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Electronic Theses and Dissertations

UC San Diego

Peer Reviewed

Title:

Computation modeling of drill bits : a new method for reproducing bottom hole geometry and asecond-order explicit integrator via composition for coupled rotating rigid bodies

Author:

Endres, Lanson Adam

Acceptance Date:

2007

Series:

UC San Diego Electronic Theses and Dissertations

Degree:

Ph. D., UC San Diego

Permalink:

https://escholarship.org/uc/item/63r062vf

Local Identifier:

b6635562

Abstract:

A new, highly adaptable, framework for the simulation of bit dynamics is introduced. The newsoftware, dubbed Three Dimensional Bit Dynamics (TDBD), is implemented as object- orientatedcode and designed to meet current and future research needs. The software uses a triangulatedmesh to represent the bit and rock surface geometry. A novel method for updating the rock surface

as the bit drills is presented. This method uses information from both the bit and rock surfaceto allow for general motion of the bit (including whirl). It also limits the number of operationsrequired to maintain the shape quality of the elements in the rock surface mesh. This methodavoids the costly operation of removing elements and remeshing the resulting hole. A derivationof the equations of motion of a roller cone bit is presented as an example of coupled rotating rigidbodies. The equations are derived using the virtual power method which naturally handles theconstraint between the bit body and the cones. A state-of-the-art numerical integrator is appliedto the equations to produce an algorithm suitable for use in a bit dynamics software application.The integrator is a composition of adjoint first order integrators (reminiscent of the approach usedearlier in Reference [47] to derive an explicit midpoint Lie method). It maintains the propertiesof the original three degree-of-freedom integrator: second order convergence, symplecticness,remarkable accuracy, and momentum conservation. This algorithm can be applied to otherapplications where rotating rigid bodies are coupled through an axis that allows rotation. The forcemodel from Reference [33], used for calculating the forces on a bit resulting from rock removal,

is implemented within the framework and used to evaluate the software. The weight-on-bit andtorque-on-bit are compared to laboratory data. Two polycrystalline diamond compact (PDC) bitsare used to calibrate the force model and a third is run in the software to "predict" the laboratorydata
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UNIVERSITY OF CALIFORNIA, SAN DIEGO

Computation Modeling of Drill Bits: A New Method for Reproducing Bottom

Hole Geometry and a Second-Order Explicit Integrator via Composition for

Coupled Rotating Rigid Bodies

A dissertation submitted in partial satisfaction of the

requirements for the degree Doctor of Philosophy

in

Structural Engineering

by

Lanson Adam Endres

Committee in charge:

Professor Petr Krysl, Chair

Professor David Benson

Professor Hidenori Murakami

Professor P. Benson Shing

Professor Chia-Ming Uang

2007


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Copyright

Lanson Adam Endres, 2007

All rights reserved.


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The dissertation of Lanson Adam Endres is approved, and it is acceptable in

quality and form for publication on microfilm:

Chair

University of California, San Diego

2007

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Dedication

To my mom, for being the personification of what it means to be a great parent.

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Table of Contents

Page

Signature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Overview of a Rotary Drilling System . . . . . . . . . . . . . . . . . . 1

1.2 Drill Bits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Dynamics and Dynamic Instabilities . . . . . . . . . . . . . . . . . . . 5

1.3.1 Roller Cone Dynamics . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Objective and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 PDC/drag Bit Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Roller Cone Bit Modeling. . . . . . . . . . . . . . . . . . . . . . . . . 13

Chapter 3 Bit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Class Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Bit Coordinate System . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Cutter Coordinate System. . . . . . . . . . . . . . . . . . . . . 22

3.3 Bit Meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 4 Rock Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 Updating the Rock Surface (Cutting Rock) . . . . . . . . . . . . . . . . 27

4.1.1 Finding Bit-Rock Interference . . . . . . . . . . . . . . . . . . 28

4.1.2 Update Directions. . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.2.1 PDC Bit Update Directions . . . . . . . . . . . . . . 30

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4.1.2.2 Roller Cone Bit Update Directions . . . . . . . . . . 35

4.1.2.3 Update Directions Reference Position . . . . . . . . . 36

4.1.3 Updated Position of Rock Vertex . . . . . . . . . . . . . . . . . 36

4.1.3.1 Rock Vertex Position Updating Algorithm. . . . . . . 37

4.1.3.2 Mapping Update Directions to Rock Vertices . . . . . 384.1.3.3 Moving Rock Vertices . . . . . . . . . . . . . . . . . 39

4.1.3.4 Checking the Neighborhood of a Triangle for a Rock

Vertex. . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.4 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Rock Surface Maintenance . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Triangle Deletion by Edge Collapsing . . . . . . . . . . . . . . 44

4.2.2 Triangle Addition by Bisection . . . . . . . . . . . . . . . . . . 47

4.2.3 Laplacian Surface Smoothing. . . . . . . . . . . . . . . . . . . 48

4.2.4 Remarks on Surface Maintenance . . . . . . . . . . . . . . . . 51

4.2.4.1 Bisection . . . . . . . . . . . . . . . . . . . . . . . . 514.2.4.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.4.3 Maintenance Frequency . . . . . . . . . . . . . . . . 52

4.2.4.4 Shape Quality Criteria . . . . . . . . . . . . . . . . . 53

4.3 Volume of Rock Removed . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.1 Bottom Hole Patterns . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.2 Long Term Behavior . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.3 Whirl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 5 Roller Cone Integrator . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Kinematics of a Roller Cone Bit . . . . . . . . . . . . . . . . . . . . . 71

5.2 Principle of Virtual Power. . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.1 Virtual Power of the Inertial Forces . . . . . . . . . . . . . . . 76

5.2.2 Virtual Power of the Applied Forces . . . . . . . . . . . . . . . 79

5.3 Equations of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1 Arbitrary Number of Cones. . . . . . . . . . . . . . . . . . . . 80

5.4 Rotational Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4.1 Body Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4.2 Cone Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.3 Angular Momentum Midpoints . . . . . . . . . . . . . . . . . . 86

5.4.4 Integral Form and Explicitness . . . . . . . . . . . . . . . . . . 875.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.7 Acknowledgement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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Chapter 6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.1 Laboratory Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 Computer Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Calibration of the Force Law . . . . . . . . . . . . . . . . . . . . . . . 98

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.4.1 Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4.2 High Frequency Comparison . . . . . . . . . . . . . . . . . . . 106

Chapter 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.1 Rock Updating Method . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2 Roller Cone Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.3 Force Law Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Appendix A Descriptions of Bits . . . . . . . . . . . . . . . . . . . . . . . . 113

Appendix B Class Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 118

B.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

B.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

B.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

B.6 Surface Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B.6.2 . . . . . . . . . . . . . . . . . . . . . . . 128

B.7 Simulation Control Classes . . . . . . . . . . . . . . . . . . . . . . . . 132

B.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

B.8 Force Law Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B.8.2 . . . . . . . . . . . . . . . . . . . . 141

Appendix C Force Law of Hanson and Hansen (1995) . . . . . . . . . . . . 142

C.1 Cutter Face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

C.2 Cutter Chamfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

C.3 CSEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

C.4 Force Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Appendix D Alternate Calibrations of the Force Law . . . . . . . . . . . . . 146

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

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Nomenclature

Every attempt has been made to prevent a variable from having multiple defini-

tions. However, where this does occur the meaning should be clear from the context.

Symbols / Operators Skew-symmetric matrix of.0 Base, reference, or initial frame or time of.

Time differentiation of ddt.

Time differentiation of

done twice d

2

dt

2

.[] A matrix or vector, a ; is used to denote the end of one row and the

beginning of the next.

A matrix or vector norm.

det Determinate of matrix.

exp[] Exponential of.

proj Projection ofonto.

Diameter.

Latin Variables

a,b,c Coordinates in the spatial frame.

a Vector determined by context.

A Area.

A# Force law coefficient where a subscript of # denotes a general coeffi-cient(s) and an integer subscript refers to a specific coefficient.

Acut Area of rock cut by a bit part.

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At Area of a triangle.

A Angular acceleration vector.

B# Calculated value in force law where a subscript of # denotes a generalcoefficient(s) and an integer subscript refers to a specific coefficient.

c0/c Vector to the center of gravity of the bit (and origin of the bits coordinatesystem) in the reference/current configuration.

cp Vector to the top of the drill string.

ct Vector of the center of the area of a triangle.

cv Vector of the area weighted average center of the triangles connected to

a vertex.

d0/d An arbitrary unit vector in the reference/current configuration such thats0 d0= 0/s d= 0.

d0 The vector d0rotated byR()used to describe a position on the cone inthe reference configuration.

D Scalar distance along d0/d.

De Distance along an edge of a cutter face.

Dr/Dr Rotational damping matrix / rotational damping scalar coefficient.

Ebr Effective back rake of cutter, angle of attack.

E1/E2 Vectors used in force law.

f Scalar force.

f Force vector.

fb Force vector applied to the body.

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fc Force vector applied to cone.

Ffn /Fc

n/Fs

n Normal force for a cutter face/chamfer/CSE used in a force law.

Fft /Fct/Fst Tangent force for a cutter face/chamfer/CSE used in a force law.

g Constant parameter.

h Time step size.

H Body frame virtual angular velocity about the reference axis of the cone.

H Virtual velocity.

Hc Translational virtual velocity.

H Body frame virtual angular velocity.

i Summation variable / spacial frame angular momentum scalar.

Ic Body frame moment of inertia of cone about s0.

I Body frame tensor of inertia of bit body and attached cone(s).

J Determinate of the Jacobian matrix.

J Jacobian matrix.

k Summation variable.

Kr/Kr Rotational stiffness matrix / rotational stiffness scalar coefficient.

Le Edge length of a triangle.

M Total mass (scalar) of bit body and attached cone(s).

n Discretized time,n is the current time step, n1 is the previous timestep, et cetera.

NG Number of Gauss Points.

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nt Normal to a triangle.

nv Normal of a vertex calculated as an area weighted average of the normals

of the triangles connected to the vertex.

NT Number of triangles.

N Basis function.

NC Number of cones.

NFb Number of forces on the bit.

NFc Number of forces on the cone.

p0/p A vector to a position on the bit in reference/current configuration.

p0 A vector to a position on the bit relative to c0in reference configuration.

P Pressure.

P A vector to a position on the bit in the bits coordinate system.

Pe Virtual power of the applied forces.

Pi Virtual power of the inertial forces.

q0 r0+Ss0.

r0/r Vector to the origin of a bit parts coordinate system in the refer-ence/current configuration; in the case of a cone it is also the attachment

point of the cone.

r0/r Vector attachment point of the cone relative to c0in the reference/currentconfiguration.

Rs Force law coefficient.

R Rotation matrix.

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R Rotation matrix that rotates a point to theX-Zplane in the bits coordi-nate system.

Rr Rotation matrix that rotations to the plane defined by the location of therock vertex and the Z-axis of the bit.

R() Rotation matrix that describes the rotation about the direction s0throughthe angle.

R(),R Rotation matrix that describes the rotation of the cone about s0/s throughthe angle.

s0/

s Unit vector along the axis of the cone in the reference/current configura-

tion.

S Scalar distance along s0/s.

Scc Confined compressive strength.

t Time / spacial frame scalar torque.

t0 Unit vector completing an orthonormal triad in the cone body-attached

frame,t0 = s0d0.T Torque vector applied to the cone.T Vector of total torque applied to bit, body and cone.

u0 Eitherp0orq0+DRd0.U Preliminary update direction (in bits coordinate system).

U Update direction as defined for a bit part.

U Update direction mapped to a reference position (theX-Z plane of the

bits coordinate system).

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Ur Update direction rotated about the Z-axis of the bit, from the reference

configuration, to the location of a rock vertex.

v Vector of the location of a vertex.

V Volume.

Vb Volume of the bit body.

Vc Volume of the cone.

W Gauss point weight.

x,y,z Coordinates in the spatial frame.

xp, yp, zp Coordinates of a point in spatial frame.

x,y, z Spacial frame basis vectors.

X, Y, Z Bit coordinate system basis vectors.

X, Y, Z Bit part coordinate system basis vectors.

Greek Variablest Time step size.

Time variable / scalar rotational degree-of-freedom.

, , Isoparametric coordinates.

,, Isoparametric basis vectors.

Spacial frame angular momentum vector of the bit.

Angular momentum of the cone about its axis. Body frame angular momentum vector of the bit.

Mass density.

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Time variable.

An angle about s0 used, in part, to describe the position of a point on the

cone.

Angle the cone has rotated about its axis. Incremental rotation vector of bit body.

Body frame angular velocity vector.

Angular velocity scalar of cone about its axis.

Angular velocity vector of cone about its axis.Typeface/Notation

bold Vector or matrix.

italic Scalar.

Computer code.

String in computer code.

% Commenting character in computer/psuedo code.

Class name in TDBD.

Abstract class name in TDBD.

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List of Figures

Page

Figure 1.1: Drilling rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Figure 1.2: The two types of bits, roller cone bits and drag bits . . . . . . . 3Figure 1.3: Three types of dynamic instabilities . . . . . . . . . . . . . . . 5

Figure 2.1: Spoke and node type of rock surface . . . . . . . . . . . . . . . 12

Figure 3.1: Bit class diagram . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 3.2: Bit A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 3.3: Bit B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 3.4: Bit C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 3.5: Bit R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 3.6: Bit coordinate system . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 3.7: Cutter coordinate systems. . . . . . . . . . . . . . . . . . . . . 22Figure 3.8: Exposure profile. . . . . . . . . . . . . . . . . . . . . . . . . . 23

Figure 3.9: Typical cutter meshes from Bit A and Bit C . . . . . . . . . . . 24

Figure 3.10: Typical teeth meshes from Bit R . . . . . . . . . . . . . . . . . 25

Figure 4.1: Rock surface class diagram . . . . . . . . . . . . . . . . . . . . 27

Figure 4.2: Activity diagram of rock updating . . . . . . . . . . . . . . . . 28

Figure 4.3: Activity diagram for finding the rock vertices within the volume

of a bit part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 4.4: Update directions for a PDC bit . . . . . . . . . . . . . . . . . 30

Figure 4.5: Update directions for a roller cone bit . . . . . . . . . . . . . . 31

Figure 4.6: Update directions for inside and outside cutters . . . . . . . . . 32

Figure 4.7: Vertex divergence in two-dimensions for the method chosen . . 33

Figure 4.8: Vertex divergence in two-dimensions for update directions that

follow the bits Z-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 4.9: Vertex divergence in two-dimensions for update directions that

follow the bits exposure profile. . . . . . . . . . . . . . . . . . . . . . 34

Figure 4.10: Update directions for teeth on a roller cone bit . . . . . . . . . . 35

Figure 4.11: Activity diagram for rock vertex position updating. . . . . . . . 37

Figure 4.12: Mapping update directions to rock nodes. . . . . . . . . . . . . 39

Figure 4.13: Activity diagram for moving rock vertices . . . . . . . . . . . . 40

Figure 4.14: Two dimensional representation of searching the neighborhoodof a triangle to find the relocated rock vertex . . . . . . . . . . . . . . . 42

Figure 4.15: Activity diagram for checking the neighborhood of a triangle for

a rock vertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 4.16: Edge collapsing sequence. . . . . . . . . . . . . . . . . . . . . 44

Figure 4.17: Activity diagram for edge collapsing of triangles . . . . . . . . 45

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Figure 4.18: Triangles that cannot be deleted . . . . . . . . . . . . . . . . . 46

Figure 4.19: Triangle bisection sequence. . . . . . . . . . . . . . . . . . . . 47

Figure 4.20: Activity diagram for bisecting triangles . . . . . . . . . . . . . 49

Figure 4.21: Two dimensional example of surface smoothing . . . . . . . . . 49

Figure 4.22: Potential volume loss by the movement of a vertex during sur-face smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 4.23: Prism formed by the previous and current locations of a triangle 54

Figure 4.24: Mapping of a prism from global to isoparametric coordinates . . 54

Figure 4.25: Bottom hole pattern of Bit A from the laboratory . . . . . . . . 58

Figure 4.26: The bottom hole pattern of Bit A from TDBD . . . . . . . . . . 59

Figure 4.27: Bottom hole pattern of Bit R from the laboratory . . . . . . . . 60

Figure 4.28: The bottom hole pattern of Bit R from TDBD . . . . . . . . . . 61

Figure 4.29: Simulations as they were initialized for drilling . . . . . . . . . 62

Figure 4.30: Bit B after drilling several inches in TDBD . . . . . . . . . . . 62

Figure 4.31: Bit R after drilling several inches in TDBD . . . . . . . . . . . 63Figure 4.32: Side view of mesh after drilling several inches with a PDC bit . 64

Figure 4.33: Bottom view of mesh after drilling several inches with a PDC bit 64

Figure 4.34: Side view of mesh after drilling several inches with a roller cone

bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 4.35: Bottom view of mesh after drilling several inches with a roller

cone bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 4.36: Position of the center of mass of Bit B in thex-yplane while it

underwent whirl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 4.37: Rock surface resulting from Bit B undergoing whirl in TDBD . 67

Figure 4.38: Side view of mesh after a bit underwent whirl . . . . . . . . . . 68

Figure 4.39: Bottom view of mesh after a bit underwent whirl . . . . . . . . 68

Figure 5.1: Roller cone bit coordinate systems . . . . . . . . . . . . . . . . 72

Figure 5.2: Body frame angular velocities resulting from an applied torque . 90

Figure 5.3: Convergence for free rotation with different initial conditions

and body configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Figure 5.4: Kinetic energy and angular momenta for the slow top problem . 92

Figure 5.5: Components of the unit vector along the axis of the slow top

problem projected into the x-y plane . . . . . . . . . . . . . . . . . . . 93

Figure 6.1: HCC high-pressure down hole drilling simulator. . . . . . . . . 95Figure 6.2: Sample output from the down hole drilling simulator . . . . . . 96

Figure 6.3: Force law calibration using Bit B and Bit C using both bottom

hole pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure 6.4: Predicted values for Bit A when calibrated using both bottom

hole pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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Figure 6.5: Force law calibration using Bit B and Bit C with different coef-

ficients for each BHP . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 6.6: Predicted values for Bit A when calibrated using separate bottom

hole pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 6.7: High frequency down hole drilling simulator (512 Hz) and TDBD(1000 Hz) data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure A.1: Four bits modeled in TDBD . . . . . . . . . . . . . . . . . . . 113

Figure B.1: class diagram . . . . . . . . . . . . . . . . . . . 117

Figure B.2: class diagram . . . . . . . . . . . . . . 119

Figure B.3: class diagram Part 1 . . . . . . . . . . . . . . . . . . . . . 120

Figure B.4: class diagram Part 2 . . . . . . . . . . . . . . . . . . . . . 121

Figure B.5: class diagram Part 1. . . . . . . . . . . . . . . . . . 122

Figure B.6: class diagram Part 2. . . . . . . . . . . . . . . . . . 123

Figure B.7: class diagram Part 1. . . . . . . . . . . . . . . 124Figure B.8: class diagram Part 2. . . . . . . . . . . . . . . 125

Figure B.9: class diagram Part 1 . . . . . . . . . . . . . . . . . . 126

Figure B.10: class diagram Part 2 . . . . . . . . . . . . . . . . . . 127

Figure B.11: class diagram Part 1 . . . . . . . . . . . . 129

Figure B.12: class diagram Part 2 . . . . . . . . . . . . 130

Figure B.13: class diagram Part 1 . . . . . . . . . . . . . . . . . 131

Figure B.14: class diagram Part 2 . . . . . . . . . . . . . . . . . 132

Figure B.15: class diagram . . . . . . . . . . . . . . . . . . . . . . 133

Figure B.16: class diagram Part 1 . . . . . . . . . . . . . . . . . . . . 134

Figure B.17:

class diagram Part 2 . . . . . . . . . . . . . . . . . . . . 135Figure B.18: class diagram Part 1 . . . . . . . . . . . . . . . . 136

Figure B.19: class diagram Part 2 . . . . . . . . . . . . . . . . 137

Figure B.20:

class diagram Part 3 . . . . . . . . . . . . . . . . 138

Figure B.21: class diagram . . . . . . . . . . . . . . . . . . . . 139

Figure B.22: class diagram . . . . . . . . . . . . 140

Figure D.1: Force law calibration using bits Bit A and Bit C . . . . . . . . . 147

Figure D.2: Force law calibration using bits Bit A and Bit B . . . . . . . . . 148

Figure D.3: Predicted values for bit Bit B . . . . . . . . . . . . . . . . . . . 149

Figure D.4: Predicted values for bit Bit C . . . . . . . . . . . . . . . . . . . 149

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List of Tables

Page

Table 5.1: Matrix of terms corresponding to integrals of products of virtual

and actual kinematic quantities . . . . . . . . . . . . . . . . . . . . . . 78Table 5.2: Variations of initial conditions for free rotation convergence tests . 91

Table 6.1: Initial values of force law coefficients . . . . . . . . . . . . . . . 99

Table 6.2: Values of force law coefficients when calibrated for 3000 psi and

6000 psi together in TDBD . . . . . . . . . . . . . . . . . . . . . . . . 102

Table 6.3: Values of force law coefficients when calibrated for 3000 psi and

6000 psi separately in TDBD . . . . . . . . . . . . . . . . . . . . . . . 104

Table 6.4: Values of the WOB and TOB objectives for separate and com-

bined bottom hole pressure calibrations. . . . . . . . . . . . . . . . . . 105

Table A.1: Bit A cutter and chamfer sizes . . . . . . . . . . . . . . . . . . . 114Table A.2: Bit B cutter and chamfer sizes . . . . . . . . . . . . . . . . . . . 114

Table A.3: Bit C cutter and chamfer sizes . . . . . . . . . . . . . . . . . . . 114

Table D.1: Values of force law coefficients when calibrated for 3000 psi and

6000 psi in TDBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Table D.2: Values of force law coefficients when calibrated for 3000 psi and

6000 psi in TDBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

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Acknowledgements

The author would like to thank the many individuals whose efforts have made

this work possible.

First and foremost, I would like to thank Dr. Petr Krysl for his support, patience,

and guidance.

I am grateful to the Hughes Christensen Company (HCC) for its support, as well

as the numerous HCC employees (past and present) for their contributions. In partic-

ular, thanks are given to Dr. Thomas Black, Van Brackin, Dr. David Curry, Dr. Olivier

Hoffmann, Roy Ledgerwood, Matt Meiners, Allen Sinor, Tim Marvel, Anupam Singh,

and Dr. David Trevas. Their effort, support, and contributions are much appreciated.

My gratitude is also extended to Dr. Jonathan Hanson. He has offered invaluable

experience and insight into bit dynamics and bit dynamics software applications.

I would also like to extend my gratitude to family members who have provided

support and educational opportunities for me: my mom, Janet Endres, my grandparents,

Phyllis and Harold Gillihan, and my great uncle, Jim Evans.

I am eternally indebt to the professors and classmates who I have learned from

in both my undergraduate and graduate work. Although they are too numerous to list,

they have had, and continue to have, a great influence on me and my education.

Chapter5, in full, has been submitted for publication of the material as it may

appear in Communications in Numerical Methods in Engineering, 2007, Endres, L.;

Krysl, P., John Wiley & Sons, 2007. The dissertation author was a primary investigator

and author of this paper.

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Vita

EDUCATION

December 2007 Ph.D. in Structural Engineering

University of California, San Diego La Jolla, CA 92093

December 2004 M.S. in Structural Engineering

University of California, San Diego La Jolla, CA 92093

May 1999 B.S. in Civil Engineering

Lawrence Technological University Southfield, MI 48075

PUBLICATIONS

[1] L. Endres and P. Krysl. Refinement of Finite Element Approximations on Tetrahe-

dral Meshes Using Charms. In Seventh U.S. National Congress on Computational

Mechanics. Albuquerque, NM, July, 2003.

[2] L. Endres and P. Krysl. Octasection-based refinement of finite element approxima-

tions on tetrahedral meshes that guarantees shape quality. Int J for Numer Meth in

Eng. 59(1) (2004): 69-82.

[3] P. Krysl and L. Endres. Explicit Newmark/Verlet algorithm for time integration of

the rotational dynamics of rigid bodies. Int J for Numer Meth in Eng. 62(15) (2005):

2154-2177.

[4] L. Endres and P. Krysl. Second-order explicit integrator via composition for coupled

rotating rigid bodies applied to roller cone drill bits. Commun Numer Meth Eng. To

appear.

xx


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ABSTRACT OF THE DISSERTATION

Computation Modeling of Drill Bits: A New Method for Reproducing Bottom Hole

Geometry and a Second-Order Explicit Integrator via Composition for Coupled

Rotating Rigid Bodies

by

Lanson Adam Endres

Doctor of Philosophy in Structural Engineering

University of California, San Diego, 2007

Professor Petr Krysl, Chair

A new, highly adaptable, framework for the simulation of bit dynamics is in-

troduced. The new software, dubbed Three Dimensional Bit Dynamics (TDBD), is

implemented as object-orientated code and designed to meet current and future research

needs.

The software uses a triangulated mesh to represent the bit and rock surface ge-

ometry. A novel method for updating the rock surface as the bit drills is presented. This

method uses information from both the bit and rock surface to allow for general motion

of the bit (including whirl). It also limits the number of operations required to main-

tain the shape quality of the elements in the rock surface mesh. This method avoids the

costly operation of removing elements and remeshing the resulting hole.

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A derivation of the equations of motion of a roller cone bit is presented as an ex-

ample of coupled rotating rigid bodies. The equations are derived using the virtual power

method which naturally handles the constraint between the bit body and the cones.

A state-of-the-art numerical integrator is applied to the equations to produce

an algorithm suitable for use in a bit dynamics software application. The integrator

is a composition of adjoint first order integrators (reminiscent of the approach used

earlier in Reference [47] to derive an explicit midpoint Lie method). It maintains the

properties of the original three degree-of-freedom integrator: second order convergence,

symplecticness, remarkable accuracy, and momentum conservation. This algorithm can

be applied to other applications where rotating rigid bodies are coupled through an axis

that allows rotation.

The force model from Reference [33], used for calculating the forces on a bit

resulting from rock removal, is implemented within the framework and used to evaluate

the software. The weight-on-bit and torque-on-bit are compared to laboratory data. Two

polycrystalline diamond compact (PDC) bits are used to calibrate the force model and a

third is run in the software to predict the laboratory data.

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Chapter 1

Introduction

Roller cone drill bits have been used for exploration and retrieval of oil and nat-

ural gas since 1909 when Howard Hughes Sr. introduced the first rotary drill bit. Roller

cone bits and their counterpart, polycrystalline diamond compact (PDC) bits (more gen-

erally known as drag bits) form the two predominate bit types in use today. Both types

are pictured in Figure1.2on page3.

Continued improvements to bit technology depend, in part, on the dynamic be-

havior of the drilling system. The dynamics involved in drilling have been shown to

affect the rate of penetration (ROP) and life of bits ([36], [65]). To understand the dy-

namics of drill bits, computer models have been built and have shown to provide useful

insight on bit design ([20]). Many models focus on the effects of a single cutter or tooth.

However, more recent models have incorporated an entire bit ([58], [33], [38]).

1.1 Overview of a Rotary Drilling System

The bit is just one part of a complex system required to drill a well. A brief

overview of a few of the components that compose a common drilling system is pre-

sented below.

A diagram of a system for onshore drilling is pictured in Figure1.1.The system

shown is a rotary rig, the most common type of rig currently in use ([3]). The basic

1


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2

Figure 1.1: Drilling rig. The primary components are a power system, lifting compo-

nents, a circulatory system, and a rotary system which includes the drill bit.

components are a power system, a superstructure which holds lifting components, a

drilling fluid circulatory system, and a rotary system ([4]). The names of the power

system and lifting components imply their functions. The power system supplies power

to the drill string, lifting components, mud pumps, et cetera. The lifting components

hoist the drill string. Amongst other things, the rotary system contains the drill string

and the bit at the end of the string. The bit is part of the bottom hole assembly (BHA).

The BHA may contain motors, steering equipment, and electronics to supply dynamic

and positional data. The circulatory system pumps a drilling fluid, commonly referred

to as mud, through the center of the drill string and out specially designed nozzles

located in the bit. The fluid carries rock cuttings up the well between the drill string and

borehole wall as well as cools and lubricates the bit.

A rotary system uses the weight of the drill string to apply a downward force to

the rock. Heavier sections of pipe, called drill colors, are a part of the BHA and attached


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3

Figure 1.2: The two types of bits, roller cone bit (left) and drag bit (right). Images are

courtesy of Hughes Christensen Company.

at the bottom of the drill string to supply weight on the bit. The hoisting system lifts up

the drill string to balance the remaining weight of the drill string. That is, the weight

on the bit is the difference between the weight of the drill string and the weight held by

the hoisting system. This allows the majority of the drill string to remain in tension to

prevent buckling. While this force is applied, the drill string and bit are rotated at the

surface by the power system.

1.2 Drill Bits

Drill bits come in two basic forms, roller cone bits and drag bits (shown in Fig-

ure1.2). The components that are referenced in this work are described below.

Roller cone bits contain freely rotating (rolling) bodies (cones) attached to the

bit body. The cones contain the cutting structures called teeth, arranged in rows, which

break up rock by pushing into the rock until it is crushed. Rows consist of teeth on the

same cone that are at the same position (distance from an arbitrary point) along the axis

of rotation of the cone. The teeth are either machined from the same material as the

cone or are of the tungsten carbide insert (TCI) form.

The cones of a roller cone bit are not true cones. Instead, they are a combination


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4

of cones of different pitches. This is meant to keep them from rolling true. That is, the

different pitches cannot all roll at the speed which they naturally desire to roll so some

compromise between different rolling speeds must occur.

Using different pitches for the cones is a design feature meant to prevent track-

ing. Tracking occurs when teeth from the cone tend to fall within an impression pre-

viously created by another tooth. An action of this type prevents cutting of the entire

bottom hole and limits rates of penetration.

Another anti-tracking design feature consists of using an off-center axis. The

axes of the cones typically do not pass through the center of rotation of the bit. Instead,

the axis of rotation of a cone is typically a few degrees off of a line connecting the center

of the back of the cone with the axis of rotation of the bit.

In addition, the teeth in a row typically are not spaced equally around the cir-

cumference of the cone. This is meant to prevent a repetitive pattern of teeth as the cone

rotates and comes in contact with the rock, and therefore, hopefully, limit tracking.

Drag bits use a scraping, or shearing, action to remove rock. The primary drag

bit in use today is a PDC bit. PDC bits have cutters that are composed of a manufactured

diamond table (black part of inset on the right hand side of Figure1.2) joined to a metal

body (grey part in the inset). The cutters are the part of the bit designed to shear the

rock.

Certain areas of cutters have specific names. The flat front of the diamond table

(this is the top of the cutter in the inset) is called the cutter face. The cylindrical portion

is called the body (for both the diamond and metal portions). The edge between them is

typically beveled and, in that case, is called a chamfer.

Roller cone bits and drag bits have some common features. Both types may con-

tain rubbing structures call gage pads at the outer edge. Gage pads may have diamonds

(in the case of PDCs) or hard facing material (in the case or roller cones) to prevent

excessive wear. The bits also have junk slots which allow the drilling fluid to pass by


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5

Figure 1.3: The three types of dynamic instabilities of a bit are torsional, vertical, and

lateral. Lateral vibration is also called whirl. When torsional and vertical vibrations

become severe enough they are referred to stick-slip and bit bounce, respectively. Image

is courtesy of Hughes Christensen Company.

the bit and up the borehole.

The body and cones of a roller cone bit are steel and PDC bit bodies are typically

manufactured from tungsten carbide. However, steel body PDC bits are also manufac-

tured.

1.3 Dynamics and Dynamic Instabilities

In terms of bit dynamics, three main instability modes exist: torsional, vertical,

and lateral (or whirl). If torsional and vertical vibrations are severe enough, they are

referred to as stick-slip and bit bounce, respectively. These modes are shown as red

arrows in Figure1.3.

Stick-slip is a torsional vibration characterized by alternating phases where bit

stops rotating and then accelerates to a velocity which is faster than the rate input by

the surface drive. This behavior may cause increased wear and can impact drilling

efficiency. Drag bits are more susceptible to stick-slip because they typically require

more torque to drill than roller cone bits.

Because the cones can rotate, roller cone bits exhibit stick-slip far less than PDC

bits ([11]) which rely on torque for their cutting action. It has been shown that the design


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6

of PDC bits influences the tendency for stick-slip to occur ([35], [67]).

Bit bounce occurs when the bit comes off of the bottom hole (loses contact with

the rock) and reengages the rock (usually violently). This behavior is generally very

harmful to the bit as well as the bottom hole assembly. Bit bounce may accompany

stick-slip.

Whirl is a lateral vibration mode where the instantaneous center of rotation is

not the geometric center of the bit. Moreover, the center of rotation is moving during

whirl. Forward whirl and backwards whirl refer to whether the geometric center of the

bit is orbiting about the center of the hole in the same direction as the bit is rotating. If

the orbiting and rotating directions coincide it is forward whirl; if the directions oppose

each other then it is backwards whirl. Synchronous forward whirl occurs when the bit

is whirling forward at the same rate as the rate of rotation.

While both forward synchronous and backwards lateral vibrations are called

whirl, they are distinctly different. Forward synchronous whirl is not an ideal behav-

ior, but is far less harmful than backwards whirl. Backwards whirl can cause cutters on

a PDC bit to travel backwards and sideways; cutters are not designed for these motions.

Whirling (whirl will be used to refer to backwards whirl unless otherwise specified)

has been identified as a primary cause of premature PDC failure in hard formations

bits ([10], [12]). Proper design can limit a bits tendency to whirl ([76]), but static anal-

ysis alone cannot determine if a bit will whirl.

1.3.1 Roller Cone Dynamics

It is known that the cones of a roller cone do not maintain a constant angular ve-

locity while drilling. Instead, they oscillate (or vibrate or chatter) as they rotate ([56]).

The dynamics of roller cone bits is more complex than drag bits because of the extra de-

grees of freedom added by the cones ([44]). While roller cone bits are less likely to

backwards whirl, they do have a tendency to forward synchronous whirl. As previously


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7

mentioned, this is not generally damaging to the bit. However, it can produce unfavor-

able wear patterns on the bit and limit the rate of penetration.

1.4 Objective and Outline

The current work focuses on a new, highly adaptable, framework for a bit dy-

namics simulation application. The new software application, dubbed Three Dimen-

sional Bit Dynamics (TDBD), introduces many new and enhanced features and ad-

dresses issues associated with previous models. The application is implemented as

object-orientated code in C++. This allows for future research requirements to be met

without redesigning the software; this is a limitation in current codes. (References tocode are minimized in the main body and left for an appendix. Where code is presented,

it is given in Unified Modeling Language (UML) 2.0 notation.)

A new bit and rock surface design were developed. Both utilize a triangulated

surface mesh in three dimensions to represent the geometry. This allows any shape to

be approximated as closely as desired by controlling the element size in the mesh. This

method also prevents the resolution issues of the spoke and node rock surface system

(discussed in Chapter2). That is, the new rock design incorporates a mesh that is equally

fine at all locations.

Chapter2outlines previously developed methods and applications for predicting

the forces on, and stability of, bits while they are drilling.

Chapter3 presents the design of the bit model. While the design is unique and

provides a more flexible research tool than previously available, the primary purpose of

this chapter is to introduce concepts used in later chapters.

The first part of Chapter4 introduces the design of the rock surface. Following

that, the method used to update the rock as the bit drills is presented. Updating the rock

(also referred to as rock cutting) allows the mesh to evolve with the drilling, replicating

the drilling process. A novel and robust system for updating the rock is presented. This


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8

method exploits information about both the bit and rock to allow the reproduction of

bottom hole patterns without the need for remeshing. Moreover, this method limits the

amount of operations required to maintain a reasonable shape quality in the triangles

of the mesh. Details important to both the evolution of the mesh and calculating the

forces from the cutting process are presented. The end of Chapter 4presents results

from simulations.

In Chapter5, a state-of-the-art numerical integrator for coupled multi-body ro-

tational dynamics is presented. The integrator was developed with a bit dynamics ap-

plication in mind and is highly suitable for roller cones. Of primary importance is that

it is explicit in the forcing evaluation. The cost of calculating forces prohibits an im-

plicit method. In addition, it is symplectic, momentum conserving, and has second order

convergence.

As a precursor to the realization of the integrator, the equations of motion for

roller cone bits are derived. The assumptions of these equations are fully stated and can

be quantified. These equations are fully three-dimensional (three degrees-of-freedom

for the body, plus one degree-of-freedom for each cone). Unlike previous equations

derived using complicated trigonometry expressions base upon roller cone geometry,

these equations nicely parallel the equations of motion of a single rigid body.

The penultimate chapter presents a comparison between laboratory data and the

software. This chapter discusses the laboratory tests, calibration process, and software

runs used to validate the software.

Conclusions are presented in Chapter7.


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Chapter 2

Literature Review

Historically, bit models have ranged from those that describe general bit behavior

based upon laboratory and field data to those that model the (nearly) exact bit geometry

and its interaction with the rock surface. Classical bit models tend to fall into the former

category, while newer models lean toward the latter.

Models for the bit generally make a few of the same assumptions: the bit is rigid

(no deflections are considered), the cones rotate on frictionless bearings, and perfect

cleaning takes place (rock is immediately removed from bottom hole once broken or

cut). Other assumptions very greatly from model to model.

2.1 PDC/drag Bit Modeling

Warren and Sinor ([75]) present a drag bit model that calculates the WOB, TOB,

imbalance force, and volume of rock removed. This model uses the kinematics for

bit motion and does not allow lateral vibrations. Therefore, the rock surface can be

represented as a single plane because the areas removed by the cutters is constant. The

geometry of cutters was determined by using a three axis coordinate measuring machine.

This model was later expanded by Behr, Warren, Sinor, and Brett ([6]) to have a

three-dimensional rock surface. This surface uses system of radial planes at 5intervals

discretized in the radial direction. The model can replicate whirl motion kinematicly.

9


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10

Glowka proposed a model for the forces on a bit based upon the results of single-

cutter tests ([28], [29], [30]). This model calculates forces by determining the area cut

by a cutter when cutter interaction is taken into account (surface in not assumed virgin).

The model attempts to predict cutter wear, WOB, TOB, the side force, and bending

moments. However, it is not a dynamic model.

In References [49] and [50], Langeveld presents a bit dynamics application for

PDC bits that accounts for formation removal. The rock surface is modeled as 120

spokes radiating from the center of the borehole. Each spoke is discretized with 500

points. It is not mentioned how the cutters are modeled. Bit gauge was assumed to be

a full circle; junk slots are not accounted for. Four degrees-of-freedom are used: x-axis

translation, y-axis translation, z-axis translation, and z-axis rotation. A fourth order

Runge-Kutta method is used to integrate the four equations-of-motion associated with

the four degrees-of-freedom. The drill string is assumed to be a simple mass-spring

system. The force law used is a very simple depth-of-cut model.

Hanson and Hansen ([33]) developed a model based on the work of Langeveld.

Impact with the borehole is accounted for by allowing energy loss from the contact.

Their model also includes a variable number of spokes and nodes along those spokes.

In addition, the surface has a general shape in the plane of a spoke. That is, in the plane

of a spoke, there may be multiple z values for a given x value. This is important to

accurately represent the bottom hole during whirl. For example, a whirling cutter can

remove a semi-circular portion of the borehole wall. A vertical line drawn through such

a cut would intersect the borehole surface twice, once at the top and once at the bottom

of the feature. The equations of motion are given as

Mc + Dtc + Kt(c cp) = f (2.1)

Mrockcrock+Dtcrock+Ktcrock = fz (2.2)

i+Dr+Kr( p) = t (2.3)


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11

where

crock = translational displacement of rock along global zdirection;

c = translational displacement vector of the bit;

cp = translational displacement of the top of the drill pipe;

Dt/Dt = translational damping matrix / translational damping scalar coef-

ficient;

Dr = rotational damping scalar coefficient;

fz = third component off;

f = force vector;

Kt/Kt = translational stiffness matrix / translation stiffness scalar coeffi-

cient;

Kr = rotational stiffness scalar coefficient;

Mrock = mass of rock;

M = mass of bit and a portion of the drill string;

t = spacial frame torque scalar;

= rotational displacement of bit;

p = rotational displacement of the top of the drill pipe;

for a system that has four degrees-of-freedom for the bit and one for the rock (Equa-

tion (2.2)). This allows the rock to move in the globalz direction to simulate drilling

in the laboratory where the rock may vibrate. This software was later modified to allow

use with roller cones as well (see Section2.2).

Two cutter force models were proposed in Hanson and Hansen. The first is based

on single point cutter tests and the second is based on matching the performance of an

entire bit in the simulation to laboratory data. The results from the second model are

considered a closer match to laboratory data. This model is discussed more in Chapter6

and AppendixC.

All the models discussed in References [6], [33], [49], and [50] use a similar ap-


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12

Figure 2.1: Spoke and node type of rock surface. The black lines represent the spokes;

the black dots (only in enlarged view) are the nodes. This type of rock surface results in

a low resolution mesh between spokes near the gauge of the borehole.

proach to represent the rock surface. The system uses a series of spokes that resemble the

spokes of a bicycle wheel when looking down the bore hole. This is shown in Figure2.1.

Each of these methods discretizes the spokes, although each method accomplishes this

in a slightly different manner. The advantage of this method is that it simplifies updating

the rock surface. Because the nodes all lie in the planes of the spokes, the process of

updating the rock surface is effectively reduced to a two-dimensional problem.

While relatively computationally efficient, this system of representing the rock

surface leads to poor resolution at the gauge of the borehole. The primary issue with

this method is shown in the enlarged view on the right side of Figure 2.1. The spokes

diverge from one another when moving from the center of the hole to the gauge. If nodes

are connected to form elements at the edge of the borehole, the resulting element

has a poor shape quality. Because of the moment arm, errors in forces at the edge

of the borehole are much more significant than errors at the center. Adding spokes may

increase the resolution at the edge, but it also adds resolution at the center of the borehole

where it is not required.


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13

2.2 Roller Cone Bit Modeling

Briggs ([13], [14]) uses an idealized bit and rock model to calculate the forces

and torque expected on a real bit. The bit is represented as a two dimensional sprocket

which is driven across a flat rock which is assumed to behave as a Coulomb plastic

material. Using this model, forces and torques are calculated for the bit.

Winters, Warren, and Onyia ([77]) presented a bit model to calculate the rate of

penetration (ROP) for a bit or, given the ROP, estimate the rock strength. The model

accounts for cone offset, rock ductility, tooth indention, and tooth sliding. Confined

compressive strength is used to characterize the rock. Bit or tooth geometry is not

introduced. This model allows for estimation of rock strength from the ROP. While thismodel may give reasonable estimates to the ratio of the rate of penetration to the weight

on bit (WOB), it cannot predict the dynamic instabilities that are particularly damaging

to bits. Moreover, it does not allow for comparing one bit design to another to determine

which bit is better in some regard.

Dekun Ma, in a sequence of papers, developed a theory and a computer appli-

cation for roller cone bits. The first paper by Ma and Yang ([56]) derives kinematic

equations for a roller cone based on a cylindrical coordinate system for the bit body and

a separate coordinate system for each cone. These equations rely heavily on both the

geometry of the bit and trigonometric expressions. As a result they are rather cumber-

some.

Mas equations of allow for rotation of the bit, translation of the bit along the

z-axis, and rotation of the cones. They do not permit translation or rotation of the bit

with respect to the x-axis or the y-axis. That is, they are not fully three-dimensional and

instead rely on a fixed orientation and path of travel of the bit.

Derivatives of the kinematic expressions are taken to give velocity and accelera-

tion equations, however, no attempt to produce equations of motion is made.

Ma and Azar ([54]) expanded upon the work in [56] to produce a computer


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14

algorithm for determining the position of the teeth are and which ones are in contact

with the hole bottom. Statistical arguments are used as justification for reducing the

dynamic problem to kinematics. The average rate of rotation for the bit and cones are

used as input to the kinematic equations for a simulation using this method.

In the derivation of the kinematic equations several assumptions are made, mainly

that the cones are symmetric, the cones are at equal angles from one another (for exam-

ple, on a three cone bit the angle between each cone is 2/3), the cones are all located

at the same elevation, and the teeth are uniformly arranged in a row (no busted pitches).

In Reference [58] Ma, Zhou, and Deng presented a method for the modeling of

roller cone bit-rock interaction. The kinematic equations of References [56] and [54]

are used to specify locations on the bit. Accordingly, the allowable degrees of freedom

are limited to the rotation of the bit about the z-axis, vertical position of the bit on the

z-axis, and rotations of the cones about their bearing axis.

Building upon the work in References [56] and [54], the model is advanced in

Reference [58] by adding some dynamics and control features. While the bit rotation

is assumed constant, the cone rotations are calculated dynamically. Cone rotations are

updated according Equation (2.5), which results from Equation(2.4).

T = Ic d2(t)dt2 (2.4)(t+ t) =+TIc(t)2 (2.5)

where

= rotation of cone,

T = torque on the cone,Ic = moment of inertia of cone about its rotation axis,

t = t ime.


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15

Equation (2.5) is written as printed in the source, however, it would seem thatTIc is atypographical error and should instead be replaced byT /Ic.

The vertical position of the bit along the z-axis is calculated using an iteration

process and the assumption that the WOB is equal to the vertical force resulting from

bit interaction with the rock.

The equations used to describe the bit are used in a computer application to

simulate the interaction between the bit and rock. An important feature of this software

is that the teeth and bottom hole are discretized to represent the actual geometry instead

of an idealized geometry. However, there is no mention of how, or if, the discretization

is updated for the rock surface as it evolves with the drilling process. The software

is used to predict the force on individual teeth, the force on rows of teeth, the rate of

penetration, and more.

The patent of Huang and Cawthorne ([38]) claims two methods for determining

cone movement. The first is to use the method of Dekun Ma, et al. in Reference [58].

The second method simply states that the moment of inertia of the cone and the forces on

the cone can be used to update the cone position. However, no further explanation of the

second method is given and equations are not supplied. The patent of Reference [39],

a continuation of [38], offers no further information related to the dynamics of roller

cones.

Sheppard and Lesage ([72]) present a model intended to predict the rate of pen-

etration and torque on a bit for a given bit rotation rate and WOB. This model can also

predict instantaneous rates of cone rotation. The equations presented are geometric and

trigonometric based, as with Mas. However, the axis of the cone (journal bearing axis)

is represented as a unit vector. Because of this, Sheppard and Lesages equations are

somewhat simpler than Mas. The model does not account for the inertial forces or al-

low for a history dependent bottom hole pattern. It also depends on the average volume

of rock removed by the bit and cannot account for individual teeth.

While originally developed for PDC bits, the bit dynamics application of Hanson


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and Hansen ([33]) was later extended to roller cone bits ([20]). Each cone has a separate

degree-of-freedom which is independent from all other degrees-of-freedom. The equa-

tions of motion for the bit and each cone are integrated using a standard Runge-Kutta

method1. Teeth on the cones are represented as spheres and the cones are limited to

surfaces of revolution.

1Communication with Jonathan M. Hanson, author of the software, October, 2003


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Chapter 3

Bit Model

This chapter describes the bit model used in TDBD. Many of the features intro-

duced in this chapter are referenced in later chapters.

The primary design requirement of the bit model was flexibility of bit features

to meet current and future research needs. Behavior and properties of the bit vary from

area to area. Variations include material, force laws, rock removal rates, and wear rates.

The bit model is required to supply these functions and allow for future requirements.

3.1 Class Design

The class diagram for the class is shown in Figure3.1. The class

is an abstract base class used by several classes in TDBD. It is discussed in

AppendixB.

From the class an abstract class is de-

rived. It serves as a container to hold and operate on s. The

also serves as a base class for both the and classes. These two

classes form the foundation of the bit model.

Results from rock cutting are stored in the structure that is a

member of the class. Section 4.1 discusses how the rock surface

evolves with time to model to drilling process. The results return from this process are

17


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18

Figure 3.1: Bit class diagram.

stored in the and used in calculating forces.

The class serves as the top level container for s. The abstract

class serves as the base class for many classes which implement specific

behavior for parts of the bit. The specific classes, not presented, simply override default

behavior of the class. Namely, they determine if, when, and how a bit part

cuts rock. However, some other modifications occur as necessary. The class for a cone

adds the rotation of the cone to the bits movement before passing the move instruction

to the teeth on the cone.The four bits used for the examples in this work are shown in Figures 3.2-3.5.

These four bits were selected by an experience petroleum engineer to cover a range

of drilling behaviors. The bits these models were produced from, and the properties of

those bits, are discussed in AppendixA.

is a structure that contains specific information, such as

the update directions, used in the rock surface updating. The update directions and their

function are covered in Section4.1.

The class is abstract. A class derived from a imple-

ments a specific cutting model. It determines the magnitude and direction of forces

applied to the bit from cutting rock. Every bit model discussed in Chapter2contains


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Figure 3.2: Bit A used in force law calibration from TDBD. On the left is the bottom

view on the right is an isometric view.

Figure 3.3: Bit B used in force law calibration from TDBD. On the left is the bottom



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Figure 3.4: Bit C used in force law calibration from TDBD. On the left is the bottom


Figure 3.5: Bit R as modeled in TDBD. On the left is the bottom view on the right is an

isometric view. The cone with the red teeth is cone one, green toothed cone is cone two,

and the blue teeth are on cone three.


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Figure 3.6: Bit coordinate system located at the center of mass of the bit. The blue,

green, and red arrows are the x, y, and z-axis, respectively. Parts of the bit are shown as

partially transparent to permit visualization of the bit coordinate system.

some form of a force model. The force model implemented for purpose of code valida-

tion is that of Hanson and Hansen ([33]) and is discussed in AppendixC.

3.2 Coordinate Systems

To facilitate the rock updating and for dynamics, several coordinate systems are

used. The first coordinate system is for the bit. Cutter and cones also have their own

coordinate system. The coordinate system of the bit and cutters is described below.

However, because the cone coordinate system is closely related to the equations of mo-

tion for roller cone bits it is described in Chapter5.

3.2.1 Bit Coordinate System

The bit coordinate system is a body frame Cartesian system located at the center

of mass of the bit. A partially transparent bit and its coordinate system is shown in Fig-

ure3.6. The bitsZ-axis is parallel to the direction the bit drills. The X-axis is located


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Figure 3.7: Cutter coordinate systems. Blue, green, and red arrows are the x-axis, y-

axis, and z-axis, respectively, of each cutter.

in the plane perpendicular to the Z-axis. Its location in the plane is arbitrarily. However,

for convenience, the bits coordinate system is assumed to align with the global coordi-

nate system at timet = 0. The Y-axis completes the orthonormal triad (the right hand

rule is assumed).

The bit coordinate system is used for internal book keeping in the application

and is the reference configuration used for the dynamics (discussed in Chapter5).

3.2.2 Cutter Coordinate System

To determine the update directions (discussed in Section 4.1.2) for PDC bits

and for internal record keeping a local coordinate system for each cutter is es-

tablished. Figure3.7 shows the cutter coordinate systems for a typical bit. The blue,

green, and red vectors are the X-axis, Y-axis, and Z-axis, respectively, of a Cartesian

coordinate system.

A cutters coordinate system is defined by its geometry. The origin of the system

is centered on the surface of the cutter face. TheX-axis of the coordinate system is


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Figure 3.9: Typical cutter meshes from Bit A (left) and Bit C (right). The mesh of Bit

A is much courser at the chamfer (outer ring in both images) and face edge (inner circle

in both images). Cutters on Bit C are similar to Bit B.

between them.

Bit A was the first mesh created. It was exported from the mesher to an inter-

mediate file which was then processed to produce an input bit geometry file for TDBD.

Because each bit part must be identified for TDBD, the intermediate file included a

designation for each triangle and vertex to indicate which surface it belonged to.

Bit R was the next mesh created. It was created in a similar manner to Bit A, but

a different mesher was used.

Bit B and Bit C were created using a more automated method using a mesher

that was integrated into the CAD software used to generate the three-dimensional solid

models. Different bit parts, such as cutter chamfers and gage pads, were mesh indepen-

dently. They were then written directly to an input file for TDBD by custom software

that interfaced with the mesher and solid modeler.

The differences in the meshes are seen in Figures3.9and3.10. In Figure3.9

the meshes from a cutter face and chamfer are shown for Bit A and Bit B. The cutter

face mesh of Bit A consists of triangles of approximately equal size. Triangles of the

chamfer are meshed so that the edges match those of triangles on the face. This creates

a course mesh at the chamfer and edge of the face. Contrast this with the mesh of Bit

B the mesh of Bit C is similar to that of Bit B which is refined from the center of


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Figure 3.10: Typical teeth meshes from Bit R. The teeth on the left are shown as meshes

on the right. For clarity the mesh of the cone has been removed from the right image.

the face to the chamfer. This provides a fine mesh at the points most likely to contact

rock, the chamfer and edge of the face. Where contact is not likely, the mesh can be

coarsened to save on computational costs. The coarseness of the mesh at the chamfer

and face edge of Bit A creates significant numerical noise in the dynamics output. This

is discussed in Chapter6.


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Chapter 4

Rock Surface Model

The definition of the rock surface and its evolution as the bit drills are two of

the most complex issues in modeling the drilling process. Regardless of how the rock

is discretized, several issues have to be addressed in order to repeat the bottom hole

patterns made by bits as they drill.

An individual bit is a complex shape and there is large variation between bit de-

signs. These shapes undergo large translations and even larger rotations during a typical

simulation. Moreover, dynamic instabilities like whirl create an even greater challenge.

Consequently, the rock surface must accurately represent the elaborate surface patterns

resulting from large movements of complex geometry, while maintaining element shape

resolution and shape quality.

The approach took uses a triangulated surface mesh in three dimensions to rep-

resent the rock. This method allows for shared code between the bit surface and the rock

surface geometry. An additional advantage is that additional memory is not required to

store volume information. However, this method does require some maintenance of the

mesh to ensure required element size and quality is maintained. This is accomplished

through the use of a few standard operations on triangulated meshes with some special-

ization to the problem at hand.

The class diagram for the class is shown in Figure4.1.A

26


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Figure 4.2: Activity diagram of rock updating.

shown in the full class diagram of Figures B.7andB.8in AppendixB. This function

implements everythingwithinthe loop shown in Figure4.2.

This model assumes that rock spalling does not occur. For ductile failure of rock

(drilling under pressure) this method is a good representation of actual failure ([29], [60],

[61]). Implementation of a spalling model for brittle failure, if it is required, should not

pose difficult in the framework of the software.

4.1.1 Finding Bit-Rock Interference

In order to update the rock surface, the interference between the bit and rock

must be located after each bit movement. This is the action labeled

in Figure4.2and shown in Figure4.3.

To start, each bit part defines its motion box. The motion box is the bounding

box of the bit part in its previous location and in its current location. A search is then

performed on the rock surface vertices to locate any that are within the region defined

by the motion box. This search is performed by a specialized library (PiRaT by Petr


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Figure 4.3: Activity diagram for finding the rock vertices within the volume of a bit part.

This is the action in Figure4.2.

Krysl). The results of the search are returned to the bit part.

Note that this process searches for nodesof the rock surface within the volume

swept out by the movement of the bit. That is, the triangles on the rock surface are

allowed to penetrate the surface of the bit. Not requiring triangles of the rock surface

to always lie outside of the bit volume simplifies coding and is less expensive computa-

tionally.

The function

in the full class diagram shown in FiguresB.7and B.8in AppendixB implements this

algorithm.

4.1.2 Update Directions

After a rock surface vertex is identified as interfering with part of the bit, the

location of the node is updated to a position on the surface of the bit. The final position

of the node is a function of both the location of the nodes position in space and the

bit geometry acting on it. The bit model in TDBD is a surface and does not retain

volumetric information. Moreover, the bit is allowed general movements, as opposed


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Figure 4.4: Rock update directions (magenta arrows) for a PDC bit.

to say, always drilling down the translational global z-axis. Hence, determining when a

node is inside the bit and which direction to move to find the bit surface requires careful

consideration.

The solution adopted defines update directions on local regions of the bit. An

update direction is a unit vector, defined for some region of the bit, which determines

the direction rock nodes are moved when that region of the bit contacts rock. Figures 4.4

and4.5show the update directions used for a PDC bit and a roller cone bit, respectively.

Different (but similar) approaches are used to determine the update directions for each

bit type.

4.1.2.1 PDC Bit Update Directions

To find the update direction for a cutter, the cutter, and its coordinate system, are

rotated so the origin of the cutters coordinate system is located on the positive X-axis

portion of theX-Zplane of the bits coordinate system. (See Figure3.6on page21for

the bit coordinate system definition). The negative of the cutters Z-axis is projected


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Figure 4.5: Rock update directions (magenta arrows) for a roller cone bit.

into the bits X-Z plane by

U= projX-Zplane

RZ

(4.1)

where

U = the preliminary update direction (in bits coordinate system);

R = the rotation matrix that rotates the cutters coordinate system

so that the origin is located on the positiveX-axis of theX-Z

plane of the bits coordinate system;

projX-Zplane = vector projection onto the X-Z plane.

The update direction is then determined by

U =U

ifU(1)>0,

[1;0;0] otherwise(4.2)


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Figure 4.6: Update directions for inside and outside cutters. The local z-axis of an inside

cutter point towards the bit gage and outside cutters have a z-axis that points towards

the center of the bit. Left image is courtesy of Hughes Christensen Company.

whereU = is the update direction as defined for a bit part;

U(1) = is the firstX

coordinate of the U vector.

This divides cutters into two categories which are referred to as outside cutters

and inside cutters. Figure4.6 shows how the update directions divide the types. The

names refer to the cutters position on the bit. Because of the shape of the profile, the

cutters that fall into the first case of Equation (4.2) are, typically, located closer to the

outside diameter of the bit. Cutters that are closer to the center of the bit tend to fall into

the latter case of (4.2).

The purpose of arranging the update directions in this manner is to minimize

concentrated distortions of the mesh as the bit drills. Large local distortions cause an

element or elements to change size or shape beyond allowed tolerances within a sig-

nificantly shorter time space than otherwise would be required. This requires surface

maintenance (covered in Section4.2) to occur more frequently, therefore adding cost.

To demonstrate this, consider a two-dimensional case. Take a single vertical

slice of the borehole mesh which intersects the center of the borehole. Three cases of

different update directions are presented in Figures4.7,4.8,and4.9.


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Figure 4.7: Vertex divergence in two-dimensions for the method chosen. On the top is

the original mesh with the update directions that give the bottom mesh.

Figure 4.8: Vertex divergence in two-dimensions for update directions that follow the

bits Z-axis. On the top is the original mesh with the update directions that give the

bottom mesh.


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Figure 4.9: Vertex divergence in two-dimensions for update directions that follow the

bits exposure profile. On the top is the original mesh with the update directions that

give the bottom mesh.

The first case is the method used in TDBD. The original mesh in the

computation model of drilling bits.pdf

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