International Journal of Petroleum and Geoscience Engineering (IJPGE) 2 (2): 120- ISSN 2289-4713 Academic Research Online Publisher Research paper Predicting Porosity through Fuzzy Logic from Well Log Data Shirko Mahmoudi a a Mining Engineering Department, Isfahan University of Technology, Isfahan, Iran * Corresponding author. Tel.: 09187769902; E-mail address: s.mahmoudi@mi.iut.ac.ir A b s t r a c t Keywords: Porosity, Fuzzy Logic, Sugeno System, Subtractive Clustering. Accepted:08 June 2014 Porosity is one of the most important characteristics for modeling reservoir. In recent years, some new methods for estimation have been introduced, which are more applicable and accurate than old methods. Fuzzy logic has shown reliable results in petroleum modeling area for describing reservoir characteristics. In this study, a Sugeno fuzzy model has been formulated to predict porosity. In order to select the number of membership function, subtractive clustering method was utilized through Gaussian membership functions. Another technique for predicting porosity was multiple linear regression to compare with fuzzy logic technique. Results indicated that correlation between real value from core data and the predicted value by fuzzy logic was more accurate than multiple linear regression technique in this scope. Academic Research Online Publisher. All rights reserved. 1. Introduction Determining Reservoir properties is a complex process, which uses all accessible data to provide precise models in order to estimate reservoir performance. It is difficult to predict core parameters through well log responses and is usually associated with error [1]. One of the most important characteristics in reservoir is porosity that relates to the amount of fluid in reservoirs [2]. There are some methods to determine this property, such as statistical methods and novel techniques like intelligent methods. The common statistical technique is multiple regression analysis. The simplest form of regression analysis is to find a relationship between the input logs and the petrophysical properties [3]. One of the reliable intelligent techniques is fuzzy logic theory, which was first introduced by Zadeh, 1965. A fuzzy inference system (FIS) is a method to formulate inputs to an output using FL [4].
A fuzzy set allows for the degree of membership of an item in a set to be any real number between 0 and 1[5]. Once the fuzzy sets have been defined, it is possible to use them in constructing rules for fuzzy expert systems and in performing fuzzy inference. This approach seems to be suitable to well log analysis as it allows the incorporation of intelligent and human knowledge to deal with each individual case. However, the extraction of fuzzy rules from the data can be difficult for analysts with little experience [6]. This could be a major drawback for use in well log analysis. If a fuzzy rule extraction technique is made available, then fuzzy systems can still be used for well log analysis [7]. The process of fuzzy inference involves setting the membership functions and establishment of fuzzy rules. There are three Fuzzy modeling methods that are belong to three categories, namely Mamdani, the relational equation, and the Takagi, Sugeno and Kang (TSK) and TSUKAMOTO fuzzy model. Takagi and Sugeno, 1985, is a FIS, in which output membership functions are constant or linear and are extracted by clusters within clustering process [8]. Each of these clusters describe a membership function (A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value or degree of membership between 0 and 1) [9]. Each membership function generates a set of fuzzy if then rules for formulating function. Each membership function generates a set of fuzzy if then rules for formulating inputs to outputs. Another method which was utilized in this study is multiple linear regression for finding relationship between target and inputs data. 2. Fuzzy Logic First step for predicting porosity after removing outliers is to normalize data between [0, 1] in order to transform numeric data into fuzzy domain using following formula: X normal = ( ) (1) Then, inputs and output data are clustered using subtractive clustering method. Since each data points is a candidate for cluster centers, a density measure at data point Xi is defined as: = (- ) (2) Where r a is a positive constant. Hence, a data point will have a high density value if it has many neighboring data points. The radius r a defines a neighborhood; data points outside this radius contribute only slightly to the density measure. After the density measure of each data points has been calculated, the data point with a highest density measure is selected as the first cluster center. Let X c1 be the point 121 P a g e
selected and D c1 its density measure. Next, the density measure for each data point Xi is calculated by the formula: =.exp (- ) (3) Where r b is a positive constant. Therefore, the data points near the first cluster center X c1 will have significantly reduced density measures, thereby making the points unlikely to be selected as the next cluster center. The constant r b defines a neighborhood that has measurable reduction in density measure. The constant r b is larger than r a because cluster centers should not be close to each other; normally r b is equal to1.5r a [8]. After recalculating the density measure for each data points, the next cluster center X c2 is selected and all of the density measures for data points are revised again. This process is repeated until a sufficient number of cluster centers are generated. An important parameter in subtractive clustering, which controls the number of clusters and fuzzy if then rules, is clustering radius. This parameter could take values between the range of [0, 1]. The use of a smaller cluster radius led to more and smaller clusters and also more rules. In contrast, a large cluster radius yielded a few large clusters in the data resulting in few rules. The inputs are combined logically using the AND or OR operator to produce output response values. Minimum operator for the AND operator and the maximum operator for the OR operator have been suggested. In this study, Probor and Prod operators are used as or and and operators, respectively [8]. Since decisions are based on the testing of all rules in FIS, the rules should be combined to make a decision. Aggregation is the process, by which the fuzzy sets that each of these represents the output of each rule, are combined into a single fuzzy set. There are several aggregation methods, such as: max method (maximum), (probabilistic or), and sum. Max method is chosen in this paper. The last step to build fuzzy system is deffuzzification, which is optional and involves conversion of the derived fuzzy number to the numeric data in real world domain. In this paper, weighted average method was selected for defuzzification. In this method, the output is obtained by the weighted average of each output in a set of rules stored in knowledge base of the system. In this study, six parameters were used to estimate porosity, such as: caliper log (CALI), sonic log (DT), neutron log (NPHI), bulk density (RHOB) and effective porosity (PHIE), and water saturation (SW). Figure.1 shows graph of each input. 122 P a g e
Fig.1: normalized variables graph upon depth In this study, Sugeno fuzzy inference system was used with Gaussian membership functions according to subtractive clustering. The optimal radius for demonstrating clusters was 0.5. Gaussian membership function that is described by following formula: Gaussian(x; c, ơ) = (4) Where c and ơ are mean and standard deviation for each cluster, respectively. Gaussian membership functions are constructed from mean and σ values of the clusters. The mean represents the cluster centers and σ is derived from [7]: σ = (radii maximum data minimum data))/sqrt (5) Through subtractive clustering method, seven clusters were demonstrated. Figure 2 shows each cluster center. Fig.2: clusters center using subtractive clustering Figure 3 describes Gaussian membership functions for each input. 123 P a g e
Fig.3: Gaussian membership functions for input variables Seven rules were set for estimating porosity from inputs. These fuzzy if-then rules are formulated according to following expressions: 1. if (CALI is in1mf1)and(dt is in2mf1)and(nphi is in3mf1)and(phie is in4mf1)and(rhob is in5mf1)and(sw is in6mf1) then (porosity is out1mf1) 2. if (CALI is in1mf2)and(dt is in2mf2)and(nphi is in3mf2)and(phie is in4mf2)and(rhob is in5mf2)and(sw is in6mf2) then (porosity is out1mf2) 3. if (CALI is in1mf3)and(dt is in2mf3)and(nphi is in3mf3)and(phie is in4mf3)and(rhob is in5mf3)and(sw is in6mf3) then (porosity is out1mf3) 4. if (CALI is in1mf4)and(dt is in2mf4)and(nphi is in3mf4)and(phie is in4mf4)and(rhob is in5mf4)and(sw is in6mf4) then (porosity is out1mf4) 5. if (CALI is in1mf5)and(dt is in2mf5)and(nphi is in3mf5)and(phie is in4mf5)and(rhob is in5mf5)and(sw is in6mf5) then (porosity is out1mf5) 6. if (CALI is in1mf6)and(dt is in2mf6)and(nphi is in3mf6)and(phie is in4mf6)and(rhob is in5mf6)and(sw is in6mf6) then (porosity is out1mf6) 7. if (CALI is in1mf7)and(dt is in2mf7)and(nphi is in3mf7)and(phie is in4mf7)and(rhob is in5mf7)and(sw is in6mf7) then (porosity is out1mf7). 124 P a g e
Fig.4: generated rules according to fuzzy inference system The set rules to map inputs to output and their graph are depicted in Figure 4: Real porosity from core data and predicted output using fuzzy logic has been shown as cross plot in Figure 5. Fig.5: cross plot of real and predicted porosity using fuzzy logic In Figure 8, the predicted output has been mapped on real output for well 1using fuzzy logic technique. 125 P a g e
Fig.8: mapped graph of predicted porosity on real porosity using FL 3. Multiple linear regression (MLR) model A Linear Regression estimates the coefficients of the linear equation involving one or more independent variables, which predict the value of dependent variable [3]. A simple linear regression illustrates the relation between dependent variable y and independent variable x based on the regression equation: y i = β 0 + β 1 x i + e i i= 1, 2, 3,, n (5) According to the multiple linear regression model, dependent variable is related to two or more independent variables. The general model for k variable is of the form: y i = β 0 + β 1 x i1 + β 2 x i2+...+ β k x ik + e i i= 1, 2, 3,,n (6) Dependent variable is porosity value and independent variables are CALI, DT, NPHI, PHIE, RHOB and SW values. Table.1 shows statistical parameters of each input. There are some methods for selecting linear regression variables, which allow specify how independent variables enter the analysis. The simple linear regression model is used to find the straight linear that fits the data. Models that involve more than two independent variables are more complex in structure but can still be analyzed using multiple linear regression techniques [10]. Using different methods, a variety of regression models can be constructed from the same set of variables. 126 P a g e
Table. 1: Statistical parameter of each input CALI DT NPHI PHIE RHOB SW Mean 0.775 0.520 0.566 0.964 0.570 0.550 Median 0.779 0.540 0.580 0.975 0.602 0.560 Mode 0.732 0.463 0.563 0.954 0.795 0.000 Std. Deviation 0.070 0.232 0.173 0.038 0.205 0.265 Variance 0.005 0.054 0.030 0.001 0.042 0.071 Skewness -0.212-0.070-0.617-2.333-0.303-0.356 Std. Error of Skewness 0.053 0.053 0.053 0.053 0.053 0.053 Kurtosis 0.003-1.230 0.558 6.221-1.086-0.536 Std. Error of Kurtosis 0.105 0.105 0.105 0.105 0.105 0.105 Five methods have been described below: Enter: A procedure for variable selection, in which all variables in a block enter a single step. Stepwise: At each step, independent variable with the smallest probability of F enters that does not exist in the equation. Previous variables in the regression equation are removed if their probability of F becomes sufficiently large. The method terminates when no more variables are eligible for inclusion or removal. Remove: A procedure for variable selection, in which all variables in a block are removed in a single step. Backward Elimination: A variable selection procedure, in which all variables enter an equation and then sequentially removed. The variable that has the smallest partial correlation with the dependent variable is first considered for removal. If it meets the criterion for elimination, it is removed. After removal of the first variable, then the residual variable with the smallest partial correlation in the equation is considered. The procedure stops when there are no variables in the equation that satisfy the removal criteria. Forward Selection: A procedure on the stepwise variable selection, in which variables sequentially enter the model. The first variable considered to enter the equation is the one with the largest positive or negative correlation with the dependent variable. This variable enters the equation only if it satisfies the criterion for entry. If the first variable enters, the independent variable not in the equation that has the largest partial correlation is then considered. The procedure stops when there are no variables that meet the entry criterion. 127 P a g e
Result of all methods was similar to each other for the above-mentioned variables. Correlation between predicted porosity using multiple regression method and real core data has been depicted in figure 6. Fig.6: cross plot of real and predicted porosity using multiple regression Correlation coefficient was obtained for porosity by fuzzy logic and multiple linear regression in Figure 7 for five wells. Fig.7: Correlation coefficient for FL & MLR in 5 wells 4. Conclusion In this study, fuzzy technique was utilized with Gaussian membership functions and sugeno inference system to estimate porosity for five wells. Subtractive clustering method was used to select the number of membership function, and probabilistic defuzzification method was used to transform our result into real domain world. Other technique was multiple linear regression, which was obtained by five methods. Results indicated that fuzzy logic had a high accuracy in all five wells compared to multiple linear regression. 128 P a g e
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