1. International Journal of Petroleum and Geoscience Engineering (IJPGE) 2 (2): 120-129, 2014
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.
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.
Accepted:08 June 2014 © 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].

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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 ra is a positive constant. Hence, a data point will have a high density value if it has many
neighboring data points. The radius ra 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 Xc1 be the point

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selected and Dc1 its density measure. Next, the density measure for each data point Xi is calculated by
the formula:
= .exp (-
| |
) (3)
Where rb is a positive constant. Therefore, the data points near the first cluster center Xc1 will have
significantly reduced density measures, thereby making the points unlikely to be selected as the next
cluster center. The constant rb defines a neighborhood that has measurable reduction in density measure.
The constant rb is larger than ra because cluster centers should not be close to each other; normally rb is
equal to1.5ra [8].
After recalculating the density measure for each data points, the next cluster center Xc2 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.

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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.

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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).

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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.

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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:
yi= β0 + β1xi + ei 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:
yi= β0 + β1xi1 + β2xi2+...+ βkxik +ei 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.

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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.

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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.

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