In subtractive clustering, cluster radius determines the range of influence of a cluster. The optimum clustering radius can be determined by trial and error. As discussed, the teaching–learning-based optimization algorithm is needless of any algorithm-specific parameter. However, common controlling parameters such as the population size and number of iterations can be modified in order to enhance the model accuracy. TLBO considers the number of learners as the population size (Rao 2015).
Optimizing the fuzzy system using TLBO
By training an initial fuzzy inference system, its membership function parameters are adjusted until the optimal solution is achieved. This tuning process can be performed utilizing various optimization algorithms. Sugeno fuzzy inference system trained with the neuro-adaptive learning was introduced by Jang (1993) as the adaptive neuro-fuzzy inference system (ANFIS). In this study, the optimization of initial Sugeno fuzzy system was performed using TLBO, rather than the classic neuro-adaptive method. The optimization process was repeated until a solution with desired satisfaction was achieved or the maximum number of iterations was reached. Figure 7 displays root-mean-square errors (RMSE) obtained by each iteration of hybrid TLBOFIS. Accordingly, RMSE declined mostly in the early iterations.
Fig. 7 Acquired RMSE by each iteration epoch of TLBOFIS (clustering radius = 0.6) in training data
Results and discussion
By applying subtractive clustering, a Sugeno fuzzy inference system was initialized. The cluster radius was varied gradually from 0.1 to 1 by intervals of 0.1, in order to find the optimal radius. As given in Table 1, clustering with a radius of 0.6 provides the hybrid TLBOFIS with the highest accuracy. For evaluating the method, several optimization algorithms were employed to train the fuzzy structure using the same database. These algorithms contain Neuro-Fuzzy (ANFIS), Genetic Algorithm (GA), Artificial Bee Colony (ABC), and Ant Colony Optimization (ACO). For all these algorithms, including TLBO, the population size and maximum number of iterations were considered as 800 and 350, respectively. Associated algorithm-specific parameters and the highest accuracy of each method are presented in Table 2.
Table 1 Correlation coefficients (R), RMSE, and number of fuzzy rules achieved by each clustering radius
Table 2 Algorithm-specific parameters and the highest accuracy achieved by each method, in test data
Estimated hydraulic aperture (by TLBOFIS method) and measured values (from EMI log) are compared in Fig. 8. Integration of the fuzzy inference system with TLBO resulted in a correlation coefficient of 0.8735 and RMSE equal to 0.0044 that significantly improved compared to the other methods. Cross-plots of all methods are illustrated in Fig. 9.
Fig. 8 A comparison between the measured hydraulic aperture and TLBOFIS predicted values for fractures in the test datasets
Fig. 9 Cross-plots showing correlation between the measured and estimated values of aperture size, for each methods
In the next step, a new wellbore from a different hydrocarbon field (the third field) was selected to evaluate the model performance in an oil-based environment. The field was located in Fars Province in southern territories of Iran. Mud system used for drilling was oil based, with an oil–water ratio of 70/30. The studied well contained both the OBMI and UBI image logs and a full set of conventional well logs. The RCAL and SCAL core test data, also, were available for recovered cores from a 50-m interval. The cored interval lies in the Upper Dalan Formation (of Upper Permian age) with a lithology consisting of carbonate and anhydrite.
By processing available image logs on the cored interval, a total number of seven intersected fractures were detected. Afterward, well log records (RHOB, NPHI, DT, and LLD) at the exact depth of each fracture were specified. By normalization of input data, the developed TLBOFIS model was applied and the hydraulic aperture size of intersected fractures was estimated. The core permeability to the air (from SCAL data) was employed in order to relate estimated values to real measurements. Using Eq. (1), the estimated hydraulic aperture was converted to permeability values. Figure 10 illustrates the processed image logs and petrophysical well logs in the studied interval. The estimated hydraulic aperture and permeability for each fracture plane are also presented. Comparing the measured permeability (from core test data) and estimated permeability (from cubic law) showed a coefficient of determination (R2) equal to 0.82 that indicates a strong relationship.
Fig. 10 Processed image logs, conventional well logs, and estimated hydraulic aperture for each fracture plane. The last column shows the comparison between core permeability and derived values
Aperture size is a key parameter to indicate the influence of natural fractures on reservoir performance, and borehole imaging is the basic method for its measurement. In oil-based mud environment, however, image logs are unable to specify this parameter. In this paper, a novel method was introduced to estimate hydraulic aperture of detected fractures using conventional well logs. Required well logs are usually available in most of the drilled wells. The proposed method utilized the TLBO algorithm, in order to optimize an initial Sugeno fuzzy inference system.
The TLBO does not need of any algorithm-specific parameter, and this feature makes it a useful tool for optimization problems. Examination of the developed model, in both conductive and resistive mud environments, confirmed that the estimated values are in a good agreement with real measurements. The proposed hybrid method, as an easygoing tool, may also be employed to estimate other reservoir parameters.
The authors would like to appreciate Dr. Ali Kadkhodaie-Ilkhchi and Christian Klimczak for helping during the research.
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