Bottom with

External Vibrating Force Excitation

by

@ Lei Wa ng

A thesis ubmittcd to the School of Graduate Studies in partial fulfillment of the requirements for the d egree of

St. John 's

Master of Engineering

D e partme nt of Engineering and Applie d Science

Memorial University of Newfoundland October 2013

ewfo undland

AUTHOR'S DECLARATION

I hereby declar e that I am the sole author of this thesis . This is a true copy of the

including any r equired final revisions ,

accepted by my examiners. I understand that my t hesis may be mad e electronically availa ble to t he public.

II

As a Vibration Assisted Rotary Drilling (YARD) tool , the newly developed Down-hole O::;cillating Device (DOD) is de::;igned for the purpo::;e of improving drilling efficiency.

The performance of the DOD was studied theoretically in this t hesis. Computational Fluid Dynamics (CFD) simulation is cond ucted to th oroughly investigate t he axial vibration force generated by the DOD. P arametric analysis is then conducted to study whether differ en t drilling fluids can vary the DOD ou tpu t . Afterwards, d yn amic Bot- tom Hole Assembly (BHA) models are built to evalua te how t his add itional vibration force affect::; drilling. In t his thesis , b oth a single rigid body model and fini te element model of the B HA are est a blished. Some other simulations are conducted with t he finite element model to investigate how variou::; external vibration forces influence drilling enha ncement.

K eywords : Dy namic analysis, CFD simulation, Bit-rock interaction , Finite element model, Vibration Assisted Rotary Drilling

lll

Acknowledgements

The author would like to acknowledge the input and suppor t from many people.

First of all , the author would like to express h er

Tatit ude to the faculty, st aff, and students of t he Depar tment of Engineering a nd Applied Science for making her stu dy at MUN such a great experience .

Then , the a uthor would like to express her thanks to her supervisor , Dr. Ste ve But t , for his great guid ance; and her co-supervisor, Dr. J ames Ya ng, for his constant advice during the last t wo years.

Also, the author would like to thank other members of the Advanced Drilling Group who were really helpful during her research . They a re Faricl Arvani, t he project man- ager; Brock Gillis, the Lead Engineer; and all the Colleagues in Adva nced Drilling Group . · without all the assis tance, she could n ot get over the diffic ult p roblems th a t easily.

Fina lly, the a uthor would like to express her sp ecial than ks to her family for support ing her all the way here.

lV

The drilling inc! us t ry normally uses imp er ial units to repor t the relative para meters while SI uni t system is employed in sorne research. In this thesis, SI uni ts are ma inly used and other units will be conver ted. Table 1 shows the conver ion between CGS and SI unit sy. tcms for some quantitie related in this r easearch.

Table 1: Conversion between CGS and SI units in Mechanics

Qua ntity Symbol CGS uni t Equivalen t in SI uni ts Leng th / Position L /x em 10-

m

l\Ia.'3s m g 10-

Kg

Time t s ls

Force F dyne 10-

N

Pressure p _{BaT 10-}

_Pa

Dy nalllic Viscosity

Po' ise 10-

Pa · s

v

Table of Contents

A UTHOR 'S DECLARATION

Abs tract

A cknowled g m e nts

Notes on Units of Dime n s ions

Ta ble of Conte nts

List of T a bles

List of Fig ures

List of Symbols, Nom encla ture or Abbrevia tions

1 Introduction

1.1 Overview of Rotary Drill Rig System 1.1.1

1.1.2

1.1.3 1.1. 4 1.1.5

Power Generation Syst em Hoistiug Syst em . . . . . . Drilling Fluid Circulating System Rotar y System . . . . . . . . Well Blowout Control Sys tem

Vl

ii

iii

iv

v

Vlll

lX

Xlll

xiv

1

2 5 5

6

6

10

1.2 Vibra tion Assisted Rotary Drilling (YARD ) Introduction 1.3 Thesis Background . . . .

1.4 Obj ective and Significance 1.5 Me thodology . . . . .. .

2 Literature Review

2.1 YARD Tool Working Mechanisms R eview 2.1.1 Hydra ulic Cavitation . . . . . . ..

11 12 12 13

15

15 15 2.1.2 Cha ngeabl e Communicating Are a - Flow restrictor 17 2.1.3 l\!Iecha nical Vibration Mechanism . . . . . . . . . . . 20 2.2 Computational F luid Dynami cs (CFD) Simulation in Drilling Application 22

2.2.1 CFD in Coiled Tubing (CT) . . . 22

2.2.2 CFD of drilling fluid in well bore 2.3 Dynamic Analysis of BRA/ Drill String

2.3 .1 Lumped P ara meter Model 2.3.2 Finite Element Model 2.3.3 Bit-rock Interaction Model .

2.3.3.1 Zener Model . . . 2.3.3.2 Foundation Mod el

2.4 Field Tests and Lab-scale Experiments on YARD Tools

3 Drilling Oscillating Device (DOD) Introduction

4 Computational Fluid Dynamics (CFD) Simulation

4.1 Introduction of the Software . 4.2 Simulation Setup and Results

4.3 Parametric Analysis on DOD's Performance

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23 25 25 26 29 29 32 35

38

40

41

42

45

5 Dynamic Models of BHA and Simulat ions

5.1 Simulation with Single Rigid Body Model 5. 1.1 Model Introduction .. .. . .

5.1.1.1 5 .1.1.2

Bit-rock Interaction

Overall Axial Excitation Force

5 .1. 2 Simulation Results Analysis and Discussion

5.2 Simulation with Finite Element Model 5.2. 1 Model Introduction .

5.2.1. 1 Bit-rock Interact ion 5.2. 1.2

5.2 .1.3 5.2.1.4

Overall Axial Excitations Overall Torsional Excitations Calculation of Stress Distribution

48

49 49 51 53 54 59 59 63 65 65

66 5.2.2 Centr al Differen tial Method for Mu lti-DOF System 67 5.2 .3 Simulation Results Analysis and Discussion 69 5.2.4 Vibration Force Optimization with Finite Element Model . 78

6 Conclusion and Discussion

6. 1 Conclusions . 6.2 Future Work .

Bibliography

Appendix A: 2-Dimensional Drawing of the Valve Assembly

Appendix B: CFD Simulation Results

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84

84 86

88

95

97

1 Conversion between CGS and SI units in Mechanics

2.1 Equivalent P ara meters fo r Analysis of Circular Founda tions on Elastic Halfs pace [37]

4.1 Boundary Conditions in CFD Simulations . . .. . . . 4 .2 Va riation of Drill Mud Properties in CFD Simulations

5. 1 Pa rame ters used in Dy namic Simula tions . . . . . . . 5.2 Additional Para meters Used in Dyna mic Simulations

lX

v

33

42

46

55

70

List of Figures

1.1 Basic Elements of a Drill Rig System [?] 3

1.2 Rotary Drilling Rig System[8] 4

1.3 Hoisting System [ 9] . . . . . . 5

1.4 Drilling Fluid Circulating System [10] 7

1.5 Typical Kelly Dri ve Arrangernen t[ll] 8

1.6 Typical Top Drive Arrangement [ll] 9

1.7 Blowo ut Preventer Stack Cornponents[15] 10

2.1 Structur e of Hydra ulic Pulsed Cavitating J et Generator [17] 16 2.2 Pressure Pulse Modulator [18] . . . . . . . . . . . . . . . . 17 2.3 Two Differ ent Configurations of th e Modulators with their Pulse Shap e[18]

18

2.4 Sch ematic Diagram of a Drill String with a Pulsa ting Device Attached to the Drill Bi t[19] . . . . . . . . . . . . . . . . . . . 19 2. 5 A Vibratory Unit and Associated Drilling System [20] 20 2.6 A Complete Circulation of the Centrifugal Force Direction Gener ated[20]

21

2. 7 Sketch of the Drill Rig System[ 4]

2.8 BHA System for Numerical Studies[29]

2.9 Drill String Mecha nicall\!Iodel[31] . . .

X

25

26

28

2.11 Model ofVibro-Impact Penetration into Visco-Elasto-Plastic Ma terial[34] 31 2.12 General Schemmatic of the Drill-String System[ 39] . 34 2.13 VARD Laboratory Scale Experiment Setup [42] . . . 36 2.14 Voltage Signa ls Collected from LVDT Showing Variable Vibration Am-

plitudes a t a Fixed Frequency[42] . . . . . . . . . . . . . . . . . . . . 36

3.1 Downhole Oscillating Device (DOD) Configuration

4.1 Mesh of DOD Valve Part

4.2 Top Vie w of Mesh of DOD Valve Pa rt 4.3 Speed Pattern of the Oscillating Valve Plate 4.4 Press ure Pulsation from the DOD . . . . . .

4.5 Press ure Pulsation Corresponding to Various Drill mud

5.1 General Schematic of the BHA Dynamic System . . . .

39

43 43 44 45 46

49 5.2 Dis plare!lJP.nt Profile Wh en Drilling with ou t the DOD ou Flat Surface 56 5.3 Weight on Bit Profile When Drilling without t he DOD on F lat Surface 56 5.4 Dis placf'JJJellt Profile When Drilliug without the DOD

Non-flat Surface 57 5.5 Weight on Bit Profil e When Drilling without the D OD on on-flat

Surface . . . . . . . . . . . . . . . . . . . .. . 57 5.6 BHA Displacement Profile When Drilling with the DOD on Flat Sur-

face 58

5. 7 Weight on Bit Profile When Drilling with the DOD on Flat Surface 58

5.~

General Configuration of the Drill Rig System

5.9 Finite Element Model of the Dynamic BHA System 60

5 .10 Three Dimensional Bernoulli Linear Beam Element wit h Two DOF at Each End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xi

5.11 Regula rization Function[39]

5.12 Grid Points in Central Differen ce Method . 5.13 WOB R esult When Drilling without the DOD 5.14 WOB Res ult When Drilling with the DOD ..

5.15 TOB Res ult When Drilling without the DOD 5.16 TOB Res ult When Drilling with the DOD ..

5.17 Axial Stress Result in the Bottom Element of BHA When Drilling 64 68 71 71 72 72

without the DOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.18 Axia l Stress Result in the Bottom Element of BHA When Drilling with

the DOD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.19 Axia l Stress Result on the Top of BHA When Drilling without the DOD 74 5.20 Axial Stress Result on the Top of BHA When Drilling With the DOD 75 5.21 Torsional Stress Result in the Bottom Elernent of BHA When Drilling

without the DOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.22 Torsional Stress Result in the Bottom Element of BHA When Drilling

with the DOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.23 Torsional Stress Result in the Top Element of BHA When Drilling

without the DOD . . . . . 77

5.24 Torsiona l Stress Result in the Top Element of BHA When Drilling with the DOD. . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.25 Comparison of the Axial Stress Distribution a long BHA . 78 5.26 Comparison of the Torsional Stress Distribution along BHA 79 5.2 7 Average WOB/ ROP vs. Mean V a lue of External Vibration Force. 80 5.28 Average WOB/ ROP vs. Amplitude of Externa l Vibration Force . 81 5.29 Average WOB/ ROP vs. Angul ar Freq uen cy of External Vibration

Force in BHA . . . . . . 82

Xll

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List of Symbols, Nomenclature or Abbreviations

rn = BHA mass

g = Gravity acceler ation

NIA , CA , K A = :VIass , Damping , Stiffness ma trices of the BHA element for axial d i- rection

FA ⁼ Axial excitation force

z, [Z ] = Axial displacement (matrix) of the system

Jr, K r, Cr = E quivalent rotar y iner tia, torsional stiffness, a nd torsional damping matrices of this d yna mic system

T = Torsional exc itation torque

¢, [ ¢] = Torsional rotation (matrix) of the syste m

M , C, K = Equivalent mass, d amp ing, stiffness matrix of the system F = Equiva lent excitation for ce rna trix

kp

Equivalent spring constant of drill pi pe k:r

Eqnivalent s pr ing consta nt of rock c,.

Equi valent damping of rock v,. = P oisson 's ratio of rock p,. = Density of rock

XlV

To = Radius of rock

Fvibration

= External vibration force

F,ncan

= Mean value of the external vi bration force

Famp =

Amplit ude of t he externa l vibration force

Wvib

= Angular frequency of the externa l vibra tion force Fh = Hook load

N = Hook load propor tionali ty G

Effective 13HA weigh t

AP = Vibration Force Acting Area

P ooD = Pressure pulsation fr om the DOD

AcToss

= Cross sectional a rea of drill pipes

Dco

= Drill colla r outer diameter

Dei

= Drill collar inner d iamete r

Hie = Drill colla r unit weigh t

E s = Youug's modulus of drill pipes Ec

Young's modulus of drill colla rs LP = Length of dr ill pipes

L

Total Leugth of the BHA

J = Momeu t of iner tia of each BHA element l

Element length in the finite element model G,. = Shear Mod ulus of t he rock

Gc = Shea r Modulus of the BHA s

Surface elevation profile function so = Initia l surface elevation

nb =

Bit factor

XV

a l , a2, a3, a4, a5 = Constants in bit- rock interaction mod el

wbit

= Bit rotating p eed

wb

= BHA Ini tial Rotating Speed Z = Regula ri zation function e = Regularizatiou pa rameter

f.l , A = l\I ass, damp ing proportional Rayl igh damping coeffi ients

Cc

= Critical dampiug Lm = Motor iuductau ce I = Motor arlllat ure current

Rm = Motor resistauce K m = l\Iotor con taut n = gear box ratio

~

= Input voltage to the mo tor T,.t = Torqu from the rotary table

c/Jrt

Rotation of the rotary table

f

⁼ Buoyancy factor

a

Axial stre. · in the BHA element

O'q,z

= Torsional s tress in the BHA I rnent for th e outer surface

f:::.. z = Axial

differ ence between th e two ends of t he DHA element

6.¢ = Torsional rotation difference b etween th e two ends of th e BHA elem ent

f:::..t , h = T ime

in central difference method

f:::..tcri = Critical time step in central differ ence method

Tn

= Smallest period of a multi-degr ee-of-freedom system WOB = Wei ght on bit

ROP = Rate of Penetration TOD = Torqu ou bit

xvi

BHA = Bottom Hole Assembly

CFD = Computational F luid Dynamics PDM = Positive Displacement Motor DOF = Degree-of-freedom

YARD = Vibration Assisted Rotary Drilling DPD = Downhole P ulsating Device

BOP = Blowout preventer

XVll

Chapter 1

Introduction

Nowadays, more and more advanced technologies are applied to p etroleum industry to improve drilling in different asp ects: extending bit life, improving drilling effi- ciency a nd accuracy, and reducing the cost simultaneously. Normally, vibration in a ny drilling application is reckoned as a drawback which will bring negative effects to t he drilling operation and da mage the drill string components [l , 2, 3, 4] . Lots of studi es are focused on eliminating the vibration to avoid its side effects. However , li ttle research have been conducted for the attempts to take adva ntage of those vibra- tions. The technology of utilizing vibration to enha nce drilling is called the Vibration Assisted Rotary Drilling (YARD) technique. Tools designed based on this technology are called YARD tools. The tools are capable of generating axial vibration forces.

Tlwse forces will the11 geutly oscillate the drill strillg or the DHA to fiually improve Rate of Penetration (ROP) of the drilling. As a result, the vibrations will be benign to drilling process a nd assist with penetration.

Basically, the YARD tools work with a rota ry drill rig system. In this cha pter , a bri d. desrripti ou of a rouveutioual rotary dri ll rig systelll is presellted first. The YARD concept is introduced as well. Then , the thesis background, the objective

1

and significance of this research work , along with the methodology employed in the research will be pres nted at the end of this chapter.

1.1 Ove rvie w of Rotary Drill Rig S ystem

Drill rig system in petroleum industry is norrnally the massive structure housing equipment used to drill oil-wells, or natural gas extraction wells. Drilling types vary with different a pplications : for examp le, auger drilling, p er cussion rotary air blast drilling, a ir core drilling, cable tool drilling, reverse circulation drilling, diamond core drilling, hydraulic rotary drilling, and sonic (vibratory) drilling[5] . However , this thesis is focusing on rotary drilling. During rotary drilling, a sharp, rotating drill bit is employed to p enetrate geologic formations. Rotary drilling became popular from the success of the well in Texas in 1901. Since theu , rotary drilling gai ns more at tention[6] .

In a ny process of drilling, the basic procedures are the same. Whet her the well is vertical or direc tional, oil well or na tural gas well, several elementary points are needed for successful a ud economical drilling. They are: a force acting on a drill bit downwardly, rotation of the drill bit, and circulation of the drilling fluid . The clear explana tion is shown in Figure 1.1.

Rotary drill ri g for oil and na tural gas is comprised of two major constituents: man-

power system and hardware system. Manpower r efers to those drilling engineers a nd

the operat ion taff. Basically, the hardware system of a rotary drill rig is made up

of the following six ma in systems: a power genera tion system, a hoisting system, a

drillin g fluid circulatin g system, a rotary system, well blowout control systems, and a

drilling data acquisition and monitoring system. Figure 1.2 schematically shows the

composition of a rota ry drill rig system. Following that the subsystems are introduced.

Ax iul Fon;c

jJ. Force on bit Figure 1.1: Basic Elements of a Drill Rig System[7]

3

Crown

Miud-mi.JCJng st1ack Derrick

F igure 1.2: Rota ry Drillin g Rig Sys tem [ 8]

5

1.1.1 Power G eneration S ystem

The power syst em of a rotary drill rig is designed to supply power for several prime comp onents, especially t he rotary system , h ois ting system, a nd drilling fluid circula- tion system. Furth nnore, a uxiliari es like the blowout preventer, wa t er pumps, rig light ing system, et c. also have to be power ed. The power is gene1ated eit her at the rig site using intern al-combustion die el en gines, or taken as electric power supply from existing power lillcs[7].

1.1.2 Hoisting Syst e m

The hoisting system in rotary drilling is designed to hoist th drill pipes and casing strings duri11g drilling and casing operations[?]. The m aj or components of this s ystem a re the dra w-works, the crown block , the traveling block , the hook, the drilling line, a nd t he ele v ator. Figure 1.3 describes how all th e parts a re assembled a hoisting system.

Crawworks

bloek

Drilling line .,.i~ r.ope}

\

\

Trilv~lling

\ \ I

Figure 1.3: Hoisting System [9]

1.1.3 Drilling Fluid Circulating System

Norma lly, heat and rock cuttings ar e continuously produced in drilling operations.

During any drilling operations, drilling fluid circul ates through the drill bit when it cuts through rock It. is proven th at drilliug fluid is able to lubricate th e bit , remove rock cutting , stabilize the wall around the hole, a nd control the pressure in th e well b ore[lO] . T l w drilliug fluid is u sually a suspeusim1 of clwtuicals aud miuerals such as b entouite clay in water or some times oil. This mixture is blended in the mud-mixing sh ack. Afterwa rds, the mixed fluid is pump ed up to the staudpipe and into th e drill pipe throu gh the kelly by a mud pump. The drilling fluid fin ally reaches the drill bit, cools down the drill Lit awl flushes rock cuttings. After t hat, the circulation flows back to the s urface th rough the well bore annulus. When r eaching the surface a nd then goiug through the shale shaker , the cuttings can b e separ ated from the drilling fluid . T he drilling fluid is r ecirculated after b eing clesilted [lO]. As so, the circulation of t he drilling fluid is rep eated continuously. Figure 1.4 is a picture of the circulating system. The maiu components are marked out in the figure.

1.1.4 Rotary System

T here are two types of rotary systems: the kelly system and the top drive system.

Kelly drive system is the older one of the two rota ry systems. As shown in Figure 1. 5,

the components include swivel and swivel bail , the kelly, kelly bus hing, rotary table,

a nd master bus hiug[ ll]. A kell y drive employs a section of pipe with a polygona l or

spliued outer urface . This part of pipe then passes through the matching polygon al

or spliued kelly bushiug a ud rotary table[l2]. Since they a re ma tched to each other ,

when the rotarv t able on th e rig fl oor rotates, th e kelly can b e rotated with th e drill

pipes connected by kelly bushing. T his rotation leads to t he r otation of t he following

Hose

Kelly Sand pipe BIOWOUI erevenler (BOP) .

Annulus Drill p1pe

Drtl! collars Ortll bit

D1S<:harge line Mud pumps

snale 5haker

IAud-mtl<lng shack

Mud tan~s Mud mjxer

Lined pit

Figure 1.4: Drilling Fluid Circulating Sys tem[lO]

7

drill pipes a nd drill bit.

OniRng hn.e

Master bushc~ng

Figure 1.5: T ypical Kelly Drive Arra ngemen t[ll]

Another rotary system is t he top drive system. F igure 1.6 s hows the top dri ve

syst.em [ll]. In this system , hydr aulic or electric motors susp ended ab ove the drill

pipe ar e employed to ena ble top drives to rotate and pump continuously while drilling

or during the removal of drill pipe fr om the hole[ll]. Compared to the kelly system,

the mech anical equipment of a top drive is located at the swivel and allows a vertical

movement up and down a long the derrick [13]. With the usage of motors, improve-

ments of the efficiency are observed. Ther efore, most offshore units and an increasing

number of la ud rigs use top drives t hese days.

I \

' I

Hydraulic motor

F igure 1.6: T ypical Top Drive Arra ngement [ll]

9

1.1.5 Well Blowout Control System

A blowout in an oil-well is the uncontrolled release of formation fluids after pressure cont rol systems have failed [14]. Blowou ts can induce severe consequences. Ther efore, the well control system becomes essent ial in well drilling a pplications. The basic components of a blowout prevent er (BOP) a re illustra t ed in F igure 1.7. As shown iu t lw fi gurP-, t hP- blowout preveuter stack comprises of an annular pr eveuter, ra m prevent ers , spools, internal prevent er s, a casing head , fl ow and choke lines a nd fittings , kill lines a nd connections, mud-a nd gas-h andling facilit ies, a nd accumulators [7].

Injector head

_ _ _ _ Stripper

D rill floor

Mud returns ~

... Annular

preventer

B lind

Shear rams

Ki.ll _ _ _ _..

IIIIIIII Ground

Figure 1. 7: Blowou t Preventer Stack Components[15]

11

1.1.6 Drilling Data Acquisition and Monitoring System

Duriug t he drilling application , various parameters should be

asured or monitored to a sure it · nonual operation. All these pa rameters are valuable information for drilling. The critical varia bles includes drilling rate, hook load, hole depth, pump pressure, flow rate. torque, rotary sp eed. mud d ensity. temperature, salinity, fl ow properties, mud tank level, pump tt·okes, w eight on bit, hoisting sp eed, etc. [7] These variables can be monitored, analyzed, displayed , and recorded by certain devices, the so-called data acquisition and monitoring equipment. This system can assist in d etecting lots of drilling problems; lo s of circ ulation , w ell kicks, pipe ticking, drilling breaks, bit b a ring failure, high concentration of drilling cuttiugs, a ud even blowout.

1.2 Vibration Assisted Rotary Drilling (VARD) In- troduction

As mentioned before, YARD t echnique is the technology of u tilizing vibration to

enha nce drilling . It is important is to create b enign axial vibration forces to gently

oscillate the drill string or t he BHA. The objectives of the improvement in drilling a re

iu ma ny aspect·, such as ROP, buckling, sliding, erratic reactive torque, poor tool face

cont rol, oriented drilling with steerable motors, high tortuosity a nd extended reach

drilling applicatious[l6]. YARD technique is now considered to be a quite promising

technology. A. · a result , the designs of YARD a r e rapidly developed. Later in th is

thesis , the design of YARD tools and their field/ lab scale tests and imulations will

be introduced.

1 .3 Thesis B a ckground

This research work concentrates on the computer simulations of t he BHA to predict th e additional YARD tool's infl uence on drilling. As ment ioned before, the YARD tool is d esigned to enhance drilling. A few field / laboratory scale tests have been implemented. However , no deta iled theoretical investigation has b een conducted on the tool's performance. As so, both single rigid b ody model and fi nite elemen t model of the whole BHA system are esta blished. Ther efore, this research work will cover the insufficient study on the YARD tool evaluat ion . Besides, a dditional simulat ions are conducted to investi gate how various external vibrat ion forces affect dr illing.

1. 4 Obj e ctive and Significa n ce

The main objective of t his thesis is to develop appropriat e dynamic models of the

BHA to evaluate a YARD tool's influence on drilling. The results obtained from the

simulations will have to be clearly analyzed and discussed to achieve t he evaluation

on t he tool's p erformance. The p erformance should include its influence on rate of

penetration , the weight on bit, and even the axial a nd torsiona l stress inside the

whole BHA. Therefor e, simply with a PC, fast predictions and evaluations of the tool

performance are availa Lle. Compared to Iield tests , ti me and cost will Le dramatically

reduced . Also , from the simulation results , the critical parameters to optimize the

tool's design will be known. T his is extremely good for the tool's future design .

Besides, the successful eva luation of a ty pical tool sets a n example for oth er similar

work. As a result, the evaluation process can b e applied to other addition al drilling

tools.

13

1. 5 M e thodology

In this thesis, the typical YARD tool employed in t he research is the Down-hole Oscil- lating Device (DOD) . To investigate how it assi sts in drill ing, a series of simulations are coud ucted in the resear ch.

Based ou the working mechani sm of the DOD, a 3-dimensional model of its core component, the va lve assembly, is built in SolidWorks 2010 software . This model is saved as STL files and then imported into FLOW 3D 20 10 softwar e to simula te t he assembly's operation. Thi s is Compu tational F luid Dynamics (CFD ) simulat ion. The simulatiou res ults will predict the pressure p ulsations created by the DOD. The data i s stored for later simulations with t he dynamic BHA system.

Th e next step is to investigate how the tool affects drilling p erformance. For this purpose both single rigi d body model and finite element mod el for the BHA are built. The former model only considers the axial t ranslation motion. Therefore, the simulat ion results can only tell the tool's influ ence on the rate of pen etra tion and w eight on bit. On the oth er hand , the axia l translation and torsiona l rotation of the BHA are coupled togeth er in the fin ite element model. As so, the t orsional influence can also h e known from the simulations with this model. Besides , the stress distributions in both axia l a nd torsional directions a re available too . All these simulatious a re conducted wi th MATLAB.

To inves tigate how the character istics of the external vibrat ion for ces affect drilling, sum P otlw r siumlat ious with the

eleHJPJ J t lllodel ar e coudu cted . The vibration force in this case is assumed as a simple ha rmonic force a nd the parameters of this vibration force ar e varied iu the simulations. T hose parameters include th e force mean value, amplitude, a ugular frequency, and even the location where the force is applied.

T his chapter briefl y introduces the comp osition of a rotary drill rig system as well

as th e develo pment a ud application of YARD tools . After wards , the resear ch work

backgr ound , its resear ch object and significan ce, and also t he methodology for t he

investigation are all provided . In t he next ch apter , thP li ter ature review on different

YARD tool mechanisms , Computational Fluid Dy na mics a pplication in drilling, a nd

the dynamic a na lysis of the BHA is described in detail.

Chapter 2

Literature Review

2 .1 VARD Tool Working Mechanisms Review

T here are sever al mecha nisms which can be applied to YARD tools . These mecha- nisms include hy draulic cavitation , changeable communication area in flow line, and even mechanical vibration mechanisms like percussive hammer. Tools may b e de- signed on those differen t mechanisms. However , the purpose is quite similar : gen- era tion of a ppropria te oscillations along the drill st ring. T he oscillations are used to gently vibrate the drill string to help improve drilling efficiency by improving ROP. In the following ::;ection, di ffer en t working mech ani::;m::; wh ich can be employed in VARD tools are discussed .

2 .1.1 H y draulic C avit ation

Cavitation is the formation and then immediate implosion of cavities in a liquid . Also, cavitation is the consequence of forces acting upon the liquid. T his kind of phenomenon usually occurs when a liquid is subjected to ra pid ch anges of pressure which may cause the formation of cavities where the pressure is relatively low. Iner tial

15

cavitation is the process where a void or bubble in a liquid rapidly collapses, producing a shock wave. It can occur in control valves, pumps, propellers and impellers.

In view of these benefits, hydro dynamic cavitation tools ar e proposed to be a kind of vibration-r otary drilling method. These tools can cr eate longitude vibration ac- celerations. Actually, the tools are p owered by pressurized well bore fluids which are supplied through the drill string in combination with a generic pumping system.

The strong pulses which are generated hydro-dynamically are by means of resonance waves. Therefore, with a ppropriate adjustment, they can be tuned for optimum p er- formance in varying drilling conditions . An example of a down-hole, liquid driven cavitation tool[17] is illustrated in Figure 2.1. T he drilling tool is designed to cou ple a dvi:111te1ges of both pulsed jet and cavitatiu g jet. When drilling fl uid flows through, fluid is modulated to pulse and cavitate. Because of jet pulsation, cavitating erosion a ud loce1l

pres:mre dfect, bottom cuttings

efficiency is euh anced aud ROP is improved .

2

5 6 7

1- Uody; 2 -Eiastic collar: 3- Divcrting dc1·ice: -!- Impeller;

5 -lmpcllcr shaft: 6- lmpcllcr and shaft sleeve; 7 -Cavity resonator

Figure 2.1: Structure of Hydra ulic Pulsed Cavitating Jet Genera tor[17]

As shown iu Figure 2. 1, this hydraulic pulsed cavitating jet generator is made up of

body, elast ic collar , diverting device, impeller bed , impeller shaft, impeller and shaft

w earing sleeve, and cavity resonator which is a lso the resonating chamber.

17

2.1.2 Changeable Communicating Area - - Flow restrictor

The a pparatus based on this rnechauisrn usually includes a housing providing a passage for a flow of drilling fluid toward the bit, a stator and a rotor. The rotor oscillates in the axial direction relatively to a stator. Therefore the oscillation periodically restricts the fl ow through the passage and creates pulsations in the flow and finally a cyclica l fluid ha111l11er effect occun; to vibrate the housiug aml the drill bit during usc.

Figure 2.2 is a perspec tive view of the pressure pulse modulator [l8]. I n use, drilling fluid flow:-: into th· top of the housing through tlw annular

lwtw<-'Pll thP extenml wall of the, upporting structure and the inner walls of the hou ing and flows through ports of the stator awl th e rotor. The fiuid fiows continue p ast the r ear standoff and on to the drill bit. The shaft drives the rotor to interr upt the fiuid jets p assing through the ports of the tator to generate a coded acoustic signal that travels upstream .

ACOUSTIC Sl<li'<AL

Figure 2.2: Pressure Pulse Modulator [l 8]

\.Vith different configurations of the modulator, different shapes of waves a re gener ated

by the appar at u . Figure 2.3 just shows two different configurations of the modulators

with the pul eel sha pes gener a ted.

-41

Figure 2.3: Two Differ en t Configura tions of the Modulator::> with their Pulse Slmpe[18]

Although severa l embodiments of this kind of modulator have been sh own and de- scribed aboY e, it is understa ndable that various cha nges and modifications can be mad e. The rotor an d/ or the stator could have several orifices, the stator and rotor orifices could be differ ent , and of cour e the shap es shown in the previous fi gures are not the only possible sha pes .

Another example is a Down-hole Pulsating Device (DPD) which was investigated by Nagib et al.[19]. When optimizing a drilling system , 4 different p arameters would be considered : bit selection, rotary spe d , WOB , and hydra ulic system. In this case, the a u thor r latecl one of those para meters to two variations in the operation of the DPD: force generated and transferred to the bit, a nd the opening ratio of the DPD.

The working mechanism here was to partially shut off fl ow to the tool-face to r aise the pressure

the drill string, as a result, exerting more WOB on the bit momentarily.

The composition of the DPD is hown in Figure 2.4. The stationary valve assembly

19

i s fixp,d to thP. b ottom sub. whilP- thP. oscill ating va]vp, assP-mbly was fixed to thP- rotor of the Positive Displacement Motor (PDlVI). The connection, zigzag rod, could move the oscillating valve into horizontal displacement with the frequency determined by the rotating speed of RPM. Therefore, pulsating loads were applied on the rotary bit.

Nagib et a.l. [19] found a two-parameter, second degree polynomial to b e considered as the model of the additional jarring for ce exerted on the bit which was generated by the hydraulic pulsation action. As a res ult, it w as proven that drilling with a DPD is technically feasible and efficient . It resulted in a higher ROP (72% incr ease) th an conventional drilling technique[ 19].

Drill String

Drilled hole Connecting rod

O:;cii.Jating vah~ a~~crnbly

l:lorwm Sllh

Drill hit a,sembly

Figure 2.4: Schematic Diagram of a Drill String with a Pulsating Device Attached to

the Drill Bit[19]

2.1.3 Mechanica l Vibration M echanism

The down-the-hole vibratory unit [20] for a drilling system includes a casing a nd a plurality of eccentrically weighted rotor assemblies positioned at least partially with the casi 11g and iJJ Auid cmumm1icatiou with the iulet. The

rot or assemblies can be unbalanced relative to a central axis. Additionally, the assemblies can lw configur<:'d to

iu respm1se to a Auid flow directed then')to to apply centrifugal forces to the casing. F igure 2.5 schema tically shows h ow the vibratory uni t looks .

Figure 2.5: A Vibratory Unit and Associated Drilling System[20]

When in the first and t hird position (from left to righ t) as shown in Figure 2.6 , the

21

axial components will decrease and the r adial components will increase a nd cancel each other out. When in the second position, there will be the maximum impact for ce on the bit end while in the last position; there should b e the minimum impact force on the bit end. Therefor e, the rotation of the rotor assembJies results in cyclical axial forces . Those for ces can also be describ ed as vibratory for ces. T h erefore, cyclical impact force on the drill bit is generated .

.

··

.'·'r

·'

'

_.;er

- _{,. . ...}

(;:,p ,

,. ..

Figure 2.6: A Comple te Circula tion of the Centrifugal Force Direction G ener ated [20]

In this research work, the changeable communication a rea mechanism is employed

in the tool of iuterest. Based on this mecha nism, this tool is more simply designed

thau the tools with other mechanisms. Also, t he There a re already some available

commertial tools based on this specific mechani sm. Also, some simple research has

been done on this kind of tool. However , there is not enough theoretical investigation

iuto this kiud of tool before . Also, having enough knowledge about this tool can help

better develop it in the future.

2.2 Computational Fluid Dynamics ( CFD) Simu- lation in Drilling Application

Nowadays, CFD technology has been applied to vari ous situations . Lots of investiga- tions in drilling opera tions are conducted by CFD. Some of them ar e introduced in this section.

2.2.1 CFD in Coiled Tubing (CT)

Rosin e et al.[21 ] investigated Coiled Tubing with CFD . CT is a practical a nd cost- P-ffed. ive mean s wlwn bei ng operated in well intervP-nt ion in servicing wells. Based on the fact that the act ual fl ow through CT has still not been revealed clearly, Compu- ta tional Fluid Dynamics is becoming widely used in this investigation [ 21] . The CFD softwar e employed was FLUENT. Also, SolidW orks was used to build the 3D solid model, and then export the model into GAMBIT softwar e in ACIS for mat to p re- process. In GAJ VIBIT, geometry is created , meshing is performed, and the boundary conditions a re established. As a r esult , execu ting a full-scale CFD model can cost significantly less than full-scale testing. Also, the research investigated the fl uid flow in both stra ight a nd cur ve sections and the detailed res ults were t hen used to evaluate the ftow-velocitv profile and the flow patterns [21].

Zhou et al. [22] also conducted some investigations into CT with CFD software. In their research , ther e were three d ifferent categories of studies : numerical method ,

fully-rlc~vP.loped

coiled pi pe fl ow, a11d bouudary

approxirnaticm nwth orlt>. In th e

las t, they concluded t hat CFD could be used in b oth Newtonian and non-New tonian

fl ow iu tlw cur vP.O sections. Tn add ition , cmuparison between th eoretical calculat ion

a nd numerical simulation are ma de. T he results showed that they agree very well for

differcut flow pattems[22].

23

The previous successful research on the CT by t he C FD softwar e shows the gr eat potent ial of this research method. And this also set a gr eat emample for par t of this resear ch.

2 .2 .2 CFD of d rilling fluid in we ll bore

F luid flow iu <:mnular spaces has received a lot of atte1 1tiou frmu oi l industries, bot h in drilling operations and in petr oleum artificial lift [23]. The drilling fluid perfor- m ance in t he well bore could be intricate when taking t he rotating or non-rota ting motion into accoun t. Basically, the inner cylinder rotation would result in two kinds of aunuli :

aud eccentr ic

about fluid cau be of great complication due to its various properties like density, viscosi ty, and the state of m oticm. Detter um lerstaudiug; how the drilliug; fluid p erforms in t he am mlus co uld help know the well condition . Also, using CFD technique, i t is possible to make predictions of t he pressnre pr ofiles and veloci ties of t he drilling fluid . Therefore, opti- m izat ion of drilling could be conducted m or e economically, and efficiently compared with experiments .

Neto et al.[24] conducted r esearch on the turbulent flows in concen t ric and eccentric annuli with a nd without rotating im1er cylinder. This r esearch was conducted beca use of t he simi la rity of th e flows occur in drilling op erations of oil wells. Five different turlmlence modeb wi th

Average

approach were em ployed in the simulations. Experimen tal data from t he li terature was compared with the predictions[24] . It was found t hat t he simulat ion resul ts agree well with available experiment data[25, 26].

Pereira et al. [23] inve::;tigated t he fl ow of non-Newtonian flui d::; through t he annulus. In

their resear ch , differen t aspects for example, viscositv of the drilling fluid , eccentrici ty

of the well bore, flow statu~ , and also the shaft rota tion were taken into consideration.

The results obtained in the research were also compar ed to those in ot her reported works[23] . At the end of t he study, it was proven t hat the usage of CFD provided satisfactory informa tion or pl ot s a bout the cont ours a nd velocity profiles . Also, the agreem ent of the numerical metho d with exp erimental da ta from other resear ch w ork on dimensionless velocity profiles t urned out quite good[23].

There w ere also other comments about t he CFD for t he drill string section. As ment ioned by Sa leh et al. [27], bot h mechanical and hydra ulic forces ac ting on the wellbor e will influence t he drilling operation , for ex am ple, drilling efficien cy. It was fouml t hat the j <:-'t

and iu<:-'rti al force

su ch siguificaut effect th at m ade t he hydraulic force b ecame unimportant.

Cui et a/.[28] developed tJw govemiug equatious of the helical flow of 11011- Newtoui au fluid in eccent ric annuli. Those equations could be solved by using the finite differen ce Hl et.hod. A S <:-'Co11d ary fl ow was formed

t he outer cylinder in t he largest clearau ce area. And t he region would get larger along with the incr ease of the eccentricity of the ecce11t ric annulus. The a uthors a lso validated t he governing equations t hrough the compari son between the computa tional flow rate of the helica l flow and the flow ra te measured from an experiment.

All t he above resear ch showes tha t CFD soft ware is a p ower ful tool to invest igat e t he

perform ance of drilling to ols wh en drilling fluid is involved in. Wit h t he software , t he

fluid performa nce can be observed clearly. The resul ts can even sh ow t he movem ent

of fluid when the resea.rch object is in operation. This is t he reason why CFD software

is employed in this research.

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2.3 Dynamic Analysis of BRA/Drill String

2.3.1 Lumped Parameter Model

Yigit and Christoforou [4] investigated the stick-slip and bit-bounce interaction in a n oil- well drill string. It is well known that stick-slip a nd bit-bounce have negative

~ignificant

efi'ect on drill

and equipment. To

the

on

a nd bit-bounce interactions under vario us operating conditions, a lumped parameter model with coupled axial and tors ional mo tions was b uilt. Figure 2. 7 is a schematic view of the drill rig system studied in this research . It is assumed that the rotary table of the system was driven by an armature controlled DC motor through a gearbox.

~

'rioisting Sysi.em

Rotary table Gearbox. DC Motor

I,

-L---1--" drillbit

Figure 2.7: Sketch of the Drill Rig System [ 4]

As mentioed before, this dynamic model contained the mutua l dependence of axial

a nd torsional v ibrations, as well as the bit-rock interaction. It was proven that bit

spPecl allrl fonu at.io11 stiffuPss wPre t he ma jor factors affPctiug clyu ami c resp ouse. Also,

the a ut hors discovered that rotational control alone might not b e sufficient to assure

smooth drilling under various oper ating conditions . An additiona l active controller

for the axial motion is needed for effectively suppressing stick-slip vibrations and

bit- bounce once they a re initiated.

2.3.2 Finite Ele m ent Model

Spa nos and P ayne[ 29] studied the dynamic response of a BRA model. In this research ,

to d evelop the dynamic model, finit e- element technique was employed. The mass 1nat.rix was set. as frequency-d epell(lellt to accouut for the ftuid added-m ass effect.

F igure 2.8 shows the BRA sys tem a nd on which the dy na mic mod el was built.

Tl'llaiiHIA l..tngtb • 168'

10' S'

lfiO'

s·

5' 3(1'

5'

3~

===

Figure 2.8: BRA System for Numerical Studies[29]

The dynamic BRA model was assumed under monochromatic ha rmonic excitation.

When building the mass matrix of the dynamic BRA syst em, the added mass effect

was considered . Euler-B ernoull i heam theory was used in deriving th e stiffness of a

27

BHA section. The damping was ass umed to include Rayleigh damping, structura l damping, and viscous damping. The excitation in drilling may come from different sources; for example, there might be bit forces, drill collar mass imbalan ce, sta bilizer loads, and drill pipe kinema tics[30]. In this r esearch , on ly a lateral force applied at the bit with an ap propriate magnitude was consider ed. The lateral stiffness of the drill pipe a t the top was assumed to be zero. And the stabilizers were regarded as pinned bounda ry couditions which would rest rict lateral displacement but not rotation. T he bit was treated as th excita tion node and it is where the lateral force is applied . The authors[29] pointed out that in order to model BHA system mo re precisely, stiffness of special components a nd bit excitation spectr a should b e addressed through laboratory-based testings. And the best resul t would be achieved from the collection , an alysis. and modeling of full-scale field d at a .

Clayer and V a udiver [31] conducted au investigation ou th( - ! efl:"ed of surface and clown- hole bounda ry conditions on the vibration of drill strings. The vibrations were speci- fi.ed as torsional and axial ones . A case study Lased on a well in south ern France was preseuted [ 31] . Also , the direct impedance measurements of the drill rig in torsiona l and axia l motion were presented as w ell. They estimated the boundary condition at t he bi t in torsional a nd axial motions. This r esearch work also made comparison betweeu the simulated a nd measured impedance functions.

The mechani cal model of Clayer and V a ndiver [31] study is shown in Figure 2.9. The excitation at the drill bit is assumed to be either an axial force or axia l displacement applied ou the bit.

In tl1is model, th e drill string is firstly di vided into multiple sections with the same

proper ties. Having the r esponse of each simple section, it is not difficult to get the

response of the whole drill string model. The power swivel or top drive was simply

modelled with a mass-spriug-clamper system. The torsional bottom boundary condi-

K e

Excitation

Swivel oquivalenl cylinder

Saver sub Measuremenl point

5" Drill pipe

Heavy wall drill pipe

8" Ori!l collars

9.5" Drill collars

1 1" Drill collars

811 ---·-~ Relative Displacement

Excitation

Rock

Rr

Figure 2.9 : Drill String 1 \! Iecha nical Model [31]

29

tion of this model was a free stiffness end with only d amping. As for the axial bottom boundary condition , it is presented by a simple equivalent linear spring and damper model. This model su ccessfully simulated the stiffness and the d amping properties of the rock. It was stated that stable BHA resonan ces might only occur under certain conc litions[31].

2.3.3 Bit-rock Inte raction Model

Bit-rock inter action is a n importa nt topic in the r esearches abou t d rill ing. It is an important com position of a dyna mic BHA mod el. Lots of models have been proposed and a few of them have been applied in dynamic research. In this section, two typical bit-rock interac tion models are introduced in detail.

2.3.3. 1 Zen er Model

Luiz and Franca[ 32] introduced a bit-rock interaction model for rotary-percussive drilling as well as for conventional rotary drilling. Since the roller-cone bits are char- acterized by moving roller cones supported by bearings, a relat ively high axial com- pliance is generated. To accoun t for this compliance, the roll er-cone bit was modeled as a mass- s pring- dashpot system and the rock beneath th e bit was modeled as a si mple spring. However, this model was not easy to deal with , it is transformed to a more common model called Zener Model or Standa rd Linear Solid Model. Figure 2.10 shows the origina l model (left) and the Zener model (right) .

Brenna n and Carrella[33] presented a consisten t a nd concise analysis of the free a nd

forced vibration of the Stand ard Linear Solid JVIodel. The a uthors studied the opti-

mum damping values for the syst em which was subjected to different types of exci-

tation and also made sever al comparisons. In the case of free vibration , there were

three roots to its characteristic equation. They were two complex conjugates and

Figure 2.10: Bit-Rock Indentation During Loading (Zener Model)[32]

one purely rea l root. From their study, achieving critical damping of the complex rout::; wa::; quite impu:::;::;ible except th e case when th e additional stiffnes:::; wa:::; at lea:::;t eight times th at of the main spring. For the forced vibration response, two different kinds of excitation were considered, harmonic excitation and white noise excitation.

The studies show that the additional spring offers no advantages when the system is excited by white noise.

Batako and Babitsky[34] illustrated a differ ent bit-rock interaction for impact pen- etration drilling. Figure 2.11 schema tically describes this vibro-impact p enetration model into a visco-elasto-plastic medium. This model is comprised of a spring, a dry friction element (plug), and a viscous element. In this system, the spring was mounted in series with the plug, and the viscous element was set parallel to the elasto-plastic element. When t he overall forces sustained by the system exceeded the threshold of the force, the bit would have a stepwise downward displacement .

The working mecha nism of this model is expla ined as follows . The striker 4 would

fir stly impart a blow onto the drill bit 2. Afterwards, the spring 9 and th e dashpot 11

would deform gr adually due to the visco-elastic properties of the medium. Therefore,

31

Figure 2. 11: l\! Iodel of Vibro-Impact Penetration into Visco- Elasto-P lastic Material[34]

the bit would oscillate around its initial position of equilibrium defined by the position

of t he dry friction element 10. Howe ver, once over all force became equa l or gr eater

t han the threshold for ce, the footing resistance of the medium will instan tly change

its na ture to be plastic without changing t he instant position and velocity of the bit .

Next, there would be a full compression stage during which the dry friction element

slips only in the positive direction of x4 in Figure 2.11. RO P was defin ed as the drill

bit downward displace ment in this stage. Also , t his full compression s tage is ended

with a plastic slip wh n the bit executed a backward restitution stage because of the

accumulated elastic energy in the medium [34] . To valida te t his model, preliminary

test drilling w as conducted . T he results showed that the a pplication of vibration to

the drilling process brought a n obvious incr ement in the ROP. T herefor e, t his result

tota lly supported the concep t of percussive-rotary drilling. Furthermore, the test

result showed tha t even a t 10- 15 Hz the superimp osed vibrat ion was able to r esult in

a significant in cr ease in the ROP [34].

2 .3.3.2 Foundation Model

Hsieh and Lys mer [35] discovered th at when ther e was external ver tical force applied on a massive found ation , its reaction can be r epresented by a single-degree-of-freedom llJ ass-spriu g-dashpot. oscill u.t.or. Later, Lvs nJer[ 3G] suggested th at th e

and damping coeffici ents were both frequency-independent . T heir use was acceptable in the low a nd medium frequency ra nge. This is called 'Lysmer's Analog'. Table 2.1 shows the calibration equations for analysis of circular founda tion on elastic half-

spac<~[3 7] .This table shows th~ sriffuess and dmupi1 1 g cakul atioJJ method for auy kiud of foundation.

Richart and Whitma n[38] extended Lysmer 's Analog and a lso described tha t a ll mod es

be studied by employing mass-spring- dashpot systems with frequ ency- independent para meters which were properly selected. Ver tical and torsional oscil- lations are both axisymme tric. For either of those two oscillations, the cylindrica l foundation is investigated with a single degree-of-freedom (SDOF) system. When studying t he two anti-symmetric mod es of oscillation which are h orizontal translation and rocking oscillations, they are coupled wi th each other. Therefore, the system is ch aracterized as the 2-DOF system. The stiffness and d amping values ar e determined by Table 2.1. A::; shown in the table, the stiffness ancl clamping ratio ar e related to the fo undation proper ties. R is the radius of the circular rigid loading area, G, v, a nd p are a ll foundation proper ties a nd they are shear modulus, Poisson 's ratio a nd density resp ectively. Besides, Ia_., Iz are mass moments of inertia a round a horizontal , vertical ax1 s.

T he math emat i cal model Ritto[39] used for the bit-rock interaction w as developed by

Tucker and Wang [40]. In this model, the axial and torsional motions a re coupled

together . Also, the calcula tion equations for ROP and Torque on Bit (TOB) are

prese nted. It is clarifi ed in the model that t he ROP clep en cls linearly on WOB ancl

33

Table 2.1: Equivalent Parameters for Analysis of Circular Foundations on Elastic Halfspace [ 37]

Mode Ver tical Horizontal Rocking Torsion al

Stiffness

Mass ratio

Da mping ratio

(l+m)m¹1² 1+2m

Fictit ious add ed mass

m

m

m .

the bit ro tation speed. However, Ri tto[39] also put forwar d the regularization fu nc- tion which is r elated to the bit rotation speed and a r egularization para meter. This regularization is created to avoid the situation when bit speed reaches zero. With this, the TOB would be continuous even when bit speed vanishes[39]. Based on this regularization function, the calculations of TOB and WOB are modified .

However, the bit- rock interaction in r eal drilling application is very complex. It con- ta ins lots of uncert ainties. To take those uncerta int i es into consideration, Rit to[39]

developed a stochas tic computational model. T he general sch eme of the drill string

system used in t his study is shown in Figure 2.12. In this model, a non-para metric

proba bilistic approach was employed to model the uncertainties in the bit- rock inter-

action, which is represen ted by a non-linear operator. The mean model a lso took the

fiuid-structure interaction and the impact forces into acco unt. When establishing the

mod el, the a uthors employed the non-linear Timoshenko beam theory. T he non-linear

dyna mical equations were discretized by means of the finite element method. The

probabilistic model presented can successfully work when conducting the simulations

in which a bit loses contact with the bottom-hole r ock or t he drill string impact on the w ell-bore. The pa rametric numerical ana lysis showed tha t th e non-linear dynam- ical responses were qui te sensitive to uncertainties in the bit-rock interaction model.

Further more, it w as proven that the uncertainties played a n important role coupling the axial, tors ional, a ud later al responses[39].

Figure 2.12: General Schemmatic of the Drill-String System [ 39]