TY - JOUR
AU - Moghadam, Rasoul, Hamidzadeh
AB - Abstract Routine core analysis provides useful information for petrophysical study of the hydrocarbon reservoirs. Effective porosity and fluid conductivity (permeability) could be obtained from core analysis in laboratory. Coring hydrocarbon bearing intervals and analysis of obtained cores in laboratory is expensive and time consuming. In this study an improved method to make a quantitative correlation between porosity and permeability obtained from core and conventional well log data by integration of different artificial intelligent systems is proposed. The proposed method combines the results of adaptive neuro-fuzzy inference system (ANFIS) and neural network (NN) algorithms for overall estimation of core data from conventional well log data. These methods multiply the output of each algorithm with a weight factor. Simple averaging and weighted averaging were used for determining the weight factors. In the weighted averaging method the genetic algorithm (GA) is used to determine the weight factors. The overall algorithm was applied in one of SW Iran’s oil fields with two cored wells. One-third of all data were used as the test dataset and the rest of them were used for training the networks. Results show that the output of the GA averaging method provided the best mean square error and also the best correlation coefficient with real core data. ANFIS, NN, GA, permeability, porosity, well logs, core data 1. Introduction The nature of reservoir rocks containing oil and gas dictates the quantities of fluids trapped within the void space of these rocks. The measure of the void space is defined as the porosity of the rock, and the measure of the ability of the rock to transmit fluids is called the permeability. Knowledge of these two properties is essential to study petroleum reservoirs (Tiab and Donaldson 2003). Estimation of permeability and porosity in un-cored but logged wells is a generic problem common to all reservoirs. Any field-scale reservoir characterization study inevitably requires knowledge of petrophysical properties at drilled wells for its starting point. Therefore, scientifically sound and geologically compatible procedures must be sought to allow for reliable calculation of permeability and porosity distributions in wells. Different methods were used to determine porosity and permeability in un-cored wells. Artificial intelligence (AI) tools such as neural networks (NN) and fuzzy logic are the most common tools for this purpose. Arpat et al (1998) used the neighbourhood approach for the prediction of permeability from wireline logs and limited core plug analysis data using back propagation artificial neural networks (ANN). Cuddy (1998) implemented fuzzy logic for lithofacies and permeability prediction. Taghavi (2005) used three methods to predict permeability from well log data for a heterogeneous carbonate reservoir in an Iranian oil field. These methods are permeability from effective porosity, multilinear regression and fuzzy logic. A method to estimate the initial pressure, permeability and skin factor of oil reservoirs using ANN was proposed by Jeirani and Mohebbi (2006). Their results show that the application of ANN in a pressure build-up test reduces the cost of the test and it is also a valuable tool for well testing. Karimpouli et al (2010) obtained an appropriate correlation coefficient between real and predicted permeability using a supervised committee machine NN for permeability prediction. Fuzzy logic and neuro-fuzzy modelling were used for permeability prediction too. Finol et al (2001) have proposed the use of fuzzy logic for the prediction of petrophysical rock parameters. Shokir (2004) proposed a novel method for permeability prediction in an un-cored well and use of the Gustafson–Kessel clustering algorithm for partitioning input–output data. Abdulraheem et al (2007) prepared a graphical user interface program to predict permeability from well log data. Olatunji et al (2010) prepared a comprehensive report of modelling the permeability of a carbonate reservoir using type-2 fuzzy logic systems. It is obvious that all the above studies focus on one or more methods to predict permeability. The proposed methods use NN, adaptive neuro-fuzzy inference system (ANFIS) and genetic algorithm (GA) to predict permeability and porosity from conventional well logs. Labani et al (2010) used these tools as a committee machine with intelligent systems to predict NMR log parameters from conventional well log data. 2. Theory and methodology The application of AI tools such as fuzzy logic and NN is evolving as an oil field technology. Over the last few years several studies have been conducted in the field of petroleum engineering by applying AI. The major applications are seismic data processing and interpretation, well logging, reservoir mapping and engineering (El Ouahed et al2005). The aim of this study is the application of these tools to predict permeability and porosity in un-cored wells. The proposed methodology consists of two major steps: in the first stage the permeability and porosity are predicted using ANFIS and NN separately. In the next stage the average of both previous methods is obtained using simple averaging and GA methods. This methodology combines the results of ANFIS, NN and GA. The results show that the final GA output is more appropriate than the output of each individual method. Here a brief description of each used intelligent system is provided. 2.1. Neural network NN is defined as a computer model that simulates certain aspects of human rational thinking processes. Neurons are interconnected by synapses (Malki et al1996). They are composed of simple elements operating in parallel employing a set of linear and nonlinear activation functions that do not require a prior selection of a mathematical model. A network can be trained to perform a particular function by adjusting the values of the connection's weights between the elements (Demuth and Beale 2000). Usually NN are adjusted or trained so that a particular input leads to a specific target output (El Ouahed et al2005). Back propagation, or propagation of error, is a common method of training ANN to learn how to perform a given task (Labani et al2010). This method is used to predict the core parameter from well logs here. A typical NN architecture is shown in figure 1. Figure 1. Open in new tabDownload slide A typical NN. Figure 1. Open in new tabDownload slide A typical NN. 2.2. Adaptive neuro-fuzzy inference system In recent years, considerable attention has been devoted to the use of hybrid NN–fuzzy logic approaches (Jang 1991, 1992) as an alternative for pattern recognition, clustering, statistical and mathematical modelling. It has been shown that NN models can be used to construct internal models that capture the presence of fuzzy rules. Neuro-fuzzy modelling is a technique for describing the behaviour of a system using fuzzy inference rules using a NN structure. The model has a unique feature in which it can express linguistically the characteristics of complex nonlinear systems (Nikravesh et al2003). The formulation between input and output data is performed through a set of fuzzy if–then rules. Normally, fuzzy rules are extracted through a fuzzy clustering process. Subtractive clustering (Chiu 1994) is one of the effective methods for constructing a fuzzy model. The effectiveness of a fuzzy model is related to the search for the optimal clustering radius, which is a controlling parameter for determining the number of fuzzy if–then rules. Fewer clusters might not cover the entire domain, and more clusters (resulting in more rules) can complicate the system behaviour and may lead to lower performance (Kadkhodaie-Ilkhchi et al2010). Depending on the case study, it is necessary to optimize this parameter for construction of the fuzzy model. The basic idea behind the neuro-adaptive learning techniques is very simple. These techniques provide a method for the fuzzy modelling procedure to learn information about a dataset, in order to compute the membership function parameters that best allow the associated fuzzy inference system to track the given input/output data. The Sugeno fuzzy model was proposed by Tagaki, Sugeno and Kang in an effort to formulate a systematic approach to generalized fuzzy rules from an input–output dataset. A typical fuzzy rule in a Sugeno fuzzy model has the following format: where A and B are the fuzzy sets in the antecedent; z = f(x, y) is a crisp function in the consequent. When f(x, y) is a constant, then we have the zero-order Sugeno fuzzy model, which can be viewed as a special case of the Mamdani fuzzy inference system. If f(x, y) is a first-order function, then we have a first-order Sugeno fuzzy model. Consider a first-order Sugeno fuzzy inference system which contains two rules. Rule 1: if X is A1 and Y is B1, then f1 = p1x + q1y + r1. Rule 2: if X is A2 and Y is B2, then f2 = p2x + q2y + r2. Figure 2(a) illustrates graphically the fuzzy reasoning mechanism to derive an output f from a given input vector [x, y]. The firing strength w1 and w2 are usually obtained as the product of the membership grades in the premise part, and the output f is the weighted average of each rule's output. To facilitate the learning or adaptation of the Sugeno fuzzy model, it is convenient to put the fuzzy model into the framework of adaptive networks that can compute gradient vectors systematically. The resultant network architecture is called ANFIS, and is shown in figure 2(b), where nodes within the same layer perform functions of the same type. Figure 2. Open in new tabDownload slide (a) First-order Sugeno fuzzy model; (b) corresponding ANFIS architecture (Jang 2000). Figure 2. Open in new tabDownload slide (a) First-order Sugeno fuzzy model; (b) corresponding ANFIS architecture (Jang 2000). Layer 1: each node in this layer generates membership grades of a linguistic label. Parameters in this layer are referred to as the premise parameters. Layer 2: each node in this layer calculates the firing strength of a rule via multiplication. Layer 3: node i in this layer calculates the ratio of the ith rule's firing strength to the total of all firing strength. Layer 4: node i in this layer computes the contribution of the ith rule towards the overall output. Layer 5: the single node in this layer computes the overall output as the summation of contribution from each rule (Jang 2000). The constructive adaptive network shown in figure 2(b) is functionally equivalent to the fuzzy inference system shown in figure 2(a). There are two methods that ANFIS learning employs for updating membership function parameters: back propagation (which calculates error signals (the derivative of the squared error with respect to each node's output) recursively from the output layer backward to input nodes. This learning rule is exactly the same as the back propagation learning rule used in the common feed forward NN) for all parameters (a steepest descent method) and a hybrid method consisting of back propagation for the parameters associated with the input membership functions, and least-squares estimation for the parameters associated with the output membership functions (Matlab user's guide 2007). 2.3. Genetic algorithm GAs were first introduced in the field of AI by Holland (1975). These algorithms mimic processes from the Darwinian theories of natural evolution in which winners survive to reproduce and pass along ‘good’ genes to the next generation, and ultimately, a ‘perfect’ species is evolved. Hence the term ‘genetic’ was adopted as the name of the mathematical algorithms (Huang et al1998). The GA works by firstly encoding the parameters of a given estimator as chromosomes (binary or floating-point). This is followed by populating a range of potential solutions. Each chromosome is evaluated by a fitness function. The better parent solutions are reproduced and the next generation of solutions (children) is generated by applying the genetic operators (crossover and mutation). The children solutions are evaluated and the whole cycle repeats until the best solution is obtained (Tamhane et al2000, Labani et al2010). A general flowchart of the GA is shown in figure 3. Figure 3. Open in new tabDownload slide A general flowchart of the GA. Figure 3. Open in new tabDownload slide A general flowchart of the GA. 3. Data preparation In this research, the dataset came from one of the SW Iranian oil fields. Coring is performed in two wells of the field. Two-thirds of all data were used as the training set and the rest of them were used as the test set. We use two methods for input data selection: cross-plot analysis and NN, considering several combinations of inputs and selecting the best data combination. The relationship between permeability and well log data including sonic log (DT), deep resistivity log (Rd), neutron porosity log (NPHI) and bulk density log (RHOB) is shown in the cross-plots of figure 4. The data used in figure 4 are normalized between 0 and 1. As shown, a strong and direct relationship between permeability and sonic logs is seen (CC = 0.5757). Figure 4. Open in new tabDownload slide Cross-plot of normalized permeability with different well logs: (a) sonic, (b) neutron porosity, (c) bulk density and (d) logarithm of deep resistivity. Figure 4. Open in new tabDownload slide Cross-plot of normalized permeability with different well logs: (a) sonic, (b) neutron porosity, (c) bulk density and (d) logarithm of deep resistivity. Data scaling is necessary here for two reasons. First, it is desired to account for essential variability in the filtered log data and, without some type of scaling process, those logs with the largest original variance would dominate the subsequent analysis. Second, it is desired to have all logs measured in similar units because it will be easier to compare them in the FIS structure of the fuzzy model and also in the NN model and the analysis will not be biased towards those with higher absolute values. In this study, a linear scaling method that maps the maximum log value to 1 and the minimum log value to zero was used. The linear scaling has the following form: 1 where zi is the scaled value, xi is the original value, xmin is the minimum log value and xmax is the maximum log value. Figure 5 shows the cross-plot of different log data with porosity of core. A great direct relationship with sonic log and great indirect relationship with bulk density log are presented. The other logs have weaker relationships with porosity. This is the first method which was used for input data selection. Figure 5. Open in new tabDownload slide Cross-plot of normalized porosity with different well logs: (a) sonic, (b) neutron porosity, (c) bulk density and (d) photoelectric index. Figure 5. Open in new tabDownload slide Cross-plot of normalized porosity with different well logs: (a) sonic, (b) neutron porosity, (c) bulk density and (d) photoelectric index. According to this method inputs with higher correlation coefficients are appropriate. However, we have performed the next data selection test to select the appropriate inputs. Table 1 reports the errors of the test dataset of porosity and permeability for different input combinations. Results of this table show that for both parameters the combination of sonic, neutron porosity, bulk density, deep resistivity, photoelectric index and gamma ray log leads to least test error. So this combination is selected as an appropriate input set for this study. A schematic of input and output dataset versus data number is illustrated in figure 6. In this figure the x-axis is the normalized value of input and output data and the y-axis is the data number. The two right ones show the porosity and logs used for porosity prediction and the left ones show the permeability and logs used for permeability prediction. Figure 6. Open in new tabDownload slide A schematic of input and output data for porosity and permeability. Figure 6. Open in new tabDownload slide A schematic of input and output data for porosity and permeability. Table 1. Performance of NN for predicting core porosity and permeability in the test data using different input data combinations. Inputs . MSE (porosity) . MSE (permeability) . DT, NPHI, RHOB, 0.015 0.015 Log(Rd), PEF, GR DT, NPHI, RHOB, 0.017 0.027 Log(Rd) DT, NPHI, RHOB 0.022 0.023 DT, NPHI 0.019 0.044 DT 0.016 0.019 Inputs . MSE (porosity) . MSE (permeability) . DT, NPHI, RHOB, 0.015 0.015 Log(Rd), PEF, GR DT, NPHI, RHOB, 0.017 0.027 Log(Rd) DT, NPHI, RHOB 0.022 0.023 DT, NPHI 0.019 0.044 DT 0.016 0.019 Open in new tab Table 1. Performance of NN for predicting core porosity and permeability in the test data using different input data combinations. Inputs . MSE (porosity) . MSE (permeability) . DT, NPHI, RHOB, 0.015 0.015 Log(Rd), PEF, GR DT, NPHI, RHOB, 0.017 0.027 Log(Rd) DT, NPHI, RHOB 0.022 0.023 DT, NPHI 0.019 0.044 DT 0.016 0.019 Inputs . MSE (porosity) . MSE (permeability) . DT, NPHI, RHOB, 0.015 0.015 Log(Rd), PEF, GR DT, NPHI, RHOB, 0.017 0.027 Log(Rd) DT, NPHI, RHOB 0.022 0.023 DT, NPHI 0.019 0.044 DT 0.016 0.019 Open in new tab 4. Data processing 4.1. ANFIS model For porosity and permeability data a multiple input–single output adaptive neuro-fuzzy inference model (ANFIS) with DT, NPHI, RHOB, Rd, PEF and GR was designed. For this purpose, the clustering radius was changed gradually from 0.1 to 1 with increments of 0.1 and the model performance was measured at each stage. The number of epochs for each stage was 500 epochs. Then the obtained model was applied on the test data. The results of the ANFIS model on test data are summarized in table 2. Table 2. MSE of ANFIS model on test dataset for core porosity and permeability prediction. Clustering radius . Porosity . Permeability . 0.1 0.0201 0.0181 0.2 0.0198 0.0221 0.3 0.0334 0.0212 0.4 0.0167 0.0182 0.5 0.0203 0.0131 0.6 0.0171 0.0169 0.7 0.0183 0.0163 0.8 0.0144 0.0165 0.9 0.0147 0.0141 1 0.0147 0.0135 Clustering radius . Porosity . Permeability . 0.1 0.0201 0.0181 0.2 0.0198 0.0221 0.3 0.0334 0.0212 0.4 0.0167 0.0182 0.5 0.0203 0.0131 0.6 0.0171 0.0169 0.7 0.0183 0.0163 0.8 0.0144 0.0165 0.9 0.0147 0.0141 1 0.0147 0.0135 Open in new tab Table 2. MSE of ANFIS model on test dataset for core porosity and permeability prediction. Clustering radius . Porosity . Permeability . 0.1 0.0201 0.0181 0.2 0.0198 0.0221 0.3 0.0334 0.0212 0.4 0.0167 0.0182 0.5 0.0203 0.0131 0.6 0.0171 0.0169 0.7 0.0183 0.0163 0.8 0.0144 0.0165 0.9 0.0147 0.0141 1 0.0147 0.0135 Clustering radius . Porosity . Permeability . 0.1 0.0201 0.0181 0.2 0.0198 0.0221 0.3 0.0334 0.0212 0.4 0.0167 0.0182 0.5 0.0203 0.0131 0.6 0.0171 0.0169 0.7 0.0183 0.0163 0.8 0.0144 0.0165 0.9 0.0147 0.0141 1 0.0147 0.0135 Open in new tab The results of this table show that the best test error takes place at 0.8 and 0.5 clustering radius for porosity and permeability, respectively. Figures 7(a) and (b) show the results of the ANFIS model. Figure 7. Open in new tabDownload slide Real core porosity and permeability versus ANFIS predicted data. Figure 7. Open in new tabDownload slide Real core porosity and permeability versus ANFIS predicted data. 4.2. NN model The back propagation method was selected to predict the porosity and permeability. The network consists of three layers. In this network, nodes are organized in input, hidden and output layers. The network was trained using the Levenberg–Marquardt training algorithm (Train LM). TANSIG was used as the transfer function between input and hidden layers and between output layer and hidden layer PURELIN was used as the transfer function. The input data set that is used for the NN model is the same as the input data set used in the ANFIS model for predicting porosity and permeability. Finally the obtained network was applied on test data and mean square errors (MSE) of 0.0151 and 0.0158 were obtained for porosity and permeability, respectively. Figures 8(a) and (b) show the result of the NN model. Figure 8. Open in new tabDownload slide Real core porosity and permeability versus ANFIS predicted data. Figure 8. Open in new tabDownload slide Real core porosity and permeability versus ANFIS predicted data. 4.3. Final prediction of porosity and permeability To obtain better results we combine the output of both ANFIS and NN models. Weighted average is used for this step. In the first stage the weight factors are determined using simple averaging. The porosity and permeability were obtained from equations (2) and (3): 2 3 Applying this method leads to errors of 0.013 88 and 0.012 58 for porosity and permeability, respectively, for the test dataset. In the next step the weight factors are determined using the GA. The objective function for the GA is defined as 4 w1 and w2 are the weight factors of NN and ANFIS, respectively, and Ti is the target (real) value. The initial population size is set to 20 which specifies the number of individuals in each generation, and the initial range is [0, 1] which specifies the range of the individuals in the initial population. The initial population is generated with a random process covering the entire problem space. The maximum number of generations that specifies the maximum number of GA iterations is set to 100. The next generation is produced using crossover and mutation operators with default functions in the GA toolbox (MATLAB 2007, Labani et al2010). After running the GA which uses different values for w1 and w2 to determine the best values to fit the predicted and real values, the following equations were obtained for porosity and permeability: 5 6 The MSE obtained for porosity and permeability are 0.013 85 and 0.012 34, respectively. Figures 9(a) and (b) show the result of the GA model. Figure 9. Open in new tabDownload slide Real core porosity and permeability versus final predicted data using the GA. Figure 9. Open in new tabDownload slide Real core porosity and permeability versus final predicted data using the GA. Plots of real and GA predicted porosity and permeability are illustrated in figures 10 and 11, respectively. These plots show how the trained network could trace the real target data in the test set. The results of all the steps are summarized in table 3. This table shows how MSE values changed in each model and also reveals the preference of the proposed model with respect to each individual model. Figure 10. Open in new tabDownload slide Core and GA predicted porosity of test data versus data number. Figure 10. Open in new tabDownload slide Core and GA predicted porosity of test data versus data number. Figure 11. Open in new tabDownload slide Core and GA predicted permeability of test data versus data number. Figure 11. Open in new tabDownload slide Core and GA predicted permeability of test data versus data number. Table 3. MSE of each method for porosity and permeability prediction. Method . MSE (porosity) . MSE (permeability) . ANFIS 0.0144 0.0131 NN 0.0151 0.0158 Simple average 0.013 88 0.012 58 GA 0.013 85 0.012 34 Method . MSE (porosity) . MSE (permeability) . ANFIS 0.0144 0.0131 NN 0.0151 0.0158 Simple average 0.013 88 0.012 58 GA 0.013 85 0.012 34 Open in new tab Table 3. MSE of each method for porosity and permeability prediction. Method . MSE (porosity) . MSE (permeability) . ANFIS 0.0144 0.0131 NN 0.0151 0.0158 Simple average 0.013 88 0.012 58 GA 0.013 85 0.012 34 Method . MSE (porosity) . MSE (permeability) . ANFIS 0.0144 0.0131 NN 0.0151 0.0158 Simple average 0.013 88 0.012 58 GA 0.013 85 0.012 34 Open in new tab 5. Conclusion This paper proposes integration of ANFIS, NN, GA and a simple averaging method for determining core porosity and permeability from conventional well logs. The results show that the output of the GA has the best correlation coefficient with real data. Tables and figures show that the MSE of the proposed method output is less than the MSE of each individual method, so it is a better method to predict porosity and permeability. This method can be used to predict porosity and permeability in all other wells of the field with great reliability. The NN input data selection was used to identify the best input combination. This method selects the best input combination for the model according to the MSE of predictions. The proposed method is fast, more accurate than individual methods, and cost effective for predicting core porosity and permeability. This method can prepare a complete data collection of porosity and permeability in all wells of the field. Finally this data collection can be used to obtain a thorough model of the reservoir. This model can be used for future reservoir studies and simulations. Acknowledgment The authors thank the National Iranian Oil Company for data preparation and permission to publish the results of this paper. 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TI - Integration of ANFIS, NN and GA to determine core porosity and permeability from conventional well log data
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/9/5/473
DA - 2012-10-01
UR - https://www.deepdyve.com/lp/oxford-university-press/integration-of-anfis-nn-and-ga-to-determine-core-porosity-and-VxPZiRgPf0
SP - 473
VL - 9
IS - 5
DP - DeepDyve
ER -