The Comprehensive Computation Model of Gas Permeability Based on Fuzzy Complementary Judgment Matrix

International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 6 | June. 2017 | 56 |
The Comprehensive Computation Model of Gas Permeability
Based on Fuzzy Complementary Judgment Matrix
ChunfuWei
School Of Mathematics &Information Science, Henan Polytechnic University, Jiaozuo, 454003- China
Corresponding author. E-mail address: mathwcf@163.com (Chunfu Wei).
I. INTRODUCTION
Recently, with the depth and intensity of coal mining increasing, the progressively larger volume of gas
pour out in coal seam has always been a serious threat to coal mining safety and productivity, especially in high
gassy coal mines. And gas drainage is the important means to the prevention of coal and gas dynamic failure.
Therefore, it is of significance to study the permeability law of gas drainage, and reasonably assign the
boreholes.
Among the work, reviews on the controlled factors of permeability of coal containing gas were made
by some researchers at early stage. Zhao [1] studied the coal-gas coupled mathematical model and presented the
experimental relations among gas permeability of coal samples, pore gas pressure and in situ stress under the
action of triaxial stress meter. Li and Zhao [2-3] conducted detailed researches on the controlled factors of
permeability of coal containing gas, strain-stress curve of gas, and physical and mechanical properties of soft
and hard coal.
Recently, considerable efforts have been made to study the evolution of gas permeability, the gas flow
law in coal seam, and the coupled effect of coal and gas during coal seam mining through laboratory
experiments, numerical simulations, and field tests. Yin [4-5] studied gas permeability in the process of strain-
stress, suggesting that pour out quantity of gas is closely related to deformation of coal samples, that is to say,
gas pour out quantity decreases as the coal samples compress and confining pressure increases. From the
physical simulation of industrial production of biogas, the evaluation and identification method of methane
permeability mechanism was studied in [6]. As discussed in [7], through three axis servo control permeability
experimental platform, permeability and mechanical properties of coal samples was studied under unloading
condition. Through the self-made permeability experimental platform, permeability of carbon dioxide gas to
flow through the anthracite was studied and the gas permeability evolution model was built [8]. Different
mathematical models of gas permeability were built under different conditions in [9-10]. For the permeability
mechanism of gas in the coal seam extraction, Liang [11] carried out studies on the interaction law of coal and
rock deformation and gas permeability.
ABSTRACT: In this paper, in order to reveal the gas migration law of loaded coal under multi-factor
coupling, the researches on gas permeability were carried out under different influencing factors,
namely effective stress, gas pressure, confining pressure and moisture content, with the self-developed
experimental platform of gas permeability. Meanwhile, the function relationship of each influencing
factor and permeability was established by use of the mathematical least squares principle. In this paper,
the comprehensive expression of gas permeability was established, which is based on fuzzy
complementary judgment matrix. And the comprehensive expression was drawn from the experimental
conclusions of the loaded coal under multi-factor coupling.
Keywords:Coupling, Gas permeability, fuzzy complementary judgment matrix, Gas permeability
experimental platform

The Comprehensive Computation Model of Gas Permeability Based on Fuzzy..
Although great progress has been made, it is difficult to study the evolution of gas permeability. In this
paper, with the self-developed experimental platform of coal containing gas multi- coupling at Henan
Polytechnic University, the influencing law of gas permeability was studied under various conditions of gas
pressure, confining pressure, axial stress and moisture content, and the fitting equation of each influencing
factor and permeability was established by use of the mathematical least squares principle. The notion of
coupling factor was introduced to build gas permeability model, which provides the theoretical guidance for
revealing the law of gas flow in the loaded coal containing gas.
II. SCHEMATIC OF FLUID-SOLID COUPLING PERMEABILITY EXPERIMENTAL
SYSTEM AND ITS PRINCIPLES
2.1 Experimental system
In this paper, the gas permeability experimental platform is designed and self-developed by Henan
Polytechnic University, which consists of axial pressure, confining pressure load control system, high pressure
cylinder pressure chamber, methane gas source control system, heated water systems and data acquisition
system, as shown in Fig.1.
y
d
e
f
g
i
c
k
h
jx
b
a
Fig.1 Schematic of fluid-solid coupling permeability experimental system. (a) axial loading system; (b) stress
sensor; (c) triaxial cell; (d)upper pressure head; (e) breathable version; (f) coal sample; (g) under pressure head;
(h) the oil inlet; (i) Hydraulic pump station; (j) The oil outlet; (k) Constant temperature water bath; (x) the gas
outlet; (y) High pressure gas cylinders.
The main technical indicators of the permeability experiment system is as follows: ① axial displacement: 0 ~
60mm; ② confining pressure control: 0 ~ 40MPa; ③ axial pressure control: 0 ~ 150MPa; ④ large water temperature
control accuracy: ± 0.01 ℃; ⑤ gas pressure control range: 0 ~ 13.5MPa; ⑥ temperature control range: 0 ~ 100 ℃;
⑦ sample size: φ30mm × 100mm, φ50mm × 100mm, φ80mm × 100mm; ⑧ stress, displacement, pore
pressure, deformation, temperature and flow are automatically collected by a data acquisition instrument.
2.2 Principles of gas permeability
Gas permeability measurement is usually divided into two kinds: one is the scene direct determination,
the other is the determination of physical laboratory experiment. The former is of high investment cost, long
testing time, large deviations, and the disadvantages of on-site construction difficulties. In view of this, the
latter, the determination of physical laboratory simulation, was adopted in this paper to measure the gas
permeability of loaded coal containing gas, laying the experimental foundation for further exploration of the gas
migration law inside the coal seam. The experiments are carried out based on the “routine core analysis method”
(SY/T t5336-1996 standard) and American ASTM standard [12]. Gas permeability of coal samples is calculated
through the gas pressure gradient of the samples at two ends and gas volume flow through. The calculation
formula is as follows:

)(
2
2
2
2
1
0
ppA
LQp
k


 (1)
Where, Q denotes the gas flow through the unit of time in cm3
/s; K is gas permeability in mD; μ is for
the gas viscosity coefficient in Pa·s, its value is （10.26+0.0305t）×10-6
Pa·s; t ranges from 0℃ to 100℃; L is
the length of coal sample in mm; A is the cross section of the specimen in cm2
; and p0is atmospheric pressure of
0.1MPa, p1 is the inlet gas pressure in MPa, p2is the outlet for gas pressure in Mpa，
III. ANALYSIS OF EXPERIMENTAL RESULTS
Gas molecules in coal medium have two states-adsorption and free, usually free-state gas rich in pore
and fissure space of the coal and rock skeleton, showing that gas pressure affects the permeability. The rules of
gas permeability withvarious gas pressure are studied by analyzing and regressively analyzing the permeability
under various gas pressure, and constant axial stress and confining pressure. As shown in Fig.2(a), each curve
represents the permeability curve of the same coal sample under different moisture content, the average
correlation coefficient of 0.995, the general expression of the gas pressure p and permeability k is obtained by
the least square method is:
11
2
1   ppk (2)
Where, 1 ， 1 ， 1 arethe fitting coefficient.
0.6 0.9 1.2 1.5 1.8
0.12
0.15
0.18
0.21
0.24
0.27
0.30
0.33
0.36
0.39 moisture content 0.00%
moisture content 0.96%
0.00% fitted curve
0.96% fitted curve
1.98% fitted curve
2.88% fitted curve
3.99% fitted curve
permeability（mD）
gas pressure（ MPa）
Model Parabola
Equation
y = A + B*x + C*x^2
Reduced Chi-Sqr 2.71723E-5 1.75191E-5 1.23083E-5 5.97286E-5 3.63657E-6
Adj. R-Square 0.99448 0.99325 0.9954 0.96555 0.99788
Value Standard Error
C1 A 0.78798 0.02072
C1 B -0.97964 0.03755
C1 C 0.41413 0.01548
E1 A 0.6271 0.01664
E1 B -0.71854 0.03016
E1 C 0.30159 0.01243
G1 A 0.60684 0.01395
G1 B -0.73066 0.02528
G1 C 0.30643 0.01042
I1 A 0.5114 0.03072
I1 B -0.58855 0.05568
I1 C 0.24516 0.02295
B A 0.48972 0.00758
B B -0.59585 0.01374
B C 0.2446 0.00566
0.6 0.9 1.2 1.5 1.8
0.12
0.15
0.18
0.21
0.24
0.27
0.30
0.33
0.36
0.39
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.00% fitted curve
1.12% fitted curve
2.10% fitted curve
2.97% fitted curve
3.89% fitted curve
permealibity(mD)
confining pressure(Mpa)
Equation y = a*exp(b*x)
Adj. R-Square 0.98427 0.99199 0.99272 0.98589 0.98182
C1 a 0.56115 0.0203
C1 b -0.2158 0.01426
E1 a 0.50375 0.01392
E1 b -0.23104 0.01112
G1 a 0.41254 0.01008
G1 b -0.21046 0.00956
I1 a 0.34586 0.01135
I1 b -0.20057 0.01268
B a 0.29874 0.01043
B b -0.18403 0.0132
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
(a) Permeability and gas pressure (b) Permeability and confining pressure
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
mositure content 3.12%
mosisure content 1.98%
3.84% fitted curve
3.12% fitted curve
1.98% fitted curve
1.07% fitted curve
0.00% fitted curve
permealibity（mD）
axial stress（ MPa）
Model Exp2PMod1
Equation y = a*exp(b*x)
Reduced
Chi-Sqr
7.24317E-7 2.07522E-
6
4.29227E-6 1.03052E-
5
3.50861E-
5
Adj. R-Square 0.99845 0.99786 0.99838 0.9989 0.99853
Value Standard Erro
r
a 0.07966 0.00159
含水率3.84% b -0.63707 0.01772
a 0.11447 0.00271
含水率3.12% b -0.64348 0.02119
a 0.18843 0.00382
含水率1.98% b -0.62415 0.01768
a 0.35693 0.00595
含水率1.07% b -0.63028 0.01468
a 0.5693 0.01104
含水率0.00% b -0.6342 0.01714
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
axial stress 0.5(MPa)
0.5(MPa) fitted curve
permeability（mD）
moisture content（ %）
Model Parabola
Equation
y = A + B*x + C*x^
2
Reduced Chi-Sqr 1.51821E-4 3.30317E-5 7.35222E-6 4.91668E-6 3.05473E-6
Adj. R-Square 0.99308 0.99428 0.99608 0.99109 0.97291
C1 A 0.41886 0.0118
C1 B -0.18055 0.0146
C1 C 0.0227 0.00362
E1 A 0.21464 0.0055
E1 B -0.09142 0.00681
E1 C 0.01131 0.00169
G1 A 0.1227 0.0026
G1 B -0.05234 0.00321
G1 C 0.00651 7.97327E-4
I1 A 0.066 0.00212
I1 B -0.02816 0.00263
I1 C 0.00347 6.52023E-4
B A 0.02954 0.00167
B B -0.0126 0.00207
B C 0.00154 5.13941E-4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
(c) Permeability and axial pressure (d) Permeability and moisture content
Fig. 2The function relationship between permeability and each influencing factor
Based on the original experimental data and permeability calculation principle , the corresponding
permeability K is obtained; and then regressively analyze the experimental data, shown in Fig.2b, the average
correlation coefficient is 0.988, the function relationship between permeability and confining pressure is obtained by
the principle of least square method:

 322 exp  k (3)
Where, 2 ， 2 are the fitting coefficient.
The regression analysis of experimental data on permeability and axial stress is made, asshown in Fig.2c. The
average correlation coefficient is 0.995, a function relationship between permeability and axial stress is:
 133 -exp k (4)
Where, 3 ， 3 are the fitting coefficient.
The regression analysis of experimental data on permeability and moisture content is made, asshown in
Fig.2d.The average correlation coefficient is 0.994, a function relationship between permeability and moisture content
is:
44
2
4   wwk (5)
Where, 4 ， 4 ， 4 are the fitting coefficient.
IV. GAS PERMEABILITY MODELl
With different vertical stress, horizontal stress, gas pressure and moisture content and other factors
suffered by coal containing gas, gas permeability will also be changed, and therefore the prediction of the
effective permeability of coal containing gas should also take the impact of these factors into account. For the
convenience and maneuverability, people often set other factors constant, study the function relation between
the permeability and a certain influencing factor. In fact, the size of gas permeability in the coal seam at some
stage is the combined effect of the moisture content, the vertical stress, gas pressure, the horizontal stress and
temperature acting together; While most experiments are to set certain factor as variables, and other influencing
factors in this procedure are assumed to be constant; At this point, no matter which experimental factors
resulting equation uses to represent the scene, the experiments will have a certain one-sidedness. Therefore, by
introducing the coupling factor, neglecting the influence of temperature and other factors, the author established
a comprehensive expression of gas permeability under multi-factor coupling:















1,2,3,4)(i1
)()(
)()(
1
433
211
1
n
i
i
n
i
ii
pff
wffkk



（6）
Where, it is assumed that )( 1f , )(pf , )( 3f and )(wf respectively represent the functions between
permeability and axial stress, gas pressure, containing pressure and moisture content. i is coupling factor, which is
the weight of a certain influencing factor under the multi-factor coupling effects.
Due to the uncertainty of the coupling system itself as well as our ambiguous understanding of the system, it
is difficult for researchers to give concrete numerical coupling coefficient. But we can qualitatively compare pairwise
relative to the sensitivity of the various factors affecting the permeability,building fuzzy complementary judgment
matrix for solving the coupling coefficient. Then in this paper, based on fuzzy complementary judgment matrix, a
general method for solving coupled vector
T
n ),( 21   is proposed.
First, as discussed in the reported works [13], fuzzy judgment matrix and the definition of fuzzy
complementary judgment matrix are introduced: Set judgment matrix nnijaA  )( , if 10  ija , the matrix

A is called fuzzy judgment matrix; Set judgment matrix nnijaA  )( , if 1 jiij aa ， 1iia , the
matrix A is called fuzzy complementary judgment matrix.Tab.4-1 shows the three complementary scale value [14-15],
according to the definition of fuzzy complementary judgment matrix, thus matrix composed of the previous three
scale value is fuzzy complementary judgment matrix.
Tab.4-1 Complementary language scale value
0-1
scale
five
scale
nine
scale
meaning
0 0.1 0.1 element B is extremely sensitive to element A
0.138 element B is highly sensitive to element A
0.3 0.325 element B is more sensitive than element A
0.439 element B is slightly sensitive to element A
0.5 0.5 0.5 element A is equally sensitive to element B
0.561 element A is slightly sensitive to element B
0.7 0.675 element A is more sensitive than element B
0.862 element A is highly sensitive to element B
1 0.9 0.9 element A is extremely sensitive to element B
Then, assume that the target layer of permeability k is related to the factors ,,,, 21 NBBB  of the next
level B (a collection of permeability influencing factors: gas pressure, confining pressure, axial stress, moisture
content),so every influencing factor iB has a coupling coefficient ),,2,1()( niBi   ink , which
can be expressed as:
T
n ),,,( 21   ，（7）
called coupled vector
Set the numerical ratio ji  / of the elements iB and jB of layer B as the elements, the judgment matrix
M is as follows:
nnjiM  )/(  ，（8）
By left multiplying the judgment matrix M with coupling coefficient vector , we have
 nM  . （9）
By the matrix theory, the largest eigenvalue n of matrix M is denoted as max , the corresponding
eigenvectors to eigenvalue n is  , which is also the desired coupling vector.
The eigenvectors is obtained by the following procedures. By establishing the judgment Matrix to make
pairwise comparison on the degree of permeability affected by certain factors, we can obtain the fuzzy language value
ijb , and then get the estimation matrix B of numerical ratio matrix M , called the judgment matrix of coupling
factors. The steps are as follows:
① Determine the fuzzy judgment value )/( ji BBf by pairwise comparing the influencing factorsin Tab.4-2.
Tab.4-2Coupling factor sensitivity judgment value table
Sensitive degree of comparing
iB with jB
)/( ji BBf )/( ij BBf
iB is equally sensitive to jB 1 1
iB is slightly sensitive to jB 3 1
iB is more sensitive than jB 5 1
iB is highly sensitive to jB 7 1
iB is absolutely sensitive to jB 9 1
Note: the sensitive degree of comparing iB with jB is among 2,4,6,8.

Take any two elements iB and jB , comparing the sensitivity of iB to jB , make )/( ji BBf as the discriminant value
of the "sensitivity"; Conversely, )/( ij BBf represents the discriminant value of the "sensitivity" of jB to iB . The
discriminant value of )/( ji BBf and )/( ij BBf can be established by fuzzy language discriminant value, shown
in Tab.2.For example,during the experiment, if the factor iB affecting the permeability k ismore sensitive than the
factor jB , the discriminant value is 5)/( ji BBf .
② Establish fuzzy complementary judgment matrix B ,
If
),,2,1,(
)/(
)/(
nji
BBf
BBf
b
ij
ji
ij  ，（10）
we have the following complementary judgment matrix















nnn
n
n
bbb
bbb
bbb
B
121
22221
11211




, （11）
Where, 1ijb , 1 jiij bb ， },,2,1{, nji  .
The largest eigenvalue max of matrix B andthe corresponding eigenvectors are obtained. The eigenvectors is also the
corresponding coupling vector },,,{ 21 nBBBB  .
③Compute the eigenvectors. Common methods of computing the eigenvectors are, among others,the following: ANC,
square root algorithm, eigenvalue method and the least squares method. Theoretically, it should solvethe exact values
of the each coupling factor, that is, the eigenvalue of matrix B . In order to calculate conveniently, in the practical
applications,approximate solution to the eigenvalue of matrix B usually is taken as the coupling coefficient. “Square
root algorithm” is adopted in this paper to compute the eigenvalue of matrixB
Normalize matrix B by row, we have
),,2,1,(
1
nji
b
b
c n
i
ij
ij
ij 

（12）
The normalized matrix is denoted as C , then sum the matrix by line, we have
),,2,1,(
1
njicm
n
j
iji  
（13）
Then normalize vector T
nmmmm ),,,( 21  ，the desired eigenvector is obtained. Each component of the
eigenvector is the coupling factor corresponding to the factors affecting permeability. That is:
T
n ),,,( 21   . (14)
Where， }),,2,1{(
1
nimm
n
i
iii  

Verification of the matrix for consistency.The fuzzy complementary judgment matrix needed to be verified for
consistency,because the experts may have been subjective and biased in the process of establishing the
complementary judgment matrix, given thecomplexity of being objective. The procedure for verification of
consistency of the judgment matrix is as follows:
Step 1, compute the largest eigenvalue mzx .


n
i i
i
mzx
n
B
1
)(


 , (15)
Where, B is afuzzy complementary judgment matrix,  is the eigenvector of matrix B , mzx is the largest
eigenvalue. The value of the ith element of the eigenvector is denoted as i ; n is the matrix order of matrix B .

Step 2, compute the consistency indexCI ：
1
max



n
n
CI

. (16)
Step 3, compute the consistency proportionCR ：
RICICR / . (17)
Tab.4-3 The average random consistency index value
n 1 2 3 4 5 6 7 8
RI 0 0 0.52 0.89 1.12 1.26 1.36 1.41
n 9 10 11 12 13 14 15
RI 1.46 1.49 1.52 1.54 1.56 1.58 1.59
Where, RI is the mean max consistency index, the data is shownin Tab.4-3[15]
. When CR < 0.1, we can accept
that the estimation matrix has satisfied consistency, otherwise we must adjustthe factors of the estimation matrix until
it has satisfied consistency.
V. CONCLUSIONS
Experiment results showed that with gas pressure increasing, permeability of coal samples was in non-
linear increase (phases of increase were rather obvious), and a function of permeability and gas pressure
presented a "V"; That is, the gas pressure existed a turning point, smaller than this value, the permeability
decreased with gas pressure increasing, larger than this value, the permeability increased with the increase of
gas pressure. The experimental data were fitted by the function relationship between permeability and various
influencing factors. The permeability and the axial stress, confining pressure showed negative exponential
function relationship. And permeability and moisture content showed quadratic function. Finally, based on
fuzzy complementary judgment matrix, model of coal gas seepage was established under the coupling effect of
the influencing factors such as gas pressure, axial stress, confining pressure and moisture content, etc.
ACKNOWLEDGEMENTS
This paper was partially supported by science and technology project of Henan Province under Grant
No. 152102210317，by the Key Research Project for the University of Henan Province under Grant No.
15A440008, by the Fundamental Research Funds for the Universities of Henan Province under Grant No.
NSFRF140138, and by the Doctoral Foundation at Henan Polytechnic University in China under Grant No.
B2015-52.
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