27

Revista de Geologia, Vol. 21, nº 1, 27-34, 2008 www.revistadegeologia.ufc.br

Associating fuzzy logic, neural networks and multivariable statistic methodologies in the automatic identification of oil reservoir lithologies through well logs Abel Carrasquillaa, Jadir da Silvab (in Memorian) & Roosevelt Flexac Recebido em 13 de dezembro de 2007 / Aceito em 30 de maio de 2008

Resumo Neste artigo, apresentamos uma nova abordagem para a identificação automática de litologia utilizando dados de perfis geofísicos de poço, a qual associa lógica difusa ou nebulosa (fuzzy), redes neurais e métodos de estatística multivariada. Em primeiro lugar, escolhemos os perfis que representam tipos litológicos tipos, como raios gama (GR) e densidade (RHOB), e, imediatamente, aplicamos um algoritmo de lógica difusa para determinar o número ótimo de agrupamentos ou conglomerados. Na etapa seguinte, uma rede neural competitiva é desenvolvida, com base na regra de aprendizagem de Kohonen, na qual, a camada de entrada é composta de dois neurônios que representam o mesmo número de perfis utilizados. Por outro lado, a camada competitiva é composta por vários neurônios, que têm o mesmo número de conglomerados, conforme determinado pelo algoritmo da lógica difusa. Finalmente, alguns elementos do banco de dados dos tipos litológicos são selecionados aleatoriamente para discriminar variáveis, o que corresponde à entrada de dados do programa de análise discriminante multi - grupo. Desta forma, com a aplicação desta metodologia, os tipos litológicos foram automaticamente identificados ao longo de um poço do Campo de Namorado, Bacia de Campos, o que apresentou certa dificuldade nos resultados, principalmente, devido a complexidade geológica deste campo. Palavras-Chave: Lógica fuzzy, Redes neurais, Estatística multivariada, Identificação automática de litologia, Reservatórios de petróleo, Perfis geofísicos de poço Abstract In this article, we present a new approach to the automatic identification of lithologies using only well log data, which associates fuzzy logic, neural networks and multivariable statistic methods. Firstly, we chose well log data that represents lithological types, as gamma rays (GR) and density (RHOB), and, immediately, we applied a fuzzy logic algorithm to determine optimal number of clusters. In the following step, a competitive neural network is developed, based on Kohonen’s learning rule, where the input layer is composed of two neurons, which represent the same number of used logs. On the other hand, the competitive layer is composed by several neurons, which have the same number of clusters as determined by the fuzzy logic algorithm. Finally, some data bank elements of the lithological types are selected at random to be the discriminate variables, which correspond to the input data of the multigroup discriminate analysis program. In this form, with the application of this methodology, the lithological types were automatically identified throughout the a well of the Namorado Oil Field, Campos Basin, which presented some difficulty in the results, mainly because of geological complexity of this field. Keywords: Fuzzy logic, Neural networks, Multivariable statistic, Lithology automatic identification, Oil reservoirs, Geophysical well logs Revista de Geologia, Vol. 21 (1), 2008

28 Carrasquilla et al., Associating fuzzy logic, neural networks and ... a

Laboratorio de Engenharia e Exploração de Petróleo, Universidade Estadual do Norte Fluminense Darcy Ribeiro, Rodovia Amaral Peixoto Km 163, Av. Brennand s/n, Imboacica, 27925-310, Macaé - RJ. E-mail: [email protected] b Departamento de Geologia, Universidade Federal do Rio de Janeiro, Av. Brigadeiro Trompowski s/n, Ilha do Fundão, 21.949-900, Rio de Janeiro - RJ. c Baker Hughes do Brasil Ltda, Rodovia Amaral Peixoto s/n, km 184, Balneario Lagomar, 27970-020, Macaé - RJ.

1. Introduction The identification of the lithologies crossed by the well can be obtained through the interpretation of well log data, which is cheaper and efficient than core analysis. Thus, the act of obtaining lithological identification, in this way, is one of the most important contributions of borehole geophysics to the geological studies into the petroleum industry. However, the identification of lithologies through logs may, in some cases, becomes a difficult task, due to the occurrence of ambiguities in the measurements, as well as, the mineralogical complexity of the geological formations. Thus, it is important to put in mind that the information derived from the well logs is affected by the variation on rocks physical properties. In this sense, diverse authors have applied conventional computational algorithms and statistical methods that have shown good results in geological and geophysical applications. For example, Serra & Abbot (1989), Bucheb (1991) and Hsieh (2005) made use of various statistical methods in their determination of lithologies and electrofacies. However, in such applications, the participation of an interpreter was always necessary to provide a priori information to define patterns in the well log data. Another studies have used neural networks to identify lithologies (Chang et al., 2002), while others have used multivariable statistics and neural networks in reservoir engineering problems, but always in an independent and separate way (Nitters et al., 1995). In our study, we present a singular and inexpensive technique that combines information derived, exclusively, from two well logs data, making an association of fuzzy logic, a competitive neural network and the technique of multi-group discriminate analysis. To test this innovate technology, we used data belonged to the Brazilian National Petroleum Agency (ANP), which were obtained off shore in Namorado Oil Field, Campos Basin, Brazil, by Brazilian National Petroleum Company (Petrobras). Revista de Geologia, Vol. 21 (1), 2008

The new methodology is considered relatively cheap and, to carry out it, we divided the whole process into three stages. Thus, this study brings, in addition to propose a new approach with a more efficient form to the automatic identification of lithologies, relevant points for the petroleum industry, mainly, the easy implementation and low cost of the method, besides the possibility to aggregate the method to existing well profiling services, providing, in this way, a fast interpretation during the acquisition of logs without the presence of an interpreter. 2. Methodology Initially, some effort was done, in a simplified way, in the lithology identification procedures, making use only on GR and RHOB well log measurements, which are considered sensitive to lithology (Serra & Abbot, 1989). The utilization of only two well logs as input vectors had the goal to adjust the neural network and the multivariable statistic programs for future applications. The competitive neural network used in our study makes part of an unsupervised learning network, which, as its name suggests, there is no supervisor to accompany. 2.1. The learning process This type of learning only becomes possible when there is redundancy in the input data. Without this redundancy, it would be impossible to find any special patterns or characteristics in the input data (Braga et al., 2000). Normally, the competitive neural networks, which operates with learning under very slight or any supervision, requires two phases to be executed: 1) the formation of unit groups, which correspond to the entry groups in the entry space, and, 2) the labeling of the groups (Nascimento & Yoneyama, 2000). The neural network used in our study is com-

29 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

posed, fundamentally, of two layers (Andrade et al., 2002): the entry layer, which has sensorial elements responsible for the input signals in the network, and, the competitive layer, which characterizes the network and is composed by neurons, which are encouraged to compete amongst themselves, such that only one neuron remains active, becoming the winner neuron. The architecture of a neural network with a competitive layer is displayed in Figure 1. Following the proposed methodology, we first sought how to define the number of clusters, which occur in the well log data. To overcome this problem, we applied a computational algorithm based on Fuzzy Logic (Nascimento & Yoneyama, 2000). The definition of the number of clusters really existing in the original data is of great importance to the implementation of the competitive neural network.

starts by minimizing the following objective function (Sun et al., 2004): n

2

k =1 i =1

(1)

where n is the total number of data vectors in the data set and c is the number of clusters. The input data set vector is given by Zk = {zk,1; zk,2; : : : ; zk,n} and V = {v1; v2; : : : ; vc}, which are the cluster centers; and U = (mik)c×n is a fuzzy partition matrix composed of the membership grade of each input vector zk , such that: c

∑µ i =1

ik

= 1 for k = 1,2,..., n and µ ik ≥ 0.

(2)

According to Das Gupta (2001), to minimize Jm(U, V), the partition matrix and the cluster centers have to assume the following iterative formula:

2.2. Number of clusters selection The main goal of this section is to develop an effective fuzzy algorithm to determine the optima number of clusters in the data set concerned with the natural gamma-ray and density logs. Before to do this, we introduce a brief review of the basic fuzzyc means algorithm (FCM); and the general model selection algorithm for determining the number of clusters and a new validity index for measuring the validity of clustering. The FCM algorithm is the most widely used fuzzy clustering algorithm in practice. It

c

J m (U , V ) = ∑∑ ( µ ik ) m z k − ν i ,

⎡ ⎢⎛⎜ c z − ν i µ ik = ⎢⎜ ∑ k ⎢⎜ j =1 z k − ν j ⎝ ⎣⎢

2 2

⎞ ⎟ ⎟⎟ ⎠

1 m −1

−1

⎤ ⎥ ⎥ , ⎥ ⎦⎥

(3)

with n

νi =

∑ (µ k =1 n

ik

∑ (µ k =1

)m zk . ik

)

m

(4)

where k = 1, 2, …, n and i = 1, 2, …, c.

Fig. 1. Model of a neural network with competitive layer architecture. Revista de Geologia, Vol. 21 (1), 2008

30 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

The basic fuzzy algorithm is as follows: Step 1: Enter the number of clusters c > 2, and the fuzzier exponent m = 2;

For c varying from cmin=2 to cmax=10 do: Step 1: Initialize cluster centers (Eq. 3); Step 2: Apply the basic algorithm to update µik and

Step 2: Initialize the cluster centers ν i( 0 ) at random; Step 3: Calculate µik according to equation (3);

ν i( 0 ) ; Step 3: Test for convergence; if no, go to Step 2; Step 4: Compute a validity value S(c) of Equation 5. Choose c such that the cluster validity index S(c) is a minimum. As shown in Figure 2, the measure of fuzzy do not varies hardly after 6 clusters, which means this is the selected number of clusters.

Step 4: Calculate ν i(1) according to equation (4). Step 5: Calculate the new µik according to equation (3) and the centers of Step (4); Step 6: If ||µik – new (µik)|| < 10-5 stop, else do ν i( 0 ) = ν i(1) , and go to Step (3). As in our case no a priori knowledge of the number of clusters exists, an automatic procedure of assessing the validity of a fuzzy partition is required. To determine correctly a validity measure that assigns to the final partition matrix or the optimum number of clusters, we follow the known fuzzy validity measure proposed by Sugeno &Yasukawa (1993): n

c

S (c) = ∑∑ ( µik ) m ( z k − ν i − ν i − z ), 2

k =1 i =1

2

(5)

where z is the average of the data. The optimum number of clusters is determined as a minimum or a convergent value of the functional S(c) as c increases. To choose this number of clusters we introduce an algorithm that applies the basic FCM clustering algorithm to the input logs for making c varying among a minimum and maximum value and chooses the best value of c based on the validity criterion given by Equation 5. So, the proposed algorithm contains the following steps:

2.3. Neural network architecture The competitive neural network was developed (Haykin, 1999), with two neurons, called as sensory elements in the input layer, which will accommodate the utilization of only GR and RHOB well logs. On the other hand, for the competitive layer, the neural network was composed of a number of neurons exactly equal to the number of clusters defined by the Fuzzy Logic algorithm. During the training phase, we applied Kohonen’s training rule (Kohonen 1989), where the synaptic weights are adjusted by the training process, with the objective to find the centers of the clusters in the input entry vectors. The synaptic weight of the winner neuron, which represents a column of the entry weight matrix, is adjusted with Kohonen’s learning rule. Supposing that the ith neuron wins, the element of the ith column of the input weight matrix is adjusted,

Fig. 2. An example of the application of the fuzzy logic algorithm to determine the number of clusters for the data of Well NA02. Revista de Geologia, Vol. 21 (1), 2008

31 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

as shown below:

[

]

Wij ( new) = Wij (old ) + α × X i − Wij (old ) , (6)

where Wij (new) is the matrix of initial synaptic weights, Wij (old) is the matrix of updated weights and α is the learning rate, (Xi, i= 1,…n) is the input vector (input layer) and (Yi, i= 1,…n) is the output layer, where n denotes the number of input nodes and m stands for the maximum number of clusters (see Fig. 1). In the case of this study, n = 2 and m = 6. Kohonen’s rule considers, as an important characteristic, what the weights of a neuron learn from the entry vectors, and due to this fact, it is of great utility in applications of recognition. After grouping the clusters, it is necessary to label the different groups, in order to indicate which lithological type of each group has the greatest possibility to represent. This labeling was done through the application of a computational program based on the interpretation of GR and RHOB crossplots. In this procedure, the classical conception of well log interpretation was considered (Ellis, 1987), and, thus, at the end of the process, the labeled groups were assumed to be lithological types present in the well. Following the labeling process, once established the definition of the lithological types present in the well, we chose at random some elements of each lithological type to compose data banks that were used as entry data, known as discriminate variables, in a program of multi-group discriminate analysis. In our study, the well log data were applied to the discriminate functions, resulting in the automatic identification of the lithologies in each point of the well log reading. The discriminate functions may be mathematically defined as follows: R = λ1ψ1 + λ2ψ2 + ... + λmψm,

(7)

where 1 < k < m, with m is the number of variables used in the analysis, R is the discriminate index of the function, λk are the coefficients of the discriminate function, and ψk are the values of the kth variable of a determined object, which represents the well log data. Thus, we have performed the association of competitive neural networks with the technique of discriminate analysis, conjugating the distinct

characteristics of the two methods, with the supporting of the fuzzy logic in the relation to clusters. Such an association permits the identification of lithological types utilizing only well log data, and minimizing the role of the interpreter in the provision of a priori information (patterns in the well logs) used in the discriminatory process. It should take into consideration that the identification of lithology based only on two logs – GR and RHOB – is not the most suitable, as it is known that a more precise interpretation would require a joint interpretation with more logs, such as NPHI (neutronic porosity), ILD (resistivity), and DT (transit time). 3. Results The first results of this study were derived from the application of this methodology to well log data, namely, wells NA02 and NA04 from Namorado Oil Field in Campos Basin/Brazil. In the first stage of the study, a competitive neural network was constructed, with two neurons in the entry or input layer. Well log data on GR and RHOB were then analyzed, corresponding to an extension of 225 m along the well. The competitive layer of the neural network was composed of six neurons, according to the quantity of clusters defined by the fuzzy logic algorithm (Fig. 2). In the training phase, 200 interactions, known as epochs, were used to adjust the synaptic weights, locating the centers of the clusters in the entry data (Fig. 3). After this stage, it became necessary to label the groups found, as well as to indicate which type of lithology each group has the greatest possibility to represent. With this goal, a computational program was developed and applied, which took into consideration the classic conception of well log interpretation. At the end of this process, the labeled groups were assumed to be the lithological types present in the well: sandstone, limestone, and shale – which are the lithological types expected in Namorado Field (Figs. 4 and 5). For other petroleum fields with more complex lithologies, one should take into account the occurrence of other lithological types. In the last step of the used methodology, some elements of the lithological type were selected at random to compose several data banks, used as Revista de Geologia, Vol. 21 (1), 2008

32 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

Fig. 3. Log data from well NA02 used as input vectors in the neural network composed of six neurons in the competitive layer, with 200 interactions, where the location of the cluster centers may been seen as bigger circles.

Fig. 4. Results of the labeling of lithological types present in Well NA02: sandstones, limestones and shales.

Fig. 5. Well NA02: (A) GR and RHOB logs; (B) lithological section; (C) actual results, with the association of neural networks, fuzzy logic and multivariable statistics; and, (D) previous results, using only discriminate analysis (Flexa et al., 2004).

entry data (discriminate variables) in a discriminate analysis program, which permitted discrimination among various groups. The same methodology used for Well NA02 was applied in another well in the Namorado Field (Well NA04). The results obtained for Well NA04 are shown in Figure 6B, which may be compared Revista de Geologia, Vol. 21 (1), 2008

with the GR and RHOB logs (Fig. 6A). A detail of a section of the actual results is shown in figure 6C, which may be compared with the results of previous studies shown in Figure 6D (Flexa et al., 2004). These results are also verified through analysis of core data from Well NA04 provided by ANP, also showing excellent concordance (Fig. 6E).

33 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

Fig. 6. Well NA04: (A) GR and RHOB logs; (B) lithological section; (C) actual results, with the association of neural networks, fuzzy logic and multivariable statistics; and, (D) previous results, using only discriminate analysis (Flexa et al., 2004).

Taking the core data analysis from Well NA02 as baseline or true information, the percent value of concordance for the association of competitive neural networks with multi-group discriminate analysis was on the order of 82.43%, while in the earlier studies, in which only discriminate analysis was applied, the success rate was only on the order of 71.62%, as shown in Table 1. For Well NA04, also taking the core data analysis as a reference point, the percent value of concordance for the association of competitive neural networks with multi-group discriminate analysis was on the order of 71.53%, while in the earlier studies, in which only discriminate analysis was applied, the success rate was only on the order of 87.59%, as shown in Table 1. The low performance of the association of competitive neural network with multi-group

discriminate analysis, for the Well NA04 is due for the fact that, in this well, the lithology is more complex, there are occurrence of marls, what affect your prediction, and mainly, because the neural network used in this study is composed of only two neurons (two well log) in the input layer. The performance becomes better with the use of more neurons in the input layer. 4. Conclusions Our work has shown, satisfactory, promissory results when only two well logs (GR and RHOB) are combining with an association of techniques such as fuzzy logic, neural networks and discriminate analysis, toward the automatic identification of lithologies along wells. These results indicate the possibi-

Tab. 1. Performance of the association of competitive neural networks and discriminate analysis when compared to the well core data and earlier studies. We lls

Core analys is (%) Earlie r s tudie s (%)

Curre nt s tudy (%)

NA02

100

71.62

82.43

NA04

100

87.59

71.53

Revista de Geologia, Vol. 21 (1), 2008

34 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

lity of developing more complex neural networks, composed with a bigger number of well logs, which should improve the process. Another possibility would be to extend this methodology to the automatic identification of fluids present in the reservoirs, such as oil and gas. In this scenario, the logs sensitive to lithology utilized in this work (GR and RHOB), would be used in conjunction with those more sensitive to fluids, such as resistivity (ILD), neutronic porosity (NPHI) and DT (transit time) logs. Finally, this study, additionally to propose a new approach in the automatic identification of lithologies along the wells, presents relevant points for the petroleum industry, mainly the facility of implementation and low cost of the method. Besides this, it exists the possibility to aggregate the method to existing well log services, providing, in this form, a fast interpretation during the acquisition of logs without the need of an expert interpreter. Acknowledgements This research was supported by funds donated by UENF and Brazilian National Science Foundation (CNPq), through PhD and research fellowships granted to authors. References Andrade, A. & Fischetti, A.I., 2002, Porosity images from well logs. Journal of Petroleum Science and Engineering. Elsevier, 36 (2): 149-158. Braga, A.P., Carvalho, A.P.L.F. & Ludermir, T.B., 2000, Redes neurais artificiais: teoria e aplicação. LTC, Rio de Janeiro, 262p.

Revista de Geologia, Vol. 21 (1), 2008

Bucheb, J.A., 1991, Aplicação de tratamento estatístico multivariante em dados de perfis de poços da Bacia de Sergipe-Alagoas. Dissertação de Mestrado, UFPA, Belém, 136p. Chang, H.C., Kopaska-Merkel, D.C. & Chen, H.C., 2001, Identification of lithofacies using Kohonen self-organizing maps. Computers and Geosciences, 28: 223-229. Das Gupta, S.P., 2001, Application of a fuzzy pattern recognition method in borehole geophysics. Computer and Geosciences, 27: 85-89. Ellis, D.V., 1987, Well logging for earth scientists. Elsevier, USA, 243p. Flexa, R.T., Andrade, A. & Carrasquilla, A., 2004, Identificação da litologia nos perfis de poço do Campo de Namorado (Bacia de Campos) e do Lago Maracaibo (Venezuela) aplicando estatística multivariada. Revista Brasileira de Geociências, 34 (4): 571-578. Haykin. S., 1999, Neural networks: a comprehensive foundation. Prentice-Hall, New Jersey, 180p. Hsieh, B.Z., Lewis, C. & Lin, Z.S., 2005, Lithology identification of aquifers from geophysical well logs and fuzzy logic analysis: Shui-Lin Area, Taiwan. Computers and Geosciences, 31: 263-275. Kohonen, T., 1989, Self-organization and associative memory. Springer-Verlag. Berlin, 312p. Nascimento, C.L. Jr. & Yoneyama, T., 2000, Inteligência Artificial em Controle e Automação. E. Blucher, FAPESP, São Paulo, 104p. Nitters, G., Davies, D.R. & Epping, W.J.M., 1995, Discriminant analysis and neural nets: valuable tools to optimize completion practices. SPE Drilling & Completion, 127133. Serra, O. & Abbot, H.T., 1989, The contribution of logging data to sedimentology and stratigraphy. SPE 9270, 18p. Sugeno, M. & Yasukawa, T., 1993, A Fuzzy-Logic-Based Approach to Qualitative Modeling. IEEE Transactions on Fuzzy Systems, 1 (1): 7-31. Sun, H., Wang, S. & Jiang, Q., 2004, FCM-based model selection algorithms for determining the number of clusters. Pattern Recognition, 37: 2027-2037.

Revista de Geologia, Vol. 21, nº 1, 27-34, 2008 www.revistadegeologia.ufc.br

Associating fuzzy logic, neural networks and multivariable statistic methodologies in the automatic identification of oil reservoir lithologies through well logs Abel Carrasquillaa, Jadir da Silvab (in Memorian) & Roosevelt Flexac Recebido em 13 de dezembro de 2007 / Aceito em 30 de maio de 2008

Resumo Neste artigo, apresentamos uma nova abordagem para a identificação automática de litologia utilizando dados de perfis geofísicos de poço, a qual associa lógica difusa ou nebulosa (fuzzy), redes neurais e métodos de estatística multivariada. Em primeiro lugar, escolhemos os perfis que representam tipos litológicos tipos, como raios gama (GR) e densidade (RHOB), e, imediatamente, aplicamos um algoritmo de lógica difusa para determinar o número ótimo de agrupamentos ou conglomerados. Na etapa seguinte, uma rede neural competitiva é desenvolvida, com base na regra de aprendizagem de Kohonen, na qual, a camada de entrada é composta de dois neurônios que representam o mesmo número de perfis utilizados. Por outro lado, a camada competitiva é composta por vários neurônios, que têm o mesmo número de conglomerados, conforme determinado pelo algoritmo da lógica difusa. Finalmente, alguns elementos do banco de dados dos tipos litológicos são selecionados aleatoriamente para discriminar variáveis, o que corresponde à entrada de dados do programa de análise discriminante multi - grupo. Desta forma, com a aplicação desta metodologia, os tipos litológicos foram automaticamente identificados ao longo de um poço do Campo de Namorado, Bacia de Campos, o que apresentou certa dificuldade nos resultados, principalmente, devido a complexidade geológica deste campo. Palavras-Chave: Lógica fuzzy, Redes neurais, Estatística multivariada, Identificação automática de litologia, Reservatórios de petróleo, Perfis geofísicos de poço Abstract In this article, we present a new approach to the automatic identification of lithologies using only well log data, which associates fuzzy logic, neural networks and multivariable statistic methods. Firstly, we chose well log data that represents lithological types, as gamma rays (GR) and density (RHOB), and, immediately, we applied a fuzzy logic algorithm to determine optimal number of clusters. In the following step, a competitive neural network is developed, based on Kohonen’s learning rule, where the input layer is composed of two neurons, which represent the same number of used logs. On the other hand, the competitive layer is composed by several neurons, which have the same number of clusters as determined by the fuzzy logic algorithm. Finally, some data bank elements of the lithological types are selected at random to be the discriminate variables, which correspond to the input data of the multigroup discriminate analysis program. In this form, with the application of this methodology, the lithological types were automatically identified throughout the a well of the Namorado Oil Field, Campos Basin, which presented some difficulty in the results, mainly because of geological complexity of this field. Keywords: Fuzzy logic, Neural networks, Multivariable statistic, Lithology automatic identification, Oil reservoirs, Geophysical well logs Revista de Geologia, Vol. 21 (1), 2008

28 Carrasquilla et al., Associating fuzzy logic, neural networks and ... a

Laboratorio de Engenharia e Exploração de Petróleo, Universidade Estadual do Norte Fluminense Darcy Ribeiro, Rodovia Amaral Peixoto Km 163, Av. Brennand s/n, Imboacica, 27925-310, Macaé - RJ. E-mail: [email protected] b Departamento de Geologia, Universidade Federal do Rio de Janeiro, Av. Brigadeiro Trompowski s/n, Ilha do Fundão, 21.949-900, Rio de Janeiro - RJ. c Baker Hughes do Brasil Ltda, Rodovia Amaral Peixoto s/n, km 184, Balneario Lagomar, 27970-020, Macaé - RJ.

1. Introduction The identification of the lithologies crossed by the well can be obtained through the interpretation of well log data, which is cheaper and efficient than core analysis. Thus, the act of obtaining lithological identification, in this way, is one of the most important contributions of borehole geophysics to the geological studies into the petroleum industry. However, the identification of lithologies through logs may, in some cases, becomes a difficult task, due to the occurrence of ambiguities in the measurements, as well as, the mineralogical complexity of the geological formations. Thus, it is important to put in mind that the information derived from the well logs is affected by the variation on rocks physical properties. In this sense, diverse authors have applied conventional computational algorithms and statistical methods that have shown good results in geological and geophysical applications. For example, Serra & Abbot (1989), Bucheb (1991) and Hsieh (2005) made use of various statistical methods in their determination of lithologies and electrofacies. However, in such applications, the participation of an interpreter was always necessary to provide a priori information to define patterns in the well log data. Another studies have used neural networks to identify lithologies (Chang et al., 2002), while others have used multivariable statistics and neural networks in reservoir engineering problems, but always in an independent and separate way (Nitters et al., 1995). In our study, we present a singular and inexpensive technique that combines information derived, exclusively, from two well logs data, making an association of fuzzy logic, a competitive neural network and the technique of multi-group discriminate analysis. To test this innovate technology, we used data belonged to the Brazilian National Petroleum Agency (ANP), which were obtained off shore in Namorado Oil Field, Campos Basin, Brazil, by Brazilian National Petroleum Company (Petrobras). Revista de Geologia, Vol. 21 (1), 2008

The new methodology is considered relatively cheap and, to carry out it, we divided the whole process into three stages. Thus, this study brings, in addition to propose a new approach with a more efficient form to the automatic identification of lithologies, relevant points for the petroleum industry, mainly, the easy implementation and low cost of the method, besides the possibility to aggregate the method to existing well profiling services, providing, in this way, a fast interpretation during the acquisition of logs without the presence of an interpreter. 2. Methodology Initially, some effort was done, in a simplified way, in the lithology identification procedures, making use only on GR and RHOB well log measurements, which are considered sensitive to lithology (Serra & Abbot, 1989). The utilization of only two well logs as input vectors had the goal to adjust the neural network and the multivariable statistic programs for future applications. The competitive neural network used in our study makes part of an unsupervised learning network, which, as its name suggests, there is no supervisor to accompany. 2.1. The learning process This type of learning only becomes possible when there is redundancy in the input data. Without this redundancy, it would be impossible to find any special patterns or characteristics in the input data (Braga et al., 2000). Normally, the competitive neural networks, which operates with learning under very slight or any supervision, requires two phases to be executed: 1) the formation of unit groups, which correspond to the entry groups in the entry space, and, 2) the labeling of the groups (Nascimento & Yoneyama, 2000). The neural network used in our study is com-

29 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

posed, fundamentally, of two layers (Andrade et al., 2002): the entry layer, which has sensorial elements responsible for the input signals in the network, and, the competitive layer, which characterizes the network and is composed by neurons, which are encouraged to compete amongst themselves, such that only one neuron remains active, becoming the winner neuron. The architecture of a neural network with a competitive layer is displayed in Figure 1. Following the proposed methodology, we first sought how to define the number of clusters, which occur in the well log data. To overcome this problem, we applied a computational algorithm based on Fuzzy Logic (Nascimento & Yoneyama, 2000). The definition of the number of clusters really existing in the original data is of great importance to the implementation of the competitive neural network.

starts by minimizing the following objective function (Sun et al., 2004): n

2

k =1 i =1

(1)

where n is the total number of data vectors in the data set and c is the number of clusters. The input data set vector is given by Zk = {zk,1; zk,2; : : : ; zk,n} and V = {v1; v2; : : : ; vc}, which are the cluster centers; and U = (mik)c×n is a fuzzy partition matrix composed of the membership grade of each input vector zk , such that: c

∑µ i =1

ik

= 1 for k = 1,2,..., n and µ ik ≥ 0.

(2)

According to Das Gupta (2001), to minimize Jm(U, V), the partition matrix and the cluster centers have to assume the following iterative formula:

2.2. Number of clusters selection The main goal of this section is to develop an effective fuzzy algorithm to determine the optima number of clusters in the data set concerned with the natural gamma-ray and density logs. Before to do this, we introduce a brief review of the basic fuzzyc means algorithm (FCM); and the general model selection algorithm for determining the number of clusters and a new validity index for measuring the validity of clustering. The FCM algorithm is the most widely used fuzzy clustering algorithm in practice. It

c

J m (U , V ) = ∑∑ ( µ ik ) m z k − ν i ,

⎡ ⎢⎛⎜ c z − ν i µ ik = ⎢⎜ ∑ k ⎢⎜ j =1 z k − ν j ⎝ ⎣⎢

2 2

⎞ ⎟ ⎟⎟ ⎠

1 m −1

−1

⎤ ⎥ ⎥ , ⎥ ⎦⎥

(3)

with n

νi =

∑ (µ k =1 n

ik

∑ (µ k =1

)m zk . ik

)

m

(4)

where k = 1, 2, …, n and i = 1, 2, …, c.

Fig. 1. Model of a neural network with competitive layer architecture. Revista de Geologia, Vol. 21 (1), 2008

30 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

The basic fuzzy algorithm is as follows: Step 1: Enter the number of clusters c > 2, and the fuzzier exponent m = 2;

For c varying from cmin=2 to cmax=10 do: Step 1: Initialize cluster centers (Eq. 3); Step 2: Apply the basic algorithm to update µik and

Step 2: Initialize the cluster centers ν i( 0 ) at random; Step 3: Calculate µik according to equation (3);

ν i( 0 ) ; Step 3: Test for convergence; if no, go to Step 2; Step 4: Compute a validity value S(c) of Equation 5. Choose c such that the cluster validity index S(c) is a minimum. As shown in Figure 2, the measure of fuzzy do not varies hardly after 6 clusters, which means this is the selected number of clusters.

Step 4: Calculate ν i(1) according to equation (4). Step 5: Calculate the new µik according to equation (3) and the centers of Step (4); Step 6: If ||µik – new (µik)|| < 10-5 stop, else do ν i( 0 ) = ν i(1) , and go to Step (3). As in our case no a priori knowledge of the number of clusters exists, an automatic procedure of assessing the validity of a fuzzy partition is required. To determine correctly a validity measure that assigns to the final partition matrix or the optimum number of clusters, we follow the known fuzzy validity measure proposed by Sugeno &Yasukawa (1993): n

c

S (c) = ∑∑ ( µik ) m ( z k − ν i − ν i − z ), 2

k =1 i =1

2

(5)

where z is the average of the data. The optimum number of clusters is determined as a minimum or a convergent value of the functional S(c) as c increases. To choose this number of clusters we introduce an algorithm that applies the basic FCM clustering algorithm to the input logs for making c varying among a minimum and maximum value and chooses the best value of c based on the validity criterion given by Equation 5. So, the proposed algorithm contains the following steps:

2.3. Neural network architecture The competitive neural network was developed (Haykin, 1999), with two neurons, called as sensory elements in the input layer, which will accommodate the utilization of only GR and RHOB well logs. On the other hand, for the competitive layer, the neural network was composed of a number of neurons exactly equal to the number of clusters defined by the Fuzzy Logic algorithm. During the training phase, we applied Kohonen’s training rule (Kohonen 1989), where the synaptic weights are adjusted by the training process, with the objective to find the centers of the clusters in the input entry vectors. The synaptic weight of the winner neuron, which represents a column of the entry weight matrix, is adjusted with Kohonen’s learning rule. Supposing that the ith neuron wins, the element of the ith column of the input weight matrix is adjusted,

Fig. 2. An example of the application of the fuzzy logic algorithm to determine the number of clusters for the data of Well NA02. Revista de Geologia, Vol. 21 (1), 2008

31 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

as shown below:

[

]

Wij ( new) = Wij (old ) + α × X i − Wij (old ) , (6)

where Wij (new) is the matrix of initial synaptic weights, Wij (old) is the matrix of updated weights and α is the learning rate, (Xi, i= 1,…n) is the input vector (input layer) and (Yi, i= 1,…n) is the output layer, where n denotes the number of input nodes and m stands for the maximum number of clusters (see Fig. 1). In the case of this study, n = 2 and m = 6. Kohonen’s rule considers, as an important characteristic, what the weights of a neuron learn from the entry vectors, and due to this fact, it is of great utility in applications of recognition. After grouping the clusters, it is necessary to label the different groups, in order to indicate which lithological type of each group has the greatest possibility to represent. This labeling was done through the application of a computational program based on the interpretation of GR and RHOB crossplots. In this procedure, the classical conception of well log interpretation was considered (Ellis, 1987), and, thus, at the end of the process, the labeled groups were assumed to be lithological types present in the well. Following the labeling process, once established the definition of the lithological types present in the well, we chose at random some elements of each lithological type to compose data banks that were used as entry data, known as discriminate variables, in a program of multi-group discriminate analysis. In our study, the well log data were applied to the discriminate functions, resulting in the automatic identification of the lithologies in each point of the well log reading. The discriminate functions may be mathematically defined as follows: R = λ1ψ1 + λ2ψ2 + ... + λmψm,

(7)

where 1 < k < m, with m is the number of variables used in the analysis, R is the discriminate index of the function, λk are the coefficients of the discriminate function, and ψk are the values of the kth variable of a determined object, which represents the well log data. Thus, we have performed the association of competitive neural networks with the technique of discriminate analysis, conjugating the distinct

characteristics of the two methods, with the supporting of the fuzzy logic in the relation to clusters. Such an association permits the identification of lithological types utilizing only well log data, and minimizing the role of the interpreter in the provision of a priori information (patterns in the well logs) used in the discriminatory process. It should take into consideration that the identification of lithology based only on two logs – GR and RHOB – is not the most suitable, as it is known that a more precise interpretation would require a joint interpretation with more logs, such as NPHI (neutronic porosity), ILD (resistivity), and DT (transit time). 3. Results The first results of this study were derived from the application of this methodology to well log data, namely, wells NA02 and NA04 from Namorado Oil Field in Campos Basin/Brazil. In the first stage of the study, a competitive neural network was constructed, with two neurons in the entry or input layer. Well log data on GR and RHOB were then analyzed, corresponding to an extension of 225 m along the well. The competitive layer of the neural network was composed of six neurons, according to the quantity of clusters defined by the fuzzy logic algorithm (Fig. 2). In the training phase, 200 interactions, known as epochs, were used to adjust the synaptic weights, locating the centers of the clusters in the entry data (Fig. 3). After this stage, it became necessary to label the groups found, as well as to indicate which type of lithology each group has the greatest possibility to represent. With this goal, a computational program was developed and applied, which took into consideration the classic conception of well log interpretation. At the end of this process, the labeled groups were assumed to be the lithological types present in the well: sandstone, limestone, and shale – which are the lithological types expected in Namorado Field (Figs. 4 and 5). For other petroleum fields with more complex lithologies, one should take into account the occurrence of other lithological types. In the last step of the used methodology, some elements of the lithological type were selected at random to compose several data banks, used as Revista de Geologia, Vol. 21 (1), 2008

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Fig. 3. Log data from well NA02 used as input vectors in the neural network composed of six neurons in the competitive layer, with 200 interactions, where the location of the cluster centers may been seen as bigger circles.

Fig. 4. Results of the labeling of lithological types present in Well NA02: sandstones, limestones and shales.

Fig. 5. Well NA02: (A) GR and RHOB logs; (B) lithological section; (C) actual results, with the association of neural networks, fuzzy logic and multivariable statistics; and, (D) previous results, using only discriminate analysis (Flexa et al., 2004).

entry data (discriminate variables) in a discriminate analysis program, which permitted discrimination among various groups. The same methodology used for Well NA02 was applied in another well in the Namorado Field (Well NA04). The results obtained for Well NA04 are shown in Figure 6B, which may be compared Revista de Geologia, Vol. 21 (1), 2008

with the GR and RHOB logs (Fig. 6A). A detail of a section of the actual results is shown in figure 6C, which may be compared with the results of previous studies shown in Figure 6D (Flexa et al., 2004). These results are also verified through analysis of core data from Well NA04 provided by ANP, also showing excellent concordance (Fig. 6E).

33 Carrasquilla et al., Associating fuzzy logic, neural networks and ...

Fig. 6. Well NA04: (A) GR and RHOB logs; (B) lithological section; (C) actual results, with the association of neural networks, fuzzy logic and multivariable statistics; and, (D) previous results, using only discriminate analysis (Flexa et al., 2004).

Taking the core data analysis from Well NA02 as baseline or true information, the percent value of concordance for the association of competitive neural networks with multi-group discriminate analysis was on the order of 82.43%, while in the earlier studies, in which only discriminate analysis was applied, the success rate was only on the order of 71.62%, as shown in Table 1. For Well NA04, also taking the core data analysis as a reference point, the percent value of concordance for the association of competitive neural networks with multi-group discriminate analysis was on the order of 71.53%, while in the earlier studies, in which only discriminate analysis was applied, the success rate was only on the order of 87.59%, as shown in Table 1. The low performance of the association of competitive neural network with multi-group

discriminate analysis, for the Well NA04 is due for the fact that, in this well, the lithology is more complex, there are occurrence of marls, what affect your prediction, and mainly, because the neural network used in this study is composed of only two neurons (two well log) in the input layer. The performance becomes better with the use of more neurons in the input layer. 4. Conclusions Our work has shown, satisfactory, promissory results when only two well logs (GR and RHOB) are combining with an association of techniques such as fuzzy logic, neural networks and discriminate analysis, toward the automatic identification of lithologies along wells. These results indicate the possibi-

Tab. 1. Performance of the association of competitive neural networks and discriminate analysis when compared to the well core data and earlier studies. We lls

Core analys is (%) Earlie r s tudie s (%)

Curre nt s tudy (%)

NA02

100

71.62

82.43

NA04

100

87.59

71.53

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lity of developing more complex neural networks, composed with a bigger number of well logs, which should improve the process. Another possibility would be to extend this methodology to the automatic identification of fluids present in the reservoirs, such as oil and gas. In this scenario, the logs sensitive to lithology utilized in this work (GR and RHOB), would be used in conjunction with those more sensitive to fluids, such as resistivity (ILD), neutronic porosity (NPHI) and DT (transit time) logs. Finally, this study, additionally to propose a new approach in the automatic identification of lithologies along the wells, presents relevant points for the petroleum industry, mainly the facility of implementation and low cost of the method. Besides this, it exists the possibility to aggregate the method to existing well log services, providing, in this form, a fast interpretation during the acquisition of logs without the need of an expert interpreter. Acknowledgements This research was supported by funds donated by UENF and Brazilian National Science Foundation (CNPq), through PhD and research fellowships granted to authors. References Andrade, A. & Fischetti, A.I., 2002, Porosity images from well logs. Journal of Petroleum Science and Engineering. Elsevier, 36 (2): 149-158. Braga, A.P., Carvalho, A.P.L.F. & Ludermir, T.B., 2000, Redes neurais artificiais: teoria e aplicação. LTC, Rio de Janeiro, 262p.

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Bucheb, J.A., 1991, Aplicação de tratamento estatístico multivariante em dados de perfis de poços da Bacia de Sergipe-Alagoas. Dissertação de Mestrado, UFPA, Belém, 136p. Chang, H.C., Kopaska-Merkel, D.C. & Chen, H.C., 2001, Identification of lithofacies using Kohonen self-organizing maps. Computers and Geosciences, 28: 223-229. Das Gupta, S.P., 2001, Application of a fuzzy pattern recognition method in borehole geophysics. Computer and Geosciences, 27: 85-89. Ellis, D.V., 1987, Well logging for earth scientists. Elsevier, USA, 243p. Flexa, R.T., Andrade, A. & Carrasquilla, A., 2004, Identificação da litologia nos perfis de poço do Campo de Namorado (Bacia de Campos) e do Lago Maracaibo (Venezuela) aplicando estatística multivariada. Revista Brasileira de Geociências, 34 (4): 571-578. Haykin. S., 1999, Neural networks: a comprehensive foundation. Prentice-Hall, New Jersey, 180p. Hsieh, B.Z., Lewis, C. & Lin, Z.S., 2005, Lithology identification of aquifers from geophysical well logs and fuzzy logic analysis: Shui-Lin Area, Taiwan. Computers and Geosciences, 31: 263-275. Kohonen, T., 1989, Self-organization and associative memory. Springer-Verlag. Berlin, 312p. Nascimento, C.L. Jr. & Yoneyama, T., 2000, Inteligência Artificial em Controle e Automação. E. Blucher, FAPESP, São Paulo, 104p. Nitters, G., Davies, D.R. & Epping, W.J.M., 1995, Discriminant analysis and neural nets: valuable tools to optimize completion practices. SPE Drilling & Completion, 127133. Serra, O. & Abbot, H.T., 1989, The contribution of logging data to sedimentology and stratigraphy. SPE 9270, 18p. Sugeno, M. & Yasukawa, T., 1993, A Fuzzy-Logic-Based Approach to Qualitative Modeling. IEEE Transactions on Fuzzy Systems, 1 (1): 7-31. Sun, H., Wang, S. & Jiang, Q., 2004, FCM-based model selection algorithms for determining the number of clusters. Pattern Recognition, 37: 2027-2037.