## 4 Evolution Characteristics of the Dilation Parameter of Gas Hydrate-Bearing Sediments
\(\gamma_{\psi,r}^{\mathrm{P}}\) refers to the value of the plastic shear strain corresponding to the residual value of the dilation angle. The above-mentioned physical meanings of these parameters are demonstrated in the sensitive analyses presented in Appendix B. Table 3 shows the values of the fitted parameters in the evolution equation of the dilation angle. According to Table 3, it can be seen that the parameters \(\psi_{\mathrm{i}}\), \(\psi_{r}\), and \(\gamma_{\psi,r}^{\mathrm{P}}\) are dependent on the hydrate saturation \(S_{\mathrm{h}}\) and effective confining pressure \(\sigma_{3}^{\prime}\). The functions of \(\psi_{\mathrm{i}}\), \(\psi_{r}\), and \(\gamma_{\psi,r}^{\mathrm{P}}\) are dependent on \(S_{\mathrm{h}}\) and \(\sigma_{3}^{\prime}\) can be analyzed by fitting the data in Table 3. Taking the parameters \(\psi_{\mathrm{i}}\) as an example, the fitting procedure is introduced briefly as follows:  
1. For each certain hydrate saturation \(S_{\mathrm{h}}\), the relation between \(\psi_{\mathrm{i}}\) and the effective confining pressure \(\sigma_{3}^{\prime}\) is fitted first. Based on the data in Table 3, it is found that this relation generally follows a linear equation for the samples with different hydrate saturations \(S_{\mathrm{h}}\), and this equation can be expressed as: \[\psi_{\mathrm{i}}=m+n\sigma_{3}^{\prime}\] (8) where \(m\) and \(n\) are two fitting parameters, and their fitted values are presented in Table 4.
2. The hydrate saturation \(S_{\mathrm{h}}\) is incorporated in the fitting. According to Table 4, it can be found that both \(m\) and \(n\) can be defined as linear functions of the hydrate saturation \(S_{\mathrm{h}}\) as follows: \[m=a_{1}+b_{1}S_{\mathrm{h}}\] (9) \[n=a_{2}+b_{2}S_{\mathrm{h}}\] (10) where \(a_{1}\), \(a_{2}\), \(b_{1}\), and \(b_{2}\) are fitting parameters, and their fitted values are presented in Figure 3.0 It should be noted that the medium values of 30% and 43% are adopted to represent the saturation domains of 27-34% and 41-45% in Table 3 during the fitting process. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(m\) & \(n\) \\ \hline
0\% & \(-\)4.07 & \(-\)16.9 \\
27–34\% & \(-\)15.25 & \(-\)6.3 \\
41–45\% & \(-\)18.23 & \(-\)7.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitted values of parameters \(m\) and \(n\) for determining \(\psi_{\mathrm{i}}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(S_{\mathrm{h}}\) & \(\sigma_{3}^{\prime}\) & \(\psi_{\mathrm{i}}\) (\({}^{\circ}\)) & \(\psi_{r}\) (\({}^{\circ}\)) & \(\gamma^{\mathrm{P}}_{\psi,r}\) (\%) \\ \hline  & 0.5 MPa & \(-\)8 & 23.2 & 10 \\
0\% & 1 MPa & \(-\)23 & 30 & 15 \\  & 2 MPa & \(-\)45 & 31.24 & 20 \\  & 3 MPa & \(-\)50 & 32.7 & 20 \\ \hline  & 0.5 MPa & \(-\)23 & 25.4 & 10 \\
27–34\% & 1 MPa & \(-\)15 & 32.7 & 12 \\  & 2 MPa & \(-\)40 & 26 & 21 \\  & 3 MPa & \(-\)37 & 28.7 & 20 \\ \hline  & 0.5 MPa & \(-\)37 & 36.8 & 16 \\
41–45\% & 1 MPa & \(-\)6 & 29.5 & 15 \\  & 2 MPa & \(-\)33 & 28.5 & 18 \\  & 3 MPa & \(-\)43 & 31 & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitted parameters for the evolution model of the dilation angle. 3. Substituting Equations (9) and (10) into Equation (8), we can obtain the function of \(\psi_{1}\) dependent on hydrate saturation \(S_{\text{h}}\) and effective confining pressure \(\sigma_{3}^{\prime}\) as: \[\psi_{1}=a_{1}+b_{1}s_{\text{h}}+(a_{2}+b_{2}s_{\text{h}})\sigma_{3}^{\prime}\] (11) According to Table 3, with the same fitting method, it can be found that the functions of \(\psi_{\text{r}}\) and \(\gamma_{\psi,\text{r}}^{\text{P}}\) follow the same form as Equation (11); therefore, they can be expressed as a uniform equation, as shown in Equation (12): \[k=a_{1}+b_{1}s_{\text{h}}+(a_{2}+b_{2}s_{\text{h}})\sigma_{3}^{\prime}\] (12) where \(k\) can represent any of the above-mentioned parameters \(\psi_{1}\), \(\psi_{\text{r}}\), and \(\gamma_{\psi,\text{r}}^{\text{P}}\).