INTRODUCTION
Optimization drilling techniques succeed to reduce drilling costs noticeably.
An essential part of these techniques is drilling rate prediction (Kaiser,
2007). Therefore, drilling engineers have been concerned about this issue
considerably during last decades because it results in optimum drilling
parameters selection, which leads to minimize drilling cost per foot (Bourgoyne
et al., 2003).
Rate of penetration is affected by many parameters such as hydraulics,
weight on bit, rotary speed, bit type, mud properties, formation characteristics,
etc. (Akgun, 2007). There exists no exact mathematical relationship between
drilling rate and different drilling variables because not only a large
number of uncertain drilling variables influence drilling rate, but also
their relationship is nonlinear and complex (Ricardo et al., 2007).
However, scientists have tried to suggest some simplified models to create
a mapping between drilling rate and its major variables. One of the most
successful one is Bourgoyne and Young model (Bourgoyne and Young, 1999).
In this model, there are some unknown parameters or coefficients, which
must be determined based on previous drilling experience in the field.
Therefore, the method of determining these coefficients has a significant
impact on model accuracy.
In order to determine unknown coefficients, Bourgoyne and Young suggested
multiple regression technique (Bourgoyne and Young, 1999). Nevertheless,
applying multiple regression method does not guarantee reaching physically
meaningful result. Furthermore, this method is limited to the number of
data points.
To reach meaningful result, some new mathematical methods have recently
been applied to find out these unknown parameters. For instance, Nonlinear
least square data fitting with trustregion method is a mathematical technique,
which is applied to this problem recently (Bahari and Baradaran Seyed,
2007). This method is one of the optimization algorithms, which minimizes
the sum of square errors function. The method is based on the interiorreflective
Newton method. In each of iterations, the approximate solution of a large
linear system is estimated using the method of Preconditioned Conjugate
Gradients (PCG) (Coleman and Li, 1994, 1996). This technique makes it
possible to determine lower and upper bounds for results and limit them
to be in the reasonable ranges (Coleman and Li, 1996). 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 (2006)
applied GA to minimize harmonics in PWM inverters or Ikramullah But and
HouFang (2006) used it in scheduling flexible job shops. In this research,
we applied GA to determine the optimal values of Bourgoyne and Young model
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 trustregion
method.
KHANGIRAN GAS FIELD
As mentioned, 9 wells of Khangiran gas field were considered to apply
and test the new scheme. Khangiran gas field is located in the northeast
of Iran. This field was surveyed in 1937.

Fig. 1: 
Stratigraphy column of a typical well in Khangiran field
and formations description 
In 1956, the stratigraphy plan
was prepared and it was named in 1962. Figure 1 shows
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 H_{2}S 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. 
BOURGOYNE AND YOUNG DRILLING RATE MODEL
Bourgoyne and Young have proposed the following equation to model the
drilling process when using roller cone bits (1):
where, Rop is rate of penetration (ft/hr). The function f_{1}
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
drillability. The functions f_{2} and f_{3} symbolize
the effect of compaction on penetration rate. The function f_{4}
signifies the effect of overbalance on penetration rate. The functions
f_{5} and f_{6} respectively, model the effect of bit
weight and rotary speed on penetration rate. The function f_{7}
represents the effects of tooth wear and the function f_{8} characterizes
the effect of bit hydraulics on penetration rate (Bourgoyne et al.,
2003). The functional relations in Eq. 1 are as follows:
a_{1}
to a_{8} 
= 
Bourgoyne
and Young model constant coefficients 
D 
= 
True
vertical depth (ft) 
d_{b} 
= 
Bit diameter (in) 
F_{j} 
= 
Jet
impact force (lbf) 
g_{p} 
= 
Pore
pressure gradient (lbm gal^{1}) 
h 
= 
Fractional bit tooth wear 
ñ_{c
} 
= 
Equivalent
mud density (lbm gal^{1})c 
N 
= 
Rotary
speed (rpm) 
W 
= 
Weight
on bit (1000 lbf) 
(W/d_{b})_{t} 
= 
Threshold
bit weight per inch of bit diameter at which the bit begins to drill 
As mentioned a_{1} to a_{8} are dependent to local drilling
conditions and must be determined for each formation using prior drilling
data sets obtained from the drilling area (Bourgoyne et al., 2003).
Bourgoyne and Young recommended specific bounds for each of eight coefficient
based on reported ranges for the coefficients from various formations
in different areas (Bourgoyne et al., 2003; Bourgoyne and Young,
1999) 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 

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 (a_{5}) 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.
DETERMINING BOURGOYNE AND YOUNG CONSTANT COEFFICIENTS USING GA
As mentioned, here employed GA to determine optimal value for constant parameters
of Bourgoyne and Young model. 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. Figure 2 shows architecture
of the predictor system.
To find constant parameters of the aforementioned model for each formation,
the following procedure was performed.
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.
We constructed a database from available data of nine wells. The database
includes quantities of D, W, N, g_{p}, ρ_{c}, h,
F_{j}, d_{b} 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.

Fig. 2: 
Architecture
of predictor system 
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.
In each formation, by applying inputs (D, W, N, g_{p}, ρ_{c},
h and F_{j}) and output (Rop) to the abovementioned model, we
use GA to find out optimum values of eight unknown coefficients. GA was
run in the following steps.
Step
1 
: 
Set
the initial parameters for GA: population size, crossover type and
probability and mutation probability. 
Step
2 
: 
Set
all bounds recommended by Bourgoyne and Young for each of eight parameters,
particularly. 
Step
3 
: 
Generate
the initial population randomly. 
Step
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. 
Step
5 
: 
Selection
of the subjects that will mate according to their share in the population
global fitness. 
Step
6 
: 
Apply
the genetic operators (crossover, mutation...). 
Step
7 
: 
Repeat
Steps 3 to 6 until the generation number is reached. 
RESULTS AND DISCUSSION
As mentioned, constants a_{1} to a_{8} were computed
for each of Khangiran field formations utilizing GA. Table
3 shows the results obtained, using multiple regression method, trustregion
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 trustregion
method and GA are all physically meaningful and in recommended bounds.
In Table 4 Standard Deviation (STD) error of drilling
rate estimation by these two methods is illustrated.
Table
2: 
A
sample of required data, obtained from wells daily drilling progress
reports 

Table
3: 
Computed
coefficients with the use of multiple regression method, trustregion
method and proposed scheme for some Khangiran gas field formations 

Table
4: 
Estimation
accuracy of TrustRegion method in comparison with proposed scheme 

It can be interpreted
that the proposed scheme is more efficient than Trustregion method in
determining coefficients of Bourgoyne and Young model which leads to more
accuracy in drilling rate prediction.
ACKNOWLEDGMENTS
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.
CONCLUSION
Accurate drilling rate prediction is highly demanded for drilling cost
optimization. A simplified model of drilling is called Bourgoyne and Young
model, which represents a general mapping between drilling rate and some
drilling variables. That model can be used in drilling rate prediction.
However, there are eight unknown parameters in this model, which must
be determined by using previous drilling experiences. Although, several
methods have been suggested to determine these coefficients in last decades,
it is hard to reach a predictor system with satisfactory accuracy. In
this study we applied GA to determine constant coefficient of Bourgoyne
and Young model. Simulation results confirm that suggested approach not
only provides meaningful result but also leads to more accuracy in comparison
with conventional methods.