Expert Systems with Applications 37 (2010) 5353–5363Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa Hybrid computational models for the characterization of oil and gas reservoirs Tarek Helmy *,1, Anifowose Fatai, Kanaan Faisal Information and Computer Science Department, College of Computer Science and Engineering, King Fahd University of Petroleum and Minerals, Mail Box. 413, Dhahran 31261, Saudi Arabia a r t i c l e i n f o a b s t r a c t The process of combining multiple computational intelligence techniques to build a single hybrid model has become increasingly popular. As reported in the literature, the performance indices of these hybrid models have proved to be better than the individual components when used alone. Hybrid models are extremely useful for reservoir characterization in petroleum engineering, which requires high-accuracy predictions for efficient exploration and management of oil and gas resources. In this paper, we have utilized the capabilities of data mining and computational intelligence in the prediction of porosity and permeability, two important petroleum reservoir characteristics, based on the hybridization of Fuzzy Logic, Support Vector Machines, and Functional Networks, using several real-life well-logs. Two hybrid models have been built. In both, Functional Networks were used to select the best of the predictor variables for training directly from input data by using its functional approximation capability with least square fitting algorithm. In the first model (FFS), the selected predictor variables were passed to Type-2 Fuzzy Logic System to handle uncertainties and extract inference rules, while Support Vector Machines made the final predictions. In the second, the selected predictor variables were passed to Support Vector Machines for training by transforming them to a higher dimensional space, and then to Type-2 Fuzzy Logic to handle uncertainties, extract inference rules and make final predictions. The simulation results show that the hybrid models perform better than the individual techniques when used alone for the same datasets with their higher correlation coefficients. In terms of execution time, the hybrid models took less time for both training and testing than the Type-2 Fuzzy Logic, but more time than Functional Networks and Support Vector Machines. This could be the price for having a better and more robust model. The hybrid models also performed better than a combination of two of the individual components, Type-2 Fuzzy Logic and Support Vector Machines, in terms of higher correlation coefficients as well as lower execution times. This is due to the effective role of Functional Networks, as a best-variable selector in the hybrid models. Ó 2010 Elsevier Ltd. All rights reserved. Keywords: Hybrid computational intelligence Petroleum reservoir characteristics Fuzzy Logic Support Vector Machines Functional Networks 1. Introduction Petroleum reservoir characterization is a process for quantitatively describing various reservoir properties in spatial variability by using available field data. It plays a crucial role in modern reservoir management: making sound reservoir decisions and improving the reliability of the reservoir predictions. The ultimate goal is a reservoir model with realistic tolerance for imprecision and uncertainty. Porosity and permeability are the two fundamental reservoir properties which relate to the amount of fluid con- * Corresponding author. E-mail addresses:
[email protected] (T. Helmy),
[email protected] (A. Fatai),
[email protected] (K. Faisal). 1 Tarek Helmy is on leave from College of Engineering, Department of Computers Engineering & Automatic Control, Tanta University, Egypt. 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.01.021 tained in a reservoir and its ability to flow. These properties make significant impacts on petroleum field operations and reservoir management (Lim, 2005). A good number of studies have been carried out on the use of various Computational Intelligence (CI) schemes, such as Logistic Regression (LR), K-Nearest Neighbor (KNN), Multilayer Perceptrons (MLP), Radial-Basis Function (RBF), Bayesian Belief Networks (BBN), Naïve Bayes (NB), Random Forests (RF), Functional Networks (FunNets), Support Vector Machines (SVM), Artificial Neural Networks (ANN), Probabilistic Networks (PN), Adaptive-Neuro Fuzzy Systems (ANFIS) and Decision Trees (DT) (Duch, Adamczak, & Jankowski, 1997; Giovanni & Vincenzo, 2005; Guojie, 2004; He, 2008; Hosmer & Lemeshow, 2000; Lauría and Peter Duchessi, 2006; Salah, Rahman, & Nath, 2005), to predict the characteristics of oil and gas reservoirs such as depth, temperature, pressure, volume, drive mechanism, structure and seal, well spacing, well-bore Functional Networks and Hybrid Systems. FL. The data are valuable for linking permeability to porosity. (He.S. 2005. We attempt to combine the individual capabilities of FL. with better performance indices. / Expert Systems with Applications 37 (2010) 5353–5363 integrity. . Li. Other hybrid systems include different combinations of several techniques such as SVM. Taboada. and Lacruz (2001a) gave a comprehensive demonstration of the application of FunNets in Statistics and Engineering. with very promising results. Artificial Neural Networks. and analysis and improvement of gas well production (Ali. The results showed that the proposed approach is a practical methodology to map the fracture network. Porosity is very important in evaluating the potential volume of hydrocarbons it may contain. Chikhi and Batouche (2004) combined a neural method with radial-basis functions to enhance the classification of lithofacies of certain wells. University. 2005). Goda. along with the methodology used in this work. 2008. The combination of two or more Computational Intelligence schemes as a single model is called Hybrid Computational Intelligence (HCI) and is becoming increasingly popular. the reported success of these systems in many real-world complex problems. Tiab. and García (2007) used different kinds of SVMs: SVM classification (multi-class one-against-all). In our case. such as oil (Excellence in Educational Development. it is not possible to have accurate solutions to many petroleum engineering problems without having accurate permeability value (K. This was followed by a similar hybridization effort by Chikhi (2006). and they found that the SVMs are perfectly comparable to kriging (a statistical model) and have better control of outliers. 2008). Nong. Section 4 explains the Well log data and tools. The results obtained by the hybrid are close to those obtained by the fuzzy Adaptive Resonance Theory (ART) approach applied to the same borehole with the same well-logs. 2007. ordinal SVM and SVM regression. 2008. Genetic Algorithm. identification of sandstone lithofacies. The rest of this paper is organized as follows: Section 2 reviews the literature on how Artificial Intelligence (AI) can be employed in Petroleum Engineering. Sun et al. In fact. drill bit diagnosis. demonstrating the superiority of the hybrid over the conventional SVMs. to predict two characteristics of oil and gas reservoirs. and these both serve as standard indicators of reservoir quality in the oil and gas industry. tolerant of outliers. Permeability is the ease with which fluid is transmitted through a rock’s pore space. 2. and Ramon (2005) developed a hybrid genetic programming and fuzzy/neural network inference system to estimate the permeability of reservoirs. The result demonstrated the generalizing capability of the neural network. as applied in various fields including reservoir characterization (Chikhi. Matías. Maier. Ali and Chawathe (2000) used neural networks to predict permeability from petrographic data while using Fuzzy Logic to screen and rank the predictor variables with respect to the target variable. It also embraces techniques that use Swarm Intelligence. especially in reservoir characterization. 2005). & Behrenbruch. & Pericles Mitkas. the recently increasing popularity of hybrid intelligent systems. Although a rock may be very porous. El Ouahed. A fuzzy linear programming Support Vector Machine (LP-SVM) was used for multiclass classification problems by Shigeo (2004). Porosity is the percentage of voids and open space in a rock or sedimentary deposit. it is not necessarily very permeable. Like the other AI techniques described in previous sections. Section 5 discusses the criteria of quality measurement used to evaluate the validity in this work while conclusion. A key prerequisite for the merging of technologies is the existence of a ‘‘common denominator” to build upon (Andreas. 2007. Section 3 presents a brief overview of Fuzzy Logic. Chikhi & Batouche. Support Vector Machines. and Varjani (2007) when they proposed a new method for the auto-design of neural networks based on genetic algorithm (GA).. & Mazouzi. Permeability is a key parameter associated with the characterization of any hydrocarbon reservoir. Fuzzy Systems. who combined Neural Networks with Hidden Markov Models to obtain the lithological identification of the same wells. A similar study was done by Saemi. porosity and permeability predictions. 2007). When compared with contemporary estimation approaches. Our motivations for this work include the quest for higher performance accuracy in the prediction of oil and gas characteristics. Peddabachigari. the greater its ability to hold water and other earth materials. 2006. & Thomas. 2003. and Zhang (2007) to better handle uncertainties existing in real classification data. Helmy et al. the need to complement the weaknesses of one algorithm with the advantages of the others and hence to combine the cooperative and competitive characteristics of the individual techniques and the existing theoretic and experimental justifications (Munakata. A hybridization of SVMs and the Interval Type-2 FL System (FLS) was performed by Chen. Permeability is a measure of how interconnected the individual pore spaces are in a rock or sediment. Ichalkaranje. 2005) that hybrids produce more accurate results than the individual techniques used separately. SVMs have been used extensively in many areas. Grosan. 2006. Extreme Learning Machines. Abrahamb. 2007).. 1994. CI covers many existing branches of science viz. Hidden Markov Model and Radial-basis Functions. Most hybrids found in the literature usually contain Neural Networks fused with one other technique due to its wide use in the computational intelligence. including oil and gas (Jian & Wenfen. Wilkinson. part of the ‘‘common denominator” for Fuzzy Logic (FL). Evolutionary Computing and Hybrid Systems. now referred to as Type-1 FL has featured in a number of research efforts. They found that fuzzy modeling is assumption-free. 2006). Wavelets. Deny. Porosity and permeability measurements are frequently made on plugs extracted from the core of wells drilled for oil and gas exploration. Ahmadi. Fractals and Chaos Theory. El Ouahed et al. Literature survey The application of the capabilities of Artificial Intelligence (AI) has been widely appreciated in petroleum engineering. Phillips-Wren. with a detailed plan of future work is described in Section 6. Decision Trees. Wang & Fu. as well as in other fields. SVM and FN in an HCI scheme. (2005) produced 2-D fracture intensity and fracture network maps in a large block of field using Artificial Neural Network and FL. 2007). The greater the porosity of a rock. Artificial Immune Systems.5354 T. The traditional FL. the hybrid yielded more consistent and robust estimated results. 2005). This increased popularity lies in the extensive success of hybrid systems in many real-world complex problems (Giovanni & Loia. porosity and permeability. 2004). etc. Hadi. Yu. & Jain. Castillo (1998) and Castillo. Triantaphyllou & Felici. Support Vector Machines (SVM) and Functional Networks (FN) is the inference procedures they deploy and their excellent predictive capabilities. namely porosity and permeability. Some of the areas of petroleum technology in which AI has been used with success include seismic pattern recognition. FunNets have also featured in a number of research studies. One of the earliest references to the application of FL in the petroleum industry was by Fang and Chen (1997) who presented a fuzzy modeling for predicting porosity and permeability from the compositional and textural characteristics of sandstones. Symeonidis. Ordóñez. and capable of making both linguistic and numeric predictions based on qualitative and quantitative data. Harrison. Type-2 FL has also featured in many recently published articles in various fields and especially in reservoir properties modeling (El Ouahed et al. Hadi. pressure and temperature) the membership is a function (not just a point value). Fig. but it can perform better when the number of training prototypes is large (Fang & Chen. Karnik & Mendel. 1997.g. Figs. Regression in Primal Y Training Dataset X Fig.1. Support Vector Machines. 1998. Hence. This is shown in Fig. in the sense that the output from one unit can serve as part of the input to another neuron or to the units in the output layer. Mendel. Once the input values are given. is fixed and the weights are learned from data using some well-known algorithms. associated with the neurons. . a layer of output units containing the output data.3. For each value of a primary variable (e. 3. The function f. Functional Networks and Hybrids 3.. 2007). Lacruz. 3. The secondary Membership Function (MF).2. one or several layers of neurons or computing units which evaluate a set of input values. 1998. Fuzzy Logic.. which can be defined by a function (Castillo. 1999. Fuzzy Logic Type-2 FLS was introduced as an extension of the concept of Type-1 FLS. 3. Castillo et al. 1]. 2003. The computing units are connected to each other. there is a need to combine the individual capabilities of each technique to obtain a more versatile and robust technique. Type-2 FL does not obtain good performance when the number of training data is small. 1]. Support Vector Machines Support Vector Machines are a set of related supervised learning methods used for classification and regression.. Type-2 FLS has membership grades that are themselves fuzzy. the MF of a Type-2 FLS is three dimensional. 1 and 2 show the structure of a Type-2 FLS and Gaussian MF with uncertain mean. 3. The structure of a Type-2 FLS. Littman.4. 2001b. SVMs map input vectors to a higher dimensional space where a maximal separating hyperplane is constructed (Burges. respectively. Helmy et al. and whose range may also be in [0. 4 shows the structure of a Functional Network and its simplification. & Neshat. 1. and which give a set of output values to the next layer of neurons or output units. 2001a. whose domain is in the interval [0. Turksen. 3. It has also been defined as an approach that combines different theoretical backgrounds and algorithms such as data mining and soft computing methodologies. 2003). Zarandi. Functional Networks Functional Networks are extensions of Neural Networks which consist of different layers of neurons connected by links. 2. coming from the previous layer. Gaussian MF with uncertain mean. / Expert Systems with Applications 37 (2010) 5353–5363 5355 Fig. the output is determined by the neuron type. They belong to a family of generalized Linear Classifiers. Each computing unit or neuron performs a simple calculation: a scalar typically monotone function f of a weighted sum of inputs. A Functional Network consists of: a layer of input units containing the input data. Castillo. Chen et al. and it is the new third dimension that provides new degrees of design freedom for handling uncertainties. 2007). Mapping input vectors to a higher dimensional space in SVM. Fig. Rezaee.T. Hybrids An approach resulting from the combination of two or more approaches is called a hybrid. Gutiérrez. Since no single technique is good for everything and in all situations. The main idea behind hybridization is to complement the weaknesses of one technique with the strength of other techniques. They can also be considered as a special case of Tikhonov Regularization. 3. It is also strong in the process of extracting rules. to be used for inferencing. 2003). and hence to combine the cooperative and competitive characteristics of the individual techniques. directly from the input data. Mendel. and time series prediction (Jian & Wenfen. while the dataset from site 2 has eight predictor variables for permeability. This weakness is intended to be complemented with the ability of SVM to handle small a dataset. but it performs better when the number of training prototypes is large (Mendel. Table 3 presents the areas of strengths and weaknesses of the techniques. Also. The datasets from site 1 have six predictor variables for porosity. Support Vector Machines (SVM) and Functional Networks (FN). These are shown in Tables 1 and 2. 5. 2003). Fig. hence. application developers have already reported state-of-the-art performances in a variety of applications in pattern recognition. It is also known to have the capability of using a small training dataset (Jian & Wenfen. Helmy et al. 2006). In order to show the reasons for the choices of FL. regression estimation. Experimental design The methodology in this work is based on the standard Computational Intelligence approach to hybridization of Artificial Intelligence technique using Fuzzy Logic (FL). The strengths of SVM lie mainly in its relative ease of training. 2003). The choice of Type-2 FLS lies in its ability to determine an exact membership function for a fuzzy set. FN and FL. 2006). SVM and FN. Predictors for permeability 1 2 3 4 5 6 7 8 GR PHIE RHOB SWT RT MSFL NPHI CALI Full meaning Gamma ray log Porosity log Density log Water saturation Deep resistivity Microspherically focused log Neutron porosity log Caliper log 4. Type-2 FL does not obtain good performance when the number of training data is small. Description of data Well-logs for porosity and permeability from six wells were used for the validation of this work. Predictors for porosity 1 2 3 4 5 6 Core Top interval Grain density Grain volume Length Diameter Table 2 Predictor variables for site 2 well log for permeability. compared to other techniques. unlike in neural networks It scales relatively well to high dimensional data Ability to explicitly control the tradeoff between complexity and error It is also known to have the capability of using small training dataset There is no need to include weights associated with links Functional approximation Ability to select the best among functions by minimizing the sum of square errors Weakness Complexity of implementation Too much time spent in tuning the parameters used for inferencing during the training process Does not obtain good performance when the number of training data is small It is weak in the sense that it needs a ‘‘good” kernel function SVM FN . Technique Type-2 FLS Strength Ability to determine an exact membership functions for a fuzzy set Ability to handle uncertainties Ability to extract rules directly from input data Ease of training No local optima. SVM and FN.1. / Expert Systems with Applications 37 (2010) 5353–5363 form in North America (site 1) and the three for permeability from a drilling site in the Middle East (site 2). 4. and to complement the weaknesses of one technique with the advantages of the others. The hybrid models were designed to benefit immensely from the strength of the individual techniques. there is no local optimal similar to neural networks. ability to explicitly control the tradeoff between complexity and error. However. it is weak in the sense that it needs a ‘‘good” kernel function (Littman. scalability to high dimensional data. its disadvantage lies in its complexity of implementation and hence too much time spent in tuning the parameters used for inferencing during the process of training. This weakness is intended to be complemented with the ability of FunNet and FL to learn directly from the input data. The hybrid models were implemented by using mainly MATLAB codes and MATLAB toolboxes for SVM. Experimental design and model validation 5. and ability to handle non-traditional data like trees as input to the system. The three well-logs for porosity were obtained from a drilling site in the Northern Marion Plat- Table 3 Strengths and weaknesses of FL. Table 1 Predictor variables for site 1 well log for porosity.5356 T. 1999. it is useful for handling uncertainties (Karnik & Mendel. However. Although the use of SVMs in applications has only recently begun. instead of feature vectors. Structure of Functional Networks and its simplification. Functional Networks–Fuzzy Logic–SVM (FFS). / Expert Systems with Applications 37 (2010) 5353–5363 5357 Least Square Fitting Handling Uncertainties from best-variable data Transformation to higher dimensional space for regression Input Data FunNets Fuzzy Logic SVM Predicted Output Data with best input variables Rule Extraction from best-variable data Fig. Functional Networks–SVM–Fuzzy Logic (FSF). The conceptual design framework of the FFS hybrid model.2. The conceptual design framework of FSF hybrid model. for regression. several iterations were made.. To further ensure fairness and integrity of the results obtained. 6 shows the conceptual design framework of this model. Wells Site 1 (Porosity) 1 Data size Training (70%) Testing (30%) 415 291 124 2 285 200 85 3 23 16 7 Site 2 (Permeability) 1 355 249 106 2 477 334 143 3 387 271 116 works. Helmy et al. This is due to its functional approximation capability to select the best of the predictor variables directly from the training data. Fig. In the first model. This reduced architecture was chosen because the two hybrid models have similar configurations and demonstrate very competitive levels of performance. 2001b). 5. 6. Least Square Fitting Handling Uncertainties from best-variable data Input Data FunNets SVM Fuzzy Logic Predicted Output Data with best input variables Transformation to higher dimensional space for regression Rule Extraction from best-variable data Fig. Support Vector Machines are used to transform the best of the predictor variables from Functional Networks to a higher dimensional space suitable for training the Type-2 Fuzzy Logic that will handle uncertainties. and the remaining 30% goes for testing. extract inference rules and make the final predictions. and the average of the runs was obtained. 5. another set of experiments was performed. using only two of the components namely: Type-2 Fuzzy Logic and SVM. Fig. Seventy percent of the entire data goes for training.3. The learning process of a Functional Network consists of obtaining the neural functions from a set of training data based on minimizing the sum of squared errors between the input and the target output by suggesting an approximation to each of the functions and selecting the best among them (Castillo et al. 5. Conceptual framework of the hybrid models A Functional Network was used as the base for the two hybrid models.T. Table 4 Division of datasets into training and testing. The output of this is passed to Support Vector Machines that have previously been trained with the best predictor variables from Functional Net- . Type-2 Fuzzy Logic is used to handle uncertainties and to extract inference rules directly from the best predictor variables obtained from the Functional Networks block. Table 4 shows the two well-logs with their sizes and divisions into training and test sets. 5 shows the conceptual design framework of this model. In the second model.4. Model implementation and validation The available data for each of the wells are divided into training and test data by using a stratified sampling technique. 5. Control experiment: removing functional from the Hybrid models In order to appreciate the role performed by the FN block in the hybrid models. 5.526042 28.18052 9.980730 0.914020 0.599756 0. and Execution Time (ET) as criteria for measuring the performance.75723 6.890625 SVM FunNet Fuzzy Logic Hybrid-FFS Hybrid-FSF 0.798557 0.17685 6.09860 9. computed by taking the square root of the average of the squared differences between each predicted value xn and its corresponding actual value yn.72873 5.129688 0.15040 7.67851 Execution time Training 15.5358 Table 5 Result of the porosity prediction site 1.835855 0. / Expert Systems with Applications 37 (2010) 5353–5363 Correlation coefficient Training Testing 0. It gives the error value the same dimensionality as the actual and predicted values The formula is : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1 Ày1 Þ2 þðx2 Ày2 Þ2 þÁÁÁþðxn Àyn Þ2 n : The formula is : P ðx À x0 Þðy À y0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P ðx À xÞ2 ðy À y0 Þ2 where n is the size of data The ET is simply the total time taken for a technique to run from the beginning to the end.920961 Testing 0.09662 Testing 7.06432 6. well 1.26323 6.957491 0. while x0 and y0 are the mean of the actual and predicted values.775465 0. .411458 4.18608 7.64195 4.442708 31.937500 9.81454 Testing 2. Criteria for performance evaluation In order to establish a valid and reasonable evaluation of this work.04103 3.916415 Table 6 Result of the porosity prediction site 1.17699 7.000000 0. we have used Correlation Coefficient (CC).82147 0.94502 0.000000 25.77203 7.34832 8.969179 0.45112 Execution time Training 7.75388 4.54717 6. The RMSE is one of the most commonly used measures of success for numeric prediction.471094 Testing 0.093750 71.395833 60. 5.000000 0.803918 0.26507 7.062500 0.518750 0.61915 Testing 2.31978 9.093750 8. Model Correlation coefficient Training SVM FunNet Fuzzy Logic Hybrid-FFS Hybrid-FSF 0. well 2.562500 Testing 0.567708 Testing 0. well 3.85104 7. Root MeanSquared Error (RMSE).315625 0.458333 0. Correlation coefficients comparisons for porosity training and testing. The CC measures the statistical correlation between the predicted and actual values.69430 7.723302 0.95456 6.840697 0. A value of ‘‘1” means perfect statistical correlation and a ‘‘0” means there is no correlation at all. where x and y are the actual and predicted values.051042 Fig.098958 155. 7.920608 0.93055 0.957003 RMSE Training 7.80758 0. where T2 is the CPU time at the end of the run and T1 is the CPU time at the beginning of the run. Helmy et al.775851 0.296875 Table 7 Result of the porosity prediction site 1. Model T.354167 0. Model Correlation coefficient Training SVM FunNet Fuzzy Logic Hybrid-FFS Hybrid-FSF 0.000000 0.816021 0.8147 Testing 0.000000 0.52180 3. and it is computed as: T 2 À T 1 .803017 0.000000 55.817708 4.928830 RMSE Training 1.813319 RMSE Training 6.042188 0.910924 0.493750 0.820348 0.824680 0.32623 Execution time Training 0.806304 0.898397 0.979167 61.80199 4.37047 6. 739583 12. The results of the simulations for the three Porosity and three Permeability wells are shown in Tables 11 and 12.000000 68. Fig. . These are summarized graphically in Figs.833204 0.868102 0. Execution time comparison for well 3 porosity training and testing. Execution times comparisons for porosity training and testing.697917 Testing 0. well 2.333333 44.62187 1.869565 0.852724 0.63803 0.182292 355. / Expert Systems with Applications 37 (2010) 5353–5363 5359 Fig.62760 0. 9.000000 128.61685 0.866123 0.854109 0. Tables 5–7 present the result of the prediction of porosity for the three wells in site 1.72800 Testing 0.832785 0.88142 0. These are also summarized graphically in the plots shown in Figs. In order to appreciate the role performed by the FN block in the FFS and FSF hybrid systems.68113 0.661458 0. 7–9. Table 8 Result of the permeability prediction for site 2. and the average of results was taken.872843 0.895119 0.64289 Execution time Training 10. Helmy et al.72689 Execution time Training 18.T.844982 0. 8.67214 0.881033 0.903620 RMSE Testing 0. respectively. 10 and 11 for both training and testing.494792 43.88144 Testing 0.860039 0.140625 187.60509 0.828125 6.62799 0.10615 0.864894 0.026042 79.901920 RMSE Testing 0.858610 0.64724 0. Model Correlation coefficient Training SVM FunNet FL Hybrid-FFS Hybrid-FSF 0. another set of experiments was performed by using only two of the components namely: Type-2 Fuzzy Logic and SVM. Experimental results We implemented and validated the individual techniques as well as the two hybrid models in the prediction of porosity and permeability by using the training and testing data described in the previous section.11178 0.140625 6.64785 0.73102 Testing 0.000000 0.052083 0.062500 87.531250 Table 9 Result of the permeability prediction for site 2.66946 0.630208 Testing 0.827372 Testing 0.65863 1.463542 14. well 1.68445 0.231250 6. Model Correlation coefficient Training SVM FunNet Fuzzy Logic Hybrid-FFS Hybrid-FSF 0.005208 0. Several iterations were made.66580 0. Tables 8–10 present the result of the prediction of Permeability for the three wells in site 2.877959 0. 18849 4.447917 Testing 0.789864 0.767553 0.32429 Execution time (s) 297.708821 RMSE 7. which extracts from the input variables .159375 0. followed by SVM. where SVM demonstrated its ability to withstand a shortage of training data but FL demonstrated otherwise. or competitively equal to.86444 0. Fig.20909 5.811098 0.741249 0. Helmy et al.411458 26.61251 8.036458 7. the FFS and FSF Hybrids also proved to be better in terms of both correlation coefficient and execution time.700457 0.815213 0.789960 0.380208 56. Table 11 Results of the simulations for the three porosity wells. Wells Site 1. 12–15.177083 241. the three individual techniques used separately. the results showed that Functional Networks are the fastest in terms of both training and testing.000000 89. well 3 CC 0.787706 0.381250 1. the hybrid models performed better than the individual components due to the cooperative spirit that was been built into them.125000 Comparisons of the performance of the Type-2-SVM above with the FFS and FSF Hybrids are shown in Figs.76140 0. but not for SVM and FN.768380 0.769102 Testing 0.74712 0.1 and Table 3. Still.78078 Testing 0. in terms of their correlation coefficient. This is the price for obtaining better models in terms reliability and robustness. well 2 Site 1.73154 0. 11. When compared to the Type-2-SVM Hybrid.801785 RMSE Testing 0. The better performance in correlation coefficient can be attributed to the role of Functional Networks in the FFS and FSF hybrids. Correlation coefficients comparisons for permeability training and testing. 6.76481 0.64583 8.730623 0.73625 0.5360 T. Discussion of results In the prediction of porosity and permeability. it is clear from the results that the hybrid models performed better than. A special characteristic was observed in the result of site 1 well 3 for Porosity. The hybrid models proved to be faster than the Fuzzy Logic component. / Expert Systems with Applications 37 (2010) 5353–5363 Table 10 Result of the permeability prediction for site 2.098958 Fig.83188 1.802083 0. The Functional Networks block serves as a best-variable selector. Type-2 Fuzzy Logic took the most time for both training and testing.753439 0. 10.800820 0.78106 Execution time Training 12.791667 11.1. due to its complexity as described in Section 3. In terms of execution time.010417 0.33084 7. well 1 Site 1.864583 137. Execution time comparisons for porosity training and testing.781196 0.10064 0.822917 51.885417 59. well 3 Model Correlation coefficient Training SVM FunNet Fuzzy Logic Hybrid-FFS Hybrid-FSF 0. 817343 0. 13. well 3 CC 0.86452 Execution time (s) 358.03125 109. The better performance of the FFS and FSF hybrids in terms of execution times is also due to the above reason.68116 0.90577 0.64555 0. Helmy et al. Fig.715730 RMSE 0.317708 125.015625 Fig. only those variables that are most relevant to the prediction system.823580 0.831428 0. In the process of . Execution time comparisons for porosity training and testing. Fig.61174 0.093750 564.T. 12. Wells Site 2. well 1 Site 2.942708 5361 69. well 2 Site 2. 14.728086 0. Correlation coefficient comparisons for permeability training and testing.61296 0. / Expert Systems with Applications 37 (2010) 5353–5363 Table 12 Results of the simulations for the three permeability wells. Correlation coefficient comparisons for porosity training and testing.395833 644.814240 0. Amsterdam. Jakarta. robust and effective. J. the hybrid will be able to select the best model to use. S. 49. & Pericles Mitkas. A. 2006. . Journal of Geophysics and Engineering. J. Fuzzy modeling and the prediction of porosity and permeability from the compositional and textural attributes of sandstone. N. Adamczak. S. and perform the required prediction. & Lemeshow. X. Symeonidis. More experiments will be done on these hybrids if data is available with uncertain numerical measurements. (1997). 122–141.. 14. In Society of Petroleum Engineers Asia Pacific Oil & Gas Conference and Exhibition. M. Chikhi. the design. 171–182. The results showed that the hybrids performed better than the individual techniques used separately for all the datasets in the prediction of porosity and permeability. Acknowledgments We would like to thank both KACST and King Fahd University of Petroleum and Minerals for providing the computing facilities and support. 185–204. This works in favor of the Type-2 FLS that does not work well with an input data of very high dimensionality. 7(6). M. A. ISSN: 0973-1873. given the reported and observed good performance of the individual components. & Jankowski. Castillo. Deny. He. K. 15. (2007). (2007). The Netherlands. (2005).ijcir. 1(2). L. Karnik. implementation and validation of the models were based on well-logs and not on expert knowledge.edu.019. S. 10–24.. Fang. (2006). USA. (1997). M. best-variable selection by the Functional Networks block. and simulated organizations series. Excellence in Educational Development. and handle uncertainties that might be present. Q. N. Motivated by the success of this work. Castillo. Gutiérrez.. Technometrics 43.htm>. Applied Soft Computing Journal.. (2001). E. Lecture Notes in Computer Science (Vol.ac. 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