Intelligent approaches for the synthesis of

IOP PUBLISHING

JOURNAL OF GEOPHYSICS AND ENGINEERING

doi:10.1088/1742-2132/5/1/002

J. Geophys. Eng. 5 (2008) 12–26

Intelligent approaches for the synthesis of petrophysical logs M Reza Rezaee1, Ali Kadkhodaie-Ilkhchi2 and Pooya Mohammad Alizadeh3 1

Department of Petroleum Engineering, Curtin University of Technology, Perth, WA, Australia School of Geology, University College of Sciences, University of Tehran, Tehran, Iran 3 Department of Petroleum engineering (Petroleum Exploration), Sciences & Research Branch, Tehran, Iran 2

E-mail: [email protected], [email protected] and [email protected]

Received 2 March 2007 Accepted for publication 19 July 2007 Published 28 November 2007 Online at stacks.iop.org/JGE/5/12 Abstract Log data are of prime importance in acquiring petrophysical data from hydrocarbon reservoirs. Reliable log analysis in a hydrocarbon reservoir requires a complete set of logs. For many reasons, such as incomplete logging in old wells, destruction of logs due to inappropriate data storage and measurement errors due to problems with logging apparatus or hole conditions, log suites are either incomplete or unreliable. In this study, fuzzy logic and artificial neural networks were used as intelligent tools to synthesize petrophysical logs including neutron, density, sonic and deep resistivity. The petrophysical data from two wells were used for constructing intelligent models in the Fahlian limestone reservoir, Southern Iran. A third well from the field was used to evaluate the reliability of the models. The results showed that fuzzy logic and artificial neural networks were successful in synthesizing wireline logs. The combination of the results obtained from fuzzy logic and neural networks in a simple averaging committee machine (CM) showed a significant improvement in the accuracy of the estimations. This committee machine performed better than fuzzy logic or the neural network model in the problem of estimating petrophysical properties from well logs. Keywords: petrophysical logs, synthesizing, artificial neural networks, fuzzy logic, committee

machine

Nomenclature ANN FIS TS FIS NPHI RHOB MSFL MSE Train LM FL CL MF CM

artificial neural network fuzzy inference system Takagi–Sugeno fuzzy inference system neutron log density log micro spherical focused log mean squared error Levenberg–Marquardt training algorithm fuzzy logic crisp logic membership function committee machine

1742-2132/08/010012+15$30.00

DT LLD LLS µ

sonic transit time deep laterolog shallow lateralog grade of membership

1. Introduction Petrophysical logs are one of the most important tools for the evaluation of hydrocarbon reservoirs. Parameters such as porosity, volume of shale, formation water saturation, lithology, fluid contacts and productive zones are obtained from well logs. In many cases, a complete set of log data may not be available—hole conditions, instrument failure, loss of data due to inappropriate storage and incomplete logging

© 2008 Nanjing Institute of Geophysical Prospecting

Printed in the UK

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Intelligent approaches for the synthesis of petrophysical logs

Figure 1. Four types of fuzzy membership functions: Gaussian, sigmoid, triangular and trapezoidal (Matlab user’s guide 2001).

are some of the reasons. In recent years, intelligent systems have been used as powerful tools for modelling and prediction in the petroleum industry. For example, Lim (2003, 2005), Huang et al (2001), Mohaghegh (2000), Tamhane et al (2000), Cuddy (1998), Soto et al (1997), Wong et al (1997), Malki et al (1996) and other researchers have applied intelligent systems to estimate several reservoir parameters from well log responses. The present study focuses on the following: (a) The application of intelligent systems including fuzzy logic (FL) and artificial neural networks (ANN), to the synthesis of petrophysical well log data including neutron, density, sonic and deep resistivity. (b) The comparison and evaluation of the results of the systems used in the synthesis of well logs. (c) The verification of the basic concepts and the validity of the applied intelligent systems in solving problems with different methodologies. (d) The combination of fuzzy logic and neural network estimations in a simple averaging committee machine for the evaluation of the accuracy of the committee machine.

Figure 2. Schematic diagram of information flow in a simple fuzzy inference system (Matlab user’s guide 2001). (a )

(b)

The dataset in this study came from three wells of lower cretaceous Fahlian formation, Southern Iran. The well log data were acquired in 1982 by Schlumberger Well Services Co.; the reservoir is predominantly limestone and the play type is oil.

2. Methods Figure 3. Membership functions for porosity cut-off in (a) the crisp logic (CL) and (b) the fuzzy logic (FL) approaches.

2.1. Fuzzy logic Fuzzy logic was initiated in 1965 by Lotfi Zadeh at the University of California. Fuzzy logic starts with the concept of fuzzy sets. Crisp sets only allow full-membership or nonmembership, whereas fuzzy sets allow partial membership which can take values ranging from 0 to 1: µA (x) : X → [0, 1]

(1)

where X refers to the universal set defined in a specific problem and µA (x) is the grade of membership for element x in fuzzy

set A (Yagar and Zadeh 1992). In other words, crisp sets are special cases of fuzzy sets, where the membership function for each element has only two values, 0 or 1 (Saggaf and Nebrija 2003). Although fuzzy logic is sometimes used as a synonym for multivalent logic, its more common use is to describe the logic of fuzzy sets (Zadeh 1965). The most important concept in fuzzy logic is definition of membership functions for input 13

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(a )

(b)

Figure 4. (a) Schematic diagram of an artificial neural network structure which consists of an input layer, one hidden layer and an output layer (Lee and Datta-Gupta 1999). (b) A mathematical model of an artificial neural network cell (Aminzadeh and de Groot 2006).

(a )

(b)

Figure 5. Crossplots showing the correlation coefficient between calculated and ANN predicted (a) porosity and (b) water saturation in the Sarvak formation, SW Iran (Rezaee et al 2003). Table 1. Appropriate inputs that were obtained from the cross-plot method for synthesizing petrophysical logs by ANN and FL. Predicted well log

NPHI

RHOB

DT

LLD

Inputs

DT, RHOB

NPHI, DT

NPHI, RHOB

LLS, MSFL, NPHI

and output data sets. For constructing a model based on fuzzy logic, it is important to select proper types and parameters for membership functions that best fit the dataset. There are several types of membership functions such as Gaussian, sigmoid, triangular, and trapezoidal functions (figure 1). In recent years, fuzzy logic has been used as a tool to construct petrophysical models in hydrocarbon reservoirs. 2.1.1. Fuzzy inference system (FIS). Fuzzy inference is the process of formulating from a given input to an output using fuzzy logic. There are two types of fuzzy inference systems: those of Mamdani and Assilian (1975) and Takagi and Sugeno (1985). Mamdani’s method attempts to control a system by synthesizing a set of linguistic control rules obtained from experienced human operators. The Takagi–Sugeno (TS) method is similar to the Mamdani FIS. The main difference between them is that the output membership functions are only constant for the TS-FIS. Figure 2 shows a schematic structure 14

of FIS. In the TS-FIS, membership functions are defined by a clustering process. Each cluster centre has a range of influence (cluster radius) for each input and output dimension. Assuming a smaller cluster radius will usually yield many small clusters (resulting in many rules) and specifying a large cluster radius yields a few large clusters in the data resulting in a few rules (Chiu 1994). In FIS, a set of if–then rules provides a relationship between input and output data sets. The closer a given input is to the ‘if’ part of the rule; the more the ‘then’ part will be influenced. Then, the fuzzy system adds up all of the ‘then’ parts and uses a defuzzification method to give the final output (Kosko 1994). 2.1.2. Why use fuzzy sets? Generally, definitions in the Geosciences are not clear-cut and most of the time are associated with uncertainty. Regarding the imprecise nature of fuzzy sets, it is appropriate to use fuzzy reasoning for solving problems that accompany vagueness and imperfection. The following simple example can clarify the subject.

Intelligent approaches for the synthesis of petrophysical logs 50

20

0 0.0

0.1

0.2

NPHI (v/v) 70

(b) Histogram of RHOB with Normal Curve

60 Frequency

50 40 30 20 10 0 2.35

2.45

2.55

2.65

RHOB (gr/cm3)

90

(c ) Histogram of DT with Normal Curve

80 70 Frequency

An artificial neural network (ANN) is a computational tool for solving difficult problems. The components of an ANN are neurons (or nodes) and connections (which are weighted links between neurons). In general, an ANN has three layers: an input layer, a hidden or inner layer and an output layer (figure 4(a)). The input layer receives the information. The hidden layer, which can consist of several layers, analyses the information and the output layer receives the results of the analysis and provides the output (Zeidenberg 1990). Figure 4(b) shows a schematic model of the processing procedure in a neuron element cell. The body of this cell consists of two parts. The first part is the ‘combination function’ that sums up all inputs and produces a value. This figure illustrates that for a neuron i all inputs Ij are weighted by a factor Wj and then summed to the final output ui: ui =

30

10

2.2. Artificial neural networks (ANNs)

i

(a ) Histogram of NPHI with Normal Curve

40 Frequency

The cut-off value of porosity for the Ziveh reservoir in the Moghan block (NW Iran) is considered to be 5%. This means if an interval of the reservoir has more than 5% porosity, it will be considered as economical. Figure 3 shows the membership functions for porosity cut-off from the crisp logic (CL) and the fuzzy logic (FL) approaches, respectively. According to the CL approach (figure 3(a)); the porosity value of 4% will not be economically viable for the Ziveh reservoir. However, FL proposes that it will be economic up to the degree of 0.7 (figure 3(b)). Therefore, fuzzy reasoning is very close to reality and can be a suitable tool for the prediction of reservoir properties.

60 50 40 30 20 10

Wj Ij .

(2)

0

j =1

60 DT (µs/ft)

70

100

(d ) Histogram of LLD with Norm al Curve Frequency

The second part is the ‘activation function’ or ‘transfer function’. When the weighted sum reaches a threshold, the transfer function is activated to produce the output. Transfer functions are of several types including sigmoid, tansig and logsig (Zeidenberg 1990). Usually, in a neural network cell there is an additional ‘bias’ input. The bias increases or decreases the weighted sum (Chen and Sidney 1997). The bias factor acts as a compensator and helps the network to best recognize the patterns. Learning in a neural network occurs by adjustment of the weights via a training rule (Zeidenberg 1990). Training algorithms are categorized as supervised or unsupervised. An error-back propagation algorithm is a supervised training technique that sends the input values forward through the network and then computes the difference between the calculated output and the corresponding desired output from the training dataset. This error is then propagated backward through the net and the weights are adjusted during a number of iterations. The training stops when the calculated output values best approximate the desired values (Bhatt and Helle 1999). Similar to fuzzy logic, ANNs have many applications in the petroleum industry. For example, one of the most important uses is their application in reserve estimation. Rezaee et al (2003, 2006) used a feed forward error back-propagation network for predicting porosity and water

50

50

0 0

100 LLD (Ohm.m)

200

Figure 6. Gaussian membership function fit to (a) NPHI, (b) RHOB, (c) DT and (d) LLD log data in the studied reservoir.

saturation in the Sarvak reservoir of Zagros Basin (SW Iran) which are two important parameters for reserve estimation. They found a very good agreement between calculated and predicted porosity and water saturation parameters (figure 5).

3. Summary of petrophysical logs used In this section, we briefly describe the wireline logs used in our petrophysical analysis. 15

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(a )

µ

(b)

Figure 7. DT and RHOB membership functions for synthesizing the NPHI log. Table 2. The role of the clustering radius on fuzzy model performance (MSE) and number of the fuzzy ‘if–then’ rules in generating NPHI, RHOB, DT and LLD logs. No. of fuzzy ‘if–then’ rules

Clustering Radius

NPHI

RHOB

DT

LLD

NPHI

RHOB

DT

LLD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

9 6 5 5 4 4 3 3 3 2 2 2 2 2 1 1 1 1 1 1

25 21 17 14 10 8 7 5 4 3 3 2 2 2 2 2 1 1 1 1

22 15 11 9 8 7 5 4 3 3 2 2 2 2 1 1 1 1 1 1

33 27 22 17 13 9 7 5 4 4 3 3 3 2 2 2 1 1 1 1

0.0097 0.0082 0.0076 0.0056 0.0025 0.0025 0.0039 0.0039 0.0039 0.0103 0.0103 0.0103 0.0103 0.0103 0.0469 0.0469 0.0469 0.0469 0.0469 0.0469

0.0185 0.0166 0.0111 0.0109 0.0092 0.0071 0.0071 0.0062 0.0065 0.0022 0.0022 0.0180 0.0180 0.0180 0.0180 0.0180 0.0504 0.0504 0.0504 0.0504

0.0212 0.0137 0.0102 0.0102 0.0090 0.0074 0.0059 0.0026 0.0018 0.0018 0.0077 0.0077 0.0077 0.0077 0.0338 0.0338 0.0338 0.0338 0.0338 0.0338

0.0441 0.0314 0.0222 0.0119 0.0097 0.0055 0.0043 0.0022 0.0018 0.0018 0.0015 0.0015 0.0015 0.0122 0.0122 0.0122 0.0561 0.0561 0.0561 0.0561

Neutron log (NPHI). In the neutron tool, high-energy neutrons are continuously emitted from a radioactive source. As the neutrons collide with a nucleus of equal mass (hydrogen atoms), their energy decreases. The slowing down of the neutrons and subsequent loss of energy is proportional to the volume of water or hydrocarbon contained in the formation which is itself directly related to the porosity of rock, if 100% fluid saturation is assumed (Rezaee 2001). Sonic log. The sonic tool measures the time required for transmission of an acoustic wave through a unit of formation thickness. Sonic transit time (DT) is used both in porosity determination and to compute secondary porosity in carbonate reservoirs. The sonic log is also an aid in lithology 16

MSE of fuzzy model

No. FIS

determination particularly when combined with the neutron logs or density logs (Serra 1984). Sonic and density logs can be combined to produce synthetic seismograms. Density log. The density log performs on the basis of the scattering of gamma rays within the formation from a radioactive source. The reflected (Compton scattered) gamma rays are recorded. This tool measures formation bulk density (RHOB) and is used as a primary indicator of porosity (Bateman 1985). The accuracy of the porosity determination is a function of measurement accuracy and knowledge of the matrix density (Patchett and Coalson 1982). A combination of this log and neutron or sonic logs helps to determine lithology and provides a more reliable porosity value.

Intelligent approaches for the synthesis of petrophysical logs (a )

(b)

µ

Figure 8. Shows the NPHI and DT membership functions for synthesizing the RHOB log. (a )

(b)

Figure 9. Shows the NPHI and RHOB membership functions for synthesizing the DT log.

Formation resistivity logs (LLD, LLS, and MSFL). Formation resistivity is a key element for identifying hydrocarbonbearing intervals and quantifying water saturation. There are numerous tools for making resistivity measurements in a borehole. Dual laterolog tools comprise a shallow reading measurement tool, lateral log shallow (LLS) and a deep measurement tool, lateral log deep (LLD). The microspherically focused log (MSFL) is a microresistivity tool with a shallow depth of investigation (Ellis 1987).

4. Synthesizing petrophysical logs 4.1. Fuzzy logic In this study, a Takagi–Sugeno fuzzy interference system (TS-FIS) was used for creating synthetic petrophysical logs in the Matlab programming environment. For each of the constructed models, the data sets were divided into two groups

including the modelling data (1986 data points) from two wells and the test data (748 data points) from the third well. Appropriate inputs to construct intelligent models based on FL are determined from crossplot analysis (table 1) of log data from the two wells (known input). All input and output membership functions and their parameters were extracted by a subtractive clustering method and then a set of fuzzy ‘if–then’ rules were generated for the formulation of input data of the target well log. In the first step, it is necessary to obtain the best type of membership functions (MFs) and an optimal number of MFs and fuzzy ‘if–then’ rules. Too few rules cannot cover the entire domain completely, and too many may complicate the system, causing low performance of the model. Membership function types were determined by fitting the proper function to the distribution of the used well log data in the studied reservoir. For this purpose, several frequency plots were generated showing the distribution of the data. Then, the best membership functions were fitted to the generated 17

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(b)

(c )

Figure 10. Shows the LLS, MSFL and NPHI membership functions for synthesizing the LLD log.

Figure 11. Processing steps in using TS-FIS for creation of a sonic log from neutron and density log inputs. This FIS consists of three rules with the antecedent of each rule separated by the ‘and’ operator.

18

Intelligent approaches for the synthesis of petrophysical logs (b )

(c )

(d )

µ

(a )

µ

Figure 12. Crossplots showing the correlation coefficient for synthesizing (a) NPHI, (b) RHOB, (c) DT and (d) LLD utilizing FL for the test well.

plots. According to figures 6(a)–(d), the Gaussian membership function is the best fit for the well log data used. So, the Gaussian membership was selected for constructing a fuzzy model. This is in agreement with the general rule for natural data which usually have a normal (Gaussian) distribution. In order to obtain the optimal number of rules and MFs, by specifying a set of values for the clustering radius which differs between 0 and 1, several numbers of rules were generated. Then, the mean square error (MSE) of the generated models was measured and the models with the highest performance (lowest error) were chosen as the optimal fuzzy inference systems for generating petrophysical logs. The effect of the clustering radius on fuzzy model performance and number of the fuzzy ‘if–then’ rules in generating petrophysical logs is shown in table 2. Generated MFs and fuzzy ‘if–then’ rules for synthesizing petrophysical logs are as below: Neutron log. By specifying 0.25 for the clustering radius, four Gaussian type membership functions were extracted for DT and RHOB inputs which were classified as very low, low, moderate and high (figure 7). Generated ‘if–then’ rules are below: (1) If (DT is very low) and (RHOB is high) then (NPHI is very low). (2) If (DT is low) and (RHOB is moderate) then (NPHI is low). (3) If (DT is moderate) and (RHOB is low) then (NPHI is moderate).

(4) If (DT is high) and (RHOB is very low) then (NPHI is high). Density log (RHOB). By specifying 0.5 for the clustering radius, three Gaussian type membership functions were extracted for NPHI and DT inputs which were classified as low, moderate and high (figure 8). Generated ‘if–then’ rules are below: (1) If (NPHI is low) and (DT is low) then (RHOB is high). (2) If (NPHI is moderate) and (DT is moderate) then (RHOB is moderate). (3) If (NPHI is high) and (DT is high) then (RHOB is low). Sonic log (DT). By specifying 0.5 for the clustering radius, three Gaussian type membership functions were extracted for NPHI and RHOB inputs which were classified as low, moderate and high (figure 9). Generated ‘if–then’ rules are as below: (1) If (NPHI is low) and (RHOB is high) then (DT is low). (2) If (NPHI is moderate) and (RHOB is moderate) then (DT is moderate). (3) If (NPHI is high) and (RHOB is low) then (DT is high). Deep laterolog. By specifying 0.65 for the clustering radius, three Gaussian type membership functions were extracted for LLS, MSFL and NPHI inputs which were classified as low, moderate and high (figure 10). Generated ‘if–then’ rules are as below: 19

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(a )

(c )

(b)

µ

(d )

Figure 13. Synthesized (a) NPHI, (b) RHOB, (c) DT and (d) LLD logs (dotted lines) utilizing the FL technique for the test well. Real logs are shown by solid lines.

(1) If (LLS is low) and (MSFL is low) and (NPHI is high) then (LLD is low). (2) If (LLS is moderate) and (MSFL is moderate) and (NPHI is moderate) then (LLD is moderate). (3) If (LLS is high) and (MSFL is high) and (NPHI is low) then (LLD is high). 20

According to table 2, the MSEs for optimal NPHI, RHOB, DT and LLD fuzzy models were equal to 0.0025, 0.0022, 0.0018 and 0.0015, respectively. After preparation of fuzzy models, the following steps were carried out for the estimation of log values in the third well (the test well) of the field by using FIS:


Applying the fuzzy operators gives a value to the antecedent of each rule and then the output membership function is truncated by this value. In this study and is used.

(a )

Step 3. Apply the aggregation method: in this step, outputs of each rule that fit into a fuzzy set are combined into a single fuzzy set.

(b )

Figure 14. (a) Purelin and (b) tansig transfer functions (Matlab user’s guide 2001).

Step 1. Fuzzify inputs: the FIS takes the inputs and determines the degree to which the inputs belong to each membership function. Step 2. Apply the fuzzy operator and the truncation method: for the case that the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one rule that represents the result of the antecedent for that rule. The most common operators are shown below: ‘and’ = use the minimum of the options. ‘or’ = use the maximum of the options. ‘not’ = use 1-option.

Step 4. Defuzzify: the input for the defuzzification process is the results of the aggregation method. Then FIS uses a defuzzification method (in this study, a weighted average) for the resulting output which is a crisp numerical value. Figure 11 shows an example of the processing steps in using TS-FIS for the creation of a sonic log from neutron and density log inputs. The measured MSEs for FL predicted NPHI, RHOB, DT and LLD in the test well were equal to 0.134, 0.128, 0.122 and 0.108, respectively. Figures 12(a)–(d) show the correlation coefficient between real data and FL predicted results for neutron, density, sonic and deep resistivity logs in the test well. Comparisons between the real and predicted neutron, density, sonic and deep resistivity logs utilizing TS-FIS are shown in figures 13(a)–(d).

(a )

(b)

(c)

(d )

µ

Figure 15. Crossplots showing the correlation coefficient for synthesizing (a) NPHI, (b) RHOB, (c) DT and (d) LLD for the test well utilizing ANNs.

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(a )

(b)

µ

(c )

(d )

Figure 16. Synthesized (a) NPHI (b) RHOB (c) DT and (d) LLD logs (dotted lines) utilizing the ANN technique for the test well. Real logs are shown by solid lines.

4.2. Artificial neural network In the artificial neural network approach, a three-layered errorback propagation algorithm was used for the construction 22

of intelligent models. The network was trained using the Levenberg–Marquardt training algorithm (Train LM); the details of which can be found in Boadu (1997, 1998) and Bishop (1995).

Intelligent approaches for the synthesis of petrophysical logs (a )

(b)

sonic and deep resistivity, respectively. Comparisons between real and predicted neutron, density, sonic and deep resistivity logs utilizing the ANN approach are shown in figures 16(a)–(d). 4.3. Construction of committee machine

(c )

(d )

µ

Figure 17. Crossplots showing the correlation coefficient for synthesizing NPHI, RHOB, DT and LLD utilizing combination of FL and ANN in a simple averaging CM for the test well.

Generally, a CM consists of a group of experts which combines the outputs of each system and thus reaps the benefits of all of the work, with little additional computation. So, performance of the model can be better than the best single expert (Haykin 1991, Sharkey 1996, Chen and Lin 2006). The simple ensemble-averaging method is the most well-used (Naftaly et al 1997, Chen and Lin 2006): It is assumed that there are N algorithms with an output vector oi, which are used to predict the target vector T (Chen and Lin 2006). The prediction error can be written as (3) ei = oi − T . The sum of the squared error for the ith algorithm oi is (4) Ei = ξ [(oi − T )2 ] = ξ ei2 . in which ξ [·] is the expectation. The average error for each of the algorithms acting alone is N N E avg = 1/N (5) E i =1/N ξ e2i . i=1

Four networks were generated for synthesizing NPHI, RHOB, DT and LLD logs using the same inputs which were used in the fuzzy model construction. For each of the networks, the data set was divided into three groups including training (1359 data points) and validation data (627 data points) from two wells and test data (748 data points) from the third well. The first layer included 2, 2, 2 and 3 neurons for the NPHI, RHOB, DT and LLD networks, respectively. The hidden, or inner, layer included 12, 7, 5 and 8 neurons for the NPHI, RHOB, DT, and LLD networks, respectively, and the third layer had one neuron for all the generated networks. The first and second transfer functions used were tansig and purelin, respectively. Purelin is a linear transfer function which calculates a layer’s output (y) from its net input (x) by a y = x relationship (figure 14(a)). Tansig is the hyperbolic tangent sigmoid transfer function which calculates the output using y = 2/(1 + exp(−2n)) − 1 relationship (figure 14(b)). More details on the mentioned transfer function can be found in Vogl et al (1988). The selected performance function for each of the networks was the average (mean) squared error between the network outputs and target outputs and was equal to 0.0036, 0.0042, 0.0021 and 0.0016 for the NPHI, RHOB, DT and LLD networks, respectively. Similar to the FL approach, the third well of the field was chosen to evaluate the reliability of ANN predictions. The input data for synthesizing each set of log data was passed through its related network and target logs were created successfully. The measured MSEs for ANN predicted NPHI, RHOB, DT and LLD in the test well were equal to 0.156, 0.164, 0.130 and 0.091 respectively. Figures 15(a)–(d) show the correlation coefficient between real and ANN predicted values for neutron, density,

i=1

Applying the averaging method, the output vector oi of the CM is N oi . (6) O CM = 1/N i=1

Therefore, the CM has the prediction squared error:  2  N E CM = ξ [(O CM − T )2 ] = ξ  1/N oi − T   = ξ  1/N

N

2  ei

i=1

.

(7)

i=1

Considering Cauchy’s inequality:

(a 1 b1 + a 2 b2 + · · · + a n bn )2 a12 + a22 + · · · + an2

. b12 + b22 + · · · + bn2 (8) and applying it to theE CM  2  N N E CM = ξ  1/N ei  1/N ξ e2i = E avg , (9) i=1

i=1

which indicates that the CM gives more accurate and reliable estimations than that of any one of the individual algorithms. In this study, the fuzzy logic and neural network estimations for each of the used petrophysical logs were combined in a CM to produce an overall estimation. Small additional computations were carried out as below to construct the CM and improve the results: NPHICM = 0.5 × (NPHIFL + NPHIANN ) RHOBCM = 0.5 × (RHOBFL + RHOBANN ) DTCM = 0.5 × (DTFL + DTANN ) LLDCM = 0.5 × (LLDFL + LLDANN ) 23

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(a )

(b)

µ

(c )

(d )

Figure 18. Synthesized (a) NPHI, (b) RHOB, (c) DT and (d) LLD logs (dotted lines) utilizing a combination of FL and ANN in a simple averaging CM for the test well. Real logs are shown by solid lines.

Crossplots of figures 17(a)–(d) show the correlation coefficient between measured and CM predicted results of NPHI, RHOB, DT and LLD logs, respectively. Comparisons between the real and predicted neutron, density, sonic and 24

deep resistivity logs versus depth utilizing CM are shown in figures 18(a)–(d). The MSEs of CM predicted NPHI, RHOB, DT and LLD are 0.112, 0.097, 0.092 and 0.085, respectively.


5. Discussion

6. Conclusions

The comparisons between measured and predicted petrophysical log data using fuzzy logic (figure 12) and neural networks (figure 15) show both techniques were successful to generate well logs. However, there is some disagreement between measured and predicted values as below:

The results of this study show that both fuzzy logic and artificial neural networks were successful for synthesizing petrophysical logs. Predicted and real well logs for a test well of the study field show a good correlation. The MSE of predicted NPHI, RHOB, DT and LLD logs was equal to 0.134, 0.128, 0.122 and 0.108 for FL models and was 0.156, 0.164, 0.130 and 0.091 for ANN models. The methods used for the estimation of petrophysical logs have different concepts and methodologies on which they base model construction. Despite this, the results of this study show they are very reliable and consistent with each other (especially for the LLD log), confirming their basic concepts and validity to solve the problem. However, judgment on the reliability of fuzzy logic or neural network models for predicting well logs is not recommended. Irrespective of rock heterogeneities, if the necessary parameters for constructing each of the mentioned models are adjusted precisely, they will reach similar results. Fuzzy logic predicted neutron, density and sonic logs are respectively lower, higher and higher than real logs. But the prediction for the same logs using ANN gives respectively higher, lower and lower values than real logs. Problems such as mineralogical heterogeneities, errors in modelling or test data, changes in fluid type and saturation may cause these kind of inconsistencies. The combination of the results obtained from fuzzy logic and neural networks in a simple averaging committee machine (CM) showed a significant improvement in accuracy of the estimations. This committee machine performed better than the best fuzzy logic or neural network model for the well logs estimation problem.

NPHI log. According to the crossplot of figure 15(a), ANN predictions of NPHI have generally overestimated real values. As the plot shows, for NPHI values between ∼0.12 and ∼0.18 v/v, the predictions are flattened around the value of 0.15 v/v. However, FL results have generally underestimated real values of NPHI (figure 12(a)). RHOB log. According to the crossplot of figure 15(b), ANN predictions of RHOB have generally underestimated real values (especially for rocks with real RHOBs above ∼2.45 g cm3). Underestimation could be as high 0.12 g cm3 for a real RHOB of 2.6 g cm3 which could result in a significant error in porosity estimation. On the other hand the predictions for RHOB values below 2.4 g cm3 are overestimated by ANN, albeit the amount of overestimation is relatively small. However, considering the crossplot of figure 12(b), FL predictions of RHOB have predominantly overestimated real values. DT log. According to the crossplot of figure 15(c), ANN predictions of DT have systematically underestimated the real values. The ANN DT prediction can be too low by up to 5 µs/ft (figure 16(c) at 11 670 to 11 700 ft and 11 220 to 11 300 ft). However, FL predictions of DT have generally overestimated real values (figure 16(c), especially at 11 850 to 11 950 ft). LLD log. Comparison between the crossplots of figure 12(d) and figure 15(d) shows FL and ANN predictions of LLD log are close to the real values. ANN predictions of LLD values below 5 m have to some extent overestimated the real values. However, as the scale of the resistivity log is logarithmic, a minor deviation between the real and synthetic logs can be seen in figure 13(d) (at 11 500 to 11 650 ft) and figure 16(d) (at the depth below 11 800 ft). These inconsistencies occur due to several reasons, including reservoir complexity, mineralogical changes, changes in fluid content and unavoidable errors in model or test data. For example, any unwanted error in text data can lead to the construction of a model which performs well on the test set but performs poorly on out-of-sample test data. Moreover, changes in mineralogy or fluid content and saturation may have different effects on FL or ANN models. A comparison among CM, FL and ANN results (figures 12, 15 and 17) shows that CM has had a significant improvement on predicting NPHI, RHOB, DT and LLD logs so that most of the disagreement between real and FL or ANN predicted results have been removed or minimized. According to figure 18, there is good agreement between CM and measured log data. However, some mis-fits can be seen in NPHI (below 11 850 ft), RHOB (11 260 to 11 310 ft) and DT (below 11 900 ft) which can be waived when the correlation is being considered in the entire interval.

Acknowledgments The first author acknowledges the Curtin University of Technology, Department of Petroleum Engineering, and the Western Australian Energy Research Alliance (WA : ERA) for their support.

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