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Side Force and Directional Tendency of BHA with
Eccentric Components
A Thesis
Presented to
The Faculty of the Department of Petroleum Engineering
University of Houston
In Partial Fulfillment
of the Requirements for the Degree of
Master of Science
in Petroleum Engineering
by
Yuan Zhang
December 2013
Side Force and Directional Tendency of BHA with
Eccentric Components
__________________________
Yuan Zhang
Approved:
____________________________________
Chairman of the Committee,
Dr. Michael Nikolaou, Professor,
Chemical and Biomolecular Engineering.
Committee Members: ____________________________________
Dr. Robello Samuel, Adjunct Professor,
Petroleum Engineering.
____________________________________
Dr. Gangbing Song, Professor,
Mechanical Engineering.
_________________________ ____________________________________
Dr. Suresh K. Khator, Dr. Michael P. Harold,
Associate Dean, Professor and Chair,
Cullen College of Engineering. Chemical and Biomolecular Engineering.
iv
Acknowledgements
I am very grateful to my advisor Dr. Robello Samuel for leading me into the field
of drilling engineering, for imparting his drilling engineering expertise, and for bringing
me career growth opportunities. With his experience in the energy industry, I was provided
much guidance when I worked on the project. Dr. Samuel also led me to be both a technical
writer and a technical speaker.
My sincere appreciation is extended to my co-advisor, Dr. Gangbing Song, for his
precious suggestions, support and encouragement. I also appreciate my thesis committee
chair Dr. Michael Nikolaou for his many suggestions on this study. I would like to thank Dr.
Tom Holley and Mrs. Anne Sturm for their continuous guidance and support in the project
and throughout my graduate program at the University of Houston.
Finally, sincere gratitude is expressed to my wife, Dr. Yilei Gong, for her
unwavering support, and for taking care of everything in home.
v
Side Force and Directional Tendency of BHA with
Eccentric Components
An Abstract
of a
Thesis
Presented to
The Faculty of the Department of Petroleum Engineering
University of Houston
In Partial Fulfillment
of the Requirements for the Degree of
Master of Science
in Petroleum Engineering
by
Yuan Zhang
December 2013
vi
Abstract
In directional drilling, both the side force and the tilt angle influence the dog leg
severity (DLS) of the wellbore. The side force and the tilt angle of the drill bit also
influence each other. It is imperative to calculate the two factors to predict the drilling
direction. Three new analytical models are developed based on force analysis in this
project in order to calculate the side force and DLS for Push-The-Bit BHA and Point-
The-Bit BHA. In the models, the BHAs are assumed to be a continuous beam supported
by the drill bit, the pivot pad and the stabilizer. The side force and tilt angle can be
calculated when the BHA is in static force balance status. The DLS of the wellbore can
be calculated with the side force and the tilt angle.
Two models, one model with rigid drill pipe assumption and one model with
elastic drill pipe, are used to analyze Push-The-Bit. The models show that the side force
is the main factor to influence the walk direction of the drill bit for Push-The-Bit. The
stronger the drill pipe, the higher the efficiency of directional drilling. One model is used
to calculate the Point-The-Bit. The Point-The-Bit model shows that the tilt angle is the
main factor that controls the walk direction for the Point-The-Bit.
The models in this project uncover the interaction between the side force and the
tilt angle. They also provide a way for drillers to estimate the drilling direction during
operation that is a great advantage in controlling the drilling direction.
vii
Table of Contents
Acknowledgements ............................................................................................................ iv
Abstract ........................................................................................................................ vi
Table of Contents .............................................................................................................. vii
List of Figures .................................................................................................................... ix
List of Tables .................................................................................................................... xii
Nomenclature ................................................................................................................... xiii
Chapter 1 Introduction ......................................................................................................1
1.1 Overview ....................................................................................................................1
1.2 Problem Statement .....................................................................................................2 1.3 Objectives ..................................................................................................................3
Chapter 2 Literature Review.............................................................................................5
2.1 Directional Drilling ....................................................................................................5
2.1.1 Walk Angle, Tilt Angle and Push Angle ......................................................7
2.1.2 Push-The-Bit and Point-The-Bit .................................................................12
2.2 Theory for Continuous Beam ..................................................................................15
2.2.1 Statically Indeterminate Structure ...............................................................16 2.2.2 Displacement method ..................................................................................17
Chapter 3 Geometry and Coordinate System .................................................................20
3.1 Assumptions .............................................................................................................20
3.2 Geometry of the system ...........................................................................................20
3.2.1 Baseline .......................................................................................................20 3.2.2 Walk Angle and DLS ..................................................................................23
3.2.3 Coordinate System ......................................................................................23 3.2.4 Coordinate of the Wellbore Axis in the Curved Shape ...............................24
Chapter 4 Force Analysis for Push-The-Bit ...................................................................28
4.1 BHA of Push-The-Bit ..............................................................................................28
4.2 Analytical Model for the Rigid BHA of Push-The-Bit ............................................30
4.2.1 Assumption .................................................................................................30 4.2.2 Force balance analysis of the model ...........................................................30 4.2.3 Geometry analysis of the model..................................................................32
4.3 Case Study for Rigid BHA Push-The-Bit ...............................................................35
4.3.1 Data and Calculation ...................................................................................35 4.3.2 Results Analysis ..........................................................................................35
4.4 Analytical Model for the Elastic BHA of Push-The-Bit ..........................................38
viii
4.4.1 Analytical Model for the Rigid BHA of Push-The-Bit ...............................39 4.4.2 Force balance analysis of the elastic BHA Push-The-Bit model ................40 4.4.3 Geometry analysis of the model..................................................................45
4.5 Case Study for Elastic BHA Push-The-Bit ..............................................................49
4.5.1 Data and Calculation of Elastic BHA Push-The-Bit .................................49
4.5.2 Results Analysis ..........................................................................................49
Chapter 5 Force Analysis for Point-The-Bit...................................................................54
5.1 Introduction ..............................................................................................................54 5.2 Assumption ..............................................................................................................54
5.2.1 Analytical Model for Point-The-Bit in the Straight Wellbore ....................56
5.2.2 Force Model for Point-The-Bit in the Curved Wellbore.............................62
5.3 Case Study for Point-The-Bit Model ......................................................................65
5.3.1 Data and Calculation ...................................................................................65 5.3.2 Results Analysis ..........................................................................................66
Chapter 6 Conclusions & Recommendations .................................................................71
6.1 Summary ..................................................................................................................71 6.2 Future Work & Recommendations ..........................................................................72
Reference ........................................................................................................................73
Appendix A .....................................................................................................................75
ix
List of Figures
Figure 2-1 Rotate sequences in a directionally drilled well, Lesso et al.(2001). ................ 6
Figure 2-2 The push angle caused by side cutting. ............................................................. 8
Figure 2-3 The walk angle, push angle and tilt angle. ........................................................ 9
Figure 2-4 Typical deviations observed in the lab according to the bit design, Menand et
al. (2012). .......................................................................................................................... 10
Figure 2-5 Experiment side cutting rate and calculate side cutting rate. .......................... 12
Figure 2-6 External steering pads on a rotating section, Sugiura(2008). .......................... 13
Figure 2-7 External steering pads on a non-rotating section , Sugiura(2008). ................. 13
Figure 2-8 Internally bending driveshaft on a non-rotating housing, Sugiura(2008). ...... 14
Figure 2-9 Geo-Stationary unit keeps bit tilt angle in a rotating section, Sugiura(2008). 14
Figure 2-10 RSS in point-the-bit configuration, Sugiura(2008). ...................................... 14
Figure 2-11 2-span continuous beams. ............................................................................. 17
Figure 2-12 The released moment continuous beams. ...................................................... 17
Figure 2-13 3-span continuous beams. ............................................................................. 18
Figure 3-1 Different baseline in small deformation condition.......................................... 21
Figure 3-2 Relationship between different baseline. ........................................................ 22
Figure 3-3 The dls angle and walk angle. ......................................................................... 23
Figure 3-4 Local coordinate system. ................................................................................. 24
Figure 3-5 Direction in element. ....................................................................................... 24
Figure 3-6 Baseline in a curved wellbore. ........................................................................ 25
Figure 4-1 Rotating housing push-the-bit. ........................................................................ 28
Figure 4-2 Non-Rotating housing push-the-bit. ................................................................ 28
x
Figure 4-3 Model for rigid push-the-bit. ........................................................................... 29
Figure 4-4 Force model for rigid push-the-bit. ................................................................. 31
Figure 4-5 DLS for rigid push-the-bit. .............................................................................. 33
Figure 4-6 Tilt angle ratio for rigid BHA Push-The-Bit. .................................................. 35
Figure 4-7 Push angle ratio for rigid BHA Push-The-Bit. ................................................ 36
Figure 4-8 Side force on bit for rigid BHA Push-The-Bit. ............................................... 37
Figure 4-9 DLS of wellbore for rigid BHA Push-The-Bit. ............................................... 37
Figure 4-10 Model for elastic BHA Push-the-bit. ............................................................ 39
Figure 4-11 Deflection in the elastic Push-The-Bit. ......................................................... 40
Figure 4-12 Deformation model for elastic BHA Push-The-Bit. ..................................... 40
Figure 4-13 Force model for rigid Push-The-Bit. ............................................................. 42
Figure 4-14 Geometry of elastic BHA Push-The-Bit. ...................................................... 46
Figure 4-15 Vertical tilt angle of the wellbore.................................................................. 47
Figure 4-16 Tilt angle of wellbore for elastic BHA Push-The-Bit. .................................. 50
Figure 4-17 Push angle of wellbore for elastic BHA Push-The-Bit. ................................ 50
Figure 4-18 Side force on the bit for elastic BHA Push-The-Bit. .................................... 51
Figure 4-19 DLS for elastic BHA Push-The-Bit. ............................................................. 52
Figure 4-20 Tilt-push ratio for elastic BHA Push-The-Bit. .............................................. 52
Figure 4-21 Relationship between side force and DLS for Push-The-Bit. ....................... 53
Figure 5-1 Point-The-Bit with a non-rotating housing. .................................................... 54
Figure 5-2 Difference between Push-The-Bit & Point-The-Bit. ...................................... 55
Figure 5-3 Force model for BHA of Point-The-Bit. ......................................................... 56
Figure 5-4 Force model for global system. ....................................................................... 57
xi
Figure 5-5 Force model for the shaft inside the housing. ................................................. 58
Figure 5-6 Deformation model for the shaft inside the housing. ...................................... 60
Figure 5-7 Deflection of the wellbore axis. ..................................................................... 63
Figure 5-8 Tilt angle ratio for Point-The-Bit model. ........................................................ 66
Figure 5-9 Push angle ratio for Point-The-Bit model. ...................................................... 67
Figure 5-10 Side force for Point-The-Bit model............................................................... 68
Figure 5-11 DLS for Point-The-Bit model. ...................................................................... 69
Figure 5-12 Relationship between side force and dls for Point-The-Bit. ......................... 69
xii
List of Tables
Table 1 Experiment data for side cutting .......................................................................... 11
xiii
Nomenclature
= Coefficient of Side Cutting
= Distance between the Wellbore Axis and the Bottom of the Wellbore
= Dog Leg Severity
= Distance between the Wellbore Axis and the Top of the Wellbore
= Young’s Modulus
= Eccentric Distance of the Eccentric Component
= Deflection of the BHA
= Deflection of the BHA in Curved Wellbore
= Deflection of the Wellbore Axis
= Side Force between the BHA and the Wellbore
= Inertia moment
= Linear Stiffness of the Drill Pipe
= Stiffness of the Drill Pipe
= Length of Drill Pipe between Two Components
= Moment in the Section of the Drill Pipe
= Degree of Indeterminacy for an Indeterminate Structure
= Number of Unknown Displacements
= Number of Rotations of the Joints
= Shear Force in the Section of the Drill Pipe
OP = Rate of Penetration
= Rate of Side Cutting
= Radius of the Drill Bit
xiv
= Radius of the Drill Pipe
= Radius of the Pad
= Radius of the Stabilizer
= Radius of the wellbore
= Dimensionless rock strength
= Tilt Angle for Baseline in Global Coordinate System
= Direction Angle Change of Wellbore Axis between Two Components
= Bending Angle of the BHA in Curved Wellbore
= Push Angle in Curved Wellbore
= Tilt Angle in Curved Wellbore
= Walk Angle in Curved Wellbore
= Bending Angle of the Drill Pipe
= Bending Angle of the Wellbore Axis
= Push Angle Caused by Side Force on the Bit
= Tilt Angles of the Drill Bit
= Walk Angle of the Drill Bit
1
Chapter 1 Introduction
1.1 Overview
Directional drilling is the operation of drilling a curved well to a position that is
not directly beneath the drill site. For many years, people can only drill a well straight
down into the ground. However, this is not always economical or achievable. For
example, to reach the reservoirs exist under the protected area, buildings, or any other
area that the drilling rig is impossible to set up or too much difficult to set up, drillers
can’t drill a straight well. In last few decades, drillers tried to improve the drilling
technology, and developed many ways to change the drilling direction during the drilling
process. Now, directional drilling becomes a routine operation.
Even though there are many types of directional drilling, the horizontal well and
the offshore well are the two main ones. In conventional vertical well, the flow region is a
column body, and the fluid flows from the outer face of the column to the inner face of
the column. In this mode, the skin factor is high and the well is very easy to be damaged.
To enhance the production and to produce in low permeability area, engineers developed
the technology to drill a horizontal well in the reservoir that can extend to thousand feet.
In horizontal direction, the flow region in the reservoir changes to be linearly flow, and
the skin factor decreases greatly.
The offshore well is normally installed on an offshore platform. It costs much
money to build such a platform that engineers have to drill to multiple positions from
single platform to save the money. Drillers normally drill one large well to the position
close to the reservoir, and then they drill many branch wells from this large well. The
2
branch wells are directional wells that change the direction from the vertical to the
inclined direction until they reach the destination.
Most of the conventional wells had been depleted for many years and the
reservoirs will be exhausted soon. And most new wells are unconventional wells that
were built on the low permeability reservoirs. This will make directional well more and
more popular in future.
1.2 Problem Statement
An important problem in directional drilling is how to control the drill direction.
The dog leg severity (DLS), the index of curvature of the wellbore, is defined by the
turning angel of the drill direction over the length of the wellbore. Drillers design the path
of the wellbore by controlling the DLS. However, when drilling under the surface of the
earth, it is not easy to detect the drilling direction. If the DLS in drilling does not follow
the designed DLS, the wellbore may deviate far away from the target.
Samuel (2009) proposed a method for well-path designs that incorporate
curvature bridging that uses curvature bridging and calculation of total strain energy.
Lesso et al. (1999) proposed a model used a finite element model to calculate the
direction of drilling. These models were very inspiring the driller’s mind and provided
some possible way to control the direction. However, a more accurate model is still
needed to improve the operation. The finite element model seems to be an accurate model,
but it is difficult to use it in an uncertain boundary environment. In this project, a new
model based on continuous beam theory is proposed to calculate the drilling direction. It
provides an easy way to understand and use the model to predict the drilling direction.
3
Drill direction is mostly influenced by the direction of the drill bit and the side
force applied on the drill bit. When the direction of the drill bit is inclined to the wellbore
axis, the axis force, or weight on bit will add an extra turning angle to the wellbore axis.
Similarly, the side force can also cause an extra angle to the wellbore axis. To control the
drill direction, we must find a method to calculate the influence of the side force and the
tilt angle.
There are two types of steering mechanism to control the drilling direction: Push-
The-Bit and Point-The-Bit. In Push-The-Bit, the side force is predominant. In Point-The-
Bit, the tilt angle is the dominant factor. In both mechanisms, the tilt angle and the side
force work together to control the walk direction of the drilling. In most cases, the two
factors can influence each other. Normally, the drillers control the deformation of the
drill string to apply the side-force on the drill bit. At the same time, the deformation of
the drill string will also change the tilt angle of the drill bit. When the side force is
changed, the tilt angle will be changed too.
1.3 Objectives
This thesis is intended to build 3 models to calculate DLS at some deformation
conditions. The aim of this project is to investigate the relationship between the
deformation of the drill string, the side force and the tilt angle. The walk angle can be
calculated using the tilt angle and the side force. Then DLS is easy to be calculated.
These models are also useful when designing the directional well and directional
tools. The distance between components on the drill string is the important factor to
influence the deformation and the side force between the drill string and the wellbore. In
4
directional tools designing, engineers can use this model to check the behavior of the
BHA, and choose the best distance.
5
Chapter 2 Literature Review
2.1 Directional Drilling
Directional drilling can be defined as the technology of drilling a wellbore along a
predetermined trajectory, normally a curved path, to a subsurface target. It is the
operation of drilling a curved well to a predetermined position.
In Early 1930’s, two wells were drilled unintentionally from land to offshore at
Huntington Beach, California. Then drillers began to intentionally drill a directional well
to an inaccessible location. In 1932, some directional wells were intentionally drilled
along the coast of Pacific Ocean. In 1933, the first record of controlled directional well in
Signal Hill field was drilled. In 1934, the first relief well in history, a controlled
directional well was drilled to control a blowout in Conroe, Texas. In 1937, the first
horizontal well was drilled in the Yarega heavy-oil field to product the heavy oil, in
Soviet Union.
From then on, the directional drilling techniques have been improved constantly
and made drilling more efficient. Now, directional drilling is a vitally important
technology in energy industry. Most of the new wells, especially the unconventional
wells are drilled with directional drilling techniques. In many cases, directional drilling is
the only method for the economic development of an oil field. Nowadays, directional
drilling operations has been proven to be successful under almost all conditions and
environments, and has been widely accepted in the energy industry.
Lesso et al.(2001) found that most directional drilling was a series of rotary
drilling followed by a section of oriented or slide-drilling with a steerable motor. Each
6
section was typically 10-20 ft. in depth. Figure 2-1 shows a series of slide-rotate
sequences, and the associated tool face angle (TFA) settings, and the length of each
section for a typical BHA run.
Figure 2-1 Rotate sequences in a directionally drilled well, Lesso et al.(2001).
Eastman (1950) had concluded that the applications of controlled directional
drilling were:
1) Deflecting from accessible locations to inaccessible locations;
2) Deflecting around and under salt dome overhangs that are difficult to drill
through;
3) Deflecting out of salt dome deposits to adjacent oil sands;
4) Deflecting relief wells into wild, burning, or cratered wells to bring them under
control;
7
5) Deflecting a hole from an unproductive to a productive portion of a lease;
6) Deflecting an old depleted well to a new location in producing territory;
7) Deflecting across faults or out of fault zones into a productive area;
8) Deflecting a hole back into the lease after it has deviated over a lease line;
9) Deflecting a plurality of wells from one location, or from a physically restricted
area such as an island;
10) Deflecting a series of holes for sub-surface geological exploration;
11) Deflecting a hole so sand can be cored after it was first drilled through
without coring;
12) Sidetracking fish or obstructions in the well;
13) Deflecting a well back into a hole that has caved and was lost;
14) Deflecting off at an angle in oil formations to give greater penetration.
2.1.1 Walk Angle, Tilt Angle and Push Angle
Normally, the deflecting methods used today are motor systems and rotary
steerable systems that may create side force and tilt angle of the drill bit. The side force
will push the bit to cut the wall of wellbore, and the tilt angle of the drill bit will change
the direction of the bit away from the direction of the wellbore axis. Before calculating
the DLS, we need figure out how to calculate the side force and tilt angle of the drill bit.
If we set a baseline at the drill bit, the walk angle, the tilt angle and the push angle
are possible to be defined as following: the walk angle, the resultant movement direction
of the drill bit, is the angle between the axis of the wellbore and the baseline; the tilt
angle, the inclination direction of the drill bit, is the angle between the axis of the drill bit
8
and the baseline, the push angle is the angle cause by the side cutting. The push angle can
be expressed by the equation
ROP
ROS
ROP
ROSs
tanarg . (1 )
Figure 2-2 The push angle caused by side cutting.
In the equation, ROS is the side cutting rate, and ROP is the penetration rate that
is created by the weight on bit (WOB). The walk angle is the summation of the tilt angle
and the push angle, which is expressed as
stw . (2 )
Figure 2-3 shows the relationship between the walk angle, the push angle and the
tilt angle. Both the tilt angle and the push angle create the deviation of the drilling
direction. The real movement direction of the drill bit is the result of the two angles
working together.
The tilt angle reflects the position and the direction of the drill bit. Drillers
developed different ways to control the tilt angle. Some of the methods are using a gear
system to control the tilt angle. The most popular method to control the tilt angle is to
control the deformation of the drill string. The change of the bend angle of the drill string
will also change the tilt angle of the drill bit.
9
Figure 2-3 The walk angle, push angle and tilt angle.
The side force is the main reason that cause push angle. The method to apply the
side force on the drill bit is to deform the drill string. When the wellbore stops the
deformation of the string, the deformed string will apply a side force on the wall of the
wellbore. This process seems just like a deformed spring pushing the wall.
The property of the rock is another important factor that influences the cutting
rate. The harder the rock is, the lower rate of the side cutting is. Even though the axial
penetration rate (ROP) is also lower when the rock is harder, the change of the ROS and
ROP may be different due to the anisotropy of the rock. Therefore, the model to calculate
the push angle must include the proper rock strength.
Some researcher also found that the shape of the drill bit will also influence the
side cutting speed. Now, the most popular drill bit used in directional drilling is PDC bit.
Menand et al. (2012) observed that long and smooth gauge bits tend to deviate in the
same direction as the tilt angle; short or aggressive gauge bits tend to deviate in the same
10
direction as the side force (as shown in Figure 2-4). Applying the same side force on the
drill bit, the cutting rate will be higher if the gauge of bit is shorter.
Brett et al. (1986) found that except the strength of the rock and the profile of the
bit, additional factors that influence the directional response of the BHA are the weight-
on-bit, the bit torque and the mud weight, which are used in modeling the buoyant force
acting on the BHA in the wellbore. The rotary speed is also a factor that can influence the
side cutting rate.
Figure 2-4 Typical deviations observed in the lab according to the bit design,
Menand et al. (2012).
Brett et al. (1986) proposed a model to calculate the side force. Brett observed
that the side cutting rate varies approximately as a constant times the square of the side
11
force, that the constant depends on rock type, and that there is not a strong dependence on
the rotary speed of the bit. This data also shows that an upper limit of approximately 2.0
ft/hr exists for the side cutting rate. These observations are summarized in the following
equation for lateral penetration rate
r
s
S
FAROS
21)(
, (3 )
where ROS is the side cutting rate in ft/hr, Fs1 is the total side force at the bit in lbs. Sr is
the dimensionless rock strength. Onyia (1987) and Warren (1987) introduced a method to
calculate Sr. The constant A is an empirically determined factor that models the
directional response of building, dropping and holding assemblies.
Table 1 Experiment data for side cutting
Rock-type RPM ROP Sideforce
ROS/ROP Field ROS Estimate
ROS ft/Hr Lb
Bedford 100 60 500 0.055 0.28 0.625
Bedford 100 60 2000 0.375 1.88
10
Bedford 100 100 480 0.085 0.71
0.576
Bedford 100 100 520 0.075 0.63
0.676
Bedford 100 100 800 0.23 1.92
1.6
Bedford 100 150 800 0.18 2.25 1.6
In the equation, the difference of the profile of the bit can cause great changes to
the constant A. However, in this study, the value of the constant A and dimensionless
rock strength Sr are based on the experiment data provided by Millhiem et al.(1978),as
shown in the Table 1. Choose parameters as hrftROPSA r /100;2;105 6 , the side
cutting rate versus the side force is plot on the Figure 2-5. From the plot we can see, the
calculated ROS matched the experiment data very well except an outlier.
12
Figure 2-5 Experiment side cutting rate and calculate side cutting rate.
In this project, the parameters to calculate side cutting rate includes all the factors
discussed in previous statement. It will not change the effectiveness of the model in this
project. However, a new set of parameter should be confirmed if applying the model to
other drill bit profiles.
2.1.2 Push-The-Bit and Point-The-Bit
In history, the whipstock once was the mainly method to direct the drill bit. Today,
the Rotary-Steerable-Systems (RSS) is the most widely used directional drilling tool in
directional drilling. There are two types of RSS tools: Push-The-Bit and Point-The-Bit.
Sugiura (2008) classified that the Push-The-Bit consists of two major subcategories of
0.0
0.5
1.0
1.5
2.0
2.5
0 500 1000 1500 2000
RO
S, f
t/h
r
Side Force, Lbs
ROS Vs Side Force
Field ROS
Calculated ROS
13
driving mechanisms: 1) applying dynamic side force from a rotating housing as shown in
Figure 2-6 and 2) applying static side force from a non-rotating housing as shown in
Figure 2-7.
Figure 2-6 External steering pads on a rotating section, Sugiura(2008).
Figure 2-7 External steering pads on a non-rotating section , Sugiura(2008).
The first type of Push-The-Bit consists of a pivot pad that can rotate with the drill
string. The second type of Push-The-Bit consists of a pivot pad that stays non-rotating
status when drilling. Samuel (2007) had pointed out that typical BHA hookups have one
or more stabilizers. The steerable stabilizer that can be adjusted to change the drilling
direction is the pad in this study. In directional drilling, drillers extend the foot of the pad
and push it towards the wellbore wall. The drill bit and stabilizer are pushed back to the
other side of the wellbore, and the force between the drill bit and the wellbore generate
the side force on the bit.
For Push-The-Bit BHA, the main factor that changes the drilling direction is the
side force on the bit. The bending direction of the wellbore for Push-The-Bit is decided
14
by the direction of the side force. If the side force is between the top of the wellbore and
the bit, it will drill build-up. Otherwise, it will drill drop-down.
Sugiura (2008) also classified that the Point-The-Bit mode consists at least 3
major and distinctive ways to tilt the bit; 1) bending a drive shaft inside the non-rotating
housing as shown in Figure 2-8, 2) holding a predetermined bias with a geo-stationary
unit inside a rotating housing as shown in Figure 2-9, and 3) positioning a non-rotating
housing with three pads to tilt a drill bit as shown in Figure 2-10.
Figure 2-8 Internally bending driveshaft on a non-rotating housing,
Sugiura(2008).
Figure 2-9 Geo-Stationary unit keeps bit tilt angle in a rotating section,
Sugiura(2008).
Figure 2-10 RSS in point-the-bit configuration, Sugiura(2008).
15
The first type of Point-The-Bit consists of a housing, a shaft and an eccentric
component within the housing. The housing is a steel pipe that the shaft can bend within
the housing. When drilling directionally, drillers increase the eccentric distance of the
component in the housing that bends the shaft and change the bending angle of the drill
bit. With proper operation, drillers can control the tilt angle of the drill bit to be the
predetermined value. The second type of Point-The-Bit consists of a housing and a
predetermined bias unit inside the housing. Drillers use the bias unit to control the tilt
angle of the drill bit when drilling directionally. The third type of Point-The-Bit is an idea
to put the pivot pad behind the stabilizer that drillers can operate the pivot pad to control
the tilt angle of the drill bit.
For Point-The-Bit BHA, the main factor to change the drilling direction is the tilt
angle of the drill bit. The bending direction of the wellbore when using Point-The-Bit is
the direction of the tilt angle. Even if there is no side force as in the second type Point-
The-Bit, the tilt angle is still under the control.
In this project, the non-rotating pad Push-The-Bit and the rotating housing Point-
The-Bit are two types of RSS to be analyzed. For the other types of RSS, similar analysis
can be applied to calculate the deformation, side force and DLS.
2.2 Theory for Continuous Beam
A continuous beam is a multi-span beam on hinged support. The end spans may
be cantilever, freely supported or fixed supported. Continuous beam structure is widely
used in buildings and bridges. Beams are made continuously over the supports to increase
16
structural integrity. The main advantages of a continuous beam comparing to a simply
supported beam are as follows:
1) With the same span, section and load, the force and moment in section are less.
2) Mid span deflection is less.
The analysis of continuous beams is based on elastic theory. To analyze the
continuous beam, axis force is normally ignored because the deformation of the beam and
movement of the support is assumed to be small enough that the deformed continuous
beams can be taken as a straight line.
2.2.1 Statically Indeterminate Structure
A continuous beam is a statically indeterminate structure. The most effective
(efficient) primary system for continuous beam is proposed by Clapeyron (French
engineer and physicist 1799-1864). The structure is an indeterminate structure if the
support forces and internal forces cannot be uniquely determined from the equations of
equilibrium. The degree of indeterminacy for an indeterminate structure is the number of
redundant involved in the structure. The degree is n if the indeterminate structure turns to
be a determinate structure after release n restraints.
When we do static analysis, the degree of the indeterminacy is the number of the
supplementary equations that are required to determine the unknown force or
deformation. A 2-span continuous beam structure is shown Figure 2-11. We can easily
write two force equilibrium equations for the structure:
0321 sss FFF and (4 )
2311 LFLF ss . (5 )
17
Figure 2-11 2-span continuous beams.
We cannot solve the equations because the number of unknown forces is more
than the number of the force balance equations. If we release the bending restrain at the
support 2, the continuous beams turns into two simple beams that are determinate
structures. So we can now the degree of the indeterminacy for this structure is 1 and we
can have 1 supplementary equation. Assuming the bending angle of the continuous beam
at the node 2 is 2 , we can write the moment balance equation at the node 2
02223221 MKK , (6 )
Figure 2-12 The released moment continuous beams.
where 2321,KK are the bending stiffness for element1 and element2. The equation is also
called as Canonical equation. Solve the 2 from the equation and then all the other
moments and force can be solved.
2.2.2 Displacement method
In the previous analysis, we have used a method that is called displacement
method. Displacement method is a good way to solve the statically indeterminate
18
structures. In displacement method, the displacement of the structure is assumed to be
unknown variables, and the total number of unknown variables (n) is equal to the number
of unknown displacements (nd) and rotations of the joints (nr):
rd nnn . (7 )
For every displacement or rotation, we can write an equation to calculate the
internal force at the end of the elements (beams). Then, from the force and moment
balance at the each node, we can write a set of Canonical equations.
A 3-span continuous beams structure is shown in the Figure 2-13. The bending
angle at the node 2 and node 3 are unknown. If we assume the unknown bending angles
at the two nodes are 1 , 2 , 3 and 4 , we can write the element equations as
Figure 2-13 3-span continuous beams.
21
12
2
1
11
11
42
24
M
M
ii
ii
, (8 )
32
23
3
2
22
22
42
24
M
M
ii
ii
, and (9 )
43
34
4
3
33
33
42
24
M
M
ii
ii
, (10 )
where iii LEIi / .Then we can write 4 supplementary equations for the moment balance
at the node 2 and node 3 as
19
024 122111 Mii , (11 )
223213222111 242 MMMiiii , (12 )
334324333222 242 MMMiiii , and (13 )
022 434333 Mii . (14 )
20
Chapter 3 Geometry and Coordinate System
3.1 Assumptions
To analyze the force on the BHA, we have to simplify the shape of the wellbore,
the drilling tools and the components on the drilling pipe. The following is the geometric
assumptions for this study:
1. The section of the wellbore is a circle, with radius of wr .
2. The route of the wellbore is a straight line or part of a circle.
3. The drill bit and the stabilizer are ring shape, with radius of br and sr .
4. The distance from the center of the BHA to the foot of the pivot pad is pr .
5. The deformations in all the parts are small enough to be a linearly
deformation.
3.2 Geometry of the system
The wellbore has two kind of shape in the directional drilling. At the start of
directional drilling and the end of the directional drilling, the wellbore is straight. In the
stable status of directional drilling, the wellbore is a part of a circle.
The BHA has also two kind of status. When drilling straight, the BHA is keep in
straight shape. When drilling directionally, the BHA changes to be a curved shape except
the rigid BHA.
3.2.1 Baseline
Baseline is a straight line that all the deflections and bending angles are measured
from the line. From physics view, changing baseline will not influence the force
21
calculation if the baseline is the original shape for the structure. However, the result of
the force will be different for a different baseline because of the direction of the baseline.
In this study, because the deformation is assumed to be very small, the different baselines
through the BHA will also be small deviation from each other. As shown in Figure 3-1.
The angle between different baselines can be expressed as
L
ee
L
eeare
*
1
*
2
*
1
*
2* tan
. (15 )
Figure 3-1 Different baseline in small deformation condition.
Rajasekaran (2001) had listed the force equation for each element as
2
2
1
1
2
2
1
1
22
22
46
26
612612
26
46
612612
M
Q
M
Q
e
e
iL
ii
L
iL
i
L
i
L
i
L
i
iL
ii
L
iL
i
L
i
L
i
L
i
, (16 )
where L
EIi . Using another baseline, the equation sets will change to be
0
0
0
0
46
26
612612
26
46
612612
*
*
2
*
*
1
22
22
e
e
iL
ii
L
iL
i
L
i
L
i
L
i
iL
ii
L
iL
i
L
i
L
i
L
i
. (17 )
22
So, the force and moment value base on the new baseline is the same value to the value
of the old baseline, and the force balance equations are
2
2
1
1
2
2
1
1
*
2
*
22
*
1
*
11
22
22
*
2
*
2
*
1
*
1
0
0
0
0
46
26
612612
26
46
612612
M
Q
M
Q
M
Q
M
Q
ee
ee
iL
ii
L
iL
i
L
i
L
i
L
i
iL
ii
L
iL
i
L
i
L
i
L
i
M
Q
M
Q
. (18 )
Figure 3-2 Relationship between different baseline.
Figure 3-2 shows the relationship between two baseline. The previous equations
show that the force calculation will not be influenced by using different baseline if the
baseline is limited in the region of the BHA. In this study, the original shape of the BHA
is a straight line, so, we should use a straight line as a baseline. For convenient, we draw
a straight line through the point of the axis of wellbore at the head of the BHA and at the
end of the BHA. This is the baseline use for this study, except extra explanation. All the
side deflection value and bending angle are the value according to this baseline.
23
3.2.2 Walk Angle and DLS
After drilling the distance of the BHA length, the walk angle change will be two
time of the walk angle, as shown in Figure 3-3. The DLS can be express as
L
DLS w2 . (19 )
Figure 3-3 The dls angle and walk angle.
3.2.3 Coordinate System
From the previous analysis, we know the best coordinate system in this study is
the local coordinate that uses the baseline as the X axis. The Figure 3-4 shows the local
coordinate system. The direction of the X axis, Y axis and rotation is also shows in the
Figure. The positive direction of the X axis is from the end point to the first point; the Y
axis is the X axis after 90° clock wise rotation; the positive direction of the moment and
angle are clockwise. There is some angle between this local coordinate and the global
coordinate system. The value calculated from this study is only base on this local
coordinate system.
To write the equation for each element, we will also define the direction of the
element. In this study, the direction of the element is show in Figure 3-5. The direction of
the force and the deflection are vertical to the element axis. The rotation angle and
moment direction are same to the local angle direction. Due to the small deformation
24
assumption, the direction of the force and the deflection are approximately similar to the
local Y axis direction.
Figure 3-4 Local coordinate system.
Figure 3-5 Direction in element.
3.2.4 Coordinate of the Wellbore Axis in the Curved Shape
If a wellbore is straight, the baseline is a straight line overlapping the axis of the
wellbore. In this case, the value of the deflection is easy to calculate. If the wellbore is
curved, as shown in Figure 3-6, extra amount of the deflection and the bending angle of
the drilling pipe should be added to the value of the straight axis status. Figure 3-6 shows
25
how to calculate the extra amount of the deflection and bending angle. The angle b is the
angle between the baseline and the horizontal line. The extra deflection at every point ie*
is the distance of the point to the baseline. The extra walk angle i* is the tangent angle
between the tangent line of the wellbore axis and the baseline.
When the structure is in small deformation status, the arc between two points will
be very close to the length of the chord. That is
ii LDLS * and (20 )
1** iii . (21 )
Figure 3-6 Baseline in a curved wellbore.
Rearrange it, the walk angle and DLA are expressed as
iii *
1*
and (22 )
26
1
1
*1
*1
1
1
1
//N
i
iN
N
i
i
N
i
i LLDLS . (23 )
Because the wellbore part of a circle, the walk angles 1* and N
* art symmetric to
each other. The following equation describe the symmetric relationship as
N*
1* . (24 )
Combine the equation (23), the walk angles at the first point and the last point are
1
1
*1
* *2
1 N
i
iN LDLS . (25 )
All the walk angles are expressed by the following equations,
NiLDLS
NiLDLS
iLDLS
N
i
iN
iii
N
i
i
1
1
*
*1
*
1
1
1*
,*2
1
1,*
1,*2
1
. (26 )
On the other side, we know that the baseline is a straight line through the first
point and the last point of the wellbore axis, that is
0** Ni ee . (27 )
Between the first point and the last point, from the small deformation, the walk angle can
be expressed by the deflection of the wellbore axis as
iiii Lee /1*** . (28 )
The equation (28) can be rearranged as
iiii Lee **1
* . (29 )
Therefore, the deflections of the curved wellbore axis at all the points can be expressed
by
27
Ni
NiLee
ie
N
iiii
,0
1,
1,0
*
**1
*
1*
. (30 )
When calculate the deformation of the BHA in a curved wellbore, the deflection
of the wellbore axis will be an extra deflection to the BHA. The equation (30) shows how
the wellbore shape impacts the deformation of the BHA.
28
Chapter 4 Force Analysis for Push-The-Bit
4.1 BHA of Push-The-Bit
Sugiura (2008) classified that there are two major kinds of driving mechanisms
for Push-The-Bit BHA; 1) applying dynamic side force from a rotating housing as in
Figure 4-1; 2) applying static side force from a non-rotating housing as in Figure 4-2. In
this study, the analytical model is a static force model bases on the non-rotating housing
mechanism. In this kind of BHA, the housing, which is also called as non-rotation pad,
doesn’t rotate when drilling. This mechanism keeps the direction of the drilling in stable.
Figure 4-1 Rotating housing push-the-bit.
Figure 4-2 Non-Rotating housing push-the-bit.
A typical kind of Push-The-Bit BHA consists of a bit, a non-rotation pad and a
stabilizer. A large section pipe goes through the 3 components. A force model for Push-
The-Bit is shown in Figure 4-3.The stabilizer is a concentric component installed on the
drilling pipe. The non-rotation pad is a hydraulically actuated pad installed near the drill
bit that generates eccentricity by extending its foot to push the wellbore when drilling
directionally. The drill bit is normally PDC bit that can drill both forward and lateral.
29
Figure 4-3 Model for rigid push-the-bit.
When the BHA is used to drill straight, the non-rotation pad is controlled to
retreat its foot, staying in small size status. In this status, the size of non-rotation pad is
less than that of the drill bit that it can stay in the center of the wellbore without touching
anywhere on the wall of the wellbore. The pipe has no side deformation and will keep
straight.
When the engineers decide to drill directionally, they extend the foot of the non-
rotation pad to the wall of the wellbore, and push the wellbore. The reaction force from
the wellbore pushes the pipe to the other side of the wellbore and cause the side force on
the stabilizer and the bit. In this process, the stabilizer and the drill bit may have
movement from the center of the wellbore to the side until they touch the wall of the
wellbore.
When the engineers decided to drill straight, they release the side force at the non-
rotation pad by retreating the extension foot. When the foot leaves the wellbore wall, the
deformation inside the drill pipe disappears and the side force on the non-rotation pad,
the drill bit and stabilizer will disappear. The pipe is recovered to its original straight
shape. Then the drilling direction will return to a straight line, the tangent direction of the
release point.
30
4.2 Analytical Model for the Rigid BHA of Push-The-Bit
A rigid BHA of Push-The-Bit is a force model that assumes all the components
and drill pipe in BHA are rigid body. All the deformation is neglected in the model. Now,
many researchers suggested making models for Push-The-Bit with rigid body assumption.
4.2.1 Assumption
To set up the model of side force for the rigid Push-The-Bit, the following
assumptions are required to use a continuous beam theory. The assumptions are:
1. The rock is rigid body, without any deformation.
2. The non-rotation pad, the bit and the stabilizer are rigid body.
3. The pipe is rigid body due to the small side deformation of the pipe.
4. The influence of the weight of the BHA is ignored.
5. The stiffness of drilling string beyond of the BHA is small enough to be
ignored.
4.2.2 Force balance analysis of the model
From the above assumption, the drilling pipe will keep in straight shape when the
BHA is in static balance status. The BHA has a tilt angle according to the baseline. The
baseline is a straight line goes through the axis of the wellbore at the drill bit and the axis
of the wellbore at the stabilizer. As shown in Figure 4-4, because the size of the stabilizer
and the size of the drill bit are different, when they are pushed to the wall of the wellbore,
the axis of the pipe will tilt to the baseline. This is the reason that even if the drill pipe is
rigid body, the drill bit will still have the tilt angle.
31
The drill bit, the non-rotation pad and the stabilizer are supported by the wall of
the wellbore. These supports are same to the support in the continuous beam model that
the pin support supports the beam. So, the force model of Push-The-Bit can be simplified
to be a 3-support-continuous-string model, as shown in Figure 4-4.
Figure 4-4 Force model for rigid push-the-bit.
The position of the axis of the drilling pipe is expressed by the following equations:
)(1 bw rre , (31 )
pw rre 2 , and (32 )
)(3 sw rre . (33 )
The tilt angle of the drilling pipe is calculated by
)()(
21
31
LLee
t
. (34 )
In this model, when the system is in force balance status, the force balance equations are
expressed as
312 sss FFF and (35 )
2311 ** LFLF ss . (36 )
32
From the equation (3), the direction change caused by the side force s is calculated as
r
sss
SROP
FAF
*
)( 211
. (37 )
When 1sF is in positive direction, the drilling direction is build-up, and s should
be negative because the walk angle should be in anti-clock-wise direction. The walk
angle is the summation of the tilt angle and the push angle,
stw . (38 )
Then, from the equation (19), the dog leg severity (DLS) of the BHA is expressed as
21
2LL
DLS w
. (39 )
Replacing the st & into this equation, the DLS is
21
211
2
21
31
*
)(22
LLSROP
FAF
LL
eeDLS
r
ss
. (40 )
4.2.3 Geometry analysis of the model
When it is at the start point of the directional drilling, there will be no solution
from the above equations because there are 3 unknown side forces but only 2 force
balance equations. In this situation, only if we can measure the side-force at the non-
rotation pad that we can calculate the other values we want to know.
However, if it is in a stable directional drilling process, we can have more
equations by geometry analysis. From the geometric view, 3 points determine a circle. In
the rigid Push-The-Bit model, if the distances between the drill bit and non-rotation pad
and stabilizer are fixed and if the radius of the wellbore is not changed, the curve of the
33
wellbore will be determined by these 3 points, as shown in Figure 4-5. Thus, the DLS of
the wellbore is possible to be calculated by the geometry analysis.
Figure 4-5 DLS for rigid push-the-bit.
Figure 4-5 shows DLS for a rigid Push-The-Bit. The BHA is a straight tool in the
curved wellbore. The distance between the bits, non-rotation pad, and stabilizer is fixed.
And the distance between the center of the bit, the non-rotation pad and the stabilizer d1,
c2, d3 are also known. So, the distances between the axis of the pipe and the top wall of
the wellbore, d1, d2, d3 are expressed by the following equations,
brd 1 , (41 )
pw rrd 22 , and (42 )
srd 3 . (43 )
The direction angle between the drill bit and the non-rotation pad is 1 , and the
direction angle between the non-rotation pad and the stabilizer is 2 . The angles between
the chord lines and the tangle line at the point 2(the point of the top of the wellbore at the
34
non-rotation pad) should be 1 and 2 . The angles between these chord lines and the
axis of the BHA are 1 and 2 . The angles are expressed as
11 5.0 , (44 )
22 5.0 , (45 )
1
21011
L
dd , (46 )
2
32022
L
dd , and (47 )
2
32
1
212121
L
dd
L
dd
. (48 )
Then, from the geometric view, the dog leg severity (DLS) of the BHA is expressed as
21
2
32
1
21
21
21
2
LLL
dd
L
dd
LLDLS
. (49 )
In the equation (44)-(49), ii & are the scalar angles, and i is the vector angle.
The equation (49) can express the direction of DLS. Notice that, the distance difference
between d1 and d3 is expressed as
1331 eerrrrrrdd wswbsb . (50 )
If we combine the DLS equation from the force balance analysis and DLS equation from
geometric analysis, the side force on the drill bit can be calculated by
21
31
2
23
1
211
*
LL
dd
L
dd
L
dd
A
SROPF r
s
. (51 )
35
4.3 Case Study for Rigid BHA Push-The-Bit
4.3.1 Data and Calculation
The case study presented here consists of a directional well and a Push-The-Bit
BHA. The size of the wellbore, drill bit, drill string, stabilizer, non-rotation pad and
distances have been listed in Appendix A-1.
The distances between the drill bit and non-rotation pad has been varied from 2ft
to 12ft to show the influence of the distance. The eccentric distance of the pad is varied
from 0.1 to 1 times of the gap between the wellbore and the drill pipe to show the
influence of the eccentricity. For each case, the side force on the bit, the tilt angle, the
push angle, the walk angle and the DLS are calculated with the rigid Push-The-Bit model.
4.3.2 Results Analysis
Figure 4-6 Tilt angle ratio for rigid BHA Push-The-Bit.
-1
-0.5
0
0 0.2 0.4 0.6 0.8 1
Ɵt/Ɵ
w
Eccentricity of Non-rotation, ec/(Rw-Ri)
Tilt Angle Ratio Vs. Eccentricity
L=4ftL=6ft
L=8ftL=10ft
L=12ft
36
Figure 4-6 shows the change of the tilt angle ratio when the bit-pad distance and
the eccentricity of the eccentric component vary. The tilt angle ratio is the ratio of tilt
angle to the walk angle. This plot shows that the tilt angle ratio is nearly 0, which means
tilt angle contributes little percentage to walk angle.
Figure 4-7 shows the change of the push angle ratio when different bit-pad
distance and the eccentricity of the eccentric component are used. The push angle ratio is
the ratio of push angle to the walk angle. This plot shows that the push angle ratio is
nearly 1 in most cases that means the push angle contributes most to walk angle.
Figure 4-7 Push angle ratio for rigid BHA Push-The-Bit.
Figure 4-8 shows the side force values of the cases. The plot shows that the side
force increases when the bit-pad distance decreases and the side force increases when the
eccentricity increases. It implies that higher eccentricity and stronger drill pipe will create
more side force. The plot also shows that the side force is positive value in most of cases.
1
1.5
2
0 0.2 0.4 0.6 0.8 1
Ɵs/Ɵ
w
Eccentricity of Non-rotation, ec/(Rw-Ri)
Push Angle Ratio Vs. Eccentricity
L=4ftL=6ft
L=8ft
L=10ftL=12ft
37
The side force is applied between the drill bit and top of the wellbore when the side force
is a positive value. So the side force will create a build-up wellbore.
Figure 4-8 Side force on bit for rigid BHA Push-The-Bit.
Figure 4-9 DLS of wellbore for rigid BHA Push-The-Bit.
0
100
200
300
0 0.2 0.4 0.6 0.8 1
Sid
eFo
rce
, lb
f
Eccentricity of Non-rotation, ec/(Rw-Ri)
SideForce Vs. Eccentricity
L=4ft
L=6ft
L=8ft
L=10ft
L=12ft
0
4
8
12
16
0 0.2 0.4 0.6 0.8 1
-DLS
, °/1
00
ft
Eccentricity of Non-rotation, ec/(Rw-Ri)
DLS Vs. Eccentricity L=4ftL=6ftL=8ftL=10ftL=12ft
38
Figure 4-9 shows the value of dog leg severity in the cases. The plot shows that
the DLS increases when the eccentricity of the eccentric component increases. That is,
the increasing deflection in the non-rotation pad will contribute higher DLS to the
wellbore shape. This plot also shows that when the bit-pad distance decreases, the DLS
increases. Almost all the DLS values are negative values that are the build-up DLS.
4.4 Analytical Model for the Elastic BHA of Push-The-Bit
In the rigid BHA model, we assume the drilling pipe is a rigid body because
normally the side deflection of the pipe is very small that we can ignore it. Many
researchers also suggested that the pipe of the BHA can be considered as a rigid body.
However, the deflection of the drilling pipe is too much to be ignored if the radius of the
pipe is small enough or the distances between the drill bit, the non-rotation pad and the
stabilizer are long enough. To calculate side force considering side deflection of the BHA,
we need an elastic BHA model.
As shown in Figure 4-10, an Elastic Push-The-Bit BHA has side deflection in
directional drilling. This BHA also consists of a bit, a non-rotation pad and a stabilizer.
When drilling straight, it is same as the previous model, rigid BHA Push-The-Bit model.
When drilling directionally, the foot of the non-rotation pad is extended to the wall of the
wellbore, pushing the non-rotation pad moving upward. After the bit and the stabilizer
are stopped by the top of the wellbore, the side force from the non-rotation pad will push
the pipe to deflect. The deflection of the pipe at the non-rotation pad will cause the
deformation of the BHA, including bending angle and deflection at every point of the
BHA. The bending angle at the Bit would be the tilt angle t of the drill bit.
39
Figure 4-10 Model for elastic BHA Push-the-bit.
At the same time, the deformation also causes the side force between the bit, the
stabilizer and the wellbore. The side force at the bit will cause the side cutting in the
process of the drilling. Similar to the previous analysis, the effect of the side cutting
equals adding an extra angle s to the walking angle. The walk angle w is the
summation of the tilt angle and the push angle, which is expressed as
stw . (52 )
The drill bit will go along the direction of the walk angle w .
When the engineers retreat the extension foot of the non-rotation pad, the
deflection of the BHA is disappeared. Then the BHA returns to be straight and the tilt
angle t becomes zero. At the same time, the side force on the drill bit also decreases to
zero that the push angle t will also become zero. Thus the walking angle w , the
summation of t and s , also becomes zero and the drilling direction changes to be
straight.
4.4.1 Analytical Model for the Rigid BHA of Push-The-Bit
The assumption for elastic BHA of Push-The-Bit is similar to the rigid BHA of
Push-The-Bit, except for the side deflection of the drilling pipe. The assumptions are:
1. The rock is a rigid body, without any deformation.
2. The non-rotation pad, the bit and the stabilizer are rigid bodies.
40
3. The drilling pipe is an elastic body.
4. The influence of the weight of the BHA can be ignored.
5. The stiffness of drilling string beyond of the BHA is small enough to be
ignored.
4.4.2 Force balance analysis of the elastic BHA Push-The-Bit model
Compared with the rigid BHA Push-The-Bit model, there is an extra extension 2c for the
non-rotation pad foot in the elastic BHA Push-The-Bit model, as shown in Figure 4-11.
2c is the distance between the center of the pipe at the non-rotation pad to the baseline.
This extra extension is the key reason that causes side force on the drill bit.
Figure 4-11 Deflection in the elastic Push-The-Bit.
Figure 4-12 Deformation model for elastic BHA Push-The-Bit.
41
As we can see in Figure 4-11, when drillers drill to build up the wellbore, they
extend the foot of the pad downward, making the side force on the bit upward. This
operation will cause a clockwise tilt angle at the drill bit, which is the opposite direction
of the walk angle. The softer the drilling pipe is, the larger the tilt angle will be, if the
side force keeps in same value. That means, for Push-The-Bit, it is more difficult to drill
directionally with softer drilling pipe.
The Figure 4-12, the straight line drawn through the center of the wellbore at the
drill bit and the center of the wellbore at the stabilizer is the baseline. Even though there
is more deflection of BHA at non-rotation pad, the positions of the drilling pipe at the
drill bit and the stabilizer are same to that of the rigid BHA Push-The-Bit model. In
Figure 4-12, c1, c2, c3 are the distances between the center of the BHA to the bottom of
the wellbore; d1, d2, d3 are the distances between the center of the BHA to the top of the
wellbore; the eccentricities e1, e2, e3 are the distances between the center of the BHA to
the baseline. The position of the axis of the drilling pipe is expressed by
)(1 bw rre , (53 )
)(3 sw rre , (54 )
and e2 is unknown. Similar to the rigid BHA Push-The-Bit, the supports from the
wellbore to the drill bit, the non-rotation pad and the stabilizer are simplified to 3 pin
supports. The force model of elastic BHA Push-The-Bit is a 3-support-continuous-string
model, as shown in Figure 4-13.
In this model, we named the support at the drill bit as support 1, the support at the
non-rotation pad as support 2, and the support at the stabilizer as support 3. The drilling
pipe is divided into two elements by the 3 supports. The element 1 is the pipe between
42
support 1 and the element 2 is the pipe between support 2 and support 3. When the
system is in static force balance status, we can use the following equations to describe the
relation between the deformation and internal force for both the elements:
LQL
EIe
L
EI
L
EIe
L
EI122
1
123
1
112
1
113
1
1 612612 , (55 )
LML
EIe
L
EI
L
EIe
L
EI12
1
122
1
11
1
112
1
1 4626 , (56 )
RQL
EIe
L
EI
L
EIe
L
EI122
1
123
1
112
1
113
1
1 612612 , and (57 )
RML
EIe
L
EI
L
EIe
L
EI12
1
122
1
11
1
112
1
1 2646 . (58 )
This can also be written as
R
R
L
L
M
Q
M
Q
e
e
iL
ii
L
i
L
i
L
i
L
i
L
i
iL
ii
L
i
L
i
L
i
L
i
L
i
1
1
1
1
2
2
1
1
1
1
11
1
1
1
1
2
1
1
1
1
2
1
1
1
1
11
1
1
1
1
2
1
1
1
1
2
1
1
26
46
612612
46
26
612612
. (59 )
Figure 4-13 Force model for rigid Push-The-Bit.
Similarly, for the element 2, the force-deformation equations are
43
R
R
L
L
M
Q
M
Q
e
e
iL
ii
L
i
L
i
L
i
L
i
L
i
iL
ii
L
i
L
i
L
i
L
i
L
i
2
2
2
2
3
3
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
2
26
46
612612
46
26
612612
. (60 )
Considering the boundary condition, at the support 1, the summation of the
moments is zero. That is
01
1
R
S
i MM . (61 )
Simplifying it, the equation (61) becomes
2
1
11
1
12111
6624 e
L
ie
L
iii . (62 )
Similarly, at the support 2, the summation of the moments is
021
2
RL
S
i MMM . (63 )
Simplifying it, the equation (63) becomes
32
2
221
1
13222111
66242 ee
L
iee
L
iiiii . (64 )
Similarly, at the support 3,
02
3
L
S
i MM . (65 )
Simplifying it, the equation (65) becomes
3
2
22
2
23222
6642 e
L
ie
L
iii . (66 )
Combining the equation (62), (64), (66), 1 2 3 can be expressed as
44
32
2
2
32
2
221
1
1
21
1
1
3
2
1
22
2211
11
6
66
6
420
242
024
eeL
i
eeL
iee
L
i
eeL
i
ii
iiii
ii
. (67 )
Solve this equation set,
21
2
2
32
21
21
1
21221
1
12
32
2
3
2
1
ii
i
L
ee
ii
ii
L
eeee
Lt
, (68 )
212
322
211
2112
iiL
eei
iiL
eei
, and (69 )
21
21
2
32
21
1
1
21232
2
3
23
22
3
2
1
ii
ii
L
ee
ii
i
L
eeee
L
. (70 )
Replacing 1 2 3 back into the element equations, we calculate the side forces as
2121
3221
21
2
1
21212
1
1212
1
111
3333
iiLL
eeii
iiL
eeii
L
iee
L
iQF Rs
, (71 )
2
2
2322
2
223
33
L
iee
L
iQF Ls , and (72 )
322
2
2212
1
1
1
2
1
12312
333 ee
L
iee
L
i
L
i
L
iFFF sss
. (73 )
Therefore, the push angle is
r
sss SROP
FAF*
)( 211 . (74 )
When 1sF is positive, the drilling direction is building up, and s should be
negative because the walk angle should be in anti-clock-wise base on the baseline. Then
the walk angle is,
45
21
2
2
32
21
21
1
21
211
2
32
2*
)(
ii
i
L
ee
ii
ii
L
ee
SROP
FAF
r
ss
stw
. (75 )
When the walk angle is positive, 0w , the drilling direction is drop-down; when the
walk angle is negative, 0w , the drilling direction is build-up. The DLS is for this walk
angle is expressed by,
21212
232
21211
2121
21
211 32
*
)(2
iiLLL
iee
iiLLL
iiee
LLSROP
FAFDLS
r
ss
. (76 )
Similarly, when the DLS is positive, the drilling direction is drop-down; when the DLS is
negative, the drilling direction is build-up.
4.4.3 Geometry analysis of the model
In the rigid BHA Push-The-Bit model, we can use the axis line of the BHA as the
baseline to calculate the position of the points and angles. However, in elastic BHA Push-
The-Bit model, the axis of the BHA is not a straight line because of the deformation.
Therefore, in this model, we will use the baseline through the axis of the wellbore, as we
defined for local coordinate system in Chapter3. As shown in Fig Figure 4-14.
In Figure 4-14, c1, c2, c3 are the distances between the baseline and bottom point
of the wellbore at the drill bit, the non-rotation pad and the stabilizer. d1, d2, d3 are the
distances between the baseline and top point of the wellbore at the drill bit, the non-
rotation pad and the stabilizer. α1, α2 are the DLS angles between the 3 components. The
angles between the chord lines and the tangle lines at the point 2(the point of the top of
the wellbore at the non-rotation pad) should be and . The angles between these
chord lines and the axis of the BHA are 1 and 2 .
1 2
46
Figure 4-14 Geometry of elastic BHA Push-The-Bit.
As shown in Figure 4-15, the walk angle change in the BHA part is very small in
real life, and the radius of the wellbore is also very small, we can ignore the vertical tilt
angle when calculating the vertical distances like d1, d2, d4. For example, for a 12 feet
BHA drilling in a 10°/100ft wellbore, the vertical angle will be less than 0.6°. This can be
proved by
6.012*100
10*5.0)(5.0 21 LLDLSvi . (77 )
The distance di is almost similar to the wellbore radius that can be calculated by
wwwi rrrd 9999.06.0cos . (78 )
Then, d1, d2, d3 are expressed by
wrd 1 , (79 )
222 22 errcrd pww , and (80 )
wrd 3 . (81 )
47
Figure 4-15 Vertical tilt angle of the wellbore.
Because e2 is a value with direction, we need add a negative factor before it in
equation (80). The angles between the chord lines and the tangle line are,
and11 5.0 (82 )
22 5.0 . (83 )
The angles between these chord lines and the baseline are,
1
211
L
dd , (84 )
2
322
L
dd , and (85 )
2
32
1
212121
L
dd
L
dd
. (86 )
Then, from the geometric view, the dog leg severity (DLS) of the BHA is expressed as
21
2
21
21
21
2122
LL
rer
LLLLDLS
wp
. (87 )
Simplifying the equation, the DLS is expressed as
48
21
22
LL
rerDLS
wp . (88 )
When the drilling direction is build-up, we can know 02 wp rer . The negative
factor in the equation (88) represents the walk angle is anti-clockwise when drilling
build-up. Notice that in the DLS equation from force analysis, only e2 is unknown value.
If we combine the equation (71), (76) and (88) together, we can solve the e2 , Fs1 and DLS.
The e2 is solved by
0232
*
)(2
21
2
21212
232
21211
2121
21
211
LL
rer
iiLLL
iee
iiLLL
iiee
LLSROP
FAF wp
r
ss. (89 )
To solve these equations, we assume the deflection at the non-rotation pad is ie )( 2 .
Then, from the force analysis, the side-force on the bit and the dog leg severity will be
andiiLL
eeii
iiL
eeiiF ii
is
2121
3221
21
2
1
21211
33
(90 )
21212
232
21211
2121
21
211 32
*
)(2
iiLLL
iee
iiLLL
iiee
LLSROP
FAFDLS ii
r
isisi
. (91 )
From the geometry equation, the deflection of the BHA at the non-rotation pad is,
wpiirrDLSLLe
*
2
12112 . (92 )
Replacing the 12 i
e back into the force equations, we can calculate iteratively till the
difference of ie2 and
12 ie is small enough. Then, replacing
12 ie into the force
equations, we can solve the side-force and the dog leg severity.
49
4.5 Case Study for Elastic BHA Push-The-Bit
4.5.1 Data and Calculation of Elastic BHA Push-The-Bit
The case study presented here consists of a directional well and a Push-The-Bit
BHA. The size of the wellbore, drill bit, drill string, stabilizer, non-rotation pad and
distances have been listed in Appendix A-2.
The distance between the drill bit and non-rotation pad varies from 2ft to 12ft to
show the influence of the distance. The eccentric distance of the pad is varied from 0.1 to
1 times of the gap between the wellbore and the drill pipe to show the influence of the
eccentricity. In each case, the side force on the bit, the tilt angle, the push angle, the walk
angle and the DLS are calculated with the elastic Push-The-Bit model.
4.5.2 Results Analysis
Figure 4-16 shows the change of the tilt angle when we vary the value of the bit-
pad distance and the eccentricity of the eccentric component. This plot shows that the tilt
angle is clockwise direction (positive); also, the tilt angle increases when eccentricity
increases; the tilt angle increases when bit-pad distance increases. The clockwise
direction is a drop-down direction that means the tilt angle for an elastic BHA of Push-
The-Bit will always be a negative factor for the directional drilling purpose.
Figure 4-17 shows the change of the push angle when we vary the value of the
bit-pad distance and the eccentricity of the eccentric component. The push angle is
always in anti-clockwise direction; the push angle increases when the eccentricity
increases; the push angle decreases when the bit-pad distance increases. Comparing with
the tilt angles in plot, the push angle is significantly larger than the tilt angle at the same
50
condition and they have different direction. It means that the side force is a main factor to
form the directional angle.
Figure 4-16 Tilt angle of wellbore for elastic BHA Push-The-Bit.
Figure 4-17 Push angle of wellbore for elastic BHA Push-The-Bit.
0
0.2
0.4
0 0.2 0.4 0.6 0.8 1
Ɵt, °
Eccentricity of Non-rotation, ec/(Rw-Ri)
Tilt Angle Vs. Eccentricity
L=4ftL=6ftL=8ftL=10ftL=12ft
-1
-0.5
0
0 0.2 0.4 0.6 0.8 1
Ɵs,
°
Eccentricity of Non-rotation, ec/(Rw-Ri)
Push Angle Vs. Eccentricity
L=4ftL=6ftL=8ftL=10ftL=12ft
51
Figure 4-18 shows the side force values of the cases. The plot shows that the side
force increases when the bit-pad distance decreases and the side force increases when the
eccentricity increases. It implies that higher eccentricity and stronger drill pipe will create
more side force. The plot also shows that the side force is positive value in most of the
cases. So the side force will likely contribute to form a build-up wellbore.
Figure 4-18 Side force on the bit for elastic BHA Push-The-Bit.
Figure 4-19 shows the value of dog leg severity of the cases. The plot shows that
the DLS increases when the eccentricity of the eccentric component increases; and the
DLS increases when distance decreases. Most of the DLS values are negative values and
they are the build-up DLS. But there are a few cases of positive DLS when distance is
long enough and eccentricity is low enough. It shows that the DLS can be drop-down
direction if the drill string is too soft, even if drillers try to push it to the build-up
direction.
0
400
800
1200
0 0.2 0.4 0.6 0.8 1
Sid
eFo
rce
, lb
f
Eccentricity of Non-rotation, ec/(Rw-Ri)
SideForce Vs. Eccentricity L=6ftL=8ftL=10ftL=12ftL=4ft
52
Figure 4-19 DLS for elastic BHA Push-The-Bit.
Figure 4-20 Tilt-push ratio for elastic BHA Push-The-Bit.
Figure 4-20 shows the ratio of tilt angle to push angle in the cases. The tilt-push
ratio reflects the relationship of the side force effect and tilt angle effect. When the tilt-
push ratio is -1, the two factors encounter each other, and the drilling direction will be
-4
0
4
8
0 0.2 0.4 0.6 0.8 1
-DLS
, deg
ree/
10
0ft
Eccentricity of Non-rotation, ec/(Rw-Ri)
DLS Vs. Eccentricity L=4ft
L=6ft
L=8ft
L=10ft
L=12ft
-2
-1.5
-1
-0.5
0
0 0.2 0.4 0.6 0.8 1
Ɵt/Ɵ
s
Eccentricity of Non-rotation, ec/(Rw-Ri)
Tilt -Push Ratio Vs. Eccentricity
L=4ftL=6ftL=8ftL=10ftL=12ft
53
straight. The plot shows that tilt-push ratio is always negative, and it can be either above -
1 or below -1. So, the drilling direction can be either build-up or drop-down.
Figure 4-21 shows side force versus DLS for both the rigid BHA and elastic BHA.
In the plot we can see, the elastic BHA model predicts higher side forces when drilling a
same DLS wellbore. The rigid BHA model requires less side force for the same DLS
wellbore. The plot also shows that, when the linear stiffness of the BHA increases, the
side force-DLS curve turns from vertical direction to horizontal direction. When the tilt
angle of the curve is smaller, the required side force is less, and the push efficiency is
higher.
Figure 4-21 Relationship between side force and DLS for Push-The-Bit.
54
Chapter 5 Force Analysis for Point-The-Bit
5.1 Introduction
In this study, we will propose a new force analysis model for the Point-The-Bit
that bending a shaft inside a non-rotating housing. It is easy to do the similar force
analysis for the other type of Point-The-Bit.
A typical Point-The-Bit consists of a bit, a pivot pad, a stabilizer and a non-
rotating housing, as shown in Figure 5-1. The shaft connects the drill bit with the housing
and the pivot pad. The space in the housing is large enough for the string to deform. The
bending of the shaft in the housing will change the tilt angle of the drill bit and push the
drill bit against the other side of the wellbore. The deformation of the shaft in the housing
can change not only the side force on the bit, but also the tilt angle of the bit.
Figure 5-1 Point-The-Bit with a non-rotating housing.
5.2 Assumption
In the Push-The-Bit model, the side force on the drill bit is the main reason for the
direction change of the drilling direction. The tilt angle is normally in opposite direction
of the walk angle, as shown in Figure 5-2. The walk angle is the push angle after
overcoming the tendency of the tilt angle. However, in Point-The-Bit model, the tilt angle
55
is obviously at the same direction as the walk angle, which is normally the main reason
for the drilling direction change.
Figure 5-2 Difference between Push-The-Bit & Point-The-Bit.
Sugiura (2008) founded that there were at least 3 ways to tilt the drill bit in Point-
The-Bit BHA. Figure 2-8, 2-9 and 2-10 show the 3 mechanisms for tilting the drill bit;
They are: 1) bending a shaft inside a non-rotating housing; 2) holding a predetermined
bias with a geo-stationary unit inside a rotating housing; and 3) positioning a non-rotating
housing with three pads to tilt a drill bit.
To analyze the force for Point-The-Bit, we assume that,
1. The rock is rigid body.
2. The pilot pad, the bit and the stabilizer are rigid body.
3. The housing is rigid body.
4. Only the shaft inside the housing is elastic body.
5. Ignored the weight of the BHA.
6. Ignore the drilling string beyond the BHA.
56
5.2.1 Analytical Model for Point-The-Bit in the Straight Wellbore
In Point-The-Bit model, the shaft is a continuous beam. The drill bit, the pivot
pad, the stabilizer and the housing are large size components that can be considered as
rigid bodies. The shaft deformed in the housing causes the side forces between the
wellbore and all the other components. Figure 5-3 shows a sketch of the force model of
Point-The-Bit. In the model, the shaft is pin-supported inside the housing. The housing is
a rigid body without contacting the wellbore. The whole Point-The-Bit is pin-supported
by the wellbore at the drill bit, the pivot pad and the stabilizer.
Figure 5-3 Force model for BHA of Point-The-Bit.
The eccentricities of the BHA, 1e , 2e , 3e , 4e , 5e , 6e , are the distance from the center
of the BHA to the baseline. At the start point of the directional drilling, the wellbore is a
straight hole. In this point, except 5e , the eccentricities are calculated by
bw rre 1 , (93 )
pw rre 2 , (94 )
1
1
22
1
213 e
L
Le
L
LLe
, (95 )
cco eLL
LeLeeee
43
433544 , and (96 )
sw rre 6 . (97 )
57
As shown in Figure 5-5, at the node 4, the eccentric ring, ce is the distance
between the shaft axis and the housing axis at the eccentric ring position. The 4oe is the
distance between the housing axis and the baseline at position of the eccentric ring.
The bending angle of the BHA, 1 , 2 , 3 , 4 , 5 , 6 , are the angle between the
tangent line of the BHA and the baseline. At the start point of the directional drilling,
except 4 , the other bending angles are calculated by
andL
ee
1
21321
(98 )
5
6565
L
ee . (99 )
Figure 5-4 Force model for global system.
The global BHA system is only supported at 3 points, the bit the pivot pad and the
stabilizer. Figure 5-4 shows the global force balance for BHA. The follow equations
describing the force balance for the BHA as
5432611 LLLLFLF ss , (100 )
58
0621 sss FFF , (101 )
0221 sLs FQF , and (102 )
LRss MMLFLLF 2322211 . (103 )
Solve these equations, the shear force Q2L and moment M2L are expressed as
62 sL FQ and (104 )
54362 LLLFM sL . (105 )
Figure 5-5 Force model for the shaft inside the housing.
The shear force and moment of the element 5 are expressed as
655 sLR FQQ and (106 )
565 LFM sR . (107 )
In this model, there is the elastic deformation only in the shaft. All the other parts,
including drill bit, pivot pad, housing, and the stabilizer are the rigid bodies. We isolate
the shaft as a structure; the influences of the rigid bodies to the shaft are the moments, the
shear forces and the deformations at the end of the shaft. As shown in Figure 5-5.
Similar to the previous elastic BHA Push-The-Bit model, the relationship between
the forces and deformations in the elements is expressed by the following equations. For
the element 1, the force-deformation equations are expressed as
59
R
R
L
L
M
Q
M
Q
e
e
iL
ii
L
i
L
i
L
i
L
i
L
i
iL
ii
L
i
L
i
L
i
L
i
L
i
3
3
3
3
4
4
3
3
3
3
33
3
3
3
3
2
3
3
3
3
2
3
3
3
3
33
3
3
3
3
2
3
3
3
3
2
3
3
26
46
612612
46
26
612612
, (108 )
where 333 / LEIi . Similarly, for the element 2, the force-deformation equations are
expressed as
R
R
L
L
M
Q
M
Q
e
e
iL
ii
L
i
L
i
L
i
L
i
L
i
iL
ii
L
i
L
i
L
i
L
i
L
i
4
4
4
4
5
5
4
4
4
4
44
4
4
4
4
2
4
4
4
4
2
4
4
4
4
44
4
4
4
4
2
4
4
4
4
2
4
4
26
46
612612
46
26
612612
, (109 )
where 444 / LEIi .
In the model, 1e , 2e , 3e , 6e , 1 , 2 , 3 is determined, 4e , 5 , 6 is possible to be
expressed by other variables. The only unknown variables are 5e , 4 , 6sF . When the
model is in static balance status, considering the moment balance at pin-support 3, pin-
support 4, pin-support 5, the force balance equations are written out. At support 3, the
total moment applied on the node is 0, 03 Support
M . That is
0323 RLSupport
MMM . (110 )
Simplifying the equation as
ecs eL
iee
L
ie
LL
iFLLLie
LL
i
3
321
1
33
43
36543435
43
3 6462
6
. (111 )
60
At support 3, the total moment applied on the node is 0, 04 Support
M . That is
0434 RLSupport
MMM . (112 )
Simplifying the equation as
6
5
421
1
3
3
3
4
43
43
434435
5
4
43
43 226
64
26e
L
iee
L
ie
L
i
L
ie
LL
iiiie
L
i
LL
iiec
. (113 )
At support 5, the total moment applied on the node is 0, 05 Support
M . That is
0545 RLSupport
MMM . (114 )
Simplifying the equation as
6
5
4
4
43
43
465445
5
4
43
4 4662
46e
L
ie
L
ie
LL
iFLie
L
i
LL
iecs
. (115 )
If we write equation (111), (113) and (115) together, the force balance equations can be
expressed by
6
5
4
4
43
43
4
3
321
1
33
43
3
6
5
421
1
3
3
3
4
43
43
43
6
4
5
54
5
4
43
4
5433
43
3
43
5
4
43
43
466
646
226
6
246
26
0426
eL
ie
L
ie
LL
i
eL
iee
L
ie
LL
i
eL
iee
L
ie
L
i
L
ie
LL
ii
F
e
LiL
i
LL
i
LLLiLL
i
iiL
i
LL
ii
ec
ec
ec
s
. (116 )
Figure 5-6 Deformation model for the shaft inside the housing.
61
From this equations set, we can solve the 3 variables 5e , 4 , 6sF . The bending
angles and the side forces are calculated by replacing these 3 variables back into the
element equations. That is
6
1
54321 ss F
L
LLLLF
. (117 )
The Figure 5-6 shows the relationship between the walk angle, the push angle and
the tilt angle. The tilt angle is the angle between the axis of the drill bit and the baseline,
as shown in the following equation,
1
211
L
eet
. (118 )
The push angle is the angle caused by the side force on the bit, as shown in the following
equation,
r
ss
sSROP
FAF
ROP
ROS
*
)( 211
. (119 )
The walk angle is the summation of the tilt angle and the push angle, which can be
calculated as
r
ss
wSROP
FAF
L
ee
*
)( 211
1
21
. (120 )
When the walk angle is positive, 0w , the drilling direction is drop-down; Otherwise,
when the walk angle is negative, 0w , the drilling direction is build-up. The DLS is
expressed as
5
1
211
5
1
1
21
*
)(22
k
kr
ss
k
k LSROP
FAF
LL
eeDLS . (121 )
62
Similarly, when the DLS is positive, the drilling direction is drop-down; when the DLS is
negative, the drilling direction is build-up.
5.2.2 Force Model for Point-The-Bit in the Curved Wellbore
The previous equation set is based on the start point of the directional drilling
when the wellbore is still straight. Otherwise the extra deflections ie* and bending angles
i* should be considered when calculate the deflections of BHA in a curved wellbore.
Figure 5-7 shows the extra eccentricities and bending angles of the wellbore in this model.
From the previous analysis in Chapter 3, if we assume the dog leg severity of the
wellbore is DLSi, the extra deflections and bending angles of the BHA can be expressed
by
iiiii LDLS **1
* , (122 )
6,*2
1
61,*
1,*2
1
5
1
*
1*
1*
5
1
*
jLDLS
jLDLS
jLDLS
k
ki
jjij
k
kij
j
, and (123 )
6,0
61,2
)(
1,0
*
*1
*
1**
*
je
jL
ee
je
j
jjj
jj
j
. (124 )
The deflections of the BHA in curved wellbore, 1ce , 2ce , 3ce , 4ce , 5ce , 6ce , are the
distances from the center of the BHA to the baseline. Except for 5e , the others are
calculated by
bwc rreee 1*
11 , (125 )
63
2*
2*
22 erreee pwc , (126 )
1
1
22
1
213 ccc e
L
Le
L
LLe
, (127 )
eccc
c eLL
LeLee
43
43354 , and (128 )
swc rreee 6*
66 . (129 )
Figure 5-7 Deflection of the wellbore axis.
The bending angle of the BHA in curved wellbore, 1c , 2c , 3c , 4c , 5c , 6c , are
the angles between the tangent line of the BHA and the baseline. They are calculated by
the following equations except for 4c ,
1
21321
L
ee ccccc
and (130 )
5
6565
L
ee cccc
. (131 )
64
Similar to the analysis as in the straight wellbore, the force balance equations are
6
5
4
4
43
43
4
3
321
1
33
43
3
21
1
3
3
3
4
43
43
43
6
4
5
54
5
4
43
4
5433
43
3
43
5
4
43
43
466
646
26
6
246
26
0426
cecc
ecccc
ccecc
s
c
c
eL
ie
L
ie
LL
i
eL
iee
L
ie
LL
i
eeL
ie
L
i
L
ie
LL
ii
F
e
LiL
i
LL
i
LLLiLL
i
iiL
i
LL
ii
, (132 )
where iii LEIi / . With this equation sets, we solve the 3 variables, 5ce , 4c , 6sF . The
bending angles and the side forces are calculated by replacing these 3 variables back into
the element equations. The side force on the bit is expressed as
6
1
54321 ss F
L
LLLLF
. (133 )
The push angle is the angle cause by this side force on the bit, which is expressed as
r
sscs SROP
FAF
ROP
ROS
*)( 2
11 . (134 )
The tilt angle is the angle between the axis of the drill bit and the baseline, which is
expressed as
1
211
L
ee cccct
. (135 )
The walk angle is the summation of the tilt angle and the push angle, which is expressed
as
r
sscccw SROP
FAF
L
ee
ROP
ROS
*)( 2
11
1
21
. (136 )
When the walk angle is positive, 0cw , the drilling direction is drop-down;
when the walk angle is negative, 0cw , the drilling direction is build-up; when 0cw ,
the drilling direction will keep straight, in the tangent direction of wellbore at the position
65
of the drill bit. The DLS*i+1 is dog leg severity calculated from the assumption of DLSi
and the force analysis, which is expressed as
5
1
211
5
1
1
211
*
*
)(22
k
kr
ss
k
k
cci
LSROP
FAF
LL
eeDLS . (137 )
Let
iii DLSDLSDLS 1*
12
1 (138 )
and replace the DLSi+1 back into the equation (122), we can calculate iteratively till the
difference between DLS*i+1 and DLSi is small enough. Then, DLSi+1 is the dog leg
severity when we apply ec as the eccentric distance inside the housing.
Similarly, when the DLSi+1 is positive, the drilling direction is drop-down; when
the DLSi+1 is negative, the drilling direction is build-up; when DLSi+1 =0, the drilling
direction is straight.
5.3 Case Study for Point-The-Bit Model
5.3.1 Data and Calculation
The model for Point-The-Bit is applied in a curved wellbore. The case study
presented here consists of a directional well and a Point-The-Bit BHA when the well is
drilled directionally. The size of the wellbore, drill bit, drill string, stabilizer, pivot pad,
shaft and distances are listed in Appendix A-3.
The distance between the drill bit and pivot pad varies from 2ft to 12ft to show the
influence of the distance. The eccentric distance of the pad is varied from 0.1 to 1 times
of the gap between the housing and the shaft to show the influence of the eccentricity.
66
For each case, the side force on the bit, the tilt angle, the push angle, the walk angle and
the DLS are calculated with the Point-The-Bit model.
5.3.2 Results Analysis
Figure 5-8 shows the change of the tilt angle ratio when the bit-pad distance and
the eccentricity of the eccentric component varies. The tilt angle ratio is the ratio of tilt
angle to the walk angle. This plot shows that the tilt angle ratio is nearly or above 1,
which means the tilt angle is the main part of the walk angle. In most cases, the tilt angle
ratio is close to 1, which means the tilt angle is almost equals to walk angle, therefore the
push angle is approximately 0.
Figure 5-8 Tilt angle ratio for Point-The-Bit model.
Figure 5-9 shows the change of the push angle ratio when the bit-pad distance and
the eccentricity of the eccentric component vary. The push angle ratio is the ratio of push
angle to the walk angle. This plot shows that the push angle ratio is nearly 0 in most cases,
-1
1
3
0 0.2 0.4 0.6 0.8 1
Ɵt/Ɵ
w
Eccentricity of Pilot Pad, ec/(Rw-Ri)
Tilt Angle Ratio Vs. Eccentricity
L=4ft
L=6t
L=8ft
L=10ft
L=12ft
67
which means the side force is not a main factor in Point-The-Bit BHA. The blue line
(L=4ft) shows that, when the distance decreases to certain value, the push angle will
increase significantly. The shorter the BHA is, the larger linear stiffness the BHA has.
This phenomenon suggested that the stronger BHA creates more side force on the bit.
Figure 5-9 Push angle ratio for Point-The-Bit model.
Figure 5-10 shows the side force values of the cases. The plot shows that the side
force increases when the bit-pad distance decreases. It is for the same reason as the
previous plot that the stronger BHA creates more side force on the bit. This plot also
shows that in most cases, the side force is a negative value that creates build-up DLS.
However, when the bit-pad distance becomes longer, the side force may change to be a
positive value. The positive force will create drop-down DLS. Therefore, the side force in
the Point-The-Bit can be either negative factor or positive factor impacting the directional
drilling.
-1
-0.5
0
0 0.2 0.4 0.6 0.8 1
Ɵs/Ɵ
w
Eccentricity of Pilot Pad, ec/(Rw-Ri)
Push Angle Ratio Vs. Eccentricity
L=4ft
L=6t
L=8ft
L=10ft
L=12ft
68
Figure 5-10 Side force for Point-The-Bit model.
Figure 5-11 shows the value of dog leg severity for the cases. The plot shows that
the DLS increases when the eccentricity of the eccentric component increases. That is,
the more deflection the driller make to the shaft, the higher DLS the wellbore is formed.
This plot also shows that when the bit-pad distance decreases, the DLS increases firstly
then begins to decrease. When the distance is long, the side force is small and push angle
is too small to influence the walk angle. When distance is small enough, the side force
can be large enough to decrease the walk angle, so the DLS will also decreases. The plot
shows the optimization bit-pad distance will be likely 6 ft for the data set. When the
distance is 6ft, the BHA can drill the largest DLS wellbore.
-1000
-500
0
500
0 0.2 0.4 0.6 0.8 1
Sid
eFo
rce
, lb
f
Eccentricity of Pilot Pad, ec/(Rw-Ri)
SideForce Vs. Eccentricity L=4ft
L=6t
L=8ft
L=10ft
L=12ft
69
Figure 5-11 DLS for Point-The-Bit model.
Figure 5-12 Relationship between side force and dls for Point-The-Bit.
0
5
10
0 0.2 0.4 0.6 0.8 1
-DLS
, deg
ree/
10
0ft
Eccentricity of Pilot Pad, ec/(Rw-Ri)
DLS Vs. Eccentricity L=4ft
L=6t
L=8ft
L=10ft
L=12ft
-700
-400
-100
200
0 2 4 6 8 10 12
Sid
eFo
rce
, lb
f
-DLS, degree/100ft
SideForce Vs. Eccentricity L=4ft
L=6t
L=8ft
L=10ft
L=12ft
70
Figure 5-12 shows side force versus DLS for Point-The-Bit BHA. In the plot we
can see, the force-DLS curve falls down from vertical position to horizontal position if
the distance between the bit and the pivot pad decreases. In most cases, the side forces are
negative value that the side cutting contributes drop-down DLS, different to build-up
effect contributed by the tilt angle. In short distance case like the blue line, L=4 ft, a
large anti-directional force are caused. In these cases, the red line, L=6ft, is the best case,
because it has the highest DLS and relatively lower side force. In the other cases, the anti-
directional side force is small that positive influence the directional drilling operation, but
the DLS is also small that negative influence the directional drilling operation. This
shows that, the DLS is higher with higher anti-directional side force if the BHA is harder.
The DLS is lower with the lower anti-directional side force if the BHA is softer. In
certain distance, there will be highest DLS and lower anti-directional side force.
71
Chapter 6 Conclusions & Recommendations
6.1 Summary
The analytical models for Push-The-Bit and Point-The-Bit are promising.
It will be proved useful when designing a directional or horizontal well as
well as controlling the direction of the wellbore.
The walk angle is proved to be the summation of the tilt angle and the
push angle in this project. The relationship between the two factors, side
force and tilt direction, which can/will influence the drill direction.
The two models for Push-The-Bit show that the force is the most
important factor in the directional drilling. To avoid the obstruction of the
tilt angle, the BHA should be made stronger. A soft BHA of Push-The-Bit
is inclined to be a lower efficiency directional drilling tool.
The rigid body assumption is an optimistic for Push-The-Bit. It represents
the best behavior of a Push-The-Bit BHA in directional drilling operation.
More side force and lower DLS need to be considered when using routine
Push-The-Bit BHA.
The model for Point-The-Bit shows that the tilt angle is the most important
factor in the directional drilling. To avoid the obstruction of the side force,
the BHA should not be made too strong. A soft BHA of Point-The-Bit is
inclined to create lower side force that can keep the drilling direction more
stable.
72
Distance is an important factor in directional drilling BHA design. It is
promising to focus on the optimization bit-pad distance in Point-The-Bit
design. The distance between the other components can also influence the
drilling process in similar way.
The three models show that normally larger DLS requires stronger side
force. However, damage or block problems may happen if the side force
exceeds the limitation.
6.2 Future Work & Recommendations
Side cutting model is very important when calculating the push angle. It is
also influence the accuracy of the walk angle model. More research work
of side cutting rate should will applied in future,.
The shape of the drill bit will also influence the side cutting rate that is not
considered in this project. It is also an interesting research topic in future,
The influence of the axial force has been ignored in the three models.
Some research work should be applied to uncover the role of the axial
force in direction control.
The models are based on static force analysis. The real drilling operation
is a dynamic process. More field data should be considered and compared
to the model to improve the model.
73
Reference
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Directional Behavior of Bottomhole Assemblies including Those with Bent Subs and
Downhole Motors, IADC/SPE 14767, presented at the 1986 IADC/SPE Drilling
Conference, Dallas, TX, 10-12 February.
Eastman H. 1950. The Latest Developments and Achievements of Directional Drilling in
the Exploitation of Oil Fields, presented at the Proceedings Third World Petroleum
Congress, Section II, 89-109.
Lesso, W. Jr., Chau M., and Lesso, W. Sr. 1999. Quantifying Bottomhole Assembly
Tendency Using Field Directional Drilling Data and a Finite Element Model, 52835-
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March.
Lesso, W., Rezmer-Cooper, I., and Chau, M. 2001. Continuous Direction and Inclination
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Menand S., Simon C., Gaombalet J., Macresy L., DrillScan, Gerbaud L., Ben Hamida M.,
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74
Conference and Exhibition of the Society of Petroleum Engineers of AIME,
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75
Appendix A
A-1
Diameter of Wellbore(in) 8.5
Eccentricity of Pad/ec(rw-ri) 0.1―1
Diameter of Bit(in) 8.5
Bit-Pad Distance(ft) 4―12
Diameter of Stabilizer(in) 8 Pad-Stabilizer Distance(ft) 20
Modulus of Drill Pipe(psi) ∞ ROP(ft/hr) 100
Out Diameter of Drill Pipe(in) 6 Sr 2
Inner Diameter of Drill Pipe(in) 6 A 5E-06
A-2
Diameter of Wellbore(in) 8.5
Eccentricity of Pad/ec(rw-ri) 0.1―1
Diameter of Bit(in) 8.5
Bit-Pad Distance(ft) 4―12
Diameter of Stabilizer(in) 8 Pad-Stabilizer Distance(ft) 20
Modulus of Drill Pipe(psi) 29E6 ROP(ft/hr) 100
Out Diameter of Drill Pipe(in) 6 Sr 2
Inner Diameter of Drill Pipe(in) 6 A 5E-06
A-3
Diameter of Wellbore(in) 8.5
Eccentricity of Pad/ec(rw-ri) 0.1―1
Diameter of Bit(in) 8.5
Bit-Pad Distance(ft) 4―12
Diameter of Pad(in) 8 Pad-Housing Distance(ft) 2
Diameter of Stabilizer(in) 8 Head-Component Distance(ft) 8
Modulus of Housing(psi) 29E6 Component-End Distance(ft) 8
Out Diameter of Housing(in) 6.5 Housing-Stabilizer
Distance(ft) 8
Inner Diameter of
Housing(in) 5.6
ROP(ft/hr) 100
76
Modulus of Shaft(psi) 29E6
A 5E-06
Out Diameter of Shaft(in) 3.5
Sr 2
Inner Diameter of Shaft(in) 2.7
