Applying a robust solution based on expert systems and GA evolutionary algorithm for prognosticating residual gas saturation in water drive gas reservoirs

Journal of Natural Gas Science and Engineering 21 (2014) 79e94

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Applying a robust solution based on expert systems and GA evolutionary algorithm for prognosticating residual gas saturation in water drive gas reservoirs Afshin Tatar a, Mahmood Reza Yassin b, Mohammad Rezaee c, Amir Hossein Aghajafari d, Amin Shokrollahi e, * a

Department of Chemical Engineering, Sahand University of Technology, Tabriz, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran Institute of Petroleum Engineering, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran, Iran d Department of Petroleum Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran e Young Researchers and Elite Club, North Tehran Branch, Islamic Azad University, Tehran, Iran b c

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 June 2014 Received in revised form 12 July 2014 Accepted 13 July 2014 Available online

In strong water drive gas reservoirs (WDGRs), the water encroachment in the gas zone has adverse effects on the gas mobility and causes considerable volume of gas to be trapped behind water front; therefore estimation of residual gas saturation after water influx is an important parameter in estimation of gas reservoirs with strong aquifer support. It is difficult to achieve a thorough and exact understanding of water drive gas reservoirs. It depends on several parameters of petrophysical and operational features. In majority of the previous studies about residual gas saturation, the correlations were depended on petrophysical properties such as porosity, permeability, and initial gas saturation. Most of these correlations are well applied on limited dataset that they are constructed based on, but they are not applicable to dataset from other references. In other words, they are not capable of generalization. One reason for this might be different experimental methods to determine the residual gas saturation. In the present study, the prediction of residual gas saturation is presented utilizing Committee Machine Intelligent Technique and some well-known correlations are used for comparison. The reviewed correlations in this study generally do not provide good results and some of them that exhibit reasonable results demand some experimental parameters that are usually unavailable. In this study, two different intelligent models are proposed for spontaneous imbibition and force flood. The suggested models provide good results for the two cases; however, the prediction for force flood is not as exact as the results for the spontaneous imbibition. At the end, an outlier approach detection based on Leverage method was applied to investigate the applicability domain of the proposed models as well as possible outlier data. © 2014 Elsevier B.V. All rights reserved.

Keywords: Gas reservoirs Residual gas saturation Water drive CMIS modeling Genetic algorithm

1. Introduction 1.1. Background Usually gas reservoirs may be confined with a water zone as aquifer. If a strong water drive is present, trapping of gas molecules behind water encroachment at higher pressures can considerably

* Corresponding author. Tel.: þ98 9363826336. E-mail address: [email protected] (A. Shokrollahi). http://dx.doi.org/10.1016/j.jngse.2014.07.017 1875-5100/© 2014 Elsevier B.V. All rights reserved.

reduce the gas recovery factor (Geffen et al., 1952; Keelan and Pugh, 1975). In strong WDGR, water flow into the gas zone causes unfavorable effects on the gas flow; therefore, a huge volume of gas is trapped behind the advanced water front. Detailed understanding of WDGR is complex and depends on both the petrophysical and operating factors, such as heterogeneity of reservoir, permeability, gas production rate, and so on. These parameters generally affect residual gas saturation, which controls the gas recovery factor. Permeability has a considerable effect in residual gas saturation as a petrophysical parameter (Keelan and Pugh, 1975). Since the pore structure is a representative of permeability in porous media, the

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main factors affecting the trapping of gas behind water phase are coordination number, pore-to-throat ratio, nature and degree of heterogeneity and the properties of rock surface in gas reservoirs with active aquifers (Hamon et al., 2001; Holtz, 2002; Wardlaw, 1980). Rezaee et al. (2013), investigated heterogeneity effect on residual gas saturation in strong water drive gas reservoirs and they concluded that heterogeneity is not always detrimental on gas recovery factor and the residual gas saturation depends on both heterogeneity index and permeability ratio of water to gas zone. In addition, in this kind of gas reservoirs, pore body to throat ratio, which is highly related to permeability and porosity, is responsible for the volume of trapped gas (Holtz, 2002). Since pore structure depends on rock type, some researchers investigated effect of rock type on residual gas saturation (Batycky et al., 1998). Many authors accept the independency of residual gas saturation on some parameters such as pressure, temperature and water influx rate and believe that they could be ignored (Chierici et al., 1963; Crowell et al., 1966; Delclaud, 1991; Geffen et al., 1952; McKay, 1974). On the other hand, Bull et al. (2011) monitored insitu saturation profile by CT-scanner and they concluded that the choice of laboratory method could have a large effect on the residual gas results. They revealed a dispersed water front at very low imbibition rates, while rates similar to the free spontaneous imbibition results in a piston like displacement that affects residual gas saturation. Geffen et al. (1952) experimentally concluded that remaining gases after water flooding were similar during spontaneous imbibition and forced tests. In their laboratory method, abandonment pressure was fixed equal to initial pressure, which is not a practical case for water drive gas reservoirs. Knapp et al. (1968) developed a computer modeling to demonstrate that gas recovery is a function of the gas production rate, aquifer strength, and formation heterogeneity. In addition, AlHashim and Bass Jr. (1988) worked on the effect of aquifer size on the performance of partial WDGR. They used Van Everdingen and Hurst (1949) in their computer modeling and showed that when the ratio of aquifer radius to gas zone radius is less than two (ra/ rg  2), the effect of aquifer on performance of gas reservoir is not significant. They also assumed that gasewater contact during gas production remains horizontal which is not a general rule in practice. Firoozabadi et al. (1987) observed in their laboratory experiments two distinct value of residual gas saturation for sandstone samples. The lower and higher values corresponded to the initial entrapment and the gas saturation to be mobilized under expansion, respectively. In other works, series of experiments were performed on both Berea sandstone and Kansas chalk by Li and Firoozabadi (2000) to investigate wettability alteration. They changed rock wettability in gasewater system from strong water wetting to intermediate or preferential gas wetting using some chemicals. This can result in improved control of production rate and enhanced pressure maintenance for production of gas during long period. Suzanne et al. (2003) presented sixty experimental relationships between initial gas saturation (Sgi) and residual gas saturation (Sgr) for a large set of sandstone samples and checked against former empirical SgreSgi correlations. They used water spontaneous imbibition for their experiments and measured maximum residual gas saturation for all tests to develop new relationships. They concluded that the constant Sgr region confirms that the micro-porosity does not trap the gas and also the linear Sgr/Sgi region corresponds to gas trapped in the macroporosity. None of the hyperbolic laws such as modified Land's equation (Land, 1968, 1971) were able to correctly describe the observed experimental behavior. In all gas reservoirs, including water drive types, the trapped gas is a controlling factor for gas recovery, and as it is evident from the literature, uncertainty in

estimation of residual gas saturation with petrophysical properties is still a challenge. As far as the knowledge of the authors, there is no universal relationship among permeability, porosity, initial water saturation and residual gas saturation. Several correlations have currently been developed which are not capable of including all parameters and their interaction respect to each other (Agarwal, 1967; Holtz, 2002; Land, 1971). Analytical relationships are noticeable because of their form and number of experimental data that they support. Some of the more popular correlations for estimation of Sgr has been provided in Appendix A. In majority of the previous studies about residual gas saturation, the correlations were depended on petrophysical properties such as porosity, permeability, and initial gas saturation. Nevertheless, the entire reservoir cannot be examined directly and there still exist uncertainties associated with the nature of geological/petrophysical data. Such uncertainties can lead to errors in the reservoir evaluation. To cope with uncertainties, intelligent mathematical techniques to predict the spatial distribution of reservoir properties appear as strong tools. Different intelligent methods have been widely used to recognize governing patterns on several different aspects of phenomena related to petroleum science and industry (Al-Marhoun et al., 2012; El Ouahed et al., 2005; Fayazi et al., 2013; HemmatiSarapardeh et al., 2014; Majidi et al., 2014). In this communication, our objective is to develop a committee machine intelligent system based on three different intelligent systems, namely radial basis function (RBF) neural network, multilayer perceptron (MLP) neural network, and least square support vector machine (LSSVM) for prediction of residual gas saturation in water drive gas reservoir using petrophysical data. To optimize the combination of the mentioned experts, genetic algorithm (GA) was chosen for its flexibility and well performance. Results obtained from the developed intelligent approaches were compared with the corresponding experimental data and discussed in further details throughout this study. 1.2. Details of intelligent method Artificial intelligent systems are capable of learning from experience, improving their performance and adopting to the changes in the environment (Santos et al., 2013). The main advantages of neural networks are the capability of processing a large amount of data and their ability to generalize the results. ANNs are parallel distributed systems that are composed of artificial neurons as processing units. 1.3. Multilayer perceptron networks Multilayer perceptron (MLP) neural networks comprise of three different types of layers, namely, input layer, hidden layer(s), and output layer (Fig. 1). A single MLP might have one or more hidden layers. Each layer is composed of some neurons. The number of neurons in the input and output layers are corresponded to number of input and output data, respectively. The number of hidden layers and also neurons in them is optional and can be determined either intelligently or by trial and error to achieve the best performance. Mean Square Error (MSE) indicates the performance of the developed network. In such networks, the error is back propagated through the network and the weights and biases are optimized through some iterations called epochs. The number of epochs should be such that the network neither undertrain nor overtrain. In the former, the network does not have enough time to complete the learning process. In the latter, the network does not learn but memorizes. This results in poor performance of network in prediction of test data set (Haykin, 1994).

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Fig. 1. Schematic representation of MLP neural network.

Fig. 4. Flow chart of genetic algorithm for optimization processes (Ahmadi and Shadizadeh, 2013).

Both MLP and Radial Basis Function (RBF) networks have the same applications but different internal calculation structures. The main advantage of RBF networks is easy design that it has just three layers. They are capable of good generalization, high tolerance of input noises and ability of online learning (Santos et al., 2013). From

the point view of generalization, RBFNs can respond very well to patterns that were not used for training (Hao et al., 2011). RBF networks are neural networks based on localized basis functions and iterative function approximation (Dayhoff, 1990; Huang and Hong-Chao, 1994; Lowe and Broomhead, 1988; Zurada, 1992). The RBF networks utilize supervise training technique and are a type of feed-forward neural networks (Karri, 1999). The RBFN is a universal approximator, with a solid foundation in the conventional approximation theory (Du and Swamy, 2006). RBF has a simpler structure than MLP and training process is much faster. These features make RBFN a popular alternative to the MLP. The origin of RBFN is in performing exact interpolation of a set of data points in a multidimensional space (Powell, 1987). It is proved that RBF networks can be implemented by MLP networks with increased input dimensions (Wilamowski and Jaeger, 1996). The RBFN architecture is similar to the classical regularization network (Poggio and Girosi, 1990). The regularization network has three desirable properties (Girosi and Poggio, 1990; Poggio and Girosi, 1990):

Fig. 3. Architecture of the support vector machine (Übeyli_, 2010).

Fig. 5. Diagram of applied committee machine intelligent system. The output of the RBF, MLP, and LSSVM experts are the input of the combiner on the basis of the genetic algorithm to find the best weights, which minimize the MSE.

Fig. 2. Schematic representation of RBFN (Du and Swamy, 2006).

1.4. Radial basis function networks

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Table 1 Statistical parameters of data used in model construction or correlations. Reference

Data

Porosity

Kg (mD)

Sgi

Sgr

Min

Max

Average

Min

Max

Average

Min

Max

Average

Min

Max

Average

6 1 1 1 1445 860 51 0.110 0.001 0.300 9.830

12,950 5600 550 4000 1915 4430 400 1283 0.440 3850 3234

1740 1285 108 683 1717 3235 154 336 0.131 215 756

0.54 0.59 0.20 0.63 0.81 0.80 0.52 0.36 0.01 0.01 0.68

0.86 1 0.80 0.92 1 1 0.69 1 1 1 0.94

0.75 0.82 0.61 0.80 0.91 0.91 0.60 0.87 0.79 0.56 0.84

0.12 0.23 0.12 0.32 0.28 0.24 0.14 0.01 0 0 0.13

0.44 0.46 0.69 0.49 0.31 0.27 0.41 0.79 0.56 0.69 0.48

0.23 0.33 0.37 0.39 0.30 0.25 0.28 0.41 0.20 0.26 0.33

0

1

0.63

0

0.8

0.3

Chierici et al. (Chierici et al., 1963) McKay (McKay, 1974) Keelan (Keelan, 1976) Keelan and Pugh (Keelan and Pugh, 1975) Firoozabadi et al. (Firoozabadi et al., 1987) Delclaud (Delclaud, 1991) Holtz (Holtz, 2002) Ding and Kantzas (Ding and Kantzas, 2001) Ding and Kantzas (Ding and Kantzas, 2002) Suzanne et al. (Suzanne et al., 2003) Bull et al. (Bull et al., 2011)

25 49 79 11 3 14 4 47 21 600 50

0.16 0.09 0.04 0.12 0.33 0.29 0.21 0.01 0.01 0.08 0.17

0.49 0.31 0.37 0.32 0.35 0.32 0.26 0.31 0.07 0.21 0.34

0.30 0.21 0.19 0.22 0.34 0.31 0.23 0.11 0.03 0.14 0.27

All data used

903

0.01

0.5

0.2

0.001

12,950

395

Table 2 Specifications of data used in model construction or correlations. Reference of data

Chierici et al. (Chierici et al., 1963) McKay (McKay, 1974) Keelan (Keelan, 1976) Keelan and Pugh (Keelan and Pugh, 1975) Firoozabadi et al. (Firoozabadi et al., 1987) Delclaud (Delclaud, 1991) Holtz (Holtz, 2002) Ding and Kantzas (Ding and Kantzas, 2001) Ding and Kantzas (Ding and Kantzas, 2002) Suzanne et al. (Suzanne et al., 2003) Bull et al. (Bull et al., 2011)

Water flood type

Rock type Consolidated sandstone

Un-consolidated sandstone

Carbonate

Force injection

* *

*

*

* * * * * * * * *

* * * * * *

* * *

* *

*

1. It is capable of approximating any multivariate continuous function on a compact domain to an arbitrary accuracy, given a sufficient number of units. 2. The solution is optimal in the sense that it minimizes a function that measures how much it oscillates.

Spontaneous imbibition *

* * * *

3. The approximation has the best-approximation property since the un-known coefficients are linear. The architecture of the RBFN is somehow similar to MLP (Fig. 2). It has three layers as input layer, output layer and a hidden layer. As

Fig. 6. Initial gas saturation versus residual gas saturation for all data.

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Fig. 7. MSE values for different structures of MLP networks to determine the best one. (a) Spontaneous imbibition, (b) force flood.

a nonlinear activation function, each node in the hidden layer uses an RBFN (f(r)). However, there are fundamental differences between RBF and MLP networks that the most important ones are as follows (Hao et al., 2011): 1. RBF networks are simpler than MLP networks; 2. Because of the simple and fixed three-layer architecture, RBFNs are easier to train than MLP networks.

3. RBF networks act as local approximation networks and the network outputs are determined by specified hidden units in certain local receptive fields, while MLP networks work globally and the network outputs are decided by all the neurons. 4. The classification method in RBF and MLP networks is quite different. In RBF networks clusters are separated by hyper spheres, while in MLP networks hyper surfaces separate the clusters.

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Fig. 8. A typical flowchart of CSA-LSSVM algorithm (Mesbah et al., 2014; Safari et al., 2014a).

Further information about RBF networks is available in literature (Tatar et al., 2013). 1.5. Least square support vector machine The support vector machine (SVM) model was proposed by Vapnik (1999), which has been studied for both classification and regression analysis (Baylar et al., 2009; Chen et al., 2011; Cortes and Vapnik, 1995; Shmilovici, 2010; Übeyli_, 2010; Vapnik, 1999; Yao et al., 2006). The architecture of SVM is shown in Fig. 3. The SVM uses the spirit of the structural risk minimization principle (Cortes and Vapnik, 1995). Through nonlinear mapping, SVM maps the input patterns into a higher dimensional feature space. Then, a linear decision surface is constructed in this high dimensional feature space (Cortes and Vapnik, 1995; Pelckmans et al., 2002; Suykens and Vandewalle, 1999; Übeyli_, 2010). In this paper, the objective of using a modified SVM is to develop a relationship between the input data and desired output. SVM is a powerful strategy developed from the machine-learning community (Eslamimanesh et al., 2012; Pelckmans et al., 2002; Suykens and Vandewalle, 1999). The SVM is a tool for a set of related supervised learning methods that is capable of recognizing patterns, analyzing data, and regression analysis. For solving the problem associated with SVM, it is necessary to solve a quadratic programming problem which is not easy to solve. In order to facilitate the original SVM algorithm Suykens and Vandewalle (Pelckmans et al., 2002; Suykens and Vandewalle, 1999) proposed the least square modification of

the SVM method, namely, least square support vector machine (LSSVM). The major advantage of LSSVM over the original SVM is the idea of modifying the inequality constraints to the equality constraints. Instead of solving a nonlinear quadratic programming in SVM, the parameters of LSSVM are readily obtained by solving a system of equations (Suykens and Vandewalle, 1999). Shokrollahi et al. (2013) have fully reviewed the details and formulations behind LSSVM in their modeling studies. 1.6. Committee machine intelligent system Committee machine is a method for solving supervised learning tasks which was initially introduced by Nilsson (1965). This method uses the common engineering principle: divide and conquer. Thus, in this method, a complex computational task is divided to some simpler subtasks and is then solved by combining the results of those subtasks. In computational problems, learning process is performed by a number of experts. The combination of experts results in construction of a committee machine (CM) (Haykin, 1999). Committee machine has a parallel structure. This method utilizes the results attained by each expert and combines them to reach a solution, which is superior to the results acquired by individual methods (Haykin, 1999; Sharkey, 1996). Committee machines, which are universal approximators, are classified into two major categories (Haykin, 1999): 1. Static structures;

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2. Dynamic structures. The method used here is ensemble averaging, which is a subclass of static structures. In this method, the results of different experts are combined to produce a superior output. After the construction of intelligent networks (experts), it is necessary to find a suitable method to combine the individual results. There are different methods to accomplish this task (Genest and Zidek, 1986; Jacobs, 1995; Xu et al., 1992). One of the popular methods among these is linear averaging. Using this method, combination of experts can be done either by simple averaging (Perrone and Cooper, 1992) or by weighted averaging (Hashem and Schmeiser, 1993; Perrone and Cooper, 1992). It is obvious that the best predictor should have the most contribution, so weighted averaging is preferred. This begs the question how to determine the best corresponding weights for each predictor. It is possible to couple an intelligent optimizer with CM to construct a committee machine intelligent system (CMIS). In order to guarantee the convergence toward actual optimal solution in a finite epoch, global optimization techniques may be used (Safari and Jamialahmadi, 2013). In this study, genetic algorithm was implemented mainly due to its wide applicability, robustness, and universality in solving various large scale optimization problems. Just like other virtual intelligence tools, the roots of Evolutionary computing are in nature. The evolutionary computing tries to mimic the evolutionary process using computer algorithms and instructions. Evolution is an optimization process (Mayr, 1988). Darwin's theory of survival of the fittest (Darwin, 1859), joint with selectionism of Weismann and genetics of Mendel, have constructed the set of arguments known as the evolution theory that is accepted universally (Fogel, 2006). Discovering a variety of zones in the desired area as well as identifying many routs simultaneously and randomly, are the important features of genetic algorithm (Holland, 1992; Rania et al., 2005; Schwefel, 1993). The primary operators of genetic algorithm are crossover, inversion, and mutation (Mohaghegh, 2000). These operators preserve the mandatory balance between exploration (searching new areas) and exploitation (taking advantage of information already obtained). To operate the genetic algorithm, first, an initial population is generated with each indicating a solution of problem. At the next step, the nobility of the each individual is evaluated by a so-called objective function that signifies the constraints of the problem. Next, the population is sorted based on the fitness of each individual. By applying crossover and mutation on the best individual, the next generation is constructed. This process continues until reaching the finishing criteria. Fig. 4 illustrates the flowchart of the applied genetic algorithm. Thus, this task reduces to an optimization problem to find the best weight for the results of each expert by minimizing the MSE of the final output of CMIS and experimental results. MSE of committee machine is defined as follows:

MSECMIS

1 00 1 n m X 1X netj A @ @ ¼ wj  yi þ b  Ti A n i¼1 j¼1

(1)

where n is the number of data, m is the number of experts and also the number of weights to be optimized, w is the weight for each expert, y is the output of each expert, and Ti is the target value. In order to prevent overtraining, we look for the minimum MSE from the total dataset. The diagram of the CMIS is presented in Fig. 5. Fig. 9. Comparison between the cross plots of the developed model for spontaneous imbibition and other well-known correlations. (a) CMIS for spontaneous imbibition, (b) Agarwal (Agarwal, 1967) correlation, (c) Jerauld (Jerauld, 1997) correlation.

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Fig. 10. Comparison between the relative deviation error of the developed model and other well-known correlations. (a) CMIS for spontaneous imbibition, (b) Agarwal (Agarwal, 1967) correlation, (c) Jerauld (Jerauld, 1997) correlation.

First, the optimized weights are determined using train dataset and then the performance is checked by total dataset. The overall performance, i.e. the MSE for the whole dataset is distinctively better than each individual result.

local accumulation of data points. The applied intelligent systems are MLP, RBFN, and LSSVM. After constructing these networks, the CMIS improves the results and provides a superior output. 2. Result and discussion

1.7. Computational procedure 2.1. Data acquisition To apply the proposed intelligent systems, first, the dataset was randomly divided to train dataset and test dataset in ratio of 4:1. Distribution of data to two sub-data should be such that there is no

Acquisition of valid and non-limited data is mandatory to construct a dependable model. Petrophysical parameters such as

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(a) 0.9 Target Values

0.8

Model Predictions

0.7 Sgr Values

0.6 0.5 0.4 0.3 0.2 0.1 0 0

100

200

300

400

500

600

700

Index of Data Points

(b) 0.9 Target Values

0.8

Model Predictions

0.7

Sgr Values

0.6 0.5 0.4 0.3 0.2 0.1 0 0

100

200

300

400

500

600

700

Index of Data Points

(c) 0.8 Target Values Model Predictions

0.7

Sgr Values

0.6 0.5 0.4 0.3 0.2 0.1 0 0

100

200

300 400 Index of Data Points

500

600

700

Fig. 11. Comparison between the experimental data and results of (a) CMIS for spontaneous imbibition, (b) Agarwal (Agarwal, 1967) correlation, (c) Jerauld (Jerauld, 1997) correlation.

porosity, permeability and initial gas saturation are the main factors affecting residual gas saturation values (Bull et al., 2011; Suzanne et al., 2003). Data used in this study are acquired from available data in literature (Bull et al., 2011; Chierici et al., 1963; Delclaud and

Pau, 1991; Ding and Kantzas, 2001, 2002; Firoozabadi et al., 1987; Holtz, 2005; Keelan, 1976; Keelan and Pugh, 1975; McKay, 1974; Suzanne et al., 2003). These data include porosity, absolute permeability, initial gas saturation (Sgi), type of rock and method of

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between input parameters and available experimental data for reasonable prediction of residual gas saturation in spontaneous imbibition and force flood schemes. Because of great flexibility and capability of committee machine intelligent system, this method is chosen to construct the networks. The individual experts, namely, MLP, RBF, and LSSVM networks, which are based on the machine learning approach were trained using four parameters as input using back propagation algorithm to minimize the MSE. Both constructed MLP networks had one hidden layer with 22 and 12 neurons for spontaneous imbibition and force flood, respectively. In this study, the optimum number of neurons was determined through evaluating different networks with 4e25 neurons in the hidden layer. Fig. 7 shows different constructions with MSE values for train and test data set for both spontaneous imbibition and force flood. To construct the RBFN models, the optimum values for controlling parameters of the network, namely, spread and maximum number of neurons, which result in the best performance, were determined by trial and error. The optimum values for spread and maximum number of neurons were determined 0.5 and 200, for both spontaneous imbibition and force flood. LSSVM networks have two tuning parameters, g and s2. Coupled Simulated Annealing (CSA) optimization algorithm was incorporated to find the optimum values for the two mentioned parameters. A typical flowchart of the CSA-LSSVM algorithm is shown in Fig. 8. For the best performance of the LSSVM network, g and s2 was found to be 7.1917 and 0.2286 for spontaneous imbibition and 57.1073 and 8.8147 for force flood. Finally, for the CMIS, the genetic algorithm was applied to find the optimum weights for each expert. The output of CMIS is as follows:

yiCMIS ¼ ayiRBF þ byiMLP þ cyiLSSSVM þ d

(2)

where a, b, c, and d were determined 0.0873, 0.8162, 0.0993, and 0.0021 for spontaneous imbibition and 0.3959, 0.7190, 0.1812, and 0.0011 for force flood. The step-by-step procedure to use the models is presented in Appendix B. 2.3. Model accuracy

Fig. 12. Comparison between the scatter plots of (a) CMIS for force flood, (b) Agarwal (Agarwal, 1967) correlation.

gas displacement by chase water as input parameters to the networks. The details of the utilized data and their sources are listed in Tables 1 and 2. Fig. 6 represents initial gas saturation versus residual gas saturation for all data points used in this study. As it is obvious, this dataset comprises a wide range of variables. The porosity varies from 0.01 to 0.5, absolute permeability from 0.001 to 12,960 mD, initial gas saturation from 0 to 1, and residual gas saturation from 0 to 0.8.

2.2. Model Development and computational procedure A brief review of literature indicates that residual gas saturation is a function of porosity, permeability, initial gas saturation, rock type, and the method of the performed experiment (Ding and Kantzas, 2001; Holtz, 2005; Suzanne et al., 2003). This study aims to construct two mathematical relationships

The accuracy of the proposed models is validated by comparison with two well-known correlations, namely Agarwal (1967) and Jerauld (1997) as presented in Appendix A. Agarwal (1967) correlation was applied for all data points in two class of spontaneous imbibition and force flood. Jerauld (1997) correlation needs maximum residual saturation (Sgrm) and the only data set with this property was the set derived from Suzanne et al. (2003) paper which all were for spontaneous imbibition scheme. To make a reasonable comparison, both graphical and statistical error analyses are utilized. Statistical parameters used are average absolute relative deviation (AARD), root mean square error (RMSE), R2 value and standard deviation error (STD). In order to illustrate the accuracy of the proposed models and former correlations, graphical plots were used. Cross-plots and error distribution plots show the degree of reliability of the models and whether the data has an error trend or not, respectively. First, spontaneous imbibition is investigated. To this aim, three different predictors are presented, which are CMIS, Agarwal (1967) correlation, and Jerauld (1997) correlation. As mentioned above, for using Jerauld (1997) correlation, maximum residual saturation (Sgrm) is needed. Therefore, in the case of spontaneous imbibition, to compare the performance of mentioned correlation with developed CMIS, Jerauld (1997) correlation was applied only on Suzanne et al. (2003) data set, because maximum residual gas

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Fig. 13. Comparison between the relative deviation error of (a) CMIS for force flood, (b) Agarwal (Agarwal, 1967) correlation.

saturation (Sgrm) property reported only for this data set. Needing maximum residual gas saturation (Sgrm) property for Jerauld (1997) correlation is a draw back for this correlation. In Fig. 9 the crossplots of the mentioned predictors are illustrated. As it is obvious

Table 3 Comparison between the performances of proposed model and common correlations for prediction of residual gas saturation. Specification

Method

R2

AARD

STD

RMSE

N (number of data)

Spontaneous

Agarwal (Agarwal, 1967) Jerauld (Jerauld, 1997) RBFN MLP LSSVM CMIS

0.3056

42.7842

0.5339

0.6020

659

0.9299

12.6742

0.1802

0.1917

595

0.9559 0.9420 0.8625 0.9573

10.7518 12.8021 22.8903 11.2524

0.3970 0.3421 0.6687 0.3805

0.3981 0.3439 0.6787 0.3826

655 655 655 655

0.0040

69.5797

0.6760

0.8606

243

0.8125 0.7423 0.5404 0.8404

15.7986 31.1474 62.7211 18.8805

0.2607 2.0467 3.6074 0.6534

0.2608 2.0468 3.6336 0.6535

243 243 243 243

Force

Agarwal (Agarwal, 1967) RBFN MLP LSSVM CMIS

from the data point concentration in close vicinity of 45 line, the proposed CMIS model has the best performance. Although, the Jerauld (1997) correlation provides acceptable results, it needs an experimental parameter Sgrm which is not always available and therefore is not practical. The relative deviation error and deviations of the residual gas values predicted by the model from experimentally measured data are depicted in Figs. 10 and 11, respectively. For force flood, only CMIS and Agarwal (1967) correlation are studied. Fig. 12 illustrates the cross-plot of the mentioned predictors. In this class as the former one, the proposed CMIS model has the best performance, also the results are not as exact as for the spontaneous class, as it far better than the results of Agarwal (1967) correlation. The relative deviation error and deviations of the residual gas values predicted by the model from experimentally measured data are presented in Figs. 13 and 14, respectively. Table 3 shows the statistical parameters for the proposed models and former correlations for the two classes. The used statistical parameters are listed in Appendix C. 2.4. Outlier detection Outliers are observations that show deviation from the bulk of data which all of them are achieved under the same condition (Fazavi et al., 2013; Mesbah et al., 2014; Rousseeuw and Leroy, 2005b; Yassin et al., 2013). It can be said that in all projects

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(a)

1 Target Values

0.9

Model Predictions

0.8

Sgr Values

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

50

100

150

200

250

Index of Data Points

(b) 0.8 Target Values 0.7

Model Predictions

Sgr Values

0.6 0.5 0.4 0.3 0.2 0.1 0 0

50

100

150

200

250

Index of Data Points Fig. 14. Comparison between the experimental data and results of (a) CMIS for force flood, (b) Agarwal (Agarwal, 1967) correlation.

concerning with data collection, there are outliers or doubtful data. In large data sets gathered such that one collected in this study, it is expected to encounter with this problem. Error in data entry is another source of outliers, especially when experimental data are recorded manually (Safari et al., 2014a, 2014b). In order to construct a promising mathematical model, it is mandatory to detect these doubtful points and exclude them in the proposed model. Outliers are categorized in three classes namely regression outliers, outliers with Leverage and influential observations (Rousseeuw and Leroy, 2005b). Regression outliers are data points which are far from the bulk of experimental data that are well within the range of measured values. These outliers have no influence on regression line. As for outliers with Leverage, they are far away from the mean data set, when some observations deviate significantly from the bulk of data point. Although having high Leverage values, these data points do not affect the regression line necessarily and it can be said that they follow the same pattern as the majority of the data points. Despite the two mentioned outlier types, the third type data points are not only out of the range of data points, but also deviated from the dominant pattern of data points. These observations have high Leverages and significantly influence the slope and intercept of regression line and considered as a threat to the successful modeling. For detection of doubtful data in this study, the method of Leverage Value Statistics (Gramatica, 2007; Rousseeuw and Leroy, 2005a) is utilized. Thorough description and formulation of this method is available in literature (Gramatica, 2007; Rousseeuw and Leroy, 2005a). Fig. 15 demonstrates the William's plot for the outputs of proposed models. Existence of the

majority of the data points within the proper ranges is indicative of validity and accuracy of the proposed models. About 11 data points of spontaneous imbibition and 3 data points of force flood seem to be doubtful observations and should be excluded from the models. 3. Conclusion Comprehensive and predictive models based on committee machine intelligent system were developed for prognosticating the residual gas saturation in spontaneous imbibition and force flood schemes. Huge database including 903 recorded data points which are experimental residual gas saturation values from the open literature were used for construction and evaluation of the developed models. In order to see the performance of proposed CMIS models, comparisons were done between predictions of the models developed in this study with experimentally measured residual gas saturation values and also other commonly published empirical correlations namely Agarwal (1967) and Jerauld (1997). Agarwal (1967) correlation was applied for all data points of spontaneous imbibition and force flood. Jerauld (1997) correlation was applied only for spontaneous imbibition scheme. Results of statistical quality measures show that the developed models have better performance among all the compared correlations. Constructed CMIS models can be integrated with commercial software such as ECLIPSE and CMG aiming to improve their precision and reliability while reducing the uncertainties associated with these applications. Results reveal the robustness of CMIS for modeling the residual gas saturation.

A. Tatar et al. / Journal of Natural Gas Science and Engineering 21 (2014) 79e94

(a)

91

12 valid data suspect data

9

high Leverage data suspect limit

6 Standardized Residuals

Leverage Limit 3 0 -3 -6 -9 -12 0

0.005

0.01

0.015

0.02

0.025

Hat

(b)

4 3 valid data suspect data

Standardized Residuals

2

suspect limit Leverage Limit

1 0 -1 -2 -3 -4 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Hat Fig. 15. Detection of the probable doubtful data and the applicability domain of the developed CMIS model for (a) spontaneous imbibition (b) force flood.

These finds also imply that predicting the residual gas saturation is still an active area of research.

Aissaoui (1983) worked on twelve Fontainebleau sandstone plugs and proposed a piecewise linear relationship:

Appendix A. Correlations for estimation of Sgr

if

Some of the most popular correlation for estimation of Sgr has been provided here. Land (Land, 1968, 1971), found a relationship between Swi and Sgr as follows:



1  Swi Sgr

 1¼C

Sgi < Sgr : Sgr ¼

else Sgr ¼ Sgrm

(A2)

Jerauld (1997) worked on fifty Berea and Prudhoe Bay sandstone plugs. The proposed relationship has hyperbolic form with an approximately zero slope at Sgi equal to 1:

(A1)

where C is called the trapping constant and is believed to be a function of pore size distribution. This parameter can be evaluated by experimental data in laboratory.

Sgrm $S Sgo gi

Sgi

Sgr ¼ 1þ





 1 Sgrm

: 1 1Sgrm

$Sgi

(A3)

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Sgrm is maximum residual gas saturation that is measured at maximum initial gas saturation (Sgi ¼ 1). Agarwal (1967) developed correlations for the estimation of residual gas saturations in gas reservoirs under water influx (Ezekwe, 2011). For unconsolidated sandstones:

Sgr ¼

      a1 102 Sgi þ a2 104 FSgi þ a3 102 F þ a4 100

First open the “Sample.xlsx” file that has been provided in “network” folder. After that, the following commands must be entered in the MATLAB (version 2014a) command window:

(A4)

where a1 ¼ 0.51255987, a2 ¼ 0.026097212, a3 ¼ 0.26769575, a4 ¼ 14.796539. For consolidated sandstones:

Sgr

   2 a1 102 Sgi þ a2 102 Sgi ¼ 100

(A5)

where a1 ¼ 0.80841168, a2 ¼ 0.0063869116, For limestones:

Sgr ¼

    a1 102 F þ a2 logðkÞ þ a3 102 Sgi þ a4 100

where a1 ¼ 0.53482234, a4 ¼ 14.403977.

a2 ¼ 3.3555165,

(A6) a3 ¼ 0.15458573,

Appendix B. Instruction for using the model A computer program is organized to use the developed model in this study. First of all, download the supplementary file and change the Matlab® software (Version 2014a) directory to the downloaded folder. To get the response from this program, you can follow the below instruction step by step. 1. Open the “Sample.xlsx” file. There are two sheets namely “Input” and “Output”. Open the “Input” sheet and feel the determined cells. You can enter more than one data sample. Save and close the Excel file. 2. Open the “Run.m” and run. As it will be prompted by Matlab®, for spontaneous imbibition enter “1”, otherwise, for force flood enter “2” and press “Enter” key. 3. Open the “Sample.xlsx” and turn to “Output” sheet. The results for MLP, RBFN, LSSVM, and CMIS are provided in their corresponding columns. To better understand the instruction of using the developed CMIS models, the following example has been provided.

Example: Calculate the residual gas saturation with the following data (Table C1). Type of flood Spontaneous

Rock type Sandstone

Porosity 0.12

Kg (mD) 2.3

Sgi 0.78

Solution: Because the “Rock Type” input is string and we cannot use the string as an input the following indices have been considered for the following rock types.

Rock type indices Sandstone 1

Unconsolidated sandstone 2

Carbonate 3

To simplify running the model, instead of typing the above commands in command window, please run the “Run.m” to calculate residual gas saturation.

A. Tatar et al. / Journal of Natural Gas Science and Engineering 21 (2014) 79e94

The result is 0.314591, where its experimental value is equal to 0.311494 (the relative deviation is 0.1%). Appendix C. Statistical parameters Correlation factor (R2): N P

R ¼1

ðCalc:ðiÞ=Est:ðiÞ  exp:ðiÞÞ2

i

2

N P

(C1)

ðCalc:ðiÞ=Est:ðiÞ  averageðexpðiÞÞÞ2

i

 Average Absolute Relative Deviation (AARD):

%AARD ¼

N 100 X jCalc:ðiÞ=Est:ðiÞ  exp:ðiÞj N exp:ðiÞ

(C2)

i

 Standard Deviation Error (STD)

STD ¼

N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X ðCalc:ðiÞ=Est:ðiÞ  averageðCalc:ðiÞ=Est:ðiÞÞÞ2 N i

(C3)  Root Mean Square Error (RMSE):

PN RMSE ¼

i¼1

! ðCalc:ðiÞ=Est:ðiÞ  expðiÞÞ2 1=2 N

(C4)

Appendix D. Supplementary data Supplementary data associated with this article can be found in the online version, at http://dx.doi.org/10.1016/j.jngse.2014.07.017. References Agarwal, R.G., 1967. Unsteady-state Performance of Water-drive Gas Reservoirs, Other Information (Ph.D. thesis). Texas A and M University, United States, p. 108. Ahmadi, M.A., Shadizadeh, S.R., 2013. Intelligent approach for prediction of minimum miscible pressure by evolving genetic algorithm and neural network. Neural. Comput. Appl. http://dx.doi.org/10.1007/s00521-012-0984-4.  orique et expe rimentale de l'hyste re sis des pressions Aissaoui, A., 1983. Etude the abilite s relatives en vue du stockage souterrain de gaz. capillaires et des perme Ecole des Mines, Paris, p. 223. Al-Hashim, H.S., Bass Jr., D.M., 1988. Effect of aquifer size on the performance of partial waterdrive gas reservoirs. SPE Reserv. Eng. 3, 380e386. Al-Marhoun, M.A., Nizamuddin, S., Raheem, A.A.A., Ali, S.S., Muhammadain, A.A., 2012. Prediction of crude oil viscosity curve using artificial intelligence techniques. J. Pet. Sci. Eng. 86e87, 111e117. Batycky, J., Irwin, D., Fish, R., 1998. Trapped gas saturations in Leduc-age reservoirs. J. Can. Pet. Technol. 37. Baylar, A., Hanbay, D., Batan, M., 2009. Application of least square support vector machines in the prediction of aeration performance of plunging overfall jets from weirs. Expert Syst. Appl. 36, 8368e8374. Bull, Ø., Bratteli, F., Ringen, J.K., Melhuus, K., Bye, A.L., Iversen, J.E., 2011. The quest for the true residual gas saturation e an experimental approach. International Symposium of the Society of Core Analysts, Austin, Texas, USA, pp. 1e12. Chen, T.-S., Chen, J., Lin, Y.-C., Tsai, Y.-C., Kao, Y.-H., Wu, K., 2011. A novel knowledge protection technique based on support vector machine model for anti-classification. In: Zhu, M. (Ed.), Electrical Engineering and Control. Springer, Berlin, Heidelberg, pp. 517e524. Chierici, G.L., Cixci, G.M., Long, G.,1963. Experimental research on gas saturation behind the water front in gas reservoirs subjected to water drive. 6th World Petroleum Congress. World Petroleum Congress, Frankfurt am Main, Germany, pp. 1e16. Cortes, C., Vapnik, V., 1995. Support-vector networks. Mach. Learn 20, 273e297. Crowell, D.C., Dean, G., Loomis, A., 1966. Efficiency of Gas Displacement from a Water-drive Reservoir. US Dept. of the Interior, Bureau of Mines.

93

Darwin, C., 1859. On the Origins of Species by Means of Natural Selection. Murray, London. Dayhoff, J.E., 1990. Neural Network Architectures: An Introduction. Van Nostrand Reinhold. Delclaud, J., 1991. Laboratory measurements of the residual gas saturation. Second European Core Analysis Symposium, London, England, pp. 431e451. Delclaud, J., Pau, E., 20e22 May 1991. Laboratory measurement of the residual gas saturation. Advances in Core Evaluation II: Reservoir Appraisal: Reviewed Proceedings of the Second Society of Core Analysts European Core Analysis Symposium. Taylor & Francis US, London, UK, p. 431. Ding, M., Kantzas, A., 2001. Measurements of residual gas saturation under ambient conditions. Proceedings of SCA Annual Symposium. Ding, M., Kantzas, A., 2002. Residual gas saturation investigation of a carbonate reservoir from Western Canada. SPE Gas Technology Symposium, Calgary, Alberta, Canada. Du, K.-L., Swamy, M.N.S., 2006. Radial Basis Function Networks, Neural Networks in a Softcomputing Framework. Springer, London, pp. 251e294. El Ouahed, A.K., Tiab, D., Mazouzi, A., 2005. Application of artificial intelligence to characterize naturally fractured zones in Hassi Messaoud Oil Field, Algeria. J. Pet. Sci. Eng. 49, 122e141. Eslamimanesh, A., Gharagheizi, F., Illbeigi, M., Mohammadi, A.H., Fazlali, A., Richon, D., 2012. Phase equilibrium modeling of clathrate hydrates of methane, carbon dioxide, nitrogen, and hydrogen þ water soluble organic promoters using support vector machine algorithm. Fluid Phase Equilib. 316, 34e45. Ezekwe, N., 2011. Petroleum reservoir engineering practice. Pearson Education. Fayazi, A., Arabloo, M., Shokrollahi, A., Zargari, M.H., Ghazanfari, M.H., 2013. Stateof-the-art least square support vector machine application for accurate determination of natural gas viscosity. Ind. Eng. Chem. Res. 53, 945e958. Fazavi, M., Hosseini, S.M., Arabloo, M., Shokrollahi, A., Nouri-Taleghani, M., Amani, M., 2013. Applying a smart technique for accurate determination of flowing oil/water pressure gradient in horizontal pipelines. J. Dispersion Sci. Technol. 35, 882e888. Firoozabadi, A., Olsen, G., van Golf-Racht, T., 1987. Residual gas saturation in waterdrive gas reservoirs. SPE California Regional Meeting. Society of Petroleum Engineers, Ventura, California. Fogel, D.B., 2006. Evolutionary Computation: Toward a New Philosophy of Machine Intelligence. Wiley-IEEE Press. Geffen, T.M., Parrish, D.R., Haynes, G.W., Morse, R.A., 1952. Efficiency of gas displacement from porous media by liquid flooding. J. Pet. Technol. 4, 29e38. Genest, C., Zidek, J.V., 1986. Combining probability distributions: a critique and an annotated bibliography. Stat. Sci. 1, 114e135. Girosi, F., Poggio, T., 1990. Networks and the best approximation property. Biol. Cybern. 63, 169e176. Gramatica, P., 2007. Principles of QSAR models validation: internal and external. QSAR Comb. Sci. 26, 694e701. Hamon, G., Suzanne, K., Billiotte, J., Trocme, V., 2001. Field-wide variations of trapped gas saturation in heterogeneous sandstone reservoirs. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, New Orleans, Louisiana. Hao, Y., Tiantian, X., Paszczynski, S., Wilamowski, B.M., 2011. Advantages of radial basis function networks for dynamic system design. IEEE Trans. Ind. Electron. 58, 5438e5450. Hashem, S., Schmeiser, B., 1993. Approximating a function and its derivatives using MSE-optimal linear combinations of trained feedforward neural networks. Citeseer. Haykin, S., 1994. Neural Networks: A Comprehensive Foundation. Prentice Hall PTR. Haykin, S., 1999. Neural Networks: A Comprehensive Foundation. Pearson Education, Inc. Hemmati-Sarapardeh, A., Shokrollahi, A., Tatar, A., Gharagheizi, F., Mohammadi, A.H., Naseri, A., 2014. Reservoir oil viscosity determination using a rigorous approach. Fuel 116, 39e48. Holland, J.H., 1992. Adaptation in Natural and Artificial Systems, second ed. MIT Press, Cambridge, MA. Holtz, M.H., 2002. Residual gas saturation to aquifer influx: a calculation method for 3-D computer reservoir model construction. SPE Gas Technology Symposium. Society of Petroleum Engineers Inc., Calgary, Alberta, Canada. Holtz, M.H., 2005. Reservoir Characterization Applying Residual Gas Saturation Modeling, Example from the Starfak T1 Reservoir, Middle Miocene Gulf of Mexico. University of Texas, Texas, Austin. Huang, S.H., Hong-Chao, Z., 1994. Artificial neural networks in manufacturing: concepts, applications, and perspectives. IEEE Transactions on Components, Packaging, and Manufacturing Technology Part A 17, 212e228. Jacobs, R.A., 1995. Methods for combining experts' probability assessments. Neural Comput. 7, 867e888. Jerauld, G.R., 1997. Prudhoe bay gas/oil relative permeability. SPE Reserv. Eng. 12, 66e73. Karri, V., 1999. RBF neural network for thrust and torque predictions in drilling operations. Computational intelligence and Multimedia applications, 1999. ICCIMA'99. Proceedings. Third International Conference on. IEEE, pp. 55e59. Keelan, D.K., 1976. A practical approach to determination of imbibition gasewater relative permeability. J. Pet. Technol. 28, 199e204. Keelan, D.K., Pugh, V.J., 1975. Trapped-gas saturations in carbonate formations. Soc. Pet. Eng. J. 15, 149e160. Knapp, R.M., Henderson, J.H., Dempsey, J.R., Coats, K.H., 1968. Calculation of gas recovery upon ultimate depletion of aquifer storage. J. Pet. Technol. 20, 1129e1132.

94

A. Tatar et al. / Journal of Natural Gas Science and Engineering 21 (2014) 79e94

Land, C.S., 1968. Calculation of imbibition relative permeability for two- and threephase flow from rock properties. SPE J. 8, 149e156. Land, C.S., 1971. Comparison of calculated with experimental imbibition relative permeability. Soc. Pet. Eng. J. 11, 419e425. Li, K., Firoozabadi, A., 2000. Experimental study of wettability alteration to preferential gas-wetting in porous media and its effects. SPE Reserv. Eval. Eng. 3, 139e149. Lowe, D., Broomhead, D., 1988. Multivariable functional interpolation and adaptive networks. Complex Syst. 2, 321e355. Majidi, S.M.J., Shokrollahi, A., Arabloo, M., Mahdikhani-Soleymanloo, R., Masihi, M., 2014. Evolving an accurate model based on machine learning approach for prediction of dew-point pressure in gas condensate reservoirs. Chem. Eng. Res. Des. 92, 891e902. Mayr, E., 1988. Toward a New Philosophy of Biology: Observations of an Evolutionist. Harvard University Press. McKay, B., 1974. Laboratory studies of gas displacement from sandstone reservoirs having strong water drive. APEA J. 1974, 189e194. Mesbah, M., Soroush, E., Shokrollahi, A., Bahadori, A., 2014. Prediction of phase equilibrium of CO2/cyclic compound binary mixtures using a rigorous modeling approach. J. Supercrit. Fluids 90, 110e125. Mohaghegh, S., 2000. Virtual-intelligence applications in petroleum engineering: part 2devolutionary computing. J. Pet. Technol. 52, 40e46. Nilsson, N.J., 1965. Learning Machines: Foundations of Trainable Pattern Classification Systems. Me Graw-Hill, New York, USA. Pelckmans, K., Suykens, J.A., Van Gestel, T., De Brabanter, J., Lukas, L., Hamers, B., De Moor, B., Vandewalle, J., 2002. LS-SVMlab: A Matlab/C Toolbox for Least Squares Support Vector Machines. Tutorial. KULeuven-ESAT, Leuven, Belgium. Perrone, M.P., Cooper, L.N., 1992. When networks disagree: ensemble methods for hybrid neural networks. DTIC Document. Poggio, T., Girosi, F., 1990. Networks for approximation and learning. Proc. IEEE 78, 1481e1497. Powell, M.J.D., 1987. Radial basis functions for multivariable interpolation: a review. In: Mason, J.C., Cox, M.G. (Eds.), Algorithms for Approximation. Clarendon Press, pp. 143e167. Rania, H., Babak, C., Olivier de, W., Gerhard, V., 2005. A comparison of particle swarm optimization and the genetic algorithm. 46th AIAA/ASME/ASCE/AHS/ ASC structures, structural Dynamics and Materials Conference. American Institute of Aeronautics and Astronautics, Austin, Texas, pp. 18e21. Rezaee, M., Rostami, B., Pourafshary, P., 2013. Heterogeneity effect on non-wetting phase trapping in strong water drive gas reservoirs. J. Nat. Gas. Sci. Eng. 14, 185e191. Rousseeuw, P.J., Leroy, A.M., 2005a. Robust Regression and Outlier Detection. John Wiley & Sons, Inc. Rousseeuw, P.J., Leroy, A.M., 2005b. Robust Regression and Outlier Detection. Wiley. Safari, H., Jamialahmadi, M., 2013. Estimating the kinetic parameters regarding barium sulfate deposition in porous media: a genetic algorithm approach. AsiaPacific J. Chem. Eng. 9 (2), 256e264. Safari, H., Shokrollahi, A., Jamialahmadi, M., Ghazanfari, M.H., Bahadori, A., Zendehboudi, S., 2014a. Prediction of the aqueous solubility of BaSO4 using pitzer ion interaction model and LSSVM algorithm. Fluid Phase Equilib. 374, 48e62. Safari, H., Shokrollahi, A., Moslemizadeh, A., Jamialahmadi, M., Ghazanfari, M.H., 2014b. Predicting the solubility of SrSO4 in NaeCaeMgeSreCleSO4eH2O system at elevated temperatures and pressures. Fluid Phase Equilib. 374, 86e101. Santos, R.B., Ruppb, M., Bonzi, S.J., Filetia, A.M.F., 2013. Comparison between multilayer feedforward neural networks and a radial basis function network to detect and locate leaks in pipelines transporting gas. Chem. Eng. 32, 1375e1380. Schwefel, H.-P.P., 1993. Evolution and Optimum Seeking: the Sixth Generation. John Wiley & Sons, Inc., New York. Sharkey, A.J.C., 1996. On combining artificial neural nets. Connect. Sci. 8, 299e314. Shmilovici, A., 2010. Support vector machines. In: Maimon, O., Rokach, L. (Eds.), Data Mining and Knowledge Discovery Handbook. Springer, USA, pp. 231e247. Shokrollahi, A., Arabloo, M., Gharagheizi, F., Mohammadi, A.H., 2013. Intelligent model for prediction of CO2 e reservoir oil minimum miscibility pressure. Fuel 112, 375e384. Suykens, J.A.K., Vandewalle, J., 1999. Least squares support vector machine classifiers. Neural Process. Lett. 9, 293e300.

Suzanne, K., Hamon, G., Billiotte, J., Trocme, V., 2003. Experimental relationships between residual gas saturation and initial gas saturation in heterogeneous sandstone reservoirs. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Denver, Colorado. Tatar, A., Shokrollahi, A., Mesbah, M., Rashid, S., Arabloo, M., Bahadori, A., 2013. Implementing radial basis function networks for modeling CO2-reservoir oil minimum miscibility pressure. J. Nat. Gas. Sci. Eng. 15, 82e92. Übeyli_, E.D., 2010. Least squares support vector machine employing model-based methods coefficients for analysis of EEG signals. Expert Syst. Appl. 37, 233e239. Van Everdingen, A., Hurst, W., 1949. The application of the Laplace transformation to flow problems in reservoirs. J. Pet. Technol. 1, 305e324. Vapnik, V., 1999. The Nature of Statistical Learning Theory. Springer. Wardlaw, N.C., 1980. The effects of pore structure on displacement efficiency in reservoir rocks and in glass micromodels. SPE/DOE Enhanced Oil Recovery Symposium. Society of Petroleum Engineers, Tulsa, Oklahoma. Wilamowski, B.M., Jaeger, R.C., 1996. Implementation of RBF type networks by MLP networks, Neural Networks, 1996. IEEE International Conference on, vol. 1673, pp. 1670e1675. Xu, L., Krzyzak, A., Suen, C.Y., 1992. Methods of combining multiple classifiers and their applications to handwriting recognition. IEEE Trans. Syst. Man Cybern. 22, 418e435. Yao, J., Zhao, S., Fan, L., 2006. An enhanced support vector machine model for intrusion detection. In: Wang, G.-Y., Peters, J., Skowron, A., Yao, Y. (Eds.), Rough Sets and Knowledge Technology. Springer, Berlin, Heidelberg, pp. 538e543. Yassin, M.R., Arabloo, M., Shokrollahi, A., Mohammadi, A.H., 2013. Prediction of surfactant retention in porous media: a robust modeling approach. J. Dispersion Sci. Technol., 1407e1418. Zurada, J.M., 1992. Introduction to Artificial Neural Systems. West.

Nomenclature

Abbreviations AARD: average absolute relative deviations ANN: artificial neural network CMIS: committee machine intelligent system GA: genetic algorithm MLP: multilayer perceptron N: number of data R2: correlation coefficient RBFN: radial basis function network RMSE: root mean square errors STD: standard deviation error WDGR: water drive gas reservoir Variables C: trapping constant K: absolute permeability (mD) Sgi: initial gas saturation (fraction) Sgr: residual gas saturation (fraction) Sgrm: maximum residual gas saturation (fraction) Swi: initial water saturation (fraction) jj.jj: Euclidean Norm jj.jjF: Frobenius Norm yi(x): output of the network Q: adjustable bias S: parameter that controls the smoothness of the interpolating function F: porosity (fraction) 4(r): nonlinear activation function 4(.): Gaussian function 40(r): constant activation function