## 2 Sensitivity Analysis of Structural Characteristics of Drill Bit
The sample set \(\left\{X_{i}^{1}\,,\,X_{i}^{2}\,,\,\cdots\,,X_{i}^{n}\right\}\) of each factor \(X_{i}\) (the structural characteristics of the drill bit in this study) is sorted according to the size of the value and the position of \(X_{i}^{k}\) in the sequence is the rank of \(X_{i}^{k}\), that is \(\alpha_{k}\). Rank the solution set \(\left\{Y_{1}\,,Y_{2}\,,\,\cdots\,,Y_{n}\right\}\) of the objective function (the ROP and footage of the drill bit drilling in a specific interval in this study) according to the corresponding value (\(Y_{k}\)) of its factor \(X_{i}^{k}\), and the position of \(Y_{j}\) in the sequence is the rank of \(Y_{j}\), that is \(\beta_{k}\). Calculate the rank correlation coefficient \(r_{i}\) between the factor sample set and the corresponding solution set of the objective function according to Formula (1). The coefficient \(r_{i}\) is expressed as an absolute value. (The result has nothing to do with the ranking method). It represents the sensitivity of the objective function to the factor \(X_{i}\). (In this topic, it represents the sensitivity of the ROP and footage of the bit to each bit structure characteristic). The greater the value of \(r_{i}\), the greater the sensitivity of the objective function to the factor \(X_{i}\), that is, the greater the influence of \(X_{i}\) on the objective function. \[r_{i}=\left|\frac{n\sum_{k=1}^{n}\alpha_{k}\beta_{k}-\sum_{k=1}^{n}\alpha_{k} \sum_{k=1}^{n}\beta_{k}}{\sqrt{n\sum_{k=1}^{n}(\alpha_{k})^{2}-\left(\sum_{k= 1}^{n}\alpha_{k}\right)^{2}}\sqrt{n\sum_{k=1}^{n}(\beta_{k})^{2}-\left(\sum_{k =1}^{n}\beta_{k}\right)^{2}}}\right| \tag{1}\]  
where \(r_{i}\) represents the rank correlation coefficient of \(X_{i}\), dimensionless; \(n\) represents the total number of samples; \(k\) represents the number of sample sets; \(\alpha_{k}\) represents the rank of \(X_{i}^{k}\); and \(\beta_{k}\) represents the rank of the solution set \(Y_{j}\) of the objective function. Based on the geological stratification and stratigraphic lithology characteristics of Luzhou Block in Sichuan Province, the strata are divided into seven sections, and the specific stratification results are shown in Table 2. The second spud size is 444.5 mm/406.4 mm. Section 1 mainly includes Shaximiao Formation, Liangshan Formation, and Ziliujing Formation, while Section 2 mainly includes Ziliujing Formation and Xujiahe Formation. The third spud size is 311.2 mm. Section 1 mainly includes Xujiahe Formation, Section 2 mainly includes Jialingjiang Formation, Feixianguan Formation, Changxing Formation, and Longtan Formation, and Section 3 mainly includes Maokou Formation and Qixia Formation. The fourth spud size is 215.9 mm. Section 1 mainly includes Qixia Formation, Liangshan Formation, Hanjiadian Formation, and Shiniulan Formation, while Section 2 mainly includes Shiniulan Formation and Longmaxi Formation. The program code for the Spearman rank correlation coefficient calculation was written in MATLAB R2021b. The specific program code is shown in Appendix B. The first drill bit used for each stage is analyzed, namely, the second spud Shaximiao formation-Ziliujing formation, the third spud Xujiahe Formation, and the fourth spud Qixia formation-Shiniulan Formation. According to the bit usage, the variables of structural feature input, footage, and ROP output were established, and the Spearman rank correlation coefficients of each structural feature were calculated according to Equation (1). Where \(n\) represents the number of samples of analytical bits in each stage, and the interval \(n\) values of ShaximiaoFormation to Ziliujing Formation in second spud, Xuijahe Formation in third spud, Qixia Formation to Shiniulan Formation in fourth spud are 17, 21, and 13, respectively. The factor \(X_{i}\) represents the structural characteristics of the bit, and the objective function solution set \(Y_{i}\) represents the footage and ROP of the bit. Since 12 structural parameters are analyzed in this study, the value of \(i\) is an integer ranging from 1 to 12.