INTRODUCTION
Permeability controls how fluid can migrate through the reservoir. The permeability
is a key parameter in reservoir development and management because it controls
the production rate (Altunbay et al., 1997). In
general, the permeability increases with increasing porosity, increasing grain
size and improved sorting. In carbonates connectivity between pores is the main
control for the permeability. Heterogeneity occurs in carbonate reservoirs due
to variation in depositional environments and subsequent diagenetic processes.
Depositional environment is important for creating primary porosity. Generally
high energy deposits give high porosity and permeability, while low energy deposits
give low permeable intervals. Often low energy deposits may have high porosity,
but if the pore throat sizes are too small, permeability may also be low. Diagenesis
both constructs and destruct porosity. For example, cementation decreases porosity
and permeability, but early cementation may prevent compaction and thereby preserve
primary porosity, while dissolution mainly increases porosity except in case
like some satellites which may form barriers (Altunbay et
al., 1997; Babadagli and AlSalmi, 2003; Carlos,
2004; Jennings and Lucia, 2003; Kolodzie,
1980; Lucia, 1999; Mohagheh
et al., 1997).
The permeability prediction is a challenge in formation evaluation and reservoir
modeling because of difficulty to measure it directly. Knowledge of permeability
is important in building 3D reservoir models and understanding production of
oil and gas and finally development strategy. The best method for direct permeability
measurement is obtained from core plug analysis. It is measured in both vertical
and horizontal directions, commonly every 30 cm (Altunbay
et al., 1997; Babadagli and AlSalmi, 2003;
Carlos, 2004; Cuddy and Putnam, 1998;
Hambalek and González, 2003; Jennings
and Lucia, 2003; Lim and Kim, 2004; Mohagheh
et al., 1997; Taghavi, 2005).
Coring is very expensive and time consuming limiting such measurements. In addition, in some cases such as horizontal wells it is technically impossible. Small scale heterogeneities that might not affect flow on a reservoir scale are measured and these need to be upscaled. An alternative way to estimate the permeability is from electrical logs. The challenge in permeability prediction is that permeability is related more to the pore throat size rather than pore size, which is difficult to measure by logging tools. Determining permeability from well logs is also complicated by the problem of scale, well logs having a vertical resolution of typically 1/2 m compared to the 5 cm diameter of core plugs.
Fuzzy logic was introduced by Zadeh (1965) and is an
extension of conventional Boolean logic (0 and 1) developed to handle the concept
of partial truth values between completely true and completely false values.
In Fuzzy sets, everything is a matter of degrees. Therefore, an object belongs
to a set to a certain degree. The Fuzzy logic can be used as a simple and useful
predictor method in uncored wells (Cuddy and Putnam, 1998;
Hambalek and González, 2003; Lim
and Kim, 2004; Mohagheh, 2000; Saggaf
and Nebrija, 2003; Saggaf and Nebrija, 2000; Taghavi,
2005).
In essence, Fuzzy logic maintains that any interpretation is possible but some
are more likely than others. One advantage of Fuzzy logic is that we never need
to make a specific decision. Other benefits of using Fuzzy logic is that it
can be described by established statistical algorithms; and computers, which
themselves work in ones and zeros, can do this effortlessly for us. Conventional
techniques try to minimize or ignore the error. Fuzzy logic asserts that there
is useful information in this error. The error information can be used to provide
a powerful predictive tool for the geoscientist to complement conventional techniques
(Cuddy and Putnam, 1998; Hambalek
and González, 2003; Lim and Kim, 2004; Mohaghegh,
2000; Saggaf and Nebrija, 2003; Saggaf
and Nebrija, 2000; Taghavi, 2005).
Fuzzy mathematical techniques have been applied to solve various petroleum
engineering and geological problems in the past, involving mainly classification,
identification, or clustering. Cuddy and Putnam (1998)
and Cuddy and Glover (2002) used Fuzzy logic to predict
permeability and lithofacies in uncored wells to improve welltowell log correlations
and 3D geological model building. In Hambalek and González
(2003) made some modification to the cuddy’s works in a more or less
similar study. In addition, Saggaf and Nebrija (2003)
used Fuzzy logic approach for the estimation of facies from wireline logs in
a field in Saudi Arabia. Taghavi (2005) applied Fuzzy
logic to Improve Permeability Estimation in a complex Carbonate Reservoir in
Southwest of Iran.
In this study, we apply the Fuzzy Logic inference method to determine the permeability
values in uncored wells based on data from wireline logs in two heterogeneous
sandy oilbearing reservoirs in Persian Gulf. For this purpose a Fuzzy model
based on the method proposed by Hambalek and González
(2003) was developed to predict permeability values in two Iranian huge
heterogeneous carbonate reservoirs. It should be mentioned that Hambalek
and González (2003) implemented their model in a Sandy deposit reservoir,
but in this study it has been tried to apply their concept for carbonates which
are much more complex.
In addition, during this study a modification to defuzzification stage of the Hambalek and González’s technique will be proposed. This new approach will diminish the high amount if uncertainty in predicted permeability values of carbonates. Moreover, the proposed approach will be justified by testifying the model in two studied carbonate reservoirs named Sarvak and Asmari formations.
It should be mentioned that definition of Fuzzy membership functions is fundamentally
based on the probability theory. Probability theory has been implemented in
some previous studies to quantify grayness of fuzziness. Although, the reason
behind random events might hardly be understood, but it has been shown in previous
studies that how Fuzzy logic could be beneficial for bring meaning to its concept.
All the studies have accepted the premise that any interpretation is possible
although some are more likely than others (Hambalek and González,
2003).
STUDIED RESERVOIR CASES DESCRIPTION
Sarvak formation: Sarvak formation hosts huge oil reserves in Southwest
Iran. It is composed mainly of limestone with occasional dolomite intervals.
The formation is divided into two parts by a thick interval of argillaceous
carbonates. The upper Sarvak Formation which is used in this study has been
deposited in a carbonate ramp setting during mid Cenomanian to early Turoninan
(midCretaceous) (IOOC, 2003).
Upper sarvak formation is a very heterogeneous reservoir where heterogeneity results from both depositional environment and diagenetic processes. Depositional setting forms different faces as an effect of sea level fluctuation and energy during sedimentation. Low energy environments forms low porosity sediments such as in lagoons and in the shallow open marine, outer shelf and intrashelf basin. These environments are dominated by wackestone and mud dominated packstones. High energy environments form high porosity sediments such as reefs and shoals with mainly grain dominated packstones, grainstones, floatstones and rudstones.
The diagenetic processes in upper Sarvak have both constructed and destructed
porosity. Dissolution and karstification (uppermost part of the reservoir) produce
porosity in both mudand grainsupported intervals during exposure because of
sea level fall. They occur as vuggy and more rarely moldic porosities. Cementation
diminishes primary porosity during early and burial diagenesis. Early and late
cements fill the primary intergranular porosity in the high energy deposits
and may create barrier for hydrocarbon movement. The effect of dolomitization
in altering porosity depends on dolomite types. Generally dolomitization associated
with dissolution marks high porosity and permeability, while dolomite within
lagoonal and intrashelf basin deposits did not change porosity. Compaction has
been characterized by core studies. It has destructed porosity by decreasing
pore space in both mud and grain supported deposits. Fracturing as recognized
in cores and by mud loss has increased permeability in some of the intervals
(IOOC, 2003) (Fig. 1).
In this area of study five continuous cored wells were available. Porosity and permeability data are measured from plugs of these wells and these core data were used for validation of the predicted data. However, core and log data from three wells used to derive a model and this model used to predict permeability in two blind testing wells with available core data to check the accuracy of prediction. The conventional well logs used besides the core permeability information to develop models are Sonic and Neutron logs which are available in all wells of studied cases.
Asmari formation: Another giant petroleum reservoir in Persian Gulf which was used in this study is Asmari Carbonate Reservoir with an average thickness of 395 meters in SouthWest of Iran. The reservoir is isolated from adjacent reservoirs by sealing faults in south and west flanks.
Petrographycally point of view, Asmari formation composed mainly of Carbonates
which include Dolomites, Shale and Marls. Generally, Dolomitization in Asmari
and Sarvak formations follow the Sabkha Dolomitization System of Persian Gulf.
Productivity of Asmari reservoir systems is affected mainly by secondary parameters
(Secondary Porosity and Permeability) and diagensis processes such as Dolomitization,
Techtonic activities (Fractures and Micro fractures) and salt dome activity
of underlying gachsaran (IOOC, 2003).

Fig. 1: 
Schematic Location of two studied cases; Sarvak and Asmari
formation which are two huge Iranian carbonates reservoirs in Persian Gulf
(IOOC, 2003) 
Same as the previous studied reservoir case, there are five continuous cored wells for Sarvak reservoir in this study, all with available well logs information; in which three of them used as model description whereas other two wells used as a blind testing for testifying the developed Fuzzy models. The input data are selected to be the same as the previous studied case.
Permeability prediction applyingHhambalek and González approach:
It was Cuddy in 20002003 that first used Normal Distribution concept for membership
function definition in constructing Fuzzy Logic model for the purpose of permeability
estimation. Cuddy applied his model in a Sandy Reservoir with almost acceptable
results. Later, Hambalek and González (2003) also
employed Cuddy’s approach with some modification in defuzzification stage
with much more reliable results. That study was also testified in Sandy Deposits.
Although, that technique was so flexible in sandy deposits but due to high complexity of carbonates it is not so efficient when subjects to carbonate deposits. To show that, we applied that technique in two carbonate reservoirs of Iran in Persian Gulf. In addition, a modification to this technique would be proposed in the next stage to transform this technique to a much more reliable technique in carbonate deposits.
It should be mentioned that normal distribution is completely determined by
two parameters: mean and variance. Its bell shape is very familiar to all us
and its application is often justified with symmetric histograms derived from
the sample data. The variance (the standard deviation squared) depends on the
hidden factors and measurement errors and it can be seen like the fuzziness
(spread) about a most probably value of occurrence (the mean). This interpretation
is a key to the method because it allows to consider the possibility of observing
any particular value of the analyzed variable but equally to accept that some
observations are more probable than others (Hambalek and
González, 2003).
The probability density that an is measured observation x occurred in a data set which is described by a mean μ and standard deviation σ will be determine by the normal distribution function:
This curve is used to estimate the relative probability or Fuzzy possibility that a data value belongs to a particular data set. For example if a permeability Geocategory has a porosity distribution with a mean μΦ and standard deviation σΦ (these values are simply derived from the calibrating or conditioning data set, usually core data), the Fuzzy possibility that a welllog porosity value Φ_{eff., x} is measured in this Permeability GeoCategory type can be estimated using Eq. 2:
Fuzzy logic applied in this study was based on establishing the Fuzzy relation between Permeability Geocategories (Let’s call it Bin) derived from core data and some associated electrical welllogs. Then, using this Fuzzy relation we can predict these permeability values in those wells without core information but with coincident well logs. It should be mentioned that each Permeability Geocategory (Bin) is defined in such way that have same characteristic.
For instance, one may define one bin as very high permeability values which
is belong to values of permeabilities grater than 1500 md. Defining permeability
values boundaries are of important factors in establishing a perfect Fuzzy model.
In this study log reading characteristics are considered besides the permeability
values to in defining bin boundaries. However, in this study the Permeability
geocategories with different biz sizes (number of data in each bin) are used.
Table1 shows the summarized information related to bin boundaries
used in this study for each studied case.
Where, there are several permeability geocategories (Bins) in a well, the porosity
value Φ_{eff., x} may belong to any of these Bins, but some are
more likely than others. Each of these bins has its own mean and standard deviation
such that for N Permeability Geocategory there are N pairs of μ and σ.
Table 1: 
Permeability Geocategories information used for Fuzzy model
description in each reservoir case. Permeability boundaries and corresponding
bin size also mentioned 

If the porosity measurement is assumed to belong to Bin f_{i}, the Fuzzy
possibility that porosity Φ_{eff., x} is measured (logged) can
be calculated similarly using Eq. 2 by substituting μ_{i} and σ_{i}, corresponding to porosity, the Fuzzy possibilities can
be computed for all N Bins. These Fuzzy possibilities refer only to particular bins f_{i} and
cannot be compared directly, as they are not additive and do not add up to 1.
The ratio of the Fuzzy possibility for each Permeability Geocategory with the
Fuzzy possibility of the mean or most likely observation could be achieved by
denormalizing Eq. 2.
The relative Fuzzy possibility R (x) of porosity Φ_{eff., x} belonging to f_{i}th Geocategory compared to the Fuzzy possibility of measuring the mean value μ_{Φσ} calculated as:
Each Fuzzy possibility is now selfreferenced to possible bins. To compare these Fuzzy possibilities between bins, the relative occurrence of each Geocategory in the well must be taken into account. This is achieved by multiplying Eq. 3 by the square root of the expected occurrence of geocategory f_{i}. If this is denoted by n_{fi}, the Fuzzy possibility of measured effective porosity Φ_{eff, x} belonging to permeability geocategory μ_{Φc} is:
The Fuzzy possibility Ffi, Φ_{eff}, (Φ_{eff, x}) is based on the effective porosity logs alone. This process is repeated for a second parameter for example Neutron Porosity Log. This will give:
where, Nphi_{x} is the Neutron Porosity log reading value in a specific horizon and μ_{1Nphi}, σ_{1Nphi} are respectively the Mean and standard deviation values of the Neutron Porosities Distribution belonging to Permeability Geocategory f_{i}. Therefore, F_{f1Nphi} (Nphi_{x}), will be the Fuzzy possibility of measured Neutron Porosity value Nphi_{x} belonging to Permeability Geocategory f_{i} with mean μ_{1Nphi} and standard deviation value σ_{1Nphi}.
As in this study Sonic Porosity and Neutron Porosity logging data are used as input parameters, the Fuzzy possibility of measured Sonic Porosity log values F_{f1ΔT} belonging to Permeability Geocategory f_{i} with mean μ_{1ΔT} and standard deviation σ_{1ΔT} is calculated as follow:
Finally, these Fuzzy possibilities are combined harmonically to give a “combined Fuzzy possibility”. In this example for a measurement x in a specific horizon for Neutron Porosity and Sonic Porosity, the combined Fuzzy possibility for these measurements to belong to a data set, let say Permeability Geocategory f_{i} will be:
This process is then repeated similarly for other defined Bins. Note that, in defuzzification stage a value should be assigned to precede fuzzified input values. These values are highly dependent with the method used for defuzzification. As the defuzzification technique gives much more flexible values to each fuzzified output processed value, the model will be more flexible to predict an accurate result values.
For defuzzification stage Hambalek and González (2003)
proposed a technique as follow: the two highest Fuzzy possibilities are taken
as the most probable categories for that log measurements for that depth. The
simulated horizontal permeability value is proposed as a weighted mean of the
representative values of the two most probable categories of permeability inferred
through the Fuzzy procedure (Hambalek and González,
2003).
where, KH (h) _{Simulated} is Simulated horizontal permeability value
for specific log measurements in depth, are
respectively Representative horizontal permeability values of the first and
second most probable predicted ith Geocategory, F_{T, I, (1)}, F_{T,
I, (2)} are respectively Combined Fuzzy possibilities associated with the
first and second most probable predicted ith Geocategory calculated for a specific
depth h.
Another consideration should be also taken into account in which the representative
permeability values, corresponded to each permeability bin defines (Hambalek
and González, 2003), considered each mean, median, maximum and minimum
values of each permeability bins as the representative permeability value for
each bin. Finally they proposed the minimum value as the best representative
permeability value that results in lowest amount of error (Table
2, 3).
Table 2: 
Permeability representative values for each permeability geocategory
(Bin) for studied reservoir case number 1; Sarvak Formation 

Table 3: 
Permeability representative values per each permeability geocategory
(Bin) for studied reservoir case number 2; Asmari Formation 

Determining number of Permeability Bines is directly related to the amount of the available data. As it was explained, to avoid statistical errors there should be at least 30 readings in each Geocategory (Bin Size). In this study different number of Permeability Geocategories was tested and results from the model with four bins were much more precise.
Furthermore, to quantify each point error in permeability determination Hambalek
and González (2003) introduced the Relative Absolute Error concept,
denoted by RAE. The RAE is defined by the difference between the simulated value
and the core reported one divided by the core derived referenced value in each
specific depth. This value could be presented as percentage.
Where:
RAE (h) 
: 
Relative absolute error at depth h 
KC (h) 
: 
Calculated corepermeability value at depth h 
KS (h) 
: 
Calculated simulatedpermeability value at depth h 
As it was mentioned, there are two testing wells per each studied reservoir
case; therefore the RAE in each testing well should be calculated separately
applying different representative values. Table 4 and 5
represent RAE values corresponding to apply Hambalek and
González (2003) approach in Sarvak and Asmari formation, respectively.
Based on the detailed information above (Table 4, 5)
total minimum errors occurred in case of selecting Minimum Values of each Permeability
Geocategory (Bin) as Representative Value. Therefore, in this study the minimum
values of each bin are selected as permeability representative values for the
constructed Fuzzy model. Figure 25 show
the predicted values versus core reposted ones in two reservoir cases applying
this approach by selecting minimum values as representative values.
New formula, defuzzification modification: The complexity of permeability
trends in carbonates due to secondary parameters makes it much more difficult
to estimate permeability values in comparison with Sandy reservoirs. In this
study, a modification in defuzzification stage of Hambalek
and González (2003) approach is proposed to make this technique much
more convenient and accurate for carbonates. As it was mentioned, different
Permeability Geocategory numbers were testified and finally the result of four
permeability bins was found to be much more reliable with lowest amount of errors.
Thus, in this study four permeability Geocategories was proposed.
In this stage, the permeability Geocategories are defined same as the previous stage. But during defuzzification stage, the formula below is proposed. This is in such way that guaranties all Geocategories contribution in defining final simulated permeability values. This formula could be achieved from the equation below:
In which the values of A and B are defined as below:
and the parameters are defined as:
KH (h)_{simulated} is Simulated horizontal permeability value for specific
log measurements in depth h, F_{i (k)} (h):(where: k=1, 2) are relatively
the first and second biggest combined Fuzzy possibility values (Eq.
7) associated with the first and second most probable predicted iGeocategory
for log measurement vales in depth h, G_{i (k)} (h): (where: k=1, 2)
are relatively the first and second lowest combined Fuzzy possibility values
(Eq. 7) associated with the first and second most probable
predicted iGeocategory for log measurement values in depth h,
are, respectively representative horizontal permeability values of the first
and second most probable predicted Ith Geocategory, are,
respectively the two lowest representative horizontal permeability values regarding
to Minimum values of each Permeability Bin, are,
respectively the two lowest representative horizontal permeability values regarding
to Maximum values of each Permeability Bin.
Table 4: 
Relative absolute error values in two testing wells applying
hambalek and gonzález approach by different representative values
in studied case number 1; sarvak formation 

Table 5: 
Relative absolute error values in two testing wells applying
hambalek and gonzález approach by different representative values
in studied case number 2; asmari formation 

Table 6: 
Representative permeability values for each permeability geocategory 

Note that, this formula guaranty that the most probable category receives the
heaviest weight. Moreover, using a combination of representative values from
different Geocategories with different defined weights will results much more
precise simulated permeability values in which data distribution characteristics
were comprehensively considered. However, any classification must be sure of
having enough sample observations inside each class for guarantying the statistical
robustness of the results. A reasonable statistical sample size is around 30.
The distribution of bin boundaries depends on the range of expected permeabilities,
same as described by Hambalek and González (2003),
an example:
To show the calculation procedure for a considered specific horizon h, permeability
prediction demonstration has been shown in this section. Suppose for the four
permeability Geocategories (Bins) the different representative values are as
Table 6. Thus, according to sthe formulas of ,
the values of A and B would be:
Finally, the calculated permeability value for the horizon h would be estimated.
RESULTS AND DISCUSSION
According to the Table 7, relative absolute errors of predicted
permeability values in all four blind testing wells belonging to either Sarvak
or Asmari formation show lowest amount of error when comparing to core reported
permeabilities. Thus, the precision of the model will increase significantly
by defuzzification proposed in this study. The results could be seen in Fig.
25.
Although, the predicted results have a perfect match in almost all part of
the wells sections Fig. 2 and 3, but to
see how accurate is the new model in prediction of permeability values in unlogarithmic
scales we select some sections of the wells with the predicted values by two
different approaches mentioned in this article versus core reported values.
Table 7: 
Relative absolute error values in two testing wells of each
reservoir case by new proposed Fuzzy approach RAE from the Hambalek and
Gonzalez approach are also mentioned for comparison 


Fig. 2: 
The calculated permeability values versus depth of well 1
and well 2, obtained from different methods for the Sarvak formation: (A)
Hambalek and González method, (B) New introduced Fuzzy method. In
all diagrams, continues line shows the predicted permeability and solid
circles indicate the core reported permeability. Note that, the results
obtained from new introduced Fuzzy method shows a better match in almost
all parts of the well sections than Hambalek and Gonzalez’s method 

Fig. 3: 
The calculated permeability values versus depth of well 1
and well 2, obtained from different methods for the Asmari formation: (A)
Hambalek and González method, (B) New introduced Fuzzy method. In
all diagrams, continues line shows the predicted permeability and solid
circles indicate the core reported permeability. Note that, the results
obtained from new introduced Fuzzy method shows a better match in almost
all parts of the well sections than Hambalek and González’s method 
As it is clear in Table 8, the permeability values predicted
by new proposed technique are much more close to the reported core values which
are considered as exact values.
As it was mentioned Sarvak and Asmari formations, two huge carbonate petroleum bearing reservoirs, are selected for case studies in this article. In each case, two wells with core and log information used as blind testing to testify the accuracy of each Fuzzy logic model constructed by two different approaches described above.
Simulated permeability values from the new modified Fuzzy model are plotted versus depth in Fig. 2 and 3 for both studied reservoir cases. The results for the first studied reservoir case of Sarvak formation is presented in Fig. 4, whereas Fig. 5 is related to another studied case of Asmari formation.
In addition, core reported values are also plotted which are considered to be the actual and exact values. According to the graph, although the general trend of simulated values follows the actual core values, there are some horizons in which the model is not so flexible in prediction and the simulated values tend to follow a straight line through
the average actual core reported ones.
Table 8:  Predicted
permeability values versus core reported ones applying two different approaches.
Data are belonging to some selective horizon of four blind testing wells


Predicted permeability values versus core reported ones in
Sarvak Formation applying Hambalek and González approach (Method
1) and New Proposed technique (Method 2) 

Fig. 4: 
Simulated permeability values versus core reported ones for
each applied technique. Results from the Fuzzy model mounted with new defuzzification
algorithm occur in a wider range just close to core reported range. Furthermore,
the general trend is close to line of 45 degree which represents the high
precision of the simulated values. The graph is related to the testing wells
of the first studied case, Sarvak formation 

Fig. 5: 
Simulated permeability values versus core reported ones for
each applied technique. Results from the Fuzzy model mounted with new defuzzification
algorithm occur in a wider range just close to core reported range. Furthermore,
the general trend is close to line of 45 degree which represents the high
precision of the simulated values. The graph is related to the testing wells
of the first studied case, Asmari formation 
This pattern could be observed in some
parts of both Sarvak and Asmari formations (Fig. 2, 3).
Furthermore, to have a better understanding of the accuracy of the predicted
results the simulated values versus core reported ones are plotted in Fig.
4 and 5. As it is clear, the predicted values occur by
new method in the actual range of core derived ones whereas the result from
coarser bins definition will occur in a narrower range (Fig. 4B,
5B).
CONCLUSION
In this study we introduced a modification in defuzzification stage of the
model provided by Hambalek and González (2003)
to propose a new Fuzzy based method which is much more convenient and precise
for carbonates. In addition, we used two studied carbonate reservoir cases to
justify our proposed technique and compared its results with the previous Fuzzy
technique.
The method simply uses some basic selected Porosity well log data sets such Neutron and Sonic porosity well logs rather than depending on new complicated logging technologies. The reason behind using porosity logs as input parameters is the close relationship between permeability and porosity. This relationship is a function of particle sizes, shapes, sorting, compaction and degree of cementation etc.
The results of permeability values by Hambalek and González
(2003) approach in both carbonate studied cases showed an acceptable correspondence
with measured core permeability in general trend but there were some horizon
in which the model was not so flexible in prediction. However, applying Fuzzy
model with the proposed modification increased the accuracy of predicted permeability
values and makes the predicted values to follow the more complex fluctuation
trends in almost all horizons of both case studies.
In this study, we used a spatial algorithm for assigning representative value to each permeability geocategory, in which permeability geocategories classified in four group and a combined Fuzzy possibility with special averaging was proposed in a way that characteristics of all Geocategories are considered to predict much more precise results.
In this study, two huge carbonate reservoirs used as case study which are Sarvak and Asmari formations in Persian Gulf located in south west of Iran. To develop Fuzzy logic model in each case three wells with available core and log information were used. Moreover, two wells with core and log information were used as blind testing in each reservoir case for testifying model’s predictions.
Cross correlation between simulated permeability values versus cored reported ones confirms the increase in accuracy of predictions when applying new approach in defuzzification. As it is clear, the general trend of the plotted values follow the line of 45 degree line which means that the predicted values are very close to the actual core reported ones.
ACKNOWLEDGMENTS
This study was prepared under the supervision and permission of NIOCExploration Directorate in cooperation with Amir Kabir University of Technology. The authors would like to thank Dr.A.R.Rabani, Mr. N. Sabeti, Mr. S.A. Miri, for their support and permissions to publish this paper. We are also especially grateful to Amir Kabir University staff, Dr. M. Irannajad, Dean of Mining, Metallurgical and Petroleum Engineering for help and close cooperation. The authors greatly appreciate the financial supports of the Institute of Geophysics and the Research Council of the University of Tehran which enabled the second author for this research. We appreciate the critical reading by the arbitration committee and we would greatly appreciate enlightening suggestion and insightful comments.