2018 International Conference on Sensor Network and Computer Engineering (ICSNCE 2018) 

86 

 

A Research of Perforation Plan-decision Based on Grey Cluster Relation 

Xue Jijun 

Mechanical Engineering College 

Xi’an Shiyou University 

Xi’an, 710065, Shanxi, P.R.China 

e-mail: xue_jijun@163.com 

 

 

Abstract—Perforation completion in oil and gas wells is the 

most important way of completion engineering, the 

optimization of perforation completion’s designing is 

influenced by a variety of factors. In order to get the ideal 

effect of perforation operation, in this paper, a Perforation 

plan-decision based on Grey Cluster Relation is putted 

forward. It aims to provide a scientific guidance for the 

Perforation. The simulation experimental results show that 

new models are effective, which offer one kind of science 

decision-making foundation for petroleum Perforation. 

Keywords-Perforating Operation; Grey Cluster Relation; 

Perforation Plan-decision 

I. INTRODUCTION 

Perforated well completion As the most extensive and 

major method of the well’s completion, the reasonable 

selection of parameters for the program has great meaning of 

improving efficiency and reducing costs[1][2]. By 

establishing a quantitative regression model to study the 

relationship between the parameters of the perforation and 

the production ratio, this algorithm can also analysis how 

different factors (perforation elasticity, perforation 

penetration, shot density, perforation  diameter, perforation 

phase angle) act on the production ratio and casing strength 

coefficient. It provides a reliable theoretical basis for the 

perforation parameter optimization, and gives different 

perforation completion optimization schemes [3]. 

Due to the mutual restriction of different parameters, the 

current subjective decision-making for perforation program 

can’t make all the factors to achieve the best at the same 

time. In order to solve the above problems and reduce the 

subjective influence of the decision maker, maximize the 

productivity ratio[4], a Perforation Plan-decision Based on 

Grey Cluster Relations proposed[5-7]. 

II. PERFORATION PLAN-DECISION BASED ON GREY 

CLUSTER RELATION 

Perforation optimization needs to confirm a solution to 

maximize the production capacity. This solution depends on 

many factors and the main influencing factors are hole depth, 

pore size, pore density, phase angle, formation heterogeneity, 

drilling pollution degree and depth, perforation compaction 

thickness and degree. All these factors are acting on the 

decision-making of the solution on the same time. 

Perforation Plan-decision based on Grey Cluster has 

made the model of perforation parameters and the oil well 

productivity. Gray parameters are clustered in the 

parameters of the perforation scheme, and the evaluation 

function is established to design the optimal scheme [8-10]. 

A. Building of model 

First simulating and calculating the productivity ratio of 

oil and gas, then making a non-linear regression analysis, 

According to whether perforation penetration penetrate the 

drilling zone or not, an equation can be established, it 

indicates the relationship between perforating parameters 

and capacity. 

1) The regression equation when the perforation 

penetration does not penetrate the drilling zone: 



2018 International Conference on Sensor Network and Computer Engineering (ICSNCE 2018) 

87 

 



2 2 2 2
PR=0.000156Y -0.000452 +0.000000205 +0.319 -0.103 -0.0009512 +0.000002378

2 2 2
-0.25+0.00148 -0.00000123 +0.00958 -0.0000998 -0.00467 lg( )+0.0178 -0.000296

2
+0.00195 -0.00001 -2*

W W K K r r
h h h zr zr w w

K K K K X K K K
s s m m w zr j j

X X
w w

2 2
0.000828 *lg( )+0.4 g( )+0.3488 -0.1745 +0.69 -0.35

-0.00778

X K l K Y Y W W
w zr zr c c c c

Y
h 

2) The regression equation that perforation penetration has penetrated the contaminated zone of drilling： 



2 2
PR=-0.25+0.00191 -0.00000136 +0.01665 -0.000173 0.00856 lg(K )+0.0201

zr

2 2 2
-0.000335 +0.00211 -0.0000108 -0.001774 lg(K ) +0.5lg(K )+0.512 -0.253

zr zr

2
+0.3315 -0.168 -0.00963 +0.000193

K K K K K K
s s m m m j

K X X X Y Y cj w w w c

W W Y Y
c c h

2 2 2
-0.000841 +0.000000714 +0.406K -0.135K

zr zr

2
-0.0009736 +0.00000243  

W W
h h h

r r
w w 

The quantitative relationships between parameters 

(perforation penetration KS, perforation aperture Kj, 

perforation phase Xw, perforation compaction degree Yc, 

perforation compaction thickness Yh, drilling damage 

thickness Wh, drilling pollution degree Wc, shot density 

Km, borehole radius rw, formation permeability Kzr) and 

the oil production ratio PR is the basis for the 

optimization of perforating parameters. 

B. Perforation program base on Grey Cluster Relation 

The main factors in the decision-making of the 

perforation plan are six factors: perforation ratio, 

perforation phase angle, shot density, perforation 

penetration, perforation diameter and casing strength 

decreasing coefficient, which are expressed by attributes 

1
x

, 2
x

, 3
x

, 4
x

, 5
x

 , 6
x

 respectively. Initial feature 

object matrix D is made like this:  

11 12 13 1n

21 22 23 2

31 32 33 3

41 42 43 4

51 52 53 5

61 62 63 6

n

n

n

n

n

x x x x

x x x x

x x x x
D

x x x x

x x x x

x x x x

 
 
 
 
 
 
 
 
 
 
 














In the formula, ij
x

 represents  j th attribute of the 

ith scheme; in the j  scheme 1 j
x

 represents the 

productivity ratio, 2 j
x

 is the phase angle, 3 j
x

 is the 

perforation diameter, 4 j
x

 is the hole depth, 5 j
x

 is the 

aperture, and 6 j
x

 is the casing strength reduction coefficient. 

There are n  scheme and 6 attributes. 

As the different dimensions will have an impact on 

decision-making, so the formula (4) - (6) are used to D for 

normalization. 

The normalization of attribute data based on the different 

effects caused by different attributes, the formula (4) shows the 

method to normalize production ratio, which called upper limit 

method. Inherent properties such perforation phase angle, shot 

density, perforation penetration, perforation diameter are 

concluded by extreme conversion method, shown as formula 

(5). Casing strength decreasing coefficient, as a cost-type 

attribute, calculated by the lower limit method, shown as 

formula (6). 

1j

1

1
max( )

j

j

x
r

x




max( ) min( )

ij
ij

ij ij

x
r

x x





6

6

6
min( )

j
j

j

x
r

x






2018 International Conference on Sensor Network and Computer Engineering (ICSNCE 2018) 

88 

 

In the formula, 2 5,i j n   , the normalized 

decision matrix can be calculated: 
6 n( )

ij
R r 

. 

The Grey Clustering analysis is used to classify the 

attributes and the similar factors can be classified and 

simplified. 

1) Initialize processing：  

1

2 5 ,

 

  

ij ij i
r r r

i j n


2) Calculate the gray absolute correlation degree ik


 

of any two parameter index data Ri and Rk sequence 

(1 ,1 6,k i j n    )： 
1

2

1

n
2

| || 0.5 * |

| || ( ) 0.5 *( )|

1 | | | |

1 | | | | | |

n

i ij in
j

n

k i kj ij ij k in
j

i k
ik

i k i k

S r r

S S r r r r r

S S

S S S S











 




    

  
 

   







3) Establishing attribute correlative sequence matrix 

according to the above gray absolute correlation 

degree: 

12 13 14 15 16

23 24 25 26

34 35 36

45 46

56

1

1

1

1

1

1

    

   

  


 



 
 
 
 
 
 
 
 
 
  

The critical value 
 r 0,1

, in pursuit of  accuracy 

the value of r is higher than 0.5, the higher the r value, the 

more accurate the classification is , and the accurate value 

of r is determined by actual data, the Ri and Rk classified 

as similar attributes; when ij


 ≥ r. 

4) Several attributes can be merged by the calculation 

above, and an attribute can be chosen to instead of 

other similar attributes. 

A new feature matrix D’ and new normalization matrix 

' ，
m n( )

ij
R r 

 is established according to the Grey Clustering 

analysis, where m is the number of attributes and n is the 

number of schemes.   

5) Computing information entropy iE  and 

weight i


(1 ,i m j n   ): 

'' ''

1

'

''

'

1

1
ln( )

ln(n)

n

i ij ij
j

ij
ij m

ij
j

E r r

r
r

r






 














In particular, when
'' 0
ij
r 

, let 
'' ''ln( ) 0
ij ij
r r 

.  

1

1
1

(1 )

i
i m

i
i

E
i m

E






  




And 
0 1

i
 

, 1 2 m
1     
, 1 6m  . 

Establish an evaluation function Zk: 

'

i
1

, 1 , 1
m

k ik
i

Z r i m k n


    


When the evaluation function value Z(Rk) is larger, the 

corresponding scheme is better. The program has the largest 

value of Z(k) is chosen as the final construction program. 

III. SIMULATION EXPERIMENT 

White XX well in Chang-qing Oilfield, the reservoir depth 

of middle layer is 1 884.5m, the total thickness is 9.5 m, the 

thickness of the perforated zone is 3.0 m, the porosity is 

13.41%, reservoir drainage radius is 200m, well-bore radius is 

0.111 m, the pressure of formation is 13.073 MPa, the crude oil 

saturation pressure is 9.86 MPa, drilling pollution depth is 

69.5mm, the drilling pollution degree is 0.6. The casing 

strength is 47.8MPa, reservoir heterogeneity is 0.7( vertical 

permeability / horizontal permeability), the water saturation is 

30.21%, rock Poisson's ratio is 0.5, the inclination is 5º, the oil 

viscosity is 1.03 MPa.S, the perforation optimization scheme is 

shown as Table 1. 

 



2018 International Conference on Sensor Network and Computer Engineering (ICSNCE 2018) 

89 

 

TABLE I. PERFORATION TABLE OF WHITE XX  

   Attributes      

 

Program 

productivity 

ratio 

perforation 

phase angle

（degree） 

shot density 

（holes/m） 

perforation 

penetration

（mm） 

perforation 

diameter 

（mm） 

casing strength 

decreasing 

coefficient（%） 

A1 0.5193 120 26 328.68 10.68 5.00 

A2 0.5188 90 26 328.68 10.68 6.10 

A3 0.5152 120 32 328.68 10.68 5.70 

A4 0.5150 60 26 328.68 10.68 5.70 

A5 0.5147 90 32 328.68 10.68 5.30 

A6 0.5108 60 32 328.68 10.68 5.00 

A7 0.5071 120 36 328.68 10.68 4.50 

A8 0.5065 90 36 328.68 10.68 4.20 

A9 0.5045 120 26 267.55 9.42 4.60 

A10 0.5039 90 36 267.55 9.42 4.30 

A11 0.5024 60 26 328.68 10.68 4.00 

A12 0.5001 120 32 267.55 9.42 4.00 

A13 0.4998 60 36 267.55 9.42 4.10 

A14 0.4995 90 32 267.55 9.42 3.80 

A15 0.4952 60 32 267.55 9.42 3.60 

A16 0.4912 120 26 267.55 9.42 3.10 

A17 0.4905 90 26 267.55 9.42 3.00 

A18 0.4874 120 16 328.68 10.68 2.60 

A19 0.4867 90 16 328.68 10.68 2.50 

A20 0.4861 60 26 267.55 9.42 2.90 

A21 0.4821 60 16 328.68 10.68 2.40 

A22 0.4695 120 16 267.55 9.42 1.80 

A23 0.4688 90 16 267.55 9.42 1.80 

A24 0.4637 60 16 267.55 9.42 1.70 

 

The initial feature matrix
6 24(x )

ij
D 

 can be 

constructed from the data in Table 1 and the results are 

shown in Table 2. 

The feature object matrix 6 24( )ijR r   is 

established by the above equation (4) - (6) and the initial 

feature matrix D, is shown as table 3. The index data 

association matrix is established by the above equations 

(7) and (8): 

1 0.9953 0.9499 0.9937 0.9937 0.9976

1 0.5841 0.8571 0.8571 0.9176

1 0.7747 0.7747 0.9602

1 1 0.9696

1 0.9696

1



 
 
 
 

  
 
 
 
 

 

According to the correlation degree matrix, take the critical 

value 0.8r , R2, R4 and R5 can be regarded as same class, 

then take R2 represent this class. Then the influencing 

attributes of perforation program are adjusted to: productivity 

ratio R1, perforation phase angle R2, shot density R3, casing 



2018 International Conference on Sensor Network and Computer Engineering (ICSNCE 2018) 

90 

 

strength decreasing coefficient R6. Establishing new 

normalization matrix R’=(rij)4×24, shown as table 4. 

TABLE II. ESTABLISH THE INITIAL FEATURE MATRIX D 

0.5193 120 26 328.68 10.68 5.50

0.5188 90 26 328.68 10.68 6.10

0.5152 120 32 328.68 10.68 5.70

0.5150 60 26 328.68 10.68 5.70

0.5147 90 32 328.68 10.68 5.30

0.5108 60 32 328.68 10.68 5.00

0.5071 120 36 328.68 10.68 4.50

0.5065 90 36 328.68 10.6

D 

8 4.20

0.5045 120 26 267.55 9.42 4.60

0.5039 90 36 267.55 9.42 4.30

0.5024 60 26 328.68 10.68 4.00

0.5001 120 32 267.55 9.42 4.00

0.4998 60 36 267.55 9.42 4.10

0.4995 90 32 267.55 9.42 3.80

0.4952 60 32 267.55 9.42 3.60

0.4912 120 26 267.55 9.42 3.10

0.4905 90 26 267.55 9.42 3.00

0.4874 120 16 328.68 10.68 2.60

0.4867 90 16 328.68 10.68 2.50

0.4861 60 26 267.55 9.42 2.90

0.4821 60 16 328.68 10.68 2.40

0.4695 120 16 267.55 9.42 1.80

0.4688 90 16 267.55 9.42 1.80

0.4637 60 16 267.55 9.42 1.70





T


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 
 
 
 
 
 
 

 

TABLE III. ESTABLISHMENT OF FEATURE OBJECT MATRIX R 

1 2 1.3 5.3767 8.4762 2.9412

0.9990 1.5 1.3 5.3767 8.4762 3.5882

0.9921 2 1.6 5.3767 8.4762 3.3529

0.9917 1 1.3 5.3767 8.4762 3.3529

0.9911 1.5 1.6 5.3767 8.4762 3.1176

0.9836 1 1.6 5.3767 8.4762 2.9412

0.9765 2 1.8 5.3767 8.4762 2.6471

0.9

R 

754 1.5 1.8 5.3767 8.4762 2.4706

0.9715 2 1.3 4.3767 7.4762 2.7059

0.9703 1.5 1.8 4.3767 7.4762 2.5294

0.9675 1 1.3 5.3767 8.4762 2.3529

0.9630 2 1.6 4.3767 7.4762 2.3529

0.9624 1 1.8 4.3767 7.4762 2.4118

0.9619 1.5 1.6 4.3767 7.4762 2.2353

0.9536 1 1.6 4.3767 7.4762 2.1176

0.9459 2 1.3 4.3767 7.4762 1.8235

0.9445 1.5 1.3 4.3767 7.4762 1.7647

0.9386 2 0.8 5.3767 8.4762 1.5294

0.9372 1.5 0.8 5.3767 8.4762 1.4706

0.9361 1 1.3 4.3767 7.4762 1.7059

0.9284 1 0.8 5.3767 8.4762 1.4118

0

T

.9041 2 0.8 4.3767 7.4762 1.0588

0.9028 1.5 0.8 4.3767 7.4762 1.0588

0.8929 1 0.8 4.3767 7.4762 1

 
 
 
 
 
 
 
 
 
 
 
 
 
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 
 
 
 
 
 
  

 

TABLE IV. DEALS WITH THE FEATURE MATRIX BY GREY CLUSTER 

RELATION R’ 

'

1 2 1.3 2.9412

0.9990 1.5 1.3 3.5882

0.9921 2 1.6 3.3529

0.9917 1 1.3 3.3529

0.9911 1.5 1.6 3.1176

0.9836 1 1.6 2.9412

0.9765 2 1.8 2.6471

0.9754 1.5 1.8 2.4706

0.9715 2 1.3 2.7059

0.9703 1.5 1.8 2.5294

0.9675 1 1.3 2.3529

0.9630 2 1.6 2.3529
R

0.


9624 1 1.8 2.4118

0.9619 1.5 1.6 2.2353

0.9536 1 1.6 2.1176

0.9459 2 1.3 1.8235

0.9445 1.5 1.3 1.7647

0.9386 2 0.8 1.5294

0.9372 1.5 0.8 1.4706

0.9361 1 1.3 1.7059

0.9284 1 0.8 1.4118

0.9041 2 0.8 1.0588

0.9028 1.5 0.8 1.0588

0.8929 1 0.8 1










T







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 
 
 

 

 

The attribute weight vectors  =(0.0036,0.2826, 

0.2797,0.4340) are calculated according to formulas (10) and 

(11).  

Then the evaluation function Z is established according to 

(12): 

Z={2.2090,2.2348,2.4716,2.1051,2.2282,2.0103, 

2.2211,2.0032,2.1068,2.0288,1.6710,2.0375,1.8364,1.8451,1.6

527,1.7238,1.5569,1.4562,1.2894,1.3900,1.1225,1.2519,1.1105

,0.9437}. The optimal scheme is A3 because the Z value of 

scenario A3 is the largest. It means under the existing 

formation conditions, the best perforation program is: 

perforation bullet SYD127-1, phase angle 120º, hole density 

32m, wearing depth 328.68mm, aperture 10.68mm. 

IV. CONCLUSION 

In this paper, a Perforation plan-decision based on Grey 

Cluster Relation is putted forward. This method can be widely 

used to predict the productivity of wells under different 

perforation conditions, determine the perforating efficiency of 

perforated bombs, and study how different factors (the 



2018 International Conference on Sensor Network and Computer Engineering (ICSNCE 2018) 

91 

 

perforation elasticity, perforation penetration, shot density, 

perforation diameter, perforation phase angle) impose 

influence to productivity ratio, and casing strength 

decreasing coefficient. According to the pending reservoir, 

it also let the oil production capacity to achieve the higher 

perforation operating parameters and process of excellent 

combination. It also saves a lot of manpower, materials 

and time cost, and provide the theoretical basis for the 

design of completion perforation construction. 

 

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