Bourgoyne and Young model (BYM) has been the dominant method for drilling rate prediction. It demonstrates a relation between drilling rate and the parameters affecting it. There are eight variables influencing the drilling rate, and they depend on ground formation type and must be determined based on the data gathered in advance. Bourgoyne and Young proposed multiple regression method for determining the unknown coefficients, albeit, this method has the shortcoming that may lead to an outcome, which doesn’t make sense physically in some circumstances. To dissolve this flaw, some new mathematical methods have been introduced, but utilizing these methods will confront us with a loss in the accuracy of drilling rate prediction. Our proposed method solves the two above mentioned deficiencies, physically meaningless coefficients, and the decrease in accuracy. In our method, we have employed Genetic algorithm (GA) to determine the unknown parameters of BYM. Our practical data sets were nine wells of “Khangiran” Iranian gas field. Simulation results do prove the efficiency of our new method for determining constant coefficients of Bourgoyne and Young model over the previous ones.

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Drilling Rate Prediction Using Bourgoyne and Young Model Associated with
Genetic Algorithm
Conference Paper · January 2009
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2. Drilling Rate Prediction Using Bourgoyne and Young
Model Associated with Genetic Algorithm
M. H. Bahari A. Bahari F. Nejati. M R. Rajaei B. Vosoughi-V
Electrical Engineering
Department,
Ferdowsi University of
Mashhad
+985117281586
m_h_bahari@yah
oo.com
National Iranian Oil
Company (NIOC),
Sarakhs,
Iran
+985117281586
aboozarbahari
@spe.org
Electrical Engineering
Department,
Ferdowsi University of
Mashhad
+985117281315
farzadn2i@yaho
o.com
Electrical
Engineering
Department,
Sharif University of
Technology
+982166162707
ramin_rajaei@y
ahoo.com
Electrical
Engineering
Department,
Sharif University of
Technology
+982166165001
vahdat@shari.ir
ABSTRACT
Bourgoyne and Young model (BYM) has been the dominant
method for drilling rate prediction. It demonstrates a relation
between drilling rate and the parameters affecting it. There are
eight variables influencing the drilling rate, and they depend on
ground formation type and must be determined based on the data
gathered in advance. Bourgoyne and Young proposed multiple
regression method for determining the unknown coefficients,
albeit, this method has the shortcoming that may lead to an
outcome, which doesn’t make sense physically in some
circumstances. To dissolve this flaw, some new mathematical
methods have been introduced, but utilizing these methods will
confront us with a loss in the accuracy of drilling rate prediction.
Our proposed method solves the two above mentioned
deficiencies, physically meaningless coefficients, and the decrease
in accuracy. In our method, we have employed Genetic algorithm
(GA) to determine the unknown parameters of BYM. Our
practical data sets were nine wells of “Khangiran” Iranian gas
field. Simulation results do prove the efficiency of our new
method for determining constant coefficients of Bourgoyne and
Young model over the previous ones.
Keywords
Drilling Rate Prediction, Bourgoyne and Young Model (BYM),
Genetic Algorithm (GA)
1. INTRODUCTION
Drilling engineers have been concerned about drilling rate
prediction significantly during last decades because it leads to
optimum drilling parameters selection, which is very important in
optimizing drilling cost per foot [2], [1].
Rate of penetration is affected by many parameters. For instance
hydraulics, weight on bit, rotary speed, bit type, mud properties,
formation characteristics, etc play very important role in drilling
rate [3]. Unfortunately, there exists no explicit mathematical
relationship between drilling rate and deferent drilling variables.
We can find the origins of this problem in large number of
uncertain drilling variables influencing drilling rate. Furthermore,
their relationship to each other and to drilling rate is nonlinear and
complex [4]. However, experts have put forward some
suggestions to address this issue. They have introduced simplified
models, which map important variables affecting drilling rate on
drilling rate. One of those models is BYM, which widely used in
practice [5]. In this model, there are some unknown parameters or
coefficients, which must be determined based on previous drilling
experience in the field. It is taken for granted that the method of
determining these coefficients has significant impact on the model
accuracy.
Creators of BYM suggested multiple regression method to
determine unknown coefficients [5]. However, applying multiple
regression method does not guarantee reaching physically
meaningful result. Additionally, this method is limited to the
number of data points.
To reach meaningful results, some new mathematical methods
have recently been applied to calculate these unknown
parameters. For instance, Non-linear least square data fitting with
trust-region method is a mathematical technique, which is applied
to this problem recently [6]. This method is one of the
optimization algorithms, which minimizes the sum of square
errors function. The method is based on the interior-reflective
Newton method. In each of iterations, the approximate solution of
a large linear system is estimated using the method of
preconditioned conjugate gradients (PCG) [7], [8]. This
technique makes it possible to determine lower and upper bounds
for results and limit them to be in the reasonable ranges [7].
However, computed coefficients using this scheme do not result in
sufficiently accurate models in practice.
During the last two decades, evolutionary algorithms such as GA
have been applied to many optimization problems. For example,
Justus Rabi applied GA to minimize harmonics in PWM inverters
or Ikramullah Butt [9] and Hou-Fang used it in scheduling
flexible job shops [10]. In this research, we applied GA to
determine the optimal values of BYM unknown parameters. Since
GA is able to handle linear constraints and bounds, it guarantees
to reach physically meaningful result. Furthermore, using GA
leads to a more accurate model in comparison with trust-region
method.

3. The rest of the paper is organized in the following manner. First,
we commence with the debut of Bourgoyne and Young drilling
rate model. Then, Khangiran Iranian gas field is introduced. Next,
application of GA in Bourgoyne and Young model coefficients
determination is elaborated. And we conclude with presenting the
simulation results on Khangiran Iranian gas field, and comparing
them with trust-region method.
2. BOURGOYNE AND YOUNG MODEL
Bourgoyne and Young have proposed the following equation to
model the drilling process when using roller cone bits (1):
87654321 ffffffffRop  (1)
Where, Rop is rate of penetration (ft/hr). The function 1f
represents the effect of formation strength, bit type, mud type, and
solid content, which are not included in the drilling model. This
term is expressed in the same unit as penetration rate and is often
called the formation drill ability. The functions 2f and
3f symbolize the effect of compaction on penetration rate. The
function 4f signifies the effect of overbalance on penetration
rate. The functions 5f and 6f respectively model the effect of
bit weight and rotary speed on penetration rate. The function 7f
represents the effects of tooth wear and the function 8f
characterizes the effect of bit hydraulics on penetration rate [2].
The functional relations in equation (1) are as follows.
8
1000
8
a
jF
f 







 (9)
Where,
1a to 8a are BYM constant coefficients.
D = True vertical depth (ft)
db = Bit diameter (in)
Fj = Jet impact force (lbf)
gp = Pore Pressure gradient (lbm/gal)
h = Fractional bit tooth wear
ρc = Equivalent mud density (lbm/gal)
N = Rotary speed (rpm)
W = Weight on Bit (1000 lbf)
(W/db)t = Threshold bit weight per inch of bit diameter at
which the bit begins to drill.
As mentioned 1a to 8a be dependent to local drilling conditions
and must be determined for each formation using prior drilling
data sets obtained from the drilling area [2]. Bourgoyne and
Young Recommended specific bounds for each of eight
coefficients based on reported ranges for the coefficients from
various formations in different areas [2], [5], and average values
of them. Lower and upper bounds to achieve meaningful results
have been suggested as shown in the Table 1. Using these bounds
increases the reliability of the achieved predictor system.
Table 1. Bourgoyne and Young Recommended bounds for
each coefficient
Coefficients Lower bound Upper bound
A1 0.5 1.9
A2 0.000001 0.0005
A3 0.000001 0.0009
A4 0.000001 0.0001
A5 0.5 2
A6 0.4 1
A7 0.3 1.5
A8 0.3 0.6
Bourgoyne and Young employ multiple regression method to
determine unknown coefficients. But, this scheme provides results
out of recommended bounds in some situations. To be more
precise, multiple regression method may result in negative or zero
values. It is taken for granted that negative or zero values for
coefficients are physically meaningless. For instance, if the weight
on bit constant (a5) is a negative value, it illustrates that increasing
the weight on bit leads to reduce the penetration rate or a zero
value implies that increasing the weight on bit has no effect on the
drilling rate. Therefore, it is needed to apply new methods to gain
an applicable predictor system.
Kef a
 1303.2
1
(2)
)10000(303.2
2
2 Da
ef 
 (3)
)9(303.2
3
69.0
3 
 pgDa
ef (4)
)(303.2
4
4 cpgDa
ef

 (5)
5
4
5
a
tb
tbb
d
W
d
W
d
W
f




























 (6)
6
60
6
a
N
f 





 (7)
ha
ef 7
7

 (8)

4. 3. KHANGIRAN GAS FIELD
Khangiran gas field is located in the northeast of Iran. This field
was surveyed in 1937. In 1956, the stratigraphy plan was prepared
and it was named in 1962. Fig. 1 indicates the stratigraphy
column and geological description of each formation for a typical
well in this field.
Khangiran field includes three gas reservoirs:
 Mozdouran: The existence of sour gas in this reservoir was
proved in 1968 and the production was started in 1983. It
consists of thick layer limestone. Up to now, 37 wells have
been drilled.
 Shourijeh B: This reservoir was explored in 1968 and
production was started in 1974. Shourigeh formation is
mainly formed from sandstone layers. So far, seven wells
have been drilled and completed in the reservoir. The gas
from this reservoir is sweet and H2S free.
Shourijeh D: This reservoir was explored in 1987 and after
drilling the well, production was started in the same year. Seven
wells have been drilled up to now. The gas from this reservoir is
sweet, too.
Figure 1. Stratigraphy column of a typical well in Khangiran
field and description of its formations.
4. DETRMINIG BOUGOYNE AND YOUNG
CONSTANT COEFFICIENTS USING GA
As mentioned, we employed GA to determine optimal value for
constant parameters of BYM. Since GA handles bound
constraints, using it guarantees to find optimum values of
coefficients in recommended bounds (not out of bounds).
Therefore, GA not only provides meaningful result but also is not
limited to the number of data points. Fig. 2 illustrates architecture
of the predictor system.
To find constant parameters of the aforementioned model for each
formation, the following procedure was performed.
I. For each formation in Khangiran field, the daily drilling
progress reports of 10 drilled wells (from the surface to the
final reservoir depth) in this field were gathered initially. After
the data quality control, nine wells having more accurate data
were opted.
II. We constructed a database from available data of nine wells.
The database includes quantities of D, W, N, gp, ρc, h, Fj and
achieved Rop in each formation. It must be noted that the
fractional tooth wear (h) is expressed just at the end of bit
running. Therefore, only drilling data at ending the bit run can
be used. Table 2 provides a sample of the required data, which
is included in our database.
III. In each formation, by applying inputs (D, W, N, gp, ρc, h and
Fj) and output (Rop) to the above-mentioned model, we use
GA to find out optimum values of eight unknown coefficients.
GA was run in the following steps.
1. Set the initial parameters for GA: population size,
crossover type and probability, and mutation
probability.
2. Set all bounds recommended by Bourgoyne and
Young for each of eight parameters particularly.
3. Generate the initial population randomly.
4. Reckoning of a fitness value for each subject. The
considered fitness function is Standard Deviation of
distances between real Rop and estimated Rop by
predictor system.
5. Selection of the subjects that will mate according to
their share in the population global fitness.
6. Apply the generic operators (crossover, mutation…).
Repeat Steps 3 to 6 until the generation number is
reached.
5. SIMULATION RESULTS
As mentioned, constants a1 to a8 were computed for each of
Khangiran field formations utilizing GA. Table 3 shows the
results obtained, using multiple regression method, trust-region
method, and proposed scheme for five formations of Khangiran
field. As is rendered from the table, when the multiple regression
method is applied, the resulting coefficients may be negative or
zero (which is physically meaningless). While, computed
coefficients gained by employing trust-region method and GA are
all physically meaningful and in recommended bounds.
Table 4 indicates estimation accuracy of Trust-Region method
and our proposed scheme. In this table Standard Deviation (STD)
error of drilling rate estimation by these two methods is
Khangiran
Chehelkaman
Pestehligh
Kalat
Neyzar
Abtalkh
Abderaz
Aytamir
Sanganeh
Sarcheshmeh
Tirgan
Shourijeh
Mozdouran
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
11000
Formations
Depth,ft
Silty shale and silty claystone
Description
Dolomite and anhydrite to sandy limestone
Micritic to crystaline limestone and dolomite
Silty shale and sandstone
Calcareous sandstone, siltstone and shale
Silty shale and marl
Micritic limestone, silty calcareous shale
Glaconitic sandstone
Calcareous shale, claystone and silty shale
Argilaceous micritic limestone, some shale
Oolitic argilaceous micritic limestone, marl, shale
Oolitic sandstone and Calcareous silty shale
Argilaceous limestone and oolitic dolomite
limestone

5. illustrated. It can be interpreted that the proposed scheme is more
efficient than Trust-region method in determining coefficients of
BYM, which leads to more accuracy in drilling rate prediction.
Table 4. Estimation accuracy of the proposed scheme in
comparison with Trust-region method
Figure 2. Architecture of predictor system.
6. CONCOLUTION
For drilling cost optimization, one important issue, which is
aimed at, is accurate drilling rate prediction. A simplified model
of drilling is called Bourgoyne and Young, which defines a
general mapping between drilling rate and some drilling variables.
The model is widely used in drilling rate prediction. However,
there are eight unknown parameters in this model, which must be
elicited from the previously gathered data. Albeit several methods
have been put forward to determine these coefficients in the last
decades, it is hard to reach a predictor system with satisfactory
accuracy. In this paper, we applied GA to determine constant
coefficient of Bourgoyne and Young model. Simulation results
confirm that suggested approach not only provides meaningful
results but also leads to more accuracy in comparison with
conventional methods.
7. ACKNOWLEDGMENT
The authors would like to thank the drilling staff of Iranian
Central Oil Fields Company (I.C.O.F.C) for their contribution and
cooperation in this research.
8. REFRENCES
[1] Kaiser, M. J., A Survey of Drilling Cost and Complexity
Estimation Models, International Journal of Petroleum
Science and Technology, vol. 1, pp. 1–22, 2007.
[2] Bourgoyne, A. T., Millheim, K. K., Chenevert. M. E and
Young, F. S., Applied Drilling Engineering. 9th ed., SPE,
Richardson, TX, p. 232.
[3] Akgun, F., Drilling Rate at the Technical Limit,
International Journal of Petroleum Science and Technology,
vol. 1, pp. 99-118, 2007.
[4] Ricardo, J., Mendes, P., Fonseca, T. C. and Serapiao. A. B.
S., Applying a genetic neuro-model reference adaptive
controller in drilling optimization, World Oil Magazine, p
228, 2007.
[5] Bourgoyne, A. T. and Young. F. S., A Multiple Regression
Approach to Optimal Drilling and Abnormal Pressure
Detection, Trans. SPEJ, p. 371, 1974.
[6] Bahari, A. and Baradaran Seyed. A., Trust-Region Approach
to Find Constants of Bourgoyne and Young Penetration Rate
Model in Khangiran Iranian Gas Field, in Proc. SPE Latin
American and Caribbean Petroleum Engineering Conf.,
Buenos Aires, Argentina, Apr. 2007, pp. 15-18.
[7] Coleman, T. F. and Li. Y., An Interior, Trust Region
Approach for Nonlinear Minimization Subject to Bounds,
SIAM Journal on Optimization, vol. 6, p. 418, 1996.
[8] Coleman, T. F., and Li. Y., On the Convergence of
Reflective Newton Methods for Large-Scale Nonlinear
Minimization Subject to Bounds, Mathematical
Programming Journal, vol. 67, 1994, p. 189.
[9] Justus Rabi. B., Minimization of Harmonics in PWM
Inverters Based on Genetic Algorithms, J. Applied Sci., vol.
6, pp. 2056-2059, 2006.
[10] Ikramullah But. S. and Hou-Fang. S., Application of Genetic
Algorithm in the Scheduling of Flexible Job Shop, J. Applied
Sci., vol. 7, pp. 1586-1590, 2006.
[11] Bahari, M. H., Bahari, A., and Nejati Moharrami, F.,
“Determining Bourgoyne and Young model coefficients
using genetic algorithm to predict drilling rate,” Journal of
Applied Sciences, vol.8, no.17, pp.3050-3054, 2008.
[12] Moradi, H., and Naghibi Sistani, M. B., "Drilling rate
prediction using an innovative soft computing
approach." Scientific Research and Essays, vol. 5, no. 13, pp.
1583 – 1588, 2010.
[13] Bahari, M. H., Bahari, A., Moradi, H., "Intelligent Drilling
Rate Predictor," International Journal of Innovative
Computing, Information and Control, vol. 7, no.
2, pp. 1511-1520, 2011.
Formation
Trust-Region
STD Error of
Estimation
Proposed
Scheme
STD Error of
Estimation
Improvement
(%)
Khangiran 5.72 5.48 4.2
Kalat 1.38 1.28 7.2
Abtalkh 1.76 1.33 24.43
Shourijeh 1.05 0.98 6.7
Mozdouran 1.13 1.08 4.4
Bourgoyne and Young
Model
D db Fj gp
gp
h ρc N W
Rop
Genetic
Algorith
m

6. Table 2. A sample required data, obtained from wells daily drilling progress report
Well No.
R
(ft/hr)
D
(ft)
W
(1000
lbf)
db
in.)
N
(Rpm)
ρc
(lbm/gal)
h
(%)
gp
(lbm/gal)
Fj
(lbf)
Well 50 41.5 1411 15 17.5 130 9.96 0.25 8.62 1776
Well 47 24.3 359 15 26 130 8.95 0.25 7.62 1611
Well 46 7.3 1772 7.5 17.5 110 10.3 0.25 8.95 1185
Well 42 9.5 1969 10 17.5 110 10.8 0.5 9.49 1324
Well 39 5.7 1900 9 17.5 100 10.5 0.5 9.15 1186
Well 29 25.9 1575 15 17.5 90 10.4 0.38 9.09 2196
Table 3. Computed coefficients with the use of multiple regression method, trust-region method, and proposed scheme for some
Khangiran gas field formations
Formation
Mathematical
Methods
a1 a2 a3 a4 a5 A6 a7 a8
Khangiran
Multiple
Regression
2920 -0.2930 -0.0627 0.2165 -2.406 2.0652 -19.014 -3.625
Trust-Region 1.0005 0.0002 1.00 e-06
3.87 e-
05
1.8354 0.7733 1.1612 0.3000
GA 1.5348 0.0001 1.16 e-06
9.17 e-
05
1.9999 0.9835 0.4864 0.3000
Kalat
Multiple
Regression
185.172
7
-0.0179 0.0046 0.0149 -0.6338 2.8618 -7.4979 1.8076
Trust-Region 1.0508
2.09 e-
05
0.0004
2.08 e-
06
1.5042 0.4894 0.5254 0.3000
GA 0.8739
6.01 e-
05
0.0007
3.04 e-
05
1.9999 0.4000
0.3000
3
0.3001
Abtalkh
Multiple
Regression
-3.9644 0.0005 -0.0001 -0.0004 0.9906 0.9113 0.4884 0.0417
Trust-Region 1.239
1.00 e-
06
1.00e-06 1.00e-06 0.8194 0.6794 0.7807 0.3000
GA 1.3744
1.00 e-
06
1.01 e-06
1.00 e-
06
1.9999 0.9999 1.4999 0.3000
Shourijeh
Multiple
Regression
0.9016 0.0001 -0.0005 0.0000 0.2747 0.3412 -0.4133 0.8126
Trust-Region 0.9168 0.0003 1.00 e-06
3.01 e-
05
1.1828 0.7024 0.5932 0.3003
GA 0.5000 0.0004 1.04 e-06
6.14 e-
06
1.9999 0.9999 0.7147 0.3000
Mozdoura
n
Multiple
Regression
-1.2201 -0.0039 0.0212 0.0020
-
21.521
7
23.962
9
-6.5392
16.452
7
Trust-Region 0.9076
1.00 e-
06
1.00 e-06
2.76 e-
05
0.5000 0.4000 0.9962 0.3013
GA 0.5593 1.0013 1.0073 1.0082 0.5000 0.4000 1.3403 0.3000
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