## 2 Principles and Methods
### Establishment of Calculation Model  
Most scholars in isotope geochemistry mainly focus on the abundance and variation of stable isotopes [31; 32]. In nature, carbon has two stable isotopes \({}^{12}\)C and \({}^{13}\)C, and the carbon isotope value \(\delta^{13}C\) can be defined by Equation (1) [33; 34].  
\[\delta^{13}C=\left(\frac{\left({}^{13}C/^{12}C\right)_{sq}}{\left({}^{13}C/^{1 2}C\right)_{st}}-1\right)\times 1000 \tag{1}\]  
In the equation, \(\left({}^{13}C/^{12}C\right)_{sq}\) is the isotope ratio of a test sample; \(\left({}^{13}C/^{12}C\right)_{st}\) is the isotope ratio of a standard sample, and \({}^{13}C/^{12}C\) of the PDB standard samples is (11,237.2 \(\pm\) 90) \(\times\) 10\({}^{-6}\).  
According to the analytical results in Section 2.1, the focus of CBM extraction is on hydrocarbon gas (CH\({}_{4}\)) and non-hydrocarbon gas (CO\({}_{2}\)). Taking CH\({}_{4}\) as an example, assuming that during CBM extraction from three coalbeds, according to the law of mass conservation for the chemical composition of the CH\({}_{4}\) carbon isotope, the CH\({}_{4}\) carbon isotope value of mixed CBM \(\delta^{13}C(CH_{4})_{mix}\) can be deduced according to Equation (2) [35]:  
\[\delta^{13}C(CH_{4})_{mix}=\frac{\sum\limits_{i=1}^{3}V(CH_{4})_{i}\cdot\delta ^{13}C(CH_{4})_{i}}{\sum\limits_{i=1}^{3}V(CH_{4})_{i}} \tag{2}\]  
Figure 1: Main, minor and micro components of extracted gas.  
In the equation, \(\delta^{13}C(\text{CH}_{4})_{i}\) is the CH\({}_{4}\) carbon isotope value for one of the three groups; \(V(\text{CH}_{4})_{i}\) is the CH\({}_{4}\) volume for one of the three groups, which can be calculated according to Equation (3) on the premise that CBM extraction is an ideal gas.  
\[V(\text{CH}_{4})_{i}=V_{i}\cdot\varphi(\text{CH}_{4})_{i} \tag{3}\]  
In the equation, \(V_{i}\) is the volume for one of the three groups; \(\varphi(\text{CH}_{4})_{i}\) is the CH\({}_{4}\) volume fraction for one of the three groups.  
Since the length of the layer-through borehole is usually less than one hundred meters, under the negative pressure pulling effect of the extraction system, the time for CBM to reach the mouth of borehole from the coalbed is very short. During this period, the changes in temperature and pressure are insignificant. Therefore, it can be assumed that the chemical reaction between the CH\({}_{4}\) molecules from different groups of CBM can be considered negligible, and the fractionation reaction of CBM itself is also not taken under consideration [36; 37]. Then, the total volume \(V\) of CBM and extraction ratio of each group can be calculated according to Equations (4) and (5).  
\[V=\sum\limits_{i=1}^{3}V_{i} \tag{4}\]  
\[x_{i}=\frac{V_{i}}{V} \tag{5}\]  
The CO\({}_{2}\) carbon isotope of the three groups of CBM can be united to establish a calculation model by Equation (6), on the premise that the CBM extraction is an ideal gas.  
\[\left\{\begin{array}{l}\sum\limits_{i=1}^{3}x_{i}=1\\ \\ \delta^{13}C(\text{CH}_{4})_{mix}=\frac{\sum\limits_{i=1}^{3}x_{i}\cdot\varphi (\text{CH}_{4})_{i}\cdot\delta^{13}C(\text{CH}_{4})_{i}}{\sum\limits_{i=1}^{3}x _{i}\cdot\varphi(\text{CH}_{4})_{i}}\\ \\ \delta^{13}C(\text{CO}_{2})_{mix}=\frac{\sum\limits_{i=1}^{3}x_{i}\cdot\varphi( \text{CO}_{2})_{i}\cdot\delta^{13}C(\text{CO}_{2})_{i}}{\sum\limits_{i=1}^{3}x _{i}\cdot\varphi(\text{CO}_{2})_{i}}\\ \end{array}\right. \tag{6}\]  
In the equation, \(\delta^{13}C(\text{CO}_{2})_{mix}\) is the CO\({}_{2}\) carbon isotope value of mixed CBM\({}_{2}\)\(\delta^{13}C(\text{CO}_{2})_{i}\) is the CO\({}_{2}\) carbon isotope value for one of the three groups; \(\varphi(\text{CO}_{2})_{i}\) is the CO\({}_{2}\) volume fraction for one of the three groups.