# Experimental Investigation of Mechanical Properties of Clay-Cement Slurry Containing Graphene Oxide
## 3 Results
### Anti-Permeability Performance of Grouting Reinforcement
#### 3.2.3 Multiple Regression Analysis of Permeability Coefficient Influencing Factors  
The experimental data were processed using regression analysis. Regression analysis is employed to establish a mathematical relationship between variables by utilizing mathematical expressions. This analytical approach enables the prediction of the dependent variable's value based on the independent variable's value. By doing so, regression analysis aims to unveil the underlying statistical patterns hidden within the randomness of the data. When the independent variables are multiple factors, the significance of the influence of each variable on the dependent variable can be found using variance analysis. The formation permeability coefficient K is used as the dependent variable, and the OPC content, GO content, and gradation are used as independent variables. Through regression analysis, the influence of OPC content, GO content, and gradation on the formation permeability coefficient can be obtained. The multiple linear regression analysis method is used to process the test data. It is assumed that there are m independent variables, which are recorded as \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{i}\), \(\ldots\), \(x_{m}\). The dependent variable y is the test result. If n sets of data are measured in the test,  
\[\left(x_{11},x_{21},\ldots,x_{11},\ldots,x_{\rm m1},y_{1}\right)\]  
\[\left(x_{12},x_{22},\ldots,x_{\rm i2},\ldots,x_{\rm m2},y_{2}\right)\]  
\[\ldots\ldots\]  
\[\left(x_{\rm in},x_{\rm 2n},\ldots,x_{\rm in},\ldots,x_{\rm mn},y_{\rm n}\right)\]  
The multivariate linear regression equation is expressed as follows:  
\[\hat{y}=a+b_{1}x_{1}+b_{2}x_{2}+\cdots+b_{m}x_{m} \tag{4}\]  
where \(a\) is a constant, and \(b_{i}\)(\(i\) = 1, 2, \(\cdots\), \(m\)) is a partial regression coefficient. According to the principle of the least square method, the unknown parameters can be solved by minimizing the residual sum of squares of the regression equation. The specific calculation method is as follows:  
The residual sum of squares \(Q_{e}\) can be obtained using the following formula:  
\[Q_{e}=\sum_{j=1}^{m}\left(y_{j}-\hat{y_{j}}\right)^{2}=\sum_{j=1}^{n}\left[y_{ j}-\left(a+b_{1}x_{1j}+b_{2}x_{2j}+\cdots+b_{m}x_{mj}\right)\right]^{2} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Source** & **Square of Deviance** &
\begin{tabular}{c} **Degree of** \\ **Freedom** \\ \end{tabular} & **F-Value** & \(p\)**-Value** \\ \hline
1-1 & 0.173 & 13 & \(5.00\times 10^{-9}\) & 1.11 \\
1-2 & 0.153 & 10 & \(8.21\times 10^{-9}\) & 2.68 \\
1-3 & 0.137 & 12 & \(7.66\times 10^{-9}\) & 2.33 \\
2-1 & 0.221 & 19 & \(2.94\times 10^{-8}\) & 3.49 \\
2-2 & 0.197 & 16 & \(1.73\times 10^{-8}\) & 2.74 \\
2-3 & 0.177 & 16 & \(2.41\times 10^{-8}\) & 4.23 \\
3-1 & 0.141 & 24 & \(5.11\times 10^{-8}\) & 7.54 \\
3-2 & 0.112 & 12 & \(2.52\times 10^{-8}\) & 9.36 \\
3-3 & 0.162 & 11 & \(3.45\times 10^{-8}\) & 7.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Variance analysis of permeability coefficient test results. The partial derivatives of the residual sum of squares for a and bi can be obtained:  
\[\frac{\partial Q_{e}}{\partial a}=-2{\sum_{j=1}^{n}}\left[y_{j}-\left(a+b_{1}x_{1j }+b_{2}x_{2j}+\cdots+b_{m}x_{mj}\right)\right]^{2}=0 \tag{6}\]  
\[\frac{\partial Q_{e}}{\partial b_{i}}=-2{\sum_{j=1}^{n}}\left[y_{j}-\left(a+b_{ 1}x_{1j}+b_{2}x_{2j}+\cdots+b_{m}x_{mj}\right)\right]^{2}=0 \tag{7}\]  
The above two equations form a set of (\(m+1\)) equations;  
\(a\) and \(b_{i}\) can be obtained by solving the equations. They obey the following equation:  
\[K=\text{A}\cdot\text{O}^{\text{B}}\cdot\text{H}^{C}\cdot\text{N}^{\text{D}} \tag{8}\]  
where \(O\) is OPC content; \(H\) is GO content; \(N\) is the ground level; and \(K\) is the permeability coefficient of the stratum. \(A\), \(B\), \(C\), and \(D\) are undetermined coefficients.