A predictive model of chemical flooding for enhanced oil recovery ...

Petroleum 2 (2016) 177e182

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Original article

A predictive model of chemical flooding for enhanced oil recovery purposes: Application of least square support vector machine Mohammad Ali Ahmadi a, *, Maysam Pournik b a

Department of Petroleum Engineering, Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran Department of Petroleum Engineering, Mewbourne School of Petroleum and Geological Engineering, University of Oklahoma, Oklahoma, United States

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 July 2015 Received in revised form 1 October 2015 Accepted 13 October 2015

Applying chemical flooding in petroleum reservoirs turns into interesting subject of the recent researches. Developing strategies of the aforementioned method are more robust and precise when they consider both economical point of views (net present value (NPV)) and technical point of views (recovery factor (RF)). In the present study huge attempts are made to propose predictive model for specifying efficiency of chemical flooding in oil reservoirs. To gain this goal, the new type of support vector machine method which evolved by Suykens and Vandewalle was employed. Also, high precise chemical flooding data banks reported in previous works were employed to test and validate the proposed vector machine model. According to the mean square error (MSE), correlation coefficient and average absolute relative deviation, the suggested LSSVM model has acceptable reliability; integrity and robustness. Thus, the proposed intelligent based model can be considered as an alternative model to monitor the efficiency of chemical flooding in oil reservoir when the required experimental data are not available or accessible. Copyright © 2015, Southwest Petroleum University. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Chemical flooding Enhanced oil recovery (EOR) Polymer Surfactant Least square support vector machine (LSSVM)

1. Introduction The oil and gas upstream industries have recently encountered with the difficulties and challenges of dealing with hydrocarbon resources whose productions with conventional technologies are following an upward trend of technical limitations. It is because of achieving the stage of decline phase by most of oilfields around the world. Therefore, how to postpone the abandonment of reservoirs has tuned into the priority of

* Corresponding author. Department of Petroleum Engineering, Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology, P.O. Box: 63431, Ahwaz, Iran. Tel.: þ98 912 6364936. E-mail address: [email protected] (M.A. Ahmadi). Peer review under responsibility of Southwest Petroleum University.

Production and Hosting by Elsevier on behalf of KeAi

researchers in the worldwide. Their researches normally highlight the concept of great necessities for inventions of new techniques, normally classified as tertiary oil recovery methods, having abilities of maintaining the economic production rate [1e3]. Chemical enhanced oil recovery approaches as one of the most effective subsets of tertiary methods are known as a key to unlock the exploitation of referred resources. Different methods for this process have been developed, such as: polymer, surfactant/polymer (SP), and alkaline/surfactant/polymer (ASP) flooding, microgel, foam, polymer gel and microemulsion flooding. These methods are applied to increase the rate of oil production through focussing on both lowering the interfacial tension and reducing the water mobility. In more details, it has enormously been declared in previous literatures that in order to design, manage and run a chemical enhanced oil recovery operation it is highly required to set very expensive and time-consuming but precise experimental procedures which their generated results must be gained to plan effectively the process of injecting chemical materials [4e9].

http://dx.doi.org/10.1016/j.petlm.2015.10.002 2405-6561/Copyright © 2015, Southwest Petroleum University. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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M.A. Ahmadi, M. Pournik / Petroleum 2 (2016) 177e182

The laboratorial generated outputs are then used to conclude two parameters, RF and NPV, which are used to evaluate the performance of the chemical flooding which is one of the most popular methods of chemical enhance oil recovery. Having knowledge about these two parameters are essentially vital to make decisions if it is beneficial to run the referred operation. Unfortunately, there are no global methods to interpret simultaneously about both aforementioned factors although there are numerous numbers of different software and numerical or analytical method which are capable of making very precise quantitative decisions about the amount of one the RF or NPV [10e12]. Hence, there is a great need in oilfield for having access to a solution or model which can predict the amount of these two parameters at the same time. The major aim of current study is execute new kind of artificial intelligence approaches called “least square support vector machine (LSSVM)” to suggest robust and accurate predictive method to forecast efficiency of the chemical flooding through petroleum reservoirs. To gain successfully this referred goal, LSSVM was executed on the previous literature data bases. The integrity and performance of the proposed predictive approach in estimating RF and NPV from the literature is described in details. 2. Data gathering The data utilized throughout this research have been gathered from previous attentions [9] in which chemical flooding had been simulated in Benoist sand reservoir, by executing UTCHEM simulator. That reservoir has been produced under primary and secondary processes over fifty years. The Original dataset contained 202 data. Each data had 7 inputs: Surfactant slug size, surfactant concentration in surfactant slug, polymer concentration in surfactant slug, polymer drive size, polymer concentration in polymer drive, Kv/Kh ratio, and salinity of polymer drive. In addition, the outputs were RF and NPV. The ranges of implemented data banks are reported in Table 1 [9]. 3. Least square support vector machine (LSSVM) The least square SVM theorem was proposed and developed by Suykens and Vandewalle in 1999 dedicated to the presumption that the implemented data assortment S ¼ {(x1,y1), …, (xn,yn)} that deal with a nonlinear function and decision function can be formulated as illustrated in equation (1). Through the addressed equation, w stands for the weight factor, 4 denotes the nonlinear function which correlates the input space to a highdimension characterization area and conducts linear regression

Table 1 Statistical analysis of the implemented chemical flooding data samples [9]. Parameter

Unit

Surfactant slug size PV Surfactant concentration Vol. fraction Polymer concentration wt.% in surfactant slug Polymer drive size PV Polymer concentration wt.% in polymer drive Kv/Kh ratio e Salinity of polymer drive Meq/ml Recovery factor (RF) % Net present value (NPV) $ MM

Type Input Input

Min 0.097 0.005

Max 0.259 0.03

Average Standard deviation 0.177 0.017

0.072 0.011

Input

0.1

0.25

0.177

0.067

Input Input

0.324 0.1

0.648 0.2

0.482 0.148

0.144 0.044

Input 0.01 0.25 0.129 Input 0.3 0.4 0.349 Output 14.82 56.99 39.67 Output 1.781 7.229 4.45

0.107 0.045 9.24 1.53

while b represents the bias term [13e34]. Following expression was implemented as a cost function of the least square support vector machine (LSSVM) in calculation steps [13e34]. N X 1 QLSSVM ¼ wT w þ g e2k 2

(1)

k¼1

Relate to the following restriction [13e23]:

yk ¼ wT 4ðxk Þ þ b þ ek

k ¼ 1; 2; …; N

(2)

To specify function estimation issue the structural risk minimization (SRM) approach is suggested and the optimization issue is implemented to mastermind the addressed R function while C represents the regularization constant and ei stands for the training error [13e34].

m 1 2 1 X Rðu; e; bÞ ¼ w þ C e2k 2 2

(3)

k¼1

To extract routs w and e, the Lagrange multiplier optimum programming approach is performed to solve equation (3); the addressed approach considers impartial and restriction parameters simultaneously. The mentioned Lagrange function L is formulated as following equation [13e34]:

Lðw; b; e; aÞ ¼ Jðw; eÞ

m X

n o ai wT ∅ðxk Þ þ b þ ek Yk

(4)

k¼1

Through above equation, ai denotes the Lagrange multipliers that may be either positive or negative because LSSVM has equality restrictions. Owing to the KarusheKuhneTucher's (KKT) conditions, conditions for optimum goal are demonstrated in equation (3) [13e23].

8 > > > > > > > < > > > > > > > :

vu L ¼ u

n X

ai 4ðxi Þ ¼ 0

i¼1

vb L ¼

n X

ai ¼ 0

i¼1

vei L ¼ Cei ai ¼ 0 vai L ¼ wT ∅ðxk Þ þ b þ ek yk ¼ 0

9 > > > > > > > = > > > > > > > ;

(5)

Therefore, the linear equations can be demonstrated below expression [13e23]:

2 6 4

0 1

3 1T 0 7 b 1 5 a ¼ y U þ IN g

(6)

While y ¼ (y1, …, yn)T, 1n ¼ (1, …, 1)T, a ¼ (a1; …; an)T and Uil ¼ 4 (xi)T 4 (xl) for i, l ¼ 1, …, n. Thanks to the Mercer's theorem, the resulting LSSVM model for function approximation turns to the following equation [13e23]

f ðxÞ ¼

N X

ak Kðx; xk Þ þ b

(7)

k¼1

Where a and b are the routs to equation (7) as below [13e23]:

b¼

1 1Tn U þ g1 In y 1Tn

Uþ

1 g In

(8)

1 1n


a¼

Uþ

1 In g

1

ðy 1n bÞ

(9)

Equation (10) may be executed as choice of nonlinear regression and utilize the Kernel function as below equation [13e23]:

f ðxÞ ¼

N X

ak Kðx; xk Þ þ b

(10)

k¼1

while K(x, xk) stands for the dependency of Kernel function to the inner values of two vectors x and xi in the feasible area referred to the inner products of the vectors F(x) and F(xi) as below equation [13e23]:

Kðx; xk Þ ¼ FðxÞT Fðxk Þ

(11)

Where the radial basis function (RBF) Kernel has been executed as following formulation [13e23]:

. Kðx; xk Þ ¼ exp kxk xk2 s2

problem through becoming introduced into the fitness function. Subsequently, the output of fitness function related to each chromosome is taken as a criterion to make a decision if the related string can provide a satisfying performance. After removing a number of the weakest individuals which is determined by the designer, it is the turn to operate crossover and mutation rates to produce new individuals with higher performance. Then, implementation of the crossover operation on the couple of chosen strings (chromosomes) to recombine them has to be followed. It has been suggested by the previous studies that the best performance of the GA becomes possible when the crossover point of any two chromosomes is randomly set. The process is followed by switching some random selected position to 1 if they are 0, and vice versa. The last described step is named mutation which is run to prevent the procedure to trap in any local maxima. Final step is defining as returning the generated off-springs into the first step during the next population in order to be evaluated again [24e34]. 5. Results and discussion

(12) 5.1. LSSVM results

Where s plays a role of decision variable through running least square SVM method, which is indicated by performing robust optimizer like genetic algorithm (GA). To gain optimum value of least square SVM parameters, the mean square error (MSE) of the obtained results of the final approach assigned as an objective function of the genetic algorithm (GA) which is formulated by the following equation [13e34]:

2 Pn i¼1 Reest:i Reexp:i RMSE ¼ ns

179

(13)

Where Re represents the recovery factor/net present value (NPV), subscripts est. and exp. represents the predicted and actual recovery factor/net present value (NPV), respectively, and ns stands for the number of data from the initial assigned population.

4. Genetic algorithm (GA) Genetic Algorithm (GA) as one of the best optimization methods which is basically attributed by its unique features which are searching quickly and optimizing efficiently; the two important characteristics which have been derived from the principle of “survival of the fittest” element of natural evolution with the genetic propagation of properties. In more details, GA operates through clarifying a variety of zones in the target area determined by experts and defining simultaneously and randomly a large number of possible paths. The GA has this capability of being replaced with classic optimization techniques thanks to its origination which is based on the idea of Darwinian natural selection and genetics in biological systems. Based on the supporting concept of ‘survival of the fittest’, the GA could converge towards the best point in the prepared space soon after a series of repetitive calculations. Foundations of this searching process are based on technical operations such as artificial mutation, crossover and selection. To run the mentioned algorithm, it is preliminarily required to prepare an initial population containing a certain number of so-called individuals which are representing the possible paths toward the favourite goal. The next step which is supposed to be taken is turning each chromosome, already introduced under the title of an individual, into an encoded string. After that, each string must show its suitability with nature of the

To clarify the sensitivity of the proposed models in determination of efficiency and economic aspect of chemical flooding, a robust and effective statistical method called analysis of variance (ANOVA) was employed. The results gained from analysis of variance are depicted in Fig. 1. As demonstrated in Fig. 1, the most important parameters affect the recovery factor are surfactant concentration and surfactant slug size while the most vital parameters affect on the NPV are surfactant concentration and polymer concentration in surfactant slug. To indicate the optimum values of the least square support vector machine parameters, g and s2, the genetic algorithm (GA) was executed. Consequently, the values of the global optima including s2 and g were indicated as 1.687654 and 27.578421, correspondingly. The results generated by the vector machine approach are depicted through Figs. 2e10. Fig. 2 depicts the comparison between the RF generated by LSSVM model and real one. As shown in Fig. 2, the LSSVM outputs lie over the line Y ¼ X, this index indicates that the RFs estimated by LSSVM approach are reasonably closed to actual ones. To serve better understanding about results generated by the proposed vector machine model, the comparison between predicted RFs and actual ones versus corresponding data index are illustrated in Fig. 3. As illustrated in Fig. 3, the results obtained from the LSSVM model are as close as possible to real RF data samples. The high considerable level of efficiency and accuracy related to the LSSVM approach in prediction of the recovery factor of chemical flooding has once again been certified in Fig. 3. Moreover, the robustness of the LSSVM has been demonstrated in terms of the relative deviations of the LSSVM model outputs from corresponding actual RF in Fig. 4. As can be observed from Fig. 4, the highest deviations of the LSSVM results are subjected to the early boundary of RF samples. 8% is the maximum degree of relative deviation as shown in Fig. 4. The relative deviations of the LSSVM results alongside corresponding surfactant concentration and surfactant slug size data samples are demonstrated through Figs. 5 and 6, correspondingly. The maximum deviation of LSSVM results are those related to surfactant slug size (PV) ¼ 0.1 PV and surfactant concentration ¼ 0.005 Volume Fraction. The values of the global optima which include s2 and g were indicated for estimating NPV of chemical flooding as 0.857483 and 4.186745, respectively. The comparison between NPVs predicted by LSSVM and actual ones in terms of regression plot is

180


Fig. 3. Comparison between suggested vector machine model and recovery factor versus relevant data index.

Fig. 4. Relative error distribution of the proposed approach versus actual RF. Fig. 1. Relative importance of each parameter on the a) RF, b) NPV.

Fig. 5. Relative error distribution of the proposed approach versus actual surfactant slug size (PV).

Fig. 2. Performance plot of the suggested vector machine model for determining recovery factor of chemical flooding owing to correlation coefficient (R2).

shown in Fig. 7. As shown in Fig. 7, the LSSVM outputs lie over the line Y ¼ X, this criterion reveals that the NPVs estimated by LSSVM model are acceptably closed to actual NPV data samples. The comparison between generated NPV by LSSVM method and


Fig. 6. Relative error distribution of the proposed approach versus actual surfactant concentration (Vol. Fraction).

181

Fig. 9. Relative error distribution of the proposed approach versus actual net present value (NPV).

Fig. 10. Relative error distribution of the proposed approach versus actual polymer concentration in surfactant slug (Wt%). Fig. 7. Performance plot of the suggested vector machine model for determining NPV of chemical flooding owing to correlation coefficient (R2).

this work is depicted in terms of the relative deviations between the LSSVM outputs and real NPVs as demonstrated in Fig. 9. As can be seen from Fig. 9, the highest deviations of the LSSVM outputs are subjected to the early boundary of NPV data. 7.5% is the maximum degree of relative deviation as depicted in Fig. 9. The relative deviations of the LSSVM approach alongside corresponding polymer concentration in surfactant slug (Wt%) data samples are illustrated in Fig. 10. The maximum deviation of the NPVs estimated by LSSVM model are those related to polymer concentration in surfactant slug (Wt%) ¼ 0.1 or 0.25. Finally, to summarize previous results, Table 2 reports the statistical indexes of the LSSVM approach in details. As reported in Table 2, it can be inferred that the LSSVM model suggested in this study has a high level of efficiency with low degree of uncertainty.

Fig. 8. Comparison between suggested vector machine model and net present value (NPV) versus relevant data index.

Table 2 Statistical parameters of the proposed approaches in prediction of efficiency of chemical flooding in oil reservoirs. LSSVM

real NPV data versus relevant data index is depicted through Fig. 8. As illustrated in Fig. 8, the NPVs gained from LSSVM approach are as close as possible to actual NPV data samples. Furthermore, the effectiveness of the LSSVM model proposed in

2

Correlation coefficient (R ) Mean square error (MSE)

RF

NPV

0.9931 0.012

0.9933 0.0089

182


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