Fuzzy logic only takes 0 or 1 of the specific function in the ordinary set, promotes the (0, 1) interval to obtain the membership function. The membership function is a generalization of the indicator function in a fuzzy set. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, no likelihood of some event or condition. It is the absolute belonging or not belonging to the gradual expansion of flexible relationship [7B. Li, and L. Xu, "Risk assessment of ground water inrush from underlying aquifers based on fuzzy comprehensive evaluation and its application", Open Fuels & Energy Sci. J, vol. 7, pp. 107-112, 2014.
[http://dx.doi.org/10.2174/1876973X01407010107] -9H.C. Liao, Z.S. Xu, X.J. Zeng, and D.L. Xu, "An enhanced consensus reaching process in group decision making with intuitionistic fuzzy preference relations", Inf. Sci., vol. 329, pp. 274-286, 2016.
[http://dx.doi.org/10.1016/j.ins.2015.09.024] ]. Thus, the mathematical methods to deal with the fuzzy concept of the middle transition are very convenient. Because of the uncertainty of fuzzy mathematics, it can determine the occurrence of the event or not. Therefore, it has the advantage of comprehensive evaluation in complex carbonate reservoirs. Carbonate reservoirs are complex and heterogeneous. Fuzzy mathematics can overcome many uncertain factors in the process of reservoir evaluation. As a result, the fuzzy method of reservoir evaluation can be more objective method for evaluation of complex and heterogeneous carbonate reservoirs and finding adequate distribution of reservoir parameters.

According to the principle of fuzzy mathematics, the evaluation model for carbonate reservoirs is established which is composed of three element sets including factors set, evaluation set, and weight set.

Factors set U:

(1)

Where u_m is the m-th evaluation factor.

Evaluation set V:

(2)

Where v_m is the m-th judging level of evaluation factor.

Weight set A:

(3)

Where a_m is the m-th Weight of the evaluation factor.

To make sure all the factors play a role in reservoir evaluation, the relationship between the factor set U and the evaluation set V is established. Definition fuzzy mapping R from the factor set U to the evaluation set V is a transformation matrix of the reservoir evaluation. The transformation matrix R is as follows:

(4)

Where r_ij is the membership degree of the i-th factor of the j-th evaluation level. Then, the weight set matrix A by the role of various factors in comprehensive reservoir evaluation and the transformation matrix R of the comprehensive reservoir evaluation are synthesized by fuzzy transformation and the comprehensive reservoir evaluation matrix is obtained:

(5)

Where B is the evaluation grade of V on rank fuzzy subsets, “○” is the fuzzy operator of two fuzzy matrixes [15H. Dong, L.X. Wei, and Q.N. Wang, "A study on oil pipeline risk assessment technique based on fuzzy analytic hierarchy process", Open Pet. Eng. J., vol. 7, pp. 125-129, 2014.
[http://dx.doi.org/10.2174/1874834101407010125] ], b_m is rank v_m comprehensive evaluation of the reservoir level of the resulting fuzzy subset B of a membership. On the basis of the maximum membership principle, the level vi of the maximum bi is chosen as the result of the comprehensive reservoir evaluation. Here, the choice evaluation parameter, the structure membership function, and the determination weight are the key factors in carbonate reservoir evaluation with fuzzy mathematics.

Because of complex nature of carbonate reservoirs, the factors that affect the productivity of oil and gas wells in the study area are analyzed comprehensively. The parameters which are the most closely related to the carbonate reservoir are selected as the index of comprehensive reservoir evaluation. According to the analysis of carbonate reservoirs in Middle Ordovician of Middle Tarim Basin, three parameters reflecting the macroscopic and microscopic characteristics of storage abundance, permeability, and median pore throat radius were selected. Storage abundance reflects the reserves capacity of the carbonate reservoirs. Permeability reflects the carbonate reservoir’s ability to conduct fluids. Median pore throat radius reflects the microscopic pore structure and pore throat size of the carbonate reservoir. These three parameters constitute a set of factors which can provide an accurate and proper understanding of characteristics of the carbonate reservoirs in the study area.

Because of the scattered distribution of the data there is high variance in the database. If the original data is directly used, the role of the orders of magnitude larger of indicators will be highlighted, and the role of the orders of magnitude smaller of highly sensitive indicators will be weakened. Therefore, it is necessary to normalize the original data [10M. Abdolreza, and T.M. Reza, "Cooperative spectrum sensing using fuzzy membership function of energy statistics", Int. J. Electron Commun., vol. 70, no. 3, pp. 234-240, 2016.
[http://dx.doi.org/10.1016/j.aeue.2015.11.005] -12A. Rami, M.D. Kudret, and K.G. Suresh, "Coping with uncertainties in production planning through fuzzy mathematical programming: application to steel rolling industry", Int. J. Oper. Res., vol. 22, no. 1, pp. 1-30, 2015.
[http://dx.doi.org/10.1504/IJOR.2015.065937] ], which can be normalized to the same range. The normalization formula is as follows:

(6)

Where is the data after normalization, x_ij is the j-th parameter of the i-th sample in the original data, n is the total number of samples.

How to construct the expression of the membership function is a theoretical problem that has not been solved by the fuzzy mathematics yet. After statistical analysis and preprocessing of the reservoir evaluation parameters of the study area, semi-drop trapezoid method is adopted to construct membership function of the three selected evaluation indexes to establish three evaluation grades for reservoirs: I-level reservoir, II-level reservoir, and III-level reservoir. The formula is as follows:

(7)

(8)

(9)

Where t is the actual value of the evaluation parameter. r_I, r_II and r_III are the standard values of the comprehensive reservoir evaluation of I-level reservoir, II-level reservoir, and III-level reservoir. Thus, the membership function values of the evaluation factors are obtained.

The weight reflects the role of the evaluation factors in the comprehensive reservoir evaluation. At present, in practical application, the weight uses the scoring method [13L. Xu, K. Zhu, and X.L. Yang, "Production forecasting of coalbed methane wells based on type-2 fuzzy logic system", Open Pet. Eng. J,, vol. 9, pp. 268-278, 2016.
[http://dx.doi.org/10.2174/1874834101609010268] -15H. Dong, L.X. Wei, and Q.N. Wang, "A study on oil pipeline risk assessment technique based on fuzzy analytic hierarchy process", Open Pet. Eng. J., vol. 7, pp. 125-129, 2014.
[http://dx.doi.org/10.2174/1874834101407010125] ]. To avoid the influence of subjective factors based on the statistics of the actual data in the study area reservoir factor sub-indexes are used to quantitatively determine the weights of fuzzy sets. The determined weights of the fuzzy sets are then used to weight the correlation coefficient between one factor and other factors. The calculation formula is as follows:

(10)

(11)

Where A_i is the i-th weight value of the reservoir evaluation factor, C_i is the i-th actual value of the reservoir evaluation factor, S_i is the i-th weighted average value of three standard levels of reservoir evaluation factor, A is the weight set of the reservoir evaluation factor. After each individual weight value standard, the weight value is limited to (0, 1). Thus, 3 evaluation indexes are obtained, which are composed of 1×3 order weight matrix A.

The factor set of the evaluation object is established after stepwise regression analysis:

(12)

Where E_sa is the storage abundance, K is the permeability, R_mpt is the median pore throat radius.

Taking the K32 well as an example, the actual value of the reservoir parameters shows that the reservoir energy abundance is 0.57 m², the permeability is 8.6×10^-3 μm², and the median pore throat radius is 0.53 μm. The factor set is constructed:

(13)

According to the characteristics of the carbonate reservoir in Middle Ordovician of middle Tarim area, the corresponding evaluation set is established:

(14)

Carbonate reservoir evaluation standards of the three parameters for the studied reservoir are presented in Table (1). Based on the analysis of core and test data of the carbonate reservoir, the carbonate reservoir is divided into three levels in terms of reservoir quality: I-level reservoir as good quality reservoir, II-level reservoir as medium quality reservoir, and III-level as poor quality reservoir.

Table 1
Evaluation standards of a carbonate reservoir in Middle Ordovician of middle Tarim area.

Based on the evaluation standards presented in Table (1), the actual parameter values used to constitute the factor set U_K32 are substituted in equations (7), (8) and (9) to determine the evaluation set V. The membership function value of comprehensive reservoir evaluation for each single factor is then obtained. From this, the transformation matrix R_K32 of K32 well is derived:

(15)

In the same way, the transformation matrix of the other wells can be calculated.

The weight should reflect the difference in material property and the difference between people's understanding weight or importance of different parameters. The weight value is calculated using equations (10) and (11) and the weight set is determined by the weight of each factor. Weight values for the K32 well are presented in Table (2). Thereby, the weight set of K32 wells is determined as the following:

(16)

Table 2
Weight calculations for K32 well in Middle Ordovician of middle Tarim area.

After the calculation of R and A, the results are obtained by the formula (5). The level of quality of the carbonate reservoir is evaluated according to the principle of maximum membership degree. The transformation matrix R by the single factor membership function values obtained and the weight set A by weight value of each single factor obtained for K32 well and then substituted into the formula (5) to get the fuzzy matrix. Using the fuzzy synthetic operator comprehensive evaluation of the carbonate reservoir quality in K32 is conducted:

(17)

Results obtained from reservoir quality assessment in the K32 well shows that I-level, II-level, and III-level reservoir qualities account for 58%, 37%, 5%, respectively. Depending on the principle of the maximum membership, the evaluation result of K32 well is I-level reservoir distribution area. In a same way, the results of reservoir comprehensive evaluation in other wells can be obtained.

The results obtained from comprehensive reservoir evaluation of representative wells of K17, K22, K32, K46, K53, and K67 in Middle Ordovician of middle Tarim area are presented in Table (3). The evaluation results are used to prepare a contour map for each level and different levels of regions are delineated. Figs. (1 to 3) show the I-level, II-level, and III-level reservoir qualities in the study area, respectively.

Fig. (1) I-level reservoir distribution of reservoir evaluation in Middle Ordovician, contour for I-level reservoir, filling graphical range for I-level reservoir.

Fig. (2) II-level reservoir distribution of reservoir evaluation in Middle Ordovician, contour for II-level reservoir, filling graphical range for II-level reservoir.

Table 3
Comprehensive reservoir evaluation results of selected wells in Middle Ordovician.

Fig. (3) III-level reservoir distribution of reservoir evaluation in Middle Ordovician, contour for III-level reservoir, filling graphical range for III-level reservoir.