Journal of Petroleum Science and Engineering 171 (2018) 1211–1222
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering
journal homepage: www.elsevier.com/locate/petrol
Evolving robust intelligent model based on group method of data handling
technique optimized by genetic algorithm to predict asphaltene
precipitation
T
Maryam Sadi∗, Abbas Shahrabadi
Research Institute of Petroleum Industry (RIPI), West Blvd. Azadi Sport Complex, P.O. Box: 14665-137, Tehran, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords:
Asphaltene precipitation
Group method of data handling
Genetic algorithm
SARA fractions
Leverage approach
Precipitation of asphaltene during primary production of hydrocarbon reservoirs leads to formation damage and
well bore plugging. Therefore, proposing an accurate model to estimate asphaltene precipitation under various
operating and thermodynamic conditions are crucial. In this study, a new mathematical model based on the
integrating group method of data handling (GMDH) with genetic algorithm has been developed to predict asphaltene precipitation as a function of reservoir pressure and temperature, crude oil API, bubble point pressure,
Saturated-Aromatic-Resin-Asphaltene (SARA) fractions and mole percent of non-hydrocarbon gases. Genetic
algorithm technique has been applied to optimize the most appropriate network structure of GMDH model. In
order to accomplish modeling, asphaltene precipitation of different crude oils from a number of Iranian reservoirs at wide ranges of operating conditions have been measured experimentally and applied for network
construction. The accuracy of developed model has been evaluated by both statistical and graphical error
analysis techniques. The average absolute relative deviation of the proposed model is 3.65%, which indicates
model predictions are in excellent agreement with experimental data. Also, the comparison of developed GMDH
model with scaling equation and least squares support vector machine (LSSVM) reveals the superiority of the
proposed GMDH structure in prediction of asphaltene precipitation over scaling equation and LSSVM technique.
In addition, the Leverage approach has been applied to detect suspected data. The results show that all experimental data are reliable and located within the applicable domain of developed model. Finally, a comprehensive sensitivity analysis based on the relevancy factor has been carried out which shows that percentages of
resin and saturated components have the largest direct and inverse impacts on asphaltene precipitation, respectively.
1. Introduction
Asphaltene is defined as the heaviest component of crude oil that is
insoluble in normal alkanes, but soluble in toluene and benzene
(Speight et al., 1985; Subramanian et al., 2016). Asphaltene can be
separated from crude oil under certain thermodynamic conditions and
precipitates. In the past decades, asphaltene precipitation and deposition which may occur in all components of a production system, has
been the major problem in oil industry. Asphaltene usually starters to
deposit in surface facilities and then in tubing where the pressure falls
below the onset pressure. If the pressure decline continues, this problem
will involve the reservoir zone (Zoveidavianpoor et al., 2013). The
deposited materials reduce the permeability of the rock and in some
cases plug the well bore and tubing which leads to operational problems and decreases the production efficiency. Therefore, asphaltene
∗
precipitation is an unwanted phenomenon to the level that it is called as
the cholesterol of oil (Kokal and Sayegh, 1995). To prevent such problems, it is essential to understand the phenomenon and predict it
under different conditions. So far, different thermodynamic models
have been presented for this purpose. These models are classified in
four categories, namely solubility model, solid model, colloid model,
and micellization model.
Solubility models are based on Flory Huggins (FH) theory in which
the stability of asphaltene is expressed in terms of reversible equilibrium of solution. Some investigators used the molecular thermodynamic model of FH with an equation of state to predict asphaltene
phase behavior (Hirschberg et al., 1984; Mansoori et al., 1988; Buckley,
1999; Novosad and Costain, 1990; Kokal et al., 1992; Rassamdana
et al., 1996). Later, Anderson and Stenby (Andersen and Stenby, 1996)
introduced an interaction parameter of oil mixture and asphaltene into
Corresponding author.
E-mail address: sadim@ripi.ir (M. Sadi).
https://doi.org/10.1016/j.petrol.2018.08.041
Received 7 May 2018; Received in revised form 5 August 2018; Accepted 16 August 2018
Available online 18 August 2018
0920-4105/ © 2018 Elsevier B.V. All rights reserved.
Journal of Petroleum Science and Engineering 171 (2018) 1211–1222
M. Sadi, A. Shahrabadi
volume injection data as model inputs. Ansari and Gholami (2015) used
support vector regression method optimized by imperialist competitive
algorithm to model asphaltene precipitation by considering temperature, solvent molecular weight and dilution rate as input variables.
Although several artificial intelligence-based modeling techniques
have been applied in the recent literature to solve problems of oil industry, to the best of the authors' knowledge, no research has been
performed on the application of group method of data handling
(GMDH) technique to model asphaltene precipitation as a function of
crude oil properties and reservoir conditions. The GMDH is a self organizing approach introduced by Ivakhnenko (1968) for modeling and
identification of complex systems without having prior knowledge
about the studied process. The main concept of GMDH technique is
identifying the functional structure of polynomial type model by application of feed forward networks based on a quadratic transfer
function (Farlow, 1984). Optimization methods are implemented to
define the best network structure of GMDH model and calculate the
optimal coefficients of transfer functions. Evolutionary optimization
techniques such as genetic algorithm and particle swarm optimization
can improve the prediction accuracy of GMDH model as reported in the
literature (Abbod and Deshpande, 2008; Shaghaghi et al., 2017).
Shaghaghi et al. (2017) showed that integrating GMDH with genetic
algorithm is more efficient than coupling GMDH by particle swarm
optimization.
The objective of this work is developing an intelligent model based
on the GMDH technique to predict asphaltene precipitation during
natural depletion by considering crude oil API, bubble point pressure,
reservoir temperature and pressure, SARA fractions and mole percent of
non-hydrocarbon gases as input parameters. Due to the advantages of
evolutionary optimization methods, genetic algorithm technique is
applied to obtain the best functional structure of the proposed model.
The asphaltene precipitation of different Iranian crude oils is measured
experimentally and used to construct the model structure. Then, the
accuracy and reliability of the proposed model is evaluated using various statistical parameters and graphical error analyses. Also, the performance of GMDH model is compared with scaling equation and the
published results of LSSVM technique. In addition, Leverage approach
is carried out to detect outliers and identify the applicable domain of
developed model. Furthermore, a sensitivity analysis is performed to
quantify the impact of all input parameters on asphaltene precipitation.
the theory of polymeric solution, and improved the previous thermodynamic models.
Solid model is a type of thermodynamic models that uses cubic
equation of state to predict phase behavior of asphaltene. In these
models precipitated asphaltene is treated as a pure solid phase and a
cubic equation of state is applied for calculation of equilibria parameters in oil and gas phases. Nghiem et al. (1993) developed a solid
model with splitting the heaviest component of crude oil into a precipitating and a non-precipitating (asphaltene) component. All properties of these two parts are the same; the only difference is in their
interaction coefficients with light components. The temperature and
pressure dependency of asphaltene fugacity was introduced into the
solid model of Nghiem by Kohse et al. (2000). This model can be
considered as a general method to obtain the fugacity of precipitated
asphaltene.
Leontarities and Mansoori (Leontaritis and Mansoori, 1987) developed a colloidal dispersion model based on statistical thermodynamic
and colloidal science, which was further extended by Park and
Mansoori (1988). In this model it is assumed that asphaltenes are dispersed in oil while being absorbed by resin molecules, and the repulsive
forces between resin molecules inhibit asphaltene precipitation.
Nowadays, after observing reversibility behavior of asphaltene, the
colloidal model as an irreversible model is less being used. Later, the
irreversibility assumption of asphaltenes was rejected in the proposed
model by Victorov and Firoozabadi (1996). They proposed the name of
“micells” for aggregates of asphaltene. In this method the Gibbs free
energy of the system composed of liquid and precipitate phases is
minimized to calculate the composition of equilibrium phase. It should
be noted that the micellization model is too complex for computation
and its limitation is inability to predict the maximum amount of precipitated asphaltene (Tavakkoli et al., 2009).
Due to the complexity of asphaltene behavior, none of the above
mentioned models can describe all aspects of asphaltene precipitation.
Therefore, using any of these models will cause some deviations from
the experimental data.
In addition to the thermodynamic models, scaling equation is another technique for modeling asphaltene precipitation. Rassamdana
et al. (1996) applied scaling equation approach for the first time to
estimate the onset of asphaltene precipitation based on the aggregation
and gelation behavior of asphaltene presented by Park and Mansoori
(1988). Ashoori et al. (2010) proposed a scaling equation to consider
the effect of temperature on the amounts of asphaltene precipitation
titration data. Moradi et al. (2012) developed a scaling equation to
model asphaltene precipitation under gas injection conditions. Kord
and Ayatollahi (2012) considered the effect of pressure in a five parameters scaling equation to predict asphaltene precipitation of live oil
due to natural depletion. Behbahani et al. (2013) proposed a new
scaling equation to estimate asphaltene precipitation of bottom hole
live oil during pressure depletion and gas injection.
Recently, intelligent techniques such as support vector machine,
neural network and neuro fuzzy, which are based on the empirical data,
have been applied for modeling different engineering processes
(Abghari and Sadi, 2013; Helmy et al., 2017; Yarveicy et al., 2018; Van
and Chon, 2017; Ayatollahi et al., 2016; Sadi, 2017) and prediction of
asphaltene precipitation (Alimohammadi et al., 2017; Ahmadi and
Golshadi, 2012; Taleghani et al., 2017).
Yetilmezsoy et al. (2011) developed an adaptive neuro fuzzy model
to simulate water in oil emulsion formation by considering density,
viscosity and Saturated-Aromatic-Resin-Asphaltene (SARA) fractions as
input variables. Hemmati Sarapardeh et al. (Hemmati Sarapardeh et al.,
2013) applied least squares support vector machine (LSSVM) approach
to estimate precipitated asphaltene during natural depletion as a
function of bubble point pressure, reservoir pressure and temperature,
crude oil API, and SARA fractions. Salahshoor et al. (2013) utilized
adaptive neuro fuzzy technique to model asphaltene deposition in terms
of pressure drop and permeability ratio, by considering time and pore
2. Experimental section
2.1. Experimental set-up and procedure
Asphaltene precipitation tests were carried out under the static
condition on a number of crude oil samples taken from some oilfields in
Iran. These crude oils have been selected in a way to cover wide ranges
of crude properties and reservoir conditions. So, it can be said that most
Iranian crude oils are in the selected ranges.
Gas chromatography techniques based on the ASTM D2887-16a and
ASTM D1945-14 (ASTM D2887-16a, 2016; ASTM D1945-14, 2014),
differential liberation and constant composition expansion tests
(Pedersen et al., 2015) have been implemented to characterize oil
composition as well as other properties including the molecular weight,
API and bubble point pressure. Moreover, oil samples have been analyzed using SARA test to determine the fractions of asphaltene, aromatic, resin, and saturate as described by the ASTM D3279-12e1 procedure (ASTM D3279-12e1, 2012). Fig. 1 shows a schematic of the
experimental set-up implemented in this study to determine the amount
of precipitated asphaltene during depressurization of crude oil at reservoir temperature. A visual equilibrium cell is the main part of the
experimental set-up. This cell which is mercury-free operates up to a
maximum temperature of 400 °F and a maximum pressure of 15000
psia. Other parts of the experimental set-up include a high pressure
pump, a 0.2 μm filter paper and holder, transfer vessels for crude oil and
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M. Sadi, A. Shahrabadi
Fig. 1. Schematic of the laboratory apparatus employed for asphaltene precipitation experiments.
Filtered oil received at the sampler was flashed and its asphaltene
content was measured using the IP 143 standard procedure (IP-143/90,
1985). Weight percentage of the precipitated asphaltene was obtained
from the difference between original and filtered crude oil, at each
pressure step.
Table 1
The ranges and statistical information of experimental data.
Variables
Minimum
Maximum
Average
Input Parameters
Pressure (psia)
Temperature (oF)
Crude Oil API
Bubble Point Pressure (psia)
Saturated (%)
Aromatic (%)
Resin (%)
Asphaltene (%)
H2S (mol %)
CO2 (mol %)
N2 (mol %)
715
210
19.85
1608
29.30
30.01
0.49
3.75
0.00
0.47
0.02
5021
260
28.10
2966
47.72
52.09
18.80
13.75
2.97
7.96
0.93
2503
242
22.74
2052
40.27
42.95
8.77
7.99
1.32
3.78
0.31
Target Parameter
Precipitated Asphaltene (wt %)
0.02
6.29
1.06
2.2. Experimental data
In the present study, the amounts of asphaltene precipitation for 35
different crude oils from a number of oil reservoirs located in south of
Iran have been measured experimentally. The measured experimental
data were randomly divided into training (75%) and testing (25%) data
sets. To partitioning measured data into testing and training subsets,
several distributions have been used to avoid the local accumulations of
experimental data points in the problem feasible region. Finally, the
homogeneous accumulations of data points on the feasible domain are
selected as adequate distributions (Eslamimanesh et al., 2012). The
training subset (220 data points) was applied to obtain the optimum
values of model unknown coefficients and testing subset (73 data
points) was utilized for selection of the most appropriate functional
structure of model.
Table 2
Genetic algorithm parameters.
Variable
Value
Population Size
Cross Over Probability
Mutation Probability
Maximum Generation
100
0.85
0.03
400
3. Model development
3.1. Input variables definition
Reliability of a predictive data-based model depends on the comprehensiveness of the empirical data and accurate selection of input
variables. As mentioned earlier, a wide range of experimental data
which covers many of the Iranian oil reservoir conditions has been
applied for model development. To construct the model structure, the
effective parameters on the asphaltene precipitation including crude oil
properties and reservoir conditions should be considered as input
variables.
In the previously published paper (Ansari and Gholami, 2015),
temperature, molecular weight of solvent and dilution ratio were selected as input variables to estimate the amounts of asphaltene precipitation titration data. In another research (Hemmati Sarapardeh
et al., 2013), to predict asphaltene precipitation during natural
solvent storage, pressure transducers and gauges.
The working temperature in all experiments has been set equal to
the corresponding reservoir temperature. Asphaltene precipitation experiments were later conducted at different pressures as follow. First,
the equilibrium cell has been cleaned and maintained at reservoir
temperature using an oven. Then, 150–200 cc of oil was injected into
the cell under high pressure at the single phase condition and brought
to equilibrium. The cell pressure was depleted at different pressure
intervals; at each step the oil was passed through a 0.2 μm filter paper
and then towards a sampler. In order to maintain the sample in a
monophasic condition, when it passed through the filter, high pressure
helium gas was used to exert a back pressure on the filter downstream.
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M. Sadi, A. Shahrabadi
Fig. 2. Structure of developed GMDH model.
The GMDH approach is quite simple and at the same time is a suitable
technique for modeling unstructured complex systems (Pazuki and
Kakhki, 2013). In this method, the most appropriate polynomial functions are selected by combination of two independent variables in each
layer to generate new virtual neurons in the next layer.
The formal definition of the problem in GMDH method is to approximate actual value (y) with a polynomial function ( fˆ ) to accurately
predict target variable (ŷ). The main objective of GMDH is developing a
multilayer network structure comprised of a set of quadratic neurons in
different layers to map input variables to a single target.
In GMDH technique, the relationship between input and output
variables for a network with multiple inputs and single output can be
represented by Volterra–Kolmogorov–Gabor (VKG) polynomial
(Ivakhnenko, 1971):
depletion, reservoir conditions including temperature, pressure and
bubble point pressure and crude oil properties such as API and SARA
fractions were chosen as input parameters. In the present study, to
develop a more comprehensive model, in addition to the aforementioned parameters (temperature, pressure, bubble point pressure, crude
oil API and SARA fractions), the amounts of non-hydrocarbon gases
including H2S, CO2 and N2 have been considered as model inputs to
estimate asphaltene precipitation as output parameter.
The ranges and statistical information of experimental data used for
model development including input variables and target values are
presented in Table 1.
3.2. Group method of data handling
The Group Method of Data Handling (GMDH) technique firstly
proposed by Ivakhnenko (1968) is a heuristic self organizing method
applied for modeling sophisticated non linear systems (Farlow, 1984).
m
yˆ = a0 +
i=1
1214
m
m
m
m
m
∑ ai xi + ∑ ∑ aij xi xj + ∑ ∑ ∑ aijk xi xj xk
i=1 j=1
i=1 j=1 k=1
(1)
Journal of Petroleum Science and Engineering 171 (2018) 1211–1222
M. Sadi, A. Shahrabadi
to a second order polynomial consisting of only two variables
(Onwubolu, 2009):
Table 3
Statistical results for the proposed GMDH network.
Parameters
Value
Training Data Set
R2
RMSE
AARD
0.9981
0.0622
2.9772
Testing Data Set
R2
RMSE
AARD
0.9965
0.0778
4.7798
Total Data
R2
RMSE
AARD
yˆ = fˆ (x1, x2) = a0 + a1 x1 + a2 x2 + a3 x1 x2 + a4 x12 + a5 x 22
(2)
The GMDH structure is constructed using an iterative procedure
consists of training and testing steps. During network training, the
unknown parameters of quadratic polynomials are calculated by
minimizing the errors between experimental data and model predicted
values:
Nt
Minimize ⎛∑ [yˆi − yi ]2 =
⎝
0.9976
0.0684
3.6502
i=1
Nt
∑ [fˆ (xip ,
i=1
2
x iq) − yi ]
⎞
⎠
(3)
where Nt is the number of training data points. In testing step, the most
appropriate combination of variables is selected using testing data set
(Atashrouz et al., 2014). As the algorithm iterates, new middle layers
are gradually produced and finally a tree of multilayered quadratic
functions is developed as model structure (Amanifard et al., 2008). The
detailed descriptions about GMDH technique can be found in the literature (Pazuki and Kakhki, 2013; Ghanadzadeh et al., 2012; Sadi,
2018).
3.3. Genetic algorithm
Genetic algorithm (GA), which was first introduced by Holland
(1975) and further described by Goldberg (1989), is an adaptive
heuristic search algorithm. This evolutionary optimization technique is
based on the Darwin's Theory of natural evolution. According to this
theory, the less adapted species tend to disappear while the fittest individuals survive and create more offspring. The advantages of evolutionary optimization techniques such as GA are simplicity, flexibility
and self-adapt ability which leads to find the global optimum (Fogel
et al., 1997).
As the first step in GA, a population of possible solutions is generated randomly and then the fitness values of all population members
are evaluated by calculating the objective function. After computing the
fitness value, a particular group of individuals is selected to generate
offspring by the defined genetic operators. Reproduction, cross over
and mutation are the main operators applied in GA (Goldberg, 1989). In
reproduction step individual pairs are selected on the basis of fitness
values and propagate in the next generation as parent members (Sadi
et al., 2008). Cross over is a recombination step which creates two new
offspring by exchanging information between parents with a predefined
rate named as cross over probability. Mutation operator which keeps
GA diversity introduces a minor change into the children with a small
rate called mutation probability (Goldberg, 1989; Sadi et al., 2008).
The above mentioned steps, including parents selection and children production, are repeated during each iteration, until a desired
termination criterion such as predefined number of generations is
reached.
As mentioned earlier, genetic algorithm technique has been applied
to calculate optimum values of quadratic functions parameters and
obtain the best structure of GMDH model. The objective function is
defined as minimization of the differences between GMDH predictions
and experimental values:
Fig. 3. Comparison between experimental data and model predictions for
training data set.
Fig. 4. Comparison between experimental data and model predictions for
testing data set.
n
OF = Minimize
∑ (yexpi − ycali )2
i=1
(4)
In the above equation, ycal, yexp and n are model prediction, experimental value and number of data points, respectively.
The genetic algorithm parameters used to optimize the structure of
GMDH model are listed in Table 2.
In the above equation, a0, a1, …, aijk and x1, x2, …, xm are the unknown coefficients and input variables, respectively; and m denotes the
number of input variables.
In most cases, the general equation of VKG series can be simplified
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M. Sadi, A. Shahrabadi
Fig. 5. Experimental data and model predictions versus data points.
Fig. 6. Relative differences between GMDH predictions and experimental values.
4. Results and discussion
oil have been considered as model input parameters to estimate precipitated asphaltene as target value.
To evaluate the reliability and accuracy of the developed GMDH
model in predicting asphaltene precipitation, graphical analysis including cross plot, relative error diagram and error histogram in addition to the various statistical parameters such as coefficient of determination (R2), root mean square error (RMSE) and average absolute
relative deviation (AARD) have been employed. These statistical
4.1. Model performance evaluation
In the present study, the GMDH technique has been utilized to
predict precipitated asphaltene due to natural depletion. In the developed model crude oil API, reservoir pressure and temperature, bubble
point pressure, SARA fractions, and H2S, CO2 and N2 contents in crude
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M. Sadi, A. Shahrabadi
Fig. 7. Errors histogram for developed GMDH model.
3. Average Absolute Relative Deviation
Table 4
Experimental data and predicted values for validation data.
Sample
No. 1
No. 2
No. 3
API
SARA
Fractions
(%)
31.28
27.86
24.49
P (psia)
Asphaltene Precipitation
(wt %)
Experimental
GMDH
model
Relative
Difference
(%)
Sat: 52.49
Ar: 41.04
Res: 5.48
Asp: 0.99
2000
3000
4000
4500
5000
0.1805
0.2574
0.3699
0.3418
0.2854
0.1793
0.2630
0.3576
0.3400
0.2910
−0.66
2.18
−3.33
−0.53
1.96
Sat: 63.61
Ar: 20.85
Res:
10.54
Asp: 5
1015
1515
2016
2517
3018
3518
0.1009
0.2501
0.4796
0.6608
0.1982
0.0621
0.1006
0.2419
0.5075
0.6279
0.2034
0.0619
−0.30
−3.28
5.82
−4.98
2.62
−0.32
Sat: 58.23
Ar: 29.89
Res: 7.08
Asp: 4.69
1000
1500
2000
3000
5000
0.6700
0.9300
0.8600
0.5600
0.0700
0.6975
0.9613
0.8321
0.5602
0.0698
4.10
3.37
−3.24
0.04
−0.29
Relative Difference (%) =
AARD =
n
∑i = 1 (yexpi − ycali )2
(5)
2. Root Mean Square Error:
RMSE =
1
n
n
∑ (yexpi − ycali )2
i=1
yexpi
∗100
(7)
The statistical results of the optimum structure of GMDH network
are presented in Table 3. These results show the high accuracy of developed GMDH model in prediction of the precipitated asphaltene
during natural depletion.
As mentioned earlier, graphical error analysis has been done to
better evaluation of model performance as shown in Figs. 3–7. The
predicted values of asphaltene precipitation during natural depletion
for training and testing data sets are plotted against experimental data
in Figs. 3 and 4, respectively. The accumulation of data points in close
to the diagonal line indicates the robustness and perfect accuracy of the
developed model. Therefore, the excellent prediction capability of the
proposed model is proven.
Moreover, Fig. 5 depicts the model predictions including training
and testing subsets and experimental values versus data number. This
figure confirms that the developed GMDH model can estimate asphaltene precipitation during natural depletion by high accuracy.
In addition, the relative errors between model predictions and experimental data are plotted in Fig. 6. As can be seen, the maximum and
average values of relative deviations for training subset are 7.54% and
2.98%, respectively. These values for testing set are 11.51% and 4.78%,
respectively. These results reveal that model predictions are in excellent
agreement with experimental data, which is another evidence for accuracy of the developed model.
Finally, in order to present further authentication of model performance, histogram of errors between predicted and actual values is
1. Coefficient of Determination (R2):
n
i=1
yexpi − ycali
- Eleven variables as model input parameters at input layer.
- Five middle layers including five different groups of virtual variables (W1-W12, Z1-Z8, V1-V5, O1-O3 and U1-U2).
- A single variable at output layer as target value.
Prec Aspcal − Prec Aspexp
∗100 .
Prec Aspexp
∑i = 1 (yexpi − yexp )2
n
∑
where yexp is the mean value of experimental data points.
The optimal configuration of the proposed GMDH model is shown in
Fig. 2. As observed, the developed network structure is as follow:
parameters which are a combination of relative and absolute errors are
defined as follows:
R2 = 1 −
1
n
(6)
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Fig. 8. Experimental data and predicted values for three samples applied for model validation.
Fig. 8. As depicted, asphaltene starts to precipitate at onset pressure
which is higher than crude oil bubble pressure. During natural depletion of hydrocarbon reservoir, the pressure decreases gradually and
asphaltene precipitation increases up to a maximum value at bubble
pressure. Below the crude oil bubble point, the amount of precipitated
asphaltene decreases. So, the bubble point pressure plays an important
role in asphaltene precipitation. Above the bubble point, by decreasing
reservoir pressure, the solubility of asphaltene in crude oil decreases
which leads to an increasing in precipitated asphaltene. By decreasing
reservoir pressure, the asphaltene solubility reaches to a minimum
value at the bubble pressure. Below this point, the solubility of asphaltene in crude oil increases with decrease in reservoir pressure
which results in reducing asphaltene precipitation (Wang, 2000).
According to this figure and relative differences between predicted
values and experimental data reported in Table 4, the developed GMDH
model has acceptable reliability in prediction of asphaltene precipitation during natural depletion for different crude oils.
Table 5
Experimental data and LSSVM technique results.
Sample P (psia) Asphaltene Precipitation (wt %)
Relative
Difference (%)
Experimental LSSVM Technique (Hemmati
Sarapardeh et al., 2013)
No. 1
2000
3000
4000
4500
5000
0.1805
0.2574
0.3699
0.3418
0.2854
0.181
0.268
0.3529
0.3396
0.296
0.28
4.12
−4.59
−0.64
3.71
No. 2
1015
1515
2016
2517
3018
3518
0.1009
0.2501
0.4796
0.6608
0.1982
0.0621
0.0778
0.302
0.493
0.5199
0.3364
0.0264
−22.89
20.75
2.79
−21.32
69.73
−57.49
No. 3
1000
1500
2000
3000
5000
0.6700
0.9300
0.8600
0.5600
0.0700
0.7152
0.8844
0.9109
0.5584
0.0835
6.75
−4.90
5.92
−0.29
19.29
4.3. Comparison of GMDH model with other techniques
In this section, the performance of the proposed GMDH model in
prediction of asphaltene precipitation has been firstly compared with a
well-known scaling equation.
Kord and Ayatollahi (2012) proposed two separate equations to
predict asphaltene precipitation at upper and lower the bubble point by
considering pressure (P), bubble point pressure (Pb), gas oil ratio
(GOR), temperature (T) and asphaltene content of crude oil (Asp) as
input parameters:
demonstrated in Fig. 7. The bell shape of error distribution indicates the
normal behavior of the proposed GMDH model.
4.2. Model validation
Y = A∗Ln (X ) + B
In this section, to check the validity of the proposed GMDH model,
the amounts of precipitated asphaltene for three samples of Iranian
crude oils have been extracted from the literature (Hemmati
Sarapardeh et al., 2013) and applied as validation data set. It should be
noted that these data, have been also employed in the next section to
compare the accuracy of the proposed GMDH model with other techniques in prediction of asphaltene precipitation.
The information of these data points including crude oil properties,
experimental values and model predictions for asphaltene precipitation
accompanied by relative differences between predicted and experimental data are reported in Table 4.
Also, model predictions and experimental values for precipitated
asphaltene of validation data set are plotted versus reservoir pressure in
for P < Pb
Y = A∗exp (B∗X )
for P > Pb
(8)
(9)
where A and B are the scaling coefficients and X and Y are two new
scaling parameters defined as:
X=
(
(P − Pb)
Pb
) GOR
z
(10)
T z"
−z ′
Y=
AspPer ⎛ P − Pb ⎞
Asp ⎝ Pb ⎠
⎜
⎟
(11)
where Aspper represents the amount of asphaltene precipitation and z, z'
and z” are the adjustable parameters.
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Journal of Petroleum Science and Engineering 171 (2018) 1211–1222
M. Sadi, A. Shahrabadi
Fig. 9. Comparison of the developed GMDH model with scaling equation and LSSVM technique for validation data a. Sample 1, b. Sample 2 and c. Sample 3.
amounts of asphaltene precipitation for some experimental data. Three
sets of these empirical data have been selected as validation data in the
present study and utilized for both validation and comparison purposes.
These data sets have been presented in Table 5.
The results of GMDH model, scaling equation and LSSVM technique
for validation data are compared in Fig. 9a and c. As observed, the
In addition to the scaling equation, the results of LSSVM approach,
which were previously published (Hemmati Sarapardeh et al., 2013),
have been utilized to compare the prediction accuracy of the proposed
GMDH model. As mentioned earlier, Hemmati Sarapardeh et al.
(Hemmati Sarapardeh et al., 2013) applied LSSVM method to estimate
precipitated asphaltene during natural depletion and reported the
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Journal of Petroleum Science and Engineering 171 (2018) 1211–1222
M. Sadi, A. Shahrabadi
Sarapardeh et al., 2016). The detailed information about Leverage
method has been presented in the literature (Mohammadi et al., 2012).
The Williams plot of the proposed GMDH model in prediction of the
precipitated asphaltene during natural depletion has been illustrated in
Fig. 10. As observed, all experimental data are in the ranges of
0 ≤ H ≤ H ∗ = 0.1229 (as critical Leverage value) and − 3 ≤ R ≤ 3 (in
the limits of green and red lines). This figure confirms that the measured experimental data are reliable and located within the applicable
domain of developed model which consequently leads to a statistically
acceptable and correct model.
Table 6
The statistical criteria of GMDH model, scaling equation and LSSVM model for
validation data.
Parameters GMDH
model
Scaling Equation
(Kord and Ayatollahi, 2012)
LSSVM model (Hemmati
Sarapardeh et al., 2013)
R2
RMSE
AARD
0.9084
0.0803
23.8133
0.9547
0.0565
15.3416
0.9958
0.0171
2.3060
proposed GMDH model can predict asphaltene precipitation more accurately in comparison to the scaling equation and LSSVM approach.
Moreover, the values of statistical parameters including R2, RMSE
and AARD for GMDH model, scaling equation and LSSVM technique are
reported in Table 6. The calculated R2 for GMDH model, LSSVM technique and scaling equation are 0.9958, 0.9547 and 0.9084, respectively. Also, the value of AARD for the proposed GMDH model is 2.31%,
whereas these criteria for LSSVM technique and scaling equation are
15.34% and 23.81%, respectively, which indicate the better performance of GMDH model than both LSSVM approach as an intelligent
technique and scaling equation as a regression method.
Therefore, based on the results summarized in Table 6, it can be
concluded that the developed GMDH model can estimate asphaltene
precipitation behavior with more accuracy compared to the scaling
equation as well as LSSVM technique.
4.5. Sensitivity analysis
A sensitivity analysis based on the relevancy factor has been conducted to quantify the effect of input parameters on asphaltene precipitation in natural depletion. The mathematical definition of relevancy factor (r) is as follow (Chen et al., 2014):
r=
n
∑i = 1 [(xk, i − xk )( yˆi − yˆ )]
n
n
∑i = 1 (xk, i − xk )2 ∗ ∑i = 1 (yˆi − yk )2
(12)
Where xk, i and xk are the ith and average values of the kth input
parameter, respectively; ŷi and ŷ are the ith and average values of the
predicted asphaltene precipitation, respectively; and n represents the
number of experimental data.
The value of relevancy factor ranges from −1 to +1. The higher
absolute value of relevancy factor between an input variable and target
value implies that input parameter has a greater influence on the model
prediction. Also, positive or negative sign of relevancy factor for each
input parameter represents the increasing or decreasing impact of that
variable on the model target (Ayatollahi et al., 2016).
The values of relevancy factor for all input parameters on asphaltene precipitation have been depicted in Fig. 11. It can be observed
from this figure that reservoir temperature, bubble point pressure,
fractions of asphaltene and resin and H2S and CO2 contents have positive r values, which mean that an increase in these parameters increases the asphaltene precipitation. While API, percentages of saturated and aromatic components and N2 content with negative r values
4.4. Outlier detection
The accuracy of applied experimental data has a great effect on the
prediction capability of the proposed model (Rousseeuw and Leroy,
1987). Accordingly, various methods have been proposed for identification of the suspected data, among which the Leverage approach is
recognized as a reliable algorithm for outlier detection purpose
(Hemmati Sarapardeh et al., 2016; Mohammadi et al., 2012). Therefore, this approach has been employed in the present study to detect
suspected data and identify the applicable domain of developed GMDH
network. In this technique, outliers are defined graphically through
calculating Hat matrix (H) and sketching the Williams plot (Hemmati
Fig. 10. Detection of the probable outliers and the applicable domain of developed GMDH model.
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Journal of Petroleum Science and Engineering 171 (2018) 1211–1222
M. Sadi, A. Shahrabadi
Fig. 11. Relative effect of input parameters on asphaltene precipitation.
Nomenclature
have an inverse effect on the asphaltene precipitation, which indicate
that by increasing these parameters, asphaltene precipitation decreases
during natural depletion.
Moreover, this figure shows that the percentages of resin and asphaltene components with positive r values of 0.59 and 0.56, respectively, have the highest direct effect on asphaltene precipitation.
Whereas, the percentage of saturated components with a negative r
value of −0.48, has the highest reverse impact on asphaltene precipitation. Finally, it can be concluded that all input parameters except
percentage of aromatic components by −0.07 r value, have a significant effect on asphaltene precipitation.
a
A
AARD
API
Ar
Asp
B
CO2
fˆ
FH
GA
GMDH
GOR
H
H∗
H2S
LSSVM
m
n
N2
Nt
O
OF
P
Pb
Prec Asp
r
R
R2
Res
RMSE
SARA
Sat
T
U
V
VKG
W
x
xk
X
y
y
5. Conclusion
In the present study, GMDH technique has been applied to model
asphaltene precipitation during natural depletion as a function of reservoir pressure and temperature, crude oil API, bubble point pressure,
SARA fraction and the content of non-hydrocarbon gases in crude oil.
For this purpose, the amounts of precipitated asphaltene for 35 different
crude oils from a number of Iranian oil reservoirs have been measured
experimentally and applied in model development. Genetic algorithm
technique has been utilized to optimize the model unknown coefficients
and select the most appropriate functional structure of GMDH model.
The calculated R2, RMSE and AARD in prediction of asphaltene precipitation were 0.9976, 0.0684 and 3.65%, respectively. These results
reveal that the proposed GMDH network can be accurately applied to
predict asphaltene precipitation. Also, the comparison of GMDH model
with scaling equation and the previously published results of LSSVM
technique confirms the better performance of developed GMDH network in prediction of asphaltene precipitation. In addition, an outlier
analysis based on Leverage approach has been performed to detect
suspected data and identify the applicable domain of developed network. The results of outlier detection reveal that all experimental data
are reliable and located within the applicable domain of the proposed
GMDH model which consequently leads to an acceptable and correct
model. Finally, the sensitivity analysis based on the relevancy factor has
been done to evaluate the effect of input variables on asphaltene precipitation. The values of relevancy factor for all input parameters show
that the percentages of resin and asphaltene components have the
highest direct effect on asphaltene precipitation, while, the percentage
of saturated components has the largest reverse impact.
Declarations of interest
None.
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unknown coefficient of polynomial
scaling coefficient
average absolute relative deviation
crude oil API gravity
percentage of aromatic components in SARA test
percentage of asphaltene components in SARA test
scaling coefficient
CO2 concentration in crude oil, mol%
approximated function
Flory Huggins
genetic algorithm
group method of data handling
gas oil ratio, SCF/STB
Hat matrix
critical Leverage value
H2S concentration in crude oil, mol%
least squares support vector machine
number of input variables
number of experimental data points
N2 concentration in crude oil, mol%
number of training data set
fourth middle layer function
objective function
reservoir pressure, psia
bubble point pressure, psia
precipitated asphaltene, wt%
relevancy factor
standardized residual
coefficient of determination
percentage of resin components in SARA test
root mean square error
Saturated-Aromatic-Resin-Asphaltene
percentage of saturated components in SARA test
temperature, oF
fifth middle layer function
third middle layer function
Volterra–Kolmogorov–Gabor
first middle layer function
input parameter
average value of the kth input parameter
scaling parameter
output value
average value of output
Journal of Petroleum Science and Engineering 171 (2018) 1211–1222
M. Sadi, A. Shahrabadi
ŷ
ŷ
Y
z
z'
z"
Z
Subscripts
predicted output
average value of the predicted output
scaling parameter
adjustable parameter of scaling equation
adjustable parameter of scaling equation
adjustable parameter of scaling equation
second middle layer function
cal
exp
per
model predictions
experimental value
precipitated
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