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Water Practice & Technology Vol 8 No 2
doi: 10.2166/wpt.2013.031
Optimization of Denser Nonaqueous Phase Liquids-contaminated
groundwater remediation based on Kriging surrogate model
Wenxi Lu*, Haibo Chu, Ying Zhao and Jiannan Luo
College of Environment and Resources, Jilin University, No. 2519, Jiefangdalu Road, Changchun 130021, PR China
*Corresponding author. E-mail: [email protected]
Abstract
Spillage of large amounts of Denser Nonaqueous Phase Liquids (DNAPLs) had resulted in serious pollution of
groundwater resources throughout the world; a large number of studies had demonstrated surfactant-enhanced
remediation is a more effective approach to remediate DNAPLs contaminations. In this paper, the remediation
optimization process was carried out in three steps. Firstly, a water-oil-surfactant simulation model had been
ﬁrstly established to simulate a surfactant enhanced aquifer remediation process. The Kriging surrogate model
had been developed to get a similar input–output relationship with simulation model. In the ﬁnal, a nonlinear
optimization model was formulated for the minimum cost, and Kriging surrogate model had been embedded
into the optimization model as a constrained condition. What is more, simulated annealing method was used
to solve the optimization model and give the optimal Surfactant-Enhanced Aquifer Remediation strategy. The
results showed Kriging surrogate model had reduced computational burden and make the optimization model
easy to solve, and the optimal strategies gave an effective guide to contaminants remediation process.
Key words: DNAPLs, groundwater remediation, Kriging, optimization, surrogate model
INTRODUCTION
Spillage of large amounts of Denser Nonaqueous Phase Liquids (DNAPLs) had resulted in serious
pollution of groundwater resources throughout the world. Due to their low aqueous solubilities,
DNAPLs may remain in aquifers for many years, and lead to a risk for drinking water resources,
thereby have detrimental effects on both human health (Qin et al. 2007). Therefore, it was essential
to take measures for the cleanup of DNAPLs contaminants. DNAPLs that are typically immiscible
with water and denser than water are widely used in industrial and manufacturing operations.
Once spilled into the subsurface, DNAPLs would migrate through the unsaturated zone and keeping
on moving downward through the water table under the inﬂuence of gravity, and ﬁnally tended to
remain at the bottom of the aquifer (Dekker & Abriola 2000). Because of low solubility, high interfacial tension, and the sinking tendency of DNAPLs, as well as the complex subsurface condition
of remediation site, it was difﬁcult to remediate the groundwater contaminated by DNAPLs (Schaerlaekens et al. 2006).
Surfactant-Enhanced Aquifer Remediation (SEAR), which developed on the basis of pump-andtreat techniques, is widely considered as one of the most promising techniques to remediate
DNAPL contaminations (Carrero et al. 2007). With injecting surfactants into groundwater, SEAR
had greatly enhanced the remediation efﬁciency and reduced the remediation duration through
increasing the solubility of the solvent contaminants in groundwater and decreasing the interfacial
tension between the DNAPL and aqueous phases to make the residual DNAPL more mobile.
The optimization remediation process was composed of three parts: simulation model, optimization model and the coupled integration technology which combine simulation model with
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Water Practice & Technology Vol 8 No 2
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optimization model. Simulation model was used to forecast the fate of contaminants in subsurface
under various conditions, which had described the response relationship between the inputs and outputs of the groundwater system. University of Texas Chemical Compositional Simulator, UTCHEM,
was applied to simulate the processes of DNAPLs transport and surfactant-enhanced remediation in
heterogeneous DNAPL-contaminated groundwater. Optimization model was used to identify the optimal strategies, which describe the decision-making environment of groundwater system and provide
the optimal design strategy that guarantee to reduce remediation costs and improve remediation efﬁciency (Mayer & Endres 2007). The coupled integration technology has a main role in embedding
simulation model within optimization model. The traditional methods each have limitations in practical applications. In recent years, surrogate method had received extensive attention in the
groundwater remediation management (Zerpa et al. 2005; Sreekanth & Datta 2010). Many research
works showed that effective surrogates can replace the complex simulation equations, efﬁciency alleviating the huge computational efforts, and get the most optimal remediation strategies on the basis of
the transport law of groundwater contaminants (Huang et al. 2003; Jin 2005). He et al. presented an
integrated simulation, inference, and optimization method for optimizing groundwater remediation
systems. The stepwise quadratic surface analysis was employed as surrogate model. The results
demonstrated that computational efﬁciency could be largely improved (He et al. 2009). Qin et al. presented integrated simulation-optimization systems as a general framework for supporting decisions of
SEAR. The surrogate models were the multivariate regression tool. He et al. proposed a non-linear
chance-constrained programming model for optimizing SEAR processes, where the stepwise cluster
analysis was chosen as the surrogate model (He et al. 2008a,b; Qin et al. 2009).
Over the past few decades, Kriging models had been widely used in many areas such as spatial
interpolation, forecasting, engineering design, in the recent years, it also had drawn much attention
in the application of optimization (Kleijnen & van Beers 2005). Kriging act as surrogate models
were applied in the shape optimization of an aeroengine turbine disc to analyze the input–output
data of the simulation model, the results showed that Kriging surrogate model had the ability to adequately approximate to the simulation function (Huang et al. 2011). In the research, kriging surrogate
model were introduced to the application of DNAPLs-contaminated groundwater remediation
optimization.
In this study, the ﬁeld remediation optimization process was carried out in three steps. Firstly, the
water-oil-surfactant model had been ﬁrstly established to simulate a surfactant enhanced aquifer remediation process. And then the Kriging surrogate model was developed to get a similar input–output
relationship between the injection and extraction rates and remove rate with simulation model. In
the ﬁnal, a nonlinear optimization model embedded with the completed surrogate model was formulated for the minimum cost with the remove rate satisfying the remediation requirements. The
completed remediation optimization process was presented to give an effective guide to groundwater
contaminants remediation process.
METHODS
Simulation model
UTCHEM has been popular used to simulate a three dimensional, multi-component, multiphase,
compositional surfactant ﬂooding process, which was developed on the basis of enhanced oil recovery, in recent years, great achievements also had been made to use in groundwater contaminant
remediation ﬁeld. The fundamental mathematical equations included the mass-balance equation for
each species, the energy-balance equation, and the pressure equation (Dennis et al. 2010).
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Mass conservation equations of multiphase simulation model are
"n
#
p
~ kr Þ
X
@ðfC
~
k
~klrC
~
~ kl Þ ¼ Rk
ul fSl K
þr
rk ðCkl~
@t
l¼1
(1)
where k is the component index, l is the phase index, include aqueous phase, oleic phase, and micro~ k is overall volume of component k per unit pore
emulsion phase, f is the porosity of the aquifer, C
volume, rk is the density of pure component k[ML3], np is the number of phases, Ckl concentration
~
Dkl is
of species k in phase l, ~
ul is Darcy velocity of phase l [LT1], Sl is saturation of phase l, ~
dispersion tensor, Rk is the total sources and sinks of component k.
Energy-balance equation of multiphase simulation model: The energy-balance equation is derived
by assuming that energy is a function of temperature only and energy ﬂux in the aquifer or reservoir
occurs by advection and heat conduction only:
"
#
np
X
@
~
ð1 fÞrs Cvs þ f
rl Sl Cvl T þ r
@t
l¼1
np
X
!
~
ul T lT rT
rl C pl~
¼ qH QL
(2)
l¼1
where T is the reservoir temperature; Cvs , Cvl are the soil and phase l heat capacities at constant
volume; C pl is the phase l heat capacity at constant pressure; and lT is the thermal conductivity
(all assumed constant). qH is the enthalpy source term per bulk volume.QL is the heat loss to overburden and under burden formations or soil computed using the Vinsome and Westerveld heat loss
method.
Pressure equation of multiphase simulation model: The pressure equation is developed by summing
the mass balance equations overall volume-occupying components, substituting Darcy’s law for the
phase ﬂux terms, using the deﬁnition of capillary pressure:
fCt
np
np
ncv
X
X
X
@P1 ~ ~
~
~ ~
~ ~
~
~
~ P1 ¼ r
~
~
þ r KlrTCr
lrlcr h þ r
lrlcr Pcl1 þ
QK
K
K
@t
l¼1
l¼1
k¼1
(3)
The total compressibility is the volume-weighted sum of the rock or soil matrix and component
compressibility’s (Wilson & Clarke 1991; Delshad et al. 1996).
The other formulation such as equations of non-equilibrium dissolution of NAPL, well models, ﬂuid
and soil properties, adsorption of surfactant and organics, cation exchange, phase behavior and phase
saturation, and equations of capillary pressure and relative permeability has been described in the
simulator.
Kriging
The Kriging method was named after that D.G. Krige ﬁrstly used the statistics based technique to analyze mining data in the 1960s, and then developed by Matheron (Krige 1951). It was denoted as the
sum of two components: the linear model and a systematic departure. The basic formulation can be
expressed as follows:
Y ðxÞ ¼ f ðxÞ þ ZðxÞ
(4)
where f (x) represents the regression model, and Z(x) is the realization of a stochastic process with
mean zero and non-zero covariance to represent a local deviation from the regression model,
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307
which can be written as
Y ðxÞ ¼
n
X
bj fj ðxÞ þ ZðxÞ
j¼1
T
f ðxÞ ¼ ½ f1 ðxÞ, f2 ðxÞ, . . . , fn ðxÞ b ¼ ½b1 , b2 , . . . , bn T
(5)
where bj denotes the matrix of regression coefﬁcients to be estimated from the samples.
And the correlation between xi and xj is deﬁned as
d
X
cov Zðxi Þ, Z xj ¼ exp ul jxi xj j pl
!
(6)
l¼1
where ul correlation parameters used to ﬁt the model, pl represent the smoothness of the function in
coordinate direction (Kleijnen et al. 2012; Janusevskis and Le Riche 2013.).
The Kriging predict model is expressed as
YðxÞ ¼ bþrT ðxÞR1 ðy F bÞ
(7)
where
b¼
FT R1 y
FT R1 F
where R denotes n n, whose entry is cov Zðxi Þ, Z xj , r is the correlation vector
whose element is cov ZðxÞ, Z xj , and F denotes an n-vector of ones (Forrester & Keane 2009).
The unknown correlation parameters u can be estimated by maximizing the log-likelihood
Function.
n
lnðuÞ ¼ ln s
2
2
1
lnðjRjÞ 2
ðy F bÞT R1 ðy F bÞ
2s2
(8)
where
s2 ¼
T
y F T b R1 y F T b =n
CASE STUDY
In this section, three numerical tests were completed with the Kriging surrogate model. One example
was a one- dimensional function, and another example was a two-dimensional function, they showed
the Kriging surrogate model can approximate to the true function visually. The last example is the
optimization of DNAPLs-contaminated groundwater remediation.
Tests on a one-dimensional function
The ﬁrst true function was expressed as:
f ðxÞ ¼ ð6x 2Þ2 sinð12x 4Þ x [ ½0, 1
(9)
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The sampling is performed with the Latin hypercube sampling (LHS) method. The surrogate model
is built with nine sample points in Table 1. The comparison between calculated and predicted value
was shown in Figure 1. A good ﬁt curve was presented to demonstrate the model can be instead of the
ﬁrst true function.
Table 1 | The nine sample points of ﬁrst true function
1
2
3
4
5
6
7
8
9
x
0.22
0.33
0.41
0.05
0.59
0.51
0.80
0.92
0.67
y
0.47
0.00
0.16
0.66
0.03
0.96
5.06
8.53
3.20
Figure 1 | Calculated and predicted value for the ﬁrst true function.
Tests on a two-dimensional function.
The second true function was expressed as:
f ðxÞ ¼ 4x21 2:1x41 þ x61 =3 þ x1 x2 4x22 þ 4x42 x1 , x2 [ ½2, 2
(10)
The surrogate model is built with forty sample points, and validated with twenty sample points by
the LHS method in Table 2.
The comparison between calculated and predicted value was shown in Figure 2. The photo on the
left was the second true function, and the photo on the right was the predicted value by the Kriging
surrogate model, and they performed almost the same. The coefﬁcient of efﬁciency (CE) of the
Table 2 | The validated sample points of second true function
1
2
3
4
5
6
7
8
9
10
x1
0.73
0.84
0.57
0.10
1.65
0.94
1.04
0.18
0.27
1.54
x2
0.33
1.34
1.61
0.92
0.62
0.49
1.60
1.41
0.34
1.71
8.75
18.64
0.59
0.93
16.43
7.76
0.02
22.25
y
0.97
11
12
13
14
0.07
15
16
17
18
19
20
x1
0.31
1.81
1.21
1.95
1.66
0.58
0.66
1.37
1.15
1.49
x2
1.02
0.68
1.39
1.98
0.10
0.12
1.06
0.93
0.43
1.81
0.24
2.54
11.45
52.96
2.18
1.00
2.58
3.14
2.28
29.06
y
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Figure 2 | Calculated and predicted value for the second true function.
validated data set was 0.99; the result indicated that the surrogate model can be very approximate to
the true function.
SEAR numerical simulation model
The study area was 49 m in the x direction and 18 m in the y direction, and the total area covered
being 882 m2. The thickness of aquifer in porous layers is 24 m, and the aquifer is conﬁned. A PCE
spillage point was in the center of study area (25 row, 5 column), the spillage amount of PCE is
1m3/d, sodium dodecyl sulfate is chosen as surfactant for aquifer remediation. The aquifer system
is heterogeneous anisotropic aquifer which is composed of sandstone and clay sedimentary layers,
hydraulic conductivity changes from 0.035 to 1.05 m/d in the x direction, which equals to hydraulic
conductivity in the y direction and is two times greater than hydraulic conductivity in the z direction.
Initial conditions were speciﬁed such that the groundwater ﬂowed from left to right, with a hydraulic
gradient of 0.004706, and the west and east boundaries were set as given pressure boundaries, south
and north were set as impervious boundaries. The other main parameters are listed in Table 3.
Table 3 | Physical and chemical parameters in the research domain
Parameter
Value
Parameter
Value
Porosity
0.32
hydraulic gradient
0.004706
Longitudinal dispersivity
5m
Transverse dispersivity
0.5 m
PCE solubility in water
240 mg/L
PCE/water interfacial tension
44.67 dyn/cm
Water density
0.998 g/cm3
PCE density
1.623 g/cm3
Water viscosity
1.00cp
PCE viscosity
0.89 cp
Residual water saturation
0.24
Residual PCE saturation
0.17
The ﬁrst step is spatial and time discretization of study area, the number of horizontal grids is 49 9
with a spatial grid size of 1 m, each cell size is 1 2 2 m3. The temporal step size was divided into
two phases: the contaminated stage (0–90 d) and the remediation stage (91–150 d). After that PCE
spillage lasts for 30 days at the rate of 1 m3/d and the natural movement lasts for 60 days, the concentration distribution are shown in Figure 3.
According to the obtained contaminant distributions, a SEAR system with 3 extraction wells and 3
injection wells at a constant ﬂow rate of 80 m3/d with a 4% surfactant solution is designed. The
Water Practice & Technology Vol 8 No 2
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Figure 3 | PCE concentration horizontal and vertical distribution after 90 days (layer 12).
extraction injection wells location is respectively at (29, 5) (i.e. 25 row, 3 column), (28, 7), (27, 3). The
injection wells location is respectively at (21, 7), (32, 8), (34, 5).
Kriging surrogate model
In the study, Kriging was used to establish the surrogate model of multiphase ﬂow numerical
simulation model. Choose LHS method combining simulation model to obtain the input (pump
rate)-output (average removal rate) sample datasets, and then datasets were used to train Kriging surrogate model (Atanassov & Dimov 2008; Kastner 2010).
1. The stage of getting the input–output samples
Due to the nonlinearity and complexity of the remediation processes, numerical modeling is
used as tools for simulating the process responses under various operating conditions, and the
data collection process includes identiﬁcation of the related inputs and outputs, determination
of the ranges of inputs, and design of surrogate model. In the ﬁnal, two subsets of training and testing are prepared. Training data set was used for optimizing the parameters of the surrogate model.
Testing data set was used for the forecasting accuracy to test the performance of the model. Totally,
60 sets of operating conditions are randomly generated and the corresponding system responses
are obtained through running the multiphase ﬂow numerical simulation model. To test the developed Kriging models, 20 additional datasets including operating conditions (the pumping rates)
and system responses (the average removal rates) are randomly generated.
The pump rates were chosen as the main control input variables in the optimization model.
Therefore, the injection rates (Q1–Q3) and the extraction rates (Q4–Q6) as input variables, and
the datasets of the average removal rates as output variables.
2. The stage of training parameters
Matlab was used to carry out the Kriging model. The computing program for Kriging was completed with the daceﬁt and predictor function. It was expressed as [dmodel, perf] ¼ daceﬁt (S, Y,
regr, corr, theta0). S and Y are respectively the input and output data set, second order polynomial
and Gaussian function were selected as regression models and correlation models, theta0 was
initial value of the parameter u. The Kriging model parameters corresponding to the initial
sample can be found in Table 4.
Table 4 | Kriging model parameters corresponding to the initial sample
u1
u2
u3
u4
u5
u6
s2
0.1242
0.1130
0.1655
0.1879
0.1000
0.1809
3.8592
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3. The stage of testing the forecasting accuracy
In the test stage, test datasets are used to evaluate the performance of the trained Kriging model,
the comparison between the model prediction and the actual observation results is shown in
Figure 4.
Figure 4 | Observed and modeling average removal rates for the testing data set.
MAE and RMSE and CE are employed to assess the ﬁtted accuracy between surrogate models and
simulation models, the formulas were expressed as:
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u n
uX
ðPi Oi Þ2 =n
RMSE ¼ t
(11)
i¼1
CE ¼ 1 n
X
n X
2
ðO i P i Þ2 =
Oi Oi
i¼1
i¼1
(12)
where O and P are the observed and predicted data respectively; O is the mean observed value (Yang
et al. 2009).
CE indicates the initial uncertainty explained by the model, and the better ﬁt between observed and
predicated values, and the bigger CE value would be, until it reached 1. In addition, the RMSE and
MAE showed the discrepancy between the observed and predicated values. The lowest the RMSE
and MAE, the more accurate the prediction is.
The input–output samples are used to generate the Kriging model that can be embedded into the
process-optimization models.
Optimization model
In the study, the objectives is to minimize the total cost of the remediation when meet the remediation
effectiveness.
min Z ¼ Q1 þ Q2 þ Q3 þ Q4 þ Q5 þ Q6
(13)
8
0 Qi 80 ði ¼ 1, 2, . . . , 6Þ
>
>
<
y ¼ f ðQi Þ ði ¼ 1, 2, . . . , 6Þ
> Qin ¼ Qex
>
:
y 0:75
(14)
where y denotes as DNAPLs removal rate, y ¼ f ðQi Þ was the expression of Kriging surrogate models.
Qin, Qex denoted as the total injection rate and the total extraction rate.
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The constraints given in Equation (14) deﬁne the feasible region which contains all the admissible
solutions. Any solution outside this region is inadmissible since it violates one or more constraints.
The vector Q denotes an optimal solution in the feasible region.
RESULTS AND DISCUSSIONS
One example used in this paper is to illustrate the optimization process, UTCHEM has been used as
the simulator, and Kriging surrogate model was build. In the test stage, the good ﬁt curve between the
model prediction and the actual observation results demonstrated the model was suitable to forecast
the system response. In Table 5, the maximum relative percentage error is 4.05 per cent, the minimum
relative percentage error is 0.29 percent, the average relative percentage error is 1.80 percent. CE is
calculated as 0.09, RMSE is calculated as 1.30, and MAE is calculated as 1.13, which means the good
performance of the Kriging surrogate model, and then the Kriging surrogate model is extremely
approximate to the simulation model and is sufﬁcient as the alternatives of the simulation model.
Table 5 | The RMSE, MAE and CE of ﬁtting (training) for the Kriging models
1
2
3
4
5
6
7
8
9
10
Oa
57.02
63.82
71.17
57.29
46.45
77.65
52.05
73.74
70.19
56.66
Pb
55.78
61.44
70.82
59.43
45.38
76.83
49.94
72.84
71.62
55.95
c
AE
1.24
2.38
0.35
2.14
1.07
0.82
2.11
0.90
1.43
0.71
REd
0.02
0.04
0.00
0.04
0.02
0.01
0.04
0.01
0.02
0.01
11
12
13
14
15
16
17
18
19
20
Oa
82.72
73.33
86.42
80.34
80.33
69.36
52.42
75.02
62.99
35.05
Pb
80.87
72.47
84.68
81.11
81.98
69.56
53.32
74.49
62.72
34.25
AEc
1.85
0.86
1.74
0.77
1.65
0.20
0.90
0.53
0.27
0.80
0.02
0.01
0.00
0.02
d
RE
0.02
0.01
0.02
0.01
0.02
0.00
RMSE
1.30
CE
0.99
MAE
1.13
a
The observed value.
b
c
The predicated value.
Absolute error.
d
Relative error(%).
In this case, simulated annealing method was used to solve the optimization model. Deﬁne Q ¼ [80,
80, 80, 80, 80, 80] as the initial solutions, and the initial temperature, annealing constant, Markov
length, the maximum tolerance error, and step factor were identiﬁed as 300, 0.80, 16, 0.01, and
1 103, respectively, and through the iterations, the optimal pumping rates Q1–Q6 respectively
was 73.54, 24.72, 26.06, 79.93, 12.31, 32.07 m³/d, the remediation effectiveness was 77.07%, the concentration distribution results are shown in Figure 5, the total extraction rate and injection rate was
248.63 m³/d. If the total extraction rate and injection rate remains constant, that is, the remediation
cost remains constant, the extraction rate and injection rate were randomly generated, the optimal
pumping rates Q1–Q6 respectively was 53.54, 34.72, 36.06, 69.93, 17.31, 37.07 m³/d, the remediation
effectiveness was 74.92%. It was found that the optimization model based on surrogate model effectively improved DNAPLs removal rate.
It will take 1–2 h to assess the solution feasibility if the simulation model was called in the optimization process, and it was hard to look for the best solution with the huge computation burden.
What’s more, the surrogate model was easier to embed into optimization process in the computer program because the simulation model included many sophisticated mathematical formulas. However,
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Water Practice & Technology Vol 8 No 2
doi: 10.2166/wpt.2013.031
Figure 5 | PCE concentration distribution after 150 days (layer 12).
SA can assess approximately 2000 solutions feasibility in a second by calling the surrogate model, and
it can approximate to the best solution quickly.
It is an effective tool for simulation-surrogate-optimization model to solve the complex system
optimization problem, and large amounts of studies have shown the potential of saving the huge computational cost and remediation cost may reach tens of millions dollars.
CONCLUSION
In this paper, the optimization process design of SEAR in DNAPLs-contaminated subsurface systems
was presented in the form of simulation-surrogate-optimization combined model. Kriging surrogate
model can make use of their own characteristic to reﬂect complex nonlinear relationships between
system inputs and responses; meanwhile it was more suitable for surrogate model because of good
accuracy as well as fast training. The results showed Kriging surrogate model had reduced huge computational burden caused by repeatedly calling simulation model in the process of optimization model
and make the optimization model easy to solve. Compared to the other surrogate models, the Kriging
surrogate model can a better understanding of input–output response relationship, that is, more
approximate to the simulation model. What’s more, SA algorithm required less computation times
compared to the other traditional optimization techniques and get the global optimal solution. The
optimal strategies gave an effective guide to contaminants remediation process.
ACKNOWLEDGEMENT
This work was supported by ‘The National Natural Science Funds’ (41072171) and Key Laboratory of
Groundwater Resources and Environment, Ministry of Education, Jilin University, Changchun,
China.
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