## 3 Analytical Model
### Deformation Compatibility on Rock-Lining Interface  
As there is no slip at the rock-lining interface (\(t\geq t_{0}\)), the equation of the deformation compatibility on the rock-lining interface can be written as follows:  
\[u_{r}^{\mathrm{RP}}(\tau+t_{0})\Big{|}_{r=R_{2}}-\left.u_{r}^{\mathrm{RP}}(t_ {0})\right|_{r=R_{2}}-\left.u_{r}^{\mathrm{RQ}}(\tau)\right|_{r=R_{2}}=\left. u_{r}^{\mathrm{L}}(\tau)\right|_{r=R_{2}} \tag{27}\]  
Substituting Equations (17)-(18) and Equation (25) into Equation (27), Equation (27) can be rewritten as follows:  
\[h(\tau)-\left[Q(\tau)J(0)+\int_{0}^{\tau}Q(\xi)\frac{dJ(\tau-\xi)}{d(\tau-\xi )}d\xi\right]=A_{0}E(R_{2})Q(\tau)+A_{0}F(R_{2})q(\tau) \tag{28}\]  
Based on the Laplace transform method [41], the expression of the supporting pressure, provided by the lining about parameter of time \(\tau\), can be obtained as follows:  
\[Q(s)=\frac{h(s)-A_{0}E(R_{2})q(s)}{sJ(s)+A_{0}F(R_{2})} \tag{29}\]  
where \(Q(s)\), \(h(s)\), \(q(s)\), and \(J(s)\) are the expressions of the Laplace transform of \(Q(\tau)\), \(h(\tau)\), \(q(\tau)\), and \(J(\tau)\), respectively. The expressions of the \(h(s)\) and the \(J(s)\) can be given as follows:  
\[\left\{ \begin{array}{l}h(s)=\left[\frac{1}{2\eta_{m}s^{2}}+\frac{b_{1}}{2} \big{(}\frac{1}{s}-\frac{1}{s+\beta}\big{)}+\frac{b_{2}}{2}\big{(}\frac{1}{s}- \frac{1}{s+1/t_{\mathrm{k}}}\big{)}\right]P_{0}\\ J(s)=\frac{1}{2\xi}\Big{[}\frac{1}{\mathcal{C}_{m}}+\frac{1}{\eta_{m}s}+\frac{ 1}{\eta_{m}(s+1/t_{\mathrm{k}})}\Big{]}\end{array} \right. \tag{30}\]  
in which  
\[\left\{ \begin{array}{l}b_{1}=a(\frac{1}{\xi_{\mathrm{K}}(1-\beta t_{\mathrm{ k}})}+\frac{1}{\mathcal{C}_{m}}-\frac{1}{\mathcal{C}_{m}})e^{-\beta t_{0}}\\ b_{2}=(\frac{1}{\xi_{\mathrm{K}}}-\frac{1}{\xi_{\mathrm{K}}(1-\beta t_{\mathrm{ k}})})e^{-t_{0}/t_{\mathrm{k}}}\end{array} \right. \tag{31}\]  
Substituting Equation (30) into Equation (29), Equation (29) can be rewritten as follows:\[Q(s)=\frac{\frac{1}{\eta_{m}}(s+\beta)(s+\frac{1}{t_{k}})P_{0}+b_{1}\beta s(s+ \frac{1}{t_{k}})P_{0}+b_{2}s(s+m)\frac{1}{t_{k}}P-2A_{0}E(R_{2})q(s)s^{2}(s+ \beta)(s+1/t_{k})}{s^{2}(s+\beta)(s+1/t_{k})\Big{\{}\Big{[}\frac{1}{\zeta_{m}}+ \frac{1}{\eta_{m}s}+\frac{1}{\eta_{h}(s+1/t_{k})}\Big{]}+2A_{0}F(R_{2})\Big{\}}} \tag{32}\]  
Equation (32) can be further simplified as follows:  
\[Q(s)=\frac{\frac{1}{\eta_{m}}(s+\beta)(s+\frac{1}{t_{k}})P_{0}+b_{1}\beta s(s+ \frac{1}{t_{k}})P_{0}+b_{2}s(s+m)\frac{1}{t_{k}}P_{0}}{s(s+\beta)a_{11}(s-x_{1 })(s-x_{2})}-f(s) \tag{33}\]  
in which  
\[\left\{ \begin{array}{l}a_{11}=\frac{1}{\zeta_{m}}+2A_{0}F(R_{2})\\ b_{11}=(\frac{1}{\zeta_{m}}+\frac{1}{\zeta_{k}}+2A_{0}F(R_{2}))\frac{1}{t_{k}}+ \frac{1}{\eta_{m}}\\ c_{11}=\frac{1}{\eta_{m}^{2}k_{k}}\\ \Delta=(\frac{b_{11}}{a_{11}})^{2}-4\frac{c_{11}}{a_{11}}\\ x_{1,2}=\frac{-b_{11}/a_{11}x\wedge\zeta}{a_{11}(s-x_{1})(s-x_{2})}\\ f(s)=\frac{2A_{0}F(R_{2})q(s)s(s+1/t_{k})}{a_{11}(s-x_{1})(s-x_{2})}\\ -Q_{4}(\tau)\end{array} \right. \tag{34}\]  
Using the Laplace transform inversion [42; 43; 44] about parameter \(s\) in Equation (33), the expression of the \(Q(\tau)\) can be given as follows:  
\[ \begin{array}{ll}Q(\tau)&=\frac{P_{0}}{a_{11}\eta_{m}}\Big{[}\frac{c_{1}^{ \tau_{1}}-c_{2}^{\tau_{2}}}{\lambda_{1}-2s}+\frac{1}{k_{1}\lambda_{2}}(1+ \frac{\gamma_{2}e^{\tau_{1}}-\gamma_{1}e^{\tau_{2}}}{\lambda_{1}-\lambda_{2}} )\Big{]}\\ &+\frac{\rho_{0}b_{1}\beta}{a_{11}}\Big{[}\frac{(x_{1}+1/t_{k})e^{\tau_{1}}}{ (x_{1}+\beta)(x_{1}-x_{2})}-\frac{(x_{2}+1/t_{k})e^{\tau_{2}}}{(x_{2}+\beta)(x _{1}-x_{2})}+\frac{e^{-\beta}f(1/t_{k}-\beta)}{(x_{1}+\beta)(x_{2}+\beta)} \Big{]}\\ &+\frac{P_{0}b_{2}(e^{\tau_{1}}-e^{\tau_{2}})}{a_{11}k_{1}(x_{1}-x_{2})}\\ &-Q_{4}(\tau)\end{array}  \tag{35}\]  
where \(Q_{4}(\tau)\) is the expression of Laplace transform inversion of \(f(s)\) (see Equation(34)). When the inner surface pressure of the lining \(q(\tau)\) in the radial direction is defined, \(q(s)\) can be deduced by the Laplace transform about parameter \(\tau\).