Prediction and Analysis of PDC Bit Wear in Conglomerate Layer with Machine Learning and Finite-Element Method

Abstract

Polycrystalline diamond compact (PDC) bits experience a serious wear problem in drilling tight gravel layers. To achieve efficient drilling and prolong the bit service life, a simplified model of a PDC bit with double cutting teeth was established by using finite-element numerical simulation technology, and the rock-breaking process of PDC bit cutting teeth was simulated using the Archard wear principle. The numerical simulation results of the wear loss of the PDC bit cutting teeth, such as the caster angle, temperature, linear velocity, and bit pressure, as well as previous experimental research results, were combined into a training dataset. Then, machine learning methods for equal-probability gene expression programming (EP-GEP) were used. Based on the accuracy of the training set, the effectiveness of this method in predicting the wear of PDC bits was demonstrated by verifying the dataset. Finally, a prediction dataset was established by a Latin hypercube experiment and finite-element numerical simulation. Through comparison with the EP-GEP prediction results, it was verified that the prediction accuracy of this method meets actual engineering needs. The results of the sensitivity analysis method for the gray correlation degree show that the degree of influence of bit wear is in the order of temperature, back dip angle of the PDC cutter, linear speed, and bit pressure. These results demonstrate that when an actual PDC bit is drilling hard strata such as a conglomerate layer, after the local high temperature is generated in the formation cut by the bit, appropriate cooling measures should be taken to increase the bit pressure and reduce the rotating speed appropriately. Doing so can effectively reduce the wear of the bit and prolong its service life. This study provides guidance for predicting the wear of a PDC bit when drilling in conglomerate, adjusting drilling parameters reasonably, and prolonging the service life of the bit.

Full text

Research Article
Prediction and Analysis of PDC Bit Wear in Conglomerate
Layer with Machine Learning and Finite-Element Method
Li-qiang Wang ,
1
Ming-ji Shao ,
2
Wei Zhang ,
2
Zhi-peng Xiao ,
2
Shuo Yang ,
2
and Ming-he Yang
3
1
Department of Petroleum Engineering, Shengli College, China Petroleum University, Shandong Dongying 257061, China
2
Exploration and Development Research Institute of TuHa Oilfield Company, CNPC, Xinjiang Hami 839009, China
3
Department of Petroleum Engineering, Yangtze University, Hubei Wuhan 430000, China
Correspondence should be addressed to Li-qiang Wang; 281035925@qq.com
Received 26 September 2021; Revised 21 November 2021; Accepted 6 December 2021; Published 10 January 2022
Academic Editor: Jinjie Wang
Copyright © 2022 Li-qiang Wang et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Polycrystalline diamond compact (PDC) bits experience a serious wear problem in drilling tight gravel layers. To achieve efficient
drilling and prolong the bit service life, a simplified model of a PDC bit with double cutting teeth was established by using finite-
element numerical simulation technology, and the rock-breaking process of PDC bit cutting teeth was simulated using the
Archard wear principle. The numerical simulation results of the wear loss of the PDC bit cutting teeth, such as the caster
angle, temperature, linear velocity, and bit pressure, as well as previous experimental research results, were combined into a
training dataset. Then, machine learning methods for equal-probability gene expression programming (EP-GEP) were used.
Based on the accuracy of the training set, the effectiveness of this method in predicting the wear of PDC bits was demonstrated
by verifying the dataset. Finally, a prediction dataset was established by a Latin hypercube experiment and finite-element
numerical simulation. Through comparison with the EP-GEP prediction results, it was verified that the prediction accuracy of
this method meets actual engineering needs. The results of the sensitivity analysis method for the gray correlation degree show
that the degree of influence of bit wear is in the order of temperature, back dip angle of the PDC cutter, linear speed, and bit
pressure. These results demonstrate that when an actual PDC bit is drilling hard strata such as a conglomerate layer, after the
local high temperature is generated in the formation cut by the bit, appropriate cooling measures should be taken to increase
the bit pressure and reduce the rotating speed appropriately. Doing so can effectively reduce the wear of the bit and prolong its
service life. This study provides guidance for predicting the wear of a PDC bit when drilling in conglomerate, adjusting drilling
parameters reasonably, and prolonging the service life of the bit.
1. Introduction
With increasing oil and gas exploration in China, many glu-
tenite reservoirs have been discovered, among which a rep-
resentative oilfield is the Mahu oilfield in the Xinjiang oil
region. At the bottom of the Badaowan Formation in the oil-
field, the gravels are well developed, with a thickness of 100–
350 m, and the ability to drill is poor. The formation lithol-
ogy of the Karamay Formation changes greatly in the hori-
zontal direction, there are many longitudinal intercalations,
the glutenite particle size is uneven, and bit selection is diffi-
cult. Other glutenite reservoirs share this feature. The hard
gravel makes it difficult to drill [1, 2].
Although polycrystalline diamond compact (PDC) bits
have the advantages of high rock-breaking efficiency, strong
wear resistance, and a long service life, when drilling in a
conglomerate formation, the wear rate of the cutting teeth
increases sharply, easily leading to bit failure. Therefore, to
enhance the rock-breaking ability, accurately predicting the
bit wear and reducing the adverse effects is important.
In research on the PDC bit wear law in conglomerate
layers, experimental methods and numerical simulation
Hindawi
Geofluids
Volume 2022, Article ID 4324202, 10 pages
https://doi.org/10.1155/2022/4324202

methods, such as the finite-element method, have mainly
been used. In experimental research, standard wear parts
or composite chips are generally used for grinding tests to
obtain the wear laws of diamond bits in different rock media,
such as the influences of rock properties, normal pressure,
cutting line speed, and wear chord length on the amount
of wear of the composite [3–7]. In comprehensive research
through experiments and numerical simulations, the experi-
mental method is generally used to study the influences of
different factors, such as the cutting angle of the PDC coring
bit, outcrop, linear velocity, and rock sample properties, on
the wear law of composite cutting teeth. Then, the feasibility
of wear law is verified using the numerical simulation
method [8, 9]. The above research mainly addressed the
wear law of the drill bit using the experimental method,
which has played a positive role in promoting research in
this field. However, the influence of temperature on bit wear
has not been considered, and the experimental method or
combined experimental and numerical simulation verifica-
tion cannot accurately predict the wear of the PDC drill
under the combined action of various factors in high-
temperature and high-pressure environments. Therefore,
not only is in-depth study of the bit wear law under different
temperatures needed, but also it is particularly important to
select appropriate prediction methods to predict bit wear
under different working conditions to achieve efficient dril-
ling and prolong the service life of bits.
Among the many prediction methods, machine learning
has developed rapidly in recent years and has good develop-
ment prospects [10–13]. Gene expression programming
(GEP) is based on the genetic algorithm (GA) and genetic
programming (GP), which has excellent performance in
knowledge mining, function discovery, optimization, and
prediction [14]. Using a machine learning modeling tool,
an explicit model with a simple structure and high predic-
tion accuracy can be obtained through evolution without it
being necessary to know the structure and parameters of
the model in advance, thereby reducing the difficulty of
establishing the prediction model and avoiding the preset
model structure based on the regression method. Then,
the subjectivity of parameters is determined using a statis-
tical method [15]. At present, the GEP method has been
Initialization population
Initialize candidate set
Expression chromosome
Perform the chromosome
program
Evaluation of fitness
Meet
termination
conditions ?
END
Premature
convergence ?
Evolution of next generation
population
Recombination
Transposition
Variation
Choice
Evolutionary population
Recombination
Transposition
Variation
Choice
Variation Variation
Evolutionary candidate set
Transposition Transposition
Recombination Recombination
Iteration
restart
Y
N
Y
N
Equal
probability
genetic
operation and
evolution
Figure 1: Flowchart of EP-GEP [19].
b
⁎
+/
a b Q a
Figure 2: Expression tree corresponding to Equation (2).
2 Geofluids

successfully applied in many disciplines and fields [16–18].
It has not been reported that the GEP machine learning
method has been used to predict bit wear. To predict the
bit wear accurately under different working conditions,
realizing the purpose of efficient drilling and prolonging
the service life of the bit, it is necessary to study the
GEP machine learning modeling method and its predic-
tion effectiveness.
2. Equal-Probability Gene Expression
Programming Algorithm (EP-GEP)
GEP combines the advantages of GAs and GP. In the form
of expression, it inherits the simple and fast characteristics
of the fixed-length linear coding of the GA. In terms of gene
expression (semantic expression), it inherits the flexible and
changeable characteristics of the GP tree structure, and it
solves complex problems with simple coding two to four
orders of magnitude faster than the traditional machine
learning evolutionary algorithm [15].
However, the traditional GEP method has some prob-
lems, such as nondirectional evolution and premature con-
vergence in the process of knowledge mining, which can
easily fall into the local optimum and reduce the efficiency
and quality of the overall optimal solution. Therefore, it is
necessary to mitigate these defects. The proposed EP-GEP
method can improve the convergence efficiency and solution
quality of the algorithm.
The flow of the EP-GEP optimization calculation is
shown in Figure 1. First, a certain number of chromosome
individuals are randomly generated to form the initial pop-
ulation. Second, the candidate set is established by selecting
excellent individuals in the initial population. According to
the bit wear analysis, the best fitness function of the indi-
vidual in the group suitable for the problem expression is
selected. The responsiveness of each individual in the
group is assessed. Then, the individual is selected, mutated,
inserted, and recombined, and other genetic operations are
carried out to produce new offspring and form new groups.
Then, they enter the next round of the optimization pro-
cess. If local precocious convergence occurs, the algorithm
enters the calculation process of equal-probability gene
expression optimization (taking three equal probability
individuals as an example in Figure 1), generating new off-
spring to form a new population, and they enter the next
round of optimization calculation. Then, the above optimi-
zation calculation process is repeated until the iteration ter-
mination condition is satisfied.
2.1. Gene Structure. The target of EP-GEP is a chromosome
(genome) composed of a single gene or multiple genes. The
gene in EP-GEP is a simplification of the gene principle in
biology. The gene in EP-GEP is composed of a head and tail.
The head can be composed of a function symbol (F) and a
terminal symbol (t), whereas the tail can only be composed
of terminal symbol t. The chromosome (or individual) in
EP-GEP is composed of one or more genes of equal length,
and multiple genes are connected by a connection function.
Each individual represents a candidate solution to the prob-
lem to be solved. Several of these chromosomes constitute
the entire population.
The relationship between tail length tand head length
his
t=h×n−1
ðÞ
+1, ð1Þ
where nrepresents the maximum number of variables
required by the function character (for example, open-
ended operation, n=1; multiplication or addition opera-
tion, n=2).
For example, the expression tree corresponding to Equa-
tion (2) is shown in Figure 2.
a+b
ðÞ
∗b1/2
a
!
:ð2Þ
The parsing rule of the expression tree is from top to
bottom and from left to right, until the node is the termina-
tor. The gene after the termination point is the noncoding
Figure 3: Simplified model of PDC bit with double cutting teeth.
3Geofluids

region of the chromosome, so it is no longer in operation.
Here, the function character set is f∗,+,/,Qg(Qis the
square-root operation), and the terminator set is fa,bg.If
the head length hof the gene is 6, then, according to Equa-
tion (1), the tail length tis 7, and the total length of the gene
is 13 [20].
2.2. Genetic Operator. EP-GEP creates an initial population
in the algorithm, and each chromosome in the population
represents a solution to the problem. Then, a series of
genetic operations are carried out to generate new high-
fitness offspring individuals to obtain better solutions. The
basic genetic operators of EP-GEP include selection, muta-
tion, inversion, insertion, root insertion, gene transforma-
tion, single point recombination, two-point recombination,
and gene recombination [19].
2.3. Fitness Function. To obtain the best solution, it is neces-
sary to evaluate the environmental adaptability of the newly
generated chromosomes. Similar to other machine learning
evolutionary algorithms, EP-GEP uses the fitness function
value (i.e., fitness) to evaluate the quality of chromosomes.
Sometimes, the fitness function can be defined according
to the solution of the problem. The commonly used fitness
functions in the symbolic regression are the complex corre-
lation coefficient method, relative hits, absolute hits, mean
square error (MSE), root MSE (RMSE), absolute mean dif-
ference, relative variance, relative root mean square error,
and relative absolute value difference.
In this study, the RMSE was obtained using the fitness
function expressed in the following equation.
RMSE = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
m〠
m
j=1
yj−y
∧
j

v
u
u
t:ð3Þ
2.4. Finite-Element Simulation Model
2.4.1. Simplified Model of Cutting Teeth. Among the main
parameters affecting the rock-breaking efficiency of the
PDC bit, the back dip angle mainly represents the cutting
ability of the cutting teeth for the formation. The role of
the side angle is to produce a pushing force on the cuttings
to discharge them, and the circumferential angle determines
the distribution of the cutting teeth. The finite-element soft-
ware used was MSC Marc. When finite-element software is
used to model and simulate the all the PDC bit cutting teeth,
there are two problems. First, the model is more complex.
Because there are many factors affecting the rock breaking
by a bit, it is difficult to highlight the role of the back dip
angle of the cutting teeth of the bit in rock breaking. Second,
0
5
10
15
20
25
30
35
40
3456789101112
Relative error (%)
Grid number (× 104)
Figure 5: Influence of grid number on simulation accuracy.
Figure 4: Mesh generation of finite-element model for rock
breaking of PDC bit with double cutting teeth.
Table 1: Attribute parameters of polycrystalline diamond and conglomerate.
Parameter name Polycrystalline diamond Conglomerate
Density (kg·m
−3
) 3520 2540
Elastic modulus (MPa) 8:9×10
55:4×10
4
Poisson’s ratio 0.07 0.27
Thermal conductivity (J·m
−1
·s
−1
·
°
C
−1
) 543.0 3.5
Specific heat capacity (J·kg
−1
·
°
C
−1
) 790.0 800.0
Coefficient of thermal expansion (10
−6°
C) 2.5 52.0
Compressive strength (MPa) 270.0 67.6
Wear coefficient 3×10
−73×10
−7
4 Geofluids

calculation is difficult, leading to the nonconvergence phe-
nomenon, which affects the stability and reliability of the
simulation results. Based on previous research and the above
two considerations, the entire cutting tooth model is simpli-
fied to a double cutting tooth model (see Figure 3). The basic
parameters of the simplified model are as follows: the side
angle is 25
°
, the diameter of the composite chip is 13.4 mm,
and the maximum diameter of the bit is 60 mm [21].
2.4.2. Material Parameters and Basic Assumptions
(1) Material Parameters. The material property parameters
of the conglomerate and PCD are shown in Table 1.
(2) Basic Assumptions. The classical Archard wear model
was used to simulate the wear of the PDC bit [22]. Because
the maximum diameter of the PDC bit with double cutting
teeth is 60 mm, to reduce the influence of rock side loading
on rock breaking, according to Saint Venant’s principle
[23], a cylindrical rock sample with a diameter of 180 mm
and a height of 40 mm was used to simulate the actual for-
mation rock. In addition, the formation rock was assumed
to be isotropic, and the influence of the drilling fluid on
the cutting tooth wear was ignored. The failure criterion of
the rock was the linear Mohr–Coulomb criterion. The con-
fining pressure was loaded on the side of the rock in the
form of stress, and thermal/structural analysis was selected
for the analysis task.
2.4.3. Grid Size and Accuracy Control. A 10-node tetrahedral
mesh was adopted, and the mesh was refined. The mesh
division of the finite-element model for the rock breaking
of the PDC bit with double cutting teeth is shown in
Figure 4. When the rock sample is broken and deformed,
mesh redivision technology is used to solve the subsequent
simulation problems.
When the PDC bit crown top (r= 100 mm) had an off-
cutting tooth loading pressure of 1:5×10
3Nand rotating
speed of 120 r/min, the wear volume of the PDC bit was
19.70 mm
3
when the composite was scrapped. The errors
in the simulation results under different grid numbers were
compared according to the results of the physical simulation
experiment. It was found that, when the mesh number was
greater than 8:00 × 104, the wear of the cutting teeth tended
to be stable. Considering the calculation accuracy and simu-
lation time, the mesh size was 1.6 mm, the mesh number was
9:44 × 104, and the cutting tooth wear was 18.98 mm
3
. Com-
pared with the experimental data in a previous report [5],
the relative error was 3.67% (Figure 5).
2.5. PDC Bit Wear Dataset
2.5.1. Single-Factor Wear Dataset. The finite-element
numerical simulation method was used to fix three of the
four variables of the bit cutting teeth, such as the back angle,
temperature, linear velocity, and bit pressure, to simulate the
change law of bit wear when the other variable changes. For
example, when the inclination angle, temperature, and linear
speed of the cutting teeth are fixed, different bit pressures are
set, and the wear amount of the bit is determined by simula-
tion. This can be expressed as
y=α,T,v,x
fg
,ð4Þ
where yis the bit wear, αis the back angle of the cutting
teeth, Tis the temperature, vis the linear speed, and xis
the weight on the bit.
Equation (4) can be further expressed as the following
vector form:
α,T,v,x,y
ðÞ
:ð5Þ
y = 1.0039x − 0.0116
R2 = 0.9922
0
1
2
3
4
5
6
7
0123456
Actual wear training value (10−3 mm3)
EP-GEP wear fitting value (10−3 mm3)
Figure 6: Comparison between wear amount of training dataset
and EP-GEP fitting value.
Table 2: Experimental parameters of EP-GEP.
Parameter Value Parameter Value
Population size 78 Length of head 9
Gene number 6 Mutation rate 0.00138
Recombination rate 0.00277 One-point recombination rate 0.00277
Two-point recombination rate 0.00277 Transposition rate 0.00277
Root insertion sequence transposition rate 0.00546 Insertion sequence transposition rate 0.00546
Link function fitness function ∗Fitness function RMSE
5Geofluids

The effects of other single variables on bit wear can be
studied similarly.
The dataset in the form of Equation (5) and the previ-
ous experimental data were combined into a single-factor
wear dataset, which was a part of the machine learning
training set.
2.5.2. Multifactor Wear Dataset. The finite-element numeri-
cal simulation method was used to fix one of the four vari-
ables of the bit cutting teeth (the back angle, temperature,
linear velocity, and bit pressure) to simulate the change law
of bit wear when the other three variables change. For exam-
ple, when the back angle of the cutting teeth was fixed as a
variable, different temperatures, linear velocities, and bit
pressures were set to determine the wear amount of the bit
through simulation. This can be expressed as
y=α,x1,x2,x3
fg
,ð6Þ
where yis the bit wear, αis the back angle of the cutting
teeth, x1is the temperature, x2is the linear speed, and x3
is the weight on the bit.
Equation (6) can be further expressed in the following
vector form:
α,x1,x2,x3,y
ðÞ
:ð7Þ
The influences of other variables on the bit wear can be
studied similarly.
The dataset of Equation (7) and that of single-factor
wear were combined to establish a complete machine learn-
ing training set.
3. Results and Analysis
3.1. EP-GEP Time Series Model Training. Fifty-four groups
of data from 80 groups were used for the EP-GEP time series
training. The experimental parameters are shown in Table 2.
The remaining 26 sets of data were used for validation.
After training, the R2value of the model was 0.9922, as
shown in Figures 6 and 7.
The expression tree structure of individual genes is
shown in parts (1)–(6) in Figure 8.
3.2. EP-GEP Model Validation. The model was compared
with the validation dataset to verify the prediction ability
of the EP-GEP model. A comparison between the wear
amount of the validation set and the EP-GEP prediction
value is shown in Figure 9, and the relative error comparison
results are shown in Figure 10.
As shown in Figure 9, the validation set is basically con-
sistent with the EP-GEP prediction results, and the gap is
small. Figure 10 reveals that the relative error between the
prediction results of the EP-GEP model and the verification
set is small, with a maximum relative error of 9.49%, a min-
imum of 0.13%, and an average of 3.64%. This shows that
the model established by the EP-GEP method can accurately
fit the bit wear.
3.3. EP-GEP Model Prediction. To highlight the generality of
the model, an irregular real number was selected for the
value of the influencing factors. All parameters were covered
according to the Latin hypercube experimental design
method. In Table 3, the predicted value of the EP-GEP
model and the results of finite-element simulation are com-
pared. The maximum relative error is −7.01%, the minimum
0
1
2
3
4
5
6
7
13 5 7 9 1113151719 212325 2729313335 3739414345 47 495153
Wear amount (10−3 mm3)
Number of training sets
Training set
EP-GEP fitting value
Figure 7: Comparison of training set wear and EP-GEP fitting value.
6 Geofluids

error is −0.47%, and the average relative error is −0.99%.
From the perspective of prediction accuracy, the model can
meet the demand of PDC bit wear prediction research.
3.4. Sensitivity Research Based on Deng’s Gray Relational
Analysis. The sensitivity of each influencing factor to the
wear amount in Table 3 was analyzed by Deng’s correlation
degree method and the gray correlation theory [24, 25].
If Xiis a system factor and its observation data at the
k-th moment is xiðkÞ, then the behavior sequence of the
factor Xiis Xi=ðxið1Þ,xið2Þ⋯⋯xiðnÞÞ. Here, X0is the
reference sequence, and X1is the comparison sequence.
Gene 1 Gene 2
Gene 3
(1) (2)
(3) (4)
(5) (6)
Gene 4
Gene 5 Gene 6
/
3Rt d1
3Rt
⁎
d3
+
Log2 X4
d3 d0 c9
+
Inv d0
X4
+
−
⁎
Exp /
c3 d2
d0
d2 c3
−
d1 /
X2
X4
/ d0
3Rt
x4
d3
/
c0 d1
+
c9 Inv
+
/
Inv +c3 d3
Exp
−
c4 c7d3
+
+Inv
⁎
−
X5 c1 X4 d3
+
d3
d2 d1
d3
+
d1 X4
Inv
Inv
Inv
⁎
X4 4Rt
c9 d0
Figure 8: Expression trees of genes 1–6.
7Geofluids

If γðx0ðkÞ,xiðkÞÞ is the real number, then the calculation
formula of Deng’s correlation degree is as follows:
γx0k
ðÞ
,xik
ðÞðÞ
=
min
imin
kx0k
ðÞ
−xik
ðÞ
jj
+ξmax
i
max
kx0k
ðÞ
−xik
ðÞ
jj
x0k
ðÞ
−xik
ðÞ
jj
+ξmax
i
max
kx0k
ðÞ
−xik
ðÞ
jj
,
ð8Þ
γX0,Xi
ðÞ
=1
n〠
n
k=1
γx0k
ðÞ
,xik
ðÞðÞ
:ð9Þ
Equation (9) is the average value of γðx0ðkÞ,xiðkÞÞ
when the four axioms of the gray relation are satisfied
[25]. When γðX0,XiÞis the gray relational degree of X1
to X0,γðx0ðkÞ,xiðkÞÞ is the gray correlation coefficient of
X1to X0.
The results show that Dun’s correlation degrees of the
cutter inclination angle α, temperature T, linear velocity V,
and bit pressure Pon wear are 0.7032, 0.7208, 0.7159, and
0.7138, respectively. According to Deng’s correlation degree,
the temperature has the greatest influence on the bit wear
during rock breaking, followed by the rake angle of the
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 26
Wear amount (10−3 mm3)
Number of validation sets
Validation set
EP-GEP tting value
Figure 9: Comparison of wear amount of verification set and EP-GEP prediction value.
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Relative error (%)
Number of validation sets
Figure 10: Comparison of relative error between verification set and EP-GEP.
8 Geofluids

cutting teeth, linear velocity, and bit pressure [21]. There-
fore, when the PDC bit is drilling in hard formations, such
as gravel, reducing the local high temperature generated by
bit cutting and maintaining the high-bit-pressure and low-
rotation-speed mode can reduce bit wear and prolong ser-
vice life to a certain extent.
4. Conclusions
(1) Through the verification of the experimental results
and the sensitivity analysis of the mesh number
and on the basis of verifying the accuracy of the
numerical simulation model, finite-element predic-
tion results of wear under different cutting tooth
caster angle, temperature, linear velocity, and bit
pressure were introduced into a machine learning
training dataset
(2) The EP-GEP machine learning method was used to
carry out modeling and prediction research.
Through a comparative analysis of the model effec-
tiveness and prediction ability, it was proven that
the EP-GEP model has good prediction accuracy
(3) The results of EP-GEP wear prediction and gray cor-
relation sensitivity analysis show that, after the actual
PDC bit cuts the formation to produce a local high
temperature, taking appropriate cooling measures,
appropriately increasing the bit pressure, and reduc-
ing the rotating speed can effectively reduce the bit
wear and prolong the service life
Data Availability
The data used to support the findings of this study are
available from the corresponding author upon request.
Conflicts of Interest
The authors declare no potential conflicts of interest with
respect to the research, authorship, or publication of this
article.
Acknowledgments
This study was supported by the ScientificResearchDevelop-
ment Plan of Shandong Province (j18ka205), the Scientific
Research Start-Up Fund for Introducing High-Level Talents
Table 3: Comparison of relative error between EP-GEP model predicted value and finite-element simulation result.
Serial number α(
°
)T(
°
C) V(m·s
−1
)P(N) EP-GEP (V/mm
3
)Relative error with the simulated value
of finite-element method (%)
1 12 120 0.07 1580 1.722 −2.433
2 12 220 0.13 1870 2.601 −3.302
3 12 340 0.23 2060 4.343 1.592
4 12 430 0.27 2265 5.730 −2.873
5 12 550 0.29 2775 8.023 2.368
6 17 120 0.13 2060 2.599 2.757
7 17 220 0.23 2265 3.741 3.885
8 17 340 0.27 2775 5.156 −6.776
9 17 430 0.29 1580 5.855 2.073
10 17 550 0.07 1870 2.907 3.174
11 22 120 0.23 2775 4.155 3.243
12 22 220 0.27 1580 3.712 −2.722
13 22 340 0.29 1870 4.979 −6.980
14 22 430 0.07 2060 2.515 −7.010
15 22 550 0.13 2265 4.263 −0.585
16 27 120 0.27 1870 2.826 0.626
17 27 220 0.29 2060 3.650 −6.791
18 27 340 0.07 2265 2.226 −5.063
19 27 430 0.13 2775 3.390 −2.726
20 27 550 0.23 1580 5.346 6.506
21 33 120 0.29 2265 2.138 −5.392
22 33 220 0.07 2775 1.776 4.279
23 33 340 0.13 1580 1.810 −0.501
24 33 430 0.23 1870 3.220 −1.587
25 33 550 0.27 2060 4.639 −0.468
9Geofluids

from the Shengli College of China University of Petroleum
(Beijing) (kq2019-005), and the Key R&D Program of Dongy-
ing City in Shandong Province (2018kjcx).
References
[1] L. Zongyu, Z. Fei, and L. Ming, “Key drilling technology for
glutenite tight oil horizontal wells in Mahu oilfield, Xinjiang,”
Petroleum Drilling Technology, vol. 47, no. 2, pp. 9–14, 2019.
[2] L. Li, Z. Lubin, and Z. Lingzhan, “Drilling and completion
technology for efficient development of Mahu low permeabil-
ity reservoir,”Xinjiang Petroleum Science and Technology,
vol. 27, no. 2, pp. 1–6, 2017.
[3] C. A. Cheatham and D. A. Loeb, “Effects of field wear on PDC
bit performance,”in SPE/IADC Drilling Conference, Louisiana,
USA, 1985.
[4] D. A. Glowka, “Implications of thermal wear phenomena for
PDC bit design and operation,”in SPE Annual Technical Con-
ference and Exhibition, Nevada, USA, 1985.
[5] Z. Deyong, Y. Jun, and X. Cheng, “A modified determination
method of rock’s abrasiveness for well drilling using diamond
bits,”Journal of China University of Petroleum, vol. 40, no. 4,
pp. 73–80, 2016.
[6] Z. H. Shaohe, X. Xiaohong, and F. Hhaijiang, “PDC abrasion
rule affected by height of protrusion and linear velocity,”Jour-
nal of Central South University, vol. 41, no. 6, pp. 2173–2177,
2010.
[7] I. H. Michaels, M. Mostofi, and T. Richard, “An experimental
study of the wear of polycrystalline diamond compact bits,”in
SPE 53rd U.S. Rock Mechanics/Geomechanics Symposium,
New York, USA, 2019.
[8] X. Xiaohong, The study on the cutter’s abrasion rule of PDC
coring bit, Central South University, Changsha, China, 2011.
[9] G. M. Gouda, M. Maestrami, and M. A. A. Saif, “A mathemat-
ical model to compute the PDC cutter wear value to terminate
PDC bit run,”in SPE Middle East Oil and Gas Show and Con-
ference, Manama, Bahrain, 2011.
[10] D. Weber, T. F. Edgar, and L. W. Lake, “Improvements in
capacitance-resistive modeling and optimization of large scale
reservoirs,”in Paper presented at the SPE Western Regional
Meeting, San Jose, California, 2009.
[11] J. Li, Z. Lei, and S. Li, “Optimizing water flood performance to
improve injector efficiency in fractured low-permeability res-
ervoirs using streamline simulation,”in Paper presented at
the SPE Kingdom of Saudi Arabia Annual Technical Sympo-
sium and Exhibition, Dammam, Saudi Arabia, 2016.
[12] M. Khan, S. Alnuaim, and Z. Tariq, “Machine learning appli-
cation for oil rate prediction in artificial gas lift wells,”in Paper
presented at the SPE Middle East Oil and Gas Show and Confer-
ence, Manama, Bahrain, 2019.
[13] C. Noshi, M. Eissa, and R. Abdalla, “An intelligent data driven
approach for production prediction,”in Paper presented at the
Offshore Technology Conference, Houston, Texas, 2019.
[14] C. Ferreira, “Gene expression programming: a new adaptive
algorithm for solving problems,”Complex Systems, vol. 13,
no. 2, pp. 87–129, 2001.
[15] Z. Kisi, “Comparative analysis of ozone level prediction
models using gene expression programming and multiple lin-
ear regression,”Geofizika, vol. 30, no. 30, pp. 43–74, 2013.
[16] C. Zhou, W. Xiao, and T. M. Tirpak, “Evolving accurate and
compact classification rules with gene expression program-
ming,”IEEE Transactions on Evolutionary Computation,
vol. 7, no. 6, pp. 519–531, 2003.
[17] N. D. Hoang and D. Tien Bui, “Spatial prediction of rainfall-
induced shallow landslides using gene expression program-
ming integrated with GIS: a case study in Vietnam,”Natural
Hazards, vol. 92, no. 3, pp. 1871–1887, 2018.
[18] J. Jedrzejowicz, P. Jedrzejowicz, and I. Wierzbowska, “Imple-
menting gene expression programming in the parallel envi-
ronment for big datasets' classification,”Vietnam Journal of
Computer Science,vol. 6, no. 2, pp. 163–175, 2019.
[19] L. Wang, M. Shao, G. Kou et al., “Time series analysis of pro-
duction decline in carbonate reservoirs with machine learn-
ing,”Geofluids, vol. 2021, 8 pages, 2021.
[20] C. Ferreira, Gene Expression Programming: Mathematical
Modeling by an Artificial Intelligence, Springer, Berlin, 2006.
[21] H. Peng, G. Feng, Z. Liwei, W. Lili, and Y. Minghe, “Study on
the wear law of PDC bits in conglomerate,”China Petroleum
Machinery, vol. 48, no. 7, pp. 1–6, 2020.
[22] J. F. Archard, “Contact and rubbing of flat surfaces,”Journal of
Applied Physics, vol. 24, no. 8, pp. 981–988, 1953.
[23] R. A. Toupin, “Saint-Venant’s principle,”Archive for Rational
Mechanics and Analysis, vol. 18, no. 2, pp. 83–96, 1965.
[24] J. L. Deng, “Figure on difference information space in grey
relational analysis source,”Journal of Grey Systems, vol. 16,
no. 2, pp. 121–139, 2004.
[25] J. L. Deng, Course of Grey System Theory, Huazhong Univer-
sity of Technology Press, 1990.
10 Geofluids