Fuzzy Inf. Eng. (2011) 1: 23-33 DOI 10.1007/s12543-011-0063-z ORIGINAL ARTICLE

Optimizing Fracture Design Based on a Novel RBF Neural Network Hong Liu · Zhen Huang · Hong-yi Gao · Qing-heng Zeng

Received: 11 January 2010/ Revised: 15 December 2010/ Accepted: 15 January 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2011

Abstract The factors affecting performance of fractured wells are analyzed in this work. The static and dynamic geologic data of fractured well and fracturing treatment parameters obtained from 51 fractured wells in sand reservoirs of Zhongyuan oilfield are analyzed by applying the grey correlation method. Ten parameters are screened, including penetrability, porosity, net thickness, oil saturation, water cut, average daily production, and injection rate, amount cementing front spacer, amount sand-carrying agent and amount sand. With the novel Radial Basis Function neural network model based on immune principles, 13 parameters of 42 wells out of 51 are used as the input samples and the stimulation ratios as the output samples. The nonlinear interrelationship between the input samples and output samples are investigated, and a productivity prediction model of optimizing fracture design is established. The data of the rest 7 wells are used to test the model. The results show that the relative errors are all less than 7%, which proves that the novel Radial Basis Function neural network model based on immune principles has less calculation, high precision and good generalization ability. Keywords Radial basis function neural network · Grey correlation method · Artificial immune system · Hydraulic fracture · Optimum design

1. Introduction The basic objective of fracturing design is to match construction parameters with stratum properties so as to meet the required productivity and economic benefit. Widely Hong Liu () · Qing-heng Zeng Chongqing University of Science and Technology, Chongqing, 401331, P.R.China email: [email protected] Zhen Huang Northwest-Sichuan Gas Field, Jiangyou, Sichuan, 621700, P.R.China Hong-yi Gao CNOOC-ETS-Oilfield Technology Services Co., Zhanjiang, 524057, P.R.China

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Hong Liu · Zhen Huang · Hong-yi Gao · Qing-heng Zeng (2011)

used methods and theories of fracturing design are mainly based on various mechanical models. In actual applications, however, the geometry size of complex crack under complex stress conditions in the fracturing process still can not be effectively simulated and analyzed with current technologies; on the other hand, currently many important parameter sites required by fracturing design models, such as ground stress distribution, are difficult to acquire, which often results in great difference between theoretical design and actual construction. It is difficult to implement items proposed in design in actual construction, and the predicted productivity after fracturing is far from actual results. The problem mentioned is exactly the “bottleneck” in hydraulic fracturing optimization design, which involves limited data, inaccurate model and parameters as well as uncertainty of mechanisms of many related issues. From the mathematical point of view, the essence of fracturing optimization design is how to map vector input in M dimensions (properties of stratum and liquid, scale of fracturing operation, construction parameters and geometric shape of crack, etc.) to vector output in N dimensions (fracturing effect). Due to the extreme complexity of the connections in these parameters for decision-making, the mapping must be highly nonlinear, and it is very difficult to solve this type of problem with conventional mathematical approaches. Therefore, artificial intelligent technology found wide applications in fracturing optimization design [1, 2, 3, 4]. Radial Basis Function (RBF) neural network [5] is a three-layer feedforward neural network which has been widely used in recent years. It provides a novel and effective approach to learning multi-layer feedforward network. It not only avoids the complicated calculation in back propagation network and improves the learning progress, but also overcomes the problem of local minimum in gradient descent algorithm in BP network. So it is successfully used in fields of signal processing, system modeling, process control and fault diagnosis, etc. In RBF neural network, the number and locations of RBF centers in hidden layers directly affect the approximation capability of network, and it is required that RBF centers can cover the whole input space. However, too many RBF centers may cause the amount of calculation in the network to spurt and the network generalization to decrease. Therefore, the key to RBF network model establishment lies in selecting appropriate RBF centers correctly. At present, methods of RBF center selection mainly include clustering algorithms and orthogonal least square algorithms. In clustering algorithms such as K-means, the clustering number needs pre-assigning; in orthogonal least square algorithms, ill-conditioned matrix may emerge easily when large amounts of data are inputted. Special importance was attached to the studies on information processing system, which were aroused by biology. Among them Artificial Immune System (AIS) is a new research field. As a highly-complicated distributed-coordination adaptive system, AIS has a series of distinctive functions such as self and non-self recognition, self organization and learning, immune memory and immune tolerance. It has been applied and developed in the fields of computer and network security, optimization, pattern recognition and robot control, etc. In the novel RBF neural network model based on the principle of immune system, the centers of RBF network model are chosen and optimized according to the

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principles of memory, learning as well as self organization and regulation of artificial immune system. Moreover, the model along with recursive least square method forms a new mixed learning algorithm of RBF neural network. In the algorithm, the input data represent antigens, and antibodies are the contraction mapping of antigens and serve as the hidden layer centers of the RBF neural network model. Based on this, the number and locations of the hidden layer centers of the RBF neural network are determined, and the weights of the network output layers are determined with the recursive least square algorithm [6,7]. This paper presents the analysis on the parameter indices affecting fracturing effects. These indices include geological static data (such as the permeability of fractured layers, effective thickness, oil saturation and skin factors), dynamic parameters in oil-well production (such as water content, bottom hole flowing pressure and daily output) and fracturing operation parameters (such as displacement, sand charge and sand ratio in fracturing operation). According to the principle of comprehensive, independent and universal factor selection, methods like grey correlation and essential component analysis were combined to carry out analysis on fifty-one fractured wells in thirteen sandstone reservoirs in the Central Plain Oil Field as well as the static and dynamic geological data and fracturing operation parameters of their fractured layers. Twelve research parameters such as the permeability, porosity, effective thickness, oil saturation, water content, stratum temperature and average daily oil output of fractured layers as well as the displacement, pad fluid quantity, quantity of sand-carrying fluid, sand charge and sand ratio in fracturing operation were screened out. AIS-RBF neural network was employed for quantitive study on the complex highly-nonlinear relationship between the twelve-parameter group (input samples) of the forty-two fractured wells chosen from the total number of fifty-one and the increase ratio of daily yield after fracturing (output sample), and a prediction model of fracturing optimization design was built. Twelve-parameter groups of seven wells were used to test the model, with all absolute errors below 7%, which indicated the high precision and generalization of the AIS-RBF algorithm. Actual applications showed that better effects were achieved with the AIS-RBF neural network method in comparison with conventional fracturing optimization methods, and it is worthy of extensive application. 2. RBF-network-model Learning Algorithm Based on Immune Principle Immune system possesses the capabilities of memory, learning as well as self-organization and regulation. In literature [8], by introducing these properties in data clustering, a data compression algorithm emerged. In literature [6,7], the above algorithm was used for selecting data centers in the hidden layers of RBF network after improvement and simplification, while the connection weights of the output layers of RBF network were learned with recursive least square algorithm. Thus, a new mixed learning algorithm based on RBF network was formed. This algorithm obviates the necessity of mentioning the number of data centers in RBF network to be reassigned. It is assumed that there are input data in number N, i.e., X = {x1 , x2 , · · · , xN }, T  wherein, xi = xi1 , xi2 , · · · , xip ∈ R p (i = 1, · · · , N ) represents the input of RBF

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neural network. A new data set C = {c1 , c2 , · · · , ch } needs to be found to determine T  RBF data centers, wherein, c j = c1 j , c2 j , · · · , c p j ∈ R p and j = 1, · · · , h. h << N represents the clustering of input data, and C is not necessarily the subset of X. When the principle of immune system is applied for selecting data centers in RBF network, stochastic initialization at a certain number scale is performed on data set C. The selection process includes the following steps: 1) With the input data being recognized by all elements in data set C, these elements compete with one another. Winners will propagate, i.e., duplicate and bring about mutations, while the data failing in recognition will be removed. 2) The elements recognized by themselves will be removed from data set C if inhibition occurs due to mutual recognition between two elements in C. 3) New element is added in data set C, and steps 1) and 2) are repeated. 4) When certain judging condition is satisfied, the selection process is terminated, and the data centers of the RBF network model are obtained. In immune system, the interaction intensity between antigens and antibodies is represented by the affinity of their bonds, and antigen-antibody interaction is described by their similarities. Here the affinity between No.i inputted data xi and No. j data center c j is denoted by ai j , and the similarity between No.i data center ci and No. j data center c j is denoted by S i j , ai j =

1 1   = 1 + D(xi , c j ) 1 +  xi − c j 

(i = 1, · · · , N, j = 1, · · · , h),

(1)

1 1  (i, j = 1, · · · , h).  (2) = 1 + D(ci , c j ) 1 + ci − c j    In the expressions, D(xi , c j ) =  xi − c j  represents the Euclidean distance between xi and c j . Thus when xi equals to c j , the affinity ai j between them equals to 1. If D(x  i , c j) does not equal to zero, then ai j is greater than zero and smaller 1. D(ci , c j ) = ci − c j  represents the Euclidean distance between ci and c j . When ci equals to c j , their similarity S i j equals to 1, which is the maximum. If D(ci , c j ) does not equal to zero, then S i j is greater than zero and smaller than 1. The details of the learning algorithm represent as follows: In each recursive step: S ij =

1) Following operations are performed for each input data xi : (i) The affinity ai j is calculated between all RBF data centers and xi . The greater the distance is, the smaller the affinity becomes; (ii) The data center with the maximum affinity is chosen to be duplicated (cloned) Nc times, thus a clone set L of the RBF data center is created. Mutation processing

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is conducted on these identical data centers according to expression (3) to form a  mutation set L , and βk is the mutation rate. The essence of the process is to search in the vicinity of the data center c j with the maximum affinity so as to obtain data centers with even higher affinities; 

ck = ck − βk (ck − xi )

(k = 1, · · · , Nc )

(3) 

(iii) The affinity is recalculated between the Nc mutated data centers in L and the input data xi , and ζ% of the data centers with the maximum affinity are chosen to generate the data-center memory set m; (iv) Those centers are removed in m with affinities lower than threshold σd , with a  trimmed data-center memory set m formed;  (v) The similarities S i j are calculated among memory data centers in m and those data centers removed with similarities which is greater than threshold σ s The process embodies the clone inhibition function of immune system and is used for eliminating clone effects and selecting the self-recognized data centers inside;  (vi) The data-center memory sets m formed from all input data are merged to M; 2) The similarities S i j are calculated among all data centers in M, and those data centers with S i j greater than σ s are removed. The process embodies the network inhibition function of immune system, i.e., searching similar data centers from different clone sets and removing them; 3) New data centers in number r are used to replace the data centers with lower affinities in C, and these new data centers can be chosen randomly. The process embodies the self-organization function of immune system, i.e., adding new generated B lymphocytes and removing useless B lymphocytes. The termination of the above recursive process can be controlled through the preassigned recursive steps. When the recursive process is terminated, M is the obtained set of RBF data centers. By combining the above RBF data-center selection algorithm based on immunity principle with the recursive least square algorithm, the mixed learning algorithm is formed on the basis of RBF neural network model. The process of the algorithm is as follows: 1) The data centers are determined in RBF network model with the algorithm of RBF data-center selection based on immune principle. 2) Recursive Least Square (RLS) algorithm is used to compute the weights of network output layers. 3. Application of RBF Network Model Based on Immune Principle in Optimizing Fracture Design 3.1. Determination of Influential Factor

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Production performance of oil/gas well after being fractured synthetically reflects the geology of layer section of fracturing well, remaining reserve, formation energy, rock fluid property, ground stress field distribution, natural fracture, porosity, percolation rate, degree of saturation, fracturing fluid, support agent, level for constructive project et al. Therefore, it includes geology status,(such as percolation rate of fractured layer, effective thickness, oil saturation, skin factor etc.), dynamic parameter of well production (such as Moisture Content, flowing bottom hole pressure, daily output etc.) and fracturing parameter (such as discharge capacity, sand content and sand production ratio in fracturing operation, etc. see Table 1)[1]. Taken the production change pre or post fracture as evaluation index, and aimed at basic reasons result in problems, the decision parameters were determined which can make the parameter more scientific. Table 1: Decision parameter table of optimizing fracture design. category

decision parameter table

geology status of monocandidate fracturing well

effective thickness, porosity, effective percolation rate, gas saturation, initial formation pressure, present formation pressure, ground stress difference between upper and lower oil layer, Remaining Recoverable Reserves, original Recoverable Reserves

Last fracturing parameter

discharge capacity, sand content, sand ratio, pad fluid volume, sand carrier volume

The present diagnosis of fracture status

Static pressure of gas well, flowing bottom hole pressure, width of the propped fracture, fracture conductivity, skin factor

Production performance data of fracturing well

Output before constructionˈOutput after construction (daily increasing gas quantity, monthly increasing gas quantity, yearly increasing gas quantity, the effective stimulation period

According to a great deal of researches for evaluating the effect of fracturing well at home and abroad in recent years, meanwhile, integrated the geology, experience of expert and repeatedly analyzing and evaluating, combined with general, independent and generalization principles of selected factors, the static and dynamic geologic data of fracturing well and fracturing treatment parameters obtained from 200 fracturing wells in 13 sand reservoirs of Zhongyuan oilfield were analyzed by applying the grey correlation and principal component analysis method. Twelve parameters were screened, including percolation rate, porosity, effective thickness, oil saturation, water ratio, formation temperature, average daily production as well as discharge capacity, pad-fluid ratio, sand carrier volume, sand content in fracturing construction. Decision output is daily production-increasing ratio of fractured well.

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3.2. To Establish Decision Parameter Data Decision parameter data on fracturing project optimization includes modeling data table, testing data table and candidate fractured well table. All the wells mentioned in the tables have been implemented fracturing operation in research block or above research block. The parameters in data table have been shown above. 1) Data preparation. A preliminary classification of evaluation result was made in accordance with the effect of fracturing well in research block. The quantitative indication was determined by actual condition of oil field, usually determined by the enhancement in oil productivity, validity duration of the enhancement and economy assessment results, as for categorical limit and quantity, choosing according to actual condition of oil field. Daily production-increasing ratio was adopted as an evaluation index in the research. Corresponding proportion was chosen to set up modeling data after evaluation parameter and index was determined. 2) Modeling data table. A key to establish modeling data table is to select the generalization and representation of the well used in the research. Meanwhile, in order to evaluate the effect of each decision parameter of fracturing well, data obtained from fracturing well should also be used as more as possible. If the few well in the block, adjoining wells may be considered, and the numbers of modeling well can account for 85 percent of all the wells in research area. Composition of the table: 12 parameters selected as input index, daily production-increasing ratio as evaluation index. 3) Testing data table. The wells mentioned in the tables is implemented fracturing operation in the block which can be selected from wells in modeling data table and account for 15 percent of all the wells, not including evaluation index. 4) Candidate fracturing well table. Candidate fractured well can be found in modeling data table or testing data table, see Table 2. 3.3. Decision Parameter Pretreatment Analysis and application of decision parameter are largely determined by precision of data. Therefore, decision parameter should be making standardized treatment by normalization method to put the decision parameter or prediction parameter between 0 and 1, which can reduce various systematic errors influencing data quality to a minimum level.

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Table 2: The comparison between actual data and output data of RBF neural network and BP neural network. well number

Pu 77

Tang 4

Wen 260

Wen164

Actual production ratio output of RBF neural network output of BP neural network

4.205 3.981 4.514

2.349 2.459 2.572

4.345 4.012 4.579

4.129 4.259 3.870

well number

Pu 80

Bai 17

Hu 110

Actual production ratio output of RBF neural network output of BP neural network

31.579 29.021 34.132

5.944 6.216 6.195

2.215 2.435 2.007

3.4. Application and Analysis of Examples The data, which was obtained from 51 fractured wells were analyzed in 13 sand reservoirs of Zhongyuan oilfield. Eight centers were randomly selected from the initial data of RBF network with its width being 10. The parameters of immune learning algorithm are: σd = 1ˈσ s = 0.32ˈζ%=10%, r = 10 with five recursive time being five. RBF network model has 15 centers.

  5%)Ёᖗ᭄

The number of RBF center

    





   ‫⭿ܡ‬㔥㒰ᡥࠊⱘ䯜ؐ







The threshold value ıs of Immune Inhibition Network

Fig.1 The threshold value σ s of inhibition network and corresponding central number of RBF According to Fig.1, the selection of threshold value σ s of inhibition network in immune learning algorithm has great influence on identifying precision of quantity and model identifying of RBF center. The number of RBF center decreased as σ s increased. When the number reached a certain value, the precision of model identi-

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fied changes slightly, which showed that a similar center existed. While the number was smaller than the value, the precision of model identifying abruptly increases, which showed that the RBF Network has reached the smallest number of center. The characteristics can be used as judgment criteria of algorithm to RBF Network data. In order to detect the generalization ability and precision of RBF model based on immune principle, data of seven fracturing wells was input to establishing model of RBF neural network. The comparison between output data and actual data can be seen from Table 3. Table 2 showing the comparison between the output data of RBF and BP neural network. The output data of RBF neural network more closely meet with the actual data than BP, which indicates that RBF model has strong generalization ability. Table 3: The comparison between output result of RBF neural network model and actual effect of project on site implementation. Well

project

sand proportion (m3 )

pad fluid (m3 )

discharge capacity (m3 /min)

sandladen fluid(m3 )

Wei 319

1 2* 3 4 5

12 14.5 16 18 20

50 54 58 60 62

2.7 2.7 3.0 3.0 3.5

62.5 65 65 67.5 67.5

Pu78

1 2 3* 4 5

7 7 8 9 9

75 78 81 83 83

2.8 30 3.25 3.5 3.8

51 53 55 57 58

Well

pump pressure (MPa)

Wei 319

50 51.8 52 54 56

Pu78

43 45 46.3 48 50

pre-fracturing production (t/d)

production prediction (t/d)

0.65

4.9 5.3 4.6 5.1 6.2

3.4

11.62 11.51 12.67 13.21 12.89

production being fractured (t/d)

5.5

12.3

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Note: * indicate project on site implementation. Two fracturing wells data were selected from 51 fracturing wells in 13 sand reservoirs to optimize the fracture design. Based on actual implementing projects of Wei 319 and Pu 78, five primary election projects were produced respectively (see Table 3). Then using established RBF neural network model to anticipate each candidate fracturing well. The comparison between output result of RBF model and actual effect of project on site implementation can be seen from Table 3. As can be seen from the Table 3, fracturing effect of project on site implementation are the same as the results of corresponding RBF neural network, and the minimum error value is obtained, which indicate that RBF neural network based on Immune Principle has good adaptability. 4. Conclusion 1) For the oil field or block where a certain number of fractured wells have been built, when fracturing designs are optimized by using AI technique. Various factors influencing the effect of fracture should be taken into consideration and parameters selection and quantity can be adjusted by the actual condition and specific requirement in oil field, which can make the fracturing design more scientific. 2) An algorithm based on Immune Principle is adopted to the selecting of the number and positions of RBF centers and the output weights are decided with the recursive least squares algorithm. The novel RBF neural network hybrid algorithm has less calculation, high precision and strong generalization ability. 3) Many questions are common in the field of petroleum engineering research, such as limitation of the data, failure to match parameter with the model, as well as the uncertainty of the specific mechanism. So Artificial Immune System, RBF neural network and any other artificial intelligent technology have good application prospects in petroleum engineering. Acknowledgments This project was funded by Natural Science Foundation of China (50604022) and Natural Science Foundation of Chongqing (2008BB6068) and Chongqing Universities Innovation Team Plan (201027). References 1. Liu H, Zhao J Z, Hu Y Q, et al (2004) Study on application of support vector machine for repetitive fracturing. Natural Gas Industry 3: 75-77 2. Liu H, Zhao J Z, Hu Y Q, et al (2002) Post-frac effect forecasting using the fuzzy neural network. Fault-block Oil & Gas Field 3: 35-38 3. Liu H, Zhao J Z, Hu Y Q, Zhang S L, Liu Y J (2002) Application of the fuzzy neural network system in the selection of wells or layers for fracturing. Drilling & Production Technology 5: 34-37 4. Huang Z W (2009) Hydraulic fracturing evaluation methods and analysis of objectives. Inner Mongolia Petrochemical Industry 14: 43-45

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