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Qihong Feng
School of Petroleum Engineering,
China University of Petroleum (East China),
Qingdao 266580, Shandong, China
Ronghao Cui
1
School of Petroleum Engineering,
China University of Petroleum (East China),
Qingdao 266580, Shandong, China
e-mail: ronghao.cui1993@gmail.com
Sen Wang
1
School of Petroleum Engineering,
China University of Petroleum (East China),
Qingdao 266580, Shandong, China
e-mail: fwforest@gmail.com
Jin Zhang
School of Petroleum Engineering,
China University of Petroleum (East China),
Qingdao 266580, Shandong, China
Zhe Jiang
School of Petroleum Engineering,
China University of Petroleum (East China),
Qingdao 266580, Shandong, China
Estimation of CO
2
Diffusivity
in Brine by Use of the Genetic
Algorithm and Mixed
Kernels-Based Support
Vector Machine Model
Diffusion coefficient of carbon dioxide (CO
2
), a significant parameter describing the
mass transfer process, exerts a profound influence on the safety of CO
2
storage in
depleted reservoirs, saline aquifers, and marine ecosystems. However, experimental
determination of diffusion coefficient in CO
2
-brine system is time-consuming and complex
because the procedure requires sophisticated laboratory equipment and reasonable inter-
pretation methods. To facilitate the acquisition of more accurate values, an intelligent
model, termed MKSVM-GA, is developed using a hybrid technique of support vector
machine (SVM), mixed kernels (MK), and genetic algorithm (GA). Confirmed by the sta-
tistical evaluation indicators, our proposed model exhibits excellent performance with
high accuracy and strong robustness in a wide range of temperatures (273–473.15 K),
pressures (0.1–49.3 MPa), and viscosities (0.139–1.950 mPas). Our results show that
the proposed model is more applicable than the artificial neural network (ANN) model at
this sample size, which is superior to four commonly used traditional empirical correla-
tions. The technique presented in this study can provide a fast and precise prediction of
CO
2
diffusivity in brine at reservoir conditions for the engineering design and the techni-
cal risk assessment during the process of CO
2
injection. [DOI: 10.1115/1.4041724]
Keywords: carbon dioxide, diffusion coefficient, support vector machine, multivariate
regression, artificial neural network
1 Introduction
Due to global energy crisis and environmental pollution, recent
years have witnessed the accelerating exploitation and progressive
utilization of new and clean energy. In this field, carbon dioxide
(CO
2
) exhibits extensive and valuable applications. Replacement
of methane hydrate with CO
2
is recognized as one of the most
promising approaches to recovering methane from its hydrates,
avoiding damage to the seabed environment [1,2]. It is also found
that using CO
2
as the heat transmission fluid has a stronger
capacity of mining heat from hot fractured rock than water [3–5].
As an ancillary benefit, injecting CO
2
into marine ecosystems and
geothermal reservoirs can reduce the CO
2
emission into the
atmosphere and be conducive for the carbon storage and seques-
tration [6–12]. CO
2
, regardless of its physical states when
injected, always comes into contact with water and dissolves into
brine through molecular diffusion. Diffusion coefficient, which is
commonly employed to characterize fluid diffusivity, plays an
essential role in the storage safety in the geologic strata and the
sea bed [13,14], because it can dominate the rates of interfacial
mass transfer and heterogeneous chemical reactions related to
saline solution and porous media [15,16].
Considerable investigations have been conducted on the labora-
tory measurement of CO
2
diffusion coefficients [16–51]. Among
these studies, commonly used experimental methods can be clas-
sified as two types, i.e., the direct methods and the indirect meth-
ods [40]. The direct methods, such as the Taylor–Aris dispersion
method, start with precise analysis of the concentration of gas in
solvent [21,41], while indirect methods need to measure the
variance of a certain property that is related to gas diffusivity
[22,51]. Such a property includes liquid volume and shape
[42,45], gas volume and pressure [49], and interfacial tension
[51]. However, at elevated pressures, the convection can strongly
enhance the mass transfer of CO
2
into water, leading to an inaccu-
rate estimation of CO
2
diffusion coefficients [14,43]. Because of
the difficulties in overcoming the effects of convection and con-
ducting in situ measurements of gas concentrations at high pres-
sures, only a few studies reported the experimental results at
elevated pressures (up to 49.3 MPa) [19,40,41]. Also, note that
experimentally determining the diffusion coefficients of CO
2
through direct and indirect methods is both time-consuming and
complex, because it requires sophisticated laboratory equipment
and reasonable interpretation procedures. For ease of application,
a few empirical correlations (Table 1) verified by experimental
data have been developed [16,40,52–54].
Pressure and temperature, as the main operating conditions dur-
ing the mass transfer process, can exert prominent impacts on the
molecular diffusion of CO
2
. Therefore, the diffusivity of CO
2
under the sedimentary condition, i.e., high pressure and high tem-
perature, is significantly different from that under normal condi-
tion. Moreover, in contrast to the pure water, groundwater in a
typical reservoir is brine with high salinity, which inevitably influ-
ences CO
2
diffusion. Because different salinity results in various
viscosities of brine [55], in this work, we account for the depend-
ence of CO
2
diffusivity on the groundwater salinity by including
the solvent viscosity in our model.
In 1955, Wilke and Chang [53] put forward a general model for
diffusion coefficients in various dilute solutions and offered an
improved diffusion-factor chart for simplification. In their model,
the CO
2
diffusivity varies as a function of temperatures and vis-
cosity. Then, Lu et al. [40] ignored the effect of pressure and pro-
posed an equation for CO
2
diffusion in pure water (268 T
473 K). Using molecular dynamics simulations, Moultos et al.
1
Corresponding authors.
Contributed by the Petroleum Division of ASME for publication in the JOURNAL
OF ENERGY RESOURCES TECHNOLOGY. Manuscript received July 13, 2018; final
manuscript received October 6, 2018; published online November 19, 2018. Assoc .
Editor: Daoyong (Tony) Yang.
Journal of Energy Resources Technology APRIL 2019, Vol. 141 / 041001-1Copyright V
C2019 by ASME
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[54] extended Lu’s expression and developed a phenomenological
correlation to estimate the diffusion coefficient in CO
2
–H
2
O mix-
tures at ultrahigh temperatures and pressures. Because the viscos-
ity of saline solution should be taken into account to precisely
determine the CO
2
diffusivity in brine, the correlations developed
by Lu et al. and Moultos et al., although exhibit remarkably accu-
rate results for the solvent of pure water, cannot be directly uti-
lized for the CO
2
-brine system. On the basis of the experimental
results for CO
2
diffusion coefficients D
CO2
versus brine viscosity
at T¼298 K, Cadogan et al. [16] reported a modified
Stokes–Einstein relation; however, the validity of this correlation
at higher temperatures has not been substantiated.
In order to overcome aforementioned limitations, accurate pre-
diction of CO
2
diffusion coefficients in brine is a key issue for
urgent solution. With the extensive application of artificial intelli-
gence, machine learning exhibits excellent performance in diverse
engineering domains and provides better solutions for uncertain
problems [56–61]. Here, coupling support vector machine (SVM),
mixed kernels (MK), and genetic algorithm (GA), we present a
hybrid intelligent model (termed MKSVM-GA model) to predict
the molecular diffusion of CO
2
in brine. As a machine learning
technique derived from statistics, SVM emanates an overwhelm-
ing superiority in solving problems concerning small-scale sam-
ple, nonlinearity, and high-dimensional pattern recognition
[62,63]. The comparison with laboratory data obtained from pre-
vious literature manifests that the proposed MKSVM-GA model,
representing a large range of temperatures (273–473.15 K) and
pressures (0.1–49.3 MPa), is proven to be reliable and high-
precise (R
2
>0.99). Our model can not only furnish great signifi-
cance to the engineering design and the technical risk assessment
during CO
2
injection, but also shed light on the further research
involved with mass transfer process in CO
2
-brine system.
2 Methodology
2.1 Data Collection. From previous literatures, we collected
92 reliable experimental data points of CO
2
diffusion in brine
[16,19–21,24,27–30,32,33,35,38–41,51]. The data set is summar-
ized in the supplementary material, which is available under the
“Supplemental Materials” tab for this paper on the ASME Digital
Collection. We consider temperature, pressure, and viscosity as
the main correlation parameters influencing CO
2
diffusion in brine
and obtain the viscosity of pure water from National Institute of
Standards and Technology’s (NIST) Chemistry Webbook [64]. As
shown in the set, the gathered data for temperature ranges from
273 to 473.15 K, pressure from 0.1 to 49.3 MPa, viscosity from
0.139 to 1.950 mPas, and diffusion coefficient from 3.1 10
10
to 16.1 10
9
m
2
/s.
2.2 Fundamentals of Algorithm
2.2.1 Support Vector Machine. By using SVM, the potential
rules can be concluded from sampling data and then utilized to
predict unknown parameters [65–68]. Particularly, in the small-
sample cases, SVM shows excellent generalization properties
[69–72]. In SVM, the training samples are given as {(x
1
,y
1
), (x
2
,
y
2
), …, (x
n
,y
n
)}, where x
1
,x
2
,…, x
n
denotes the space of input
variables, i.e., pressure, temperature, and viscosity in this work;
y
1
,y
2
,…,y
n
are the experimental diffusion coefficients of CO
2
in
brine (outputs). The objective of SVM is to find an optimal hyper-
plane that can minimize the structural risk using the SVM regres-
sion function. In other words, it aims to strike a great balance
between the learning capability and the model complexity. The
regression function can be designed as follows:
fðxÞ¼wTwðxÞþb(1)
where wand bare the weight vector and the bias, respectively;
w(x) is the map function, which transforms the n-dimensional
input vector into a high-dimensional feature vector in feature
space to make the nonlinear issues turn into linear regression
problem. In SVM proposed by Vapnik [65,68], all pairs (x
i
,y
i
) are
supposed to satisfy the following requirement:
jyifðxiÞj e8ðxi;yiÞ2Re>0(2)
where eis the given error and Ris the set of inputs. In the preci-
sion of e, searching for the regression function f(x) as flat as possi-
ble is equivalent to minimizing jjwjj
2
. However, some samples out
of the margin may exert a serious impact on the fitting results.
Therefore, the slack variables need to be introduced to address the
problem, which enable some unrealistic samples to deviate from
the optimal hyperplane in a certain extent. We reach the mathe-
matical model stated by Vapnik [65,68]
min 1
2jjwjj2þcX
n
i¼1
niþf
i
subject to yiwTwxi
ðÞ
beþni
wTwxi
ðÞ
þbyieþf
i
ni;f
i0
(3)
where crepresents the penalty coefficient, which controls the
tradeoff between the flatness of the function and the minimum
cumulative value of deviations; n
i
and n
i
*are the slack variables
that depend on the e-insensitive loss function expressed as below:
jnj¼ 0ifjyifðxiÞj <e
jyifðxiÞj eotherwise
(4)
In Fig. 1, we schematically illustrate this situation. The samples
out or on the edge of the e-tube are the support vectors. The loss
function of circular dots inside the tube is zero while the square
Table 1 Empirical correlations for the diffusion coefficients of CO
2
in water
Author Reference Year Correlation Solvents
Othmer and Thakar
a
[52] 1953 DCO2¼14 109
l1:1V0:6
m
Brine
Wilke and Chang
b
[53] 1955 DCO2¼7:4108Tffiffiffiffiffiffiffiffi
/M
p
lV0:6
m
Brine
Lu et al.
c
[40] 2013 DCO2¼D0½T=Ts1mPure water
Cadogan et al.
d
[16] 2014 DCO2¼kBT=ðnSEplaÞBrine
Moultos et al.
e
[54] 2016 DCO2¼D0ðPÞ½T=Ts1mðPÞPure water
a
D
CO2
is the diffusion coefficient of CO
2
,m
2
/s; lis the solvent viscosity, mPas; V
m
is the molar volume of the diffusing substances, cm
3
/gmol.
b
Tis the temperature, K; /is the association parameter; Mis the molecular weight of solvent.
c
D
0
¼13.942 10
9
m
2
/s; T
s
¼227.0 K; m¼1.7094.
d
k
B
¼1.38065 10
23
J/K; n
SE
is the Stokes–Einstein number; ais the hydrodynamic radius of the solute, pm; a¼a
298
[1 þa(T298)], where
a
298
¼168 pm, a¼2.0 10
3
.
e
D
0
¼a
1
ln(P)þa
2
,m¼b
1
ln(P)þb
2
, where a
1
¼2.3097 10
9
,a
2
¼2.1064 10
8
,b
1
¼0.17812, and b
2
¼2.59406; Pis the pressure.
041001-2 / Vol. 141, APRIL 2019 Transactions of the ASME
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dots outside the e-tube are penalized. The penalty coefficient c
ensures the strength of the punishment. It is worth mentioning that
because cis given artificially, finding the optimal cis a requisite
for the excellent performance of SVM. We will discuss the choice
of cin Sec. 2.3.
It has been confirmed that obtaining the solution of convex
optimization problem in Eq. (3) through its dual formulation is
much easier [73]. The corresponding dual optimization model for
Eq. (3) is [65,74]
max eX
n
i¼1
aiþa
i
þX
n
i¼1
yiaia
i
1
2X
n
i¼1;j¼1
aia
i
aja
j
wxi
ðÞ
Twxj
ðÞ
subject to X
n
i¼1
aia
i
¼0
0aic;0a
ic
8
>
<
>
:
(5)
where a
i
and a
i
*
are Lagrange multipliers.
2.2.2 Mixed Kernels. The value of w(x
i
)
T
w(x) can be calcu-
lated through the kernel function, i.e., K(x
i
,x)¼w(x
i
)
T
w(x). Then
simple computation in low dimensional space replaces the com-
plex computation in high dimensional space. Owing to the utiliza-
tion of kernel function, SVM avoids the curse of dimensionality
[67,75]. In fact, the kernel function is divided into two types, i.e.,
global kernels and local kernels. The polynomial kernel in Eq. (6)
and the radial basis function (RBF) kernel in Eq. (7) are typical
examples of global and local kernels, respectively [76]. Figure 2
depicts the difference between them. As shown in Fig. 2(a), data
points far from each other are able to affect kernel values effec-
tively in global kernels, while local kernels in Fig. 2(b)only allow
data points close to each other to exert an impact on kernel values
Kpolyðxi;xÞ¼ðhxi;xiþ1Þd(6)
KRBFðxi;xÞ¼expðcjjxixjj2Þ(7)
where dis the polynomial degree and cis the kernel parameter.
Any function which satisfies Mercer’s conditions can be desig-
nated as the kernel function, including mixtures of global and
local kernels. It was found that the mixed kernel would combine
the advantages of global and local kernels, and strike a great
balance between the interpolation and extrapolation capability of
SVM [77]. The mixed kernel is defined as below:
Kmixðxi;xjÞ¼mKpoly ðxi;xjÞþð1mÞKRBFðxi;xjÞ(8)
where mis the fraction coefficient. As shown in Fig. 2(c),
mixed kernels not only receive strong response around the test
point but also guarantee that the value of response far from the
test point wouldn’t be attenuated rapidly. However, we can see
from Fig. 2that all of these free coefficients, i.e., d,c, and m
influence the performance of SVM. In Sec. 2.3, we will find
their optimal values.
Fig. 1 Schematic showing the fundamentals of SVM. The
square dots represent support vectors. The circular dots are
normal points. The full line denotes the hyperplane and the e-
tube is the region between two dotted lines.
Fig. 2 Function curves of the polynomial kernel (a), the RBF
kernel (b), and the mixed kernel (c). x50.2 is the test point in
three types of kernels. As an example of mixed kernel func-
tions, d51 and c510 are set.
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2.2.3 Genetic Algorithm. We use GA to obtain the optimal
values of c,d,c, and m. Introduced by imitating the evolutionary
principle of biological population, GA has been one of the most
popular global optimization methods [78–80]. Generally, GA
starts with an original population in which the optimal solution
candidates may exist. The population is composed of a certain
number of individuals with gene encoding. According to the prin-
ciple of survival of the fittest, approximate optimal solutions can
be found during evolution. Depending on the evolutionary goals,
the values of fitness function are calculated. Three genetic opera-
tors, including selection, crossover, and mutation, are performed
to create a new generation in each iteration. Particularly, it is
important for GA to set appropriate initial parameters such as pop-
ulation size, genetic algebra, crossover rate, and mutation rate
[81,82].
2.3 Model Development of Genetic Algorithm and Mixed
Kernels -Based Support Vector Machine. We present the com-
bination of GA and MK based SVM in this section. Based on the
principle that both of training set and testing set cover ranges as
large as possible with no overlap, we divided the whole database
into two groups. The first group is the training set composed of 72
data points for the development of our model. The second one is
the testing set (20 data points) employed to verify our model. We
summarize the information of the two groups in Table 2.
In order to accelerate the convergence and improve the model
accuracy in the training process, both of inputs and outputs are
normalized to a range of [1, þ1]
Xn¼2XXmin
ðÞ
Xmax Xmin 1(9)
where X
n
is the normalized value; Xis the actual value; X
min
and
X
max
are the minimum and maximum values of the parameter Xin
the whole data set.
After data preprocessing, as shown in Fig. 3, the hybrid model
of GA and MK based SVM can be established through the follow-
ing steps:
(1) Design a basic SVM model that is used in the regression
analysis.
(2) Mix the polynomial kernel and the RBF kernel using a frac-
tion coefficient m.
(3) Initialize the operating parameters of GA including popula-
tion size, evolutionary generations, selection rate, crossover
rate, and mutation rate.
(4) Generate initial values of c,d,c, and mrandomly in a cer-
tain scope we preset.
(5) With given c,d,c, and m, the fitness value for training set
is calculated. If the stopping criterion is met, stop the opti-
mization process and output c,d,c, and m. Otherwise, run a
GA and optimize the values of c,d,c, and m.
(6) Repeat step 5 with new values of c,d,c, and muntil the
stopping criterion can be satisfied. The optimal c,d,c, and
mcan be output.
(7) With the optimal c,d,c, and m, the training set is used to
train SVM. Finally, diffusion coefficients of CO
2
can be
estimated by a trained SVM.
To determine the optimal c,d,c, and m, mean square error
(MSE), as the fitness value, is calculated as below:
MSE ¼1
nX
n
i¼1
Dexp
iDpre
i
2(10)
where D
i
exp
and D
i
pre
are, respectively, the diffusion coefficients
estimated from experiments and our proposed model, m
2
/s. When
the fitness value reaches its minimum value, the optimal c,d,c,
and mis determined. The stochastic property of GA makes it pos-
sible to globally search for multiple optimal solutions. But its opti-
mal solution aggregate is not fixed every time. In order to obtain
the most appropriate parameter configuration for diffusivity pre-
diction, we run the MKSVM-GA model ten times in different
population size and evolutionary generations, as shown in Fig. 4.
When population number and generation number are set as 200,
uptime (about 30 min for each run) is sacrificed for minimum
average and error margin of MSE. Therefore, in this work, for the
balanced performance between model precision and computa-
tional efficiency, we choose 50 as population number and genera-
tion number. Besides, the rates of selection, crossover, and
mutation are adjusted in a small scope, referred to values set in
the literature [71]. Table 3summarizes parameter configuration of
the MKSVM-GA model.
2.4 Model Evaluation
2.4.1 Quantitative Evaluation. To examine the capability of
our proposed model, we utilized various frequently used statistical
Table 2 Information of the training set and the testing set
Group Temperature (K) Pressure (MPa) Viscosity (mPas) Diffusion coefficient (10
9
m
2
/s) Number of data points
Training set 273.15–473.15 0.1–49.3 0.139–1.950 0.31–16.10 72
Testing set 273.00–423.00 0.1–48.6 0.186–1.807 0.67–12.33 20
Fig. 3 Computational procedure of the MKSVM-GA technique
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evaluation indicators to measure its accuracy, i.e., mean absolute
error (MAE), mean absolute relative error (MARE), root-mean-
squared error (RMSE), and coefficient of determination (R
2
).
These indicators are defined as follows:
MAE ¼1
nX
n
i¼1
yexp
iypre
i
(11)
MARE ¼100% 1
nX
n
i¼1
yexp
iypre
i
yexp
i
(12)
RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
nX
n
i¼1
yexp
iypre
i
2
s(13)
R2¼1X
n
i¼1
yexp
iypre
i
2.X
n
i¼1ðyexp
iyave
exp Þ2(14)
where y
i
exp
and y
i
pre
are, respectively, the experimentally deter-
mined value and prediction, m
2
/s; y
exp
ave
is the average value of
experimental results, m
2
/s.
In addition to evaluation indicators and scatter plots that are
visually intuitive methods measuring the prediction performances
of the regression model, there exists a more quantitative
method—hypothesis test. Two widely used hypothesis tests, T-test
and F-test, are conducted to further assess the robustness of our
model [83]. In T-test, null hypothesis that the data belongs to nor-
mal distributions can be determined to accept or not, while F-test
(also known as the homoscedasticity test) estimates the null
hypothesis that the variances of two samples from normal distri-
butions are identical. If the return value of the test is zero, the null
hypotheses cannot be rejected at 5% significant level.
2.4.2 Outlier Diagnostics. We also applied outlier diagnostics
to ensure the results had a goodness of fitting. This regression
analysis approach is commonly employed to identify single datum
or a series of data which probably differ from the vast majority of
the data points in the whole dataset [83–85].
We conducted Leverage approach in the Williams plot to detect
outliers as well as doubtful data. The essential components in
Williams plot, including the standardized residual (SR), the hat
values (H), and the warning leverage (H*), need to be first calcu-
lated. Interested readers could find more details of the computa-
tional procedure for this method from Refs. [85] and [86]. As
shown in Fig. 5, the data points falling into the square region of
3<SR <3 and 0 <H<H* are valid, while those points lying
out of the range will be identified as outliers [85]. Typically, there
are two kinds of outliers: good high leverage (GHL) points and
bad high leverage (BHL) points. The points located in the ranges
of 3SR 3 and HH* are defined as GHL points that can-
not be predicted by the model. BHL points, which drop into the
domain of SR <3orSR>3 (no matter what H* value is) prob-
ably result from experimental errors [85,86].
3 Results and Discussion
3.1 Performance of the Proposed Model. Using the hybrid
technique of SVM, MK, and GA, the parameters c,d,c, and m
were determined within a wide range (0 c200, 0 d10, 0
c100, 0 m1). The optimal values of c,d,c, and mare
38.4115, 88.9051, 4, and 0.3430, respectively. Then, CO
2
diffu-
sion coefficients in brine are predicted.
The cross plot exhibited in Fig. 6(a)compares the estimations
of our model and laboratory results. Evidently, all of the points in
training and testing datasets closely scatter around the bisector of
the first quadrant, i.e., the 45 deg line, which manifests a good
agreement between our predictions and the experimental data
from previous literatures. In the light of the statistical evaluation
results presented in Table 4, MAE, MARE, RMSE, and R
2
for the
total data set are 1.311 10
10
m
2
/s, 7.91%, 1.954 10
10
m
2
/s,
and 0.9960, respectively. Therefore, it is concluded that our pre-
diction model is of high accuracy and strong generalization. In
Table 3 Basic parameters used in the MKSVM-GA model
Parameters Value
Population size 50
Evolutionary generations 50
Selection rate 1.0
Crossover rate 0.6
Mutation rate 0.03
Fig. 5 Illustration of the Williams plot
Fig. 4 Mean square error and average uptime of the MKSVM-GA model in different
population size and evolutionary generations
Journal of Energy Resources Technology APRIL 2019, Vol. 141 / 041001-5
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order to highlight the performance of our model in a visually bet-
ter way, we exhibit the measured and predicted diffusion coeffi-
cients of CO
2
versus the index of data points in Fig. 6(b). The
comparison indicates that the predictions estimated from our
model are remarkably consistent with the experimental results.
Figure 7shows that the residuals follow the normal distribution
with a mean value of 4.115 10
11
m
2
/s. The return values of
T-test and F-test are both zero, which indicates that both of the
null hypotheses are accepted. Therefore, the statistical signifi-
cance of our prediction is substantiated.
In Fig. 8, Williams plot is presented to detect the outliers. Here,
the warning leverage evaluated for the whole dataset is 0.130.
Given that the majority of the predicted points lie in the square
region of 3SR 3 and 0 H0.130, the MKSVM-GA
model, establishing a credible internal relation between these vari-
ables and diffusion coefficients of CO
2
in brine, is proven to be
statistically effective. There is no GHL point and we list the three
detected BHL points in Table 5. The data point measured at 373
K, 20 MPa reported by Lu et al. [40] does not coincide with its
variation trend with the pressure reported by Cadogan et al. [41].
Therefore, the experimental value at this condition may be errone-
ous. The validity of the other outliers cannot be determined
because the diffusion coefficients of CO
2
at these conditions were
not reported by any other researchers. As reported by Lu et al.,
overestimation of diffusion coefficients at temperatures higher
than 353 K is attributed to the scarcity of data points in high tem-
peratures [40]. With more sufficient data points at high tempera-
tures, some unreasonable data could be removed.
3.2 Comparison With Other Models
3.2.1 Artificial Neural Network Model. In the field of
machine learning, artificial neural network (ANN) is another prev-
alent technique. This method can deal with complicated functional
relationships; hence, it is widely used to predict many important
chemical parameters [58,59]. Typically, an artificial neural net-
work is composed of a bunch of nodes called neurons and the con-
nection constitution. Each neuron in ANN is featured by three
elements including weight, bias, and activation function. The
weight parameter measures the connecting strength for certain
input. The bias is a nonzero constant representing the type of con-
nection weight added to the summation of inputs and correspond-
ing weights. The activation function is used to introduce a
nonlinear relation into a multilayer perception neural network
[87–89].
To examine the capability of our model, we built a typical
three-layered feed forward ANN model to forecast diffusion coef-
ficients of CO
2
in brine and compared the results with the
Fig. 6 Prediction performance of the proposed MKSVM-GA model: (a) cross plot
of the estimated results versus experimental diffusion coefficients of CO
2
in brine;
(b) comparison of each experimental and predicted data points
Table 4 Statistical evaluation results of the MKSVM-GA model
Evaluation matrices Training set Testing set Total set
MAE (10
9
m
2
/s) 0.1112 0.2028 0.1311
MARE (%) 7.17 10.55 7.91
RMSE (10
9
m
2
/s) 0.1527 0.3028 0.1954
R
2
(fraction) 0.9975 0.9910 0.9960
Fig. 7 Distribution of the residuals between predicted and
experimental diffusion coefficients of CO
2
. The columns are
instances and the solid curve is the normal distribution curve.
Std Dev represents the standard deviation of residuals.
Fig. 8 Williams plot used for the detection of outliers
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predictions of our MKSVM-GA model. Figure 9illustrates the
structure of this ANN model including an input layer, a hidden
layer, and an output layer. The input layer consists of three neu-
rons denoting attributes of these samples i.e., pressure, tempera-
ture, and viscosity. Diffusion coefficients of CO
2
in brine are
obtained from the neuron of the output layer. We determined the
number of neurons in the hidden layer through the means of trial
and error. In this work, the number of neurons was raised from 4
to 20 at intervals of two neurons. At each number of neurons, the
model was tested ten times and its MSE was traced. Aimed at the
minimum MSE for the training set, we found the optimal number
of neurons for this ANN model. The minimum MSE curve with
variation of neuron number is shown in Fig. 10. The minimal one
is situated at the neuron number of 8. Therefore, the optimal ANN
architecture is determined as 3-8-1.
The MKSVM-GA model and the ANN model were both run
ten times. Figure 11 depicts performance comparison between the
MKSVM-GA model and the ANN model in the training stage.
The convergence scope of the MKSVM-GA model, shaded with
blue, is distinctly lower and narrower than results of the ANN
training model. This phenomenon is attributed to small-scale sam-
ples. ANN is based on traditional statistics, a gradual theory that
sample size is tending to be infinite. Thus, when small-scale sam-
ples are manipulated in ANN, it is doubtful to take it for granted
that statistical properties are concluded. In fact, desired perform-
ance is not achieved in ANN in this situation. On the other hand,
the statistical learning theory is fundamental to SVM, which is a
theory on the condition of small sample size. Professionally
speaking, SVM, which embodies the structural risk minimization
principle, can minimize the upper bound of the generalization
error rather than the training error from ANN, which embodies
the empirical risk minimization principle [90–92].
The most optimal result (the largest R
2
) predicted by ANN is
shown in Fig. 12. The statistical evaluation results of ANN model
and our proposed model are summarized in Table 6. Lower MAE,
MARE, RMSE, and higher R
2
demonstrate that the MKSVM-GA
model is more precise than the ANN model in both the testing and
the training stages. These evaluation results, as well as the
Table 5 Outliers recognized in the Williams plot
Author Reference Temperature (K) Pressure (MPa) Viscosity (mPas) Diffusion coefficient (10
9
m
2
/s) Solvent Outlier type
Maharajh et al. [19] 0.1 273.00 1.807 1.00 Pure water BHL
Lu et al. [40] 20.0 373.15 0.287 6.43 Pure water BHL
Lu et al. [40] 20.0 393.15 0.237 8.13 Pure water BHL
Fig. 9 Illustration of the structure of a typical three-layered
feed forward ANN model
Fig. 10 The minimum MSE with variation of number of neurons
in the hidden layer
Fig. 11 Comparison of the training performance between the
MKSVM-GA model and the ANN model
Fig. 12 Comparison of the prediction performance between
the MKSVM-GA model and the ANN model
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absolute error curve shown in Fig. 13, substantiate the fact that
MKSVM-GA performs better than ANN on the condition of small
sample size. However, due to solution difficulty of high dimen-
sional matrices in convex quadratic programming, it is a technical
conundrum for SVM to be implemented with huge sample size,
while ANN could handle it with ease. Even so, there is no clear
dividing boundary on sample size between SVM and ANN. In
fact, most of researchers prefer to investigate the adaptability of
SVM or ANN through trial and error methods [87,93].
3.2.2 Empirical Correlations. In this section, we compared
our model with empirical correlations to examine its validity. As
mentioned above, there are five correlations that can predict the
diffusion coefficient in CO
2
-brine system, i.e., Othmer–Thakar’s
model [52], Wilke–Chang’s model [53], Lu’s model [40], Cado-
gan’s model [16], and Moultos’ model [54]. However, Moultos’
correlation cannot be used to conduct this comparison because
this formula, modified from Lu’s model, is available only in ultra-
high temperature (323.15 T1023.15 K) and pressures (200
P1000 MPa), which are beyond the scope of our training set.
Table 6 Comparison of the statistical evaluation results between the MKSVM-GA model and the ANN model
Training set Testing set Total set
Evaluation matrices MKSVM-GA ANN MKSVM-GA ANN MKSVM-GA ANN
MAE (10
9
m
2
/s) 0.1112 0.1490 0.2028 0.3172 0.1311 0.1855
MARE (%) 7.17 11.97 10.55 14.80 7.91 12.59
RMSE (10
9
m
2
/s) 0.1527 0.2275 0.3028 0.4608 0.1954 0.2944
R
2
(fraction) 0.9975 0.9944 0.9910 0.9785 0.9960 0.9908
Fig. 13 Absolute error curves of the MKSVM-GA model, the
ANN model, and four empirical correlations
Fig. 14 Comparison of the prediction performances between the MKSVM-GA model and four
empirical correlations: (a) Othmer and Thakar; (b) Wilke and Chang; (c) Lu et al.; (d) Cadogan
et al. Squares and triangles represent the values predicted using the MKSVM-GA model and
empirical correlations, respectively.
041001-8 / Vol. 141, APRIL 2019 Transactions of the ASME
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Figure 14 compares the performance of each empirical correla-
tion with our proposed model. Unlike the data points from the
MKSVM-GA model, which distribute uniformly around the
45 deg line, the predictions calculated by using the first three
correlations—Othmer’s model, Wilke–Chang’s model, and Lu’s
model, deviated up or down obviously, particularly for higher dif-
fusion coefficients. Meanwhile, the results estimated from Cado-
gan’s model are more dispersed along the 45 deg line than our
MKSVM-GA model. Therefore, the four empirical correlations
show lower accuracy than our proposed model. In addition, the
evaluation results shown in Table 7and the absolute error curves
depicted in Fig. 13 confirm that all of the four empirical correla-
tions have weaker prediction capability. MAE, MARE, and
RMSE of empirical correlations approximate to twice or more as
much as that of the MKSVM-GA model. In comparison to the
high fitting degree of intelligent algorithms, none of the empirical
correlations has a greater coefficient of determination (R
2
) than
0.99, even though the performance of Cadogan’s model is far bet-
ter than the other formulas.
Because the models proposed by Othmer and Thakar as well as
Wilke and Chang are general equations for gas diffusivity in solu-
tions, they only provide the rough guess and estimation for CO
2
in
brine. The model of Lu et al., only available in pure water, failed
to account for the effect of solvent viscosity. These empirical cor-
relations including Cadogan’s model are established on the same
assumption that the effect of pressure is negligible. As a matter of
fact, the variation in pressure can lead to fluctuations of CO
2
dif-
fusion coefficients. The ignorance of influencing parameters is
indeed the most important reason to explain why the empirical
correlations exhibit poor performances. A considerate composi-
tion of influencing parameters unleashes the advantages of artifi-
cial intelligent algorithms. With the aid of the excellent
robustness and fault tolerance, our proposed model exhibits an
incomparable superiority to the traditional empirical correlations.
4 Conclusions
We developed a hybrid technique of support vector machine,
mixed kernels, and genetic algorithm to predict CO
2
molecular
diffusivity in brine. A total of 92 experimental points obtained
from previous literatures are employed to train and test the
MKSVM-GA model. The statistical evaluation indicators of total
data set, i.e., MAE (1.311 10
10
m
2
/s), MARE (7.91%), RMSE
(1.954 10
10
m
2
/s), and R
2
(0.9960), demonstrate that our pro-
posed model exhibits an excellent performance with high accu-
racy and strong generalization. To highlight the superiority of our
model, a typical three-layered ANN model and four commonly
used empirical correlations are employed to conduct the compari-
son with the MKSVM-GA model. The results confirm that the
proposed model has significantly better prediction capability.
Potentially, our technique can be extended to other engineering
disciplines that demand fast and convenient access to basic physi-
cal parameters.
Funding Data
National Program for Fundamental Research and Develop-
ment of China (973 Program) (Grant No. 2015CB250905).
Program for Changjiang Scholars and Innovative Research
Team in University (Grant No. IRT1294).
National Postdoctoral Program for Innovative Talents (Grant
No. BX201600153).
China Postdoctoral Science Foundation (Grant No.
2016M600571).
Qingdao Postdoctoral Applied Research Project (Grant No.
2016218).
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