3D Geological Modeling of Subsurface for Drilling Purposes Using Neural Networks and Fuzzy Logic

Esta é uma pré-visualização de arquivo. Entre para ver o arquivo original

Rock Mechanics for Natural Resources and Infrastructure 
SBMR 2014 – ISRM Specialized Conference 09-13 September, Goiania, Brazil 
© CBMR/ABMS and ISRM, 2014 
 
SBMR 2014 
3D Geological Modeling of Subsurface for Drilling Purposes Using 
Neural Networks and Fuzzy Logic 
 
Alvaro Gustavo Talavera Lopez 
PUC-Rio, Rio de Janeiro, Brasil, alvaro@ele.puc-rio.br 
 
Vivian Rodrigues Marchesi 
PUC-Rio, Rio de Janeiro, Brasil, vivianrm@puc-rio.br 
 
Débora Lopes Pilotto Domingues 
PUC-Rio, Rio de Janeiro, Brasil, deborapilotto@puc-rio.br 
 
Sergio Augusto Barreto da Fontoura 
PUC-Rio, Rio de Janeiro, Brasil, fontoura@puc-rio.br 
 
Clemente José Gonçalves 
Petrobras, Rio de Janeiro, Brasil, clemente@petrobras.com.br 
 
Marcos Fonseca Alcure 
Petrobras, Rio de Janeiro, Brasil, marcosalcure@petrobras.com.br 
 
SUMMARY: 3D geological modeling is an important part of geomechanical models. Therefore, the 
knowledge of spatial distribution algorithms of geological properties is essential for the proper 
characterization of the subsurface. Generally, geostatistical and neural networks can be used as 
forecasting strategies of geological characteristics. One problem in forecasting logs and 
geomechanical properties is that they are not linearly related to several decision variables. Through 
neural networks it is possible to describe the behavior of complex systems and to estimate relevant 
correlations between a given set of inputs and another of outputs using a nonlinear function. 
Alternatively, fuzzy logic is a knowledge-based system that allows adding rules, which can be used 
to infer the result of a forecast. This work presents a methodology for 3D geological modeling of oil 
fields using artificial neural networks for forecasting geological properties, and a Fuzzy logic system 
– employed in order to improve the performance of neural network through the inclusion of geological 
rules. It was concluded that, using both computational intelligence techniques, it is possible to obtain 
reliable models based on numerical results. 
 
 
KEYWORDS: Neural networks, Fuzzy logic, properties forecast. 
 
 
 
1 INTRODUCTION 
 
One problem in predicting logs and 3D 
geomechanical properties is that they are related 
to several decision variables, such as seismic 
attributes, lithofacies, information core and well 
logs. This large database generates highly 
nonlinear relations regarding the variables to be 
forecast; thus, contributing to the low 
performance of the predictive models of standard 
statistical methods. 
 Neural networks have great potential for the 
precise description of complex systems 
behavior. Neural networks make it possible to 
approximate a relation between an input set of 
dimension n and an output set of dimension m 
through a nonlinear function. This method 
allows for an efficient representation of highly 
 
 
 
 
SBMR 2014 
nonlinear relations and performs well in 
situations in which the mathematical model 
related to the problem to be solved is very 
complex to obtain (Haykin, 1999; Mohaghegh, 
2000). 
 Artificial neural networks are becoming 
increasingly popular in areas such as geology 
and geophysics. They are used, for instance, in 
the characterization of petroleum reservoirs, for 
geological properties forecasting and well logs 
prediction, for seismic inversion and fractures 
modeling (Mohaghegh and Ameri, 1995; Van 
der Baan and Jutten, 2000; Calderón-Macías et 
al., 2000; Mohaghegh, 2005; Baddari et al. 
(2010) and Darabi et al., 2010). 
 Huang et al. (1996) used back-propagation 
artificial neural networks (BP-ANN) to model 
the interrelations between spatial position, six 
different well logs, and permeability. The BP-
ANN produced permeability values that 
compared well to measured values in the cored 
intervals. Fung et al. (1996) used neural 
networks and Learning Vector Quantization 
(LVQ) for forecasting rock properties. Saggaf et 
al. (2003) used BP-NN to estimate the 
distribution of reservoir porosity using 3D 
seismic in the field of Ghawar, Saudi Arabia. 
 Wang et al. (1998) modeled porosity 
distribution through the integration of techniques 
of neural networks and kriging in the field of 
A'nan, north of China. The association of 
geostatistics and neural network can also be 
found in Niu et al. (2000) and Nakayama (2000). 
Studies involving the relation between reservoir 
properties and seismic attributes to estimate 3D 
reservoir parameters can be found in Arzuman 
(2009). Although seismic attributes assist in 
predicting the geological properties, they add 
more complexity to the forecasting operation, 
due to nonlinearities of seismic attributes and 
large amount of seismic 3D data. 
 Most studies mentioned above use a single 
neural network for predicting properties in the 
well and 3D field. This paper proposes a model 
of multi-neural networks in which each neural 
network is designed for each geological zone of 
the subsurface. For this purpose, a study of the 
different types of input variables in each neural 
network appropriate for each zone was carried 
out. This study implied working with many 
variables. A principal component analysis was 
carried out in order to reduce the dimension of 
data space variables. The variables were chosen 
among seismic attributes, well logs and 
geological properties of reservoir. This strategy 
has the advantage of assigning neural networks 
for each spatial region of the reservoir for more 
accuracy in prediction. 
 In general, the prediction of geological 
properties by neural networks or by any other 
prediction model are subject to errors due to 
possible criteria disregarded in the forecast, such 
as geological faults, fractures, data quality, very 
thin facies layers and heterogeneity in the field. 
To correct possible errors in the prediction, a 
system with fuzzy logic is used to infer the result 
of the prediction model. 
 Lin and Cunningha (1995) have proposed the 
use of fuzzy logic to model the identification of 
nonlinear systems. Subsequently, other studies 
used this strategy for reservoir characterization 
(Bhuiyan, 2001; Weiss et al., 2001). Lim and 
Kim (2004) proposed a model that uses fuzzy 
logic to select the best reservoir properties, 
which will be used by the neural network for the 
prediction of permeability and porosity. Taghavi 
(2005) used linear regression and fuzzy logic in 
the prediction of permeability and concluded that 
the fuzzy logic produced better results. 
Kadkhodaie-Ilkhchi et al. (2009) used seismic 
attributes as decision variables in a committee 
fuzzy inference system for estimating water 
saturation and porosity parameters. The model 
proposed in this paper is divided in two stages. 
The first stage is the prediction of properties 
using neural networks and the second stage is to 
use fuzzy logic to correct possible errors in the 
prediction, using expert knowledge. Next, we 
present the theoretical framework of the theories 
used herein such as neural networks, fuzzy logic 
and principal component analysis. The proposed 
method is described afterwards followed by the 
description of the application of the method to 
predict sonic logs at a field case. Finally, we 
present some conclusions regarding the method, 
its potential and possible developments for its 
application are drawn. 
 
 
 
 
 
 
 
SBMR 2014 
2 THEORY 
 
2.1 Neural Networks 
 
The Rumelhart-Hinton-Williams multilayer 
network that we consider here is a feed-forward 
type network
with connections between 
adjoining layers only. Networks generally have 
hidden layers between the input and output 
layers. Each layer consists of computational 
units as it can be seen in Figure 1. 
The input-output relationship of each unit is 
represented by inputs 𝑥𝑖, output 𝑦, connection 
weights 𝑤𝑖, threshold 𝜃, and differentiable 
function 𝜑 as follows: 
 
𝑦 = 𝜑(∑ 𝑤𝑖𝑥𝑖 −
𝑘
𝑖=1 𝜃) (1) 
 
Usually, the function φ is called the activation 
function, which is a bounded and monotone 
differentiable function such as the sigmoid 
function represented by Equation (2), where 𝑎 is 
the slope parameter of the function (Funahashi, 
1989). 
 
𝑓(𝑥) = 
1
1+𝑒−𝑎𝑥
 (2) 
 
The learning rule of the neural network is 
known as backpropagation algorithm 
(Rumelhart et al., 1986) and consists in the use 
of the gradient descent method to find a set of 
weights so that the error between the desired 
output and the output signal network is 
minimized. 
 
 
 
Figure 1. Schematic Multi-Layer Perceptron (MLP) 
network (Darabi, et al. 2010). 
Neural network modeling process comprises 
three basic steps related to experimental data: 
training, testing and predicting. Network training 
is the process of adjusting the network 
connection weight so that explanatory and 
response values match the data as closely as 
possible. Testing data are used for checking the 
training. Finally, validating data are used for 
studing model accuracy. A comprehensive 
reference on neural networks can be found in 
Haykin (1999). 
 
2.2 Fuzzy Logic 
 
Fuzzy logic is a logic based on fuzzy set theory 
which aims to model the approximate mode of 
reasoning, mimicking human ability to make 
decisions in an environment of uncertainty and 
imprecision (Zadeh, 1976; Pedrycz and Gomide, 
1998). It allows intelligent control and decision 
support systems to deal with imprecise 
information. A fuzzy logic system is able to 
simultaneously process numerical data and 
linguistic knowledge (Mendel, 1995). In fuzzy 
set theory, each element may belong to a set of 
degree (μ), which can take values ranging from 
0 to 1. Each fuzzy set is represented by a 
membership function (MF). MFs can present 
several types, such as Gaussian, triangular, 
trapezoidal and sigmoidal, see Figure 2. 
 
 
Figure 2. Examples of four classes of parameterized MFs 
 
 
 We aim, with this work, at developing a fuzzy 
inference system FIS, which is a process that 
maps a set of input data into a set of output data 
 
 
 
 
SBMR 2014 
using fuzzy logic. A FIS has 4 main parts, 
presented in Figure 2, which are (a) 
fuzzification, (b) rules, (c) inference engine and 
(d) defuzzification. 
 
 
 
 
Figure 3. Main parts of a Fuzzy Inference Systems. 
 
The fuzzification consists in mapping out the 
accurate inputs or non-fuzzy (crips) numbers 
(resulting measurements) into fuzzy sets. This 
process is needed in order to activate rules which 
are expressed in terms of linguistic variables and 
have fuzzy sets associated to them. Rules can be 
provided by experts or extracted from the 
numerical data (this method is particularly useful 
in problems of forecasting time series) in the 
form of linguistic sentences, e.g., “IF 𝑢1 is very 
warm and 𝑢2 is quite low THEN turn 𝑣 
somewhat to the right”. The inference process 
are used to combine fuzzy IF-THEN rules from 
to fuzzy rule base into a mapping from fuzzy 
input set to fuzzy output set. Once the fuzzy 
output set is obtained through inference (modus 
ponens generalized), the stage of defuzzification 
is carried out in order to interpret the 
information. There are several defuzzification 
methods in the literature, two of the most used 
ones are the center of gravity and the mean of the 
maximum. 
In this work, the fuzzy inference used is 
called the method of Mamdani in which the 
output membership functions are fuzzy sets. 
After the aggregation process, there is a fuzzy set 
for each output variable that needs 
defuzzification. For example, consider the rule: 
 
𝑅𝑖: 𝑖𝑓 𝑥 𝑖𝑠 𝐴1𝑎𝑛𝑑 𝑦 𝑖𝑠 𝐵1𝑡ℎ𝑒𝑛 𝑧 𝑖𝑠 𝐶𝑖, 𝑖 = 1,2, … , 𝑛 
 
The process of Mamdani method is 
illustrated in Figure 4 and more details on this 
method can be obtained from Kadkhodaie-
Ilkhchi (2009). 
 
 
 
 
Figure 4. Illustration of Fuzzy Inference Systems. 
 
2.3 Principal Components Analysis 
 
The principal component analysis (PCA) is a 
linear transformation of data to a new coordinate 
system (space features). This transformation is 
designed so that the data set can be represented 
by a reduced number of data ensuring higher 
intrinsic information contained in the data set. 
Therefore, the data set is reduced dimensionally. 
In this sense, PCA can be defined as an optimal 
linear transformation in order to preserve the 
subspace that has the highest variance. Principal 
component analysis is a technique commonly 
used in pattern recognition and signal processing 
applications. 
In petroleum field, it is widely used especially 
in the area of geo-modeling and in modeling 3D 
properties where the amount of data is too large 
e.g. seismic attributes. In this sense, it is used to 
reduce the dimension of the data. Hagen (1982) 
describes the application of PCA to model 
seismic stratigraphy and Brito (2010) also 
mentions seismic data and seismic attribute 
volume to improve seismic stratigraphy using 
PCA. Guo et al. (2006) describe the use of PCA 
to reduce the dimension of the spectral 
components in seismic images to delineate 
stratigraphic features of interest. 
 
 
3 PROPOSED MODEL 
 
This paper proposes a model using neural 
networks and fuzzy logic for predicting 
geological properties and electric logs in 
 
 
 
 
SBMR 2014 
petroleum reservoirs. Figure 5 shows the block 
diagram of the proposed model. The model is 
divided into three steps. The first step is called 
data preparation; it includes the division of the 
model field in 𝑛 geological zones and the 
evaluation of decision variables that the neural 
network will use in predicting properties in each 
zone. The decision variables include the seismic 
attributes due to its 3D nature (seismic cube), 
which helps predicting properties (Taner, 2001), 
the facies observed at the drilled wells that 
provide the geological lithology in each zone, the 
well log at drilled wells and the normalized 
coordinates x, y, z for each well. Finally, data 
dimensionality is reduced using PCA. 
 
 
 
Figure 5. Proposed Model. 
 
 The second step is the development of the 
model using neural networks and fuzzy logic. In 
this step, neural networks are designed for each 
geological zone in order to predict some property 
�̂�. The advantage of using this strategy is that it 
is easy to identify which neural network had the 
lowest or highest performance in predicting and 
then change one decision variable or the 
structure of the neural network. On the other 
hand, choosing the input variables for each 
neural network is complex because each network 
will not necessarily have the same entries. The 
strategy used for setting is shown in Figure 6. 
 
 
 
 
Figure 6. Neural Networks Strategy. 
 
Once the prediction by neural network is 
carried out, a fuzzy logic system is applied. Since 
it is possible to enter if-then rules with a geologic 
sense in fuzzy system in order to eliminate errors 
due to possible criteria disregarded in the 
forecast, such as geological faults, fractures,
low 
data quality noises, very thin facies layers, there 
is a reduction of errors in forecasting, especially 
in thinner layers of zones where the neural 
network does not have good performance in 
prediction, as found by Tang et al. (2011). 
The third step is to regroup all the results of 
neural networks in each zone to generate the 3D 
model of the field. To evaluate the predicted 
outcome the results of all networks in each zone 
are added together and compared to the same 
property in well validation (i.e. a well whose data 
were not used at all stages). 
 
4 APPLICATION OF THE PROPOSED 
MODEL 
 
In this section, the proposed model is applied for 
the studied field. The methodology used is as 
follows: (i) the studied field has eight wells that 
will be referred to as 𝑝1, 𝑝2, 𝑝3, 𝑝4, 𝑝5, 𝑝7 and 𝑝8, 
(ii) well 𝑝4 is chosen as well validation, i.e. it 
will only be used to compare with the final result 
of the prediction made by the proposed model. 
The studied field is divided into seven geological 
zones so seven neural networks will be 
projected. 
 
 
 
 
SBMR 2014 
In this first example, the objective is to design 
the proposed model to predict the cube of 
compressional sonic log, DTC, in the field. As 
explained in the previous section, the set of input 
variables for each neural network is selected 
using studies such as Taner (2000), Arzuman 
(2009) and Azevedo et al. (2009) that show the 
usefulness of each seismic attribute for detecting 
geological properties. The selected variables are 
presented in Table 1, where A is amplitude, AI is 
the acoustic impedance, IV is interval velocity, F 
is the facies and ‖𝑧‖ is the normalization of 
depth. 
The input variables of each network is shown 
in Table 1 and the results of the proposed model 
of DTC prediction in the 3D field is shown in 
Figure 7, where the vertical arrow is the location 
of the well validation 𝑝4 
 
 
Table 1. Selecting inputs by neural networks - DTC. 
Zone NN A AI 
 
IV F ‖𝑍‖ 
1 1 X - X X X 
2 2 X - X X X 
3 3 X - X X X 
4 4 X X - X X 
5 5 - X X X X 
6 6 - X X X X 
7 7 X X - X X 
 
 
 
 
Figure 7. Prediction of DTC at coordinate J. 
 
 To validate this model the Mean Absolute 
Percentage Error (MAPE) is calculated 
according to equation 3, where 𝑥 is the value of 
property DTC in well validation 𝑝4, �̂� is the DTC 
estimated by the proposed model. Using this 
error measure, a MAPE of 6.10% was obtained. 
 
𝑀𝐴𝑃𝐸 =
|𝑥−�̂�|
𝑥
× 100 (3) 
 
 The same exercise was done for the shear 
sonic log, DTS, following the same procedure. 
Table 2 shows the input variables for the neural 
networks. The result of the prediction model is 
shown in Figure 8, with a MAPE regarding well 
validation of 6.80%. 
 
Table 2. Selecting inputs by neural networks - DTS. 
Zone NN A AI 
 
IV F ‖𝑍‖ 
1 1 X - X X X 
2 2 X X - X X 
3 3 - X X X X 
4 4 X X - X X 
5 5 - X X X X 
6 6 - X X X X 
7 7 X X - X X 
 
 
 
 
Figure 8. Forecast DTS by Neural Network at coordinate 
J. 
 
 It can be seen in Figure 8, the formation of 
pockets, which can be interpreted as errors in 
forecasting, especially in very thin layers where 
the neural network cannot get a good prediction, 
as in Tang et al. (2011). One way of solving this 
problem is to use a filter to eliminate these 
disturbances. In this case, it is advisable to use 
Fuzzy Logic. To do this, rules for the behavior of 
properties of well profiles such as DTC, Gamma 
 
 
 
 
SBMR 2014 
Ray and DTS, were generated as follows: 
 
1. If RG is low and RHOB is high then DTS is 
low. 
2. If RG is high and RHOB is medium then 
DTS is medium. 
3. If RG is low and RHOB is medium then DTS 
is medium. 
 
 It is important to highlight that these rules are 
just examples and should not be applied as 
general truth. 
 The fuzzy sets were designed according to the 
expert's knowledge and statistics of each profile, 
as shown in Figure 9. The fuzzy sets and fuzzy 
inference system were developed using the 
Petrel calculated by a program *. mac. 
 
 
Figure 9. Fuzzy Sets Gamma Ray, respectively low, 
medium and high GR. 
 
 The fuzzy system accepts two inputs: the 
Gamma Ray GR and Resistivity RHOB. Then, 
the input data are normalized (fuzzification 
step), followed by the application of fuzzy 
inference system and denormalization of data 
output (desfuzzication), resulting in a DTS real 
value. This process can be seen in Figure 10. 
 
 
Figure 10. Fuzzy inference system. 
 
 Applying the fuzzy system in prediction made 
by the neural network in thinner layers can 
reduce the approximation error of the neural 
network. This can be seen in Figure 11. Using 
this strategy, the MAPE is reduced to 5.94%. 
 
 
Figure 11. Filtered using Fuzzy Logic. 
 
It can also be seen in Figure 11 that the neural 
network has a lower performance on thin layers 
due to the limited information in these regions. 
 
 
5 CONCLUSIONS 
 
This paper presents a method for 3D geological 
modeling of oil fields using artificial neural 
networks for forecasting geological properties 
and a system of fuzzy logic to improve the 
prediction performance. 
The proposal for designing one neural 
network for each different geological unit 
improves the evaluation of the performance of 
each neural network as well as facilitates the 
selection of input parameters for each geological 
zone. The use of fuzzy logic in order to make the 
prediction of properties more accurate indicated 
that the technique is quite promising. As future 
work, it is proposed to study de fuzzy inference 
rules that can help predict specific properties; 
possibly rules that include the degree of rock 
fracturing. 
 
 
ACKNOWLEDGEMENTS 
 
The authors thank Petrobras for the data 
provided and Schlumberger for the academic 
 
 
 
 
SBMR 2014 
license of the Petrel software. 
 
REFERENCES 
 
Azevedo, L. (2009). Seismic Attributes in Hydrocarbon 
Reservoirs Characterization. MSc. Dissertation. 
Departamento de Geociencias, Universidade de 
Oviedo. 
Arzuman, S. (2009). Comparasion of Geoestatistics and 
Artificial Neural Networks in Reservoir Property 
Estimation. Turkey: Theses-Geologial Engineering 
Deparment. The Midle East Technical University. 
Baddari, K., Noureddine D., Tahar A., and Jalal F. (2010). 
"Acoustic impedance inversion by feedback artificial 
neural network." Journal of Petroleum Science and 
Engineering 71, no. 3–4 (April 2010). 
Bhuiyan, M. H. (2001). An Intelligent System’s Approach 
to Reservoir Characterization in Cotton Valley. Master 
theses. Department of Petroleum and Natural Gas 
Engineering, 2001. 
Calderón-Macías, C.; Mrinal K. S., and P. L. Stoffa 
(2000). "Artificial neural networks for parameter 
estimation in geophysics." Geophysical Prospecting 
48 21–47. 
Darabi, H., Kavousi A., M. Moraveji, and M. Masihi. 
(2010). "3D fracture modeling in Parsi oil field using 
artificial intelligence tools." Journal of Petroleum 
Science and Engineering 71, no. 1-2: 67–76. 
Funahashi, K. (1989). "On the approximate realization 
ofcontinuous mappings by neural networks." Neural 
Networks 2: 183-192. 
Fung, C. C., Kok W. W., Halit E., Charlebois, R. and 
Crocker, H. (1996). "Modular Artificial Neural 
Network for Prediction of Petrophysical Properties 
from Well Log Data." IEEE Instrumentation and 
Measurement Technology Conference. Brussels. 
Haykin, S. (1999). "Neural Networks: A comprehensive 
foundation"Second
Edition, Prentice Hall, New 
Jersey. 
Kadkhodaie-Ilkhchi, A., M. R. Rezaee, H. Rahimpour-
Bonad, and A. Chehrazi. (2009). "Petrophysical Data 
Prediction from Seismic Attributes Using Committee 
Fuzzy Inference System."Computers & Geosciences: 
2314-2330. 
Lin, Y. and G. A. Cunningha. (1995). "A New Approach 
to Fuzzy-Neural System Modeling." IEEE 
Transactions on Fuzzy Systems 3, no. 2: 190-198. 
Mohaghegh , S. and S. Ameri. (1995). "Artificial Neural 
Network As A Valuable Tool For Petroleum 
Engineers." Society of Petroleum Engineers. SPE 
29220. 
Mohaghegh, S. (2000). “Virtual-Intelligence Applications 
in Petroleum Engineering: Part I - Artificial Neural 
Networks.” Journal of Petroleum Technology, 
Distinguished Author Series, September: 64-72. 
Mohaghegh, S. (2005) "Recent Developments in 
Application of Artificial Intelligence in Petroleum 
Engineering." Journal of Petroleum Technology, 
Distinguished Author Series, April: 86-91. 
Nakayama, K. (2000). "Estimation of Reservoir Properties 
by Monte Carlo Simulation ." Yokohama: SPE Asia 
Pacific Conference on Integrated Modelling for Asset 
Management. 
Niu, Y., P. M. Wong, and L. Chen (2000). "Sequential 
Neural Simulation: A New Approach for Stochastic 
Reservoir Modelling, 24-25 Outubro ." Paris: SPE 
European Petroleum Conference. 
Pedrycz W. and Gomide F. (1998). “An Introduction to 
Fuzzy Sets: Analysis and Design”. MIT Press Complex 
Adaptive Systems. 
Rumelhart, D. E., G. E. Hinton, and R. J. Williams. (1986); 
Learning internal representations by error propagation. 
Vol. 1, in Parallel distributed processing: explorations 
in the microstructure of cognition, by D. E. Rumelhart, 
J. L. McClelland, & PDP Research Group (Eds.), 
edited by Rumerlhart, 318-362. Cambridge MA: MIT 
Press. 
Saggaf, M., Nafi Toksoz M., and Mustafa H. (2003) 
“Estimation of reservoir properties from seismic data.” 
Geophysics 68:1969–1983. 
Taghavi, A. A. (2005). "Improved Permeability 
Estimation through use of Fuzzy Logic in a Carbonate 
Reservoir from Southwest, Iran." Kingdom of Bahrain: 
SPE Middle East Oil and Gas Show and Conference. 
Taner, M. T. (2000). Attributes Revisited. Rock Solid 
Images. 
Tang, H.; Toomey, N. and Meddaugh, S. (2011). "Using 
and Artificial-Neural-Network Method To Predict 
Carbonate Well Log Facies Successfully" SPE Journal 
Paper 
Tanscheit, R. Sistemas Fuzzy. Rio de Janeiro: 
Departamento de Engenharia Elétrica, PUC- Rio, 
2004. 
Van der Baan, M., and C. Jutten. (2000) "Neural networks 
in geophysical applications." Geophysics 65: 1032-
1047. 
Wang, L., P.M. Wong , and S.A.R. Shibli . (1998). 
"Modelling Porosity Distribution in the A'nan Oilfield: 
Use of Geological Quantification, Neural Networks 
and Geostatistics." SPE International Conference and 
Exhibition in China. SPE 48884. 
Weiss, W. W. , Shaochang Wo, and J.W. Weiss. (2011). 
"Data Mining at a Regulatory Agency to Forecast 
Waterflood Recovery." Colorado: SPE Rocky 
Mountain Petroleum Technology Conference. 
Zadeh, L.A. (1976). "Fuzzy Sets as a Basis for a Theory of 
Possibility". Fuzzy Sets andSystems, Vol. 1: 3-28. 
Zehui, Huang; Shimeld, John; Williamson, Mark and 
Katsube, John. (1996)."Permeability prediction with 
artificial neural network modeling in the Venture gas 
field, offshore eastern Canada."Geophysics 61: 422-
436.