## 4 Evolution Characteristics of the Dilation Parameter of Gas Hydrate-Bearing Sediments  
The data for this study are also from the laboratory experiment results by [PERSON] et al. [15]. The axial and lateral plastic strains are also obtained based on the method introduced in Section 2.2. Consequently, a series of dilation angles (\(\psi\)) can be calculated corresponding to different values of plastic shear strain. These data are plotted as scattered data points in Figure 3 for analyzing the evolution of the dilation angle for the GHBS samples with different hydrate saturations at various effective confining pressures. The main evolution characteristics can be observed as follows: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{S_{h}}\) & \(\mathbf{c_{1}}\) (MPa) & \(\mathbf{c_{max}}\) (MPa) & \(\mathbf{\gamma^{\rm P}_{\rm c,max}}\) (\%) & \(\mathbf{c_{r}}\) (MPa) & \(\mathbf{\gamma^{\rm P}_{\rm c,r}}\) (\%) \\ \hline
0\% & 0.17 & 0.26 & 12.14 & 0.23 & 30 \\
27–34\% & 0.28 & 0.47 & 0.03 & 0.03 & 30 \\
41–45\% & 0.42 & 0.70 & 1.62 & 0.42 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitted parameters for the evolution model of cohesion (\(c\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{S_{h}}\) & \(\mathbf{\varphi_{\rm r}}\) (\({}^{\circ}\)) & \(\mathbf{\varphi_{\rm i}}\) (\({}^{\circ}\)) & \(\mathbf{\gamma^{\rm P}_{\rm c,r}}\) (\%) \\ \hline
0\% & 33.70 & 6.30 & 5.90 \\
27–34\% & 35.59 & 7.65 & 4.99 \\
41–45\% & 37.11 & 8.65 & 7.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitted parameters for the evolution model of the internal friction angle (\(\varphi\)). 1. For all the samples with different hydrate saturations at various effective confining pressures, the dilation angle increases with increasing plastic shear strain and then becomes stable at a residual value. Specifically, it can be observed that the dilation angles are negative at the early stage and then grow to be positive gradually. The negative dilation angle indicates plastic contraction, while the positive value means an irreversible increase in volume [38]. Therefore, the evolution characteristics observed in the tests on the GHBS samples show a good agreement with the deformation behaviors reported in [15], i.e., compaction occurs at the early stage of compression before expansion happens due to dilation. 2. The samples with higher hydrate saturations show higher dilation angles (\(\psi\)). This feature coincides well with the observation reported in [15] that samples with higher hydrate saturations show more obvious dilation behaviors because of a higher proportion of the solid component. However, this difference is not so significant when the effective confining pressure is higher (\(\sigma_{3}^{\prime}=2\) and 3 MPa) because the effective confining pressure has a limitation effect on the dilation behavior. Based on the data of the dilation angle, a mobilized model for the dilation angle of GHBS samples can be obtained by non-linear fitting. The fitted curves are presented in Figure 3. The fitted equation can be adopted to describe the evolution of the dilation angle as shown in Equation (7):  
\[\psi=\psi_{\mathrm{r}}+(\psi_{\mathrm{i}}-\psi_{\mathrm{r}})\left[2-\frac{2} {1+\exp\left(-5\frac{\gamma_{\mathrm{p}}^{\mathrm{p}}}{\gamma_{\mathrm{p}, \mathrm{r}}^{\mathrm{p}}}\right)}\right] \tag{7}\]  
Figure 3: _Cont._  
Figure 3: Evolutionary dilation angle \(\psi\) (the scattered data points) and the fitted curves of GHBS samples with different hydrate saturations (**(a)**\(s_{\rm h}\) = 0%, **(b)**\(s_{\rm h}\) = 27–34%, and **(c)**\(s_{\rm h}\) = 41–45%) under various confining pressures. The analyses are based on the experimental data from [15] with permission. where \(\psi_{1}\) refers to the initial value of the dilation angle, i.e., the value of the dilation angle when the plastic shear strain value is 0; \(\psi_{r}\) is the residual value of dilation angle.