- Email: [email protected]
Evolving smart approach for determination dew point pressure through condensate gas reservoirs
Fuel 117 (2014) 1074–1084
Contents lists available at ScienceDirect
Fuel journal homepage: www.elsevier.com/locate/fuel
Evolving smart approach for determination dew point pressure through condensate gas reservoirs Mohammad Ali Ahmadi a,b,⇑, Mohammad Ebadi c a
Department of Petroleum Engineering, Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology, P.O. Box 63431, Ahwaz, Iran Research Institute of Petroleum Industry (RIPI), Tehran, Iran c Department of Petroleum Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran b
h i g h l i g h t s Evolving low parameter model to predict dew point pressure in retrograde gas reservoirs. Comparing effectiveness of the conventional models versus developed LSSVM model. Handling extensive dew point pressure data in retrograde gas reservoirs by new type of intelligent based model.
a r t i c l e
i n f o
Article history: Received 8 August 2013 Received in revised form 25 September 2013 Accepted 8 October 2013 Available online 25 October 2013 Keywords: Dew point pressure Least square support vector machine (LSSVM) Condensate gas Empirical correlation Computer program
a b s t r a c t To design gas condensate production planes with low uncertainty along with robust reservoir simulation, precise estimation or monitoring of dew point pressure play a crucial role. To handle successfully the addressed issue of condensate gas reservoirs, massive attentions have been performed previously but unfortunately fail to develop accurate approach for estimation dew point pressure. Dedicated to this fact, in current study enormous attempts have been put forth to proposed revolutionary method for determining dew point pressure in gas condensate reservoirs. To gain this end the new type of support vector machine method which evolved by Suykens and Vandewalle was utilized to generate robust approach to figure dew point pressure in condensate gas reservoir out. Also, lucrative and high precise dew point pressures reported in previous attentions were carried out to test and validate support vector machine approach. To serve better understanding of the proposed support vector machine approach, the conventional feed-forward artificial neural network and couple of genetic algorithm (GA) and fuzzy logic applied to the referred data banks and the gained solutions were contrasted with each other. According to the root mean square error (RMSE), correlation coefficient and average absolute relative deviation, the suggested support vector machine approach has acceptable reliability, integrity and robustness draw an analogy with the artificial neural network model and conventional methods. Thus, the proposed intelligent based way can be considered as an alternative model to monitor the dew point pressure of condensate gas reservoirs when the required real data are not accessible. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Gas condensate reservoirs are generally known as one of the most precious types of hydrocarbon reservoirs having capability of supplementing a massive clean amount of energy [1–3]. As a result, providing efficient, multidisciplinary and detailed production plans for these reservoirs has its own technical and economic importance. To design crucial and vital schemes to exploit from these gas sources, requiring accurate, precise and specified knowledge about reservoir fluid properties has always been a matter of ⇑ Corresponding author Address: Research Institute of Petroleum Industry (RIPI), Tehran, Iran. Tel.: +98 912 6364936. E-mail address: [email protected] (M.A. Ahmadi). 0016-2361/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.fuel.2013.10.010
consideration. In other words, pressure–volume–temperature (PVT) properties, which even small errors in their estimations lead to be encountered with some serious difficulties in subsequent procedures, play the leading role in every aspect of these kinds of reservoirs simulations and developments [2,4–6]. After beginning the step named ‘‘Flow-In’’ in gas condensate reservoirs, continues reduction in reservoir pressure caused formation of liquid drops in zones vicinity of the wellbore as a direct consequence of crossing the reservoir pressure from a threshold, a pressure-type border called dew point pressure (Pd) [7–9]. The creation of referred drops gives rise to decline dramatically the gas relative permeability and also gas production rate [10–12]. As a result, exact determination of the Pd must be taken as a very important topic. Therefore, numerous numbers of theoretical or
1075
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
experimental methods have smartly been put forward to measure Pd [13–16]. The Constant Composition Expansion (CCE) and the Constant Volume Depletion (CVD) are known as laboratorial procedures which are capable of extracting the Pd factor from gathered samples. These methods which their detailed steps have fully been described in literatures are routinely concluded to have some difficulties such as their expensive and time-consuming processes and also their accuracies are highly infected by some external parameters like human errors [17–21]. Moreover, empirically derived equations and Equation of States (EOS) are mathematical inspired concepts for measurements of some critical PVT properties [22–24]. Through running a multiple regression and gaining from an extensive database, a correlation based on temperature, characterizations of C7+ and fluid compositions was developed by Nemeth and Kennedy to predict Pd working properly under specified thermo dynamical ranges [25]. Improvements in characterizations and production from gas condensate reservoirs without using PVT data were achieved by Marruffo et al. who proposed a model to predict Pd and C7+ contents of gas condensate reservoirs [26]. Also, the influence of non-hydrocarbon impurities, particularly H2S, on the Pd has been investigated by Carison and Cawston [27]. Potsch and Braeuer proposed a graphical model as strong function of Z-factor accurate reading to determine the Pd based on observations of total volume during running the CCE test [28]. To sum up, relatively easy to use and not normally considering the temperature behavior are respectively known as one of advantages and disadvantages of empirical correlations [16]. On the other hand, the noticeable level of dependency towards initial deriving data caused EOSs losing their appropriate performances in case of applying on new locations and channel operators towards calibrating again the related parameters [21]. Hence, great efforts have been made to propose more useful, exact and suitable methods. Soft computing approaches thanks to their abilities of dealing with non-linearity, uncertainty and ambiguity of supposed problems have drawn attentions of researchers to defeat obstacles of reservoir engineering problems like extracting PVT properties, ashphaltene precipitation, condensate-to-gas ratio, minimum miscible pressure (MMP) and reservoir permeability [29–36]. For instance, Akbari et al. implemented a certain type of an Artificial Neural Network (ANN) to predict Pd through taking a set of compositional and thermo dynamical factors as input [21]. Furthermore, Nowroozi et al. designed an Adaptive Neuro-Fuzzy Inference System (ANFIS) to predict Pd by regarding mostly compositional parameters [16]. In addition, The main goal of current study is execute new kind of reversed based solution approaches called ‘‘least square support vector machine (LSSVM)’’ to develop robust, lucrative and precise predictive correlation to forecast dew point pressure through gas condensate reservoirs. To beat successfully this referred hurdle, least square support vector machine (LSSVM) was carried out on the previous literature data bases. The integrity and performance of the proposed predictive approach in estimating experimental dew point pressure from the literature is described in details. Furthermore, to point out reliability of the (LSSVM) results, expensive experimental data from one of the northern Persian Gulf gas fields of Iran was implemented to draw an analogy and proves the intelligent approach versus well-known dew point pressure methods.
formulated as illustrated in Eq. (1). Through the addressed equation, w stands for the weight factor, u denotes the nonlinear function which correlates the input space to a high-dimension characterization area and conducts linear regression while b represents the bias term [37–46]. Following expression was implemented as a cost function of the least square support vector machine (LSSVM) in calculation steps [37–46].
Q LSSVM ¼
The least square SVM theorem was introduced and developed by Suykens and Vandewalle in 1999 dedicated to the presumpion that the implemented data assortment S = {(x1, y1), . . . , (xn, yn)} that deal with a nonlinear function and decision function can be
ð1Þ
Relate to the following restriction [37–46]:
yk ¼ wT uðxk Þ þ b þ ek
k ¼ 1; 2; . . . ; N
ð2Þ
To figure out 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 [37–46].
Rðx; e; bÞ ¼
m 1 1 X jjwjj2 þ C e2k 2 2 k¼1
ð3Þ
To extract routs w and e, the Lagrange multiplier optimum programming approach is performed to solve Eq. (3); the addressed approach considers impartial and restriction parameters simultaneously. The mentioned Lagrange function L is formulated as following equation [37–46]:
Lðw; b; e; aÞ ¼ Jðw; eÞ
m X
ai fwT £ðxk Þ þ b þ ek Y k g
ð4Þ
k¼1
Through above equation, ai denotes the Lagrange multipliers that may be either positive or negative because LSSVM has equality restrictions. Owing to the Karush Kuhn–Tucher’s (KKT) conditions, conditions for optimum goal are demonstrated in Eq. (3) [44–46].
8 > > > > > > > > > < > > > > > > > > > :
@xL ¼ x
n X
ai uðxi Þ ¼ 0
i¼1
@bL ¼
n X
ai ¼ 0
i¼1
@ ei L ¼ Cei ai ¼ 0 @ ai L ¼ wT £ðxk Þ þ b þ ek yk ¼ 0
9 > > > > > > > > > = > > > > > > > > > ;
ð5Þ
Therefore, the linear equations can be demonstrated below expression [44–46]:
"
1T
0
# b
1 X þ 1c IN
a
¼
0 y
ð6Þ
While y = (y1, . . . , yn)T, 1n = (1, . . . , 1)T, a = (a1; . . . ; an)T and Xil = u(xi)u(xl) for i, l = 1, . . . , n. Thanks to the Mercer’s theorem, the resulting LS-SVM model for function approximation turns to the following equation [44–46] T
f ðxÞ ¼
N X
ak Kðx; xk Þ þ b
ð7Þ
k¼1
where a and b are the routs to Eq. (7) as below [44–46]: 1
b¼ 2. Least square support vector machine (LSSVM)
N X 1 T w w þ c e2k 2 k¼1
1Tn ðX þ 1c In Þ y 1
1Tn ðX þ 1c In Þ 1n
1
a ¼ X þ In c
ð8Þ
1 ðy 1n bÞ
ð9Þ
Eq. (10) may be executed as choice of nonlinear regression and utilize the Kernel function as below equation [37–46]:
1076
f ðxÞ ¼
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084 N X
ak Kðx; xk Þ þ b
ð10Þ
have been formulated in Eqs. (14)–(16) for zmf, pimf and smf, respectively.
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 U(x) and U(xi) as below equation [37– 46]:
Kðx; xk Þ ¼ UðxÞT Uðxk Þ
ð11Þ
Where the radial basis function (RBF) Kernel has been executed as following formulation [37–46]:
Kðx; xk Þ ¼ expðkxk xk2 =r2 Þ
ð12Þ
where r plays a role of decision variable through running least square SVM method, which is indicated by performing robust optimizer like genetic algorithm (GA) on the addressed least square SVM approach. To gain optimum value of least square SVM parameters, the root mean square error (RMSE) of the obtained outcomes of the final approach assigned as an objective function of the genetic algorithm (GA) which is formulated by the following equation [37– 46]:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ðPdesti Pdexpi Þ RMSE ¼ ns
ð13Þ
where Pd represents the dew point pressure, subscripts est and exp represents the predicted and actual dew point pressure through condensate gas reservoirs, respectively, and ns stands for the number of data from the initial assigned population. It would better stressed on here that the executed LSSVM method introduced by Pelckmans et al. [46] and Suykens and Vandewalle [45] has been carried out in the current study. 3. Fuzzy logic Uncertainty, imprecision and ambiguity are the inherent features of the daily problematic issues of the real world that are supposed to be dealt with, the characteristics which influence negatively the results generated from applying of the mainstream calculating methods, complicated mathematical formulas and numerical up-to-the-minute simulations. To put it another way, manipulating and conducting the globally accepted mathematical criteria and conventional logical approaches from the very early days of sciences progressions up to now have channeled the conclusions towards being drawn and interpreted with not satisfyingly, efficiently and fulfilling quality. Accordingly, the shown figure prevents the researchers from dominating and ruling robustly, perfectly and effectively the massive dynamic world of different technologies and its related scientific aspects including multitude of multidisciplinary aspects which are supposed to be challengingly tackled. However, great efforts have been put forth by scientists and cooperation to nurture and evolve the Zadeh’s theory, Fuzzy Logic (FL), which is the key of unlocking the addressed frustrating gates [47–55]. Going into more details, within this developed version of traditional binary logic objects are supposed to be members of a particular regarded set to some degree, varies between 0 and 1, dedicated with specific distinguished mechanisms, fundamentally known as Membership Functions (MFs) which are built to make smooth transitions between real world and fuzzy models. There are different kinds of MFs whose parameters can be altered according to the nature of issues and fulfill our targets [56–58]. Among many put forward MFS, the zmf, pimf and smf are three very popular ones which have mostly been used in numerous numbers of technical aspects, particularly petroleum engineering. These MFs
zmf ðx; a; bÞ ¼
8 > > > > <
0; xa2
2 ba ; xb2 > > 1 2 ba ; > > : 0;
9 > > > > a 6 x 6 aþb = x6a
2
ð14Þ
> 6 x 6 b> > > ; xPb
aþb 2
8 9 0; x6a > > > > > > > > > xa 2 aþb > > > 2 ; a 6 x 6 > ba 2 > > > > > > > > > xb 2 aþb > > 1 2 ; 6 x 6 b > > ba 2 < = 1; b6x6c pimf ðx; a; b; c; dÞ ¼ > > > > 1 2xc 2 ; c 6 x 6 cþd > > > > > > dc 2 > > > > > > xd2 > > cþd > > 2 ; 6 x 6 d > > dc 2 > > > > : ; 0; xPd
ð15Þ
8 9 1; x6a > > > > > > xa2 > > <1 2 = ; a 6 x 6 aþb ba 2 smf ðx; a; bÞ ¼ 2 aþb > 2 xb ; > > 6 x 6 b> > > ba 2 > > : ; 1; xPb
ð16Þ
Then, designed fuzzy rules which are supposed to make communications between input and output parameters through gaining from linguistic terms are accompanied with built MFs to construct a Fuzzy Interface System which is finally capable of predicting and modeling different parameters [59]. 4. Results and discussion Before tackling the discussion part, it is highly required to be informed that the statistical parameters of the implemented data, those which have been gathered from one of the Iranian Persian gulf gas reservoirs along with reported data banks in previous attentions [60], used basically throughout this research are summarized in Table 1. A traditional type of back-propagation neural network has also been conducted to deal with the addressed issue in order to prove the ability of the evolved least square method to determine the Pd. In general, the performance of an Artificial Neural Network (ANN) is intensely infected by some parameters like number of hidden neurons. To reflect the mentioned concern and underline its importance a detailed sensitivity analysis of ANN model performance alongside the number of hidden neurons was run step-bystep. The statistical parameters of correlation coefficient (R2) and root mean square error (RMSE) of the generated results of ANN
Table 1 Statistical parameters of the implemented dew point pressure data set. Variables
Min.
Max.
Average
Dewpoint pressure, psia Temperature, °F Hydrogen sulfide, mol fraction Carbon dioxide, mol fraction Nitrogen, mol fraction Methane, mol fraction Ethane, mol fraction Propane, mol fraction Butanes, mol fraction Pentanes, mol fraction Hexanes, mol fraction Heptanes-plus, mol fraction Molecular weight C7+ Specific gravity C7+
1400 76 0 0 0 0.0337 0.0034 0.0010 0.0022 0.0005 0.0005 0.0017 109 0.737
10,800 325 0.2875 0.9293 0.4135 0.9642 0.1497 0.1075 0.2024 0.0550 0.0487 0.1331 237 0.868
4855.778 216.84 0.006547 0.018655 0.011896 0.793606 0.061711 0.031852 0.020132 0.010608 0.008054 0.038114 151.2775 0.792548
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
versus the number of hidden neurons are depicted in Figs. 1 and 2 respectively. Sensitivity of the ANN correlation coefficient towards the corresponding hidden neurons has been demonstrated in Fig. 1. Consequently it can be deduced that the best correlation coefficient is accompanied with 9 hidden neurons. Also, Fig. 2 shows dependency of root mean square error (RMSE) on number of hidden neurons. Fig. 2 proves that the optimum number of hidden neurons is 9. The acquired results of the optimized neural network are illustrated in Figs. 3–5. The comparison between ANN results and determined dew point pressure data against relevant data index has been made and performed in Fig. 3. Based on Fig. 3, the ANN results have not followed the trend of actual dew point pressure data points. Next, the correlation between generated outputs of the ANN and corresponding actual dew point pressure data has been demonstrated in Fig. 4. According to Fig. 4, there is a noticeable deviation of the network outcomes from the diagonal line, this uncovers that the accuracy and efficiency of the network method are not as satisfying as the same relevant parameters of the least square support vector machine (LSSVM). Eventually, the relative deviation of the network outputs for both training and testing sets from actual dew point pressure data points versus corresponding experimental data are graphed separately in Fig. 5. Fig. 5 makes to conclude that the maximum deviation occurred in early boundary of actual dew point pressure data points. Besides, the maximum error of 50% has roughly been calculated for the network approach which is not scientifically permissible at all. As mentioned in section before, to obtain the optimum values of the least square support vector machine parameters, c and r2, the genetic algorithm (GA) was applied to minimize the root mean square error (RMSE) of the output results generated with the developed least square SVM. Consequently, the values of the global optima which include r2 and c have been extracted as 12.68725 and 49.83642, correspondingly. The generated results of the least square support vector machine (LSSVM) method are depicted through Figs. 6–8. The existence contrasts between least square SVM outputs and related measured dew point pressure of gathered samples from condensate gas reservoirs versus relevant data index have been demonstrated in Fig. 6. As illustrated in Fig. 6, the obtained results of least square SVM are as close as possible to actual dew point pressure data samples. To put it another way, the outputs of the LSSVM approach have the same behavior as measured actual data do. As depicted in Fig. 7 which is a graphical and scatter presentation of the LSSVM results versus corresponding experimental determined dew point pressure data, the LSSVM outputs lie over the line Y = X, the fact that indicates the identity of outputs gained from least square SVM and corresponding actual dew point pressure. The high
1077
Fig. 2. Sensitivity of root mean square error (RMSE) versus number neurons for estimation dew point pressure.
Fig. 3. Contrast between proposed ANN outputs and actual dew point pressure versus data index.
Fig. 4. Scatter plot of proposed ANN output against relevant actual dew point pressure.
Fig. 1. Sensitivity of correlation coefficient versus number neurons for estimation dew point pressure.
considerable level of efficiency and accuracy related to the least square support vector machine (LSSVM) in estimation of the dew point pressure dataset of condensate gas reservoirs has once again been proved in Fig. 7. Furthermore, the robustness of the LSSVM has been demonstrated in terms of the relative deviations of LSSVM model outputs from corresponding determined dew point
1078
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
Fig. 5. Relative deviation of the ANN results versus relevant actual dew point pressure.
Fig. 6. Scatter plot of proposed least square SVM results against relevant actual dew point pressure.
Fig. 8. Relative deviation of the least square SVM method results versus relevant actual dew point pressure.
connected to dew point pressure data measured between 2500 and 5000 psi. 20% is the maximum degree of relative deviation depicted in Fig. 8. Finally, to summarize previous results, Table 2 reports in detail the statistical index of the least square SVM and ANN models along with the other conventional approaches such as Nementh and Kennedy, Elsharkawy, Humoud and Al-Marhoun. According to Table 2 it can be inferred that the least square SVM gains from a high level of efficiency with low degree of uncertainty while these are not true about the artificial neural network model, hybrid genetic algorithm (GA) and fuzzy logic and other models like as Elsharkawy, Humoud and Al-Marhoun dedicate to low value of correlation coefficient and high value of root mean square error (RMSE). To clarify the sensitivity of the evolved approach in determination of dew point pressure, a robust and effective statistical method called analysis of variance (ANOVA) was used. The gained results from analysis of variance are depicted in Fig. 9. As demonstrated in Fig. 12, the most important parameters affect the dew point pressure are molecular weight of C7+ and specific gravity of C7+. In addition the relative error distribution of the evolved least square SVM versus corresponding molecular weight of C7+, specific gravity of C7+ and reservoir temperature (F) are illustrated in Figs. 10–12, correspondingly. As shown in Fig. 10, the high deviation of the developed model results were corresponded to the MWC7+ in ranges 110–140 g/mol while lowest deviation was observed for MWC7+ greater than 200 g/mol. Fig. 11 depicts the relative deviation of our model results versus corresponding specific gravity of C7+. As depicted in Fig. 11, the maximum deviation of our model was addressed to SGC7+ = 0.84 while the minimum deviation observed for SGC7+ = 0.86. In addition the error distribution of the developed dew point pressure model versus reservoir temperature is illustrated in Fig. 12. As shown in Fig. 12, the max-
Table 2 Statistical parameters of the evolved least square SVM versus conventional approaches.
Fig. 7. Contrast between proposed least square SVM method results and actual dew point pressure versus data index.
pressure data in Fig. 8. The relative deviations of the least square SVM outputs alongside relevant experimental data have been shown in Fig. 8. As could be observed in Fig. 3, the highest deviations of the LSSVM results are subjected to the early boundary. In more details, the maximum deviation of LSSVM outputs are those
a
Approach
RMSE
R2
AARD%
Nemth and Kennedy [61] Elsharkawy [62] Humoud [63] Marruffo and Rojas [64] ACEa Model [65] ANN Model GA-Fuzzy Model LSSVM approach
770.657 903.487 768.589 823.753 889.875 546.226 446.57 323.600
0.64 0.32 0.4761 0.3844 0.3364 0.9288 0.3801 0.9575
35.55 50.13 38.75 40.76 45.68 6.79 40.01 5.02
Alternating Conditional Expectations (ACE).
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
Fig. 9. Relative importance of utilized variables on dew point pressure (Pdew).
1079
Fig. 12. Relative deviation of the least square SVM method results versus relevant reservoir temperature.
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 low H2S is low and CO2 is low and N2 is low then Pd is low (Weight3 = 1).
Fig. 10. Relative deviation of the least square SVM method results versus relevant molecular weight of C7+.
Fig. 11. Relative deviation of the least square SVM method results versus relevant specific gravity of C7+.
imum relative error was referred to temperature in ranges of 200– 250 °F. In order to infer the most compatible rules from the gathered data, it was decided to run the ANOVA test to determine the most effective parameters. As clear be seen from Fig. 9, the most important parameters are MW C7+ and SG C7+ and after that the mole percent of C1, C7+ and the amount of T are the other important factors. Also, It can be observed that parameters such as mole percent of C2, C3, C4, C5, C6, N2, CO2 and H2S have the lowest effects on the amount of Pd. Regarding the generated results caused proposing the following fuzzy linguistic rules:
Weights show that how much rules are more effective than the others. Based on the assigned weights all the rules have the same effect in comparison with each other. To develop the supposed Fuzzy Interface System (FIS) capable of predicting the amount Pd, each attribute must have its dedicational MF. The MFs types which have already discussed and presented have a number of parameters which each one must either manually or automatically be set. Therefore, it was decided to gain from the Genetic Algorithm (GA) to obtain the most optimized relevant parameters of the MFs. The GA has the abilities of searching quickly and optimizing efficiently bade on the principle of ‘‘survival of the fittest’’ element of natural evolution with the genetic propagation of properties. According to the Darwinian concept of ‘survival of the fittest’ which is the foundation of this optimization method, the GA can converge towards the best coordinates in the provided space soon after a sequence of repetitive computations. The most basic functions of the GA are known as artificial mutation, crossover and selection operators. More information and tips about the GA and its eye-catching theory can be found in recent great literatures [66–69]. The GA with its specified operators and features must be run to identify the most optimized forms of considered MFs. To reach this goal, the GA was channeled towards minimizing the MSE. But, before making the final conclusions about the regarded parameters and their values, it was decided to run a sensitivity analysis to acquire the best structure of the GA which is supposed to optimize the MFs. Moreover, the preliminary GA was set by the following characteristics, Initial Population = 8, Length of Genes = 10, Number of Generation = 80, Crossover Rate = 2 and Mutation Rate = 0.0075. Effects of altering each one of these operators on the final corresponding MSE amount of the most elite individual of the last generation have been depicted in Figs. 13–17. Based on the Generated results it can be deduced that applying a GA with Initial Population of 12, Gene Length of 14, Generation Number of 110, Crossover Rate of 4 and Mutation Rate of 0.0125 results in generation of the best results. Running the GA for optimization of the 3 weights and 45 relevant parameters of MFs lead to produce the values of 65, 50 and
1080
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
211000
200000
198000
MSE
MSE
192000 185000
184000 172000 176000 159000
Mutation Rate
Initial Population
Fig. 17. Effect of Mutation Rate on the amount of MSE. Fig. 13. Effect of Initial Population on the amount of MSE.
222000
Membership Degree
1
MSE
210000
198000
186000
174000
Medium
0.8 0.6 0.4 0.2 0 0.1
Length of Gene
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C1 Fig. 14. Effect of Length of Genes on the amount of MSE. Fig. 18. The optimized MF of the C1.
202000 1
Membership Degree
MSE
196800 191600 186400 181200
0.8 0.6 Low
0.4 0.2
176000 0
Number of Generation
0.02
0.04
0.06
0.08
0.1
0.12
0.14
C2
Fig. 15. Effect of Number of Generation on the amount of MSE.
Fig. 19. The optimized MF of the C2.
1
Membership Degree
199000
MSE
194000
189000
0.8 0.6
Low
0.4 0.2 0
184000
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Crossover Rate Fig. 16. Effect of Crossover Rate on the amount of MSE.
C3 Fig. 20. The optimized MF of the C3.
0.1
1081
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
1
0.8 0.6 Low
0.4 0.2
Membership Degree
Membership Degree
1
0
0.8 0.6
Low
0.4 0.2 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.1
0.2
0.3
C4
0.8
0.9
1
0.6
Low
0.4 0.2
Membership Degree
0.8
0.8 0.6
Low
0.4 0.2 0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055
0
0.05
0.1
0.15
0.2
0.25
H2S
C5 Fig. 22. The optimized MF of the C5.
Fig. 26. The optimized MF of the H2S.
1
1
0.8 Low
0.6 0.4 0.2
Membership Degree
Membership Degree
0.7
Fig. 25. The optimized MF of the CO2.
1
0
0.8
Medium High
0.6 0.4 0.2 0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
120
C6
140
160
180
200
220
MW C7+
Fig. 23. The optimized MF of the C6.
Fig. 27. The optimized MF of the MW C7+.
1
1
Membership Degree
Membership Degree
0.6
CO2
Fig. 21. The optimized MF of the C4.
Membership Degree
0.5
0.4
0.8 0.6 0.4 0.2
0.8 0.6 Low
0.4 0.2
Medium
0
0 0.02
0.04
0.06
0.08
C7+ Fig. 24. The optimized MF of the C7+.
0.1
0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
N2 Fig. 28. The optimized MF of the N2.
0.35
0.4
1082
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
15030
12118
0.8 0.6
Medium High
0.4 0.2
Predicted Pd (psia)
Membership Degree
1
9206
6294
3382
0 0.74
0.76
0.78
0.8
0.82
0.84
0.86
470 1400
sg C7+
3280
5160
7040
8920
10800
Measured Pd (psia)
Fig. 29. The optimized MF of the SG C7+.
0.3801 Fig. 32. The scatter plot of generated Pd versus the Measured dew point pressure. Medium
80
0.8 0.6
40 0.4 0.2 0 100
150
200
250
300
Relative Error %
Membership Degree
1
0
40-
T Fig. 30. The optimized MF of the T.
801405
5159
3282
7036
8913
10790
Measured Pd (Psia) Fig. 33. Relative error distribution alongside of the measured Pd.
0.8 16000
Low Medium High
0.6 0.4 0.2 0 2000
3000
4000
5000
6000
7000
8000
9000 10000
Pd Fig. 31. The optimized MF of the Pd.
17 for Weight1, Weight2 and Weight3 respectively and form Figs. 18–31 for MFs. By forming the MFs, the supposed FIS is developed. Firing of the FIS for the gathered database, results in generation of outputs which have been depicted by the scatter form in Fig. 32. The Rsquare of 0.3801 and MSE of 0.85 have been calculated for generated results. Also, the relative error distribution alongside of the measured Pd has been plotted in Fig. 33. Although the generated results are not statistically satisfying at all, the error distribution along the Pd is roughly the same for all points. In terms of making another comparison between generated data, the measured Pd and the corresponding have been plotted against the data index in Fig. 34.
Dew Point Pressure (Psi)
Membership Degree
1
12000
8000
4000
0 0
101
202
303
404
Data Index Fig. 34. The predicted Pd in comparison with the measured dew point pressure.
5. Conclusions Performance monitoring of retrograded gas condensate reservoirs is highly depends on the accurate and precise determining of dew point pressure. Due to this fact, in this work massive efforts have been put forth to evolve intelligent based solution to figure out dew point pressure through condensate gas reservoirs. In
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
addition, expensive and accurate dew point pressure data banks utilized to evolve and test the proposed dew point pressure through condensate gas reservoirs correlation. Dedicated to the gained results of this contribution following major conclusions can be drawn: 1. Adequate agreement between gain dew point pressure from the developed least square SVM model versus corresponding experimental dew point pressure values observed however; the correlation between the outputs of the traditional approaches like as Elsharkawy and houmud, hybrid genetic algorithm and fuzzy loc approach, etc. and relevant actual dew point pressure data are unacceptable. In the other words, the conventional approaches fail to monitor dew point pressure through condensate gas reservoirs dedicated to the gained statistical criteria such as root mean square (RMSE) and correlation coefficient. 2. The evolved intelligent least square SVM model for monitoring dew point pressure through condensate gas reservoirs is user friend, fast and cheap for implementation. Moreover, it is very useful and user friend for evolving the accuracy and robustness of the commercial reservoir simulators like as ECLIPSE and computer modeling group (CMG) software for production scenarios from condensate gas reservoirs.
References [1] Mohammadi H, Sedaghat MH, Khaksar Manshad A. Parametric investigation of well testing analysis in low permeability gas condensate reservoirs. J Nat Gas Sci Eng 2013;14:17–28. [2] Mokhtari R, Varzandeh F, Rahimpour MR. Well productivity in an Iranian gas– condensate reservoir: a case study. J Nat Gas Sci Eng 2013;14:66–76. [3] Sadeghi Boogar A, Masihi M. New technique for calculation of well deliverability in gas condensate reservoirs. J Nat Gas Sci Eng 2010;2(1):29–35. [4] Alavi FS, Mowla D, Esmaeilzadeh F. Production performance analysis of Sarkhoon gas condensate reservoir. J Petrol Sci Eng 2010;75(1–2):44–53. [5] Ursin J-R. Fluid flow in gas condensate reservoirs: the interplay of forces and their relative strengths. J Petrol Sci Eng 2004;41(4):253–67. [6] Yong L et al. Water production analysis and reservoir simulation of the Jilake gas condensate field. Petrol Explor Dev 2010;37(1):89–93. [7] Brown AS et al. Measurement of the hydrocarbon dew point of real and synthetic natural gas mixtures by direct and indirect methods. Energy Fuels 2009;23(3):1640–50. [8] Elsharkawy AM, Foda SG. EOS simulation and GRNN modeling of the constant volume depletion behavior of gas condensate reservoirs. Energy Fuels 1998;12(2):353–64. [9] Nasrifar K, Bolland O, Moshfeghian M. Predicting natural gas dew points from 15 equations of state. Energy Fuels 2005;19(2):561–72. [10] App JF, Burger JE. Experimental determination of relative permeabilities for a rich gas/condensate system using live fluid. SPE Reservoir Eval Eng 2009;12(2):263–9. [11] Chowdhury NS et al. A semianalytical method to predict well deliverability in gas–condensate reservoirs. SPE Reservoir Eval Eng 2008;11(1):175–85. [12] Thomas FB, Bennion DB, Andersen G. Gas condensate reservoir performance. J Can Petrol Technol 2009;48(7):18–24. [13] Berning T. The dew point temperature as a criterion for optimizing the operating conditions of proton exchange membrane fuel cells. Int J Hydrogen Energy 2012;37(13):10265–75. [14] Louli V et al. Measurement and prediction of dew point curves of natural gas mixtures. Fluid Phase Equilib 2012;334:1–9. [15] Rahimpour MR et al. Enhancement in NGL production and improvement in water dew point temperature by optimization of slug catchers’ pressures in water dew point adjustment unit. J Nat Gas Sci Eng 2011;3(1):326–33. [16] Nowroozi S et al. Development of a neural fuzzy system for advanced prediction of dew point pressure in gas condensate reservoirs. Fuel Process Technol 2009;90(3):452–7. [17] Luo K et al. Experimental investigation into revaporization of retrograde condensate. In: SPE production and operations symposium, Oklahoma City, Oklahoma; 2001. [18] Shadizadeh SR, Rashtchian D, Moradi S. Simulation of experimental gasrecycling experiments in fractured gas/condensate reservoirs. In: SPE gas technology symposium, Calgary, Alberta, Canada; 2006. [19] Shen P et al. Experimental study of near-critical behavior of gas condensate systems. In: SPE production and operations symposium, Oklahoma City, Oklahoma; 2001. [20] Zheng X et al. Experimental investigation into near-critical phenomena of rich gas condensate systems. In: International oil and gas conference and exhibition in China, Beijing, China; 2000.
1083
[21] Jalali F, Abdy Y, Akbari MK. Dewpoint pressure estimation of gas condensate reservoirs, using artificial neural network (ANN). In: EUROPEC/EAGE conference and exhibition, London, UK; 2007. [22] Bonyadi M, Esmaeilzadeh F. Prediction of gas condensate properties by Esmaeilzadeh–Roshanfekr equation of state. Fluid Phase Equilib 2007;260(2):326–34. [23] Elsharkawy AM. Predicting the dew point pressure for gas condensate reservoirs: empirical models and equations of state. Fluid Phase Equilib 2002;193(1–2):147–65. [24] Li C, Jia W, Wu X. Application of Lee–Kesler equation of state to calculating compressibility factors of high pressure condensate gas. Energy Proc 2012;14:115–20. [25] Co KNLR-S, Falls W. A correlation of dewpoint pressure with fluid composition and temperature. SPE J 1967;7(2):99–104. [26] Marruffo I et al. Statistical forecast models to determine retrograde dew pressure and C7+ percentage of gas condensates on basis of production test data of eastern Venezuelan reservoirs. In: SPE Latin American and Caribbean petroleum engineering conference, Buenos Aires, Argentina; 2001. [27] Carlson MR, Cawston WB. Obtaining PVT data for very sour retrograde condensate gas and volatile oil reservoirs: a multi-disciplinary approach. In: SPE gas technology symposium, Calgary, Alberta, Canada; 1996. [28] Potsch KT, Braeuer L. A novel graphical method for determining dewpoint pressures of gas condensates. In: European petroleum conference, Milan, Italy; 1996. [29] Farasat A et al. Toward an intelligent approach for determination of saturation pressure of crude oil. Fuel Process Technol 2013;115:201–14. [30] Zendehboudi S, Ahmadi MA, James L, Chatzis I. Prediction of condensate-to-gas ratio for retrograde gas condensate reservoirs using artificial neural network with particle swarm optimization. Energy Fuels 2012;26(6):3432–47. [31] Ahmadi MA. Neural network based unified particle swarm optimization for prediction of asphaltene precipitation. Fluid Phase Equilib 2012;314:46–51. [32] Ahmadi MA, Golshadi M. Neural network based swarm concept for prediction asphaltene precipitation due natural depletion. J Petrol Sci Eng 2012;98– 99(11):40–9. [33] Ahmadi MA, Zendehboudi S, Lohi A, Elkamel A, Chatzis I. Reservoir permeability prediction by neural networks combined with hybrid genetic algorithm and particle swarm optimization. Geophys Prospect. 2013;61:582–98. [34] Ahmadi MA. Prediction of asphaltene precipitation using artificial neural network optimized by imperialist competitive algorithm. J Petrol Explor Prod Technol 2011;1:99–106. [35] Zendehboudi S, Ahmadi MA, Bahadori A, Shafiei A, Babadagli T. A developed smart technique to predict minimum miscible pressure—EOR implication. Can J Chem Eng 2013:1–13. http://dx.doi.org/10.1002/cjce.21802. [36] Zendehboudi S, Ahmadi MA, Mohammadzadeh O, Bahadori A, Chatzis I. Thermodynamic investigation of asphaltene precipitation during primary oil production: laboratory and smart technique. Ind Eng Chem Res. [37] Fazeli H, Soleimani R, Ahmadi MA, Badrnezhad R, Mohammadi AH. Experimental study and modeling of ultrafiltration of refinery effluents using a hybrid intelligent approach. Energy Fuels 2013;27:3523–37. [37] Ahmadi MA, Ebadi M, Hosseini SM. Prediction breakthrough time of water coning in the fractured reservoirs by implementing low parameter support vector machine approach. Fuel 2014;117:579–89. [39] Liu H, Yao X, Zhang R, Liu M, Hu Z, Fan B. Accurate Quantitative StructureProperty Relationship Model To Predict the Solubility of C60 in Various Solvents Based on a Novel Approach Using a Least-Squares Support Vector Machine. The Journal of Physical Chemistry B 2005;109:20565–71. [40] Eslamimanesh A, Gharagheizi F, Illbeigi M, Mohammadi AH, Fazlali A, Richon D. Phase equilibrium modeling of clathrate hydrates of methane, carbon dioxide, nitrogen, and hydrogen + water soluble organic promoters using support vector machine algorithm. Fluid Phase Equilib 2012;316:34–45. [41] Curilem M, Acuña G, Cubillos F, Vyhmeister E. Neural networks and support vector machine models applied to energy consumption optimization in semiautogeneous grinding. Chem Eng Trans 2011;25:761–6. [42] Cortes C, Vapnik V. Support-vector networks. Mach Learn 1995;20.
1084
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
[51] Jafari Kenari SA, Mashohor S. Robust committee machine for water saturation prediction. J Petrol Sci Eng 2013;104(0):1–10. [52] Mohaghegh SD. Recent developments in application of artificial intelligence in petroleum engineering. J Petrol Technol 2005;57(4):86–91. [53] Ozkaya SI. Using probabilistic decision trees to detect fracture corridors from dynamic data in mature oil fields. SPE Reservoir Eval Eng 2008;11(6):1061–70. [54] Shokir EME-M. Novel density and viscosity correlations for gases and gas mixtures containing hydrocarbon and non-hydrocarbon components. J Can Petrol Technol 2008;47(10). [55] Tahmasebi P, Hezarkhani A. A fast and independent architecture of artificial neural network for permeability prediction. J Petrol Sci Eng 2012;86– 87(0):118–26. [56] Ahmadi MA, Shadizadeh SR. New approach for prediction of asphaltene precipitation due to natural depletion by using evolutionary algorithm concept. Fuel 2012;102(0):716–23. [57] Nowroozi S, Ranjbar M, Hashemipour H, Schaffie M. Development of a neural fuzzy system for advanced prediction of dew point pressure in gas condensate reservoirs. Fuel Process Technol 2009;90(3):452–7. [58] Shafiei A, Dusseault MB, Zendehboudi S, Chatzis I. A new screening tool for evaluation of steamflooding performance in naturally fractured carbonate reservoirs. Fuel 2013;108(0):502–14. [59] Ahmadi MA, Ebadi M, Shokrollahi A, Majidi SMJ. Evolving artificial neural network and imperialist competitive algorithm for prediction oil flow rate of the reservoir. Appl Soft Comput 2013;13(2):1085–98. [60] Al-Dhamen MA. New models for estimating the dew-point pressure of gas condensate reservoir. King Fahd University of Petroleum & Minerals; 2010.
[61] Nemeth LK, Kennedy HT. A correlation of dewpoint pressure with fluid composition and temperature. Paper SPE 1477 presented at SPE 41st annual fall meeting held in Dallas, Texas; 1966. [62] Elsharkawy A. Predicting the dewpoint pressure for gas condensate reservoir: empirical models and equations of state. Fluid Phase Equilib 2002:147–65. [63] Humoud AA, Al-Marhoun MA. A new correlation for gas condensate dew point pressure prediction. Paper SPE 68230 presented at the 2001 SPE Middle East Oil Show, Bahrain, 17–20 March. [64] Marruffo I, Maita J, Him J, Rojas G. Correlation to determine retrograde dew pressure and C7+ percentage of gas condensate reservoirs on basis of production test data of Eastern Venezuelan fields. Presented at the SPE gas technology symposium held in Calgary, Alberta, Canada; 30 April 2002. [65] Breiman L, Friedman JH. Estimating optimal transformations for multiple regression and correlation. J Am Stat Assoc 1985(9):580–619. [66] Al-Gosayir M, Leung J, Babadagli T. Design of solvent-assisted SAGD processes in heterogeneous reservoirs using hybrid optimization techniques. J Can Petrol Technol 2012;51(6):437–48. [67] Chen S, Li H, Yang D. Optimization of production performance in a CO2 flooding reservoir under uncertainty. J Can Petrol Technol 2010;49(2):71–8. [68] Edmunds N, Moini B, Peterson J. Advanced solvent-additive processes by genetic optimization. J Can Petrol Technol 2010;49(9):34–41. [69] Willigers BJA, Weis F, Majou F. Creating portfolio insights by a practical multimethod optimization approach. SPE Econom Manage 2013;5(2).
Contents lists available at ScienceDirect
Fuel journal homepage: www.elsevier.com/locate/fuel
Evolving smart approach for determination dew point pressure through condensate gas reservoirs Mohammad Ali Ahmadi a,b,⇑, Mohammad Ebadi c a
Department of Petroleum Engineering, Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology, P.O. Box 63431, Ahwaz, Iran Research Institute of Petroleum Industry (RIPI), Tehran, Iran c Department of Petroleum Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran b
h i g h l i g h t s Evolving low parameter model to predict dew point pressure in retrograde gas reservoirs. Comparing effectiveness of the conventional models versus developed LSSVM model. Handling extensive dew point pressure data in retrograde gas reservoirs by new type of intelligent based model.
a r t i c l e
i n f o
Article history: Received 8 August 2013 Received in revised form 25 September 2013 Accepted 8 October 2013 Available online 25 October 2013 Keywords: Dew point pressure Least square support vector machine (LSSVM) Condensate gas Empirical correlation Computer program
a b s t r a c t To design gas condensate production planes with low uncertainty along with robust reservoir simulation, precise estimation or monitoring of dew point pressure play a crucial role. To handle successfully the addressed issue of condensate gas reservoirs, massive attentions have been performed previously but unfortunately fail to develop accurate approach for estimation dew point pressure. Dedicated to this fact, in current study enormous attempts have been put forth to proposed revolutionary method for determining dew point pressure in gas condensate reservoirs. To gain this end the new type of support vector machine method which evolved by Suykens and Vandewalle was utilized to generate robust approach to figure dew point pressure in condensate gas reservoir out. Also, lucrative and high precise dew point pressures reported in previous attentions were carried out to test and validate support vector machine approach. To serve better understanding of the proposed support vector machine approach, the conventional feed-forward artificial neural network and couple of genetic algorithm (GA) and fuzzy logic applied to the referred data banks and the gained solutions were contrasted with each other. According to the root mean square error (RMSE), correlation coefficient and average absolute relative deviation, the suggested support vector machine approach has acceptable reliability, integrity and robustness draw an analogy with the artificial neural network model and conventional methods. Thus, the proposed intelligent based way can be considered as an alternative model to monitor the dew point pressure of condensate gas reservoirs when the required real data are not accessible. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Gas condensate reservoirs are generally known as one of the most precious types of hydrocarbon reservoirs having capability of supplementing a massive clean amount of energy [1–3]. As a result, providing efficient, multidisciplinary and detailed production plans for these reservoirs has its own technical and economic importance. To design crucial and vital schemes to exploit from these gas sources, requiring accurate, precise and specified knowledge about reservoir fluid properties has always been a matter of ⇑ Corresponding author Address: Research Institute of Petroleum Industry (RIPI), Tehran, Iran. Tel.: +98 912 6364936. E-mail address: [email protected] (M.A. Ahmadi). 0016-2361/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.fuel.2013.10.010
consideration. In other words, pressure–volume–temperature (PVT) properties, which even small errors in their estimations lead to be encountered with some serious difficulties in subsequent procedures, play the leading role in every aspect of these kinds of reservoirs simulations and developments [2,4–6]. After beginning the step named ‘‘Flow-In’’ in gas condensate reservoirs, continues reduction in reservoir pressure caused formation of liquid drops in zones vicinity of the wellbore as a direct consequence of crossing the reservoir pressure from a threshold, a pressure-type border called dew point pressure (Pd) [7–9]. The creation of referred drops gives rise to decline dramatically the gas relative permeability and also gas production rate [10–12]. As a result, exact determination of the Pd must be taken as a very important topic. Therefore, numerous numbers of theoretical or
1075
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
experimental methods have smartly been put forward to measure Pd [13–16]. The Constant Composition Expansion (CCE) and the Constant Volume Depletion (CVD) are known as laboratorial procedures which are capable of extracting the Pd factor from gathered samples. These methods which their detailed steps have fully been described in literatures are routinely concluded to have some difficulties such as their expensive and time-consuming processes and also their accuracies are highly infected by some external parameters like human errors [17–21]. Moreover, empirically derived equations and Equation of States (EOS) are mathematical inspired concepts for measurements of some critical PVT properties [22–24]. Through running a multiple regression and gaining from an extensive database, a correlation based on temperature, characterizations of C7+ and fluid compositions was developed by Nemeth and Kennedy to predict Pd working properly under specified thermo dynamical ranges [25]. Improvements in characterizations and production from gas condensate reservoirs without using PVT data were achieved by Marruffo et al. who proposed a model to predict Pd and C7+ contents of gas condensate reservoirs [26]. Also, the influence of non-hydrocarbon impurities, particularly H2S, on the Pd has been investigated by Carison and Cawston [27]. Potsch and Braeuer proposed a graphical model as strong function of Z-factor accurate reading to determine the Pd based on observations of total volume during running the CCE test [28]. To sum up, relatively easy to use and not normally considering the temperature behavior are respectively known as one of advantages and disadvantages of empirical correlations [16]. On the other hand, the noticeable level of dependency towards initial deriving data caused EOSs losing their appropriate performances in case of applying on new locations and channel operators towards calibrating again the related parameters [21]. Hence, great efforts have been made to propose more useful, exact and suitable methods. Soft computing approaches thanks to their abilities of dealing with non-linearity, uncertainty and ambiguity of supposed problems have drawn attentions of researchers to defeat obstacles of reservoir engineering problems like extracting PVT properties, ashphaltene precipitation, condensate-to-gas ratio, minimum miscible pressure (MMP) and reservoir permeability [29–36]. For instance, Akbari et al. implemented a certain type of an Artificial Neural Network (ANN) to predict Pd through taking a set of compositional and thermo dynamical factors as input [21]. Furthermore, Nowroozi et al. designed an Adaptive Neuro-Fuzzy Inference System (ANFIS) to predict Pd by regarding mostly compositional parameters [16]. In addition, The main goal of current study is execute new kind of reversed based solution approaches called ‘‘least square support vector machine (LSSVM)’’ to develop robust, lucrative and precise predictive correlation to forecast dew point pressure through gas condensate reservoirs. To beat successfully this referred hurdle, least square support vector machine (LSSVM) was carried out on the previous literature data bases. The integrity and performance of the proposed predictive approach in estimating experimental dew point pressure from the literature is described in details. Furthermore, to point out reliability of the (LSSVM) results, expensive experimental data from one of the northern Persian Gulf gas fields of Iran was implemented to draw an analogy and proves the intelligent approach versus well-known dew point pressure methods.
formulated as illustrated in Eq. (1). Through the addressed equation, w stands for the weight factor, u denotes the nonlinear function which correlates the input space to a high-dimension characterization area and conducts linear regression while b represents the bias term [37–46]. Following expression was implemented as a cost function of the least square support vector machine (LSSVM) in calculation steps [37–46].
Q LSSVM ¼
The least square SVM theorem was introduced and developed by Suykens and Vandewalle in 1999 dedicated to the presumpion that the implemented data assortment S = {(x1, y1), . . . , (xn, yn)} that deal with a nonlinear function and decision function can be
ð1Þ
Relate to the following restriction [37–46]:
yk ¼ wT uðxk Þ þ b þ ek
k ¼ 1; 2; . . . ; N
ð2Þ
To figure out 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 [37–46].
Rðx; e; bÞ ¼
m 1 1 X jjwjj2 þ C e2k 2 2 k¼1
ð3Þ
To extract routs w and e, the Lagrange multiplier optimum programming approach is performed to solve Eq. (3); the addressed approach considers impartial and restriction parameters simultaneously. The mentioned Lagrange function L is formulated as following equation [37–46]:
Lðw; b; e; aÞ ¼ Jðw; eÞ
m X
ai fwT £ðxk Þ þ b þ ek Y k g
ð4Þ
k¼1
Through above equation, ai denotes the Lagrange multipliers that may be either positive or negative because LSSVM has equality restrictions. Owing to the Karush Kuhn–Tucher’s (KKT) conditions, conditions for optimum goal are demonstrated in Eq. (3) [44–46].
8 > > > > > > > > > < > > > > > > > > > :
@xL ¼ x
n X
ai uðxi Þ ¼ 0
i¼1
@bL ¼
n X
ai ¼ 0
i¼1
@ ei L ¼ Cei ai ¼ 0 @ ai L ¼ wT £ðxk Þ þ b þ ek yk ¼ 0
9 > > > > > > > > > = > > > > > > > > > ;
ð5Þ
Therefore, the linear equations can be demonstrated below expression [44–46]:
"
1T
0
# b
1 X þ 1c IN
a
¼
0 y
ð6Þ
While y = (y1, . . . , yn)T, 1n = (1, . . . , 1)T, a = (a1; . . . ; an)T and Xil = u(xi)u(xl) for i, l = 1, . . . , n. Thanks to the Mercer’s theorem, the resulting LS-SVM model for function approximation turns to the following equation [44–46] T
f ðxÞ ¼
N X
ak Kðx; xk Þ þ b
ð7Þ
k¼1
where a and b are the routs to Eq. (7) as below [44–46]: 1
b¼ 2. Least square support vector machine (LSSVM)
N X 1 T w w þ c e2k 2 k¼1
1Tn ðX þ 1c In Þ y 1
1Tn ðX þ 1c In Þ 1n
1
a ¼ X þ In c
ð8Þ
1 ðy 1n bÞ
ð9Þ
Eq. (10) may be executed as choice of nonlinear regression and utilize the Kernel function as below equation [37–46]:
1076
f ðxÞ ¼
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084 N X
ak Kðx; xk Þ þ b
ð10Þ
have been formulated in Eqs. (14)–(16) for zmf, pimf and smf, respectively.
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 U(x) and U(xi) as below equation [37– 46]:
Kðx; xk Þ ¼ UðxÞT Uðxk Þ
ð11Þ
Where the radial basis function (RBF) Kernel has been executed as following formulation [37–46]:
Kðx; xk Þ ¼ expðkxk xk2 =r2 Þ
ð12Þ
where r plays a role of decision variable through running least square SVM method, which is indicated by performing robust optimizer like genetic algorithm (GA) on the addressed least square SVM approach. To gain optimum value of least square SVM parameters, the root mean square error (RMSE) of the obtained outcomes of the final approach assigned as an objective function of the genetic algorithm (GA) which is formulated by the following equation [37– 46]:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ðPdesti Pdexpi Þ RMSE ¼ ns
ð13Þ
where Pd represents the dew point pressure, subscripts est and exp represents the predicted and actual dew point pressure through condensate gas reservoirs, respectively, and ns stands for the number of data from the initial assigned population. It would better stressed on here that the executed LSSVM method introduced by Pelckmans et al. [46] and Suykens and Vandewalle [45] has been carried out in the current study. 3. Fuzzy logic Uncertainty, imprecision and ambiguity are the inherent features of the daily problematic issues of the real world that are supposed to be dealt with, the characteristics which influence negatively the results generated from applying of the mainstream calculating methods, complicated mathematical formulas and numerical up-to-the-minute simulations. To put it another way, manipulating and conducting the globally accepted mathematical criteria and conventional logical approaches from the very early days of sciences progressions up to now have channeled the conclusions towards being drawn and interpreted with not satisfyingly, efficiently and fulfilling quality. Accordingly, the shown figure prevents the researchers from dominating and ruling robustly, perfectly and effectively the massive dynamic world of different technologies and its related scientific aspects including multitude of multidisciplinary aspects which are supposed to be challengingly tackled. However, great efforts have been put forth by scientists and cooperation to nurture and evolve the Zadeh’s theory, Fuzzy Logic (FL), which is the key of unlocking the addressed frustrating gates [47–55]. Going into more details, within this developed version of traditional binary logic objects are supposed to be members of a particular regarded set to some degree, varies between 0 and 1, dedicated with specific distinguished mechanisms, fundamentally known as Membership Functions (MFs) which are built to make smooth transitions between real world and fuzzy models. There are different kinds of MFs whose parameters can be altered according to the nature of issues and fulfill our targets [56–58]. Among many put forward MFS, the zmf, pimf and smf are three very popular ones which have mostly been used in numerous numbers of technical aspects, particularly petroleum engineering. These MFs
zmf ðx; a; bÞ ¼
8 > > > > <
0; xa2
2 ba ; xb2 > > 1 2 ba ; > > : 0;
9 > > > > a 6 x 6 aþb = x6a
2
ð14Þ
> 6 x 6 b> > > ; xPb
aþb 2
8 9 0; x6a > > > > > > > > > xa 2 aþb > > > 2 ; a 6 x 6 > ba 2 > > > > > > > > > xb 2 aþb > > 1 2 ; 6 x 6 b > > ba 2 < = 1; b6x6c pimf ðx; a; b; c; dÞ ¼ > > > > 1 2xc 2 ; c 6 x 6 cþd > > > > > > dc 2 > > > > > > xd2 > > cþd > > 2 ; 6 x 6 d > > dc 2 > > > > : ; 0; xPd
ð15Þ
8 9 1; x6a > > > > > > xa2 > > <1 2 = ; a 6 x 6 aþb ba 2 smf ðx; a; bÞ ¼ 2 aþb > 2 xb ; > > 6 x 6 b> > > ba 2 > > : ; 1; xPb
ð16Þ
Then, designed fuzzy rules which are supposed to make communications between input and output parameters through gaining from linguistic terms are accompanied with built MFs to construct a Fuzzy Interface System which is finally capable of predicting and modeling different parameters [59]. 4. Results and discussion Before tackling the discussion part, it is highly required to be informed that the statistical parameters of the implemented data, those which have been gathered from one of the Iranian Persian gulf gas reservoirs along with reported data banks in previous attentions [60], used basically throughout this research are summarized in Table 1. A traditional type of back-propagation neural network has also been conducted to deal with the addressed issue in order to prove the ability of the evolved least square method to determine the Pd. In general, the performance of an Artificial Neural Network (ANN) is intensely infected by some parameters like number of hidden neurons. To reflect the mentioned concern and underline its importance a detailed sensitivity analysis of ANN model performance alongside the number of hidden neurons was run step-bystep. The statistical parameters of correlation coefficient (R2) and root mean square error (RMSE) of the generated results of ANN
Table 1 Statistical parameters of the implemented dew point pressure data set. Variables
Min.
Max.
Average
Dewpoint pressure, psia Temperature, °F Hydrogen sulfide, mol fraction Carbon dioxide, mol fraction Nitrogen, mol fraction Methane, mol fraction Ethane, mol fraction Propane, mol fraction Butanes, mol fraction Pentanes, mol fraction Hexanes, mol fraction Heptanes-plus, mol fraction Molecular weight C7+ Specific gravity C7+
1400 76 0 0 0 0.0337 0.0034 0.0010 0.0022 0.0005 0.0005 0.0017 109 0.737
10,800 325 0.2875 0.9293 0.4135 0.9642 0.1497 0.1075 0.2024 0.0550 0.0487 0.1331 237 0.868
4855.778 216.84 0.006547 0.018655 0.011896 0.793606 0.061711 0.031852 0.020132 0.010608 0.008054 0.038114 151.2775 0.792548
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
versus the number of hidden neurons are depicted in Figs. 1 and 2 respectively. Sensitivity of the ANN correlation coefficient towards the corresponding hidden neurons has been demonstrated in Fig. 1. Consequently it can be deduced that the best correlation coefficient is accompanied with 9 hidden neurons. Also, Fig. 2 shows dependency of root mean square error (RMSE) on number of hidden neurons. Fig. 2 proves that the optimum number of hidden neurons is 9. The acquired results of the optimized neural network are illustrated in Figs. 3–5. The comparison between ANN results and determined dew point pressure data against relevant data index has been made and performed in Fig. 3. Based on Fig. 3, the ANN results have not followed the trend of actual dew point pressure data points. Next, the correlation between generated outputs of the ANN and corresponding actual dew point pressure data has been demonstrated in Fig. 4. According to Fig. 4, there is a noticeable deviation of the network outcomes from the diagonal line, this uncovers that the accuracy and efficiency of the network method are not as satisfying as the same relevant parameters of the least square support vector machine (LSSVM). Eventually, the relative deviation of the network outputs for both training and testing sets from actual dew point pressure data points versus corresponding experimental data are graphed separately in Fig. 5. Fig. 5 makes to conclude that the maximum deviation occurred in early boundary of actual dew point pressure data points. Besides, the maximum error of 50% has roughly been calculated for the network approach which is not scientifically permissible at all. As mentioned in section before, to obtain the optimum values of the least square support vector machine parameters, c and r2, the genetic algorithm (GA) was applied to minimize the root mean square error (RMSE) of the output results generated with the developed least square SVM. Consequently, the values of the global optima which include r2 and c have been extracted as 12.68725 and 49.83642, correspondingly. The generated results of the least square support vector machine (LSSVM) method are depicted through Figs. 6–8. The existence contrasts between least square SVM outputs and related measured dew point pressure of gathered samples from condensate gas reservoirs versus relevant data index have been demonstrated in Fig. 6. As illustrated in Fig. 6, the obtained results of least square SVM are as close as possible to actual dew point pressure data samples. To put it another way, the outputs of the LSSVM approach have the same behavior as measured actual data do. As depicted in Fig. 7 which is a graphical and scatter presentation of the LSSVM results versus corresponding experimental determined dew point pressure data, the LSSVM outputs lie over the line Y = X, the fact that indicates the identity of outputs gained from least square SVM and corresponding actual dew point pressure. The high
1077
Fig. 2. Sensitivity of root mean square error (RMSE) versus number neurons for estimation dew point pressure.
Fig. 3. Contrast between proposed ANN outputs and actual dew point pressure versus data index.
Fig. 4. Scatter plot of proposed ANN output against relevant actual dew point pressure.
Fig. 1. Sensitivity of correlation coefficient versus number neurons for estimation dew point pressure.
considerable level of efficiency and accuracy related to the least square support vector machine (LSSVM) in estimation of the dew point pressure dataset of condensate gas reservoirs has once again been proved in Fig. 7. Furthermore, the robustness of the LSSVM has been demonstrated in terms of the relative deviations of LSSVM model outputs from corresponding determined dew point
1078
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
Fig. 5. Relative deviation of the ANN results versus relevant actual dew point pressure.
Fig. 6. Scatter plot of proposed least square SVM results against relevant actual dew point pressure.
Fig. 8. Relative deviation of the least square SVM method results versus relevant actual dew point pressure.
connected to dew point pressure data measured between 2500 and 5000 psi. 20% is the maximum degree of relative deviation depicted in Fig. 8. Finally, to summarize previous results, Table 2 reports in detail the statistical index of the least square SVM and ANN models along with the other conventional approaches such as Nementh and Kennedy, Elsharkawy, Humoud and Al-Marhoun. According to Table 2 it can be inferred that the least square SVM gains from a high level of efficiency with low degree of uncertainty while these are not true about the artificial neural network model, hybrid genetic algorithm (GA) and fuzzy logic and other models like as Elsharkawy, Humoud and Al-Marhoun dedicate to low value of correlation coefficient and high value of root mean square error (RMSE). To clarify the sensitivity of the evolved approach in determination of dew point pressure, a robust and effective statistical method called analysis of variance (ANOVA) was used. The gained results from analysis of variance are depicted in Fig. 9. As demonstrated in Fig. 12, the most important parameters affect the dew point pressure are molecular weight of C7+ and specific gravity of C7+. In addition the relative error distribution of the evolved least square SVM versus corresponding molecular weight of C7+, specific gravity of C7+ and reservoir temperature (F) are illustrated in Figs. 10–12, correspondingly. As shown in Fig. 10, the high deviation of the developed model results were corresponded to the MWC7+ in ranges 110–140 g/mol while lowest deviation was observed for MWC7+ greater than 200 g/mol. Fig. 11 depicts the relative deviation of our model results versus corresponding specific gravity of C7+. As depicted in Fig. 11, the maximum deviation of our model was addressed to SGC7+ = 0.84 while the minimum deviation observed for SGC7+ = 0.86. In addition the error distribution of the developed dew point pressure model versus reservoir temperature is illustrated in Fig. 12. As shown in Fig. 12, the max-
Table 2 Statistical parameters of the evolved least square SVM versus conventional approaches.
Fig. 7. Contrast between proposed least square SVM method results and actual dew point pressure versus data index.
pressure data in Fig. 8. The relative deviations of the least square SVM outputs alongside relevant experimental data have been shown in Fig. 8. As could be observed in Fig. 3, the highest deviations of the LSSVM results are subjected to the early boundary. In more details, the maximum deviation of LSSVM outputs are those
a
Approach
RMSE
R2
AARD%
Nemth and Kennedy [61] Elsharkawy [62] Humoud [63] Marruffo and Rojas [64] ACEa Model [65] ANN Model GA-Fuzzy Model LSSVM approach
770.657 903.487 768.589 823.753 889.875 546.226 446.57 323.600
0.64 0.32 0.4761 0.3844 0.3364 0.9288 0.3801 0.9575
35.55 50.13 38.75 40.76 45.68 6.79 40.01 5.02
Alternating Conditional Expectations (ACE).
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
Fig. 9. Relative importance of utilized variables on dew point pressure (Pdew).
1079
Fig. 12. Relative deviation of the least square SVM method results versus relevant reservoir temperature.
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 low H2S is low and CO2 is low and N2 is low then Pd is low (Weight3 = 1).
Fig. 10. Relative deviation of the least square SVM method results versus relevant molecular weight of C7+.
Fig. 11. Relative deviation of the least square SVM method results versus relevant specific gravity of C7+.
imum relative error was referred to temperature in ranges of 200– 250 °F. In order to infer the most compatible rules from the gathered data, it was decided to run the ANOVA test to determine the most effective parameters. As clear be seen from Fig. 9, the most important parameters are MW C7+ and SG C7+ and after that the mole percent of C1, C7+ and the amount of T are the other important factors. Also, It can be observed that parameters such as mole percent of C2, C3, C4, C5, C6, N2, CO2 and H2S have the lowest effects on the amount of Pd. Regarding the generated results caused proposing the following fuzzy linguistic rules:
Weights show that how much rules are more effective than the others. Based on the assigned weights all the rules have the same effect in comparison with each other. To develop the supposed Fuzzy Interface System (FIS) capable of predicting the amount Pd, each attribute must have its dedicational MF. The MFs types which have already discussed and presented have a number of parameters which each one must either manually or automatically be set. Therefore, it was decided to gain from the Genetic Algorithm (GA) to obtain the most optimized relevant parameters of the MFs. The GA has the abilities of searching quickly and optimizing efficiently bade on the principle of ‘‘survival of the fittest’’ element of natural evolution with the genetic propagation of properties. According to the Darwinian concept of ‘survival of the fittest’ which is the foundation of this optimization method, the GA can converge towards the best coordinates in the provided space soon after a sequence of repetitive computations. The most basic functions of the GA are known as artificial mutation, crossover and selection operators. More information and tips about the GA and its eye-catching theory can be found in recent great literatures [66–69]. The GA with its specified operators and features must be run to identify the most optimized forms of considered MFs. To reach this goal, the GA was channeled towards minimizing the MSE. But, before making the final conclusions about the regarded parameters and their values, it was decided to run a sensitivity analysis to acquire the best structure of the GA which is supposed to optimize the MFs. Moreover, the preliminary GA was set by the following characteristics, Initial Population = 8, Length of Genes = 10, Number of Generation = 80, Crossover Rate = 2 and Mutation Rate = 0.0075. Effects of altering each one of these operators on the final corresponding MSE amount of the most elite individual of the last generation have been depicted in Figs. 13–17. Based on the Generated results it can be deduced that applying a GA with Initial Population of 12, Gene Length of 14, Generation Number of 110, Crossover Rate of 4 and Mutation Rate of 0.0125 results in generation of the best results. Running the GA for optimization of the 3 weights and 45 relevant parameters of MFs lead to produce the values of 65, 50 and
1080
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
211000
200000
198000
MSE
MSE
192000 185000
184000 172000 176000 159000
Mutation Rate
Initial Population
Fig. 17. Effect of Mutation Rate on the amount of MSE. Fig. 13. Effect of Initial Population on the amount of MSE.
222000
Membership Degree
1
MSE
210000
198000
186000
174000
Medium
0.8 0.6 0.4 0.2 0 0.1
Length of Gene
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C1 Fig. 14. Effect of Length of Genes on the amount of MSE. Fig. 18. The optimized MF of the C1.
202000 1
Membership Degree
MSE
196800 191600 186400 181200
0.8 0.6 Low
0.4 0.2
176000 0
Number of Generation
0.02
0.04
0.06
0.08
0.1
0.12
0.14
C2
Fig. 15. Effect of Number of Generation on the amount of MSE.
Fig. 19. The optimized MF of the C2.
1
Membership Degree
199000
MSE
194000
189000
0.8 0.6
Low
0.4 0.2 0
184000
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Crossover Rate Fig. 16. Effect of Crossover Rate on the amount of MSE.
C3 Fig. 20. The optimized MF of the C3.
0.1
1081
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
1
0.8 0.6 Low
0.4 0.2
Membership Degree
Membership Degree
1
0
0.8 0.6
Low
0.4 0.2 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.1
0.2
0.3
C4
0.8
0.9
1
0.6
Low
0.4 0.2
Membership Degree
0.8
0.8 0.6
Low
0.4 0.2 0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055
0
0.05
0.1
0.15
0.2
0.25
H2S
C5 Fig. 22. The optimized MF of the C5.
Fig. 26. The optimized MF of the H2S.
1
1
0.8 Low
0.6 0.4 0.2
Membership Degree
Membership Degree
0.7
Fig. 25. The optimized MF of the CO2.
1
0
0.8
Medium High
0.6 0.4 0.2 0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
120
C6
140
160
180
200
220
MW C7+
Fig. 23. The optimized MF of the C6.
Fig. 27. The optimized MF of the MW C7+.
1
1
Membership Degree
Membership Degree
0.6
CO2
Fig. 21. The optimized MF of the C4.
Membership Degree
0.5
0.4
0.8 0.6 0.4 0.2
0.8 0.6 Low
0.4 0.2
Medium
0
0 0.02
0.04
0.06
0.08
C7+ Fig. 24. The optimized MF of the C7+.
0.1
0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
N2 Fig. 28. The optimized MF of the N2.
0.35
0.4
1082
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
15030
12118
0.8 0.6
Medium High
0.4 0.2
Predicted Pd (psia)
Membership Degree
1
9206
6294
3382
0 0.74
0.76
0.78
0.8
0.82
0.84
0.86
470 1400
sg C7+
3280
5160
7040
8920
10800
Measured Pd (psia)
Fig. 29. The optimized MF of the SG C7+.
0.3801 Fig. 32. The scatter plot of generated Pd versus the Measured dew point pressure. Medium
80
0.8 0.6
40 0.4 0.2 0 100
150
200
250
300
Relative Error %
Membership Degree
1
0
40-
T Fig. 30. The optimized MF of the T.
801405
5159
3282
7036
8913
10790
Measured Pd (Psia) Fig. 33. Relative error distribution alongside of the measured Pd.
0.8 16000
Low Medium High
0.6 0.4 0.2 0 2000
3000
4000
5000
6000
7000
8000
9000 10000
Pd Fig. 31. The optimized MF of the Pd.
17 for Weight1, Weight2 and Weight3 respectively and form Figs. 18–31 for MFs. By forming the MFs, the supposed FIS is developed. Firing of the FIS for the gathered database, results in generation of outputs which have been depicted by the scatter form in Fig. 32. The Rsquare of 0.3801 and MSE of 0.85 have been calculated for generated results. Also, the relative error distribution alongside of the measured Pd has been plotted in Fig. 33. Although the generated results are not statistically satisfying at all, the error distribution along the Pd is roughly the same for all points. In terms of making another comparison between generated data, the measured Pd and the corresponding have been plotted against the data index in Fig. 34.
Dew Point Pressure (Psi)
Membership Degree
1
12000
8000
4000
0 0
101
202
303
404
Data Index Fig. 34. The predicted Pd in comparison with the measured dew point pressure.
5. Conclusions Performance monitoring of retrograded gas condensate reservoirs is highly depends on the accurate and precise determining of dew point pressure. Due to this fact, in this work massive efforts have been put forth to evolve intelligent based solution to figure out dew point pressure through condensate gas reservoirs. In
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
addition, expensive and accurate dew point pressure data banks utilized to evolve and test the proposed dew point pressure through condensate gas reservoirs correlation. Dedicated to the gained results of this contribution following major conclusions can be drawn: 1. Adequate agreement between gain dew point pressure from the developed least square SVM model versus corresponding experimental dew point pressure values observed however; the correlation between the outputs of the traditional approaches like as Elsharkawy and houmud, hybrid genetic algorithm and fuzzy loc approach, etc. and relevant actual dew point pressure data are unacceptable. In the other words, the conventional approaches fail to monitor dew point pressure through condensate gas reservoirs dedicated to the gained statistical criteria such as root mean square (RMSE) and correlation coefficient. 2. The evolved intelligent least square SVM model for monitoring dew point pressure through condensate gas reservoirs is user friend, fast and cheap for implementation. Moreover, it is very useful and user friend for evolving the accuracy and robustness of the commercial reservoir simulators like as ECLIPSE and computer modeling group (CMG) software for production scenarios from condensate gas reservoirs.
References [1] Mohammadi H, Sedaghat MH, Khaksar Manshad A. Parametric investigation of well testing analysis in low permeability gas condensate reservoirs. J Nat Gas Sci Eng 2013;14:17–28. [2] Mokhtari R, Varzandeh F, Rahimpour MR. Well productivity in an Iranian gas– condensate reservoir: a case study. J Nat Gas Sci Eng 2013;14:66–76. [3] Sadeghi Boogar A, Masihi M. New technique for calculation of well deliverability in gas condensate reservoirs. J Nat Gas Sci Eng 2010;2(1):29–35. [4] Alavi FS, Mowla D, Esmaeilzadeh F. Production performance analysis of Sarkhoon gas condensate reservoir. J Petrol Sci Eng 2010;75(1–2):44–53. [5] Ursin J-R. Fluid flow in gas condensate reservoirs: the interplay of forces and their relative strengths. J Petrol Sci Eng 2004;41(4):253–67. [6] Yong L et al. Water production analysis and reservoir simulation of the Jilake gas condensate field. Petrol Explor Dev 2010;37(1):89–93. [7] Brown AS et al. Measurement of the hydrocarbon dew point of real and synthetic natural gas mixtures by direct and indirect methods. Energy Fuels 2009;23(3):1640–50. [8] Elsharkawy AM, Foda SG. EOS simulation and GRNN modeling of the constant volume depletion behavior of gas condensate reservoirs. Energy Fuels 1998;12(2):353–64. [9] Nasrifar K, Bolland O, Moshfeghian M. Predicting natural gas dew points from 15 equations of state. Energy Fuels 2005;19(2):561–72. [10] App JF, Burger JE. Experimental determination of relative permeabilities for a rich gas/condensate system using live fluid. SPE Reservoir Eval Eng 2009;12(2):263–9. [11] Chowdhury NS et al. A semianalytical method to predict well deliverability in gas–condensate reservoirs. SPE Reservoir Eval Eng 2008;11(1):175–85. [12] Thomas FB, Bennion DB, Andersen G. Gas condensate reservoir performance. J Can Petrol Technol 2009;48(7):18–24. [13] Berning T. The dew point temperature as a criterion for optimizing the operating conditions of proton exchange membrane fuel cells. Int J Hydrogen Energy 2012;37(13):10265–75. [14] Louli V et al. Measurement and prediction of dew point curves of natural gas mixtures. Fluid Phase Equilib 2012;334:1–9. [15] Rahimpour MR et al. Enhancement in NGL production and improvement in water dew point temperature by optimization of slug catchers’ pressures in water dew point adjustment unit. J Nat Gas Sci Eng 2011;3(1):326–33. [16] Nowroozi S et al. Development of a neural fuzzy system for advanced prediction of dew point pressure in gas condensate reservoirs. Fuel Process Technol 2009;90(3):452–7. [17] Luo K et al. Experimental investigation into revaporization of retrograde condensate. In: SPE production and operations symposium, Oklahoma City, Oklahoma; 2001. [18] Shadizadeh SR, Rashtchian D, Moradi S. Simulation of experimental gasrecycling experiments in fractured gas/condensate reservoirs. In: SPE gas technology symposium, Calgary, Alberta, Canada; 2006. [19] Shen P et al. Experimental study of near-critical behavior of gas condensate systems. In: SPE production and operations symposium, Oklahoma City, Oklahoma; 2001. [20] Zheng X et al. Experimental investigation into near-critical phenomena of rich gas condensate systems. In: International oil and gas conference and exhibition in China, Beijing, China; 2000.
1083
[21] Jalali F, Abdy Y, Akbari MK. Dewpoint pressure estimation of gas condensate reservoirs, using artificial neural network (ANN). In: EUROPEC/EAGE conference and exhibition, London, UK; 2007. [22] Bonyadi M, Esmaeilzadeh F. Prediction of gas condensate properties by Esmaeilzadeh–Roshanfekr equation of state. Fluid Phase Equilib 2007;260(2):326–34. [23] Elsharkawy AM. Predicting the dew point pressure for gas condensate reservoirs: empirical models and equations of state. Fluid Phase Equilib 2002;193(1–2):147–65. [24] Li C, Jia W, Wu X. Application of Lee–Kesler equation of state to calculating compressibility factors of high pressure condensate gas. Energy Proc 2012;14:115–20. [25] Co KNLR-S, Falls W. A correlation of dewpoint pressure with fluid composition and temperature. SPE J 1967;7(2):99–104. [26] Marruffo I et al. Statistical forecast models to determine retrograde dew pressure and C7+ percentage of gas condensates on basis of production test data of eastern Venezuelan reservoirs. In: SPE Latin American and Caribbean petroleum engineering conference, Buenos Aires, Argentina; 2001. [27] Carlson MR, Cawston WB. Obtaining PVT data for very sour retrograde condensate gas and volatile oil reservoirs: a multi-disciplinary approach. In: SPE gas technology symposium, Calgary, Alberta, Canada; 1996. [28] Potsch KT, Braeuer L. A novel graphical method for determining dewpoint pressures of gas condensates. In: European petroleum conference, Milan, Italy; 1996. [29] Farasat A et al. Toward an intelligent approach for determination of saturation pressure of crude oil. Fuel Process Technol 2013;115:201–14. [30] Zendehboudi S, Ahmadi MA, James L, Chatzis I. Prediction of condensate-to-gas ratio for retrograde gas condensate reservoirs using artificial neural network with particle swarm optimization. Energy Fuels 2012;26(6):3432–47. [31] Ahmadi MA. Neural network based unified particle swarm optimization for prediction of asphaltene precipitation. Fluid Phase Equilib 2012;314:46–51. [32] Ahmadi MA, Golshadi M. Neural network based swarm concept for prediction asphaltene precipitation due natural depletion. J Petrol Sci Eng 2012;98– 99(11):40–9. [33] Ahmadi MA, Zendehboudi S, Lohi A, Elkamel A, Chatzis I. Reservoir permeability prediction by neural networks combined with hybrid genetic algorithm and particle swarm optimization. Geophys Prospect. 2013;61:582–98. [34] Ahmadi MA. Prediction of asphaltene precipitation using artificial neural network optimized by imperialist competitive algorithm. J Petrol Explor Prod Technol 2011;1:99–106. [35] Zendehboudi S, Ahmadi MA, Bahadori A, Shafiei A, Babadagli T. A developed smart technique to predict minimum miscible pressure—EOR implication. Can J Chem Eng 2013:1–13. http://dx.doi.org/10.1002/cjce.21802. [36] Zendehboudi S, Ahmadi MA, Mohammadzadeh O, Bahadori A, Chatzis I. Thermodynamic investigation of asphaltene precipitation during primary oil production: laboratory and smart technique. Ind Eng Chem Res. [37] Fazeli H, Soleimani R, Ahmadi MA, Badrnezhad R, Mohammadi AH. Experimental study and modeling of ultrafiltration of refinery effluents using a hybrid intelligent approach. Energy Fuels 2013;27:3523–37. [37] Ahmadi MA, Ebadi M, Hosseini SM. Prediction breakthrough time of water coning in the fractured reservoirs by implementing low parameter support vector machine approach. Fuel 2014;117:579–89. [39] Liu H, Yao X, Zhang R, Liu M, Hu Z, Fan B. Accurate Quantitative StructureProperty Relationship Model To Predict the Solubility of C60 in Various Solvents Based on a Novel Approach Using a Least-Squares Support Vector Machine. The Journal of Physical Chemistry B 2005;109:20565–71. [40] Eslamimanesh A, Gharagheizi F, Illbeigi M, Mohammadi AH, Fazlali A, Richon D. Phase equilibrium modeling of clathrate hydrates of methane, carbon dioxide, nitrogen, and hydrogen + water soluble organic promoters using support vector machine algorithm. Fluid Phase Equilib 2012;316:34–45. [41] Curilem M, Acuña G, Cubillos F, Vyhmeister E. Neural networks and support vector machine models applied to energy consumption optimization in semiautogeneous grinding. Chem Eng Trans 2011;25:761–6. [42] Cortes C, Vapnik V. Support-vector networks. Mach Learn 1995;20.
1084
M.A. Ahmadi, M. Ebadi / Fuel 117 (2014) 1074–1084
[51] Jafari Kenari SA, Mashohor S. Robust committee machine for water saturation prediction. J Petrol Sci Eng 2013;104(0):1–10. [52] Mohaghegh SD. Recent developments in application of artificial intelligence in petroleum engineering. J Petrol Technol 2005;57(4):86–91. [53] Ozkaya SI. Using probabilistic decision trees to detect fracture corridors from dynamic data in mature oil fields. SPE Reservoir Eval Eng 2008;11(6):1061–70. [54] Shokir EME-M. Novel density and viscosity correlations for gases and gas mixtures containing hydrocarbon and non-hydrocarbon components. J Can Petrol Technol 2008;47(10). [55] Tahmasebi P, Hezarkhani A. A fast and independent architecture of artificial neural network for permeability prediction. J Petrol Sci Eng 2012;86– 87(0):118–26. [56] Ahmadi MA, Shadizadeh SR. New approach for prediction of asphaltene precipitation due to natural depletion by using evolutionary algorithm concept. Fuel 2012;102(0):716–23. [57] Nowroozi S, Ranjbar M, Hashemipour H, Schaffie M. Development of a neural fuzzy system for advanced prediction of dew point pressure in gas condensate reservoirs. Fuel Process Technol 2009;90(3):452–7. [58] Shafiei A, Dusseault MB, Zendehboudi S, Chatzis I. A new screening tool for evaluation of steamflooding performance in naturally fractured carbonate reservoirs. Fuel 2013;108(0):502–14. [59] Ahmadi MA, Ebadi M, Shokrollahi A, Majidi SMJ. Evolving artificial neural network and imperialist competitive algorithm for prediction oil flow rate of the reservoir. Appl Soft Comput 2013;13(2):1085–98. [60] Al-Dhamen MA. New models for estimating the dew-point pressure of gas condensate reservoir. King Fahd University of Petroleum & Minerals; 2010.
[61] Nemeth LK, Kennedy HT. A correlation of dewpoint pressure with fluid composition and temperature. Paper SPE 1477 presented at SPE 41st annual fall meeting held in Dallas, Texas; 1966. [62] Elsharkawy A. Predicting the dewpoint pressure for gas condensate reservoir: empirical models and equations of state. Fluid Phase Equilib 2002:147–65. [63] Humoud AA, Al-Marhoun MA. A new correlation for gas condensate dew point pressure prediction. Paper SPE 68230 presented at the 2001 SPE Middle East Oil Show, Bahrain, 17–20 March. [64] Marruffo I, Maita J, Him J, Rojas G. Correlation to determine retrograde dew pressure and C7+ percentage of gas condensate reservoirs on basis of production test data of Eastern Venezuelan fields. Presented at the SPE gas technology symposium held in Calgary, Alberta, Canada; 30 April 2002. [65] Breiman L, Friedman JH. Estimating optimal transformations for multiple regression and correlation. J Am Stat Assoc 1985(9):580–619. [66] Al-Gosayir M, Leung J, Babadagli T. Design of solvent-assisted SAGD processes in heterogeneous reservoirs using hybrid optimization techniques. J Can Petrol Technol 2012;51(6):437–48. [67] Chen S, Li H, Yang D. Optimization of production performance in a CO2 flooding reservoir under uncertainty. J Can Petrol Technol 2010;49(2):71–8. [68] Edmunds N, Moini B, Peterson J. Advanced solvent-additive processes by genetic optimization. J Can Petrol Technol 2010;49(9):34–41. [69] Willigers BJA, Weis F, Majou F. Creating portfolio insights by a practical multimethod optimization approach. SPE Econom Manage 2013;5(2).