## 2 Materials and Methods
### Water Absorption Prediction Model of Soft Rock  
On the basis of the above analysis, a prediction model of soft rock water absorption is established. The model can be used to predict the water absorption process of rocks, given the type and content of clay minerals, porosity and properties of water solutions. Firstly, build the factor and level matrices (Equations (7) and (8)). In the factor matrix, the sum of the \(f\) values equals 7. If all factors are present, then each \(f\) value is equal to 1. If only some factors exist, for example, only VSs and pH exist, then \(f_{VSS}=f_{pH}=3.5\), and the remaining \(f\) values are equal to 0. Similarly, in a level matrix, the \(l\) values of each column sum to 3. For the first column, all \(l\) values are 1 if all three clay minerals are present. If only one clay mineral, illite, exists, then \(l_{ILL}=3\) and \(l_{MMT}=l_{KAO}=0\). Then, establish the degree of influence matrices of the factor and level matrices (Equations (9) and (10)). The values in the matrix \(DI_{F}\) are the results of the normalization of each factor \(R_{t}\) in Figure 9, and the values in the matrix \(DI_{L}\) are the results of the normalization of the degrees of influence of each level in Figure 8. Finally, the comprehensive influence factor (\(p\)) is obtained through matrix operations (Equation (11)). \[F=\big{[}f_{CMT}\quad f_{CMC}\quad f_{VSS}\quad f_{CTs}\quad f_{ATS}\quad f_{CCs} \quad f_{pH}\big{]}, \tag{7}\]  
where \(F\) is the factor matrix, \(f\) is the value of each factor and its value range is 0-7. \[L= \begin{bmatrix}l_{MMT}&l_{30\%}&l_{5 MPa}&l_{Na^{+}}&l_{Cl^{-}}&l_{0.1 mol/L}& l_{4}\\ l_{ILL}&l_{50\%}&l_{10 MPa}&l_{Ca^{2+}}&l_{5O^{2-}}&l_{1.0 mol/L}&l_{7}\\ l_{KAO}&l_{70\%}&l_{15 MPa}&l_{K^{+}}&l_{CO^{2-}_{3}}&l_{SAT}&l_{10}\\ \end{bmatrix}  \tag{8}\]  
where \(L\). is the level matrix, \(l\) is the value of each level and the value range is 0-3. \[DI_{F}=\big{[}0.156\quad 0.143\quad 0.139\quad 0.140\quad 0.194\quad 0.107 \quad 0.121\big{]}, \tag{9}\]  
Figure 9: Degree of influence of seven factors on the four parameters. \[DI_{L}= \begin{bmatrix}0.444&0.322&0.277&0.290&0.254&0.306&0.316\\ 0.278&0.309&0.457&0.302&0.246&0.322&0.330\\ 0.278&0.369&0.266&0.407&0.500&0.372&0.355\end{bmatrix} . \tag{10}\]  
\[p=diag[DI_{L}\times L]_{1\times 7}\times diag[DI_{F}\times F]_{7\times 1}, \tag{11}\]  
where \(p\) is the comprehensive influence factor, and \(diag\) is the operator that takes the main diagonal value of the matrix. Since the interaction between factors and levels is not considered, only the diagonal of the matrix is meaningful. Substituting \(p\) into Equations (3) and (5) yields a multifactor water absorption prediction model (Equations (12) and (13)). \[w=A_{1}p\times e^{-\frac{t}{pT_{1}}}+A_{2}p\times e^{-\frac{t}{pT_{2}}}+w_{0}p, \tag{12}\]  
\[v=10^{ap}\times 10^{bp\times\lg(t)}. \tag{13}\]  
In order to verify the reliability of the soft rock water absorption prediction model, this model is compared with the related test results of the hard rock in Figure 1 of the literature [50] and the soft rock in Figure 5 of the literature [51], as shown in Figure 10. In general, the water absorption prediction model can better predict the water absorption process of rocks under the premise of known clay minerals, rock porosity and water solution properties. The curve calculated by the water absorption prediction model is basically consistent with the change trend of previous research results (except for the sample \(w\)-304). Moreover, the model can not only be used to predict the water absorption process of soft rock, but can also be used to predict the water absorption process of hard rock. Due to different experimental materials, methods and conditions, the data of some water absorption processes are different, but this does not affect the judgment of the critical time point when rocks are damaged due to changes in water content. Previous researchers also studied the water absorption prediction model, such as [PERSON] et al.