A SURVEY ON THE METHODS TO PREDICT RATE OF

EASTERN MACEDONIA AND THRACE INSTITUTE OF TECHNOLOGY
FACULTY OF ENGINEERING
DEPARTMENT OF PETROLEUM AND NATURAL GAS TECHNOLOGY
MSc in OIL AND GAS TECHNOLOGY
MASTER THESIS
A SURVEY ON THE METHODS TO PREDICT RATE OF
PENETRATION IN DRILLING PROJECT
PANAGIOTIS ILIOPOULOS
B.Sc. Mechanical Engineer
SUPERVISOR: VASILEIOS GAGANIS
KAVALA 2015

A survey on the methods to predict rate of penetration in drilling project
By
Iliopoulos Panagiotis
Submitted to the Department of Petroleum and Natural Gas Technology,
Faculty of Engineering
in Partial Fulfillment of the Requirements for the Degree of
Masters of Sciences in the Oil and Gas Technology
at the
Eastern Macedonia and Thrace Institute of Technology
APPROVED BY:
Thesis Supervisor: Vasileios Gaganis
Committee member:
Committee member:
Date defended:

EASTERN MACEDONIA AND THRACE INSTITUTE OF TECHNOLOGY
FACULTY OF ENGINEERING
DEPARTMENT OF PETROLEUM AND NATURAL GAS TECHNOLOGY © 2013
This Master Thesis and its conclusions in whatsoever form are the property of the author and of the
Department of Petroleum and Natural Gas Technology. The aforementioned reserve the right to
independently use and reproduce (partial or total) of the substantial content of this thesis for teaching
and research purposes. In each case, the title of the thesis, the author, the supervisor and the
department must be cited.
The approval of this Master Thesis by the Department of Petroleum and Natural Gas Technology does
not necessarily imply the acceptance of the author’s views on behalf of the department.
--------------------------------------------------------------
The undersigned hereby declares that this thesis is entirely my own work and it has been submitted to
the Department of Petroleum and Natural Gas Technology in partial fulfillment of the requirements for
the degree of Masters of Sciences in the Oil and Gas Technology. I declare that I respected the
Academic Integrity and Research Ethics and I avoided any action that constitutes plagiarism. I know
that plagiarism can be punished with revocation of my master degree.
Signature
Panagiotis Iliopoulos

ABSTRACT
For the purpose of this thesis we will first refer to the efforts of oil and gas industries
for development. We start from the beginning of the 20th century until today with a
historical retrospection in order to classify the operating period in accordance with
the rate of penetration (ROP) prediction methods. An extensive literature survey on
drilling optimization was conducted for this research study that presents all the
drilling optimization models and the main factors that participate in the drilling
activity influenced the penetration rate. Entire models which predict the rate of
drilling penetration as a function of available parameters and which are used from
the industry such as perfect cleaning theory, cutting removal model, best constant
weight, rotary speed and multiple regression process will be analyzed extensively in
order to make clear the relationship between drilling parameters. Also the terms cost
and time, which are directly connected with the models performance, are evaluated,
offering to the reader a full understanding of the optimization process. In addition,
we will analyze a real time optimization process as this technique is going to be
widely used in future drilling activities reducing drilling cost. Finally new advanced
simulation methods used from the industry, based on real time data, are evaluated
and we should pay attention to their beneficial and economic point of view.
SUBJECT AREA: Rate of penetration prediction in drilling project
KEYWORDS: Mathematical studies, Multiple regression, Real time methods,
Simulation models

ΠΕΡΙΛΗΨΗ
Για τους σκοπούς της παρούσας διατριβής πρώτα θα αναφερθούμε στην ανάγκη για
την ανάπτυξη των βιομηχανιών πετρελαίου και φυσικού αερίου. Ξεκινάμε από τις
αρχές του 20ου αιώνα μέχρι σήμερα, κάνοντας μια ιστορική αναδρομή, θέλοντας να
χαρακτηρίστει η περίοδος λειτουργείας σύμφωνα με το ποσοστό διείσδυσης (ROP)
των μεθόδων πρόβλεψης. Μια εκτεταμένη βιβλιογραφική έρευνα για τη
βελτιστοποίηση γεωτρήσεων πραγματοποιήθηκε για αυτήν την ερευνητική μελέτη
παρουσιάζονται όλα τα μοντέλα βελτιστοποίησης γεωτρήσεων και τους κύριους
παράγοντες που συμμετέχουν σε γεώτρησης δραστηριοτητες, επηρεάζοντας το
ρυθμό διείσδυσης. Ολόκληρα τα μοντέλα που προβλέπουν το ρυθμό διείσδυσης
γεωτρήσεων ως συνάρτηση των διαθέσιμων παραμέτρων και χρησιμοποιείται από τη
βιομηχανία, όπως η τέλεια θεωρία καθαρισμού, απομάκρηνση θραυζμάτων μοντέλο,
καλύτερα σταθερές βάρος - ταχύτητα περιστροφής και πολλαπλής βελτιστοποίησης
διαδικασια αναλύονται εκτενώς, προκειμένου να γίνει κατανοητή η σχέση μεταξύ των
παραμέτρων γεώτρησης. Επίσης, οι όροι του κόστους και των χρόνου συνδέονται
άμεσα με την επίδοση των μοντέλων που αξιολογήθηκαν, δίνοντας στον αναγνώστη
μια πλήρη κατανόηση της διαδικασίας βελτιστοποίησης. Επιπλέον θα αναλυθεί μια
διαδικασία βελτιστοποίησης σε πραγματικό χρόνο διότι η τεχνική αυτή πρόκειται να
χρησιμοποιηθεί ευρέως στο μέλλον σε δραστηριότητες γεώτρησης μειώνοντας το
κόστος διάτρησης. Τέλος οι νέες προηγμένες μεθόδους προσομοίωσης που
χρησιμοποιούνται από τη βιομηχανία και βασίζονται σε πραγματικού χρόνου
δεδομένα, αξιολογούνται ενώ θα πρέπει να δώσουμε προσοχή στα ωφέλοι από
οικονομικής μεριάς.
ΘΕΜΑΤΙΚΗ ΠΕΡΙΟΧΗ: Πρόβλεψη ρυθμού διείσδυσης σε γεώτρησης
δραστηριότητα
ΛΕΞΕΙΣ ΚΛΕΙΔΙΑ: Μαθηματική μελέτες, πολλαπλή βελτιστοποίηση, μεθόδους σε
πραγματικό χρόνο, μοντέλα προσομοίωσης

This thesis is dedicated to my lovely parents.

ACKNOWLEDGEMENTS
I want to thank my supervisor Vasileio G. for his cooperation and effort in terms of
providing all the needed information and for his immediate response to my questions
during this study. I would also like to give very sincere thanks to my friends, pieces
of advice which were more than helpful for the completion and success of this
project. Above all, I want to express my gratitude to my parents who shared their
support, financially and physically through my graduate study.

TABLE OF CONTENTS
Contents
1. CHAPTER 1 INTRODUCTION.............................................................................15
1.1 INTRODUCTION ...........................................................................................15
1.2 HISTORY OF DRILLING OPTIMIZATION.......................................................17
1.3 FACTORS AFFECTING RATE OF PENETRATION............................................19
1.4 OBJECTIVE OF THIS STUDY.........................................................................20
2. CHAPTER 2 LITERATURE OVERVIEW ..............................................................21
2.1 DRILLING ACTIVITY PARAMETERS ..............................................................21
2.1.1 DESCRIPTION OF CONTROLLABLE PARAMETERS THAT
INFLUENCE RATE OF PENETRATION ........................................................... 21
2.1.2 DESCRIPTIONS OF MEASURABLE PARAMETERS................................... 23
2.2 DRILLING OPTIMIZATION RESEARCH..........................................................24
2.2.1 RATE OF PENETRATION OPTIMIZATION STUDIES............................... 24
2.2.2 RATE OF PENETRATION SIMULATION MODELS ................................... 30
2.2.3 STUDIES OF REAL TIME DRILLING OPTIMIZATION ............................. 32
3. CHAPTER 3 COMMON OPTIMIZATION MODEL THEORY.................................35
3.1 MAURERS PERFECT CLEANING THEORY ..............................................35
3.2 WARREN CUTTING REMOVAL MODEL..........................................................38
3.3 GALLE AND WOODS BEST CONTANT WEIGHT AND ROTARY SPEED...........42
3.3.1 THREE FUNDAMENTAL EQUATIONS ..................................................... 43
3.3.2 ADDITIONAL CALCULATION EQUATION............................................... 45
3.3.3 CALCULATION OF CONSTANTS............................................................. 45
3.4 BOURGOYNE AND YOUNGS MULTIPLE REGRESSION MODEL......................47
3.4.1 FORMATION STRENGTH FUNCTION ..................................................... 48

3.4.2 FORMATION COMPACTION FUNCTION................................................. 49
3.4.3 DIFFERENTIAL PRESSURE FUNCTION .................................................. 49
3.4.4 BIT DIAMETER AND WEIGHT FUNCTION ............................................. 50
3.4.5 ROTARY SPEED FUNCTION................................................................... 50
3.4.6 TOOTH WEAR FUNCTION ..................................................................... 50
3.4.7 HYDRAULIC FUNCTION......................................................................... 50
4. CHAPTER 4 ANDANCED OPTIMIZATION METHODS .....................................52
4.1 REAL TIME DATA..........................................................................................52
4.1.1 MEASURE WHILE DRILLING PIPING ..................................................... 52
4.1.2 REAL TIME TECHNICAL CENTERS......................................................... 55
4.2 REAL TIME BIT WEAR ..................................................................................55
4.3 ROP PREDICTION USING FUZZY K MEANS ..................................................58
4.4 ROP PREDICTION TECHNOLOGIES BAZED ON NEURAL NETWORK.............61
4.4.1 UNDERSTANT AND LEARN CONCEPT.................................................... 61
4.4.2 DRILLING OPTIMIZATION ANN............................................................. 62
4.4.3 ELM AND RBF TECHNOLOGIES ............................................................. 63
5. CHAPTER 5 CONCLUSIONS ...............................................................................65
5.1 TOPICS DISCUSSED .....................................................................................65
5.2 GENERAL CONSIDERATION..........................................................................66
ABBREVIATIONS – INITIALS...................................................................................68
REFERENCES...............................................................................................................69

LIST OF FIGURES
Figure 2.1: Drilling operation centers time line of significant initiatives..........................32
Figure 3.1: Crater formation mechanism .......................................................................34
Figure 3.2: Crater volume VS impact energy.................................................................35
Figure 3.3: General rate of penetration equation...........................................................47
Figure 4.1: Simplified MPT system description .............................................................52
Figure 4.2: MPD setup with wired drill pipe and the resulting control volumes..............53
Figure 4.3: Schematic shows how PDC and roller cone bit types cutters have
measured.......................................................................................................................56
Figure 4.4: Framework of prediction procedure.............................................................58
Figure 4.5: Artificial neural network ...............................................................................60
Figure 4.6: Drilling optimization ANN.............................................................................61

LIST OF TABLES
Table 2 1: Fluid type characteristics..............................................................................21

CHAPTER 1: INTRODUCTION
Panagiotis Iliopoulos - 15 - 2015
1. CHAPTER 1
INTRODUCTION
1.1 INTRODUCTION
In recent years the increasing demand for energy research from the ground has
forced operators to develop a subject of survey ensuring that well drilling is realized
in a more efficient manner. For that reason oil and gas companies tend to find
different methods with different consideration on drilling activities in order to reduce
cost, increase performance and overcome possible difficulties. There is no doubt that
energy sources are reducing day by day and the oilfield exploitation will be more
difficult in the future. These entail that the future project should improve productivity
and make well construction cost effective. New methods which improve drilling
operations have been based in technological advantages that maximize the desired
goals.
The basic principle for all operations is the relation between cost and time, which are
two interdependent amounts. It is understood that when time expands, cost
increases and vice versa. From the beginning of the 20th century, oil and gas
companies have realized how important is to minimize drilling operation cost. As a
result, all efforts aim to increase drilling speed in order to accelerate penetration rate
(ROP) [1]. It is generally accepted that there are many factors referred to as
performance qualifiers (PQ) which influence ROP. Some of them are more important,
some other less and all together make the relationship complex, as they require the
development of mathematical models in order to be determined. Consequently, only
when all parameters affecting ROP are met to the greatest extent possible, they give
the best combination of drilling operating conditions. Hence, during the drilling
process the main objective is to conduct all the activities in the most economic way
[2].

MASTER THESIS DEPARTMENT OF PETROLEUM AND NATURAL GAS TECHNOLOGY
Nowadays, many countries lack resources and hire oil and gas operator companies in
order to have energy providers. On the other hand, companies wish to have mass
production and develop techniques which give them the capability to drill many
different wells from one location, when these wells belong to the same reservoir.
Therefore with the directional drilling method, companies avoid construction and cost
for each new well, while at the same time drillers by using PC based systems have
the opportunity to collect data from close wells and gain years of knowledge [3].This
is very useful because knowing the structure from the same geological area and the
operation parameters from nearby wells, the driller can estimate and optimize all the
factors which have impact in drilling operations. The problem is that there are many
variations on drilling data and it is impossible to find the suitable combination without
using a mathematical model. There were many mathematical models attempting to
combine all relations of drilling factors. Most of these models aim to calculate the
best selection of weight on the bit (WOB) and rotary speed so as to achieve optimal
time and cost reduction [4].
It is remarkable that structure and properties formation is one of the most important
factors on drilling process. However, it is considered one of the most difficult factors
to estimate because the ground does not present a uniform geological structure. For
this reason geologists try to illustrate the real conditions of subsoil and provide as
accurate data as possible. The data do not guarantee success because in many cases
the ground presents helical structure, cracking, salt dome and other geological
phenomena [5]. It is common to have different geological allocation between two
wells which are close to each other and this is exactly the reason why we can never
be sure and we should always consider the uncontrollable factors.
There are controllable factors such as bit types, fluid properties, WOB, horsepower,
hydraulics and rotary speed. While the driller follows the good drilling practice, he
has the opportunity to select and determine the factors using suitable models which
predict the rate of penetration.
The scope of this study is to analyze all models that have been used for ROP
predictions during the drilling operation from the initial method at the beginning of
the previous century until today.

1.2 HISTORY OF DRILLING OPTIMIZATION
If we go back to the beginning of drilling activities we observe the need for
knowledge. The development of all suitable and important techniques took place in
the first 20 years, such as rotary drilling bits, fluid dynamics, casing installation,
cement. During this first period all methods and tools improved, hence it was named
development period. After this period there is a gap for about thirty years as oil
companies did not invest large amounts of money on drilling research. From 1948 to
1968 oil companies started to perceive the importance of research. During these
years the scientific period took place and consequently the total cost increased. The
thought for optimized drilling is one of the most important assumptions of the
scientific period but in reality it started in 1968. It should be mentioned that many
researchers spend endless time studying all parameters included in drilling and the
relation between them. The period after the 1970s is known as automation period. At
that time the first computer systems were created which performed operations
improving drilling. Most of oil and gas companies started to use automated rig
systems, based on closed-loop computer system that controlled drilling variables and
had complete planning of well drilling from spud to production [6].
Looking at the chronological axis some dates are worth mentioning as they are
considered landmarks of drilling optimization. The Graham and Muenh study in 1959
can be regarded as the first integrated model which approached and included the
most important drilling factors. More precisely this mathematical model evaluated the
correlation between WOB and rotary speed, as well as the shelf life of bit.
Summarizing, drilling rate was predicted combining depth, rotary speed and WOB.
Four years later, another research was carried out. In 1963 Galle and Woods created
special arranged graphs which indicated the best combination of drilling parameters
[7]. So far the most important model on which all modern studies have relied is the
linear penetration model by Bourgoyne and Young. This model uses multiple
regression analysis in order to achieve the best selection of drilling parameters.
Consequently, model’s equation is developed for different formations. Their basic
purpose was to create a model able to calculate maximum penetration rate with the
minimum cost, taking into account all technical specifications [8-9]. During the next
decade, no significant changes occurred in drilling optimization so petroleum industry

kept using past mathematical approaches in order to assess drilling activities. The
need for a more accurate project led one of the biggest industries, named SHELL, to
develop a new concept named “drill the limit” (DTL). Actually through methodology
they discovered the removal time, that is the difference between real well time and
theoretical well duration. The main objective was to increase rig time efficiency, by
finding the best critical path between parallel works [10-11]. At the end of the
millennium the real time monitoring was a very promising method. A drilling
procedure brought to light tools which had the opportunity to give additional
information from the bottom hole during the drilling operation in real time. This way,
companies had a better understanding of the factors that influence the penetration
rate and increase the drilling cost. The basic advantage of this evolution is that it
allows drilling parameters monitoring from many different locations [12,64]. Therefore
real time operation centers were established in the next years so that the companies
have a more integrated information system. The initial idea is quite simple; the
drilling data is stored and transferred in real time. In the coming years the
technological development and the new improved tools which communicate directly
with the computers, provided the possibility to have better control and optimization
service including additional measurement, such as pressure control and rock strength
[13].
Drilling optimization from 1950 until today:
Scientific period
 1950 – Expansion of drilling research
– Beginning of drilling optimization
 1952 – Jet type of roller cone bits
 1959 – First drilling optimization model by Graham and Muench
 1963 – Galle and Woods model
Automation period
 1970 – Beginning of automation period
 1974 – Multiple regression model by Bourgoyne
 1986 – Real time drilling optimization at Chevron rig site
 1999 – Real time drilling monitoring
 2003 – Real time operation centers Shell and Halliburton

 2005 – Real time monitoring at ExxonMobil rig site
 2006 – Real time transfer centers by Statoil
1.3 FACTORS AFFECTING RATE OF PENETRATION
It has been observed that the rate of penetration depends on many factors. These
factors are distinguished in two basic categories as to whether they can change or
not. So the first category is named controllable factors, such as WOB, hydraulics and
drill string rotary speed which can be influenced by the user where it is necessary.
On the other hand, there are environmental factors characterized as uncontrollable
that can be measured but not changed, that’s why we adjust our project always
based on them. The geological structure and the formation properties is an
understandable example. However, additional factors essential for a normal drilling
operation such as the bit type, downhole pressure, temperature, cutting
transportation, horsepower from pumps and general auxiliary equipment influence
drilling operations [14]. It should be mentioned that all previous factors can be applied
regardless of whether the well is horizontal or inclined, but there is no doubt that the
degree of difficulty is greater in the second case.
Cutting removal is the factor that requires particular attention in order to have the
best possible bottom hole cleaning. The efficiency of cutting removal is one of the
most important factors of drilling penetration rock, because only at that time the fluid
from the nozzle will achieve fracturing and the drilling process will continue [15].
It is remarkable that there are soft geological formations which are very easy to be
drilled but there are also hard geological formations which require more expensive
bit and more time. Since the trip time increased, the well cost also increased enough,
while the trip time takes a major part of the well operation. This is contrary to
efficiency because time is money in drilling operation and the main objective of the
industry is to operate with the lowest cost per foot. Also, it should be mentioned
that if we change some factors and improve ROP, this shall not entail improvement
of drilling efficiency [16].

1.4 OBJECTIVE OF THIS STUDY
The methods to predict rate of penetration (ROP), used from oil and gas industry,
have proved to be a valuable tool that with continuous improvement, ensure a
smoother and more economic operation. The primary objective is the models to be
as close as possible to the recording data. If this assumption is confirmed, the
method can be considered accurate for new predictions.
This survey presents a theoretical approach to the drilling problem based on the
main factors that influence the drilling process. The objective of this research is to
analyze and explain all models which have been used until now and to estimate the
efficiency of each one.
Subject area
 Presentation of the results from laboratory drilling experiments
 Determination of parameters which influence rate of penetration
 Presentation of the drilling rate equations
 Analysis of mathematical models constants
 Comparison between the most common ROP prediction models
 Presentation of the optimum drilling conditions which minimize the drilling cost
 Introduction of new advanced techniques
This survey uses existing studies and industry knowledge in order to correlate all
current parameters during the drilling process and present the effect to the rate of
penetration. This survey is also an attempt to make the term “cost per foot” more
understandable when the rate of penetration increases dramatically.

CHAPTER 2: LITERATURE OVERVIEW
2. CHAPTER 2
LITERATURE OVERVIEW
2.1 DRILLING ACTIVITY PARAMETERS
A brief reference to parameters involved in the drilling process is required in order to
explain how the rate of penetration is influenced. Another important point which
should be understood is how these parameters interact, accelerating or slowing the
drilling activity. In addition there are parameters that can be characterized as
measurement parameters as they describe the specifications and the amount of
recorded data either during the drilling process or before.
2.1.1 DESCRIPTION OF CONTROLLABLE PARAMETERS THAT INFLUENCE
RATE OF PENETRATION
WOB: By the term “weight on the bit” we refer to the total weight exerted on the bit
from the drilling string. This amount of weight in practice can be measured using the
drilling line tension. That means that a sensor is applied in drilling line, recording the
unique value that is converted to weight. With this tool we calculate the overall
weight including the weight of the block. This calculation requires particular attention
in order not to have to incorrect results. In addition, the new technology tools (MWD
collars) can measure the axial force exerted to the collars and transfer the
information [17].
RPM: This term describes the rotation speed of the drill string per minute. The
rotation motion starts from the rotation machine, which in some cases is a rotary
table, in other cases it is a top drive system and it is transferred through drill string
on the bit. The data are obtained by an electronic device and are considered quite
accurate [17-18].

PUMP PRESSURE: The pump pressure is directly linked to the term WOB while it
influences the total force which is exerted in the formation from the drill bit. When
the pump pressure increases the nozzle has more strength and as a result the rate of
penetration increases. A large amount of pump power is consumed on the bit. Flow
meters are used as pressure detection at the entrance and exit of the fluid [19].
BIT ENERGY: The three previous factors are considered bit energy parameters while
the amount of weight, speed and pressure is converted to energy strength. In other
words we are talking about the energy created between bit and rock [20].
FLUID PROPERTIES: This thesis focuses on the use of liquid drilling fluid (oil, water,
synthetic) while there are gas-liquid mixtures (foam, aerated water) and gases (air,
natural gas). The mud properties such as density, viscosity, are considered the two
most important rheological parameters for a safe drilling operation without risk for
kick and with an effective cutting removal. Also the mud properties are responsible
for other functions such as bit bailing, bit coiling, high torque and stuck pipe.
Nowadays, there are sensors which measure the mud weight and fluid viscosity in
real time, accelerating the process. The following table indicates the three different
types of liquid mud [20-21].
Table 1.1 Fluid type characteristics
HOLE CLEANING: Maybe there are the most important parameters for the bit drilling
and the increase of well depth. The cutting removal and consequently the hole
cleaning depend from many things. The most significant are the hole angle, cutting

size, annular size, bit specifications, fluid flow regime, fluid velocity and fluid
properties [15, 22-23].
2.1.2 DESCRIPTIONS OF MEASURABLE PARAMETERS
TORQUE: This is a down hole drilling measurement which describes the fatigue of
the drill string while it rotates. They are also referred to as rotational friction which
entails the interaction between the bit and the formation. This measurement
indicates if the bit is damaged and cannot be used again. It is a clear indication that
protects the bit from premature wear [17, 24].
DRILL STRING PROPERTIES: This term includes all the specifications that all parts
which constitute the drill string should have. Pipes must be designed to resist loads
such as buckling and axial forces. The parts that accept huge forces are designed to
be more durable [25].
VIBRATION: The term vibration describes the axial, torsional and lateral motions at
the bit. Such effects, such as a slip/stick, a bit whirl and bit bounce, have as a result
a faster cutter damage, shortening the life expectancy. After many experiments, it
has been proved that a diamond bit (anti-vibration bit) is able to sustain controlled
frequency vibrations increasing the ROP [26-28].
BIT BALLING: This is a phenomenon happening during the drilling process, that
occurs both on roller cone bits and PDC bits. As a consequence the rate of
penetration decreases continuously while the bit loses the performance. New
techniques such as electro-osmosis have focused on reducing bit balling and
increasing ROP [29-30].
ROCK STRENGTH: Rock strength should be defined for similar types of drilled rock
using the same bit and under the same conditions. Recorded database such as
formation drillability catalog provides a useful indication for the prediction of power
requirements for a particular drilling operation. When the rock strength is immense,
the required drilling conditions have negative effect on the penetration rate [31].
INCLINATION: This parameter referred to the directional drilling methods is
conducted with different tools. This advantageous equipment can take

measurements while drilling (MWD) that are continuously updated and send the
information in real time [32].
2.2 DRILLING OPTIMIZATION RESEARCH
When the industries perceived the main factors that influence the drilling operation it
was necessary to study the interaction between them. Following this consideration
and having as ultimate purpose to improve the process, the first research studies
were performed. For oil and gas companies the drilling optimization is accompanied
with two elements, rate of penetration increase and cost per foot reduction.
It was clear that the most significant factor which could guarantee improvement of
the drilling rate was the hydraulic maximization. For this reason companies tried to
improve the bit characteristics by doing tests and spending money.
However, we should mention that most of the trials were conducted in laboratories
and most of the studies describe the static optimization process. After that, the
advantageous communication system was a determining factor for the real time
drilling optimization period. There are two different categories of models which
optimize the drilling. The optimization was achieved using analytical methods such as
Warren model and statistical methods such as (Bourgoyne and Youngs) multiple
regression models.
Early drilling optimization models and recent real time optimization methods
constitute the subject of this assignment.
2.2.1 RATE OF PENETRATION OPTIMIZATION STUDIES
Bourgoyne and Youngs study is considered one of the most widespread models. It
became an accurate tool for the ROP predictions while it was a standard model on
which many of next researches are based. They used data from twenty five wells in
order to enact the constants. Taking into account eight different variables such as
formation strength, formation compaction, pressure differential, bit diameter- bit
weight, tooth wear, bit hydraulics and rotary speed, they established a linear

penetration model using multiple regression analysis in order to find the best WOB,
RPM and bit hydraulics characteristics. In addition, they claim that a simpler drilling
optimization model is possible to reduce the total costs by about 10% [8].
John W. Speer in 1959 wanted to create a chart which would indicate the best
correlation between five drilling parameters knowing the minimum of previous data.
Using personal experience he tried to determine the combination of WOB, RPM,
hydraulic horsepower and drillability formation which have the best result with
minimum drilling cost [33].
Garnier and Lingen focused on the formation characteristics and conducted
laboratory experiments with soft drag bits and roller cone bits on rock type with
different strength and permeability. They observed that there is a reduction of the
penetration rate when the formation strength is larger due to the dispute between
mud and pore pressure. They also supported that the cuttings due to pressure
differentiation in many cases remain at the bottom and the rocks are less drillable
[34].
Graham and Muench supported that if the optimum combination between bit weight
and rotary speed is found, the drilling cost will decrease while the rate of penetration
will increase. On the other hand, they noted that if the rate of penetration increases
due to greater bit weight and rotary speed, the cost of making round trip and bit cost
increases while the bit life expectancy decreases. It was clear that changes on WOB
and RPM, which increase or decrease the drilling efficiency, should be determined at
any drilling conditions by a mathematical analysis [7].
Galle and Woods are some of the first researchers who assumed that ROP is affected
only by two parameters and developed a mathematical relation between weight on
the bit and rotary speed in order to find the best combination of these constants.
They created a model which predicts the ROP and includes parameters such as
weight on bit, rotary speed, bit tooth wear and type of formation. They presented a
graph which indicates that the drilling cost can be minimized using the suitable
combination of drilling parameters. At the end they used the previous model and
established the drilling rate equation (2.1) rate of dulling equation (2.2) and bearing
life equation (2.3) [35].

𝑑𝐹
𝑑𝑡
= 𝐶𝑓𝑑
𝑊 𝑘 𝑟
𝑎 𝑝 (Drilling rate equation) (2.1)
𝑑ℎ
𝑑𝑡
=
1
𝐴 𝑓
𝑖
𝑎𝑚
(Rate of dulling equation) (2.2)
𝐵 = 𝑆
𝐿
𝑁
(Bearing life equation) (2.3)
Maurer studied the rate of penetration equation for roller cone bit from rock cratering
mechanisms. This equation has as general consideration the perfect cleaning which
means that all the rock scraps between teeth are removed. Also he established the
correlation of rate weight speed (RWS) and created the following formula (2.4) as a
function of depth which proves that failure in bottom hole cleaning is an important
factor reducing the rate of penetration with depth [36].
𝑑𝐹
𝑑𝑡
=
4
𝜋𝑑 𝑏
2
𝑑𝑉
𝑑𝑡
(2.4)
Where:
F = the distance drill by bit
V = is the volume of rock removed
db = is the bit diameter
Langston presented a way for allocation, recording and usage of existing information
with day to day competitive drilling circumstances. These factors are all those
involved in drilling operation and must be optimized in order to have successful
results [37].
Eckel conducted an experimental study and observed a reduction in drilling rate due
to the changes from water to mud. He claims that viscosity is one of the most
important factors affecting drilling rate while influencing the cleaning effect [38].
Subsequently, he performed a microbit study showing that the drilling rate may be
expressed as an exponential function of a pseudo Reynolds number involving flow
rate, nozzle size, fluid viscosity and density [39].
R= k (Re)0.5 , 5 a<Re a<100 (2.5)

Young invented the development of a computer control system that will collect,
record and analyze the data. The system used the data as input information to solve
minimum cost drilling formulas and control the drilling parameters. More analytically,
the minimum cost solution is accompanied by four different equations, i.e., drilling
rate, bit tooth wear rate, bit wearing rate and drilling cost [40].
Lummus supported that drilling fluid and hydraulics are the most important factors
affecting drilling rate. He also classified the data required for optimized drilling in
three categories: data needed as input, day by day data which determine the
efficiency of optimization and data for better future optimization. However, he said
that it is possible to face problems when the drilling optimization programs have
difference requirements than the ones actually supplied from the rig equipments,
i.e., rig pump is not enough to provide adequate hydraulics. As a result, the
difference in rig equipments should affect the weight, rotary speed, mud and
hydraulics programs. In order to avoid this, the drilling programs should be planned
to be as much flexible as possible so that they adapt to rig equipments and satisfy
the optimum recommendation [41].
Wilson and Bentsen in 1972 presented a drilling optimization study which has as a
primary objective to minimize the drilling cost. This model presupposed that all the
parameters which affect the drilling process are restricted to two basic ones, WON
and RPM, while all others have been preselected. Due to complexity it is necessary to
develop three different methods: firstly minimize the cost per foot, secondly minimize
a cost of a selected interval and thirdly the cost over a series of interval [42].
Reed method predicted the best combination of factors, such as weight on the bit
and rotary speed, taking into consideration two different cases, when all other
variables were constant and when they were fluctuated. This method reached the
same result as Galle and Wood method, but is considered more accurate because it
has resolved the Monte Carlo Scheme. It should also be mentioned that this method
presented effective advantages in connection with field application [43].
Bizanti and blick did many laboratory experiments because they wanted to study the
factors which influence the cutting removal. During the trials they observed that
parameters such as nozzle diameters, cutting size, mud density, mud viscosity, yield

point, flow rate, rotary speed, bottom hole pressure, pressure drop through the
nozzles and inclination angle were responsible for the variations of a regression
analysis. The previous parameters were expressed as dimensional parameters such
as Reynolds number and Froude number in regression equations, where combining
these with a chart was a useful tool for cutting removal and ROP optimization [44].
Tanseu in 1975, using heuristic approach from several bit runs regression equation,
wanted to predict penetration rate and bit life. He considered the weight on the bit,
rotary speed and bit hydraulic horse power as controllable variables. These variables
impose the maximum drilling rate while the drilling cost is minimized over these
variables. Also he introduced an online optimization scheme for updates with each
new bit run [45].
Al-Betairi applied Bourgoyne and Young model using statistical analysis system and
observed that there are parameters which are not estimated in their model. The
purpose of his study was to find the correlation between the unknown drilling
parameters from the result of statistics, but the estimation is not more accurate due
to the presence of multicollinearity [46].
Reza and Alcocer created a new mathematical drilling optimization model that
consisted of three different equations which predicted the rate of penetration (2.6),
rate of bearing wear (2.7) and the rate of bit dulling (2.8) just as Galle and Woods
had done. These equations include factors such as WOB, RPM, mud density, mud
viscosity, rock hardness, fluid flow rate, pressure differential, temperature and heat
transfer coefficient. However, the determination of coefficient is very difficult in
laboratory using data from actual deep well [47].
𝐹
𝑁𝑑 𝑝
= 0.33 [
𝑁𝑑 𝑝
2
𝑢
]
0.43
[
𝑁𝑑 𝑝
3
𝑄
]
−0.68
[
𝐸𝑑 𝑝
𝑊
]
−0.91
[
𝛥𝑝𝑑 𝑝
𝑊
]
−0.15
(2.6)
𝐵
𝑁
= 0.05 [
𝑡ℎ𝑑 𝑝
𝑊𝑁
]
0.51
[
𝑢
𝑁𝑑 𝑝
2]
0.4
[
𝑄
𝑁𝑑 𝑝
3]
−0.5
(2.7)
𝐷
𝑁𝐷 𝑏
= 0.001 [
𝑄
𝑁𝐷 𝑏
3]
0.56
[
𝑊
𝐸𝐷 𝑏
2]
0.26
[
𝐷 𝑏
𝑄
]
−0.03
(2.8)

Hoover and Middleton performed laboratory experiments wishing to determine the
bit performance and bit wear characteristics correlating the results with bit design
options. The experiments were conducted at 100 and 500 rpm with different bit
design in three types of rock: Nugget sandstone, Crab Orchard sandstone, and Sierra
White granite. They observed that as the bit wear has large wear flat, the torque
presented more variables when changed the weight on the bit [48].
In 1984, Warren wanted to determine a torque relationship based on a force balance
concept, using laboratory drilling tests and field data. He claimed that the torque was
determined by the weight on the bit and the depth of the tooth while the new model
is not influenced from parameters such as formation type, bit hydraulics and mud
characteristics [49]. Three years later he presented the rate of penetration model of
roller cone bit by cutting removal process which comprised of two terms. The first
includes the weight on the bit effect without depth of the tooth when the rate of
penetration is calculated and the other term includes the tooth effect [50].
Miska and Ziaja focused on evaluating the formation strength and formation
abrasiveness. They performed an experimental model with a verified rate of
penetration equation considering that it will confirm the reduction in penetration rate
due to the bit wear. The results were as expected while they achieved a perfect
matching with the theoretical model. This method can indicate the index of rock
strength [51].
Maidla and Ohara developed a drilling model, using previous drilling data. They
wanted to find the suitable bit bearing, weight on the bit and rotary speed. They had
as a basic ambition to reduce the drilling cost. The results from this drilling model
compared with the Bourgoyne and Young model. They supported that the drilling
rate could be predicted if we analyze the coefficient from previous drilling data and
added that the drilling model accuracy depends on the quality of these data [52].
Brett and Millheim method was a practice based on data from previous well which
had been drilled in a specific area. They created a method which was named drilling
performance curve (DPC). This method is a useful tool, while it gives all the
information from a variety of wells for the drilling process. This model was

considered a simple model because it restricted all the value of constants to three
[53].
Wojtanowicz and Kuru in 1987 developed a new mechanism model which includes all
the constants and functions of the drilling process. The validity of the constants was
tested using some field and laboratory data. Also the concept of maximum bit
performance (UBP) curve was taken into consideration. The curve indicates the
maximum values of the average drilling rates for various values while it was analyzed
for both roller cone bits and PDC bits [54].
Fear’s methods used foot based mud logging data, geological information and bit
characteristics in order to determine the correlations between controllable drilling
parameters. These correlations were used to generate recommendations for
maximizing ROP in drilling process [55].
Samuel and Miska in 1998 studied the optimization of motor performance and the
effect of drilling parameters. They performed a new test called wear off test to
establish an operating benefit from optimization of positive displacement motors
either on roller cone bit or on diamond bit. This study proves that the PDC
optimization accelerates the rate of penetration without aggravating motors
efficiency [56].
2.2.2 RATE OF PENETRATION SIMULATION MODELS
Pessier and Fear performed a full scale simulation test and developed an energy
balance model for boreholes drilling under hydrostatically pressurized conditions. The
basic elements are mechanical specific energy input, drilling efficiency and a
minimum specific energy equal to the rock strength. As a result they acquired better
and more accurate methodologies for WOB, ROP evaluation while supporting that the
drilling bearing problem is more reliable by continuously monitoring Es and μ,
equations [57].
𝐸𝑠 = 𝑊𝑂𝐵 (
1
𝐴 𝑏
+
13.33𝜇 𝑠 𝑁
𝐷 𝐵 𝑅𝑂𝑃
) (2.9)

𝜇 = 36
𝑇
𝐷 𝐵 𝑊𝑂𝐵
(2.10)
Cooper created a drilling software that gives the opportunity to a student or engineer
to simulate and optimize the drilling process. Consequently, the program should run
either as a training program or as a simulator of a real project. The simulator
contains an algorithm which determines the rate of penetration, the rate of wear of
the bit and finds other accidental conditions such as well kick as drilling continues.
Also, it provided analytical indicator for cost per foot during drilling, while the user
could perform all the functions that are included in a real project [58].
Baraggan created a program which was based on the heuristic approach in order to
find the optimum drilling conditions using Monte Carlo Simulation and developing
numerical algorithm. In this study we have analyzed five different rate of penetration
equations (Moore, Maurer, Bingham, Cunningham, Eckel, Galle) having as a main
purpose to prove that drilling optimization of well phase is more economical than the
optimization by single bit runs. It is mentioned that the heuristic approach accepts
easily the constrain values to the drilling parameters [59].
Dubinsky and Baecker performed a computerized drilling simulation study. They used
the PC based simulator to determine dynamic behavior of the bit for various drilling
conditions. This was an attempt to simulate many of the major drilling dynamic
functions such as bit bounce, vibration, bottom hole assembly, torque shocks, stick
slip and torsional oscillation. However, they supported that the model required self
learning and practical experience in order to achieve the on line drilling optimization.
On the other hand, the program should be used as a training tool for MWD operator
[60].
Millheim and Gaeble created a new concept in order to reduce drilling cost and
increase the performance, which was called Virtual Experience Simulation (VES) for
drilling. This new concept was based on heuristics and exploited unused data
accumulations which are processed from specific data sets in specific geographical
and geological environments, such as geology, tripping, cementing, logging and ROP.
Very good ROP isomeric maps as well as 3D graphs were illustrated in the outcomes
of their work. They supported that these new data are valuable while they give the

user access to virtual drilling experience. VES offers a way that can provide
knowledge from others as virtual experience [61].
2.2.3 STUDIES OF REAL TIME DRILLING OPTIMIZATION
Simmons study is considered as the first study of real time drilling optimization.
Initially, Simmons used each available technology and engineering model that could
predict and evaluate the drilling process. These two concepts open new horizons
while the drilling supervisor on location using experience and usable technology,
which provides real time drilling performance, has the ability to determine the drilling
optimization parameters. Following Simmons’ study a combination of current
technology, engineering knowledge and real time drilling optimization can improve
drilling efficiency and save on overall drilling costs [62].
Zachariah John created a new advance data transmission system named InterACT
Web Witness (IWW) and could transfer data from remote drilling well sites in real
time. This procedure is 10-20 times quicker than the conventional system FTP. The
main advantage of real time system is that drilling experts have the opportunity to
exploit real time information in order to provide more effective support to the well
site staff especially when critical decisions should be taken [63].
Rommetveit and team created an innovative system for drilling automation and
simulation which was named drilltronics. This advance system had the capacity to
collect all available drilling data in real time and therefore to optimize the drilling
process. In reality the combination of equipment contributed in this project. The
system is constituted by a software modeling based on algorithm; using models that
drive drilling data in real time. This integrated drilling simulator develops the models
simultaneously comparing the ROP. Moreover, with the operations use such as
automatic control and automatic detection it warns for problems that may arise.
Consequently, this innovative system can detect unwanted event, improve drilling
data and automate a critical process [65].
J.E. Booth describes the coordinated effort from an operator and service oil and gas
companies to establish real time operation centers (RTOC) in order to improve

drilling efficiency. The evolution of drilling centers is divided into two periods. The
first drilling operation centers focus on data management and distribution and make
ambitious attempts to change the drilling process and provide a new work process
for managing and supporting remote operations, using data from different locations
of operation. The following timeline illustrates the chronology of the events [64].
Figure 2. 1 Drilling operation centers timeline of significat initiatives
Dupriest and Koederitz performed a new system which was called Navigation
Optimization (NAVO) and was based on (MSE) mechanical specific energy theory.
This innovative system monitors all dynamic drilling parameters during drilling
operations in real time, having as basic principle the ROP maximization and drilling
cost reduction [66].
Iversen and team created an integrated drilling monitoring system which promised
better optimization in drilling operation. This new system consisted of computer
controlled machinery and advanced computer modeling which are continuously

updated in real time data. Having all information for fluid flow we can detect
unwanted occurrences which exceed the safe limit for the drilling process. Operations
such as the cover tripping and reaming, pump start up, friction tests, stick slip
prevention, bit load optimization and monitoring are included in this process [67].
Milter studied the project on real time data, transferred from offshore facilities to
land support centers. The data were collected not only from one site but from any
place/site with high speed internet communication while piped data cover all the
necessary information in order to facilitate remote support. They ascertained that
this system of real time data transmission can minimize the number of unforeseen
event during drilling process [68].
Strathman and other member of Statoil team in 2007 were able to make a step
change in drilling activities. They supported that time needed for data extraction is
the main factor for an effective analysis, unlike the usual way for optimal operation
which focused on depth.
A data system included up to 200 parameters while the data frequency were derived
every 5 seconds from 20 different wells. The basic advantage of this system was the
effective optimization without having the experts on the rig [69].
Iversen and other member of Stavanger international institute (IRIS) presented in
2008 a new drilling controlled system for real time data optimization and automation
control which was installed into the rig control mechanism in order to pipe signal
from sensor in a real time basis. However the test showed that data transfer
credibility was not sufficient but measures have been taken to solve this problem. At
the end they concluded that parameters can be calculated and verify the quality of
safeguard calculations while the system functionality depends on data and correct
system setup [70].

CHAPTER 3: COMMON OPTIMIZATION MODEL THEORY
3. CHAPTER 3
COMMON OPTIMIZATION MODEL THEORY
Many optimization studies have been performed in order to determine the parameters
which influence the drilling rate, most of these aiming to reduce the total drilling cost
and increase the efficiency. The objective of this chapter is to present and analyze the
basic principle of the most common model which has been used from the industries
and that gave the stimulus to further research. There are four common drilling rate of
penetration models which are: Maurer’s [36], Warren’s [49-50], Galle and Woods’ [35] and
Bourgoyne & Young’s theories [8].
3.1 MAURERS PERFECT CLEANING THEORY
Maurer’s drilling rate formula is based on the perfect cleaning theory whereby all of
the rock scraps have been removed between teeth. This formula for roller cone bits,
which is derived from rock catering mechanism, consists from two main operations.
Initially is created a crater under the big teeth and immediately after this, the cuttings
is removed from the craters.The following picture illustrates the crater formation
mechanism to the overall drilling operation.
Figure 3. 1 Crater formation mechanism

When the bit tooth comes in contact with rock, its surface creates a deformed force,
which develops until its power overcomes the crushing strength of the rock. Exactly at
this point, a wedge of crushed rock is formed below the tooth and the crushed
material is compressed, creating high lateral forces around it. As a result, when the
force is higher than the limit of the rock, the fracture is transmitted from the point
under the tooth to the surface of the rock [71-72].
As demonstrated in the previous picture, the volume that arises from the fracture
should be removed, in order to continue the drilling process. This volume (V0),
depends on the following equation:
𝑉0 ∝ 𝐸 𝐶 − 𝐸0 (3.1)
Where:
EC = energy imparted to the rock during formation of a single crater
E0= threshold energy required to initiate cratering
In case that there is a second free face to which crater the volume of material that
removed is larger and the relation between VO and E0 is a linear in order to slope
which presented the next picture. The new relationship is 𝑉0 ⋉ 𝐸 𝐶 .
Figure 3. 2 Crater volume VS impact energy
It is remarkable that wedge or cones with larger included angles should be considered
as more effective, because they have the ability to crush larger volume. On the other
hand, it has been ascertained that smaller included angles have a greater penetration

depth. However, after many experiments, it was obtained that the previous
consideration is verified for angle ranking from 30 degrees to about 90 degrees, which
is the most effective point. That is because it creates a larger number of deeper lying
cracks that are more difficult to complete back to the rock surface to form fragments.
After this point, for tool angles greater than 90 degrees, the effectiveness decreases
rapidly [36, 71]. The drilling rate formula can be expressed as:
𝑅 =
4
𝜋𝐷 𝑏
2
𝑑𝑉
𝑑𝑡
(3.2)
Where:
V = is the volume of rock removed
Db = is the bit diameter
When all of the broken rock is removed from the craters between impacts:
𝑉 = 𝑛 𝑉𝐶 (3.3)
Where:
n = is the number of the impacts
The total volume of each crater is independent of time and the equation (3.3)
transfomes to:
𝑑𝑉
𝑑𝑡
=
𝑑𝑛
𝑑𝑡
𝑉𝐶 (3.4)
The rate at which teeth are impacting is:
𝑑𝑛
𝑑𝑡
=IN (3.5)
Where:
I=is the number of impacts per revolution
N= rotary speed
When the effect force and the previous parameters are included, the drilling rate
formula is:

𝑅 ∝
𝐼𝑁(𝑊−𝑊0)2
𝑛 𝑖
2 𝐷2 𝑆2 for 𝑊 ≥ 𝑊0 (3.6)
𝑅 = 0 for 𝑊 ≤ 𝑊0 (3.7)
Where:
W= is the weight on the bit
W0= is the threshold weight required before the teeth penetrate the rock
Ni= is the number of teeth in contact with the rock when there is maximum
force per tooth
S= is defined as the drillability strength of the rock
W0 depends on the type of the formation, for example, we assume that W0 is small
compared to W when the formation is very soft while low strengths are observed for
this kind of formations. According to perfect cleaning theory, the penetration per
revolution (R/N) should be independent of the rotary speed [73]. When the weight on
the bit has a very high value, the R/N ratio decreases very rapidly, as the rotary speed
is increased due to cleaning problem, which is occurred from high drilling rates. This
problem is considered as one of the majors problems during the drilling process. In
this case, the equations (3.6) reduce to:
𝑅 = 𝑘
𝑁𝑊2
𝐷2 𝑆2 (3.8)
Equation (3.8) indicates the good correlation of rate, weight and speed (R, W, N)
under perfect cleaning conditions. However, it should be mentioned that the data
relationships derived, apply only to the specific conditions under which they were
obtained, while it is very difficult to create a formula for imperfect cleaning conditions
[36].
3.2 WARREN CUTTING REMOVAL MODEL
Warren’s observation doesn’t depart from Maurers theory. After laboratory test, he
remarked that the ROP reduction at high bottom hole pressure is the effect from
insufficient cleaning. Warren created a model making an effort to represent all the
parameters of the physical process in one equation. The ROP depend by either the

cutting generation process or the cutting removal process, because under steady state
drilling condition the cutting removal is equal to the rate which new chips are formed.
The perfect cleaning model includes parameters such as rate of penetration, weight
on the bit, rotary speed, rock strength and bit size. The initial equation is the
imperfect cleaning model, which is the following [50,74].
𝑅 ≡ (
𝑎𝑆2 𝑑 𝑏
3
𝑁 𝑏 𝑊2 +
𝐶
𝑁𝑑 𝑏
)
−1
(3.9)
Where:
a,c=dimensional constants
Nb=bit rotary speed
S= rock strenght
W= weight on the bit
The first fraction on the equation is based on the assumption that the WOB is
supported by a fixed number of teeth and is independent from the teeth depth. On
the other hand, the second term describes the WOB distribution for more teeth as the
WOB is increased and the teeth penetrate deeper into the rock. Equation (3.9)
includes the bit size effect, WOB, rotary speed and rock strength. However, this
equation can’t predict the ROP without modification to account imperfect cleaning,
because the cutting removal is an important obstacle to the process [50].
At this point, it should be mentioned that the relation between WOB and ROP is not
standard and present different value under different conditions. In order to make the
previous relation more understandable, it’s very important to mention the
phenomenon which conducted in the inflection point. At low WOB the ROP increases
at an increasing rate as WOB to be increased up to a point. After this point WOB
continue to increase but a decrease rate. Exactly this point is called “inflection point”,
which has been observed that it occures when using bits with long teeth, which
increase the ROP, but it doesn’t occure when using bits with sort teeth [50].

Hydraulics flow is considered the most critical factor for the bottom hole cleaning.
When the level of hydraulics is increased the bit has more weight. As a result, the ROP
is decreased due to the mud properties which contribute to poor cleaning.
At this point, it is necessary to estimate the hydraulic energy which is developed under
the bit and available for the cutting removal. Warren, relying on this assumption,
calculated the impact pressure in order to evaluate the ability of the jet stream to
transfer energy to the bottom of the hole. The next equation indicates the impact
pressure:
𝑝 𝑚 =
50
1.238,6𝑠2 𝑝𝑑 𝑛
2
𝑣 𝑛
2
(3.10)
Where:
1.238,6 becomes 7991 when expressed in SI metric values
p=fluid density
dn= nozzle diameter
vn= nozzle velocity
s= distance from jet to impact point
The impact pressure measured under the bit, indicate a part of the energy has been
lost due to the jet flow into a confined space and the counterflow, which is the return
flow of fluid from under the bit. Making theoretical approach to impact pressure,
which should be independent of the nozzle size for a fixed bit size, the calculated
impact force can be found by the following equation:
𝐹𝑗 = 0.000516𝑝𝑞𝑣 𝑛 (3.11)
Where:
0.000516 becomes 0.061S3 when expressed in SI metric values
p=fluid density
q=flow rate
vn=nozzle velocity

The maximum impact pressure is considered to be more suitable measurement for the
hydraulic cleaning ability of various hydraulic conditions, than the impact force is. The
reverse flowing fluid is a function of the ratio of the jet velocity to the return fluid
velocity while the volumetric flow rate through the jets is the same as the return flow
rate [76]. In order to calculate the velocities, it is very important to know the nozzle
cross sectional area and the cross sectional area around the bit. The function below
gives the ration between the two velocities [50, 75, 76]:
𝐴 𝑣 =
𝑉𝑛
𝑉 𝑓
(3.12)
For roller cone bit with three jets it is assumed that, the area available for fluid return
flow is 15% of the total bit area and the previous equation is transformed to:
𝐴 𝑣 =
𝑉𝑛
𝑉 𝑓
=
0.15𝑑 𝑏
2
3𝑑 𝑛
2 (3.13)
Where:
Vn=nozzle velocity
Vf=return fluid velocity
db=bit diameter
dn= nozzle diameter
In the equation (3.10), the impact pressure for the various bit calculate,
𝑝 𝑚 = (1 − 𝐴 𝑉
−0.122)
50
1.238,6𝑠2 𝑝𝑑 𝑛
2
𝑣 𝑛
2
(3.14)
and the impact force when is affected the same as the impact pressure.
𝐹𝑗𝑚 = (1 − 𝐴 𝑉
−0.122
) 𝐹𝑗 (3.15)
The improved ROP model that originates from the equation (3.9) is combined with the
impact force and mud properties in order to account the cutting removal. The
following equation describes the process from cutting generation to cutting removal as
the controlling factor to ROP [50].

𝑅 ≡ (
𝑎𝑆2 𝑑 𝑏
3
𝑁 𝑏 𝑊2 +
𝐶
𝑁𝑑 𝑏
+
𝑐𝑑 𝑏⋎𝑓𝜇
𝐹 𝑗𝑚
)
−1
(3.16)
The first and second term has been analyzed previously; the third fruction includes the
following parameters:
C=dimensional constants
db= bit diameter
⋎ 𝑓=fluid specific gravity
μ=plastic viscosity
Fjm=modified jet impact force
The equation indicates that when the cutting size is increased, an increase from the
impact force is required, to maintain a particular level of cutting removal.
Nevertheless, when the cutting size is huge, the nozzle size used generally becomes
less important. Additional to this, it should be mentioned that hydraulic cleaning can’t
be improved by increasing the fluid density, which increases the impact force [76, 77, 78].
3.3 GALLE AND WOODS BEST CONTANT WEIGHT AND ROTARY SPEED
Galle and Woods study is focused on the best selection effect of weight on the bit and
rotary speed, for lowest drilling cost on the roller cone bits. There is a consideration
which supports that, when the weight on the bit and the rotary speed are constant
during the procedure, the total cost is higher than, when the two previous factors are
varied. According the above consideration, we can distinguish the next categories:
 The best combination of constant weight and rotary speed
 The best constant weight for any given rotary speed
 The best constant rotary speed for any given weight
In the first case, the rig equipment permits the use of any WOB and rotary speed.
When there are limitations from the rig on the rotary speed we apply the second case.
The third case describes situations in which ther is the maximum weight, for example
overstress of drill string.

In order to found the best constant WOB and rotary speed for lowest drilling cost, the
following eight cases were examined for each of three categories individually. [35]
 Case 1 Teeth limit bit life.
 Case 2 Bearings limit bit life.
 Case 3 Bearings and teeth wear out simultaneously.
 Case 4 Drilling rate limits economical bit life.
 Case 5 Drilling rate and bearings limit bit life simultaneously.
 Case 6 Drilling rate and teeth limit bit life simultaneously.
 Case 7 Drilling rate, teeth and bearings limit bit life simultaneously.
 Case 8 Neither drilling rate, nor teeth, nor bearings limit bit life.
3.3.1 THREE FUNDAMENTAL EQUATIONS
Galle and Woods presented graphs for each of three procedures which indicate that
the drilling cost can be minimized using the following fundamental equations, which
are denoted by an identifying seven digit number on each of the graphs.
Drilling rate equation
𝑑𝐹
𝑑𝑡
= 𝐶𝑓
𝑊̅ 𝑘
𝑟
𝑎 𝑝 (3.17)
Where:
F=distance drilled by bit
Cf=formation drillability parameter
W=equivalent bit weight
a=0.928135D2+6.0D+1 (function of dullness)
k=1.0 for most formations and 0.6 for very soft formations
p=0.5 for self-sharpening or chipping-type bit tooth wear
r= rotary speed to a fraction power
This equation gives us detailed information about rate of penetration. We observe that
ROP increases with drillability, weight and rotary speed, while decreases with dullness.

Also, we should have in mind that the drillability parameter (Cf) includes all the effects
of bit types, hydraulics, drilling fluid and formation [35, 80].
Rate of dulling equation
𝑑𝐷
𝑑𝑡
=
1
𝐴 𝑓
𝑖
𝑎𝑚
(3.18)
Where:
D=bit tooth dullness, fraction of original tooth height worn away
Af= formation abrasiveness parameter
a=0.928135D2+6.0D+1 (function of dullness)
i= N+4348 x 10-5 N3
m= 1359.1-714.19log10W
In this equation the abrasiveness constant includes all the effects from factors such as
bit type, hydraulics, drilling fluid and formation. It is clear that the rate of wear
increases as the abrasiveness, weight and rotary speed increase. On the other hand, it
decreases as the dullness is increases [80].
Bearing life equation
𝐵 = 𝑆
𝐿
𝑁
(3.19)
Where:
S= value of drilling fluid
L= tabulated function of W used in bearing life equation
N= rotary speed
It should be noted that, when the weight and rotary speed are increased, the bearing
life is decreased. The only factor which can contribute to the bearing life increases, is
the drilling fluid factor (S) [80].
As discussed before, the use of graph is necessary in order to determine the equation
constant. There are three different sets of graphs each identified by a seven digit
number, for example (2 075 060). The first number indicates the type of tooth wear

obtained on the bit. After this, the first three digit denotes the drilling rate to rotary
speed and the second three digit indicates the drilling rate to weight [35].
3.3.2 ADDITIONAL CALCULATION EQUATION
Total rotating time equation
𝑇𝑅𝑇 = [
𝑆 𝑛∗𝐿
𝑁
] 𝐴 𝑓 When teeth or drilling rate limit bit life (3.20)
𝑇𝑅𝑇 = [
𝑆 𝑛 𝐿
𝑁
] 𝐴 𝑓 When bearings limit bit life (3.21)
Calculation cost per foot
Cost per foot =
𝐾(𝑟𝑖𝑔 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟)
𝐶 𝑓
(3.22)
Calculation of total footage
𝐹𝑓 =
𝐴 𝑛+
𝑆 𝑛∗𝐿
𝑁
𝐾
𝐴 𝑓 𝐶𝑓 When teeth or drilling rate limit bit life (3.23)
𝐹𝑓 =
𝐴 𝑛+
𝑆 𝑛 𝐿
𝑁
𝐾
𝐴 𝑓 𝐶𝑓 When bearings limit bit life (3.24)
3.3.3 CALCULATION OF CONSTANTS
Calculation of formation constants Af and Cf
Af is a constant which measures the abrasiveness and Cf is a constant which measures
the drillability of the formations. Using the values of N, W, D and the Galle and Woods
table, which indicates the correlation between constants, we can find the values of i, r,
m, L, U, V in order to calculate these constants [80].
𝐴 𝑓 =
𝑇 𝑓 𝑖
𝑚̅ 𝑈
(3.25)
𝐶𝑓 =
𝐹 𝑓 𝑖
𝐴 𝑓 𝑟 𝑊̅ 𝑚 𝑉
When using (2 075 100) or (2 043 100) (3.26)

𝐶𝑓 =
𝐹 𝑓 𝑖
𝐴 𝑓 𝑟 𝑊̅ 6 𝑚 𝑉
When using (2 075 060) (3.27)
𝑊̅ =
788𝑊
𝐻
(3.28)
H=hole or bit diameter
W= bit weight
Ff= final distance drill by bit
Tf= final rotating time
Calculation of drilling fluid constant (S)
High value of S means very good drilling fluids and the opposite low value of S entail
bad quality of drilling fluid. The determination of this factor is necessary in order to
calculate the bearing life. The bearing life is affected from S, weight and rotary speed.
The following equation gives the drilling fluid value: [35, 80]
𝑆 =
𝑇 𝑓 𝑁
𝐵 𝑥𝑓 𝐿
(3.29)
Calculation An and Sn
𝐴 =
𝑏𝑖𝑡 𝑐𝑜𝑠𝑡
𝑟𝑖𝑔 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟
+ 𝑡𝑟𝑖𝑝 𝑡𝑖𝑚𝑒, ℎ𝑟 (3.30)
𝐴 𝑛 =
𝐴
𝐴 𝑓
(3.31)
𝑆 𝑛 =
𝑆
𝐴 𝑓
(3.32)
Sn*=∫
𝑁𝑚𝑎
𝐿𝑖
𝐷𝑓
0
𝑑𝐷 (3.33)
Special attention is required when we want to determine the value of the constant,
because we should read the corresponding set of graph that applies in the right case
[35, 79].

3.4 BOURGOYNE AND YOUNGS MULTIPLE REGRESSION MODEL
Bourgoyne and Youngs’ model is the most widespread model in the industry while it is
considered to be one of the most complete mathematical drilling models for the
penetration rate prediction. It is a linear penetration model, which is consisted from
controllable and uncontrollable drilling variables. The following formula is considered
as a general linear rate of penetration equation for roller cone bit [8].
𝑑𝐹
𝑑𝑡
= 𝑒{𝑎1+∑ 𝑎𝑗𝑥𝑗8
𝑗=2 }
(3.34)
The constants can be determined by a multiple regression analysis of field data which
are caused from the formation strength effect, compaction effect, differential pressure
effect, bit diameter and bit weight effect, rotary speed effect, tooth wear effect and
bit hydraulic effect [81].
A multiple regression model is configured based on controllable variables in the
general ROP equation, such as bit weight and rotary speed, whoms function influence
the other uncontrollable data from regression cycle. The past drilling data from other
wells is essential condition in order to determined the constants which given in the
previous equation.
A complete description of the controllable and uncontrollable drilling variables is given
from the following equation and is represented by the following figure. [8, 81, 82]
𝑑𝐹
𝑑𝑡
= 𝐸𝑥𝑝 {𝑎1 + 𝑎2(8000 − 𝐷) + 𝑎3 𝐷0.69
(𝑔 𝑝 − 9) + 𝑎4 𝐷(𝑔 𝑝 − 𝑝𝑐) +
𝑎5 𝐿𝑛 {
𝑤
𝑑 𝑏
−0.02
4−0.02
} + 𝑎6 𝐿𝑛 (
𝑁
60
) + 𝑎7(−ℎ) + 𝑎8
𝑝𝑞
350𝜇𝑑 𝑛
} (3.35)

Figure 3.3 General rate of penetration equation
3.4.1 FORMATION STRENGTH FUNCTION
The constant a1 represents the effect of formation strength. This means that the value
of the constant is proportionate with ROP. For very low value of this constant, we
have low penetration rate and vise versa. Also, the formation strength function
includes the effect of other drilling parameters, such as drilling cuttings, which have
not been modeled mathematically. Additional factors could be introduced as new
function, influencing the general ROP equation. The following term of the general
equation indicates the drillability of the formation which is the same with ROP as
exponential function [81-82].
𝑓1 = 𝑒 𝑎1 (3.36)

3.4.2 FORMATION COMPACTION FUNCTION
The two next terms of the general equation (a2, a3) describe the compaction effect.
There are two different compaction effects, which are depended from the formation
properties. It is very common that the rock strength is increased as the depth is
getting larger. In this case, the formation compaction effect is called normal
compaction while it is observed an exponential decrease in penetration rate with
increasing depth [8].
𝑓2 = 𝑒 𝑎2 𝑋2 = 𝑒 𝑎2(8000−𝐷)
(3.37)
The second term of the formation compaction function, which is defined as (a3),
describes the under compaction effect. This effect is conducted when we have
abnormally pressured formations, where the rate of penetration shows an increasing
behavior to the depth. Therefore, the equation of this function indicates an
exponential increase in penetration rate because the pore pressure gradient is higher
[8].
𝑓3 = 𝑒 𝑎3 𝑋3 = 𝑒 𝑎3 𝐷0.69(𝑔 𝑝−9)
(3.38)
3.4.3 DIFFERENTIAL PRESSURE FUNCTION
The pressure differential factor is considered to be an inhibiting factor, because the
penetration rate is reduced when there is a pressure difference. The term which
includes the differential pressure is defined as (a4) and it indicates an exponential
decrease in ROP when it excesses the bottom hole’s pressure. In other words, when
the pressure between the bottom hole and the formation is zero, the effect of this
function is equal to 1 [82].
𝑓3 = 𝑒 𝑎4 𝑋4 = 𝑒 𝑎4 𝐷(𝑔 𝑝−𝑔 𝑐)
(3.39)

3.4.4 BIT DIAMETER AND WEIGHT FUNCTION
The term a5 determines the function of bit diameter and weight, since it is the term
that has direct effect on penetration rate. The exponential term is normalized to equal
1.0 for 4000lb pert inch of bit diameter which is the requirement tooth force in order
to the fracture begins. This force is called threshold force [8, 81].
𝑓5 = 𝑒 𝑎5 𝑥5 = 𝑒
𝑎5 𝐿𝑛{
𝑤
𝑑 𝑏
−0.02
4−0.02
}
(3.40)
3.4.5 ROTARY SPEED FUNCTION
The next term of the general equation (a6) represents the effect of rotary speed. The
relation is similar with the weight function while the term ea6x6 is normalized to be
equal to 1.0 for 100 rpm. Also the rotary speed reported value is ranging from 0.4 for
very hard formation to 0.9 for very soft formation [8].
𝑓6 = 𝑒 𝑎6 𝑥6 = 𝑒
𝑎6 𝐿𝑛(
𝑁
60
)
(3.41)
3.4.6 TOOTH WEAR FUNCTION
The function for the tooth wear is defined by coefficient (a7). The tooth wear function
is usually expressed as a fraction of tooth height (h) of an inch. The value of this
function depends on the bit type and the formation type. The following tooth wear
exponent equation intimates that this functions equals to 1 when the h or a7 is zero [8,
81].
𝑓7 = 𝑒 𝑎7 𝑥7 = 𝑒 𝑎7(−ℎ)
(3.42)
3.4.7 HYDRAULIC FUNCTION
The function for the hydraulic effect is defined by coefficient (a8). The effect of bit
hydraulics is based on microbit experiments performed by Eckel, who found that ROP

was proportional to a Reynolds number group raised to the 0.5 power while μ is
defined as the apparent viscosity measured at 10,000 seconds -1 [81].
𝑓8 = 𝑒 𝑎8 𝑥8 = 𝑒
𝑎8
𝑝𝑞
350𝜇𝑑𝑛 (3.43)
At this point, it should be mentioned that Bourgoyne and Youngs’ method is
considered to be the most suitable method for real time drilling optimization since it is
based on evaluation of the past drilling parameters from many wells that are
introduced continuously in linear penetration equation in order to conduct the multiple
regression analysis. The basic advantage of multiple regressions method is that it has
the capacity to estimate the rate of penetration as a function of independent drilling
parameters. In other words, the controllable variables monitored -in respect to the
depth and any deflection from the initial model due to uncontrollable variables, such
as formation characteristics- are taken into consideration in order to achieve the
regression.

4. CHAPTER 4
ANDANCED OPTIMIZATION METHODS
Many advanced optimization practices have been performed the last years from oil
and gas industries in order to minimize the drilling problems and to achieve the
reduction of drilling cost. Nowadays, all the advanced optimization methods use a
database which is based in real time information from different drilling sites. The
objective of this chapter is to present the way which drilling data are collected and
shipped and to analyze some of the modern ROP optimization models.
4.1 REAL TIME DATA
After 1980, emerged the need from petroleum industries for using innovative
systems and tools that measure petrophysical properties while drilling and monitor in
real time critical downhole parameters, which influence the penetration rate. As a
result of this necessity, the following years new advanced logging tools (LWD) which
come to cover the needs replace the wireline logging. Real time technology centers
started to be established in order to evaluate, manage, analyze and share the
informations [83-84].
4.1.1 MEASURE WHILE DRILLING PIPING
Measure While Drilling tools provide a volume of informations about the drilling
function which are very useful for the real time engineers in order to optimize the
drilling conditions. At this stage, it is very important to examine the way on which
these informations transfer from the downhole to the surface of the rig and after this
to real time technology centers. There are three ways to transmit data from the
downhole to the surface and will be analyzed below [93].

CHAPTER 4: ADVANCE OPTIMIZATION METHODS
Mud pulse telemetry (MPT)
This is a tool which transmits informations from downhole to the surface, introduced
in the industries in the early 60’s. Valves that exist in the BHA, regulate the flow of
the mud in order to produce pressure fluctuations that correspond to the transmitted
information. The pulses are propagated within the mud inside the drillstring towards
the surface where computers decode them into binary bits. The following represent
the MPT process [85-87].
Figure 4.1 Simplified MPT system description
Electromagnetic telemetry (EM)
There is a potential to have underbalanced drilling or extreme lost circulation
conditions. In this case, the mud pulse telemetry -as a way for data transmission-
could not have application, so electromagnetic telemetry is used as an alternative
solution. In practice, in an electromagnetic system the drillstring is used as an
antenna to transmit signal to the surface. This method uses low-frequency
electromagnetic waves that are transmitted through the formation, transferring the
encoded data [88-89].

Wired drill pipes
This system has higher cost from the previous telemetry systems. This technology
(WDP) is considered as the fastest way to send data to the surface. In this method,
the measurements are transmitted to the surface through electrical wires that are
well housed inside every single pipe of the drillstring [90]. Also this system contributes
to evaluation of kick, because it has pressure sensors along the string that divide the
annulus into control volumes. Using this volume technique the gas kick can be
estimated [91-92].
Figure 4.2 MPD setup with wired drill pipe and the resulting control volumes

4.1.2 REAL TIME TECHNICAL CENTERS
Drilling reports are typically created and transferred from the rig site on a daily basis.
The main operational data that are critical for real-time centers, which are disclosed
through daily reports, are:
 Mud rheology
 Drilling activity details
 Bottom hole assembly components
 Bit type and configuration
 Crew information
 Well risks
The state of the art is the connection in real time data directly with the required
report information in order to facilitate the drilling process [94]. When the drilling
measurements reach in real time centers, the data are stored and the optimization
process starts. The new advanced softwares are valuable tools for the real time
engineers, because monitoring and visualizing the data, provides them the ability to
reduce the project risk, which arises from the time that it is spended by comparing
the real-time data with the operations data. The software matches historical data
from other wells, so the users can correlate previous data as reference for current
well. When data are analyzed, regression coefficients should be determined in order
to be used in calculating the predicted rate of penetration and find the optimum
parameters [93, 95].
4.2 REAL TIME BIT WEAR
Having analyzed the real time data transfer mechanism, it is very important to study
the way on which these advanced tools contribute in ROP prediction. This module
presents a new method to combine the mechanical specific energy (MSE) and ROP
model, in order to calculate real time bit wear. The mechanical specific energy
method is defined as the work needed to destroy a given volume of the rock. On the
other hand, the rate of penetration model is used in order to predict the drilling
process measurements, such as formation drillability calculating, the effect of drilling

parameters, bit design and bit wear. The combination of these two methods can
optimize the drilling operation, showing the bit wear status during drilling, which is a
determining factor to take decisions. Bourgoyne and Youngs’ ROP prediction model
has been extensively analyzed in the previous chapter [96-97].
Bourgoyne and Youngs ROP model
We remind that this model is a linear penetration model, which includes controllables
and uncontrollables drilling variables, such as the effect of formation strength, the
compaction, the differential pressure, the bit diameter and bit weight, the rotary
speed, the tooth wear and the bit hydraulic effect. The model has been
mathematically expressed as: [8, 81]
𝑅𝑂𝑃 = 𝑓1 × 𝑓2 × 𝑓3 × 𝑓4 × 𝑓5 × 𝑓6 × 𝑓7 × 𝑓8 (4.1)
The above equation indicates that the rate of penetration is determined from eight
functions, which can be inverted in order to occure the formation drillability (f1).
𝑓1 =
𝑅𝑂𝑃
𝑓2×𝑓3×𝑓4×𝑓5×𝑓6×𝑓7×𝑓8
(4.2)
Mechanical specific energy (MSE)
Using the mechanical specific energy could optimize the drilling parameters, since it
gives the ability to monitor the process and detect changes in drilling efficiency. The
MSE is measured as input energy, which is required to destroy a given volume of the
rock to the ROP. Taking into consideration this assumption, the following MSE
equation can be expressed as: [96]
𝑀𝑆𝐸 =
𝑊𝑂𝐵
𝐴 𝐵
+
120𝜋×𝑁×𝑇
𝐴 𝐵×𝑅𝑂𝑃
(4.3)
Where:
AB=bit surface area
N=rotary speed
T= measured torque

The torque can be determined as a measurement while drilling, but, in many cases
the torque is expressed as a function of the weight on the bit and bit sliding friction
coefficient from the following formula:
𝑇 = 𝜇
𝐷 𝐵×𝑊𝑂𝐵
36
(4.4)
In order to be calculated, a bit sliding friction coefficient which is a constant with the
same value both the roller cone and the PDC bit, it is necessary to conduct
laboratory measurement using torque and WOB. From the previous equations (4.2)
and (4.3) results the modified formula [97]:
𝑀𝑆𝐸 𝑀𝑂𝐷 = 𝑊𝑂𝐵 (
1
𝐴 𝐵
+
13.33×𝜇×𝑁
𝐷 𝐵×𝑅𝑂𝑃
) (4.5)
Real time bit wear developed model
If we combine the formation drillability and MSE, we have the following relationship:
𝑀𝑆𝐸 = 𝐾1 × (
1
𝑓1
)
𝐾2
(4.6)
Fractional bit wear is simplified and it is considered as a linear decreasing trend vs
depth, using the following equation [98]:
ℎ =
(𝐷𝐸𝑃𝑇𝐻 𝐶𝑈𝑅𝑅𝐸𝑁𝑇−𝐷𝐸𝑃𝑇𝐻 𝐼𝑁)
(𝐷𝐸𝑃𝑇𝐻 𝑂𝑈𝑇−𝐷𝐸𝑃𝑇𝐻 𝐼𝑁)
×
𝐷𝐺
8
(4.7)
Where:
DG= reported bit wear dullness
Figure 4.3 Schematic shows how PDC and roller cone bit types cutters have measured

The bit wear value starts at the point of T1, at the beginning of each bit run and is
decreasing throughout the bit run. The model used to estimate bit wear is based on
the approach developed by Rashidi [98] which included rock confined compressive
strength (CCS). General form of the equation is showed below:
𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 = (
1
𝐾1
) = 1 − ℎ 𝐵
(4.8)
Where:
B= is a constant
K1= is the ratio of the MSE to the inverse rock drillability for each meter of the
drilled wells
The MSE, which incorporates the effect of bit wear, can be used in combination with
the CCS from ROP models, to back out real time fraction bit wear. Bit wear fraction
can be obtained using the following equation for the roller cone and the PDC bits: [98]
𝑊𝑓 = 1 − 𝑎 = (
𝛥𝐵𝐺
8
)
𝑏
(4.9)
Where:
ΔBG=8*h when fractional scale of bit grading from 0 to 8
This developed model is the basis of the creation of software that receives the data
from an online server and estimates real time bit wear. It should be mentioned that
the constant K1 are calculated manually for each bit run in order to have better bit
wear trend. This software improves the drilling operation while it has been observed
that achieves better match between calculating and reporting bit wear out value. As
a result, we can say that, using bit wear software, can minimize drilling cost by
reducing tripping time [95, 97].
4.3 ROP PREDICTION USING FUZZY K MEANS
It is perceived that ROP prediction is a complex phenomenon, because it depends
from many factors. Bourgoyne and Youngs ROP model which has been analyzed in
the previous chapter, it is used from the most petroleum industries during last

decades. However, it is not considered enough accurate method because it computes
coefficients using multiple regression analyses, which may have negative or zero
values that are completely absurd. For example, if the weight on the bit coefficient is
negative, it illustrates that increasing the WOB will decrease the penetration rate.
The main purpose of this section is to present a new method based on fuzzy K-
mean clustering -a computer simulation method- that predict the drilling rate
accurately. As mentioned before, there are uncertain parameters which influence
ROP. However, this simulation system receives the main variables, considering the
ROP as a nonlinear function g(x) with eight following inputs: true vertical depth (D),
weight on bit (W), bit diameter (db), rotary speed (N), pore pressure gradient (gp),
equivalent mud density (ρc), fractional bit tooth wear (h), jet impact force (Fj). For
every Fuzzy simulated annealing (SA) in real continuous functions g(x), there is a
fuzzy system f(x) such that [98]:
𝑠𝑢𝑝| 𝑓( 𝑥) − 𝑔( 𝑥)| < 𝜀 (4.10)
Fuzzy simulated annealing algorithm provides an estimator f(x) to approximate g(x)
while predict undetermined parameters with minimum error.
Figure 4.4 Framework of prediction procedure

The figure (4.4) indicates the ROP prediction procedure which can be distinguished in
the following three steps [98-99]:
Clustering the training data
The first step maybe is the most significant. The input data should separate by K-
mean clustering approach in eight different clusters. In order to achieve this
classification, rules should be set. The number of rules is equal to the number of
clusters. The basic idea is to create a group of input-output clusters and use one rule
for each cluster. At this point, it is required special attention because it has been
observed that a large number of rules can be producing a complex fuzzy system. On
the other hand, a few rules create a less powerful system.
Setting up a typical Fuzzy system
Each group of input data or cluster is accompanied from a membership function.
Using the simulated annealing (SA) this function is optimized. Example for a common
cluster rule is indicated as follows:
if x1 is A’1 and x2 is A’2……….xi is A’i Then y is B’
Where A’i and B’ are mean and standard deviation of Gaussian with the
following membership grade:
ℎ𝑖
𝑙
( 𝑥𝑖) = 𝑒𝑥𝑝 [−
1
2
(
𝑥 𝑖−𝑐𝑖
𝑙
𝜎𝑖
𝑙 )
2
] (4.11)
Where 𝑐𝑖
𝑙
and 𝜎𝑖
𝑙
are mean and the standard deviation of Gaussian
membership function for I input variable of i Fuzzy rule.
The simulated annealing (SA) is used in order to determine these two parameters of
all membership function of the Fuzzy system and find better estimator f(x).
Determining the parameters of Fuzzy system using SA
We can observe 6 steps into the required estimator f(x) process:
 Initializing the parameters of SA (initial temperature, cooling coefficient,
searching time, termination condition)

 Set a current solution X for above variables. MSE value of X can obtain by the
simplified Fuzzy inference system.
 Randomly search for a neighbor solution set X’, which equals to X augments.
ΔE = MSE(X’) − MSE(X ). If D≤ 0, then the current solution set will be replaced
by the neighbor solution set; otherwise (when ΔE > 0), the winning probability of
the neighbor solution set is F (X’) =exp (-ΔE/T)
 Compare X with optimal solution X’ and if X is better replaced with X’
 If the maximum searching time is not achieved; go back to step 2
 Check the termination condition is reached. If yes the algorithm has finished if
no return the step 3 until the termination condition is fulfilled
After many trials has been observed that new computing approach predicts the
penetration rate with more acceptable accuracy than a conventional method such as
Bourgoyne and Youngs prediction model. Using the root mean square error (MSE)
and standard deviation (SD) of ROP, we have more accurate results [98].
4.4 ROP PREDICTION TECHNOLOGIES BAZED ON NEURAL NETWORK
In the previous section we presented a new simulated method which is based on
Fuzzy K- mean clustering. At this section, we analyze another advanced simulated
method based on the artificial neural network (ANN) technologies, which predict the
penetration rate using MATLAB function codes. The ANN uses the previous data from
offset wells and runs to find the expected ROP, including any change of drilling
conditions as input.
4.4.1 UNDERSTANT AND LEARN CONCEPT
The ANN process requires comprehensive collecting data as input in order the system
to analyze the relationship between input and output. The system provides two
outputs, which are compared continuously until the errors are reduced and the
desired outputs become reasonable close. However, the system is very flexible since
it does not have a static formula that requires full set of data but finds the correlation

between input and output to extrapolate the missing data. In addition, it does not
need reprogramming every time the function is changed but it is required a correct
evaluation of input if we want to build a correct model of the offset wells. The below
figure illustrates the schematic process [101-102].
Figure 4.5 Artificial neural network
4.4.2 DRILLING OPTIMIZATION ANN
The first step in applying ANN system is to build a model of the offset wells using
formation analysis software.
The second stage is to establish the correlation (link) between drilling variables and
the results. A multiple neural network (MNN) consists from three different types of
layer: input layers, output layers and many of hidden layers. Input layers collect data
from databases, the hidden layers develop and analyze the relation between input
and output; the output layers produce the results. As is shown in the next figure
(4.6) every neuron of a layer is connected to each neuron of the next layer [101,103].
Every neuron of input represents a parameter which is received from the network as
input. On the other hand, neurons of hidden indicate the extraction output. The
number of hidden layers and neurons is unlimited while the relations between input
parameters are immeasurable. Each connected link has an associated weight which
is transmitted to a signal. This signal transfer is conducted through neurons over the
connecting links [100-101].

Figure 4.6 Drilling optimization ANN
In the next stage the simulator compares the outputs with the desired outputs.
Certainly, the first outputs will show huge errors, because the weight is calculated
with random way. The error signal transmitted back from the outputs layer to the
intermediate layer; the process is repeated layer by layer. The weight’s update is
based on the error signal until the outputs present the closest value to the desired
outputs value [101-102].
4.4.3 ELM AND RBF TECHNOLOGIES
The extreme learning machines (ELM) and radial basis function network (RBF) are
contained in artificial neural network techniques. Both ELM and RBF are single hidden
layer feedforward networks (SLFN) which use MATLAB function codes in order to find
the best results. It is interesting to examine the simulator outputs for these two
methods, in order to have a comparison between them in terms of accuracy and
processing speed. The following four terms, training time, training accuracy, testing
time and testing accuracy, validate a detailed comparison [103].

The ELM techniques training time are not influenced from small changes in the
number of hidden layers while from the five functions, one spends the most time.
Generally, ELM techniques are considered to be quicker because it is required less
training time than RBF techniques. The RBF are not affected by the speed
parameters but has the same training time at the values of MSE used.
Root mean square error (RMSE) and standard deviation (SD) is used in order to
ascertain the training accuracy of ELM which gives more accurate results comparing
with RBF. The ELM accuracy gets when the number of hidden layer is increased but
the RBF accuracy is set to be two MSE values.
ELM testing time is random and not affected by the number of hidden neurons. On
the other hand, RBF testing time is not affected by the choice of goal training
accuracy. However, the RBF testing time is higher than ELM.
Testing accuracies are compared in terms of (RMSE), of (SD) and of absolute percent
relative error. RBF testing is not accurate when training target MSE is chosen low and
very good when it is chosen close to ELM training accuracy.
The conclusion is that the ELM techniques give more accurate result in processing
time. On the other hand, RBF techniques are considered as more accurate methods
for ROP prediction, but if the speed is very important the ELM is more suitable for
use.

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