## 2 Background and Method
### Weighting Method of Eigenvector Based on the GRA  
The structure and properties of sandstone determine its water-bearing capacity. The evaluation of the water-bearing capacity of sandstone essentially refers to the coupling of all the influencing indexes. Therefore, the determination of the index weight is the key to solving the multi-attribute decision-making problem. As an objective analysis method, GRA can be used to compare various influencing factors pairwise, observe the grey relational grade (GRG) between the reference and comparison sequences, evaluate the correlation between various indexes from the data level, and sort the correlation. However, combining all indexes and obtaining quantitative evaluation is difficult in GRG [31]. Therefore, based on GRA, an improved index weighting algorithm was proposed in which the index importance can be ranked by estimating the similarity between the data, and the evaluation matrix was established based on this. Then, the eigenvector of the matrix was calculated, and the evaluation model was finally established. The detailed steps of the algorithm are as follows. First, the evaluation object and evaluation criteria are determined. The reference sequence is recorded as \(x_{j}=\left\{x_{j}(1),x_{j}(2)\cdots x_{j}(n)\right\}\). The comparison sequence is recorded as \(x_{i}=\left\{x_{i}(1),x_{i}(2)\cdots x_{i}(n)\right\}\). After standardizing the comparison sequence, according to Equation (2), the following evaluation matrix (2) is established:  
\[y_{i}=\frac{x_{i}-\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}}{\sqrt{\frac{1}{n-1} \sum\limits_{i=1}^{n}(x_{i}-\frac{1}{n}\sum\limits_{i=1}^{n}x_{i})^{2}}} \tag{2}\]  
\[Y=\left[ \begin{array}{cccc}y_{11}&y_{12}&\cdots&y_{\text{ln}}\\ y_{21}&y_{22}&\cdots&y_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ y_{n1}&y_{n2}&\cdots&y_{mn}\end{array} \right] \tag{3}\]  
Second, using Equation (3), the grave correlation coefficient (GCC) was calculated as follows:  
\[\xi_{\text{i}}(k)=\frac{\min_{i}\min_{k}\left|y_{i}(k)-x_{j}(k)+\rho\max_{i} \max_{k}\left|y_{i}(k)-x_{j}(k)\right|}{\left|y_{i}(k)-x_{j}(k)\right|+\rho \max_{i}\max_{k}\left|y_{i}(k)-x_{j}(k)\right|} \tag{4}\]\[\xi_{i}(k)-\text{The GCC of the }K\text{ in the }i;\text{ }\rho-\text{Resolution coefficient},\text{ the value is }0.54.\]  
Then, using Equation (4), we calculated the GRG of each evaluation index. \[G(n,i)=\frac{1}{n}\sum\limits_{k=1}^{n}\xi_{i}(k) \tag{5}\]  
\[G(n,i)-\text{The Grey Relational Grade}(GRG);\text{ }k=1,2,3,\cdots,n;\text{ }i=1,2,3,\cdots,m;\]  
The relational degree sequence of the evaluation object is established through GRG, and the relational degree is sorted. The larger the GRG is, the better the correlation is. Next, the evaluation matrix containing all the indexes was established. The relation degree of the level set, \(a_{ij}\), is closely related to the judgement index and is determined using \(G(n,i)\). \[A= \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1j}\\ a_{21}&a_{22}&\cdots&a_{2j}\\ \vdots&\vdots&\ddots&\vdots\\ a_{i1}&a_{i2}&\cdots&a_{ij}\ldots\end{array}  \tag{6}\]  
Next, using Equation (7), the eigenvector corresponding to each matrix is calculated as follows:  
\[W_{i}=\frac{a_{ij}}{\sum\limits_{i=1}a_{ij}}\Bigg{/}\sum\limits_{j=1}\frac{a _{ij}}{\sum\limits_{i=1}a_{ij}}i=1,2,\cdots n;\text{ }j=1,2,\cdots m \tag{7}\]  
where \(W_{i}\) is the influence intensity of a comparison sequence on the reference sequence as well as the weight value of the factors in the comparison sequence. To ensure the validity of Equation (7), the matrix completed through calculation is tested for consistency (Equation (8)). \[CI=(\frac{1}{n}\sum\limits_{i=1}^{n}[(A_{w})_{i}/W_{i}]-n)/(n-1),\text{ }CR=CI/RI \tag{8}\]  
\[RI=\frac{CI_{1}+CI_{2}+\cdots+CI_{n}}{n}\]  
The consistency test refers to determining the allowable range for judgement matrix A. The unique nonzero characteristic root of the \(n\times n\) consistency matrix is \(n\).