7.2.1 Euler’s method ......................................................................... 63 7.2.2 Geometric description .............................................................. 65 7.2.3 Step size versus error ............................................................... 66 7.2.4 Example of the step size effect in Euler’s method................... 67 7.2.5 Euler’s method calculated in a three-dimensional wellbore ................................................................................... 68

VIII DISCUSSION OF RESULTS ........................................................................ 72

8.1 Introduction .............................................................................................. 72 8.2 Mathematical Models for Three-dimensional Wellbores ......................... 72 8.2.1 Lowering the pipe into the hole ............................................... 72 8.2.1.1 Build section .......................................................... 72 8.2.1.2 Drop section ........................................................... 73 8.2.2 Pulling the pipe out of the hole ................................................ 74 8.2.2.1 Build section .......................................................... 74 8.2.2.2 Drop section ........................................................... 75 8.3 Soft-string Model for Three-dimensional T&D Calculations .................. 76 8.4 Example and Comparison in Force Calculations ..................................... 77 8.5 Example and Comparison in Torque Calculations ................................... 79 8.5.1 Example while rotating off the bottom .................................... 79 8.5.2 Example for survey calculations .............................................. 84

IX CONCLUSIONS AND RECOMMENDATIONS ......................................... 87

9.1 Conclusions .............................................................................................. 87 9.2 Recommendations .................................................................................... 88

NOMENCLATURE ......................................................................................................... 89

REFERENCES ................................................................................................................. 93

APPENDIX A .................................................................................................................. 96

APPENDIX B ................................................................................................................ 107

APPENDIX C ................................................................................................................ 115

APPENDIX D ................................................................................................................ 123

x

Page

APPENDIX E ................................................................................................................. 133

APPENDIX F ................................................................................................................. 139

VITA .............................................................................................................................. 149

xi

LIST OF FIGURES FIGURE Page

1.1 Schematic of forces acting on downhole tubular assembly ............................... 5

2.1 Illustration of forces in build-up section (vertical view) ................................... 8

2.2 Illustration of forces in inclined section (vertical view) .................................... 9

2.3 Illustration of forces in drop section (vertical view).......................................... 9

2.4 Illustration of forces while the wellbore turns to the right (horizontal view) ................................................................................................................ 10

2.5 Illustration of forces while the wellbore turns to the left (horizontal view) ................................................................................................................ 11

2.6 Soft-string T&D model schematic ................................................................... 12

2.7 Illustration of forces in build-up section (lowering the pipe into the hole, vertical view) ................................................................................................... 13

2.8 Illustration of differences between positive and negative forces in build- up section (lowering the pipe into the hole, vertical view) .............................. 14

2.9 Illustration of force in hold section (lowering the pipe into the hole, vertical view) ................................................................................................... 15

2.10 Illustration of forces in drop section (lowering the pipe into the hole, vertical view) ................................................................................................... 16

2.11 Illustration of differences between positive and negative forces in drop section (lowering the pipe into the hole, vertical view) ................................... 18

2.12 Illustration of forces when the wellbore turns right (lowering the pipe into the hole, horizontal view) ......................................................................... 19

2.13 Illustration of forces when the wellbore turns left (lowering the pipe into the hole, horizontal view) ................................................................................ 20

2.14 Illustration of forces in build-up section (pulling the pipe out of the hole, vertical view) ................................................................................................... 21

xii

FIGURE Page

2.15 Illustration of differences between positive and negative forces in build- up section (pulling the pipe out of the hole, vertical view) ............................. 23

2.16 Illustration of forces in the hold section (pulling the pipe out of the hole, vertical view) ................................................................................................... 24

2.17 Illustration of forces in the drop section (pulling the pipe out of the hole, vertical view) ................................................................................................... 25

2.18 Illustration of differences between positive and negative forces in the drop section (pulling the pipe out of the hole, vertical view) .......................... 27

2.19 Illustration of forces when the wellbore turns right (pulling the pipe out of the hole, horizontal view) ............................................................................ 28

2.20 Illustration of forces when the wellbore turns left (pulling the pipe out of the hole, horizontal view) ................................................................................ 29

3.1 Wellbore build & hold type in three-dimensional wellbore ............................ 32

3.2 Horizontal view of wellbore ............................................................................ 37

3.3 Vertical view of wellbore ................................................................................ 37

3.4 3D wellbore path by MATLAB version 7.4.0 ................................................. 38

3.5 Build type well design ..................................................................................... 38

3.6 Build and hold type well design ...................................................................... 39

3.7 Build hold and drop type well design .............................................................. 40

3.8 Horizontal well design ..................................................................................... 41

4.1 Spiral borehole as shown in 2D (Tracks 1 and 2) and 3D images ................... 44

4.2 An MWD survey tool cannot detect a tight spiral ........................................... 45

4.3 Illustrates the evidence of profound spiraling.................................................. 46

4.4 3D CAD model of 12-1/4” borehole................................................................ 47

4.5 Rippling 2D oscillation .................................................................................... 48

xiii

FIGURE Page

4.6 Spiraling 3D corkscrew ................................................................................... 49

4.7 Hour-glassing cyclic hole enlargement............................................................ 49

4.8 Showing the two-dimensional schematic of the drift equation........................ 50

5.1 Torque-turn curve ............................................................................................ 53

5.2 Representative torque-turn curve with torque ranges shown........................... 54

5.3 Cross-sectional view of a typical premium connection shown........................ 54

5.4 A comparison of a premium casing connection and a proprietary rotary shouldered connection ..................................................................................... 54

6.1 Sinusoidal buckling of the pipe in a horizontal wellbore ................................ 60

6.2 Helical buckling of the pipe in a horizontal wellbore ...................................... 60

6.3 Helical buckling in vertical wellbores ............................................................. 62

7.1 Euler’s approximations y k−1 = y k + h f (t k ,y k ) ................................................. 65

7.2 Comparison of Euler solutions with different step sizes ................................. 68

7.3 Illustrates force in the build-up section (lowering the pipe into the hole, vertical view) ................................................................................................... 69

7.4 Illustrates the differences between positive and negative forces in the build-up section (lowering the pipe into the hole) ........................................... 70

7.5 Comparison between Wu and Juvkam-Wold’s (1991) equations and numerical methods ........................................................................................... 71

8.1 Wellbore geometry for this example ............................................................... 77

8.2 Comparison between Wu and Juvkam-Wold’s (1991) equations and numerical methods ........................................................................................... 78

8.3 Axial tension plot for this example using Wu and Juvkam-Wold’s (1991) method .................................................................................................. 81

xiv

FIGURE Page

8.4 Torque plot for this example using Wu and Juvkam-Wold’s (1991) method ............................................................................................................. 81

8.5 Axial tension plot for this example using numerical method .......................... 83

8.6 Torque plot for this example using numerical method .................................... 84

8.7 3D wellbore path by MATLAB version 7.4.0 ................................................. 85

8.8 Normal contact force (lb/ft) versus measured depth (ft).................................. 86

8.9 Axial force (lb) versus measured depth (ft) ..................................................... 86

A.1 T&D calculations for 3D well planning user form .......................................... 97

A.2 Showing enlarge picture application in 3D software ....................................... 98

A.3 Build type user form ........................................................................................ 99

A.4 Build & hold type user form .......................................................................... 100

A.5 Build hold & drop type user form .................................................................. 102

A.6 Horizontal wellbore user form ....................................................................... 103

A.7 T&D calculation between survey user form .................................................. 104

A.8 3D wellbore path by MATLAB version 7.4.0 ............................................... 106

B.1 Wellbore schematic for example in Appendix B ........................................... 108

B.2 Normal contact force (lb/ft) versus measured depth (ft) for this example using Wu and Juvkam-Wold’s (1991) method .............................................. 109

B.3 Axial tension plot for this example using Wu and Juvkam-Wold’s (1991) method ................................................................................................ 110

B.4 User dorm of Appendix B example ............................................................... 111

B.5 Result user form of Appendix B example ..................................................... 112

B.6 Normal contact force (lb/ft) versus measured depth (ft)................................ 113

xv

FIGURE Page

B.7 Axial tension force (lb) versus measured depth (ft) (F > 0 referred to tensile force) .................................................................................................. 114

C.1 Wellbore Schematic for example in Appendix C .......................................... 116

C.2 Normal contact force (lb/ft) versus measure depth (ft) for this example using Wu and Juvkam-Wold’s (1991) method .............................................. 117

C.3 Axial tension plot for this example using Wu and Juvkam-Wold’s (1991) method ................................................................................................ 118

C.4 User form of Appendix C example ................................................................ 119

C.5 Result user form of Appendix C example ..................................................... 120

C.6 Normal contact force (lb/ft) versus measured depth (ft)................................ 121

C.7 Axial force (lb) versus measured depth (ft) (F < 0 referred to tensile force) .............................................................................................................. 122

D.1 Wellbore schematic for example in Appendix D........................................... 124

D.2 Force table for Appendix D example ............................................................. 126

D.3 Normal contact force (lb/ft) versus measure depth (ft) from Wu and Juvkam-Wold’s (1991) equation ................................................................... 127

D.4 Axial tension plot for this example using Wu and Juvkam-Wold’s (1991) method ................................................................................................ 128

D.5 User form of Appendix D example................................................................ 129

D.6 Result user form of Appendix D example ..................................................... 130

D.7 Normal contact force (lb/ft) versus measured depth (ft)................................ 131

D.8 Axial force (lb) versus measured depth (ft) (F < 0 referred to tensile force) .............................................................................................................. 132

E.1 Wellbore schematic for example in Appendix E ........................................... 134

E.2 Result user form with 2D wellbore trajectory (no left/right turn) ................. 135

xvi

FIGURE Page

E.3 Axial force (lb) versus measured depth (ft) (F < 0 referred to tensile force) from 2D wellbore trajectory ................................................................ 136

E.4 Result user form with 3D wellbore trajectory (10 deg/100ft left/right turn) ................................................................................................................ 137

E.5 Axial force (lb) versus measured depth (ft) (F < 0 referred to tensile force) from 3D wellbore trajectory ................................................................ 138

F.1 Comparison between push-the-bit results and point-the-bit-results .............. 141

F.2 The standard BHA configuration ................................................................... 143

F.3 Illustrates deflection in the bottom hole assembly......................................... 144

xvii

LIST OF TABLES TABLE Page 3.1 Input field data for tie-on surveys and target directions .................................. 36

5.1 Maximum SCF values and locations in shouldered connections .................... 57

7.1 Showing parameters for build section ............................................................. 69

8.1 Showing parameters ......................................................................................... 78

8.2 Input field data for tie-on surveys and target directions for 3D wellbore paths ................................................................................................................. 85

F.1 Listing of generic T&D reduction techniques ............................................... 145

F.2 Advantages and disadvantages of T&D reduction techniques ...................... 146

1

CHAPTER I INTRODUCTION Excessive torque and drag in the design of a wellbore trajectory and drillstring configuration might cause severe damage to a device that turns the drillstring (topdrive) capacity, drillpipe strength, and available lifting capacity. It can increase pipe fatigue, casing wear, and mechanical borehole problems, such as hole enlargement and can lead to an inability to slide. Moreover, a conventional steerable assembly might increase frictional forces, which can lead to failures in the tubular from excessive wear, bucking, and collapse.

If helical bucking is unavoidable, then torque and drag (T&D) models must be more robust if they are to accurately calculate the additional drag created in the post-buckled portion of the string. This is essential to predict the loss of weight on a bit, the potential for lock-up, and the impact on fatigue (Haduch, Procter, and Samuels 1994).

According to over two decades of petroleum literature that addresses Torque and Drag (T&D) software, the basic mathematical model that underlies most T&D software has not changed significantly since its original inception. Now is the right time to reflect on the state of current models and identify the future requirements because T&D software is commonly used during planning processes (Adewuya and Pham 1998).

This dissertation follows the style and format of SPE Drilling and Completion.

2

A new model will help engineers identify feasible well designs and define drilling limitations for particular field development options. A reliable mathematical model is fundamental to a true understanding of the accuracy and applicability of T&D models. Software based on a more accurate T&D mathematical model for each particular well design will be highly useful in well planning design processes and will prevent the problems caused by T&D.

1.1 Literature Review T&D calculations and other information need to be changed in T&D software including T&D equations, tortuosity effects, stress concentration factor, and buckling.

1.1.1 T&D equations for three-dimensional wellbore Mason and Chen (2007) states that T&D modeling is regarded as an invaluable process in well planning for assisting and predicting, as well as preventing, drilling problems. Although T&D software has been developed for over 20 years, some confusion still exists over the validity of the models. Meanwhile, Exxon production research has developed soft string models for T&D equations. The soft string model is so called because it ignores any effects of tubular stiffness. This means the drillstring is represented as a heavy chain that transmits axial tension and torque caused by drillstring friction resulting from normal contact forces between the pipe and the wellbore. The soft string will be used in this research for T&D calculations during surveying in three- dimensional wellbores. Moreover, Mason and Chen (2007) also provide criterion for each type of buckling (sinusoidal and helical).

3

When discussing T&D calculations in the build section, Wu and Juvkam-Wold (1993) included three activities: rotating off bottom, running in the hole, and pulling out of the hole. His paper provides analytical solutions for T&D calculations in two-dimensional wellbore design. However, it will be most beneficial if we can develop these two- dimensional equations into three-dimensional equations. This will be of great advantage for our next generation of T&D calculations.

Aston, Hearn, and McGhee (1998) discuss techniques for solving present torque and drag problems and mention many other techniques that are widely used for reducing torque and drag problems. One of the techniques mentioned is to optimize the well profile before drilling. This means that before the drilling begins, we have already acquired the information concerning optimizing the wellbore profile. As a result, the optimization will be greatly useful in facing any difficulties in the drilling process.

1.1.2 Tortuosity effects Gaynor, Chen, Stuart, and Comeaux (2001) explain how to quantify tortuosity. Their paper discusses “micro-tortuosity,” as well as the primary cause of hole spiraling that will cause poor hole quality. Spiraling can be easily eliminated. It is desirable to reduce “micro-tortuosity,” and thus it will improve hole quality. From this paper it will be useful if T&D software programs also consider reducing this effect.

Gaynor, Halmer, Chen, and Stuart (2002) discuss the information on tortuosity versus micro-tortuosity. This information has greatly assisted eliminating excessive tortuosity,

4

which is regarded as a successful factor in solving extended reach drilling operation problems. This paper also provides a mathematical model of a spiral hole and gives a change in diameter in torque equations which changes the diameter of tool joint (D tj ) to the average diameter that has been used in torque equations (D drift ).

1.1.3 Stress concentration factor Tang, Muradov, Chandler, Jellison, Prideco, Gonzalez, and Wu (2006) present the new stress concentration factor (SCF) analysis methodology for rotary shouldered connections (RSCs) by using a finite element analysis as a primary method to calculate SCF. This represents the connecting performance. In fact, this paper has application in evaluating drill string connection design. However, it will not affect T&D calculations in terms of increasing T&D.

1.1.4 Buckling Wu and Juvkam-Wold (1993) discuss helical buckling and sinusoidal buckling of pipes in horizontal wells and drilling and completion technologies. It is a highly difficult technique when associated with transmitting compressive axial loads to the bit (or the packer) on the bottom due to frictional force between pipe and wellbore. This paper provides all of the buckling types in the horizontal wellbore that are used in this research.

5

Fig. 1.1–Schematic of forces acting on downhole tubular assembly (Aston et al. 1998).

Fig. 1.1 shows a schematic of the downhole forces acting on a tubular sliding and rotating in an inclined wellbore. This will help us to choose the torque and drag model for calculating torque and drag data. This picture shows all the forces that impact the drillstring while the drillstring is downhole. Also, this research will cover more than just T&D; the parameters that will be considered from the above figure will be axial load, wall force, friction, torque, weight of pipe, and dogleg severity.

1.2 Objectives and Organization This research’s objective will start by improving the equations that are normally used in T&D software calculations, reflecting on the state of current models and identifying future requirements. This research will provide more accurate T&D models that will help alleviate helical buckling problems; normally, helical buckling causes the potential for lock-up and impacts fatigue.

6

This research’s second objective is to prevent T&D problems while drilling by trying to optimize well profiles before drilling begins. Moreover, this research will try to show the relation between well planning design and T&D calculation. This will make it easier to find out which type of well design is more suitable in each particular area. Additionally, this research will help tremendously in the design of long horizontal wellbores.

This research’s third objective is to help field personnel prepare for any unexpected trend changes in a timely fashion during the drilling process. They will be able to anticipate these changes by just inputting wellbore data and T&D parameters. Furthermore, the outcome of the data from the T&D calculation program will be more realistic because it will be based on a 3D model.

Chapter II will propose the new equations to be used in torque and drag analysis in three-dimensional wellbores. Chapter III will provide the equations that have been used in well planning; these are divided into four types of wellbore curves (build type, build & hold type, build & hold & drop type, and horizontal wellbore type). Chapter IV will explain the effect from tortuosity on torque and drag calculations. Chapter V discusses continuing processes on torque and drag analysis of each drillstring connecting joint. Chapter VI discusses buckling effects in T&D calculations. Chapter VII will emphasize numerical methods in torque and drag. Lastly, Chapter VIII will present conclusions as well as recommendations for future work.

7

CHAPTER II TORQUE AND DRAG CALCULATIONS IN THREE-DIMENSIONAL WELLBORE METHODOLOGY 2.1 Introduction The mathematical model for torque and drag calculations in three-dimensional wellbore design is based on wellbore curve design. If we look at well planning just in a vertical depth plane, we could derive each type of assumed curve without tortuosity in vertical sections. We will divide the wellbore functions and hence the torque and drag calculations into three steps: rotation off the bottom, following through running in the hole, and pulling out of the hole (Wiggins, Choe, and Juvkam-Wold 1992). Dogleg severity (δ) will be considered. The wellbore will be divided into two planes for calculation: the vertical view and the horizontal view. We will start with T&D calculations for rotation off the bottom. For rotating on bottom, the calculation will be similar to rotation off bottom, however the difference for calculation is force at the bit (F bit ). In addition, for sliding drilling, the calculation method is the same as running in the hole, however the difference will be F bit (Juvkam-Wold and Wu 1992).

2.2 Original Concept for Calculating Normal Contact Force We start with a simple concept for normally contacting force calculations (N) (Aadn Ø y and Anderson 1998) by assuming that no friction (f) exists along the wellbore (rotating off bottom).

8

Fig. 2.1–Illustration of forces in build-up section (vertical view).

From Fig. 2.1 neglecting friction (e.g., pipe rotation) will be expressed as follows (Johancsik, Friesch, and Dawson 1984): ΣF along normal : WsinI – (T+ΔT)sin - Tsin - N = 0 (2.1) WsinI – 2Tsin - ΔTsin - N = 0 (2.2) N = WsinI – 2Tsin (assuming ΔTsin 0) (2.3) Note: f - Is the force of two surfaces in contact, or the force of a medium acting on a moving object, lbf W - In this research, refers to buoyed weight of the string element, lbf/ft I,θ - A deviation or the degree of deviation from the vertical T - The tension force at the lower end of the string element, lbf Δ - A normalized estimate of the overall curvature of an actual well path between two consecutive survey stations, degrees per 100 ft

9

Fig. 2.2–Illustration of forces in inclined section (vertical view).

From Fig. 2.2 neglecting friction (e.g., pipe rotation) will be illustrated as follows: N = WsinI (2.4)

Fig. 2.3–Illustration of forces in drop section (vertical view).

10

From Fig. 2.3 neglecting friction (e.g., pipe rotation) will be illustrated as follows: ΣF along normal : WsinI + (T+ΔT)sin + Tsin - N = 0 (2.5) WsinI + 2Tsin + ΔTsin - N = 0 (2.6) N = WsinI + 2Tsin (Assuming ΔTsin 0) (2.7)

Fig. 2.4–Illustration of forces while the wellbore turns to the right (horizontal view).

From Fig. 2.4 neglecting friction (e.g., pipe rotation) by using the same calculation as drop section; however, W and I will not be considered. Nturn ≈ (T + ΔT)sin + Tsin (2.8) Nturn = 2Tsin (Assuming ΔTsin 0) (2.9)

11

Fig. 2.5–Illustration of forces while the wellbore turns to the left (horizontal view).

From Fig. 2.5 neglecting friction (e.g., pipe rotation) will be illustrated as follows: Nturn ≈ (T + ΔT)sin + Tsin (2.10) Nturn = 2Tsin (Assuming ΔTsin 0) (2.11)

2.3 Soft-string Model for Three-dimensional T&D Calculation The original soft-string T&D programs are based on a model developed by Exxon Production Research (Mason and Chen 2007). The value of N (normal contact force) depends on how the wellbore contacts with the formation and the actual amount of normal contact force (Menand, Sellami, Tijani, Stab, Dupuis, and Simon 2006):

12

N total = (2.12) Note: T - The tension force at the lower end of the string element, lbf ΔØ - The change in azimuth angle over the string element, rad. I,θ - A deviation or the degree of deviation from the vertical W - In this research, refers to buoyed weight of the string element, lbf/ft If the wellbore turns neither left nor right, will equal 0; then using the normal contact force equation, the tension and torque change can be calculated from Eq. 2.13 and Eq. 2.14.

(2.13)

(2.14)

Eq. 2.13 shows that whether it is plus (+) or minus (-) depends on which direction the friction will be, as illustrated in Fig. 2.6. The next section will discuss the case of lowering the pipe into the hole.

Fig. 2.6–Soft-string T&D model schematic (Mason and Chen 2007).

13

2.4 Lowering the Pipe into the Hole This section will show the equations that will be used in three-dimensional wellbore designs while lowering the pipe into the hole (Maidla and Wojtanowicz 1987).

2.4.1 Lowering the pipe into the hole in the build section

Fig. 2.7–Illustration of forces in build-up section (lowering the pipe into the hole, vertical view).

From Fig. 2.7 ΣF along N axis will be illustrated as follows: NRΔα = (F + ΔF)sin + Fsin + WRΔαsin(90 0 – (α+ )) (2.15) From Eq. 2.15 divide by RΔα, thus N = ( ) + ( ) + Wcos(α+ ) (2.16) When (Δα 0), then sin( ) will approach close to 0. N = Wcos(α) + (2.17)

14

From Fig. 2.7 ΣF along X axis will be illustrated as follows: Note: R - The radius of curvature of the string element while the wellbore is in the build section (vertical view), ft Α - The angle used to calculate the deviation of the wellbore, rad.

(F + ΔF)cos( ) – Fcos( ) - F RΔα + WRΔαcos(90 - (α + )) = 0 (2.18) ΔFcos( ) - F RΔα + WR Δαsin(α + ) = 0 (2.19) Eq. 2.19 divide by Δα

( ) = F R – WRsin(α + ) (2.20) When Δα 0 = F R – WRsin(α); F = μ|N| (2.21)

Fig. 2.8–Illustration of differences between positive and negative forces in build-up section (lowering the pipe into the hole, vertical view).

15

From Fig. 2.8 for N > 0

= μ{ R – WRsin(α); (α ≥ α ≥ α ) (2.22) *The term comes from wellbore turning and will be shown in the next section of this chapter. The value will be zero if there is no right or left turn of wellbore; means if compressive force > 0 and if tension force < 0 (Wu and Juvkam-Wold 1991).