Study on Application of Three Methods for Calculating the Dispersion Parameters – A Case Study in Yinchuan, China

International Journal of Science and Research (IJSR), India Online ISSN: 2319-7064
Volume 2 Issue 8, August 2013
www.ijsr.net
Study on Application of Three Methods for
Calculating the Dispersion Parameters – A Case
Study in Yinchuan, China
Zhang Zhihua1, 2
, Zhang Hanting1, 2
1
School of Environmental science and Engineering, Chang’an University, No. 126 Yanta Road, Xi’an, 710054, Shaanxi Province, P.R. China
2
Key Laboratory of Subsurface Hydrology and Ecological Effect in Arid Region of Ministry of Education, No. 126 Yanta Road, Xi’an
710054, Shaanxi Province, P.R. China
Abstract: In this paper, linear graphical method, moment method and inverse function method are first applied in the laboratory test of
one dimensional sand column device, determining the longitudinal dispersion coefficient. The longitudinal dispersions for five groups of
sand taken from 20cm below the ground surface in the Oil Refinery of China Petroleum Ningxia Filial are obtained. On this basis, the
problems within the calculation process when the three kinds of methods are applied into actual data were discussed. It can be readily
concluded that the three values of dispersion coefficients are approximate, and the errors caused by the subjective factors of artificial
mapping and numerical reading were avoided. The inverse function method is recommended to apply for the high accuracy, sample
calculation process, less known conditions and better linearity.
Keywords: longitudinal dispersion coefficient; laboratory test; linear graphical method; moment method; inverse function method.
1. Introduction
Dispersion parameter is the foundation of researching the
groundwater pollutant regular pattern and predicting the
water quality (Liu et al. 2011). The convection dispersion and
molecular diffusion in the porous medium are named as
hydrodynamic dispersion coefficient. It is a comprehensive
reflection coefficient of solute, soil, and it is not only related
to the condition of porous medium and the properties of
solute, but also depended on the effect of moisture content
and the velocity of pore water (Song et al. 1998). The
dispersion parameter is indispensable for the study of
migration law of chemical fertilizers, pesticides and heavy
metals in farmland, the monitoring of water movement and
salt in saline and alkaline land, and the protection of the
groundwater resource. The macroscopic parameters that
quantify the solute dispersion in porous medium are
longitudinal and transverse dispersion coefficients
(Aggelopoulos et al. 2007). There are many researches about
hydrodynamic dispersion in the world. Several methods based
on analytical solutions have been developed particularly in
order to determine the hydrodispersive characteristics from
the tracer experiments (Wang et al. 1987; Fried 1975; Sauty
1978). The theory of hydrodynamic dispersion in porous
materials has been developed on a series of boundary
conditions (Bachmat and Bear 1964; Fried 1975; Hibsch and
Kreft 1979; Harleman and Rumer 1963). Delgado (2007) has
studied the longitudinal and transverse dispersion in porous
media. A series of laboratory tests were carried out on
artificially produced particle mixtures (Klotz 1980; Klotz and
Moser 1974). For the measurement of the dispersion
parameter in saturated aquifer, the commonly used and
mature method is laboratory dispersion test with the column
devices. At present, the laboratory dispersion test is mostly
used in this situation: the steady flow, continuous tracer
injection of definite concentration at one side of the column.
Get the change of concentration of different time at each
section of one dimensional sand column, repeatedly (Li et al.
1983). For the calculation of dispersion coefficient, the most
widely used method is the method of breakthrough curves
(Zhang et al. 2003). In addition, Guo J Q also put forward a
series of solutions aiming at one dimensional dispersion of
groundwater. Such as linear graphical method (Guo et al.
1997), the moment method (Guo et al. 1997) and inverse
function method (Guo et al. 1999). These methods have not
been used in practice science they were put forward. Here,
five groups of actual data are calculated by the three methods.
2. Methods
The tracer of concentration c0 is continuously and steadily
poured into the top of the semi-infinite columnar aquifer.
Seepage is uniform flow. One-dimensional dispersion, and
there isn’t other source sink term. The mathematical model (1)
for this problem is listed as follows:



















00),(
0),0(
00)0,(
0,0
0
2
2
ttc
ttc
xxc
tx
x
c
V
c
t
c
c
x
DL
(1)
The solution of this mathematical model is:
)]
2
()exp()
2
([
2
)]exp(
1
[)exp(),( 1
0
2 DD
V
D
c
L
D
c
LL
x
L
o
L
Vtx
erfc
t
Vtx
erfcpbk
p
Vt
txc



 
(2)
Or it can be written as:
)]
2
()exp()
2
([
2
1),(
0 DD
V
Dc LL
x
L
Vtx
erfc
t
Vtx
erfc
txc
c



 (3)
When x is sufficiently large, the second term is too small
compared with the first term in equation (3). Then the second
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term can be neglected, and the equation (3) can be written as:
)
2
(
2
1),(
0 t
Vtx
erfc
txc
Dc L

 (4)
Where, c(x,t) - the tracer concentration at t time, x coordinate.
c0 – tracer concentration at the injection site.
x – vertical coordinate.
t – time coordinate.
V – seepage velocity.
DL – longitudinal dispersion coefficient.
erfc() – complementary error function, and its
expression is as follows.


 d
t
Vtx
erfc t
Vtx
L
DL
D
)
2
exp(
2
1
1)
2
( 2
2




 (5)
Based on the above two equations the following methods are
put forward.
2.1 Linear graphical method
Linear graphical method is put forward by Guo J Q (1997).
This solution cites the theory in literature (Wang et al. 1987)
and is obtained via proper conversion conducted for
approximate analytic solution during the dispersion test on the
condition of one dimensional, stable state and injection of
tracer with definite concentration at one side of semi-infinite
sand column. Take the injection site as the origin of
coordinates, and the groundwater flow direction as the
direction of x axis. Based on equations (4), (5) and the integral
principle of variable upper limit, the linear equation (6) can be
obtained after taking the logarithm. The linear constant is a
function of longitudinal dispersion coefficient, so it is feasible
to determine dispersion coefficient with linear diagrammatic
solution or by linear regression.
byaT  (6)
Where,
)4ln(
2
1
DLa 
(7)
DL
b
4
1

(8)
The following formulas can be used for data transformation:
t
t
y
i
i
i
Vx 

(9)
)ln(
1
1
2/1
yy
cc
T
ii
ii
i






(10)
)(
4
1
12/1 yyy iii  
(11)
Here linear diagrammatic solution or linear regression method
can be used to solve the value of DL after confirming the
value a, b with the following equations.
)2exp(
4
1
aDL 
 (12)
Or
b
DL
4
1

(13)
2.2 Moment method
The moment method (Guo et al. 1997) is derived with forepart
inference result of linear graphical method. And its final form
is the same as normal probability density function of variance
δ2
=4DL，and mathematical expectation m=0. As follows:
)
4
(
4
1
)(
2
D
y
D LL
Expyf 
 (14)
In equation (14), the expression of y is the same as equation
(9). The value of δ2
can be worked out with the formulaδ
2
=M2/M0. Then the value of dispersion coefficient (DL) can be
calculated from the formula DL= δ 2
/4. Some equations
needed in the computational process are listed as follows:
c
txc
c
0
),(

(15)
yy
cc
y
ii
ii
if






1
1
2/1 )(
(16)


 






 1
1 1
1
2
1
1
2 )(
4
)(
)(n
i ii
ii
ii
ii
yy
yy
yy
cc
M
(17)
 
 





 1
1 1
1
1
0 )()(n
i ii
ii
ii
yy
yy
cc
M
(18)
2.3 Inverse function method
The linearization idea is adopted within the process of the
inverse function method’s putting forward (Guo et al. 1999).
When data is sufficient the equation (5) can be transformed
into the following equation.
btaGt  (19)
Where,
)(2
1
yarcNtGt  (20)
The expression of y is the same as equation (9). ArcN(y)
means the inverse function of the normal function. It can be
got from pegging normal function table. The value of a and b
may be obtained through linear diagrammatic solution or
linear regression method by using the serial data Gt~t. In
equation (19), the expressions of a and b can be written as
followings:
D
x
L
a
2
0

(21)
DL
V
b
2

(22)
Once the value a and b are obtained, the value of dispersion
coefficient (DL) and seepage velocity (V) can be got with
equation (21) and (22).
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3. Study Area and Data Collection
3.1 Description of study area
The oil refinery of China Petroleum Ningxia filial is located in
the south of Yinchuan City. The study area is located in the
alluvial-proluvial plain, and the lithology of the phreatic
aquifer mainly is fine sand. Here, the groundwater flows from
southwest to northeast. Pollutants may migration with
groundwater flow to south water source protection area. In
this experiment, the sand column uses phreatic aquifer core of
the drill hole in the oil refinery. The specification of testing
apparatus depends on the sample size. The bigger size the
sample, the larger the seepage pipe diameter required. And the
seepage length must be more 3~4 times than pipe diameter
(Zhang et al. 1993). Five groups of laboratory tests are
arranged with one dimensional column device. The process of
the tests is in accordance with the description in literature (Li
et al. 2012).
3.2 Data collection
The five groups of data measured in laboratory dispersion test
are called data1, data2, data3, data4 and data5. As shown in
table1 to table5.
Table 1: Data 1
t(d) t(d) t(d) t(d) t(d)
0.00
0
0.000
6
0.12
5
0.001
6
0.22
2
0.000
0
0.31
9
0.755
7
0.41
7
0.968
5
0.02
1
0.000
2
0.13
9
0.000
1
0.23
6
0.003
2
0.33
3
0.806
9
0.43
1
1.000
0
0.04
2
0.000
9
0.15
3
0.001
4
0.25
0
0.025
9
0.34
7
0.858
1
0.44
4
0.976
4
0.06
3
0.000
2
0.16
7
0.000
8
0.26
4
0.135
0
0.36
1
0.905
4
0.45
8
0.981
9
0.08
3
0.000
5
0.18
1
0.000
6
0.27
8
0.336
0
0.37
5
0.960
6
0.09
7
0.000
9
0.19
4
0.000
5
0.29
2
0.475
9
0.38
9
0.956
7
0.11
1
0.000
6
0.20
8
0.000
6
0.30
6
0.499
5
0.40
3
0.993
7
Table 2: Data 2
0.00
0
0.001
2
0.48
6
0.003
2
0.81
9
0.006
1
1.15
3
0.134
2
1.56
3
0.735
5
0.02
1
0.001
2
0.50
0
0.002
8
0.83
3
0.008
4
1.16
7
0.173
0
1.58
3
0.752
0
0.04
2
0.004
8
0.51
4
0.001
8
0.84
7
0.009
5
1.18
1
0.191
4
1.60
4
0.765
0
0.06
3
0.003
4
0.52
8
0.002
6
0.86
1
0.011
4
1.19
4
0.197
9
1.62
5
0.784
6
0.08
3
0.003
6
0.54
2
0.004
3
0.87
5
0.012
6
1.20
8
0.214
4
1.64
6
0.788
0
0.10
4
0.004
4
0.55
6
0.001
5
0.88
9
0.013
0
1.22
2
0.245
5
1.66
7
0.814
5
0.12
5
0.003
2
0.56
9
0.002
2
0.90
3
0.015
3
1.23
6
0.276
6
1.68
8
0.826
8
0.13
9
0.005
8
0.58
3
0.001
8
0.91
7
0.016
8
1.25
0
0.300
7
1.70
8
0.848
3
0.15
3
0.001
5
0.59
7
0.003
8
0.93
1
0.019
7
1.26
4
0.315
7
1.72
9
0.865
2
0.16
7
0.004
5
0.61
1
0.003
0
0.94
4
0.023
3
1.27
8
0.337
2
1.75
0
0.880
5
0.18
1
0.004
5
0.62
5
0.004
3
0.95
8
0.025
8
1.29
2
0.367
1
1.77
1
0.830
6
0.19
4
0.002
5
0.63
9
0.003
0
0.97
2
0.030
5
1.30
6
0.380
2
1.79
2
0.907
4
0.22
2
0.003
3
0.65
3
0.003
0
0.98
6
0.033
9
1.31
9
0.400
9
1.81
3
0.917
0
0.23
6
0.002
0
0.66
7
0.001
6
1.00
0
0.036
0
1.33
3
0.429
7
1.83
3
0.930
8
0.25
0
0.003
1
0.68
1
0.002
6
1.01
4
0.048
3
1.35
4
0.452
7
1.85
4
0.943
8
0.26
4
0.003
6
0.69
4
0.003
4
1.02
8
0.052
3
1.37
5
0.491
1
1.87
5
0.959
9
0.27
8
0.001
0
0.70
8
0.003
0
1.04
2
0.056
7
1.39
6
0.516
0
1.89
6
0.959
9
0.33
3
0.002
8
0.72
2
0.009
0
1.05
6
0.065
5
1.41
7
0.533
7
1.91
7
0.974
5
0.36
1
0.003
6
0.73
6
0.005
1
1.06
9
0.080
5
1.43
8
0.582
0
1.93
8
0.971
1
0.38
9
0.002
3
0.75
0
0.000
9
1.08
3
0.084
0
1.45
8
0.588
5
1.95
8
1.002
9
0.41
7
0.005
0
0.76
4
0.004
5
1.09
7
0.103
1
1.47
9
0.619
6
1.97
9
0.987
6
0.44
4
0.005
4
0.77
8
0.005
4
1.11
1
0.115
4
1.50
0
0.694
4
2.00
0
0.997
2
0.45
8
0.004
6
0.79
2
0.005
3
1.12
5
0.124
2
1.52
1
0.695
2
2.02
1
1.006
0
0.47
2
0.003
4
0.80
6
0.005
3
1.13
9
0.130
8
1.54
2
0.713
6
2.04
2
1.006
4
Table 3: Data 3
0.00
0
0.000
9
0.20
8
0.001
4
0.37
5
0.196
0
0.54
2
0.888
0
0.70
8
0.978
7
0.02
1
0.001
4
0.22
2
0.002
1
0.38
9
0.306
1
0.55
6
0.902
5
0.72
2
0.978
7
0.04
2
0.000
6
0.23
6
0.002
1
0.40
3
0.434
8
0.56
9
0.902
5
0.73
6
0.978
7
0.06
3
0.000
8
0.25
0
0.001
4
0.41
7
0.548
3
0.58
3
0.909
4
0.75
0
0.997
7
0.08
3
0.000
8
0.26
4
0.002
3
0.43
1
0.658
7
0.59
7
0.925
4
0.76
4
0.993
9
0.10
4
0.001
2
0.27
8
0.003
7
0.44
4
0.690
7
0.61
1
0.944
4
0.77
8
0.996
2
0.12
5
0.000
8
0.29
2
0.004
8
0.45
8
0.738
0
0.62
5
0.948
2
0.79
2
0.975
6
0.13
9
0.000
0
0.30
6
0.005
2
0.47
2
0.788
2
0.63
9
0.947
4
0.80
6
1.000
0
0.15
3
0.001
3
0.31
9
0.008
6
0.48
6
0.817
2
0.65
3
0.963
4
0.16
7
0.001
8
0.33
3
0.021
6
0.50
0
0.833
9
0.66
7
0.965
0
0.18
1
0.002
0
0.34
7
0.062
4
0.51
4
0.845
4
0.68
1
0.925
4
0.19
4
0.000
2
0.36
1
0.111
8
0.52
8
0.891
1
0.69
4
0.963
4
Table 4: Data 4
0.00
0
0.000
7
0.19
4
0.000
0
0.34
7
0.456
4
0.50
0
0.942
3
0.65
3
0.970
8
0.02
1
0.001
5
0.20
8
0.001
1
0.36
1
0.542
5
0.51
4
0.946
8
0.66
7
0.965
6
0.04
2
0.001
0
0.22
2
0.001
9
0.37
5
0.629
4
0.52
8
0.939
3
0.68
1
0.935
6
0.06
3
0.000
7
0.23
6
0.002
2
0.38
9
0.714
7
0.54
2
0.955
8
0.69
4
0.976
8
0.08
3
0.000
1
0.25
0
0.007
1
0.40
3
0.778
4
0.55
6
0.963
3
0.70
8
0.988
0
0.10
4
0.000
1
0.26
4
0.021
0
0.41
7
0.842
0
0.56
9
0.955
8
0.72
2
1.000
0
0.12
5
0.001
3
0.27
8
0.055
2
0.43
1
0.840
5
0.58
3
0.992
5
0.73
6
0.985
0
0.13
9
0.000
4
0.29
2
0.117
6
0.44
4
0.924
4
0.59
7
0.946
8
0.75
0
0.991
0
0.15
3
0.000
4
0.30
6
0.180
1
0.45
8
0.909
4
0.61
1
0.946
8
0.76
4
0.994
8
0.16
7
0.000
5
0.31
9
0.272
6
0.47
2
0.931
9
0.62
5
0.973
0
0.18
1
0.001
9
0.33
3
0.349
3
0.48
6
0.935
6
0.63
9
0.970
8
Table 5: Data 5
0.00
0
0.000
8
0.18
1
0.000
7
0.31
9
0.796
2
0.45
8
0.974
3
0.59
7
0.963
8
0.02
1
0.000
0
0.19
4
0.000
6
0.33
3
0.831
7
0.47
2
0.975
8
0.61
1
0.988
7
0.04
2
0.000
8
0.20
8
0.002
3
0.34
7
0.850
6
0.48
6
0.977
4
0.62
5
0.979
6
0.06
3
0.001
0
0.22
2
0.012
8
0.36
1
0.881
5
0.50
0
0.966
0
0.63
9
0.977
4
0.08
3
0.000
6
0.23
6
0.067
5
0.37
5
0.889
8
0.51
4
0.968
3
0.70
8
1.000
0
0.10
4
0.000
4
0.25
0
0.200
4
0.38
9
0.895
8
0.52
8
0.969
8
0.72
2
0.998
5
0.12
5
0.001
3
0.26
4
0.350
2
0.40
3
0.924
5
0.54
2
0.982
6
0.73
6
1.000
0
0.13
9
0.001
1
0.27
8
0.490
6
0.41
7
0.945
7
0.55
6
0.954
7
0.15
3
0.001
5
0.29
2
0.653
6
0.43
1
0.954
7
0.56
9
0.953
2
0.16
7
0.001
1
0.30
6
0.735
8
0.44
4
0.963
8
0.58
3
0.962
3
4. Results and discussion
Table 6 shows the results of dispersion coefficient with three
methods and five groups of data. The average value of
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dispersion coefficient (DL) with linear graphical method,
moment method and inverse function method is respectively
0.068 m2
/d, 0.056 m2
/d and 0.081 m2
/d. It can be easily found
that the three values are approximate. In inverse function
method, the value of seepage velocity (V) calculated is very
approximate to that measured, which brings convenience to
the column experiment.
Table 6: Calculation results with the three methods
Data Data1 Data2 Data3 Data4 Data5 Average value
x(m) 1.3 1.29 1.3 1.3 1.29
V(m/d) 4.6656 0.948 3.3336 3.8976 5.0064
Linear graphical method
a -0.2676 0.4628 -0.029 -0.277 -0.2331
b -2.8921
-11.551
5
-6.5338 -2.8137 -2.3529
DL(m
2
/d) 0.0864 0.0216 0.0383 0.0889 0.1063 0.068
Moment method
M2 -0.1287 -0.2 -0.1595 -0.1942 -0.4358
M0 -0.9816 -1.0051 -0.9986 -0.9933 -1
DL(m
2
/d) 0.0328 0.0497 0.0399 0.0489 0.1089 0.056
Inverse function method
a 4.398 5.729 3.152 2.796 2.398
b -14.37 -4.155 -7.079 -7.751 -8.235
DL(m
2/
d) 0.0437 0.0254 0.0851 0.1081 0.1447 0.081
V(m/d) 4.2476 0.9356 2.9196 3.6038 4.43
On the condition of DL = 0.5 m2
/d, c0 = 333.33 g/m3, V = 5
m2
/d, x = 1 m, the data c/c0 ~ ti can determined with equation
(5), listed in table 7 (Guo 1997, 1999). The relative error can
be obtained with these theoretical data for every method (table
8).
Table 7: Theoretical data
t(d) c/c0 t(d) c/c0 t(d) c/c0 t(d) c/c0
0.1 0.057 0.25 0.691 0.4 0.943 0.55 0.991
0.15 0.259 0.3 0.891 0.45 0.967 0.6 0.995
0.2 0.5 0.35 0.898 0.5 0.983 0.65 0.997
Table 8: Calculation results of theoretical data
Name of method DL(m2
/d) relative error（%）
Method of breakthrough curves 0.564 12.8
Linear graphical method 0.507 1.4
Moment method 0.469 -6.2
Inverse function method 0.503 0.6
From table 8, it can be found that the relative error of four
methods from small to big is respectively inverse function
method, linear graphical method, moment method and method
of breakthrough curves. In other words, the relative error of
the commonly used method (method of breakthrough curves)
is the biggest, although this method is easy to operate. Taking
the result of inverse function method whose relative error is
the smallest as reference, the close degree of results of other
three methods relative to inverse function from small to big is
respectively linear graphical method, moment method and
break through.
According to this, the data in column 6 should be closer to
data in column 13 than data in column 9 (Table 6). But this is
not the case. In table1, the value of dispersion coefficient
(DL) with moment method is closer to inverse function
method than linear graphical method for data1, data3 and
data5, and the value of DL with linear graphical method is
closer to inverse function method than moment method for
data2, data4 and average value.
Compared with the method of breakthrough curves, the
relative error of the three methods mentioned in this paper is
smaller when using theoretic data (Guo et al. 1997, 1997,
1999). The three methods avoid the errors caused by the
subjective factors of artificial mapping and numerical reading,
and the calculation results have high accuracy, but they have
different applying conditions.
For linear graphical method, it can be found that yi is
monotone decreasing function from expression (9). In order
to guarantee that Ti+1/2 is meaningful in expression (10), value
must be in the increasing trend. But the value of metrical
fluctuate along with the time goes by. So the data which
decrease with time must be deleted when using this method.
Simultaneously, it must be guaranteed that the metrical time
intervals of the rest data are more or less the same. Because
the value yi varies with time, the value yi+1/2 may become
much larger relative to other value when the time interval is
large. Much deviation of yi+1/2 and Ti+1/2 may be caused under
this situation mentioned before, which bring about the variety
of DL. The consequence is that the waste of data resources
and fluctuant results. Especially under the condition of that
the metrical data itself is less and some data must be deleted,
the results will change greatly. In a word, this method suits for
the situation that plenty data, no fluctuation existing, and
better linearity between yi+1/2 and Ti+1/2.
The moment method needn’t considerate the linearity
compared with the other two methods. It can be completed via
some simple calculation processes. Due to the fluctuation of
measuring data have no effect on counting process. No data
need to be deleted by this method. The relative error of
theoretical data by using the moment method is smaller than
that of by using the method of breakthrough curves, though
there will be a little error among counting process when the
difference quotient is used instead of the differential quotient.
With inverse function method, it can be found that there is a
good linear relationship between Gt and t. Show in figure 1
and 2. So it becomes unimportant whether the metrical data
are more or less. And applying computer programming into
the work of pegging normal function table will bring
convenience. But the following situation must be noted.
When the data are very small or close to 0.5, they must be
deleted. Because inverse function value of y corresponding to
these data will be very approximate or the same, and which
will have great influence on the linearity of the results.
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Figure 1: Inverse function method for data1
Figure 2: Inverse function method for data2
5. Conclusions
(1) With linear graphical method, the moment method and
inverse function method, the dispersion coefficient in
Yinchuan oil refinery are obtained. They are 0.068 m2
/d,
0.056 m2
/d and 0.081 m2
/d respectively. The three results are
approximate. These methods avoid the errors caused by the
subjective factors of artificial mapping and numerical reading.
Data can be quickly and easily calculated by computer.
(2) For linear graphical method, the fluctuation and the
interval of value have great effects on the results. Adding or
deleting some data will also lead to the variety of results.
(3) Although the fluctuation of value has no effect on the
moment method, this approximate calculation which the
difference quotient is used instead of the differential quotient
brings error to the results.
(4) The fluctuation, interval and quantity of data all have no
influence on results with inverse function method. And the
value of seepage velocity (V) didn’t need to be known. So
inverse function method is a decent method. However, as to
the mountainous work of looking up the normal function
table, computer programming brings convenience.
(5) Overall, it can be observed that the inverse function
method is easy to calculate the dispersion coefficient under
one-dimensional dispersion, which has less known
conditions, smaller relative error and well linearity of results.
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Author Profile
Zhang Zhihua received bachelor degree from Chang'an University
in 2011, now studying at the Graduate School of Chang'an
University, China.
Zhang Hanting received bachelor degree from Chang'an University
in 2011, now studying at the Graduate School of Chang'an
University, China.
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