Menad Nait Amar, Noureddine Zeraibi

Laboratoire Génie Physique des Hydrocarbures, Faculty of Hydrocarbons and Chemistry, University M'hamed Bougara of Boumerdes, Avenue de l’Indépendance, 35000, Boumerdes, Algeria

Keywords: CO2- EOR process CO2-Crude oil minimum miscibility pressure Support vector regression (SVR) Artificial bee colony (ABC)

ABSTRACT Minimum miscibility pressure (MMP) is a key parameter in the successful design of miscible gases injection such as CO2 flooding for enhanced oil recovery process (EOR).MMP is generally determined through experimental tests such as slim tube and rising bubble apparatus (RBA).As these tests are time-consuming and their cost is very expensive, several correlations have been developed.However, and although the simplicity of these correlations, they suffer from inaccuracies and bad generalization due to the limitation of their ranges of application.This paper aims to establish a global model to predict MMP in both pure and impure CO2-crude oil in EOR process by combining support vector regression (SVR) with artificial bee colony (ABC).ABC is used to find best SVR hyper-parameters.201 data collected from authenticated published literature and covering a wide range of variables are considered to develop SVR-ABC pure/impure CO2-crude oil MMP model with following inputs: reservoir temperature (TR), critical temperature of the injection gas (Tc), molecular weight of pentane plus fraction of crude oil (MWC5+) and the ratio of volatile components to intermediate components in crude oil (x vol /xint).Statistical indicators and graphical error analyses show that SVR-ABC MMP model yields excellent results with a low mean absolute percentage error (3.24%) and root mean square error (0.79) and a high coefficient of determination (0.9868).Furthermore, the results reveal that SVR-ABC outperforms either ordinary SVR with trial and error approach or all existing methods considered in this work in the prediction of pure and impure CO2-crude oil MMP.Finally, the Leverage approach (Williams plot) is done to investigate the realm of prediction capability of the new model and to detect any probable erroneous data points.

1.Introduction

Inside mature oil reservoirs remains a considerable amount of residual oil even after applying the primary and secondary oil recovery mechanisms.Nowadays, several methods have been applied with the aim to decrease the residual oil and improve the oil recoverability.These techniques are categorized under on so-called “enhanced oil recovery (EOR)”.Among EOR techniques, miscible CO2injection is one of the most effective and attractive [1], since it has a great potential to increase oil recovery by enhancing microscopic sweep efficiency on one hand; and its lower cost in comparison to other methods in addition to environmental benefits [2], on the other.The right design of miscible CO2injection projects depends greatly on a key factor that must be taken into account before the execution phase: minimum miscible pressure (MMP).MMP is defined as the lowest pressure at which the flood changes from immiscible (multiple phase flow) to miscible (single phase flow) [3-5].Hence, accurate and fast calculation/estimation of the MMP is of great importance.Three main categories of methods can be distinguished in the aim to determine MMP for either pure CO2- crude oil or impure CO2-crude oil [6]: experimental tests, empirical/ analytical correlations and computational methods (artificial intelligence).

The main experimental tests are slim tube [7], rising bubble apparatus (RBA) [8], multi-contact mixing-cell experiment [5] or the vanishing interfacial tension (VIT) technique [9].Although the accuracy of these tests, all of them are very expensive and time consuming.

To overcome the limitations of experimental test, several empirical/ analytical correlations have been developed for determining the MMP of pure CO2-crude oil, impure CO2-crude oil or both pure and impure CO2-crude oil.A quick estimation of the MMP can be done by applying these correlations.However, as all these latter have been developed under specified experimental data, they have certain constraints and conditions of application.Among these correlations, we can cite Cronquist correlation [10], Yellig and Metcalfe correlation [11], Orr & Jensen correlation [12], Emera-Sarma correlation [13], Shokir correlation [14].Other correlations such as Alston et al.[15], Sebastian et al.[16], Eakin and Mitch [17], Lee [18], Dong [19], and Yuan et al.[20] have focused directly on specific oil reservoirs, and this cannot satisfy the level of comprehensiveness and generalization required by various other oil reservoirs where the characteristics are different.

A good number of studies have been carried out on the application of various artificial intelligence (AI) techniques such as artificial neural network (ANN), gene expression programming (GEP) and support vector regression (SVR) in many fields of science and engineering [21-30].In petroleum and reservoir engineering, these techniques are becoming more attractive in resolving many problems of practical importance such as well-test analysis [31], PVT parameters estimation [32-34], history matching [35], WAG optimization [36], bottom hole pressure estimation [37], reservoir characterization [38-40] and others.MMP estimation of CO2-crude oil system can also be done by applying AI methodologies.Many articles shed light the success of these methodologies in establishing reliable models to estimate MMP of CO2- crude oil systems.Huang et al.[41] applied multilayer perceptron to predict MMP for both pure and impure CO2 injection cases.Ahmadi et al.[42] combined least square support vector machine (LS-SVM) and evolutionary algorithms to the same aim.Sayyad et al.[43] proposed an ANN model based on PSO (particle swarm optimization) to estimate MMP.Ahmadi et al.[44] established a MMP model based on fuzzy logic.Three different CO2-crude oil MMP correlations using GEP have been developed by Kamari et al.[45], Fathinasab-Ayatollahi [46] and Ahmadi et al.[47], respectively.Although the accuracy showed by the aforementioned CO2-crude oil MMP based AI, it is necessary to underline that some of them were developed under a limited number of data (such as in Refs.[42] and [44] with only 68 and 58 data respectively).

Among computational intelligence tools, support vector regression (SVR) has gained considerable popularity and importance thanks to its high generalization capability that is assured by its exploitation of both structural risk minimization (SRM) and empirical risk minimization (ERM) principles [48].The prediction accuracy of an SVR model relies heavily on appropriate determination of model parameters.Therefore, to improve this selection and find the optimal values of these parameters, hybridizations of SVR with metaheuristic algorithms, which are known for their global solution to such problems, have been proposed in many studies [39,49].

In this study, we propose a new robust, fast, cheap and global model to predict MMP of both pure and impure CO2-crude oil system.This model consists in hybridization SVR with artificial bee colony (ABC).This latter is used to optimize SVR-hyper parameters.201 measured MMP data collected from published literature (105 of 201 data are pure CO2-oil while the remain 96 are impure CO2-oil) are used for SVR-ABC development.Four inputs are considered: reservoir temperature (TR), critical temperature of the injection gas (Tc), molecular weight of pentane plus fraction of crude oil (MWC5+) and the ratio of volatile components to intermediate components in crude oil (x vol/xint).Subsequently, the results obtained for the proposed SVR-ABC are compared against literature reported data, SVR-TE (support vector with trial and error method) and previously published correlations.In addition, several statistical parameters and graphical error analysis are considered.In assessing the validity of the SVR-ABC, the Leverage approach (Williams plot) is utilized to determine its prediction capability and also to identify probable erroneous data points.There are two main differences between this work and other research works in this field (such in Ref.[50]): (1) the number of data used in the developed model and their intervals that cover a big variety, this gives an excellent generalization of the established model, and (2) besides the application of SVR as an effective tool with excellent predictive and generalization ability for estimating MMP of CO2-crude oil system, this study shows the efficiency and the robustness of the ABC algorithm in the optimization of SVR hyper parameters to better estimate MMP of CO2-crude oil system.

2.Methodology

2.1.Support vector regression

Support Vector Regression (SVR) is a supervised learning technique based on statistical learning theory introduced by Vapnik [51,52].SVR trains to identify a function which computes the relationship dependency between targetsdefined onR, and inputsthatxiRnandkis the data size.This is done by mapping the original problem into a high dimensional space by introducing a kernel function to make the original problem be able to be fitted by a linear regression function and to conveniently solve non- linear regression problems.This can be formulated as follows:

To estimatewandb, first, the regression problem is transformed into a strict minimization problem of the regularized risk function that defines both the model complexity and the empirical error under ε- insensitive loss.This can be expressed as follows:

wherew2is the measure of function flatness,reflects the empirical error, the constantC＞0 is the penalty parameter that computes the trade-off between the empirical error and the model complexity (regularization factor),are positive slack variables which represent lower and upper excess deviation, respectively andis the error tolerance.

Then, the constrained optimization problem (2) is resolved after its transformation into dual space using Lagrange multipliers.To keep the work concise, the procedure is not described here.The obtained solution is shown below [53]: whereare Lagrange multipliers and are subjected to the constraintsand the termis the kernel function.Among kernel functions proposed in the literature [53], radial basis function (RBF) and Gaussian function are the popular ones.In this paper, RBF is used as the kernel function.RBF is defined as shown below:

To achieve a high performance and high accuracy of SVR, the combination of,and the kernel function parametershould be selected properly.Hence, optimizing these parameters using robust metaheuristic algorithms able to select automatically the optimum combination is a favorite alternative to overcome the use of the traditional trial and error method (TE).

Meanwhile, it should be noted that one of the main advantages of SVR is that its computational complexity does not depend on the dimensionality of the input space [54].In addition, sequential minimal optimization (SMO) [55], one of quadratic programming solvers for SVRs, has become a quasi-standard in SVR training as it reduces the computational complexity of SVR to approximatelyO(N2), whereNis the number of training data [56].

2.2.Artificial bee colony (ABC)

Artificial Bee Colony (ABC) algorithm was proposed by Karaboga in 2005 [57] for optimizing numerical problems.ABC is inspired from honey bee swarms, which represent the intelligent foraging behavior of bee swarms.This algorithm provides solution in organized form by dividing the bee objects into 3 different tasks/categories of bee: employed bees, onlooker bees, and scout bees.These latter determine the objects of problems by sharing information to others bees.Each food source discovered so far is tracked by an employed bee.Hence, the number of food sources is equal to employed bees.The food source is the solution candidate and is coded at time (iteration) t with a D dimensional vector as:

The best food source is the most qualified optimized solution for the problem.An employed bee computes a modified position from her memorized food location depending on the local information and tests the quality of food on the new source.If the new location is a better food source compared to the previous one, the bee memorizes the new position and forgets the old one.After all employed bees complete their exploration; they share new food source information and their position information with the onlooker bees on the dance area.Onlooker bees select food sources based on the dance performed by the employed bees: they watch the dance of hive bees and select the best food source according to the probability proportional to the quality of that food source [57,58] (for this purpose, a roulette wheel selection technique can be used).An employed bee is transformed into a scout bee if its food source is abandoned.Scout bees carry out random search in the model space.

Four phases are followed in ABC algorithm: initialization, employed bees phase, onlooker bees phase and scout bee phase, each of which is explained below:

(1)Initialization phase: Initialize the population of solutionsxiwhere

After producing food sources and assigning them to the employed bees, the objective function specific for the optimization problem is operated, its value is obtained, and all the fitness values of the food sources are calculated by using:

wherefiti,tis the fitness of the source i at the iteration t (t = 1 for the initialization phase), andf(.) is the objective function (the optimization problem is to minimize this objective function).

(2)Employed bees phase:during this phase, each bee modifies its current solution according to the information of individual experiences and the fitness value of the new solution.If the fitness value of the new food source is higher than that of the old, the bee updates its position with the new one and discards the old one.The position is updated by the equation below:

(3)Onlookers phase:Onlooker bee phase is started after finishing the employed bee phase, this phase operates on sharing the fitness information that has been collected by the employed bee phase, this information includes updated solutions and their position information.Onlooker bees analyze the available information and select a solution with a probability related to its fitness:

wherefitiis the fitness of ith solution.As in the case of the employed bee, an onlooker bee produces a modification in the position in its memory with the same principle to employed bees.

(4)Scout Bees Phase:In this phase, the employed bee become scout bee if the employed bee is associated with an abandoned food source or after a certain defined number of iteration, also, the food source is replaced by randomly choosing another food source from the search space.

In this paper, ABC is employed to tune SVR hyper-parameters (C, ε and γ).The manner of implementation of ABC into SVR is summarized in flowchart shown in Fig.1.

3.Model development

3.1.Data acquisition and pre-processing

The number and the variety of initial input data sets affect the accuracy and generalization of the proposed SVR-ABC model for MMP estimation of pure/impure CO2-crude oil system.In this study, 201 MMP of pure/impure CO2-crrude oil data were collected from the published literature [10-20,59-66].Among these 201 data, 105 and 96 refer to pure and impure CO2respectively.

A review on recent researches shows that [14-17,20,66]:

(1) The main factors affecting CO2-oil MMP are: the reservoir temperature, the purity of injected gas, and the components and properties of oil.

(2) As the temperature increases, the MMP increases as well and vice versa.

(3) MMP decreases when the crude oil is characterized with high content of C2～C6and low molecular weight.On the contrary, themore the heavy components in the crude oil are, the less favorable it will be for miscibility.

(4) The presence of methane and nitrogen (CH4and N2) impurities in the injected gas substantially increase the MMP, but intermediate hydrocarbons (C2-C5) or H2S impurities decrease the MMP.

Accordingly, four factors that have great effects on MMP were selected as the input variables: reservoir temperature (TR), oil composition which is generally summarized by the ratio of volatile components (C1and N2) to intermediate components (C2-C4, H2S and CO2), and the critical temperature of injection gas.Before proceeding to the model development, it is very useful to get an overview about the statistical analysis of the employed datasets.Statistical criteria including standard deviation (SD), which measures the extent of variation or dispersion within the data points of each variable, relative standard deviation (% RSD) which shows the extent of variability in relation to the mean of the data points of each variable, mean, maximum and minimum value which allow a better comprehension of the content of the dataset, are considered for this purpose.A full statistical description of the system input/output is reported in Table 1.MMP for this dataset is ranged from 6.50 to 38.52 MPa, while the inputs used varied as follows: Reservoir temperature from 307.5 to 410 K, the molecular weight of the C5+ fraction from 136.26 to 391 g/mol, the ratio of volatile components to intermediate components in crude oil from 0 to 13.07 and the critical temperature of the injection gas from 281.45 to 338.77 K.To improve the convergence conditions of SVR-ABC model, the used data are standardized.

Fig.1.SVR-ABC flowchart.

Variables Max Min Mean SD %RSD Output MMP (MPa) 38.52 6.50 16.02 6.12 38.20 Inputs MWC5+ (g/mol) 391 136.26 194.63 40.10 20.60 TR (°K) 410 307.5 345.45 24.31 7.04 xvol/xint 13.07 0 1.60 2.09 130.62 Tc (°K) 338.77 281.45 302.72 8.31 2.75

To demonstrate the correlation between MMP and the used independent variables the correlation matrix is implemented.This matrix aims to distinguish linear relationship between two different variables in multi-variables system [67,68].

Its values are between [-1, 1].Two variables are said to be positively linearly related if their correlation coefficient is close to 1, and negatively linearly if it is close to -1.For values nearby zero, it would indicate a weak linear relationship between the variables.

The obtained results are shown in the bar graph of Fig.2 and in Table 2.According to this figure, it can be seen that reservoir temperature (TR) has the highest linearly relation with MMP (0.62).Then molecular weight (MW) of the C5+and ratio of volatile components to intermediate components in crude oil (xvol/xint) by 0.57 and 0.37, respectively.Furthermore, it can be deduced that critical temperature of the injected gas (Tc) is in negative linearly relation with MMP which means that the MMP is high, if Tcis low.It is obvious from the aforementioned results that in multivariable phenomena, the relationship of the input-output variables has a weak linear relationship and cannot be handled using a linear mode, hence the need of SVR model to identify this relationship.

Fig.2.Correlation coefficient of the independent variables.

Table 2 Correlation coefficients between the descriptors and the target.

3.2.Computational procedure

The used database is divided into two parts: the first one contains 175 data (from the 201) and it is used in the training of the model, and the other part (with the remain 26 data) is devoted to test its accuracy and robustness to predict MMP of pure/impure CO2-EOR with blind data.

Table 1 Statistical description of the input/output data used for developing SVR-ABC.

All the programming tasks developed in this work are carried out using MATLAB? 2016-a computing environment [69].The corresponding parameters of ABC algorithm used in the optimization process of SVR hyper-parameters (C, ε and γ) are selected using parameter- tuning method [70], and their corresponding values are as follows: the total number of bees is 40 (whose 20 are employed and the same goes for onlookers); scout bees' max iteration (i.e.after how many iterations an employed bee becomes scout bee): 5; and the maximum number of iteration is 50.Then, the steps of the flowchart of Fig.1 which describes the process of hybridization SVR with ABC, are implemented.As a comparison of SVR-ABC performances against SVR with the classical trial and error method (SVR-TE) is done, this last is developed by manually searching a specified subset of the hyper-parameter (C, ε and γ).For this purpose, a control loop is added to the main program of the SVR, which varied the values of these constants and examined the accuracy of the model in each situation in finding the more appropriate conditions.

3.3.Performance assessment

To evaluate how well the model fits the data was used, a group of statistical indicators, including the average absolute percent error (AARD%), standard deviation (SD), the coefficient of determination (R2) and the root mean square error (RMSE) are considered.These statistical indexes are calculated by the following equations:

wherekrepresents the number of the measured information,is the experimental minimum miscibility pressure values, whileis the calculated MMP values which are predicted by the developed model.Average value of the MMP data is shown byMMP.

To better show the model ability to estimate MMP, it is compared against existing correlations and approaches.Furthermore, as the leverage technique (detection of the outlier data points) plays a significant role to identify the applicability domain of a model and detect suspect data [71-73], this technique through Williams plot is used to check whether the SVR-ABC model is statistically valid.A detailed description of computational procedure and equations for the leverage technique can be found elsewhere [71-73].

4.Results and discussions

The above-described computational steps of SVR-ABC and SVR-TE are followed to achieve best SVR hyper-parameters.Moreover, as previously mentioned, for assessing the performance capabilities, statistical error analysis, in which AARD, R2, SD and RMSE, as well as graphical error analysis, in which cross plot and APRE error distribution plot are sketched, have been implemented.As a consequence, the obtained best SVR hyper-parameters (C, ε and γ), which yield the most precise results for both trial and error method and ABC algorithm are shown in Table 3.According to this table, high values of the regularization factor (C) and small Epsilon values are achieved for both SVR-ABC and SVR-TE, while medium values of the kernel parameter (γ) are expected.

Table 3 Obtained SVR hyper-parameters for developed models.

Cross plots between output and target values for training and test data of SVR-ABC and SVR-TE models are illustrated in Fig.3.For each model, all the predicted values are sketched against the experimental values and therefore across plot is created and compared against a unit slope line that shows the perfect model line: the closer the plotted data to this line, the higher is the reliability of the model.As clear be seen, the prediction capability of the SVR-ABC is documented.For a deep comparison between the calculated MMP by SVR-ABC and SVR-TE and the experimental data, the results of the aforementioned statistical indexes are reported in Table 4.As can be observed in Table 4, the SVR model optimized by artificial bee colony (SVR-ABC) shows the higher accuracy in comparison with SVR-TE by total values of 3.16%, 0.9879, 0.78 and 0.052 for AARD, R2, RMSE and SD, respectively.In order to further evaluate the SVR-ABC accuracy, a performance distinction of this model in predicting the MMP of the cases pure and impure CO2- crude oil is presented separately in Table 5.This table highlights that the proposed SVR-ABC model predictions are in excellent agreement with experimental data with AARD of 2.75% for pure CO2case and 4.33% for impure CO2case are noticed, respectively.Fig.4 depicts the error distribution of the SVR-ABC model for estimating MMP CO2-crude oil.In this figure, the error is the percent relative error (PRE%) which measures the relative deviation of predicted data from the experimental data.The figure confirms that SVR-ABC provides favorable results so that it has a low scatter around the zero error line, and most of its predictions have a relative error between -15% and 15%.This indicates the potential of SVR-ABC model for predicting CO2-crude oil MMP with small expected errors.

The performance of the model has been compared with that of some of the most widely utilized correlations.These include seven correlations for pure CO2-crude oil cases, namely those of: Cronquist [10], Lee [18], Yellig & Metcalfe [11], Orr & Jensen [12], Emera-Sarma [13] and Shokir [14]; and two correlations for impure CO2-crude oil cases: Kamari et al.correlation [45] and Fathinasab-Ayatollahi correlation [46].Table 6 reports the corresponding results.As it is shown in Table 6, SVR-ABC model leads to very satisfactory performances for determination of MMP values either for pure or impure CO2-crude oil cases, and outperforms largely the published correlations in terms of accuracy of the predictions.

Finally, in order to check whether the SVR-ABC model is statistically acceptable, the Williams plot has been illustrated in Fig.5.The existence of the majority of data points in the ranges 0 ≤ H ≤ 0.0746 and -3 ≤ Standardized Residuals ≤3 confirms that the SVR-ABC model developed for the calculation of CO2-crude oil MMP is statistically accurate and reliable.Consequently, good high leverage data points are located in the domain of H＜0.0746 for SVR-ABC: the entire data except seven in the data set are located within the applicability domain.As a result, there are only seven points in the datasets that are not within this domain, and accordingly, these data can be stated as probable doubtful datum.

5.Conclusions

Fig.3.Cross-Plots of the Results: (a) MMP measured vs MMP SVR-TE (training + test); (b) MMP measured vs SVR-ABC (training + test).

Table 4 Statistical analysis of performance of SVR-ABC and SVR-TE.

Table 5 Statistical analysis of performance of SVR-ABC for pure CO2-crude oil and impure CO2-crude oil MMP estimation.

Fig.4.Relative deviations of the MMP values obtained by SVR-ABC model from data base values.

In this work, a new model based on optimized support vector regression (SVR) by artificial bee colony algorithm (ABC) namely SVR- ABC was proposed and applied to predict the MMP of pure/impure CO2- crude oil systems.In this model, four factors (i.e.reservoir temperature, critical temperature of the injection gas, molecular weight of C5+fraction of crude oil, and the ration of volatile components to intermediate components in crude oil) representing the most comprehensive and robust set were selected as the input variables, while MMP was considered as the output variable.A large number of data points collected from open literature and covering a wide range of independent variables was considered for developing the model.Statistical and graphical error analysis have been carried out to establish the adequacy and accuracy of the model as well as to compare it with classical SVRbased on trial and error method (TE), and with existing correlations.The results show that SVR-ABC model provides predictions in satisfactory agreement with experimental data and outperforms all the other models either SVR-TE or correlations.The precision analysis of the developed model shows its high ability to predict MMP with a low mean absolute percentage error (3.24%) and root mean square error (0.79) and a high coefficient of determination (0.9868).Furthermore, leverage approach reveals that this model is statistically correct and valid with only seven probably doubtful data points in the whole experimental data set.

Table 6 Comparison of SVR-ABC performances against available correlations for both pure/impure CO2-crude oil.

Fig.5.Detection of the probable doubtful data and the applicability domain of the developed SVR-ABC model.

Nomenclatures

EOR Enhanced Oil Recovery

ABC artificial bee colony

SVR support vector regression

Int. intermediate components

MMP Minimum Miscibility Pressure

Mw Molecular Weight

TRreservoir temperature

Tccritical temperature

R2coefficient of determination

Vol. volatile components

AARD average absolute relative deviation

RMSE root mean Squared Error

SD standard deviation

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