American Journal of Analytical Chemistry, 2013, 4, 616-622
Published Online November 2013 (
Open Access AJAC
Application of Artificial Intelligence (AI) Modeling in
Kinetics of Methane Hydrate Growth
Jalal Foroozesh1, Abbas Khosravani2, Adel Mohsenzadeh3, Ali Haghighat Mesbahi4*
1Heriot-Watt University, Institute of Petroleum Engineering, Edinburgh, UK
2Department of Computer Science and Engineering, Amirkabir University of Technology, Tehran, Iran
3School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran
4Department of Chemical Engineering, Polymer Engineering Group, Isfahan University of Technology, Isfahan, Iran
Email: *, *
Received August 13, 2013; revised September 13, 2013; accepted October 13, 2013
Copyright © 2013 Jalal Foroozesh et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Determining thermodynamic and kinetic conditions for natural gas hydrate formation is an interesting subject for many
researches. At the present, suitable information including experimental data and the thermodynamic models of hydrate
formation are available which predict the thermodynamic conditions of hydrate formation. Conversely, there is no suf-
ficient study about the kinetics of natural gas hydrate and most of experimental data and kinetic models in the literature
are incomplete. Artificial Intelligence (AI) having sub-branches such as artificial neural network (ANN), and adaptive
neuro-fuzzy inference system (ANFIS) has been proved as a novel tool with acceptable accuracy for modeling of engi-
neering systems. Therefore, this paper aims to investigate the kinetics of hydrate formation by predicting the relation-
ship of growth rate of methane hydrate with temperature and pressure using ANN and ANFIS. This goal can also be
achieved by solving complicated governing equations while artificial intelligence provides an easier way to accomplish
this goal. The result has shown that ANIFS is a more potential tool in predication relationship of kinetics of hydrate
formation with temperature and pressure in comparison of ANN in present work.
Keywords: ANFIS; ANN; Kinetic; Growth Rate; Hydrate; Simulation
1. Introduction
Hydrates are ice-like compounds that could be formed at
temperature and pressure conditions higher than ice point
as a function of gas compositions. In fact, gas hydrates
are crystal compounds which are formed by water and
gas molecules that the ratio of water to hydrate molecules
is varying from 5.67 to 17. Hydrate formation in the oil
and gas industry causes difficulties such as plugging of
offshore oil and gas production pipelines, pressure drop
and corrosion of facilities. But, it has several applications
such as storage and transportation of natural gases, de-
salination of sea water, storage of carbon dioxide for a
long time and separation of hydrogen from other light
At the present, suitable information about thermody-
namic model of hydrate formation and experimental data
are available that mostly are able to predict the condi-
tions of hydrate formation. On the other hand, there is no
sufficient information about the kinetics of hydrate
growth rate and most of experimental data and kinetic
models in the literature are incomplete.
Establishing a kinetic model for hydrate growth rate
and its quality started in the 1970s with the work of Glew
and Haggett [1] on the hydrate of ethylene oxide. Their
experimental observation showed that the hydrate forma-
tion is an exothermic process and the growth rate was
directly proportional to the temperature difference be-
tween the reactor and its cooling bath. Thus, heat transfer
rate from the reactor to the cooling bath was considered
as a controlling step. Based on experimental data, hy-
drate production rate was hence modeled by simplifica-
tion of energy balance for the reactor.
Englezos et al. [2] developed a kinetic model to de-
scribe Vysniauskas and Bishnoi’s experimental data [3]
of methane and ethane hydrates formation combining the
theories of crystallization and mass transfer at a gas-liq-
uid interface. The driving force for hydrate formation
was defined as the fugacity difference between dissolved
*Corresponding author.
gas and three-phase equilibrium fugacity. Herri and
Bishnoi [4] performed in situ measurements of crystals
size distribution with the light scattering technique at
constant pressure condition in a 1000 cm3 stirred reactor.
Their experimental data showed that the kinetics of
methane hydrate formation strongly depended on the
stirring speed (ω) which should be considered in the mo-
Gnanendran and Amin [5] predicted a kinetic model
for hydrate formation in semi-batch regime spray reac-
tors. In brief, the model utilized a dynamic representation
of the drops in the reactor to describe hydrate nucleation
and growth. The rest of this paper is structured as follows.
First, the methodology that has been used in this article is
presented. Next, the theory section extends the back-
ground to the article. Afterward, experimental results are
brought and analyzed in the results section. Finally, the
paper is discussed and concluded by a discussion and
conclusion section.
2. Material and Methods
2.1. Fuzzy Logic
By the emergence of fuzzy set theory in 1965 through the
work of Lotfi A. Zadeh [6], it has been applied to many
fields from control theory to artificial intelligence. In
contrast to classical set, fuzzy set is not constrained by
crisp boundaries. In other words the transition from “be-
long to a set” to “not belong to a set” is gradual and this
is characterized by membership functions. Membership
functions give flexibility in modeling linguistic expres-
sions such as “the weather is cold”. Fuzzy sets as Zadeh
pointed out “play an important role in human thinking,
particularly in the domains of pattern recognition, com-
munication of information, and abstraction”.
A fuzzy set A in which is referred to as universe of
discourse is defined as a set of ordered pairs,
 (1)
is called membership function (MF) for
the fuzzy set
which ranges between 0 and 1. Figure
1 illustrates MFs of linguistic values “cold”, “warm” and
“hot”. There are some well-known parameterized MFs
including Triangular MFs, Trapezoidal MFs and Gaus-
sian MFs.
In order to make decisions in an environment of un-
certainty and imprecision, we employ fuzzy if-then rules
of the following form,
Rj: if x1 is Aj1 and x2 is Aj2 and … and xn is Ajn then
,,,, 1,2,,
fxxxfX jN (2)
R is the rule label, i
is the antecedent fuzzy
is a crisp function which is usually a
polynomial function of input variables and N is the total
Figure 1. MFs of linguistic values “cold”, “warm” and
number of fuzzy if-then rules. This type of if-then rules is
call Sugeno fuzzy model proposed by Takagi, Sugeno [7].
x is a first-order polynomial; it is called a
first-order Sugeno fuzzy model and is defined as:
jj jj
Xccx cxcx  (3)
Several types of fuzzy reasoning by Lee [8] have been
proposed in literature. In the case of Sugeno’s fuzzy if-
then rules the output of each rule is a linear combination
of input variables plus a constant term, and the weighted
average of each rule output produces the final output,
 
yX W
W is the firing strength of
R , defined as
kA k
WT x
In which denotes a
T-norm operator of minimum
or product. Figure 2 illustrates fuzzy reasoning for a rule
base with two Sugeno’s type fuzzy if-then rules each
with two inputs and one output.
2.2 Artificial Neural Network (ANN)
Artificial neural networks have been developed for a
wide variety of problems such as classification, function
approximation, and prediction. A neural network is struc-
tured by a parallel full-connection computation units ar-
ranged in layers mimicking the physiologic structure of
the brain. There are multiple connections within and be-
tween the layers which indicate the strengths or weights
between neurons that are learned under an optimization
criterion. All information learnt by the network is stored
in the interconnection weights between each two succes-
sive layers.
An important class of artificial neural network (ANN)
is multilayer perceptron (MLP) which schematically de-
picted in Figure 3. This network consists of an input
layer, one or more hidden layers of computation nodes
and an output layer of computation nodes. MLP neural
network uses error back-propagation algorithm in order
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Figure 2. Reasoning from a two-input first-order sugeno
fuzzy model with two rule [7].
Figure 3. Architecture of a multi-layer perceptron (MLP)
neural network with two hidden layers.
to learn its parameters (weights). This algorithm consists
of two passes through the different layers of the network:
forward and backward pass. In forward pass, an input
pattern is applied to the network to produce a set of out-
puts while the weights are all fixed. In layer the
output is computed as follows:
1, 2,,,1, 2,,
where l
is the input to the neuron in jth
y is the output of the neuron in layer, is
the weight of the connection between neuron in
layer and neuron in
is an acti-
vation function, l is the number neurons in nlth
and N is the number of layers.
During backward pass the error (the difference be-
tween the network outputs and the true values) is propa-
gated back from the output to the connection weights and
updates the weights to minimize the prediction error. The
most common error function is mean squared error
where i is the target value for the ith
tput and t
the number of neurons in the output layer. Then the
following algorithm is used to update the weights:
 
 
1, 1,
1, 1,
ll ll
ijij ll
ll ll
ij ij
wk wkk
 
is the learning rate,
is momentum factor
which helps to improve the performance of back-propa-
gation algorithm.
2.3. Adaptive Neuro-Fuzzy Inference
System (ANFIS)
An adaptive network is a network structure with nodes
that are mainly adaptive and directional links that con-
nect them. An adaptive node is a node whose output de-
pends on some external parameters. The learning rule
indicates how these parameters should change to mini-
mize error. In 1993, R. Jang [9] proposed a class of adap-
tive networks which were functionally equivalent to
fuzzy inference systems. This architecture which is re-
ferred to as ANFIS is the combination of human-like
reasoning style of fuzzy logic by the use of fuzzy sets
and if-then rules and learning structure of neural network.
ANFIS mimics the interpretability of fuzzy logic and the
accuracy and learning power of neural networks to make
it a hybrid intelligent system that has the ability to solve
nonlinear problems.
Figure 4 illustrates the ANFIS architecture of the
fuzzy model depicted in Figure 2. We used square nodes
to indicate the adaptability. The functionality of each
node varies and depends on overall input-output function.
Note that there is no weight associated with each link.
The learning procedure involves adjusting the parameters
of adaptive nodes given training data. A gradient method
can be used as the learning rule but this method is slow
and likely to fall in local minima. We utilized a hybrid
method [10] which is a combination of gradient method
and least squares estimate (LSE) to adjust the parameters.
Hybrid method, like MLP, has two passes: forward and
backward. In the forward pass each node output goes for-
ward and the consequent parameters are identified using
least squares method. In the backward pass the errors
propagate backward and the premise parameters are ad-
justed by gradient descent.
3. Theory
Nearly all of reported experimental works and modeling
on kinetics of hydrate growth were done in stirred semi-
batch reactor under isothermal and isobaric conditions
Figure 4. Equivalent ANFIS architecture of Figure 2 [9].
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with constant gas supply for maintaining the pressure.
Thus, because of the geometric factors and agitation ef-
fects on hydrate growth rate, the kinetic data which are
obtained for a specific reactor cannot be used for another.
In a novel experimental method, Chang Feng and Chang
Yu [11] measured the rate of gas hydrate growth inde-
pendent of the factors aforementioned. In this method,
the hydrate was allowed to form at the gas-liquid inter-
face of gas bubble suspended in a stagnant water phase.
Since the dynamic factors of mass transfer in these ex-
periments have no effects on the rate of hydrate growth,
it is possible to correlate a good relationship between
growth rate and deriving force which will follow in this
paper using artificial intelligence.
Nowadays some novel modeling tools such as ANN
and ANFIS are used with acceptable accuracy for invest-
tigation of engineering and physical systems. For exam-
ple, for investigation of hydrate formation, Mohammadi
et al. [12-14] presented a mathematical model based on
feed-forward artificial neural network algorithm which
could estimate hydrate dissociation conditions for the
hydrates of following systems: hydrogen + water, hy-
drogen + tetrahydrofuran + water and hydrogen +
tetra-n-butyl ammonium bromide + water, and also for
estimation of the dissociation pressures of the binary
clathrate hydrates of tetrahydrofuran + methane, carbon
dioxide or nitrogen as a function of temperature and
concentration of tetrahydrofuran in the aqueous solution
below/equal to its stoichiometric concentration. They
demonstrated that the predicted and the experimental
data are in acceptable agreement confirming the reliabil-
ity of this algorithm as a predictive tool.
Zahedi et al. [15] estimated hydrate formation tem-
perature (HFT) using the Engineering Equation Solver
(EES) and Statistical Package for the Social Sciences
(SPSS) software for statistical analysis of the 203 ex-
perimental data points collected from literature. Also,
HFT was estimated by artificial neural network (ANN)
approach using 70% of experimental data for training of
ANN. Comparing the results of ANN model with 30% of
testing data confirmed the brilliant estimation perform-
ance of ANN. It was found that ANN was more accurate
compared to traditional methods.
In this paper efforts are conducted to using ANFIS and
ANN for predicting the relationship of temperature and
pressure with growth rate of methane hydrate using ex-
perimental data of Chang Feng and Chang Yu [11,16].
The results could be useful in controlling and managing
the methane hydrate formation for preventing or acceler-
ating its formation.
4. Results
4.1. Data Description
Chang Feng and Chang Yu [11,16] proposed an innova-
tive method to measure the rate of hydrate growth. As it
is shown in Figure 5, the hydrate was allowed to form at
the gas-liquid interface of a methane gas bubble sus-
pended in a stagnant water phase. The whole hydrate
formation process was recorded by the video camera and
the surface area b of a gas bubble was calculated by
image software. Based on the measured
t (recording
time until the surface of suspended bubble is fully cov-
ered with hydrate), the average hydrate-growth rate (r)
on the bubble surface is simply calculated as:
The calculated hydrate-growth rate data (expressed in
terms of mm2/s) at various temperatures and pressures
for methane is presented in Tables 1. Figure 6 also illus-
trates the sample data used in the experiment.
4.2. Building the Model
We designed an ANFIS model with two inputs (tempera-
ture and pressure) and one output (hydrate-growth rate)
to predict the growth rate. We used fuzzy c-means clus-
tering algorithm to extract a set of fuzzy rules and mem-
bership functions that best model the data behavior.
Gaussian membership function defined as:
 
;, exp 2
Figure 5. Schematic of hydrate formation on the surface of
a gas bubble. 1) Water inlet; 2) hydrate-covered gas bubble
atop the gas injection needle; 3) hydrate-covered hemisph-
ere formed by a previous bubble; 4) gas outlet; 5) high-
pressure window ed cell;6) gas phase; 7) hydrate layer float-
ing atop the water phase; 8) water phase; 9) gas injection
needle; 10) gas inlet; [11].
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Table 1. Experimental data of methane hydrate growth
rate [11,16].
T (K) P(*105 Pa) G R (mm2/s) T (K) P (*105 Pa) G R (mm2/s)
278.1 51.5 0.02794 281.8 103.5 0.38424
276.6 45.5 0.03143 278.1 74 0.38963
280.4 68.5 0.03385 278.1 75.5 0.35412
281.1 73.3 0.0445 278.4 79 0.48592
280.4 71 0.05983 277.9 81 0.6587
282.4 83.1 0.06448 277.9 85 0.81643
281.4 76.6 0.07096 278.1 89 0.77055
280.6 72.1 0.07535 277.8 91.2 1.61606
282.5 84.5 0.07815 277.4 95.7 1.9065
278.9 63 0.08491 277.9 103 1.8975
282.6 87.3 0.09026 277.9 107 1.86085
281.4 81.2 0.09356 276.3 87.3 3.886
281.3 80.4 0.09946 276.2 82.5 3.396
281.6 83 0.10498 276.2 78.8 3.157
278.8 67 0.12199 276.2 74.3 2.747
281.6 86.5 0.14215 276.2 70.9 2.363
281.4 85 0.14728 276.2 68.5 1.996
282.1 89.8 0.16202 276.2 65.4 1.841
281.6 88 0.14571 276.2 62.3 1.409
278.8 69 0.19019 276.2 56.2 0.864
281.6 89.5 0.23662 276.1 51.4 0.586
278.1 67 0.28314 276.1 48.3 0.38
281.6 91 0.25738 276.1 48.3 0.371
281.9 95 0.26543 276.1 47.9 0.328
278.8 75 0.37292
T: Temperature; P: Pressure; G R: Growth Rate.
Figure 6. Illustration of the sample data used in this ex-
is the mean and
is the standard deviation,
is used in the model. An optimal number of 5 rules found
by fuzzy c-means clustering were trained using hybrid
learning algorithm.
The performance index used for evaluating the model
is based on the present of average relative deviation
00 (11)
where i is the target value, i is the ANFIS output
and is the total number of test samples. A 10-fold cross-
validation method is used to evaluate the performance of
the model. We first divided the dataset into 10 distinct
sets; putting aside one set for testing, we trained the AN-
FIS model with other sets. Let i be the error of the
t y
set used as test set. By repeating this process 10
times and each time with a different set, the performance
of the model is calculated as:
E (12)
where is the number of folds. Figures 7 and 8 repre-
sent the performance of the ANFIS for training and test-
ing data, respectively.
One of the major advantageous of 10-fold cross-vali-
dation evaluation method over holdout method in which
some random experimental data are used for evaluation
is that the entire experimental data will be tested by the
model and as a result, the bias of the true error rate will
be small.
We also analyzed the results obtained by MLP neural
network with 2 layers, 7 neurons in the first layer and 5
in the second layer which performs best using try and
error. Table 2 illustrates the average relative deviation
(ARD) for MLP neural network and ANFIS model.
Finally, a plot of the output surface of the ANFIS
model trained using the entire dataset is depicted in Fig-
ure 9.
5. Discussion and Conclusions
Due to the fact that artificial intelligence (AI) provides an
Figure 7. performance of ANFIS for training data.
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Figure 8. Performance of ANFIS for testing data.
Table 2. Results of MLP and ANFIS model on testing data
calculated using ARD (%).
Train Test Train Test
ARD (%) 23.5 35.11 9.45 17.65
Figure 9. The output surface of the ANFIS model.
effective tool for many complicated engineering prob-
lems in various fields, here ANFIS and ANN are em-
ployed as its sub-branches for understanding of methane
hydrate formation rate in various pressures and tempera-
tures. In this paper, we made a model which illustrates
the relationship of the temperature and pressure with CH4
hydrate growth rate for a set of real data using ANFIS as
well as ANN. Understanding the kinetics of hydrate for-
mation can be achieved by solving complicated equation
while artificial intelligence provides an easier way to
accomplish this goal.
Although ANN is one of the remarkable tools in engi-
neering processes, some problems are associated with
ANN that ANFIS removes two problems of this.
1—since that initial weight of ANN set randomly, ob-
viously we have different modeling performances in each
run (in constant topology and configuration of network)
and consequently we should run remarkable number of
time, for example 100 run, for each designed net and
report average of this 100 time or we can run our design-
ed network so much so that we obtain the best perform-
ance (Root Mean square error (RMSE), Average relative
error (ARE), Average relative deviation (ARD)… and set
this bunch of weight as the best initial weight of network,
which both of mentioned processes are highly time-con-
suming and ANFIS removes this problem with getting
amount of input in aggregation of membership function
as fuzzification process. With this work, we don’t have a
problem of ANN (different performance in each run).
2—Although black box of ANN sometimes is a great
advantage, it may be possible that we want to know inner
structure of subjected system by utilizing fuzzy system,
instead of this black box. This problem can be removed.
Furthermore, by doing so, the problem of existing an
expert in the FIS successfully will be removed by back
propagation learning algorithm of NN.
In this paper, we applied fuzzy c-means clustering al-
gorithm to extract a set of fuzzy rules and updated their
parameters using training data so as to predict the data
behavior properly. We utilized a 10-fold cross-validation
method to evaluate the performance of the model which
produces more reliable results especially for a few num-
bers of experimental data. The data were also compared
with MLP neural network having 2 layers, 7 neurons in
the first layer and 5 in the second layer. The obtained
results indicate the effectiveness of ANFIS as a more
powerful modeling tool for the kinetics of methane hy-
drate. Therefore, ANFIS model can be of great help for
engineers to prevent methane hydrae formation during
natural gas transportation through pipelines or accelerat-
ing its formation for gas storage purposes. Also optimi-
zation of ANFIS parameter for achieving more accuracy
model with different approaches (such as try and error
method or genetic algorithm method) in future is prom-
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