Estimation of fracture aperture from petrophysical logs using teaching–learningbased optimization algorithm into a fuzzy inference system
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Abstract
Aperture, which refers to the opening size of a fracture, is a critical parameter controlling rock mass permeability. Moreover, distribution of permeability within the reservoir is commonly affected by natural fracture occurrences. In a waterbased mud environment, boreholeimaging tools are able to identify both location and aperture size of the intersected fractures, whereas in oilbased environment, due to invasion of resistive mud into the fractures, this technique is impractical. Recently, some artificial intelligence techniques facilitated reliable estimations of reservoir parameters. In this paper, a teaching–learningbased optimization algorithm (TLBO) trained an initial fuzzy inference system to estimate hydraulic aperture of detected fractures using well logs responses. Comparing the results with real measurements revealed that the model can provide reliable estimations in both conductive and resistive mud environments, wherever the aperture size is unknown. TLBO, besides of its easier application, outperformed earlier optimization algorithms, which were used to evaluate the method effectiveness.
Keywords
Aperture size Fracture permeability Fuzzy logic Image logs TLBOIntroduction
Natural fractures can affect essential reservoir characteristics tremendously. As a fluid conduit, natural open fractures can control the reservoir permeability (Boro et al. 2014). Conventional techniques of oil recovery may drain fractured reservoirs inefficiently, leading to early breakthrough in the secondary recovery (Alvarado and Manrique 2010; Speight 2016). In these cases, where production substantially takes place through fracture networks, aperture size is one of the basic parameters controlling reservoir permeability (Nelson 2001; Tiab and Donaldson 2015).
During fracture characterization, the aperture size is commonly characterized with the image logs. In conductive mud environments, imaging tools can determine fracture aperture on the order of microns (Tiab and Donaldson 2015), whereas in oilbased mud environments, due to the invasion of resistive mud into the fracture planes, the imaging tools such as OilBase Micro Imager (OBMI) display the same appearance for both the open and healed fractures (Bishop 2015). In addition, the vast majority of natural fractures have an aperture size less than the OBMI tool pixel width (Cheung et al. 2001). Statistics have shown that the most frequent range of aperture size is between 10 and 40 microns (GolfRacht 1982). Therefore, the presence of a quick and costeffective method to estimate aperture size of detected fractures can be useful, wherever the aperture size is unknown.
Nakashima and Kikuchi (2007) performed a quantitative estimation of fracture apertures, of a size higher than 0.2 mm, using NMR logging data. They showed that there is a linear relationship between fracture aperture size and bulkfluid porosity. Wu (2013) proposed a workflow to determine the relationship between aperture size and a fracture indicator value by using measurements from multiaxial electromagnetic induction well logging instruments. Ramandi et al. (2016) used Xray microcomputed tomography to measure fracture apertures in a fractured coal sample from digital images. Shalaby and Islam (2017) have shown that fractured zones can be detected by using conventional well logs.
Recently, artificial intelligence techniques were utilized successfully for prediction of different reservoir parameters (Ahmadi 2011; Wang et al. 2013; Anifowose et al. 2017). El Ouahed et al. (2005) developed a 2D fracture network map and fracture intensity map in Hassi Messaoud oil field by using fuzzy logic and artificial neural network. Ja’fari et al. (2012) suggested a model to estimate fracture intensity from conventional well logs by employing the adaptive neurofuzzy inference system. Accordingly, a fuzzy system maps input space to an output space using a set of fuzzy rules.
In this paper, a new method is introduced to estimate aperture size from well log data. The proposed method is an integration of fuzzy inference system and teaching–learningbased optimization algorithm (TLBO), in which the TLBO is utilized to train an initial fuzzy structure. TLBO is one of the most recent algorithms (Rao et al. 2011), which stands out as a powerful technique for optimization problems (Satapathy and Naik 2015; Kothavade and Deshpande 2016).
In order to evaluate the method effectiveness, various optimization algorithms available in the literature were employed to train the same fuzzy structure. Comparing the results indicated that the hybrid TLBOFuzzy Inference System (TLBOFIS) provides a higher level of accuracy. Finally, by estimating aperture size of detected fractures in both conductive and resistive mud environments, validation of the model was verified.
Methodology
Fuzzy inference system
Fuzzy set theory was first introduced by Zadeh (1965) as a method for handling data uncertainty. Unlike the Boolean logic, where variables take only a true or false value (1 or 0), fuzzy logic deals with the degree of truth and variables may have any value between 0 and 1. Capability of fuzzy sets to express gradual transitions from membership to nonmembership provides a powerful representation of measurement uncertainties (Klir and Yuan 1995).
There are three types of fuzzy inference systems: Mamdani (Mamdani and Assilian 1975), Tsukamoto (Tsukamoto1979), and Takagi–Sugeno–Kang (Sugeno 1985). Sugeno system is a more compact and computationally efficient model, in which its output membership functions are either constant or linear (Matlab Users Guide 2013). Considering x and y as the input values and z as the crisp output, a typical rule in the Sugeno model has the form of:
If x is A and y is B, then z = f(x,y)
where A and B are fuzzy sets and f(x,y) is a function in consequent. Once the fuzzy structure initialized, the membership function parameters can be trained by employing optimization algorithms.
Teaching–learningbased optimization algorithm (TLBO)
Local search of the TLBO occurs during the learner phase. This phase uses accepted function values at the end of teacher phase as its own input. In this phase, learners enhance their knowledge by interacting randomly with themselves. A learner enhances knowledge of other learners, when he or she has more knowledge than they do.
The TLBO is a populationbased evolutionary algorithm, which uses a population of solutions to proceed to the global solution (Ganesh and Reddy 2014). It is interesting to note that the TLBO does not require any algorithmspecific parameter. Common controlling parameters, such as the population size and number of iterations, are sufficient for its performance (Rao 2015). This ability of TLBO makes it an efficient tool, over the other algorithms, especially when dealing with multiobjective optimization problems (Zuperl and Cus 2016; Rai 2017).
Hybrid TLBOFuzzy Inference System (TLBOFIS)
Data preparation
Here, Electrical Micro Imaging (EMI) logs along with the conventional well logs from the same intervals were the used data for initializing fuzzy model. These data were acquired from five wells drilled into two different hydrocarbon fields. The study area lies in the Zagros Basin in the Fars Province of Iran. The first field contains three wells with recorded well logs ranging in depths between 3050 and 3500 m. In the second field, which includes the other two wells, studied depths range between 2500 and 2850 m. Lithology of the logged intervals consists of anhydrite, bedded dolomite, limestone, and shale, assigned to the PermoTriassic Dehram Group.
EMI is a waterbased formationimaging tool, which performs a qualitative microresistivity reading around the borehole. Six imaging pads of the EMI, in contrast to fourarm tools, provide more accurate hole volumes and diameter computations (Changxiong 2004).
By processing available image logs, a total number of 132 conductive fractures were detected and their hydraulic aperture sizes were calculated. In the first studied hydrocarbon field, fractures show a NEESWW dominant strike, southeast azimuth, and dip inclination ranging between 75° and 85°. Fracture orientations in the second hydrocarbon field are disordered. Nevertheless, NE–SW strike, mean azimuth of 110° and mean dip of 78°, is dominant.
In the literature, various empirical models were proposed in order to relate the mechanical and hydraulic apertures (Renshaw 1995; Zimmerman and Bodvarsson 1996; Li and Jiang 2013). The CIFLogGeoMatrix computes the average hydraulic aperture as the cubic mean value of fracture trace aperture.

LLD measures formation deep resistivity and, among other well logs, has the highest correlation with aperture size. Generally, wider apertures show lower resistivities (Fig. 5).

NPHI tracks hydrogen concentration in the formation and shows a linear relationship with measured aperture sizes, as illustrated in Fig. 5.

DT displays travel time of elastic wave through the formation rock. Generally, void spaces reduce the speed of the sound (Serra 2008). Bigger apertures accompany with slower velocity records.

RHOB is a record of formation bulk density and is related to the rock porosity. As seen in Fig. 5, lower densities imply bigger aperture sizes.
Final data sets, consisting of measured hydraulic aperture and well log responses, were merged and divided into two groups: a group including 98 sets for training the model and another group with 34 sets for testing.
Initializing fuzzy inference system
In order to generate the initial fuzzy inference system and define membership functions, a fuzzy clustering technique may be used. Subtractive clustering and Fuzzy CMeans are two basic methods of fuzzy clustering. Here, a subtractive clustering method was employed to determine the fuzzy rules. In a set of data, subtractive clustering defines the number of clusters and cluster centers and returns a fuzzy structure that contains fuzzy if–then rules. This method eliminates the need of specifying grid resolution, in which the agreement between the accuracy and computational complexity must be regarded (Bataineh et al. 2011).
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–learningbased optimization algorithm is needless of any algorithmspecific 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
Results and discussion
Correlation coefficients (R), RMSE, and number of fuzzy rules achieved by each clustering radius
Clustering radius  Fuzzy rules  RMSE  R 

0.1  92  0.0056  0.7999 
0.2  65  0.0052  0.8041 
0.3  39  0.0049  0.8357 
0.4  11  0.0050  0.8200 
0.5  7  0.0044  0.8668 
0.6  6  0.0044  0.8735 
0.7  4  0.0050  0.8546 
0.8  2  0.0045  0.8665 
0.9  2  0.0048  0.8581 
1  2  0.0055  0.7816 
Algorithmspecific parameters and the highest accuracy achieved by each method, in test data
Used method  RMSE  R  Algorithmspecific parameters 

ABCFIS  0.0055  0.7902  Number of onlooker bees = 800 
Trial limit (L) = 600  
Acceleration coefficient upper bound (a) = 1  
ACOFIS  0.0059  0.0804  Sample size = 400 
Intensification factor (q) = 0 .5  
Deviation–distance ratio (zeta) = 1  
ANFIS  0.0061  0.7556  Optimization method = hybrid 
GAFIS  0.0048  0.8414  Crossover percentage (pc) = 0.5 
Number of parents (nc) = 400  
Mutation percentage (pm) = 0.6  
Number of mutants (nm) = 480  
Mutation rate (mu) = 0.25  
Selection pressure (beta) = 6  
Gamma = 0.8  
TLBOFIS  0.0044  0.8735  Needless 
In the next step, a new wellbore from a different hydrocarbon field (the third field) was selected to evaluate the model performance in an oilbased 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 50m interval. The cored interval lies in the Upper Dalan Formation (of Upper Permian age) with a lithology consisting of carbonate and anhydrite.
Conclusion
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 oilbased 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 algorithmspecific 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.
Notes
Acknowledgements
The authors would like to appreciate Dr. Ali KadkhodaieIlkhchi and Christian Klimczak for helping during the research.
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