## 1 Introduction  
China's coalbed methane (CBM) resources are abundant, and more of these resources are developed each year. According to the 13 th five-year plan, the production of CBM will reach \(2.4\times 10^{10}\) m\({}^{3}\) by 2020. The continued expansion of exploration and development has made the selection and evaluation of CBM blocks increasingly important. The formation conditions of CBM and the evolutionary history of the reservoir structure in China are relatively complex, which leads to the differences in the geological parameters of the same region. The same strata in the same block can have profound differences and complexity, and the exact value is not sufficient to represent the block feature. The selective development of CBM blocks requires a multi-factor evaluation method that is suitable for complex conditions for which the evaluation index is an interval number. The determination of weightsis an important part of the evaluation method and should directly relate to the evaluation results. The method of selecting index weights to avoid subjective assumptions and objective randomness requires further exploration. Several researchers have proposed relative model methods for block optimization for the geological conditions that are associated with CBM in China. Based on the risk probability and system theory, [PERSON] et al. [1] proposed the idea of \"the risk evaluation + analytic hierarchy process (AHP)\" and established a \"one vote veto + hierarchical optimization\" evaluation method system. [PERSON] et al. [2] followed the dialectical thinking of geological research--quantitative sequencing--geological analysis and established a method system for the evaluation of CBM blocks, which consists of a hierarchical structure optimal selection model, an absolute weight progressive model, and a risk probability model. Areas with high yields and rich concentrations of CBM are expected in some basins of China. [PERSON] et al. [3] proposed a comprehensive evaluation model combined with the AHP and the gray clustering method. This method was used to quantitatively evaluate 121 CBM blocks in China and provide ideas for CBM block evaluation. [PERSON] [4] used the AHP to determine the index weights and comprehensively evaluated the favorable CBM blocks in the Ordos Basin by using fuzzy mathematics to identify priority developmental blocks. Several quantitative evaluations of CBM blocks have been performed; however, most of these evaluations directly used existing methods from other fields, and the evaluation indices were mostly fixed values. The gray system theory emerged as an interdisciplinary approach in 1982 [5]. The gray relational analysis (GRA) is an important aspect of the gray system theory, which measures the degree of closeness between factors according to the sameness or differences in the development trend among the factors. The advantages of GRA in quantitative forecasting and in decision-making based on uncertain and small sample size information attract a large number of researchers. GRA has a wide range of applications, such as those in information technology [6; 7], financial markets [8], and industries [9; 10; 11]. In addition to these areas, GRA has also been used extensively in the energy fields of various countries. [PERSON] and [PERSON] used 47 office buildings in Taiwan as an example to evaluate and rank their energy performance to illustrate the procedure and effectiveness of the GRA [12]. [PERSON] et al. proposed combining GRA and error propagation theory to select a biofuel and its optimal combustion conditions of South Europe (Spain) [13]. [PERSON] presented a novel effective method for the optimization of oil agglomeration process of Zonguldak/Turkey bituminous coal with multiple performance characteristics based on the GRA [14]. [PERSON] et al. aimed to identify the optimum parameters affecting the energy and exergy efficiencies for a novel design solar air heater using the gray relational analysis [15].