Comparison of neurofuzzy network and response surface methodology pertaining to the viscosity of polymer solutions
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Abstract
This study has utilized the response surface methodology (RSM) and adaptive neurofuzzy inference system (ANFIS) approaches for the modeling of polymer solution viscosity. In the absence of reports in the previous study on applying these two approaches, the main objective of this study has been to compare the performance of these methods toward the viscosity modeling of a polymer solution. By utilizing RSM technique, the effects of three independent parameters including shear rate, polymer concentration, and sodium chloride concentration on viscosity of polymer solution were examined. The RSM results showed that all the parameters were not equally important in the polymer solution viscosity. Moreover, analysis of variance (ANOVA) was also carried out and indicated that there was no evidence of lack of fit in the RSM model. As a second approach for polymer solution viscosity modeling, ANFIS was utilized with two rules constructed based on the firstorder Sugeno fuzzy approach and trained by back propagation neural networks algorithm. High coefficient of determination (R ^{2}) values ( >99%) showed that the prediction ability of both the ANFIS and RSM models was good enough for the response when the interpolation ability of the models was considered. In order to evaluate the extrapolation abilities of the two developed models, two data sets lying beyond the originally considered data were also taken into account. The results showed that their extrapolation predictive ability was poor. The reason could simply be the inherent behavior of the polymeric solution, i.e., the correlational structure seen in the sample used in the training step did not continue outside the sample space.
Keywords
HPAM Adaptive neurofuzzy inference system (ANFIS) Polymer solution viscosity Response surface methodology (RSM)Introduction
A main target in the enhanced oil recovery (EOR) process is the economic improvement of the displacement efficiency. The efficiency of the EOR process is controlled by a fundamental factor, known as mobility ratio, which is a function of relative permeability and viscosity of displacing and displaced fluids (Sheng 2011; Sorbie 1991). The main role of injecting water in water flooding process is for the sake of maintaining pressure. Because the viscosity of injected water is often much lower than that of oil, however, it may cause unstable displacement. Increasing the injected water viscosity by addition of a polymer effectively reduces the mobility ratio and results in higher Buckley–Leverett front heights and a more pistonlike displacement which ultimately leads to higher recovery efficiency (Sorbie 1991).
In the current polymer flooding process, polyacrylamide polymers with some degree of hydrolysis (HPAM) are the most commonly used polymer (Jung et al. 2013; Sheng 2011). HPAM is formed by replacing some of the acrylamide group in the polyacrylamide polymer with the carboxylic group in the hydrolyzation process. In this synthetic polymer, negative charges have been spread along the chain as a result of presenting the carboxylic group. Eventually, by using this polymer, a chain extension which leads to high solution viscosity is attained due to repulsion between those negative charges.
It has been reported that Carreau equation is more powerful in providing a better fit to the viscosity/shear rate data (Bird and Hassager 1987; Chauveteau 1982); however, compared to the two parameters in power law model, it requires four. This may result in much more complication in a definite analytical calculation which deals with viscosity function (Sheng 2011; Sorbie 1991).
Even though the abovediscussed models are powerful in terms of prediction ability, they are only useful for a specific case of reservoir application. To summarize, these models consider the impact of shear rate only while keeping other parameters in a permanent specific value. According to (Levitt and Pope 2008), other process variables, such as brine salinity and polymer concentration, also have a pronounced effect on viscosity of polymer solution.
As an example, the salinity has a significant effect on the viscosity of the polymer solution. The reason is that negative charges along the polymer chain will be screened by the existing monovalent cations in the solution. Then, the screening results in lowering the polymer solution viscosity due to coiling up of polymer chains (Jang et al. 2015; Lee et al. 2009).
One of the other factors with a significant and eminent role is the polymer concentration in the solution. In fact, it has an important role not only in economic matters but also in the determination of the polymer solution viscosity. Overall, a higher solution viscosity will be attained by providing a higher polymer concentration for almost all polymers. This is due to the fact that interaction among the polymer molecules will be increased by adding more polymer into the solution (Rashidi et al. 2010).
The shortcomings of the abovestated models have been recently taken into account by some researchers and have come up with empirical models that can consider the impacts of most process variables (Cheng et al. 2012; Hashmet et al. 2013a, b). For instance, Hashmet et al. (2013a, b) developed an empirical model which is able to predict the viscosity of a polymer solution as a function of different process variables, such as salinity, temperature, degree of hydrolysis. Unfortunately, this empirical model has been developed based on the method of studying the effect of one variable at a time (OVAT). The method of OVAT is a wasteful use of resources and is not capable of identifying the presence of quantifying the interaction among variables. Considering the existence of interaction among the variables, the main effect of one variable on the response is not consistent in the studied range of parameters. In other words, the effect of one variable on the response will be increased or decreased by the level of another variable (Mathews 2005).
By reviewing such a wide range of conditions reported in the previous study, it is clear that in the viscosity modeling of a polymer solution, determination and quantifying individual effects of each input parameter on the viscosity are essential (Sorbie 1991). In developing such a model, the previous study review has indicated that the response surface methodology (RSM) can be a good candidate (Nguyen et al. 2015; Panjalizadeh et al. 2015; Zhu et al. 2015). In this technique in order to develop a regression (empirical) model, a collection of mathematical and statistical techniques is used. And after a specific design of experiment, the goal is defined to minimize an output variable (prediction error) that is affected by several input variables. Runs, a series of tests, are designed in order to carefully asses the response (model output) as a function of changes in input variables. RSM uses a minimal amount of experimental data to develop a model for different industrial processes with high level of accuracy (Box and Wilson 1951). RSM is recognized to model the effect of different input process variables, both main and interaction effects, on the process output variable (Box et al. 2005).
Nowadays, a combination of fuzzy logic and neural network, the socalled neurofuzzy technique (FNN) developed by Jang (1993, 1996), Jang and Sun (1995), and JyhShing Roger (1996), has been also employed as a predictive tool in a wide range of disciplines, including engineering. The main reason for using this technique is the accurately finding relationships between the parameters of input and output even for nonlinear functions due to its ability to employ learning algorithms (Aghbelagh et al. 2013; Aïfa et al. 2014; Esmaeilnezhad et al. 2013; Pilkington et al. 2014). Applying fuzzy technique alone has this main drawback that rules should be written by an expert. However, neurofuzzy technique as an extension of fuzzy systems has this advantage over fuzzy that input–output relation, in the form of rules, for the presented data, is created automatically. To do so, algorithms of artificial neural networks are utilized to tune system parameters for minimizing the error of the neurofuzzy system.
The principal objective of this current study has been to compare the RSM and FNN techniques toward predicting the viscosity of HPAM solutions. A comparison has been performed for both extrapolation purposes and interpolation. The focus of this study is on evaluating the effect of only three input parameters (shear rate, polymer concentration, and salinity) on the viscosity of polymer solution.
The remainders of this paper have been constructed as follows. Firstly, it discusses the experimental work that was carried to extract the data set. The next step is presented in which modeling of the polymer solution viscosity was carried out along with a complete explanation of the procedure. Presented next are the prediction accuracies of the models which were then evaluated using several statistical indices, such as the coefficient of determination (R ^{2}) and root mean square error (RMSE). Analysis of variance (ANOVA) in the matter of RSM was employed to check any significant lack of fit. In the next step, extrapolation capabilities of both developed models, RSM and ANFIS models, were evaluated for unseen data. After performance evaluation of the developed models, the paper ended up with the main conclusions.
Materials and methods
Characterization of HPAM
The commercial polymer used in this study was partially hydrolyzed polyacrylamide (HPAM) with 5 mol. % hydrolysis provided in powder form by SNF Floerger. The polymer had an approximate molecular weight of 8–9 million Daltons (g/mol). Information about degree of hydrolysis and molecular weight was supplied by the manufacturer.
Experimental design
The input parameters of the shear rate (X _{1}), polymer concentration (X _{2}), and NaCl concentration (X _{3}) were investigated for their effects on the polymer solution viscosity. The range of the shear rate, polymer concentration, and NaCl concentration investigation was chosen to be 4–56 (s^{−1}), 700–3300 (ppm), and 2.75–22.25 (g/L), respectively. In fact, these ranges were chosen purposely and the reason is that the polymer solution has a remarkable change in its behavior outside of these ranges. Therefore, this phenomenon was planned to be used to evaluate the accuracy of the developed models in the extrapolation step.
In this current study, a specific design of the RSM, which is called central composite design (CCD), has been employed to determine the experimental conditions because the required complexity of the model was unknown for accurate predictions to be made. Thus, this design allows for a larger range of conditions to be checked compared to other RSM techniques (Box et al. 2005).
The order and extraction conditions for the central composite design (CCD) including the coded levels of each parameter
Run no.  Shear rate (s^{−1})  Polymer concentration (ppm)  NaCl concentration (g/L) 

1  30(0)  2000(0)  22.25(1.3) 
2  4(−1.3)  2000(0)  12.5(0) 
3  30(0)  2000(0)  12.5(0) 
4  30(0)  2000(0)  12.5(0) 
5  50(1)  1000(−1)  5(−1) 
6  50(1)  1000−1)  20(1) 
7  50(1)  3000(1)  5(−1) 
8  10(−1)  1000(−1)  20(1) 
9  30(0)  2000(0)  12.5(0) 
10  10(−1)  3000(1)  20(1) 
11  50(1)  3000(1)  20(1) 
12  10(−1)  3000(1)  5(−1) 
13  30(0)  3300(1.3)  12.5(0) 
14  56(1.3)  2000(0)  12.5(0) 
15  30(0)  2000(0)  12.5(0) 
16  30(0)  2000(0)  12.5(0) 
17  10(−1)  1000(−1)  5(−1) 
18  30(0)  2000(0)  2.75(−1.3) 
19  30(0)  700(−1.3)  12.5(0) 
20  30(0)  2000(0)  12.5(0) 
Neurofuzzy
Neural networks and fuzzy logic are naturally complementary implements in constructing intelligent systems. While neural networks are lowlevel computational structures that are carried out well when dealing with raw data (Jang 1993; Jang and Sun 1995), fuzzy logic handles reasoning on a higher level, using linguistic information provided by field experts (Jang 1996). Nevertheless, fuzzy systems suffer from a lack of learning ability and are not able to regulate themselves to a new condition. Besides, although neural networks are able to learn, they are not clear to the client. Fuzzy logic and neural network are combined together to make a hybrid intelligent system known as a neurofuzzy system. Integrated neurofuzzy systems can unite with the humanlike knowledge representation and clarification abilities of fuzzy systems with the parallel calculation and learning abilities of neural networks (Jang and Sun 1995).
Adaptive neurofuzzy inference system (ANFIS)

Fist rule: If x _{ 1 } is A _{ 1 } and x _{ 2 } is B _{ 1 }, then y _{ 1 } = a _{ 1 } x _{ 1 } + b _{ 1 } x _{ 2 } + c _{ 1 }

Second rule: If x _{ 1 } is A _{ 2 } and x _{ 2 } is B _{ 2 }, then y _{ 2 } = a _{ 1 } x _{ 1 } + b _{ 1 } x _{ 2 } + c _{ 1 }
First layer: Neurons in the input layer, the first layer, just send external crisp signals to the subsequent layer.
Second layer: Neurons in the second layer perform fuzzification.
ANFIS applied to polymer solution viscosity predicting
The fuzzy inference system structure in this current study was generated from the obtained data using subtractive clustering for modeling of the polymer solution viscosity. In order to implement the ANFIS approach, a type of command line function of MATLAB, ‘genfis2,’ was used and sufficiently trained to find the best ANFIS model. The Gaussian membership function, as a membership function, was utilized in this study for its good performance. This type of ‘genfis2’ does not deliver the same quality of training data results as does test data except that the radii value, a training option parameter, is selected small which may result in overfitting problems (Jang 1993). To discover the best ANFIS model, having a tolerable error value for both training and test data sets, the training options including radii value must change a great deal to reach a balance point between overfitting and underfitting scenarios.
Best ANFIS model
Type of genfis  ‘genfis2’ 

Inputs/outputs  [3 1] 
Number of input membership functions  [2 2 2] 
Number of output membership functions  9 
Number of rules  2 
Output membership function  Linear 
Input membership function  gaussian 
raddi  1.201 
Epoch  12 
Initial mu  0.6100 
Mu decrease factor  0.8 
Mu increase factor  9 
Experimental procedure
Bulk solution
Sodium chloride (NaCl) solutions of different concentrations ranging from 2.75 to 22.25 (g/l) in water were used as solvents. The temperature here was constant at 24 °C.
Solution preparation
Brine was prepared by dissolving the required weight of NaCl in demineralized water on a massbalance basis, and it was stirred by using a magnetic stirrer for 15 min. Then, it was stirred at 720 (rpm) until the creation of a vortex was achieved; at the same time, the polymer was added gently to the vortex to avoid agglomeration of the polymer particles. The beaker was covered with an aluminum foil to prevent contact with air. Then, after 30 min, the stirrer speed was decreased to 350 (rpm) and it was further stirred for at least one day to guarantee the complete hydration of the polymer and also to gain a homogeneous polymer solution. Moreover, a standard polymer solution with a concentration of 5000 ppm was prepared. The standard sample was diluted to obtain any desired concentration of polymer in a range from 700 to 3300 ppm.
Viscosity measurement
In order to perform the viscosity measurement, a rotary rheometer, Malvern Bohlin Gemini II, was utilized. The apparent viscosity of prepared polymer solution was measured at 24 °C by changing shear rates from 1 to 100 (s^{−1}) and vice versa. It is noteworthy to mention that no shear rate hysteresis was detected for any of the solutions.
Results and discussion
The result of the RSM model
Table 3 represents coefficient values for the quadratic model for the best fit. The model p_values confirm that all the three design variables had substantial main effects and/or twofactor interactions. Moreover, the quadratic terms were statistically insignificant or marginal except for the polymer concentration quadratic term. Practically, from a higher Fisher’s F test values and lower p_values, a higher significance of each term will be achieved. It can be observed from Table 3 that all of the terms in the developed quadratic model were statistically significant (p_value <0.05) in determining the polymer solution viscosity, and the regression analysis was predominantly linear with respect to the NaCl concentration and shear rate. The quadratic terms that did not show statistical significance were the NaCl concentration*NaCl concentration (p_value = 0.4999) and shear rate*shear rate (p_value = 0.3715). On the other hand, the interaction terms’ p_values (0.005, 0.01, and 007) demonstrate that they had remarkable effects on the polymer solution viscosity and ignoring them would introduce errors into the model prediction outcomes. To judge the goodness of the model fit to the data, the model was also evaluated by examining the lack of fit using ANOVA, with the results shown in Table 4. The relatively small Fisher’s F value and corresponding large probability (p_value = 0.6568) value indicate that there was no evidence of lack of fit in the quadratic model.
Regression analysis of the response surface methodology (RSM) model for the viscosity of polymer solution, with the associated statistical significance of each coefficient
Coefficient  Value  F value  p_value 

\(\varvec{\beta}_{0}\)  0.843792  –  – 
\(\varvec{\beta}_{{\varvec{X}_{1} }}\)  −0.03878  50.68892  <.0001 
\(\varvec{\beta}_{{\varvec{X}_{2} }}\)  0.281959  2679.731  <0.0001 
\(\varvec{\beta}_{{\varvec{X}_{3} }}\)  −0.07648  197.1351  <0.0001 
\(\varvec{\beta}_{{\varvec{X}_{1} \varvec{X}_{1} }}\)  0.006638  0.875249  0.3715 
\(\varvec{\beta}_{{\varvec{X}_{2} \varvec{X}_{2} }}\)  −0.02761  15.13966  0.0030 
\(\varvec{\beta}_{{\varvec{X}_{3} \varvec{X}_{3} }}\)  0.004967  0.489969  0.4999 
\(\varvec{\beta}_{{\varvec{X}_{1} \varvec{X}_{2} }}\)  −0.0231  12.64037  0.0052 
\(\varvec{\beta}_{{\varvec{X}_{1} \varvec{X}_{3} }}\)  0.019426  8.941567  0.0136 
\(\varvec{\beta}_{{\varvec{X}_{2} \varvec{X}_{3} }}\)  −0.02171  11.16972  0.0075 
Analysis of variance (ANOVA) to determine the suitability of the developed quadratic model in fitting the experimental data
Source  Sum of squares  Degree of freedom  F value  p_value 

Residual  0.0033  10  –  – 
Lack of fit  0.0013  5  0.6833  0.6568 
Pure error  0.0020  5  –  – 
The result of the ANFIS model
The experimentally obtained viscosity of polymer solution compared to that predicted by the associated RMS and ANFIS models
Run no.  Experimental viscosity*  RSM predicted  ANFIS prediction 

1  0.739968  0.75  0.734878535 
2  0.890756  0.899  0.911929147 
3  0.838849  0.849  0.850141399 
4  0.819544  0.849  0.850141399 
5  0.568202  0.562  0.559675773 
6  0.491362  0.491  0.493497656 
7  1.11059  1.123  1.115874493 
8  0.50515  0.483  0.480040713 
9  0.826075  0.849  0.850141399 
10  1.053078  1.05  1.055601774 
11  0.973128  0.965  0.97237299 
12  1.294466  1.285  1.294892926 
13  1.170262  1.173  1.166182973 
14  0.815046  0.799  0.814067544 
15  0.869232  0.849  0.850141399 
16  0.863323  0.849  0.850141399 
17  0.633468  0.632  0.655344894 
18  0.960185  0.948  0.95634432 
19  0.419791  0.44  0.430370531 
20  0.851258  0.849  0.850141399 
Decreasing the apparent viscosity due to shear rate increment demonstrated the behavior of pseudoplastic fluids. The macromolecules orientation along the flow stream line is a good justification for the observed shear thinning behavior of the solution as has been highlighted by different researchers (Ait‐Kadi et al. 1987; Ballard et al. 1988; Carreau 1972; Chagas et al. 2004; Ghannam 1999).
By adding more polymer into the solution for all polymers, interaction among polymer chains will increase and solution states will be changed from dilute region to semidilute and concentrated regions which means higher viscosity. Results are in line previous observations (Rashidi et al. 2010; Sorbie 1991). Besides, Fig. 5b shows that viscosity of polymer solutions was reduced by increasing the shear rate and NaCl concentration. These findings also were in line with previous studies (Ait‐Kadi et al. 1987; Chagas et al. 2004; Rashidi et al. 2010). By adding NaCl into the polymer solution, the negative charges distributed along the polymer chain were shielded by Na^{+} ions and eventually molecules coiled up which led to a lower viscosity for polymer solution (Ait‐Kadi et al. 1987; Chagas et al. 2004; Rashidi et al. 2010). Last but not least, it is projected in Fig. 5c that the higher polymer concentration can be achieved if the polymer and NaCl concentrations set to highest and lowest possible values, respectively.
Comparison of RSM and ANFIS
Developing an ANFIS model may be more complicated than an RSM model. After running experiments based on the designed orthogonal array, and generating a data set, the RSM model was able to be constructed by running a valid regression analysis on the data set. However, the ANFIS model could only be generated by tuning several training options, such as raddi, initial Mu, Mu increase factor, and Mu decrease factor, which was a timeconsuming stage. As an example, choosing a small raddi led to an overfitting problem, which was a good fit for the training data set but a bad fit for the test data, whereas choosing a large raddi value led to an underfitting problem, which was a poor fitting for both the training and testing data sets. In the absence of a general and precise method to achieve the optimum value of raddi, trial and error was the only option. Finding the optimum value was not a necessity only for the raddi parameter but also for other training option parameters.
Extrapolation evaluation
The outcomes obtained from the extrapolation step revealed that the models in predicting polymer solution viscosity were not successful. Figure 8a indicates that a good matching for the polymer study with the 5000ppm concentration has not been achieved with R ^{2} and MAPE as the quantitative indices were observed to be 0.6 and 21.8% for the RSM and 0.4 and 20.8% ANFIS models, respectively. To be more specific, the evaluation procedure was repeated for the 500ppm polymer concentration and the results have been demonstrated in Fig. 8(b). Similar to the 5000 ppm, there was a poor matching with R ^{2} and MAPE values of −7.4849 and 15.5% for RSM and −18.98 and 19.28% for ANFIS being achieved.
The observed results for the RSM model were not far from expectation, but for ANFIS they were. On one side, it is widely accepted by the researchers that RMS, which is a polynomial model, cannot be applied for extrapolation purposes since the approximation is local and restricted to the region of the parameter values (Pilkington et al. 2014). On the other side, a number of researchers have claimed that the ANFIS model is efficient enough to be used for extrapolation purposes (AlGhamdi and Taylan 2015) which was certainly not correct here.
In the previous study, two main reasons have been addressed for model prediction failure in extrapolation investigations: the overfitting problem of the model and unusual behavior of the process outside of the training data set. An overfitting problem in the RSM model was not observed because of no evidence of lack of fit in the quadratic model (p_value = 0.6568). In addition, for the ANFIS model, good prediction ability for unseen data, presented to the model as test data, confirmed that the model was clear from any overfitting problem.
Conclusion

A central composite design (CCD), as a specific design of RSM, was chosen to determine the experimental conditions. Regarding the prediction of the polymer solution viscosity in the design ranges, a high coefficient of determination (R ^{2}) value of 99.60% was obtained for the RSM model. This showed the high accuracy of the RSM model.

In this study, ANFIS as another approach for modeling has also been considered. The fuzzy inference system structure was generated from the obtained data using subtractive clustering. The Gaussian membership function was used in this study for its good performance. The optimum value of the raddi, one of the main training options, was determined to be 1.201, which led to a reasonable error value for both the training and test data sets and can provide a balance between overfitting and underfitting scenarios. Regarding the prediction of the polymer solution viscosity in the design ranges, a high R ^{2} value of 99.61% was obtained, which showed the high accuracy of the ANFIS model.

Generally, it can be concluded that the accuracies of RSM and ANFIS were almost same and they presented interesting results in the design ranges. Besides R ^{2} values, the root mean square error (RMSE), mean absolute percentage error (MAPE), and sum of squared errors of prediction (SSE) values for each model confirmed that the models have almost the same superiority in terms of predicting the polymer solution viscosity. These results have also been supported by almost the same small residual ranges. The residual range was from −0.46 to 0.42 for the RSM model, whereas the range was from 0.48 to 0.318 for the ANFIS model.

Based on to the RSM and ANFIS results, all the independent parameters are not equally important in polymer solution viscosity. The polymer concentration, shear rate, and NaCl concentration are the most important factors in the determination of the polymer solution viscosity, respectively.

Extrapolation results showed that although RSM and ANFIS are two powerful tools to be used in design ranges, this does not necessarily mean that they also can be used for extrapolation purposes.
A good match with the actual data was achieved by developing the RSM and ANFIS models. To develop such a model is no need for any human experience or theoretical knowledge during the learning process, which is the main feature of these two models; furthermore, good fitting results have been guaranteed.
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