Ali Kadkhodaie Ilkhchi 1, Mohammadreza Rezaee 1,3 and Seyed Ali Moallemi Introduction

Transcription

1 INSTITUTE OFPHYSICS PUBLISHING JOURNAL OFGEOPHYSICS ANDENGINEERING J. Geophys. Eng. 3 (2006) doi: / /3/4/007 A fuzzy logic approach for estimation of permeability and rock type from conventional well log data: an example from the Kangan reservoir in the Iran Offshore Gas Field Ali Kadkhodaie Ilkhchi 1, Mohammadreza Rezaee 1,3 and Seyed Ali Moallemi 2 1 School of Geology, University College of Science, University of Tehran, Tehran, Iran 2 Research Institute of Petroleum Industry/NIOC, Tehran, Iran Received 11 May 2006 Accepted for publication 10 October 2006 Published 6 November 2006 Online at stacks.iop.org/jge/3/356 Abstract Permeability and rock type are the most important rock properties which can be used as input parameters to build 3D petrophysical models of hydrocarbon reservoirs. These parameters are derived from core samples which may not be available for all boreholes, whereas, almost all boreholes have well log data. In this study, the importance of the fuzzy logic approach for prediction of rock type from well log responses was shown by using an example of the Vp to Vs ratio for lithology determination from crisp and fuzzy logic approaches. A fuzzy c-means clustering technique was used for rock type classification using porosity and permeability data. Then, based on the fuzzy possibility concept, an algorithm was prepared to estimate clustering derived rock types from well log data. Permeability was modelled and predicted using a Takagi-Sugeno fuzzy inference system. Then a back propagation neural network was applied to verify fuzzy results for permeability modelling. For this purpose, three wells of the Iran offshore gas field were chosen for the construction of intelligent models of the reservoir, and a forth well was used as a test well to evaluate the reliability of the models. The results of this study show that fuzzy logic approach was successful for the prediction of permeability and rock types in the Iran offshore gas field. Keywords: rock types, permeability, fuzzy logic, fuzzy c-means clustering, back propagation neural network, Kangan reservoir, Iran offshore gas field 1. Introduction The Iran offshore gas field, the Iranian part of the world s largest non-associated gas accumulation is located in the Persian Gulf between Qatar and Iran, some 100 km from shore. The Upper Permian to Lower Triassic Dalan and Kangan Formations are two huge, condensate rich, gas bearing reservoirs over the field. This paper focuses on application of 3 Present address: Curtin University of Technology, Department of Petroleum Engineering, Western Australia 6151, Australia. the fuzzy logic technique for the estimation of two important reservoir parameters including permeability and rock type in the Kangan Formation. So far, several studies have been carried out for the estimation of reservoir parameters by intelligent systems (e.g. Cuddy 1998, Saggaf and Nebrija 2003, Chang et al 2000, Lim 2003, 2005, Ali and Chawathe 1999, Huang et al 2001, Arpatet al 1998, Wong et al 1997 and Malki et al 1996) for rock type and permeability prediction. These studies indicate the successful role of methods such as fuzzy logic (FL), artificial neural networks (ANN), neuro-fuzzy (NF) and /06/ $ Nanjing Institute of Geophysical Prospecting Printed in the UK 356

2 A fuzzy logic approach for estimation of permeability and rock type (a) (b) Figure 1. The membership functions corresponding to classification of lithology types by V p/vs ratio in crisp logic (a) and fuzzy logic (b) approaches. Table 1. V p/vs ratio values for different lithologies. V p/vs Lithology type Clean sand Shaly sand Limestone Dolomite genetic algorithms (GA) for reservoir characterization. The present study shows another example of using fuzzy logic for reservoir characterization. Generally, geosciences disciplines are not clear-cut and most of the time are associated with uncertainties. For example, prediction of core parameters from well log responses is difficult and is usually associated with error. Statistical methods try to minimize and ignore this error, whereas fuzzy logic derives useful information from this error and uses it as a powerful parameter for increasing the accuracy of the predictions. The following simple example can clarify the subject: According to the works of Tatham (1982), Domenico (1984), Castagna et al (1985) and Ikwuakor (1988), the ratio of compressional (V p) to shear wave velocity (V s) is a key factor for lithologic identification in hydrocarbon reservoirs. Lithologies such as dolomite, limestone, shaly sand and clean sand can be identified by this ratio (table 1). The membership functions corresponding to this classification of lithologies in fuzzy and non-fuzzy approaches are shown in figure 1. According to the crisp or non-fuzzy approach (figure 1(a)), a rock with a ratio of 1.8 will be classified as limestone whereas another with a ratio of will be considered a shaly sand. Such a description is of course far from reality. In the crisp approach there is a clear cut and sharp boundary between descriptions, whereas fuzzy logic asserts that there is a traditional and plastic boundary between the descriptions. For example, according to figure 1(b), arockwithvp/vs ratio 1.83 will be classified as shaly sand to a degree of 0.85 and as limestone to a degree of Note that we should discriminate between probabilities and possibilities. The sum of probabilities is always equal to 1, whereas such reasoning will not be usually valid for possibilities. In the above example, the values of membership functions (possibilities) for a V p/vs ratio of 1.83 do not add up to 1 (e.g. 0.1 and 0.85). Saggaf and Nebrija (2003) have used a good example to clarify the difference between possibility and probability; consider this statement made before two wells are drilled: the possibility that the first well is economically viable is 0.8, whereas, the probability that the second well is economically viable is 0.8. Although the two numbers are the same, they describe different notions. The probability indicates that the likelihood of this well being economically viable is 0.8. There is an 80% chance of finding oil in sufficient quantities in this well. However, there is a chance that the well could very well turn out to be a dry hole. The possibility does not relate to likelihood or chance at all. It simply expresses the information that the well fits our definition of what we consider to be an economically viable well by 0.8. This well is going to be better than average in what we consider good wells, it is not going to be great, but neither is it going to be a dry hole. In fact, after the wells are drilled, the probability would either be 0 (in case the second well does not have oil in economic quantities) or 1 (otherwise). However, the possibility would remain the same. As it is evident from these simple examples, fuzzy reasoning is very close to reality and the description of the facts as they are. Therefore, it will be appropriate to use fuzzy logic to solve problems that are accompanied by vagueness 357

3 A Kadkhodaie Ilkhchi et al (a) (b) (c) (d ) (e) (f ) Figure 2. Crossplots showing the relationship between permeability and conventional well log data including NPHI (a), DT (b), RHOB (c), GR (d), Rlld (e) and Depth (f ). and imperfection such as those of predicting core parameters from conventional well log responses. 2. Methodology 2.1. Fuzzy inference system Fuzzy inference is the process of formulating from a given input to an output using fuzzy logic (Matlab user s guide 2001). There are two types of fuzzy inference systems (FIS) including Mamdani and Assilian (1975) and Takagi-Sugeno (1985). Mamdani s method attempts to control a system by synthesizing a set of linguistic control rules obtained from experienced human operators. The Takagi-Sugeno method is similar to the Mamdani s FIS. The output membership functions are the main difference between Mamdani and Takagi-Sugeno methods. In Takagi-Sugeno-type FIS (Matlab user s guide 2001), output membership functions are constant or linear, and membership functions are defined by a clustering process. A small cluster radius usually yields many small clusters; however, a large cluster radius yields a few large clusters in the data (Chiu 1994). Each of these clusters refers to a membership function. Each membership function generates a set of fuzzy if then rules for formulating inputs to outputs Fuzzy c-means clustering Fuzzy c-means (FCM) is a data clustering technique where each data point belongs to multiple clusters to some degree that is specified by a membership grade. This technique was first introduced by Dunn (1973) and improved by Bezdek (1981). It provides a method that shows how to group data points that populate some multidimensional space into a specific number of different clusters. FCM is based on minimization of the following objective function (Ozer 2005): n k=1 i=1 c (u ik ) m ( x k v i ) 2, (1) where n is the number of objects to be clustered, c is the number of clusters, u ik is the degree of membership of object k in cluster i, x k is a vector of h characteristics for object k, v i is a vector of the cluster means of the h characteristics for 358

4 A fuzzy logic approach for estimation of permeability and rock type Figure 3. The NPHI, DT, RHOB, GR and Rlld membership functions for permeability modelling using TS-FIS. cluster i and m is the weighting exponent varying in the range [1, ]. Equation (1) represents the sum of squared errors and is a goal function that the FCM algorithm tries to minimize. The constraints are as follows: 0 u ik 1, i, k, (2) ( c ) u ik = 1, k. (3) i=1 Condition (2) ensures that the degrees of memberships are between 0 and 1, and condition (3) means that, for a given individual, the degrees of membership across the clusters sum to one. Fuzzy partitioning is carried out through an iterative optimization of equation (1), with the update of membership u ik and the cluster centres v j by: v i = u ik = n k=1 (u ik) m x k n k=1 (u, i (4) m ik) c j=1 1 ( xk v i ), (5) 2/(m 1) x k v j 2/(m 1) for x i v j, i, k, and m>1. { The iteration will stop when max ik u (t+1) ik u (t) } ik <ε, where ε is error level for termination which varies between 0 and 1, and t are the iteration steps. In detail, this iterative procedure converges to a local minimum of equation (1). 3. Permeability modelling 3.1. Fuzzy logic In this study, a Takagi-Sugeno FIS was used to generate permeability models for the Kangan Formation. First, several crossplots showing the relationship between well log data and permeability were generated for well A (figures 2(a) (f )). The crossplots shows that neutron (NPHI), sonic (DT), and density log (RHOB) data have the strongest relationship with permeability. Five inputs including NPHI, DT, RHOB, GR, and deep laterolog resistivity (Rlld) logs were used as inputs. There is a logical relationship between the selected inputs and permeability. NPHI, DT, and RHOB logs are porosity logs. According to figure 2, generally, permeability in the Kangan Formation increases as porosity increases. A GR log is a measure of the natural radioactivity of a formation and is used to distinguish between clean and shaly reservoir. Normally, clean intervals with low GR show higher permeability than shaly intervals. The electrical resistivity of a formation depends on many parameters including fluid saturation and pore geometry. Pore geometry is a key element that controls permeability. When input data were selected, they were divided into two groups including modelling data (1076 data points) from three wells (A, B and C) and test data (125 data points) from the forth well (D). All membership functions and their parameters were extracted by a subtractive clustering method. By specifying 0.65 for the clustering radius, three Gaussian-type membership 359

5 A Kadkhodaie Ilkhchi et al Figure 3. (Continued.) functions were extracted for used inputs which were captioned by low, moderate, and high (figures 3(a) (e)). The if-then rules generated are as below: 1. If (NPHI is low) and (DT is low) and (RHOB is high) and (GR is low) and (Rlld is low) then (permeability (K) is low). 2. If (NPHI is moderate) and (DT is moderate) and (RHOB is moderate) and (GR is moderate) and (Rlld is moderate) then (permeability (K) is moderate). 3. If (NPHI is high) and (DT is high) and (RHOB is low) and (GR is high) and (Rlld is high) then (permeability (K) is high). By extraction of membership functions and fuzzy if-then rules, the Takagi-Sugeno fuzzy model was constructed. The performance of the model was measured as by using the mean squared error (MSE) function. The formulation between input well log data and permeability is shown in figure 4. After preparation of the fuzzy models, the following steps were implemented for the prediction of permeability in the forth well of the field by using FIS (the test well D): Step 1. Fuzzify inputs: The FIS takes the inputs and determines the degree to which the inputs belong to each membership function. Step 2. Apply fuzzy operator and truncation method: For the case that the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one rule that represents the result of the antecedent for that rule. The most common operators are shown as follows: and = use the minimum of the options. or = use the maximum of the options. not = use 1- option. Applying the fuzzy operators gives a value to the antecedent of each rule, and then the output membership function is truncated by this value. In this study and operator has been used. 360

6 A fuzzy logic approach for estimation of permeability and rock type Figure 4. Formulation between well log data (inputs) to permeability (output) using the TS-FIS. Figure 6. Crossplot showing correlation coefficient between measured and FL predicted permeability in test well D. Step 3. Apply aggregation method: In this step, outputs of each rule that fit into a fuzzy set are combined into a single fuzzy set. Step 4. Defuzzify: The input for the defuzzification process is the results of the aggregation method. Then FIS uses a defuzzification method (in this study, a weighted average) for the resulting output which is a crisp numerical value. Figure 5 shows an example of processing steps to use TS-FIS for permeability modelling from used well log inputs. Measured MSE for predicted permeability in the test well D is Figure 6 shows the correlation coefficient between measured and FL predicted permeability in the test well D. A comparison of measured and predicted permeability versus depth is shown in figure Back propagation neural network A back propagation artificial neural network (BP-ANN) is a supervised training technique that sends the input values forward through the network and then computes the difference between the calculated output and the corresponding desired Figure 5. The processing steps required to use TS-FIS for permeability modelling from used well log inputs. 361

7 A Kadkhodaie Ilkhchi et al Figure 7. A comparison of measured and FL predicted permeability versus depth. Figure 8. The schematic diagram of ANN for permeability modelling (output) using well log data inputs. output from the training dataset. This error is then propagated backward through the net and the weights are adjusted during a number of iterations. The training stops when the calculated output values best approximate the desired values (Bhatt and Helle 1999). In this research, a three layered BP-ANN was used to verify the validity of generated fuzzy models for permeability prediction. The data set was divided into three groups including training (785 data points), validation data (150 data points) and test data (168 data points) from three wells. The forth well D was used to evaluate the generalization capability of the ANN model. Similar to Takagi-Sugeno FIS, five inputs including NPHI, DT, RHOB, GR, and Rlld logs were used in the first layer. The number of neurons in the hidden layer was 3, and output layer included one neuron for permeability data. The schematic diagram of the BP-ANN we used is shown in figure 8. The mentioned three layered neural network was trained using the Levenberg-Marquardet training algorithm (trainlm), the details of computation process and training for which can be found in Boadu (1997, 1998) and Bishop (1995). The default MSE performance function was applied to optimize weights and default bias values. The transfer function Figure 9. Graph of mean squared error (MSE) versus training epochs for training (blue/solid), validation (green/dotted) and test data (red/dashed). 362

8 A fuzzy logic approach for estimation of permeability and rock type 4. Rock types classification Figure 10. Crossplot showing correlation coefficient between measured and ANN predicted permeability in the test well D. from layer one to two is LOGSIG and from layer two to layer three is PURELIN. After 12 epochs of training MSE performance function was set to (figure 9). When the training and optimization of the model was finished, the input well log data of the test well D was passed from it and permeability was calculated. According to figures 10 and 11, the permeability predicted values for the test well with correlation coefficient of 0.76 are similar to results of FL, so ANN verifies its reliability. The measured MSE of the predicted values is Carbonate reservoir characterization has commonly had difficulty in relating dynamic reservoir properties to a geologically consistent rock type classification system. Traditional approaches that use log and core data have not been successful for rock type definition (Boitnott et al 2006). In the Schlumberger Oilfield Glossary, each rock type is defined based on a set of characteristics that several rocks have in common. The characteristics of interest are usually those pertaining to fluid movement and fluid storage capacity. This section shows a case study of rock type classification using a FCM clustering technique in the Kangan carbonate reservoir. The methodology is described as below: First, input porosity and permeability data (a matrix of 1076 by 2) was passed from the following function of Matlab, [center, U, obj fcn] = fcm (data,cluster n) where data (input matrix of porosity and permeability), cluster n (number of cluster to be derived) and fcm (Matlab s fuzzy c-means clustering algorithm) are input arguments of the function. The output arguments are center (matrix of final cluster centres), U (membership function matrix) and obj fcn (values of the objective function during iterations). By specifying arbitrary values for cluster n, clusters were derived for the reservoir studied. Then each cluster was deemed as a unique rock type of the reservoir. In this study from five to twenty rock types were generated for the Kangan Formation. However, in order to accomplish rock typing in an applicable way, the case of specifying ten rock types was selected for further study. The porosity permeability crossplots of the ten mentioned rock types are shown in figure 12. In the FCM approach rock typing, facies type is not considered and each rock type will be defined by its production capability and will have a set of characteristics for porosity and permeability parameters which include minimum, maximum, mean and Figure 11. A comparison of measured and ANN predicted permeability versus depth. 363

9 A Kadkhodaie Ilkhchi et al Figure 12. Porosity permeability crossplot of ten rock types derived by the FCM clustering method. standard deviation. Also, they do not overlap each other in porosity permeability crossplots so that each one has a unique boundary. Table 2 shows the description of the mentioned rock types based on porosity and permeability parameters. For each rock type, a photomicrograph with its porosity and permeability parameters is shown. 5. Rock type estimation using fuzzy logic In this section, the FCM clustering derived rock types were estimated from well log data using fuzzy logic. For this purpose, the distribution of well log data for the identified rock types was first investigated. According to figure 13, which shows an example of the distribution of neutron log values for the rock types derived in previous stage, the data sets have been fitted by a Gaussian function. The normal distribution of data by a Gaussian function is as below: /σ 2 P(x,σ,c) = e (x c)2 σ (6) 2π where f(x,σ,c)is the probability function that an observation x is measured in the data set by a mean c and standard deviation σ. This curve was used to estimate relative probability or fuzzy possibility that a data value belongs to each rock type. The methodology described here is similar to Cuddy s approach (1998) for lithofacies estimation using fuzzy mathematics. Each log data value may belong to any FCM clustering derived rock type to a degree that can be calculated from a Gaussian membership function using equation (6). Each rock type has its own mean and standard deviation, namely, for n number of rock types; there are n pairs of c and σ rock type. For example, the fuzzy possibility that a neutron log data belongs to rock type 1 is obtained by substituting c rock type 1 and σ rock type 1 in equation (6): P(NPHI) = exp( (NPHI C rock type 1) 2 /(σ rock type 1 ) 2 ) (σ rock type 1 ). (7) 2π The ratio of the fuzzy possibility for each rock type with the fuzzy possibility of the mean or most likely observation is obtained by de-normalizing equation (7). The fuzzy possibility for mean of neutron in rock type 1 is obtained by substituting NPHI by c rock type 1 in equation (7): P(c rock type 1 )= exp( (C rock type 1 C rock type 1 ) 2 /(σ rock type 1 ) 2 ) (σ rock type 1 ) 2π 1 = (σ rock type 1 ) 2π. (8) The relative fuzzy possibility R(NPHI rock type1 ) of a neutron porosity NPHI belonging to rock type 1 compared to the fuzzy possibility of measuring the mean value c rock type 1 is equation (7) divided by equation (8): R(NPHI rock type 1 ) = exp(( (NPHI C rock type 1 ) 2 /(σ rock type 1 ) 2 ). (9) Each value derived from equation (9) is now indicated to possible rock types. To compare the relative fuzzy possibilities of this equation between rock types, equation (9) is multiplied by a coefficient named relative occurrence of each rock type in the reservoir interval. For rock type 1, it is noted as nrock type 1 : F(NPHI rock type 1 ) = ( n rock type 1 ) exp( (NPHI C rock type 1 ) 2 /(σ rock type 1 ) 2 ). (10) 364

10 A fuzzy logic approach for estimation of permeability and rock type Table 2. Description of 10 rock types derived by the FCM clustering method based on porosity and permeability parameters. In this approach facies type is not important, whereas production capability based on porosity and permeability is. The obtained fuzzy possibility from equation (10) is based on neutron log data only. This process should be repeated for other logs such as sonic (DT), density (RHOB),...at this point. This will give F(DT rock type1 ), F(RHOB rock type1 ),... for rock type 1. These fuzzy possibilities are combined harmonically to give a final fuzzy possibility: 1 1 = C rock type 1 F(NPHI rock type 1 ) + 1 F(DT rock type 1 ) (11) F(RHOB rock type 1 ) 365

11 A Kadkhodaie Ilkhchi et al Table 2. (Continued.) This process is repeated for other rock types and all derived fuzzy possibilities are combined harmonically. Then, the rock type with the highest combined fuzzy possibility is taken as the most possible rock type at that point. A comparison between FCM clustering derived and fuzzy predicted rock types versus depth for the test well that was not used to model construction (the test well D) is shown in figure 14. The fuzzy possibilities in fuzzy logic are combined harmonically, whereas, statistical methods such as Bayes theorem, combine probabilities geometrically. When 366

12 A fuzzy logic approach for estimation of permeability and rock type (a) (b) (c) (d ) (e) (f ) (g) (h) (i ) ( j ) Figure 13. Histograms showing the fitted Gaussian membership function to neutron log values distribution of the rock types generated by FCM clustering method. comparing rock types that are equally likely, with similar probabilities, the harmonic combination emphasizes any indicator which suggests the lithology selection is unlikely. Secondly, fuzzy logic weights the possibilities by the square root of the proportion in the calibrating data set, whereas Bayes uses the direct proportion (Cuddy 1998). 367

13 A Kadkhodaie Ilkhchi et al Figure 14. A comparison between clustering derived and fuzzy predicted rock types versus depth for the test well D. 6. Conclusions The results of this study show that the fuzzy logic technique is an instrumental tool for the prediction of permeability and the identification of permeable and non-permeable zones in the Kangan Formation. A fuzzy c-means clustering method was useful for rock type definition in Kangan Formation. In this approach, production capability based on porosity and permeability parameters was considered to classify rock types so that each rock type has a specific boundary in porosity permeability cross plots and does not overlap with another. Finally, the algorithm based on the fuzzy possibility concept was successful for the estimation of clustering derived rock types from well log data. 7. Recommendations With regard to obtaining satisfying results when using fuzzy logic in this case study, the following is recommended: (a) In order to identifying rock types to show flow performance in reservoir, it will be useful to use clustering methods for reservoir study and modelling. Such a description of rock types will lead to ease relating well log responses to each rock type. (b) With regard to high accuracy estimation of rock types form well log data using fuzzy reasoning, it is recommended to use fuzzy logic as a powerful tool for modelling and estimation of rock types over the reservoir. 368

14 Acknowledgments The authors appreciate reservoir geology department of Research Institute of Petroleum Industry (RIPI) of Iran for sponsoring, data preparation and permission to publish this paper. References Ali M and Chawathe A 1999 Using artificial intelligence to predict permeability from petrographic data J. Comput. Geosci Arpat G B, Gumrah F and Yeten B 1998 The neighborhood approach to prediction of permeability from wireline logs and limited core plug analysis data using backpropagation artificial neural networks J. Pet. Sci. Eng Bezdek J C 1981 Pattern Recognition with Fuzzy Objective Function Algorithms (New York: Plenum) Bhatt A and Helle H B 1999 Porosity, permeability and TOC prediction from well logs using a neural network approach 61 ST EAGE, Conf. (Helsinki, Finland) pp 1 4 Bishop C M 1995 Neural Networks for Pattern Recognition (Oxford: Clarendon) p 670 Boadu F K 1997 Rock properties and seismic attenuation: neural network analysis Pure Appl. Geophys Boadu F K 1998 Inversion of fracture density from field seismic velocities using artificial neural networks Geophysica Boitnott G N, Lauten W T, Jones D H, Clerke E A and Funk J J 2006 Pore space inversions for petrophysical rock type identification: application to a large carbonate reservoir GEO 2006 Middle East Conference and Exhibition Technical Program Castagna J P, Batzle M L and Eastwood R L 1985 Relationships between compressional-wave and shear-wave velocities in elastic silicate rocks Geophysics Chang H C, Kopaska-Merkel D C, Chen H C and Durrans S R 2000 Lithofacies identification using multiple adaptive resonance theory neural networks and group decision expert system J. Comput. Geosci Chiu S 1994 Fuzzy model identification based on cluster estimation J. Intell. Fuzzy Syst Cuddy S J 1998 Litho-facies and permeability prediction from electrical logs using fuzzy logic 8th Abu Dhabi International Petroleum Exhibition and Conference SPE A fuzzy logic approach for estimation of permeability and rock type Domenico S N 1984 Rock lithology and porosity determination from shear and compressional wave velocity Geophysics Dunn J C 1973 A Fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters J. Cybern Hoen Lee S and Gupta A D 1999 Electrofacies characterization and permeability predictions in carbonate reservoirs: Role of multivariate analysis and nonparametric regression Annual Technical Conference and Exhibition in Houston (Texas, 3 6 October) SPE Huang Y, Gedeon T D and Wong P M 2001 An integrated neural-fuzzy-genetic-algorithm using hyper-surface membership functions to predict permeability in petroleum reservoirs J. Eng. Appl. Artif. Intell Ikwuakor K C 1988 Vp/Vs revisited: pitfalls and new interpretation techniques World Oil (September), pp Lim J S 2003 Reservoir permeability determination using artificial neural network J. Korean Soc. Geosyst. Eng Lim J S 2005 Reservoir properties determination using fuzzy logic and neural networks from well data in offshore Korea J. Pet. Sci. Eng Malki H A, Baldwin J L and Kwari M A 1996 Estimating permeability by use of neural networks in thinly bedded shaly gas sands SPE Comput. Appl (April) Mamdani E H and Assilian S 1975 An experimental in linguistic synthesis with a fuzzy logic control Int. J. Man-Mach. Stud Matlab user s Guide 2001 Fuzzy logic toolbox, Matlab CD-ROM Mathworks, Inc Ozer M 2005 Fuzzy c-means clustering and Internet portals: a case study Eur. J. Oper. Res Saggaf M M and Nebrija Ed L 2003 A fuzzy approach for the estimation of facies from wireline logs AAPG Bull Sugeno M 1985 Industrial Applications of Fuzzy Control (Amsterdam: Elsevier) Takagi T and Sugeno M 1985 Identification of systems and its application to modeling and control. Institute of electrical and electronics engineers transactions on systems Man Cybemetics Tatham R H 1982 Vp/Vs and lithology Geophysics Wong P M, Henderson D J and Brooks L J 1997 Reservoir permeability determination from well log data using artificial neural networks: an example from the Ravva field, offshore India Proc. SPE Asia Pacific Oil and Gas Conference (Kuala Lumpur, Malaysia, April) SPE