Robust intelligent tool for estimating dew point pressure in retrograded condensate gas reservoirs: Application of particle swarm optimization

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Robust Intelligent Tool for Estimation Dew Point Pressure in Retrograded Condensate Gas Reservoirs: Application of Particle Swarm Optimization Mohammad Ali Ahmadi, Mohammad Ebadi

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Received date: 25 October 2013 Revised date: 6 February 2014 Cite this article as: Mohammad Ali Ahmadi, Mohammad Ebadi, Robust Intelligent Tool for Estimation Dew Point Pressure in Retrograded Condensate Gas Reservoirs: Application of Particle Swarm Optimization, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2014.05.023 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust Intelligent Tool for Estimation Dew Point Pressure in Retrograded Condensate Gas Reservoirs: Application of Particle Swarm Optimization Mohammad Ali Ahmadi1*, Mohammad Ebadi2 1)

Department of Petroleum Engineering, Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology, Ahwaz. Iran

2)

Department of Petroleum Engineering, Science and Research Branch, Islamic Azad University, Tehran. Iran

Abstract Liquid production from gas condensate reservoirs, which is a very important economic and technical issue, is intensely dependent to the thermo dynamical conditions governing the supposed porous media. Estimating accurately the relevant parameters have been an incentive for researchers to develop and propose a diversity of correlations although some of them do not have precision enough such as those which are routinely applied to determine the dew point pressure (Pd). Due to the numerous numbers of misunderstandings in terms of Pd estimations which are normally observed in upstream industries, in this study great efforts have been made to put forward a very high performance method to monitor the Pd. The solution has been evolved by making a hybrid of two effective and robust methods, swarm intelligence and artificial neural network (ANN). The proposed model was extended by gaining from exactly precise dew point

*)

Address to Corresponding Author: Department of Petroleum Engineering, Ahwaz Faculty of Petroleum

Engineering, Petroleum University of Technology, Ahwaz. IRANP.O.BOX:63431, Tel:(+98)912-6364936, Email: [email protected]  

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pressure data samples reported in previous literatures. Moreover, based on the same data samples, a comparison was made between the evolved intelligent approach and conventional schemes. The statistical results show the eye-catching performance of the evolved smart model to determine the dew point pressure of condensate gas reservoirs. Based on the reliable generated results which have a high level of accuracy and effectiveness, it can logically be inferred that implementation of the proposed approach, PSO-ANN, can cause the better understanding of reservoir fluid behavior through reservoir simulation scenarios. Keywords: Dew Point Pressure; Swarm Intelligence; Fuzzy Logic; Condensate Gas Reservoirs; Artificial Intelligence

1. Introduction Regarding the gas condensate reservoirs as one of the most valuable sorts of hydrocarbon sources having the ability of supplementing a huge amount of energy has turned into a popular trend.(Mohammadi et al., 2013, Mokhtari et al., 2013, Sadeghi Boogar and Masihi, 2010).Consequently, preparing effective, complete and multidisciplinary for production from the introduced reservoirs has its own practical, methodical and monetary significance. To propose fundamental and vital orders to produce from referred resources, demanding exact, accurate and definite understanding about reservoir fluid properties has continuously been taken into account. To put it another way, PVT properties, which even insignificant and infinitesimal mathematical errors in their estimations results in facing up with severe problems in successive processes, are the most important information which are highly gained in every step of reservoirs simulations and developments.(Alavi et al., 2010, Mokhtari et al., 2013, Ursin, 2004, Yong et al., 2010) The step properly named “Flow-In” is the start of reservoir pressure reduction which subsequently leads to formation of liquid drops in zones very close to the wellbore; it is because 2

of overtaking a pressure threshold called dew point Pressure (Pd). (Brown et al., 2009, Elsharkawy and Foda, 1998, Nasrifar et al., 2005)Observing the dramatic reduction of the gas relative permeability and gas production rate are the intense effects of the discussed drops formation. (App and Burger, 2009, Chowdhury et al., 2008, Thomas et al., 2009)Accordingly, careful determination of the Pd is essential to be taken as a very imperative issue. Hence, great efforts either theoretical or laboratorial methods have cleverly been made and suggested to measure Pd (Berning, 2012, Louli et al., 2012, Rahimpour et al., 2011, Nowroozi et al., 2009). The Constant Volume Depletion (CVD) is an experimental process which Pd is one of results generated from their running on the gained samples. These methodologies that their comprehensive steps have totally been cleared in already publicized literatures are routinely have some technical hitches such as their pricey and slow scenarios and also their reliabilities are massively infested by some exterior influences like human errors.(Luo et al., 2001, Shadizadeh et al., 2006, Shen et al., 2001, Zheng et al., 2000, Jalali et al., 2007) Furthermore, there are some mathematically inspired solutions and concepts such as Equation of States (EOS) and some empirically derived correlations which have all in all be presented to measure the critical PVT properties.(Bonyadi and Esmaeilzadeh, 2007, Elsharkawy, 2002, Li et al., 2012) For instance, a formula mainly based on C7+’s characterizations, temperature and fluid compositions was evolved by Nemeth and Kennedy through running a multiple regression which was supported by an extensive database to predict Pd. The applicability of the formula is reliable under specified thermo dynamical ranges. (K. et al., 1967) Also, some attempts were made to make a picture about production from gas condensate reservoirs without gaining from implementing PVT data. It is shown by the study done by Marruffo et al. who achieved an approach to estimate Pd and C7+ contents of gas condensate reservoirs. (Marruffo et al., 2001) Besides, the impact of impurities, mostly H2S, on the Pd has been chosen as a topic of research 3

Carison and Cawston. (Carlson and Cawston, 1996) Observation of total volume during running the CCE test was taken as a hint to propose a graphical model to predict the Pd according to the accurate reading of the Z-factor. In sum, comparatively stress-free and easy to use and not generally regarding the temperature influence are correspondingly taken as one of benefits and weaknesses of experiential relationships.(Nowroozi et al., 2009)Then again, the perceptible rate of reliance towards primary deriving data caused EOSs dropping their suitable presentations when being applied on new sites and make operators to calibrate related parameters.(Jalali et al., 2007) Thus, noticeable studies have been done to put forward more beneficial, careful and appropriate approaches. Soft computing methods, thanks to their inherent capabilities of dealing with nonlinearity, vagueness and uncertainty of considered issues, have drawn attentions of scientist to overcome hurdles of reservoir engineering issues such as extraction of PVT properties.(Farasat et al., 2013, Zendehboudi et al., 2012)For example, Akbari et al. applied a certain sort of an Artificial Neural Network (ANN) to calculate Pd through taking a group of thermo dynamical and compositional features as input.(Jalali et al., 2007) Likewise, Nowroozi et al. constructed an Adaptive Neuro-Fuzzy Inference System (ANFIS) to predict Pd by considering chiefly compositional factors (Nowroozi et al., 2009). Similarly, Keydani et al. (2013) proposed a conventional sort of back-propagation artificial neural network to predict the Pd of lean retrograde gas condensate reservoirs. In terms of conducting the modern optimizing algorithms, it is valuable to refer the research has been done by Rostami et al. (2012) proposed a model to predict Pd through coupling the Gaussian processes and particle swarm optimization methods.Moreover, the aim of this contribution is summarized to propose an easy-to-use, robust and sharp model to predict dew point pressure (Pd) in retrograded gas condensate reservoirs. To gain this end, hybrid of swarm intelligence and 4

neural network, fuzzy logic (FL) and coupled of genetic algorithm (GA)-Fuzzy Logic (FL) as robust type of artificial intelligent models was utilized to fire the addressed issue of this research. Owing to this fact, massive dew point pressure data banks extracted from previous literature (Ahmadi and Ebadi, 2014; Al-Dhamen, 2010) were faced to the referred approaches for testing and validating. To confirm the capability and reliability of the evolved PSO-ANN approach, conventional correlations were executed to estimate the saturation pressure of crude oils. The gained results for both intelligent and conventional correlations are demonstrated in details in further sections. Also, the descriptions of the addressed models are explicated through next parts.

2. Methodology 2.1.Artificial Neural Network (ANN) Artificial neural network, a bio-inspired approach which their initial pattern has been recognized from studying the everyday procedures of human brain, is succinctly capable of correlating numerically and inversely the relationships between inputs and outputs of each supposed system by thanks to their distinctive mathematical structures. The gathered laboratorial data are technically implemented to train the network then; the prepared network is gained to estimate the imprecise and blurred data (Bain, 1973; James, 1890; Ahmadi and Shadizadeh, 2012; Ahmadi et al., 2013a; 2013b; Zendehboudi et al., 2013a; 2013b; 2012; Ahmadi, 2011; Ahmadi and Golshadi, 2012; Ahmadi, 2012). The depicted scheme is conductible through relying on synchronous processing units, known as neurons and nods, located in layers. The layers, input, output and hidden, are the basic components of each artificial neural network (ANN) which the number of their neurons are specified by the available data, designers and target of the discussed problem, respectively. Indisputably, the back-propagation feed forward network and specifically 5

the multilayer perceptron (MLP) networks, those evaluate through considering the classical techniques in relation to their much reduced development time and their potential to make usage of related information, are the most favorable and common types of ANN in chemical engineering (Ahmadi and Shadizadeh, 2012; Ahmadi et al., 2013a; 2013b; Zendehboudi et al., 2013a; 2013b;2012; Ahmadi, 2011; Ahmadi and Golshadi, 2012; Ahmadi, 2012; Hagan et al., 1966; Vallés, 2006; Hornick et al., 1989; 1990; Garcia-Pedrajas et al., 2003). Before tackling by details to the main issue of this study which is carrying an up-to-the-minute optimizing method out to set precisely the ANN related parameters, including weights and biases, in terms of training, a conventional solution using the trial and error method has been tried out as well. The referred theme has been followed by dividing the database into two main parts apparently named training and testing sets. Regarding this division is due to determine the most appropriate network structure by applying the larger group, training ones, while the testing set which has not earlier been faced to the network in the training step is piloted to examine the reliability of the proposed network in the case of correlating the saturation pressure. Running the optimization of interconnected weights and node biases is continued up till the performance of the proposed ANN is based on some statistical criteria like Mean Squared Error (MSE) permissible and it is when the values of outputs at the neurons of output layer are very nearly close to the corresponding experimental data. The MSE is expressed as follow MSE Approach =

1 G m ∑∑ ⎡Y j (k ) − T j (k ) ⎤⎦ 2 k =1 j =1 ⎣

2

(1)

Where number of output nodes is shown by the m, training samples are represented by the G, and the expected and the actual outputs are denoted by the Yj (k) and Tj (k) , repectively. When the MSE closes gradually to the zero, the error of our developed network model starts declining (Ahmadi and Shadizadeh, 2012; Ahmadi et al., 2013a; 2013b; Zendehboudi et al., 2013a; 2013b;2012; Ahmadi, 2011; Ahmadi and Golshadi, 2012; Ahmadi, 2012). 6

2.2.Particle Swarm Optimization (PSO) Particle Swarm Optimization (PSO) is an optimization which has mathematically been inspired from studying and modeling the behavior of social organisms like a flock of birds. Similarly to the GA and PSO is initiated with a population of random routs, called particles. These particles are supposed to stir within the defined search space with an adjustable velocity to save the best position. Also, in order to keep an eye on the target, each particle has the ability to update its velocity vector as well. This is possible thanks to their own flying experience and the flying experience of the other particles in the search space as illustrated in Figure 1 (Ahmadi and Shadizadeh, 2012; Ahmadi et al., 2013b; Zendehboudi et al., 2013a; 2013b; 2012; Ahmadi and Golshadi, 2012; Ahmadi, 2012).

2.

Results and Discussion 3.1.

Fuzzy Logic Results

Running the addressed program resulted in formation of pdfs for all 14 attributes which the generated results have been depicted in Figure 2.Consequently, regarding the center of values or pick of each pdf and boundaries channeled towards building MFs shown in Figure 3.

3.2.

Hybrid of Fuzzy Logic and Genetic Algorithm Results

Concluding the best well-matched rules based on the collected data has been taken as the incentive to operate the ANOVA assessment to clear up the most influencing parameters.(Figure 4)Generated results demonstrates that the MW C7+ and SG C7+ can intensely move the Pd which is a strong function of amount of T,C7+and C1 and the as well. Additionally, it is possible to note that Pd is infinitesimally impacted by parameters such as mole percent of H2S, N2, CO2, C6, C5, C4, C3 and C2. 7

According to consider the constructed MFs and based on the previous literatures and comments of experienced experts, the following rules to predict the Pd have been proposed. •

If MW C7+ is high and SG C7+ is high then Pd is high. (Weight1 = 1)

•

If MW C7+ is medium and SG C7+ is medium and C1 is medium and T is medium and C7+ is medium then Pd is medium (Weight2 = 1)

•

.If C2 is low and C3 is low and C4 is low and C5 is low and C6 is lowH2S is low and CO2 is low and N2 is low then Pd is low. (Weight3 = 1)

Each associated weight is an indication to show the amount, effect and importance of the related rule in comparison with the others. The introduced fuzzy rules are supposed to tune in a Fuzzy Interface System (FIS). In order to improve the performance of the evolved FIS it seems necessary to gain from the GA to optimize the weights of fuzzy rules and connected parameters of MFs. The GA with its special already introduced operatives and characteristics has to be run to extract the most optimized forms of regarded MFs and rules. To meet this goal, the GA was set to minimize the MSE. In order to reach the most optimized general form of the FIS through running the GA, it is vital to conduct a sensitivity analysis test to obtain the best arrangement of the GA. Besides, the opening GA was set by the succeeding features, Crossover Rate = 2, Mutation Rate = 0.0075, Genes Length = 10, Generations Number = 80 and Initial Population = 8. Consequences of changing each one of declared operatives on the final connected MSE of the most enhanced individual of the ultimate generation have been exemplified in Figure 5. As it can be seen, a GA with feature of 110, 10, 12, 3 and 0.75 for number of generation, initial population, length of genes, crossover rate and mutation rate; respectively, has the ability of generating the results with the highest performance. The couple of optimized GA and the already 8

developed FIS were run for the gathered dataset and weights of 78, 15, 94 were formed for rule 1, rule 2 and rule 3 respectively. Moreover, the MFS were optimized as well and they have been shown in Figure 6. Eventually, the optimized FIS were run for the gathered data which the optimization trend has been plotted in Figure 7 and the generated results have graphically been shown in Figure 8. The best lines which have been fitted into the performed data have the equations of y = 1.0272 x –92.54.The relative errors of responded values have been illustrated based on the corresponding measured points in Figure 9. Based on this figure, it can be inferred that the trend of relative error distribution for recovery factor data indicates its own maximum and minimum values near the middle values and also a misbehaved peak can be seen for very high pressures, but there is no clear pattern for plot of relative error distribution of net present value. In addition, the generated results have been compared with their corresponding measured values according to their data indices which have been illustrated in Figure 10. As mentioned before, statistical analysis uncovered the high dependency of Pd to parameters such as MW C7+, SG C7+, T, C1 and C7+. In order to exhibit these situations the generated results and their corresponding measured values have demonstrated versus their relevant mentioned parameters in Figure 11.

3.3.

Hybrid PSO and ANN Results

Figure 12 demonstrates the regression plot of the swarm intelligence output versus corresponding real measured data. As depicted in Figure 12, the output of the aforementioned model both testing and training phase follow the diagonal line (Y=X). Put it another way, the outputs of the swarm model are closet to corresponding real values. Figure 13 depicts the outputs of swarm

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model and actual saturation pressure versus corresponding data index. As illustrated in Figure 13, swarm model outputs follow exactly the actual trend of the experimental saturation data. Finally, Figure 14 shows the relative error distribution of the outputs of the swarm model versus relevant dew point pressure data sample. Dedicated to Figure 14, the maximum deviation of the aforementioned model outputs referred to dew point pressure in ranges of 1500 to 2000 Psi, which is around 15%. Based on the relative deviation and correlation coefficient known as robust statistical indexes, the evolved saturation pressure correlation is rigorous than the other approaches such as fuzzy logic and GA-Fuzzy approaches in estimation of dew point pressure in retrograded gas condensate reservoirs. Finally, to wrapped previous outcomes, Table 1 reveals by details in terms of the statistical indexes of the PSO-ANN method in comparison with the other traditional solutions such as Humoud and Al-Marhoun, Elsharkawy, Nementh and Kennedy, etc. In keeping with Table 1 it can be concluded that the PSO-ANN gains from a high level of proficiency and effectiveness with low rate of vagueness whereas these are not proper about models derived from the hybrid GA and fuzzy logic, artificial neural networks and other correlations such as Humoud and Al-Marhoun, Elsharkawy having high value of root mean square error (RMSE) and low values of correlation coefficient.

4. Conclusions Accurate determination of the dew point pressure (Pd) through retrograded condensate gas reservoirs has vital impact on the simulation of gas/condensed gas flow through porous medium. Owing to this fact, in this research huge amount of efforts have been made to estimate dew point pressure (Pd) with high degree of accuracy versus low degree of uncertainty. To reach this main end, couple of swarm intelligence and neural network was used to develop the efficient approach to pass the referred hurdle successfully with acceptable precision and accuracy. Furthermore, the 10

accurate experimental dew point pressure (Pb) data samples which reported in previous surveys were introduced to the proposed model for tuning and validating suggested approach. Attributable to the generated aftermaths of this research, succeeding deductions can be made: 1. Acceptable similarity between obtained dew point pressures (Pd) from the suggested swarm intelligence model against relevant actual dew point pressure (Pd) values observed. Put it in another way, the previously evolved models fail to estimate dew point pressure (Pd) owning to the calculated statistical parameters of each aforementioned models. 2. The proposed smart model (PSO-ANN) for dew point pressure (Pd) determination through retrograded gas condensate reservoirs is easy-to-use, cheap and high-performance for execution. Furthermore, it is very much beneficial and user friend to tune the integrity and performance of the marketable reservoir simulators such as PVTi in ECLIPSE package for history matching and other aspects of condensate productions with the developed model of this research. Moreover, the proposed smart model could be conducted as a substitution when the essential dew point pressure data are unavailable.

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18

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APPENDIX A This section represents the formulation of the implemented correlations throughout this research study as follow as: Nemth & Kennedy (1967):

(

(

)

Ln ( Pd ) = A1 Z C2 + Z CO2 + Z H 2 S + Z C6 + 2 Z C3 + Z C4 + Z C5 + 0.4Z C1 + 0.2Z N 2

)

⎛ ⎞ Z C1 2 + A2 SGC 7 + + A3 ⎜ ⎟ + A4T + A5 Z C7+ MWC 7 + + A6 ( Z C7+ MWc 7 + ) ⎜ Z C + 0.002 ⎟ ⎝ 7+ ⎠ ⎡ MWC7+ ⎤ MWC7+ MWC7+ + A7 ( Z C7+ MWc 7 + )3 + A8 ⎢ ]2 + A10 [ ]3 + A11 ⎥ + A9 [ SG + 0.001 SG + 0.001 SG + 0.0 01 ⎢⎣ C7+ ⎥⎦ C7+ C7+ (A-1)

(

)

19

Table A1: values of the Nemth & Kennedy (1967) approach parameters Parameters Value A1

-2.0623054

A2

6.6259728

A3

-4.4670559×10-3

A4

-1.0448346×10-4

A5

3.2673714×10-2

A6

-3.6453277×10-3

A7

-7.4299951×10-5

A8

-1.1381195×10-1

A9

-6.2476497×10-4

A10

-1.1381195×10-1

A11

-1.0746622×10

Elsharkawy (2001):

Pd = A0 + AT 1 + A2 Z H 2 S + A3 Z Co2 + A4 Z N 2 + A5 Z C1 + A6 Z C2 + A7 Z C3 + A8 Z C4 + A9 Z C5 + A10 Z C6 + A11Z C7+ + A12 MWC7+ ⎛ MWC7+ ⎞ + A13 SGC7+ + A14 Z C7+ MWC7+ + A15 ⎜ ⎟ ⎜ SGC ⎟ 7+ ⎠ ⎝ ⎛ Z C MWC7+ ⎞ ⎛ ⎛ Z C7+ ⎞ Z C7+ + A16 ⎜ 7+ + A18 ⎜ ⎟ + A17 ⎜ ⎟ ⎜ ⎟ ⎜ ZC + ZC + ZC + ZC ⎜ ⎟ 4 5 6 ⎝ Z C1 + Z C2 ⎠ ⎝ 3 ⎝ SGC7+ ⎠

(

)

(A-2) ⎞ ⎟ ⎟ ⎠

Table A2: values of the Elsharkawy [56] approach parameters Parameters

Value

Parameters

Value

A0

4268.85

A10

691.5298

A1

0.094056

A11

40660.36

A2

-7157.87

A12

205.26

A3

-4540.58

A13

-7260.32

A4

-4663.55

A14

-352.413

A5

-1357.56

A15

-114.519

A6

-7776.10

A16

8.13300

20

A7

-9967.99

A17

94.916

A8

-4257.10

A18

238.252

A9

-1417.10

Hamoud and Al-Marhoun (2001):

Ln ( Pd ) = β 0 + β1 ln (T ) + β 2 ln ( Rm ) + β3 ln ( PSPTSP ) + Rm =

β4 Tpr

+

β5 Ppr

+

β6

γ C 7+

(A-3)

RSPγ gSP

(A-4)

γ C 7+

β 0 = 43.777183, β1 =-3.594131, β 2 = -0.247436, β 3 =-0.053527, β 4 =-4.291404, β 5 =-3.698703, β 6 =-4.590091 Alternating Conditional Expectations (ACE): T ( Pd )2

Pd = eC1 Where

2

+ C2T ( Pd ) + C3T ( Pd ) + C4

(A-5)

T ( Pd ) = ln[T (TR ) + T ( GOR ) + T ( γ g ) + T ( γ cond ) + 10] 3

2

(A-6)

T (TR) = p1TR +p2TR +p3TR+p4

(A-7)

T (GOR) = r1ln (GOR) + r2

(A-8)

T (γg) = q1γg2 + q2 γg + q3

(A-9)

T (γcond) = s1 γcond3 + s2 γcond2 + s3 γcond+ s4

(A-10)

C1= 49.1377, C2= -336.5699, C3= 770.0995, C4= -580.0322, p1= -0.35014×10-6, p2= 0.18048×10-3, p3= -0.32315 ×10-1, p4= 1.2058 r1= -0.3990, r2= 5.1377, q1= -23.8741, q2= 36.9448, q3= -12.0398 s1= -30120.78, s2= 69559, s3= -53484.21, s4= 13689.39

Marruffo & Rojas (2002): K5 K7 CGR K 2 × K 8 × API ( K 4×Tr − K 6×C7+ ) ] (A-11) K3 %C7 + K1 = 346.7764689, K2 = 0.0974139, K3 = -0.294782419, K4 = -0.047833243, K5 = 0.281255219,

Pd = K1[

K6 = 0.00068358, K7 = 1.906328237, K8 = 8.4176216 21

% C7+ = (

CGR −0.8207 ) 70680

(A-12)

APPENDIX B This section illustrates the background of fuzzy logic and genetic algorithm (GA) approaches as following:

B.1. Fuzzy Logic Dealing with daily problematic issues of the world we do live in is intensely connected with considering their natural characteristics which are uncertainties, imprecision and ambiguities, otherwise the concluded results which are resulted from conducting the conventional calculating approaches, intricate mathematical formularies and arithmetical state-of-the-art simulations are far from the expected ones and have high rate of deviations. In other words, applying the globally accepted computational standards and conventional logical solutions which have been derived from very early days of science has lead made results toward a style with not fulfilling qualities. Accordingly, the described situation blocks the researchers to dominate and govern effectively, robustly and perfectly the massive dynamic world with a diversity of technologies and their relevant methodical parts as well as multitude of multidisciplinary stumbling block which are supposed to be challengingly beaten. Even so, great attempts have been made by scientists and experts to nurture and develop the Zadeh's innovation, Fuzzy Logic (FL), which is the break to stop facing up with critical problems which have typically and traditionally been answered so that their inherent vague features have been ignored through making some simplifying assumptions.(Ahmadi et al., 2014, Ahmadi et al., 2013, Asoodeh and Bagheripour, 2012, Bagheripour and Asoodeh, 2013, El-Sebakhy et al., 2012, Ghiasi-Freez et al., 2012) According to the mechanism of FL, numbers vary between 0 and 1, known as membership degree, must be assigned to members of a considering special set based on a very unique 22

procedures under the title of Membership Functions (MFs) which have been proposed to prepare smooth transitions between real and fuzzy worlds. It is due to show that belonging of members to each group can be to some degree. There are different types of MFs whose parameters can be adjusted based on the nature of issues and fulfill our targets. According to represented MFS, piecewise linear MFs can very effectively picture problems and current constraints of the real world and shift them into the mathematical limitations of FL. Furthermore, this kind of MF has extendedly been applied in a huge number of industrial aspects. Equation 1 shows mathematically the construction of the referred MF and Figure A1 is its graphical overview.(Yu et al., 2010, Olatunji et al., Zahraie and Hosseini, 2009, Fu, 2008) ⎧ 0, ⎪ x −a ⎪ , ⎪b −a ⎪ x −b , ⎪ ⎪⎪ c − b μ j ( x ) ⎨ 1, ⎪ x −c ⎪ , ⎪d −c ⎪x −d , ⎪ ⎪e −d ⎩⎪ 0,

x ≤ a, a < x < b, b < x < c, x = c,

(B-1)

c < x < d, d < x < e, e ≤ x,

Where a, b, c, d and e are the constructing values of the MF connected with the linguitic terms of a variable. In other words, limitations of a domain relevant to a linguistic term of a supposed MF is determined with [a, e].

23

1

Membership Degree

0.8

0.6

0.4

0.2

0 0

a

b

c Support

d

e

f

Figure A1: The graphical form of the piecewise membership functions (Ahmadi et al., 2013)

Normally, the knowledge of experienced experts is gained to construct the general form of MFs and control their relevant values which can cause some misunderstandings too. To prevent happening of this technical problems, a decision was made to use methods of automotive MF generator. To attain the automatic generation of MFS for all gathered records of data related to the already introduced attributes were divided into main 3 parts, Low, Medium and High, by conducting a basic equal-frequency partitioning (clustering) methodin which the middle point between intervals in different clusters is known as the primary interval boundary point. Simply, 400 records of data have been divided into 3 main parts so that each group has almost an equal number of data.(Beynon et al., 2004) (See Table B1) Table B1: The related information about three clusters of dew point pressure. Description

Low (L)

Medium (M)

High (H)

Interval

L ≤ 4055 (133)

4055 < M ≤ 5377 (133)

H > 5377 (134)

24

After that, estimated distributions in terms of probability distribution functions (pdfs), which has been a considering topic since launching of FL by Zadeh, must be made for all intervals of thermo dynamical and compositional properties to prepare the automotive formation of MFs through linking between probability distributions and MFs (through possibility distributions). Then, instead of regarding all details of expanding the pdfs on the wide of an interval, center of areas connected with each pdf and boundary values of intervals are taken as the main parameters to set the MFS. The general formula of the pdf has been represented in equation 2, the mathematical map which has exactly been followed step by step. pdf j ( x ) =

(

1

2m jπ max ( I j ) − min ( I j )

⎡ m j exp ⎢ − ∑ ⎢ 2 i =1 ⎣ mj

)

⎛ ⎞ x − xi ⎜ ⎟ ⎜ max ( I j ) − min ( I j ) ⎟ ⎝ ⎠

2

⎤ ⎥ ⎥ ⎦

(B-2) Where mj indicates the number of values relying within Ij which is the interval relevant to the linguistic term and xi shows the value of each data in the interval and x is the variable. The detailed procedure about the programming of the above formula has fully been in our previous studies (Ahmadi et al., 2013, Beynon et al., 2004).

B.2. Genetic Algorithm (GA)

A modern optimization method such as Genetic Algorithm (GA) can be gained to improve the performance of FL method. In more details, the GA is usually applied to optimize the relevant parameters of MFs to make them more compatible with the attribute that they are representing. Searching quickly and optimizing professionally are the main attributes of the GA which is based on the concept of “survival of the fittest”, which is the foundation of the natural growth with the genetic circulation of belongings. As stated by the Darwinian opinion of ‘survival of the fittest’ 25

which is the background of the brought up optimization technique, the most optimized point in the considered zone can be detected by the GA very soon after handling a series of cyclic calculations. Selection operators, artificial mutation and crossover are routinely considered as the most elementarily functions of the GA. More material and instructions about the GA and its attention-grabbing philosophy could be found in recent, great and eye-catching literatures.(Alves et al., 2012, Bashipour and Ghoreishi, 2012, Bayat et al., da Silva et al., 2014, Jafari Kenari and Mashohor, 2013, Martins et al., 2013, Nejad Ebrahimi et al., 2013, Salmachi et al., 2013, Soleimani et al., 2013, Wali et al., 2012, Wang et al., 2012)

Research Highlights: • Developing sophisticate approach to monitor the Dew point Pressure in Retrograde Gas Condensate reservoirs. • Comparing robustness of the fuzzy approaches against suggested PSO-ANN approach • Handling extensive dew point pressure data in gas condensate reservoirs by PSO-ANN model

26

Randomly initialize population locations and velocities

Evaluate fitness of particle

YES

Meet stopping criteria NO If Particle fitness > Global best fitness, Update global best

Exit criteria (Global best satisfactory)

If Particle fitness >Particle best fitness, Update particle best

Next Particle

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Update particle velocity

Update particle position Figure 1: Flow chart of particle swarm optimization process (Ahmadi and Shadizadeh, 2012; Zendehboudi et al., 2012; 2013a; 2013b)

21

x 10

4

x 10

5 pdf Low pdf Medium pdf High

12000

pdf Low pdf Medium pdf High

4.5

4

pdf Low pdf Medium pdf High

8

7 4

10000

6 3.5

8000

5

pdf

pdf

pdf

3 2.5

4

6000 2 3

I2

4000

I3

1.5 2 1

I1

I1

2000

I1

0.5 0

0.1

0.2

0.3

0.4

0.5 C1

0.6

0.7

0.8

0

0.9

0.02

I2 0.04

x 10

0.08 C2

I2

I3

1

I3

0.06

(a)

0.1

0.12

0

0.14

0.01

0.02

0.03

0.04

4

x 10 pdf Low pdf Medium pdf High

0.05

0.06

0.07

0.08

0.09

0.1

C3

(b)

10

(c)

5

pdf Low pdf Medium pdf High

2

25

pdf Low pdf Medium pdf High

20 8

1.5

pdf

pdf

pdf

15 6

1 10

4

I1 I2 I1

0.5

I1

2

I2

I2

I3

5

I3

I3 0

0.02

0.04

0.06

0.08

0.1 C4

0.12

0.14

0.16

0.18

0

0.2

0.005

0.01

0.015

0.02

(d)

x 10

0.025 0.03 C5

0.035

0.04

0.045

0.05

0

0.055

100

150

200 Temperature

(e)

4

x 10

8

pdf Low pdf Medium pdf High

7

250

300

(f)

6

x 10

5

2.5

pdf Low pdf Medium pdf High

2

pdf Low pdf Medium pdf High

1.8 2 6

1.6 1.4

5

1.5

pdf

pdf

pdf

1.2 4

1 3

1

0.8 0.6

2

0.5

0.4

I1

1

0

I2

0.02

I3

0.04

0.2

0.06

0.08

0.1

0 -0.05

0.12

0

0.05

0.1

C7+

(g)

x 10

0.15

0.2 N2

0.25

0.3

0.35

0 -0.1

0.4

0

0.1

0.2

0.3

(h)

x 10 pdf Low pdf Medium pdf High

0.4 CO2

0.5

0.6

0.7

0.8

0.9

(i)

7

3.5

4

6

pdf Low pdf Medium pdf High

50 45

pdf Low pdf Medium pdf High

5

3 40 2.5

4

35

1.5

pdf

30

2

pdf

pdf

25

3

20 2 15

1

I1 I1

10

I2

I3

I2

I3

1

0.5 5 0 -0.02

0

0.02

0.04

0.06

0.08 H2S

0.1

0.12

0.14

0

0.16

(j)

120

140

160 MWC7+

180

200

0

220

0.74

0.76

0.78

(k)

0.8 SGC7

0.82

0.84

0.86

(l)

0.8 25

pdf Low pdf Medium pdf High

pdf Low pdf Medium pdf High

0.7

20

0.6

0.5

pdf

15

pdf

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

10

0.4

0.3

I1

I2

I1

0.2

I3

I2

I3

5 0.1

0

100

150

200 Temperature

250

0

300

(m)

2000

3000

4000

5000

6000 Pd

7000

8000

9000

10000

(n)

Figure 2:Estimated distribution related to intervals I1, I2, I3 for: a) C1 b) C2 c) C3 d) C4 e) C5 f) C6 g) C7+ h) N2 i) CO2 j)H2S k) MWC7+ l) SG C7+ m) T n) Pd 22

1

0.8

Low Medium High

0.6

0.4

Low Medium High

0.6

0.4

0.2

0 0.2

0.3

0.4

0.5 C1

0.6

0.7

0.8

0.02

0.04

0.06

0.08 C2

0.1

0.12

0.14

0.01

1

0.8

Low Medium High

0.6

0.4

0.4

0.2

0

0 0.12

0.14

0.16

0.18

Low Medium High

0.6

0.2

0.1 C4

Membership Degree

1

0.8

0.08

0.005

0.01

0.015

0.02

0.025 0.03 C5

0.035

0.04

0.045

0.05

0.005

0.06

0.08

0.1

Low Medium High

0.6

0.4

0 0.04

Membership Degree

1

0.8

Membership Degree

1

0 0.12

0.05

0.1

0.15

0.2 N2

0.25

0.3

0.35

0.4

0

0.4

0 0.2

0.25

0.05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Low Medium High

0.6

0.4

0 140

160 MW C7+

180

200

220

0.74

0.76

0.78

(k)

1

0.8

Membership Degree

1

Low Medium High

0.4

0.2

0.8 SG C7+

0.82

0.84

0.86

(l)

0.8

0.6

0.045

0.2

120

(j)

Membership Degree

Low Medium High

0.6

0 0.15 H2S

Membership Degree

Membership Degree

1

0.8

0.2

0.04

(i)

1

0.2

0.035

CO2

0.8

0.4

0.03

Low Medium High

(h)

Low Medium High

0.025 C6

0.4

1

0.1

0.02

0.6

0.8

0.05

0.015

0 0

(g)

0

0.01

0.2

C7+

0.6

0.1

(f)

0.8

0.2

0.09

Low Medium High

0.055

1

0.2

0.08

0.4

0.8

0.4

0.07

0

0.2

Low Medium High

0.06

0.6

(e)

0.6

0.05

0.2

(d)

0.02

0.04

(c)

1

0.06

0.03

C3

0.8

0.04

0.02

(b)

Membership Degree

Membership Degree

0.4

0

0.9

(a)

0.02

Low Medium High

0.6

0.2

0 0.1

Membership Degree

Membership Degree

1

0.8

Membership Degree

Membership Degree

1

0.8

0.2

Membership Degree

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Low Medium High

0.6

0.4

0.2

0

0 100

150

200 T

250

300

2000

(m)

3000

4000

5000

6000 Pd

7000

8000

9000

10000

(n)

Figure 3: Membership functions of the decision classes L, M, and H connected with: a) C1 b) C2 c) C3 d) C4 e) C5 f) C6 g) C7+ h) N2 i) CO2 j)H2S k) MWC7+ l) SG C7+ m) T n) Pd 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 4: The results of running the ANOVA test (Ahmadi and Ebadi, 2014)

24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a)

(b)

(c)

(d)

(e) Figure 5: Sensitivity analysis on GA parameters. a) Number of Generation b) Initial population c) Length of genes d) Crossover rate e) Mutation rate

25

Membership Degree

0.4

1

0.8

0.4

0.2

0.2

0

0 0.1

0.2

0.3

0.4

0.5 C!

0.6

0.7

0.8

Low Medium High

0.6

0.02

0.04

0.06

0.1

0.12

0.14

0.01

1

0.8

Low Medium High

0.6

0.4

Membership Degree

1

0.8

0.6 Low Medium High 0.4

0.2

0 0.08

0.1 C4

0.12

0.14

0.16

0.18

0.005

0.4

0.01

0.015

0.02

0.025 0.03 C5

0.035

0.04

0.045

0.05

0.005

Membership Degree

1

0.8

Membership Degree

1

0.8

0.2

Low Medium High

0.6

0.4

0.2

0 0.06

0.08

0.1

0.12

0.05

0.1

(g)

0.15

0.2 N2

0.25

0.3

0.35

0.4

0

Membership Degree

1

0.8

0.6

0.2

0 0.1

0.15 H2S

0.2

0.25

0.04

0.045

0.05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.6 Low Medium High

0.4

0 140

160 MW C7+

180

200

220

0.74

0.76

0.78

(k) 1

0.8

Membership Degree

1

Low Medium High

0.4

0.2

0.8 SG C7+

0.82

0.84

0.86

(l)

0.8

0.6

0.035

0.2

120

(j)

Membership Degree

Low Medium High

0.4

0

0.03

(i)

1

0.2

0.025 C6

CO2

0.8

0.4

0.02

Low Medium High

0.4

1

0.05

0.015

0.6

0.8

0

0.01

(h)

Low Medium High

0.1

0 0

C7+

0.6

0.09

0.2

0 0.04

0.08

(f)

1

0.4

0.07

Low Medium High

0.055

0.8

Low Medium High

0.06

0.6

(e)

0.6

0.05

0

0.2

(d)

0.02

0.04

0.2

0 0.06

0.03

(c)

1

0.04

0.02

C3

0.8

Membership Degree

Membership Degree

0.08 C2

(b)

0.2

Membership Degree

0.4

0

0.9

(a)

0.02

Low Medium High

0.6

0.2

Membership Degree

Membership Degree

0.6

1

0.8

Membership Degree

Low Medium High

1

0.8

Membership Degree

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Low Medium High 0.6

0.4

0.2

0

0 100

150

200 T

250

300

2000

(m)

3000

4000

5000

6000 Pd

7000

8000

9000

10000

(n)

Figure 6: Membership functions of the decision classes L, M, and H connected with:a) C1 b) C2 c) C3 d) C4 e) C5 f) C6 g) C7+ h) N2 i) CO2 j)H2S k) MWC7+ l) SG C7+ m) T n) Pd

26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 7: The optimization trend taken by the FIS-GA

27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 8: The scatter plot of generated results in comparison with their measured values

28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 9: Relative error distribution versus the measured points

29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 10: The generated results and the measured values base on their corresponding indices

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a)

(b)

(c)

(d)

(e) Figure 11: The generated results and their measured corresponding values versus the most effective parameters. a) T b) MW C7+ c) SG C7+ d) C1 e) C7+

31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a)

(b) Figure 12: Regression plot of the evolved PSO-ANN method for determination of dew point pressure in retrograded gas reservoirs versus relevant actual dew point pressure a)Training phase b) Testing Phase 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a)

(b) Figure 13: Draw a parallel of the PSO-ANN method outcomes and relevant actual dew point pressure against corresponding data index a)Training phase b) Testing Phase

33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 14: Error distribution of the developed PSO-ANN output versus relevant actual dew point pressure

34