INTRODUCTION
Many oil reservoirs have heterogeneity in rock properties. Understanding the
form and spatial distribution of these heterogeneities is fundamental to the
successful characterization of these reservoirs. Challenges to reservoir characterization
are to identifying what types of reservoir heterogeneities are most relevant
to fluid flows so that the right types of data can be acquired and to determining
how to build a robust reservoir models with limited subsurface information and
often times typically in a short time frame (Farzadi and
Hesthammer, 2007).
Geological reservoir modeling can be divided into two parts: structural modeling
and property modeling (Lyndon, 2001). Structural modeling
is the process of building the reservoir skeleton structure or the reservoir
3D grid. Properties will later be distributed in such a structure in the property
modeling stage which can be done with the stochastic or deterministic approach.
As it is perfectly clear, in more than one dimension, the variogram generally has directional properties. Here, variograms should be generated and compared for different directions. In this study, a complete directional variography for permeability has been mounted. Horizontal variograms in different directions are computed and compared to determine major and minor anisotropy directions for each zone. In addition, the variogram map for this purpose was mounted which could be applied as an easy and fast way for investigation of anisotropy. The variogram map is a representation of variograms that are calculated in several different orientations. The center of variogram map shows (0, 0) lag distance and the lag is increasing outward from this center in several different directions (either in + or in – direction). The color changes on this map show the variance. The changes in the variance can show the major and minor directions of anisotropy and therefore it can works instead of different directional variograms.
In this study, the petrophysical, seismic and geological interpretation results will be integrated and a structural model for the Arab formation will be determined. Then, property models for porosity, shale volume and water saturation will be generated using both deterministic and stochastic approaches. Seismic attributes will be used as a secondary variable to help property distribution. This study focuses on the permeability property so the Sequential Gaussian technique of the stochastic approach was applied for distributing permeability within the reservoir grid. In addition seismic attributes was used as a secondary variable that resulted in defining two different separated cases to distribute permeability in the reservoir Fieldwide. Finally, the oil in place will be calculated for different zones of the Arab formation applying deterministic approaches.
In another part of this study, a sensitivity analysis on permeability calculation
has been considered applying two different techniques. Permeability is one of
the fundamental rock properties, which reflects the ability to flow when subjected
to applied pressure gradients. While this property is so important in reservoir
engineering, there is no well log for permeability and its determination from
conventional log analysis is often unsatisfactory (Mohagheh
et al., 1997).
The direct measurement of permeability is carried out through the core analysis. Typically, the wells with available core data are not sufficient to drive an accurate property distribution model. On the other hand, almost every vertical or horizontal well may have electronic logs data. Thus, the challenge is to establish an explicit relation between the log behavior and core data of those wells that contain both types of information. Then, describe permeability profiles of wells with log information only which commonly have much more complete coverage of the field. This could be a key to achieving a much more precise threedimensional model either by implementing any interpolation techniques or by incorporating further sources of information such as seismic data.
To calculate permeability values two different methods were applied during
this study. In the first method, the concept of fuzzy logic was used to construct
a narrow fuzzy relationship between permeability and some selected conventional
well logs. The fundamental of the model is the same as those proposed by Cuddy
and Exploration (2000) and Hambalek and Gonzalez (2003)
but with some modifications in this study. The second method which is a common
technique in permeability analysis is the method proposed by Tixtier as a modification
of WyllieRose formula to be applied for carbonates. In each case to see the
precision, a well with both core and log data was used as a blind testing. The
more precise method will be applied later on to calculate permeability in the
wells with just available logs information.
MATERIALS AND METHODS
This study focuses on one of the giant Iranian marine oil fields which is located near Qatar waterside. The Arab and Khatya formations are the two main reservoirs in this field which are composed of an alternation of thin dolomite with anhydrite layers. After Sarvak, the carbonate Arab formation is the second largest oil bearing formation amongst the marine fields. The studied structure is a dome shape plunging toward the Northeast and linked to salt activity at a depth which is affected by a highly dense fault network.
Figure 1 shows the location of the studied reservoir in the
eastern part of the marine. The depth map shows the tops and bottoms of the
Arab and Khatya formations. The locations of the vertical and deviated wells
are also presented.
In this study, fourteen wells have been used. All of them have Resistivity, PEFZ, Gamma Ray, Neutron porosity and density logs and only three of them have sonic log (3W1, 3W2 and 1P). In addition, a 3D high resolution seismic acquisition program which covered an area of 98 km^{2} (full fold) is available. The resolution of seismic data in the studied reservoir is near 12 m.
FUZZY LOGIC TECHNIQUE APPLIED TO PERMEABILITY PREDICTION
An introduction to fuzzy logic: Fuzzy logic is an extension of conventional
Boolean logic (zeros and ones) developed to handle the concept of partial truth
values between completely true and completely false. In contrast to binaryvalued
(bivalent) logic, truth is ascribed either 0 or 1, multivalent logic can ascribe
any number in the interval (0, 1) to represent the degree of truth of a statement.
This is a normal extension of bivalent logic and it is a form of logic that
humans practice naturally. Dr. Lotfi Zadeh, an Iranian professor of UC/Berkeley
introduced it in the 1960’s as a means to model uncertainty (Brown
et al., 2000).
More common use of fuzzy logic is to describe the logic of fuzzy sets (Zadeh,
1965). These are sets that have no crisp, welldefined boundaries and which
may have elements of partial instead of full membership. For fuzzy sets, elements
are characterized by a membership function that describes the extent of membership
(or the degree of fit) of each element to the set. Such a membership function
maps the entire domain universe to the interval (0, 1).

Fig. 1: 
Location of the studied reservoir; (A) reservoir map location
in the marine and (B) depth map of the Arab and Khatya reservoirs with the
vertical and deviated wells locations 
Fuzzy mathematical techniques have been applied for permeability forecasting
in some previous studies. Cuddy (1998, 2000) used fuzzy
logic to predict permeability and lithofacies in uncored wells to improve welltowell
log correlations and 3D geological model building (Brown
et al., 2000; Cuddy and Putnam, 1998). In
2003, Hambalek and González (2003) made some modification
to Caddy’s work in a more or less similar study. These studies were implemented
on sandy deposited reservoirs and perfect predictions resulted. Taghavi
(2005) also applied fuzzy logic to improve permeability estimation in a
complex carbonate reservoir in the Southwest of Iran. To see how accurate the
simulated permeabilities by fuzzy logic in carbonates well be, in this study,
the fuzzy inference was used same as the one applied by Cuddy and Exploration
(2000) but with some modifications in a carbonate deposited reservoir in south
of Iran.
Fuzzy logic model description: In this study, the fuzzy logic inference method was applied to determine permeability values of uncored wells based on data from wireline logs in the Arab and Khataya formations. The technique makes no assumptions and retains the possibility that a particular permeability value can give any log reading although some are more likely than others.
The challenge for permeability determination is how to define permeability geocategories and which characteristics will the boundaries have. Cuddy and Exploration (2000) defined the boundaries so that the number of core permeabilities in category 1 represents the 10th percentile boundary of the permeability data. Geocategory 2 represents the 20th percentile boundary and so on. In his study, there were 10 divisions in the data but he mentioned that there is no reason why there could not be 20 or more.
Moreover, the fundamental of fuzzy inference technique which Cuddy
(2000, 2005) used to define membership functions was applied (Brown
et al., 2000; Cuddy and Putnam, 1998). In
addition, the mean value of each geocategory is used as a representative value
per each bin. During the defuzzification stage, the simulated 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 proposed
by Hambalek and González (2003).
Furthermore, the log readings of two selected cored wells and the core derived permeabilities was implemented to mount a narrow fuzzy relationship between core derived permeabilities and some conventional well logs. This model could be later used to predict permeability value based on log information only. There is no limitation on the number of input logs in this method. However, the addition of more curves may possibly not reduce the uncertainty of the determination of rock type, but, it is important to have a consistent set of logs in all wells. Gamma Ray (GR), neutron porosity (NPHI), density (RHOB) Sonic (DT) and shallow and deep resistivity logs (LLS and LLD) were used to define the model.
In order to test the uncertainty of the predicted permeability values by the fuzzy logic method the coredwell A was selected to be a blind testing well.
PERMEABILITY ESTIMATION USING THE WYLLIEROSE APPROACH
In this study another technique was also implemented for permeability determination
which is more common in carbonate deposits. This technique was implemented besides
the fuzzy logic method to have a better understanding of the accuracy of the
fuzzy logic in comparison with other common methods in carbonates. This method
is based on the practical formula which is used to predict permeability (Schlumberger,
1987). In addition to effective porosity, this practical formula incorporates
irreducible water saturation and has a general form of equation.
Where:
Based on the general formula of Wyllie and Rose, several investigators have
proposed various empirical relationships with which permeability can be estimated
from porosity and irreducible water saturation derived from well logs (Schlumberger,
1987). Tixier proposed the values of C, D and E to be more accurate for
use in carbonate. In this study, the Tixier based approach was used for predicting
permeability values in this studied reservoir case.
To perform this all available well logs were analyzed comprehensively in this
studied case. The effective porosity and water saturation were determined precisely
to calculate permeabilities. Again well A was used as a blind testing. Figure
2 shows a perfect match in general trending of the predicted permeability
values versus core derived ones so that the predicted results occurred in a
narrow range when plotted versus core measured values (Fig. 3).
Cuddy found the fuzzy model to be a very precise model in sandy deposits. But
he did not apply any comparison with other common methods for permeability determination
in such kind of reservoir. So, there was not any idea about the precision of
the fuzzy model in comparison with other usual techniques in that study. In
addition, it was mentioned out the precision of the model would be increased
when increasing the number of the wells to derive the model. In this study,
although only two wells are used to derive the fuzzy based model but the predicted
results occurred in a narrow range along the direction of a 45° angle. Figure
3 shows the predicted permeability values in testing well A versus the core
measured ones in both reservoirs for each applied method. Both techniques failed
in some intervals (for example intervals of low permeability values) but the
calculated permeability values based on effective porosity and water saturation
seemed to be more reliable in both reservoirs.
Moreover, a very simple fuzzy model was mounted in this study without any enhancements. This is besides the fact that there was a limitation on the number of cored wells when deriving the model resulting in inaccuracy of the fuzzy predicted permeability values when compared to the core values. Furthermore, the high complexity of carbonate due to cementation, dissolution and highly fractured intervals is another source of error. Due to the difference in vertical resolution between core and logs, these features will cause no or weak reflection in well log curves.
On the other hand, when just concerned the general trend of the prediction values versus depth (Fig. 2) fuzzy logic is a perfect method in a general trend match. In addition, fuzzy logic incorporates the raw well log data in a very simple and quick way which is an advantage of this technique in comparison with other techniques that require a comprehensive log analysis. Furthermore, the accuracy of the fuzzy model is highly dependent on the number of wells when deriving the model; therefore, one might increase the number of wells and derive a much more accurate model even more than other common techniques.
RESERVOIR STRUCTURAL MODELING
In order to construct a static reservoir model, the first stage is interpretation
of the well logs and determination of some important petrophysical properties
such as porosity, shale volume and water saturation (Lyndon,
2001; Davis, 2002). The well logs of 14 wells are interpreted
in this study. The next step towards defining better structural configuration
and specially fault patterns is seismic interpretation which was done comprehensively
during this study. The upper and lower seismic horizons of the Arab reservoir
(depth map) have been used as bounding surfaces of the grid. Three faults within
the Arab formation were used to govern the grid model. The grid model has been
divided into 7 geological zones and 35 layers. Figure 4 shows
a middle skeleton structural grid, the faults and their trends in 2D and 3D.
In addition, Table 1 shows statistics of the 3Dgrid generated
in this study for the Arab reservoir.

Fig. 2: 
Predicted permeability values versus depth for two different
permeability predictive techniques. The core measured permeabilities are
also plotted for comparison 

Fig. 3: 
Predicted permeability values versus core measured ones for
both testing reservoir applying fuzzy logic and Tixier modified WyllieRose
approaches. Both techniques fail to predict very low permeability intervals 
RESERVOIR PROPERTY MODELING
Property modeling is the process of filling the cells of the grid with petrophysical
properties. The point of property modeling is to distribute properties between
the wells such that it realistically preserves the reservoir heterogeneity and
matches the well data. There are two approaches for property modeling in general;
deterministic and stochastic method. The first step is defining the main direction
of anisotropy within the area of study which could be achieved through a comprehensive
variogram analysis. Then, the Sequential Gaussian Simulation approach was applied
to distribute permeability in the reservoir grid both with and without seismic
attributes. This will achieve three different cases which will be discussed
later. Note that the petrophysical properties will be modeled for each zone
of the reservoir separately.
Variogram modeling: Geological data is usually anisotropic (at least between the vertical and horizontal directions); therefore variograms should be calculated in several different directions to obtain the main directions of anisotropy when distributing reservoir properties in the grid model. In more than one dimension, the variogram generally has directional properties. Generally, two ways can be used to find anisotropy in data:
• 
Trial and error with directional horizontal variograms 
• 
Using the variogram surface (map) 
In this study, both cases was took in consideration. Their results will be
discussed later (Chambers et al., 2000;
Jeffrey, 2007; Sahin et al., 1998).
Table 1: 
Arab formation 3D grid statistics 


Fig. 4: 
Structural grid of the Arab formation. (A) a middle skeleton
grid showing in two dimensional graph with a representation of the faults
and their trends and (B) three dimensional structural model of the Arab
formation with different zones 


Fig. 5: 
Variograms for different azimuths, in the directions of 20,
40, 60 and 80 (from bc) for zone 5 of the Arab formation. The variograms
are regression between the points and may not be completely acceptable.
(a) regression curve nugget: 1.01, Sill: 1.22, Range: 1.54E+3, (b) regression
curve nugget: 0.0451, Sill: 0.677, Range: 900, (c) regression curve nugget:
0, Sill: 1.28, Range: 990 and (d) regression curve nugget: 0.295, Sill:
1.32, Range: 1.31E+3 
In this study, a comprehensive directional variography was applied for all
zones. In this study, zone 5 of the Arab formation is chosen to show as an example
for horizontal variograms with different orientations (Fig. 5ad).
Four experimental variograms for zone 5 of the Arab formation (the points on
the Fig. 5) are generated for different directions, starting
from North direction and increasing by increments of 20° azimuth angle clockwise
(i.e., N20, N40, N60 and N80). Data set anisotropy has been observed but it
is a little bit hard to define major and minor directions of it. This is might
because of the high complexity of carbonates. On the other hand, these parameters
were determined much more easily and conveniently when variogram surface maps
had been created. The variograms surface map for the previous zone example,
zone 5 of the Arab formation, is shown in Fig. 3 which shows
an azimuth angle of 40° clockwise (N40).
After doing directional variography using experimental variograms for each zone, the major directions of anisotropy were determined. It is the time to decide which variogram model to be used; exponential, spherical or Gaussian? Different variogram models where tested for each zone. The spherical model had the best matching for most of the cases. It was decided to use the spherical model for all the zones and direction. The spherical model is an average model and is located between the Gaussian and exponential models, so it is possible to use it for all the cases.
RESULTS
The final variogram characteristic for permeability is shown in Table 2. Note that the sill value for each zone should be 1, the minor direction angle is perpendicular to the major direction and the spherical variogram model is used for all the zones. These variogram properties were used for property modeling.
The variogram maps can be very helpful to determine directions of anisotropy. For the Arab formation variogram maps were generated for the permeability of each zone using an acoustic impedance map (Fig. 6). These maps were integrated with directional variogram analysis results to investigate anisotropy for each zone. The maps are very complex especially at greater distances from the center of the maps (i.e., more range values). As a result, the continuity distances (the range values) for each zone are generally less than 2000 m. However, looking at the central part of the maps, a good prediction for the direction of the anisotropy can be obtained. For example by looking at the center of the variogram map for zone 5 of the Arab formation, the major direction of anisotropy has an azimuth of N40 that matches with the directional variogram investigation results (Fig. 6).
Because the studied reservoir case in this study is a highly complex carbonate
one which has an interlaminated complex lithology of carbonate and anhydrite,
the related variograms will vary dramatically for different orientations.

Fig. 6: 
Variogram surface (map) for zone 5 of Arab formation. The
major direction of anisotropy has an azimuth of N40 which coincides with
the results of directional variography analysis 
Table 2: 
Summary of variography results for permeability 

For all the zones, the variogram type is spherical, the sill
is 1 and the minor direction azimuth is perpendicular to the minor direction 
This
became much more obvious in the Arab formation which could be attributed to
the level of digenesis in different zones of this reservoir. In other words,
the more digenesis in the formation, the more variability (less continuity)
will occur.
During the variogram analysis study the spherical model yielded the best matching for most of the cases. Therefore, it was decided to use the spherical model for all the zones and direction. In addition, the spherical model is theoretically an average model between the Gaussian and exponential ones, so it is possible to use it for all the cases.
Fieldwide permeability property modeling: In this study, the results for the permeability property model will be mentioned but porosity and water saturation were also modeled during this study. These properties are derived from interpretation of the logs of 14 wells which are then upscaled into the grid. These petrophysical properties were modeled separately for each zone. The data analysis results, including histogram analysis, data transformation and variogram analysis parameters (type, range, anisotropy, sill and nugget) were the basis of the modeling. The seismic attribute data was used to help the modeling of well data in the case where a good correlation was dominant. The stochastic approach was applied to distribute the permeability model.
Countless examples of reservoir performance prediction exist whose failure
is due to use of overly simplistic and smooth models like kriging. Stochastically
simulated models are generated to overcome this problem of reservoir performance
prediction and help flow simulation (Farzadi and Hesthammer,
2007; Marion et al., 2000). The Sequential
Gaussian Simulation (SGS) is the most commonly used and the most straightforward
type of simulation for generating images or realizations of studying variable.
This approach is applied here for the permeability property model of the Arab
formation.
Simulated permeability models are using the same variograms which have been previously generated in the data analysis part of this study. Normal score transforms were applied for permeability data in the data analysis part. Ten images of porosity at each zone were generated and all were nearly similar. The tenth image was taken as an example and is shown in Fig. 7. The left figure is a view of the simulated permeability model for all the cells of the grid and the right is a zoom into the edge of the grid.
There are many possible realizations when simulating in SGS from one dataset
depending on the seed value. In other words, for each seed value one possible
image is simulated. Three realizations for layer 20 (top of zone 5) are shown
in Fig. 8 as an example. The seed numbers for the realizations
one (left) to three (right) are13442, 24593 and 5949. The results are almost
similar but major differences can also be seen in some parts. Notice the difference
in high and low permeability concentration throughout the field in Fig.
8.

Fig. 7: 
A view of the simulated permeability model for the Arab interval
on the left and a zoom into the edge on the right 

Fig. 8: 
Three permeability realizations belonging to layer 20
(from zone 5), created with different seed numbers 
For realization 1 high permeability values are concentrated on the Eastern
areas whereas for realization 3 they are more on the Western area.
It was easily predictable that the simulated models here will not show any smoothing effect between the wells when compared to the deterministic approach resulted models. This heterogeneity of the model can be seen in all the images of SGS and in fact is an advantage of the stochastic simulation. Therefore, the models derived by SGS indicate much more variability and are more acceptable even though they may be wrong is some parts. So, there should be no judgment just based on only one realization result. In order to be more confident some uncertainty analysis should be done. For example many images should be simulated and compared, the more similar images the less uncertainty.
The SGS method almost preserves the original statistical distribution of the
data which is another strength point of this method. Figure 9
shows the upscaled permeability histograms and also the histograms of modeled
porosities (for all the cells).
These two sets of histograms are quite the same and this is a major characteristic of the SGS algorithm in contrast with other deterministic approaches that smooth the distribution of data. As seen from the simulated images and the distribution histograms, zones 5, 4 and 2 have higher permeability values and zones 1, 3 and 6 are much impermeable zones.
Sequential Gaussian simulation with seismic trend: The best option for
property modeling can be the use of stochastic methods combined with a secondary
seismic data (Farzadi and Hesthammer, 2007). Two seismic
attributes have been used in this study as secondary variables besides SGS:
(1) SIDseismic depth phase attribute and (2) Acoustic impedance.


Fig. 9: 
(a) Distribution of upscaled porosity values for zone 1 at
left to zone 6 at right and (b) distribution of modeled porosity using SGS
for zone 1 at left to 6 at right 

Fig. 10: 
Crossplot of permeability and instantaneous phase attribute
at the location of the well for zone 5. The correlation coefficient is 0.41 
Here, the
randomness of the SGS process will be much less and instead the seismic trend
will be affected in the results. But the model preserves the need for heterogeneity.
More correlation coefficient between primary data and seismic trend causes more
effect of seismic data in the result model.
The required correlation was found only in zone 5 so only this zone was modeled
using a secondary data. The crossplot for zone 5 which illustrates the correlation
between permeability and instantaneous phase has been shown in Fig.
10. The correlation coefficient is 0.41.
The seismic attribute is usually Acoustic Impedance (AI) which indicates good
correlation with permeability predicted from Tixtier approach. The best seismic
data that can be used as a secondary variable for permeability modeling is AI,
An example of Acoustic impedance modeled from Inline 2245 of the Arab formation
is shown in Fig. 11.
In this part of the study seismic acoustic impedance was being used as a secondary
variable with SGS technique to distribute permeability throughout the reservoir
grid. Figure 12 shows the correlation between permeability
and seismic acoustic impedance for zone 5 of the Arab formation which achieved
a correlation coefficient of 0.52. It is clear that using acoustic impedance
will achieve much more accurate results.

Fig.11: 
An example view of 2D Acoustic Impedance which has been
modeled from Inline 2245 of the Arab formation 

Fig. 12: 
The crossplot of permeability and Acoustic Impedance (AI)
at the location of the well for zone 5. The correlation coefficient is 0.52 
In Fig. 13, the permeability models with different SGS algorithms
are shown for layer 20 as an example. An applied SGS with Acoustic impedance
(part A) can be seen in the permeability model; the permeability values in the
middle area and east flank are greater. This is not random but is a real geological
trend deduced from acoustic impedance. These parts coincide with the location
of the producible wells which are permeable horizons. On the other hand, in
the permeability model from SGS with seismic attribute phase (part B) the middle
parts of the model show much lower permeable characteristics. This does not
confirm with the fact that there are producible wells in these parts which are
surely permeable.
Cross validation of the models: One way to validate a model is to eliminate some of the data points from the calculation and compare the modeled value with the true value of the data points. This validation is especially effective for the estimation models such as kriging or inverse distance, because such models are not risky. In other words, the estimation models calculate average smooth results and therefore the modeled values are correlated with the true values. When there is a large data set or when the points are surrounded by many other points, these models give more correlated results.
But this kind of validation is not a good way to test the models randomly i.e.,
stochastic methods, when from the beginning it was clear that the simulated
value is probably not the same or even close to the true value. The methods
such as SGS naturally produce results that are not highly correlated with the
true values but these methods preserve the heterogeneity and that is why it
is better to use them especially the data set is sparse.

Fig. 13: 
Comparison between permeability models built with different
SGS methods for layer 20 using seismic trends as a secondary variable. (A)
SGS with Acoustic Impedance and (B) SGS with seismic attribute phase 

Fig. 14: 
The cross plots between the modeled values at the well location
cells and the upscaled well values (the actual values) for the wells: b1,
b2 and b3 
Here, three wells: b1, b2 and b3 (one well from each platform) are dropped
out from the data set and the previous permeability models are calculated again.
The model algorithms used here include: SGS, SGS with a seismic depth and SGS
with acoustic impedance. All the conditions such as variogram analysis results
are kept the same. The cross plots between the modeled values at the well location
cells and the upscaled well values are generated to compare the results as Fig.
14.
Table 4: 
Deterministic results for each zone and for the whole
Arab formation 

The correlation coefficient between the actual well permeability and
the permeability modeled using different algorithms is shown in Table
3.
According to the Table 3, generally the correlation coefficients
resulting from SGS with seismic information as a secondary are much higher than
those of related to the SGS approach individually. This might be because the
seismic attributes will provide much more dense information that affect on the
randomness property of the SGS process. The result shows that the seismic acoustic
impedance result is a much more precise model compared with the seismic depth.
These results coincide with those shown in Fig. 13.
Volumetric calculations: It is time to calculate oil volume in the reservoir using the previously generated models which is the last stage of static reservoir modeling. In this stage pore volume and oil volume over the Arab interval is computed cell by cell using the property models for porosity and water saturation. There are two options, whether to use deterministic models or stochastic ones. It is more common to use deterministic models for volumetric calculations and stochastic ones for flow simulation. In this study the deterministic approach was used for calculating the oil in place at reservoir condition. There is no sign of gas in the reservoir and the computations are only done for oil. The computation results for each zone and for the total Arab formation are shown in the Table 4. In addition, water saturation per each zone was calculated and the result shown in Table 5.
The Arab is generally a dense reservoir so the pore volume is much less than
bulk volume. It can be seen from the Table 4 and 5,
that zone D contains most of the oil in place and can be a target for further
studies and future drilling. However, it should be noticed that water saturation
for most of the zones is higher than 60% and this means that most of OOIP is
not recoverable in these zones.
Table 5: 
Average water saturation, porosity for each zone of the Arab
formation 

The best oilbearing part is almost in the middle
part and eastern flank of the reservoir where most porous part with low S_{w} values is.
DISCUSSION
This study was focused on two parts both regarding permeability. First, the fuzzy logic inference was testified when building a model for predicting permeability values based on the well log readings. In addition, the Tixier modified formula of WyllieRose was applied which is a common technique in carbonates. Then, the results were compared with those derived from the fuzzy logic model. In the next part of this article the procedure for constructing a static model of our exampled carbonate oil reservoir was explained by applying the stochastic approach. Reservoir structural and property modeling derivations are discussed in detail. In addition a comprehensive variogram analysis was applied to define the major direction of permeability anisotropies. Here are some points regarding our studies:
In this study, a precise network of fuzzy rules was mounted between core measured permabilities and well logs readings from two wells in a highly complex carbonate reservoir in the South of Iran. In addition, the Tixier modified technique of WyllieRose was implemented. This is a common permeability predictive technique in carbonates. The results were compared with the fuzzy predicted technique to define its accuracy both quantitatively and qualitatively. The accuracy of each model is tested using a blind testing by comparing with core permeabilities as the exact values. Both techniques failed in low permeability intervals (<0.1 md) for the Arab formation. This is the same for all parts of the Khataya testing well which has a highly complex lithology and has a highly dispersive data.
For the Arab formation, except for the intervals of very low permeability values
the simulated results occurred in a narrow range just close to the actual core
measured values. The simulated permeabilities from the fuzzy logic model copy
exactly the actual permeability tendency. This could make a powerful tool for
the exploitation businesses. In addition, in some cases, especially in high
permeability values it shows some benefits qualitatively. This is besides the
simplicity and easy application property of the fuzzy model (fuzzy incorporate
raw log data) this is an advantage in comparison with the Tixier technique which
needs a comprehensive and precise log analysis study.
All in all, fuzzy logic in this study was found to be a good method only quantitatively
especially for permeabilities greater than 0.1 md. In addition, it is predicted
that by increasing the number of wells when defining the model the accuracy
of the fuzzy model will increase. Present results confirm those proposed by
Cuddy (2000, 2003), which was in a sandy deposit. But
to build the three dimensional geological model in this study the WyllieRose
approach was applied because of its high accuracy.
There are nine major faults in the Arab reservoir which were precisely modeled during this study. All these faults are normal (Fig. 4). Finally, the Arab reservoir has been modeled into seven zones (AG) in which the zone G in completely located below water oil contact.
In this study, the spatial variogram analysis for different zones implies that
properties are heterogenic for all the zones. Zone 5 of the Arab formation is
illustrated as an example (Fig. 5, 6).
A sensitivity analysis on the permeability distributing method was applied
in this study in which seismic attributes were applied as a secondary variable
with SGS. The seismic cubic impedance with SGS achieved perfect determination
coefficients in all wells (Table 3 and Fig.
13, 14).
From the comprehensive reservoir modeling, permeability had a direct relationship with porosity. The highest porosity and permeability observed in zone E which is a zone with 24% porosity and 250 md permeability value. In addition, permeability is reduced in zone F, A, D, B and C, respectively (Table 5). In addition, water saturation profile has its highest value in zone B. This value decreases for zones C, D, E, A and F, respectively (Table 5). Furthermore, total oil in place for this reservoir is estimated to be (10^{6} m^{3}) applying the deterministic approach, zone D has the highest amount of oil in place and would be better for future infill drilling target.
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 and 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.