Introduction

Enhanced oil recovery (EOR) is an important field of petroleum industry that is used to increase oil recovery from petroleum reservoirs. CO2 injection is a cost-effective and favorable method that is widely used as an EOR technology for oil industry. Due to low polarity and high density of CO2 at supercritical state, it has high amount of solubility in oil and can extract hydrocarbon from crude oil (Hemmati-Sarapardeh et al. 2013; Zhang et al. 2013; Abedini and Torabi 2014; Cao and Gu 2012). Because heavy fractions of oil have low solubility in supercritical CO2, EOR applications of CO2 injection have been limited in oil industry. Another limitation of applying CO2 extraction phenomenon in EOR is asphaltene precipitation that creates major problems for oil recovery and production. Therefore, finding the solubility of oil fractions in supercritical CO2 is very important (Daryasafar et al. 2018).

Due to low miscibility of heavy oil in supercritical CO2, reservoirs with light to medium oil density are more preferable for CO2 injection (Cao and Gu 2012, 2013; Luo et al. 2012). Cao and Gu investigated the effect of hydrocarbon polarity on the recovered oil density through CO2 injection process (Cao and Gu 2012). According to the results of their work, the recovered oil is lighter and hydrocarbon solubility becomes lower as hydrocarbon polarity increases. Many researchers have studied phase equilibrium behavior of n-alkane and supercritical CO2 system at high temperature and pressure condition (Ren and Scurto 2007; Gardeler et al. 2002; Yu et al. 2006; Eustaquio-Rincón and Trejo 2001). Choi et al. found dew and bubble points of some normal alkanes (hexane, heptane, octane and nonane) contacting supercritical CO2 for pressure and temperatures up to 153.41 bar and 395.7 K, respectively (Choi and Yeo 1998). By measuring volume expansion of alkane-CO2 system at different temperatures and pressures in PVT apparatus, Yang et al. demonstrated that system expansion depends on the dispersion status of CO2 molecules (Yang, et al. 2012). Chandler et al. calculated the capacity factors of solid n-alkanes (C9H20–C36H74) by estimating the solubility of these n-alkanes at temperature range of 308.2–348.2 K and pressure range of 100–200 bar (Chandler et al. 1996). Shi et al. studied the effects of system condition and solution properties on n-alkane (C9H20–C36H74) solubility at pressure and temperature range of 9.185–20.862 MPa and 318–343 K (Shi et al. 2015). They showed that CO2 density and system pressure have direct relation with n-alkane solubility; however, temperature and the length of n-alkane chain are inversely proportional to it. Wang et al. measured the solubility of polar substitute in supercritical CO2 by using compounds with aromatic nucleus (Wang et al. 2006). Furuya et al. determined the solubility of heavy n-alkane in supercritical CO2 and at pressures up to 50 MPa (Furuya and Teja 2004). They used these measured values to evaluate Krichevskii parameters and showed that these parameters were directly related to n-alkane carbon number. Eustaquio-Rincon et al. performed experimental measurements by using a semi-flow-type setup to find n-octadecane solubility in supercritical CO2 at pressure range of 10–20 MPa and temperature range of 310–353 K (Eustaquio-Rincón and Trejo 2001). The results of their work indicated that n-octadecane solubility has direct relation with pressure and it is maximum at the temperature of 313 K.

Expensive and time-consuming experimental methods for determining n-alkane solubility in supercritical CO2 demand a fast and accurate model that can predict the solubility of different n-alkane at a wide range of pressure and temperature values. Accordingly, many models have been suggested for predicting n-alkane solubility in supercritical CO2. Peters et al. proposed a model that relates logarithm of solute mole fraction to the logarithm of solvent carbon number at supercritical conditions (Peters et al. 1989). Jha and Madras predicted the solubility of n-alkane hydrocarbon in supercritical CO2 using Peng–Robinson EOS, quadratic mixing rules and an adjustable parameter (Jha and Madras 2004). They proved that there is a linear relation between the adjustable parameter and carbon number of the n-alkane largest chain. The limitation of their model is the defined adjustable parameter that needs time-consuming laboratory measurements. Some researchers (Gordillo et al. 1999; Sung and Shim 1999; Keshmiri et al. 2014) have suggested their semiempirical models for simulating n-alkane solubility in supercritical CO2. Chrastil proposed a correlation between supercritical CO2 density and the solute solubility (Chrastil 1982). The simplicity and accuracy of this model have convinced many researchers to use it in their studies. Moreover,  a number of machine learning algorithms have been used in the literature to model solubility of n-alkane in supercritical CO2 (Huang et al. 2019a, b; Sedaghat and Esfandiarian 2019; Abdi-Khanghah et al. 2018). Daryasafar et al. investigated the viability of using artificial techniques for predicting the solubility of n-alkane in supercritical CO2. Based on the results of their study, intelligent tools can be good candidates for predicting n-alkane solubility (Daryasafar et al. 2018).

In this work, least-squares support vector machine (LSSVM) technique has been utilized to predict n-alkane solubility in supercritical CO2. Modeling with LSSVM tools has some advantages as high flexibility in solving nonlinear regression, lowering the required time and cost of regression process (Gorjaei et al. 2015; Hou et al. 2009). In this study, two different LSSVM-based models including PSO optimized LSSVM (PSO–LSSVM) and cross-validation Simplex–LSSVM (CV-Simplex–LSSVM) models have been applied to model and predict n-alkane solubility in supercritical CO2. The key purpose of this study is to propose a comprehensive and accurate model that can relate the influential parameters such as molar weight of n-alkane, temperature and pressure of the system and CO2 density to n-alkane solubility in supercritical CO2.

Model development for determining solubility in supercritical CO2

Least-squares support vector machine (LSSVM)

Support vector machine is one of the artificial intelligence methods that uses statistical calculation through learning process and can be implemented for both classifying and regressing big amount of data (Mesbah et al. 2014; Vapnik 2013; Cortes and Vapnik 1995). SVM is a powerful and reliable tool that solves nonlinear problems by mapping input data from input space to a feature space (Bassir and Madani 2019; Fayyaz et al. 2019). The data transformation into the feature space updates all nonlinear relations of input space to linear relations in the feature space. Another ability of SVM that makes it unique among other artificial intelligent tools is its flexibility in using the deviated data through model training process. Actually, SVM does not discard any deviated data and allows them to make part through training process by a penalty factor (γ), which is a tuning parameter.

In case of large-sized datasets, regression with SVM demands quadratic programming that makes computation difficult and time-consuming (Suykens et al. 2002; Suykens and Vandewalle 1999). Recently, a new modification for SVM, named LSSVM, has been introduced by Suykens and Vanderwalle (1999) that changes inequality constrains of conventional SVM into equality constrains. As a result, LSSVM solves a linear equation system instead of quadratic programming. Accordingly, LSSVM can solve large-scale data with high convergence speed and accuracy.

Similar to other artificial intelligent tools, a nonlinear regression function f is determined by using map function (ϕ) and evaluating w and b in Eq. 1. LSSVM minimizes J in Eq. 2 subject to linear constrain, Eq. 3, to find w (Suykens and Vandewalle 1999):

$$f\left( x \right) = \omega \phi \left( x \right) + b$$
(1)
$$\min J\left( {w,e} \right) = \frac{1}{2}\left\| {\omega^{2} } \right\| + \frac{1}{2}\gamma \mathop \sum \limits_{i = 1}^{N} e_{i}^{2}$$
(2)
$$y_{i} = \omega \phi \left( {x_{i} } \right) + b + e_{i} ,\quad i = 1, \ldots ,N$$
(3)

where eis are slack variables, ω is weight factor, b is bias, and yis are matching data. By applying Lagrange multiplier method for minimizing Eq. 2 with regard to linear constrain of Eq. 3, the regression function is reformulated as below (Gorjaei et al. 2015; Farasat et al. 2013; Zhang and Ge 2019)

$$L_{{{\text{LSSVM}}}} = \frac{1}{2}\left\| {\omega^{2} } \right\| + \gamma \frac{1}{2}\mathop \sum \limits_{i = 1}^{N} e_{i}^{2} - \mathop \sum \limits_{i = 1}^{N} \alpha_{I} \left( {\omega \phi \left( {x_{i} } \right) + b + e_{i} - y_{i} } \right.$$
(4)

where αis are Lagrange multipliers. The optimal conditions for Eq. 4 are

$$\left\{ {\begin{array}{*{20}l} {1. \frac{{\partial L_{{{\text{LSSVM}}}} }}{\partial b} = 0 \to \mathop \sum \limits_{i = 1}^{N} \alpha_{i} = 0} \hfill \\ {2.\frac{{\partial L_{{{\text{LSSVM}}}} }}{\partial \omega } = 0 \to \omega = \mathop \sum \limits_{i = 1}^{N} \alpha_{i} \phi (x_{i} )} \hfill \\ {3.\frac{{\partial L_{{{\text{LSSVM}}}} }}{{\partial \alpha_{i} }} = 0 \to \left\{ {\left( {\omega^{T} \phi \left( {x_{i} } \right)} \right) + b + e_{i} } \right\} - y_{i} = 0} \hfill \\ {4. \frac{{\partial L_{{{\text{LSSVM}}}} }}{{\partial e_{i} }} = 0 \to \alpha_{i} = \gamma e_{i} } \hfill \\ \end{array} } \right. i = 1, \ldots , N$$
(5)

where LSSVM variables including dependent and independent are separable. LSSVM can be formulated as below (Gorjaei et al. 2015):

$$y = \mathop \sum \limits_{i = 1}^{N} \alpha_{i} x_{i}^{T} x + b$$
(6)

Equation 4 is only valid for linear regression problems. In case of nonlinear regression, ϕT(xi)ϕ(x) is substituted with xiTx. Because dot product between input maps (ϕ(xi)) makes the explicit computation complex, some kernel functions have been suggested to evaluate the kernel matrix implicitly. The mainly used kernel functions are linear, polynomial and radial basis function (RBF). In this study, regarding lower tuning parameter and better numerical verification (Keerthi and Lin 2003; Vapnik and Lerner 1963; Na'imi et al. 2014), RBF kernel has been used. RBF kernel is defined as:

$$k\left( {x_{i} ,x} \right) = \phi \left( {x_{i} } \right)\phi \left( x \right) = \exp \left( {\frac{{ - \left\| {x_{i} x} \right\|^{2} }}{{\sigma^{2} }}} \right)\quad i = 1, \ldots , N$$
(7)

where σ2 represents RBF kernel width and needs to be tuned by an external optimizer before training LSSVM model (Gorjaei et al. 2015; Zhang et al. 2009).

Parameters optimization

LSSVM at least has two tuning parameters. These parameters need to be optimized before training the model. In this study, two optimization algorithms, namely particle swarm optimization (PSO) and cross-validation Simplex algorithm, were used to optimize the tuning parameters.

Particle swarm optimization

Kennedy and Eberhart suggested an optimization algorithm named particle swarm optimization (PSO) for finding optimal value of a continuous space (Eberhart and Kennedy 1995). PSO applies insect swarm behavior and evolutionary programming theory to find the global best of a function or space. Through optimization process, particles with no volume explore the search space to discover the optimum location. Velocity and position are two specifications of the particles that are updated through each step. In the first step, PSO uses random function to define initial particle position and initial particle velocity. Through each iteration, PSO seeks the minimum value of the space of each neighborhood (local best) and the minimum all over the space (global best). Using the following equations, position and velocity of particles are updated for next step (Eberhart and Kennedy 1995; Shi and Eberhart 1999)

$$v_{i}^{n + 1} = \omega v_{i}^{n} + c_{1} r_{1}^{n} \left( {x_{{i{\text{Lb}}}}^{n} - x_{i}^{n} } \right) + c_{2} r_{2}^{n} \left( {x_{{{\text{gb}}}}^{n} - x_{i}^{n} } \right)$$
(8)
$$x_{i}^{n + 1} = x_{i}^{n} + v_{i}^{n + 1}$$
(9)

where v is the velocity of the particles, n denotes iteration number, x is particle position at i iteration, xiLb and xgb represent local and global best, ω is moment of inertia, c1 and c2 are acceleration factors and usually are about 2, and r1 and r2 represent random numbers and change in [0,1] interval.

For the next step, all particles except global best’s velocity are updated toward global best of the last step. Through optimization process, particle migration continues until all particles converge to a position with lowest cost value.

Data preparation

In this work, using data mentioned in (Eustaquio-Rincón and Trejo 2001; Shi et al. 2015; Furuya and Teja 2004), a complete data bank was gathered to generate comprehensive models for predicting the solubility of n-alkane in supercritical CO2. To generate a reliable model, it is crucial to use influential parameters as input data. According to Chrastil (1982) (Chrastil 1982), temperature and carbon dioxide density are two parameters affecting the n-alkane solubility in solubility in supercritical CO2. In addition, n-alkane carbon number (n-alkane molar weight) and pressure are influential in this regard (Yang et al. 2012; Chandler et al. 1996; Shi et al. 2015). Therefore, input data include temperature (K), pressure (MPa), normal alkane molecular weight (g/mol) and carbon dioxide density (kg/m3). A set of 180 input points are used to propose a model for predicting the mole fraction of dissolved n-alkane in supercritical CO2. The summary of the gathered data is tabulated in Table 1.

Table 1 Range of input and output parameters utilized in this study
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Optimizing LSSVM model using PSO

Determining an optimum penalty factor γ and tuning kernel parameter are the prerequisites of using LSSVM. One of the recommended kernel functions for nonlinear system is RBF which contains only one tuning parameter σ2 (Gorjaei et al. 2015). Optimizing γ and σ2 before starting training process can strongly assist LSSVM in determining the support vectors. For this reason, the data points are divided into three groups, namely training, validating and testing data. The number of these groups must be selected cautiously so that enough training data are available for model training and validating and adequate data are considered for testing the model. After several try and errors, the best division of 90% training data and 10% test data was found for this work. The swarm size of 30, iteration number of 200 and minimum MSE of 0.0001 were defined before staring the optimization.

Through optimization process and at each iteration, LSSVM is trained for all particle positions, γ and σ2. MSE of the outputs indicates the accuracy of the trained model for the considered position. Finally, the location of the particle with lowest MSE is the global best of the system. The details of optimization process are demonstrated in Fig. 1.

Fig. 1
figure 1

Flowchart of PSO–LSSVM optimization: a PSO optimization b cost function calculation

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Optimizing LSSVM model using cross-validation Simplex technique

Simplex is a popular algorithm that has widely been suggested for linear optimization (Klee and Minty 1972). Through Simplex optimization, the minimum of a function is found by searching the function in a limited interval. The interval is updated at each iteration by reflection, contraction and expansion techniques (Nelder and Mead 1965). Interval update continues until function minimum with acceptable error is found. In this study, cross-validation method has been applied as cost function evaluation technique for LSSVM function.

Cross-validation (CV) is one of the essential techniques in sample selection and optimization validation technique (Schaffer 1993; Browne 2000). To evaluate cost value, CV divides training dataset T into K subsets (T1, T2, T3,…, TK) and trains LSSVM K times. At each training iteration, one T set is reserved for validating the model and other sets participate in support vector training process. Finally, average MSE of all reserved sets will be considered as the cost value.

Results and discussion

In this study, PSO–LSSVM and CV-Simplex–LSSVM models have been trained to predict the relationship between solution properties and n-alkane solubility in CO2. A total number of 180 experimental data with division of 162 points for model training and 18 points for model testing are used in model preparation. The developed models are compared with Chrastil correlation which is as follows (Chrastil 1982):

$$S = \rho^{k} \exp \left( {\frac{m}{T} + n} \right)$$
(10)

where T(K) is temperature, ρ (kg/m3) is CO2 density, k, m, n are characteristic constants of the correlation, and S (kg/m3) is n-alkane solubility defined as:

$$S = \frac{{x\rho M_{1} }}{{\left( {1 - x} \right)M_{0} }}$$
(11)

where x represents mole fraction of dissolved n-alkenes in supercritical CO2, and M0 and M1 are, respectively, the molar weight of CO2 and n-alkane.

As mentioned before, LSSVM with RBF kernel function has two tuning parameters σ2 and γ that have been determined by two different optimization algorithms (PSO and CV-Simplex) in this work. In PSO optimization, the data were divided into three subgroups, namely training, validating and testing data. Dividing data into these groups is an important part of optimization since if the portion of training data is small, the model will be inaccurate and if it is large, the model will be over-trained. Therefore, data were selected randomly and the portion of each group was determined after multiple runs. Consequently, 59% of data were chosen for training the model, 31% were kept for model validation, and 10% were reserved for testing the model. For Simplex case model, validity was checked by applying k-fold cross-validation technique. For this purpose, a total number of tenfold were assigned to the model. K-fold CV allows the optimization algorithm to select training and validating data with alike data distribution. To be confident that the validation process is not dependent on the data points or group subdivision, the validation is repeated ten times with different validation groups and average MSE of validation data of these recurrences will be considered as cost value. Therefore, in CV-Simplex optimization, LSSVM model needs to be trained 10 times for each Simplex iteration.

PSO determined the values of tuned σ2 and γ 4.958 and 7.284 × 105; however, CV-Simplex adjusted these parameters to 2.527 × 105 and 2.034 × 107, respectively. By using these tuned parameters, two different LSSVM models were trained with training data and the accuracy of these models were compared with some statistical parameters including MSE. It is worth mentioning that the determined value for σ2 by CV-Simplex optimizer was abnormally high, but the results of CV-Simplex–LSSVM model were reasonable. Correlation coefficient (R2) is another statistical parameter defined as:

$$R^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {y_{i}^{{{\text{ac}}}} - y_{i}^{{{\text{sim}}}} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{N} \left( {y_{i}^{{{\text{ac}}}} - \overline{y}^{{{\text{ac}}}} } \right)^{2} }}$$
(12)

where \(\overline{y}^{{{\text{ac}}}}\) is average value of actual data, \(y_{i}^{{{\text{sim}}}}\) is the simulated value for data with index i, and \(y_{i}^{{{\text{ac}}}}\) is the actual data with index i.

The predicted values of dissolved n-alkane mole fraction in supercritical CO2 for both PSO–LSSVM and CV-Simplex–LSSVM models are demonstrated in Figs. 2, 3, 4 and 5 for training and testing datasets. The dotted line shown in the figures is the best linear fit for model predicted results, while the solid lines (Y = X) represent the prediction with no error. According to these figures, the simulation accuracy of CV-Simplex–LSSVM is not as high as PSO–LSSVM model. Also, the figures show that PSO–LSSVM model can better track actual data for both training and testing data; however, CV-Simplex–LSSVM cannot predict testing as well as training datasets. This difference in accuracy proves the superiority of PSO algorithm.

Fig. 2
figure 2

PSO–LSSVM predicted mole percent of dissolved n-alkane versus actual data for training dataset

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Fig. 3
figure 3

CV-Simplex–LSSVM predicted mole percent of dissolved n-alkane for training dataset

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Fig. 4
figure 4

PSO–LSSVM predicted mole percent of dissolved n-alkane for testing dataset

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Fig. 5
figure 5

CV-Simplex–LSSVM predicted dissolved mole percent of n-alkane for testing dataset

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The determined linear fit for training data of both PSO–LSSVM and CV-Simplex–LSSVM is obtained as, respectively:

$$y = 1.0x: R^{2} = 0.997$$
(13)
$$y = 0.98x: R^{2} = 0.978$$
(14)

As Eq. 13 shows, the linear fitted line for PSO–LSSVM model completely overlaps y = x line and R2 of training data for this model is 0.997; however, R2 of CV-Simplex–LSSVM is 0.978 and the fitted line for correlated outputs is slightly deviated from y = x.

The same linear fit lines are calculated and demonstrated in Eqs. 15 and 16 for testing data in PSO–LSSVM and CV-Simplex–LSSVM models, respectively:

$$y = 1.08x - 1.0: R^{2} = 0.94$$
(15)
$$y = 0.9x: \;R^{2} = 0.857$$
(16)

According to PSO–LSSVM calculated R2 of 0.997 and 0.94 for training and testing datasets, the proposed PSO–LSSVM model has high capability in predicting the solubility of n-alkane in supercritical CO2 at a wide range of system conditions.

To compare the precision of the proposed model with analytical models of literature, the predicted outputs of the model and Chrastil correlation calculated outputs have been plotted versus actual data in Fig. 6. As this figure and the linear lines show, PSO–LSSVM capability in predicting n-alkane solubility is excellent and its results are reliable.

Fig. 6
figure 6

Predicted mole percent of dissolved n-alkane by Chrastil, CV-Simplex and PSO–LSSVM models versus actual data

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The relative error percent of model outputs calculated by Eq. 17 are shown in Fig. 7. The relative error percent for both train and testing datasets for PSO–LSSVM model change in the range of − 20.02% to 20.18% with minimum absolute error of 3.88%; however, the relative error percent for CV-Simplex–LSSVM model repose in span of − 68.0% to 83.01% with minimum absolute error of 13.49%. Chrastil model can also predict the outputs with relative error of − 75.92% to 63.68%. According to Fig. 7, the superiority of PSO–LSSVM, compared to other models, in predicting n-alkane solubility in supercritical CO2 is inevitable.

$${\text{Relative}}\;{\text{Error}}\;{\text{Percent}} = \frac{{y_{{{\text{ac}}}} - y_{{{\text{pred}}}} }}{{y_{{{\text{ac}}}} }} \times 100$$
(17)
Fig. 7
figure 7

Relative error percent of a PSO–LSSVM, b CV-Simplex–LSSVM and c Chrastil models in predicting n-alkane mole percent

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To better compare and find the robustness of the developed models, a number of statistical parameters such as AARE, MSE and R2 of PSO–LSSVM, CV-Simplex–LSSVM and Charstil models are reported in Table 2. The values of these parameters for training and testing datasets prove the superiority of PSO–LSSVM model in comparison with other models, which is due to method that PSO uses to find the minimum by seeking all the space without limiting the search space.

Table 2 Comparison between statistical parameters of Chrastil, PSO–LSSVM and CV-Simplex–LSSVM models in predicting n-alkane solubility in supercritical CO2
Full size table

The developed PSO–LSSVM model can be used to accurately predict n-alkane solubility in supercritical carbon dioxide without requiring laboratory methods which are inherently expensive and time-consuming. Note that the limitation in utilizing this model is that it is solely applicable within the range of input parameters (See Table 1).

Conclusions

In this work, the capability of implementing intelligent tools for determining n-alkane solubility in supercritical CO2 was studied. For this purpose, least-squares support vector machine (LSSVM) tuned with two different algorithms, namely particle swarm optimization (PSO) and cross-validation-compensated Simplex was trained to predict the molar fraction of n-alkane solved in supercritical CO2. The considered influential factors for this prediction were temperature, pressure, n-alkane molecular weight, and carbon dioxide density. According to the results, PSO–LSSVM model was more powerful and accurate than CV-Simplex tuned LSSVM. By using evolutionary finding technique, PSO could efficiently locate the minimum of a function with local minimums and steep varying curvature. The performance of PSO–LSSVM model was compared with Chrastil model by calculating relative error percent of these models. PSO–LSSVM model with MSE value of 0.016, R2 of 0.994 and average relative error percent of 3.88% was proved to be an efficient method for calculating n-alkane solubility in supercritical CO2 at wide range of pressure and temperature values and can be used with high confidence and reliability.