## 2 Governing Equations
### Laplace Transform and Governing Equations
's study [29]):  
\[s\widetilde{u}_{\mathrm{a}}-u_{\mathrm{a}}^{0}=-s\widetilde{C}_{\mathrm{a}} \widetilde{u}_{\mathrm{w}}+\widetilde{C}_{\mathrm{a}}u_{\mathrm{w}}^{0}- \widetilde{C}_{\mathrm{v}}^{\mathrm{a}}\frac{\partial^{2}\widetilde{u}_{ \mathrm{a}}}{\partial z^{2}}+\frac{\widetilde{D}_{\mathrm{a}}}{s+b} \tag{6}\]  
Substituting Equation (3) into Equation (6) results in  
\[\frac{\partial\widetilde{J}_{\mathrm{a}}}{\partial z}=s\frac{k_{\mathrm{a}}}{ \widetilde{C}_{\mathrm{a}}g}\widetilde{u}_{\mathrm{a}}+s\frac{k_{\mathrm{a}}}{ \widetilde{C}_{\mathrm{v}}g}\widetilde{C}_{\mathrm{a}}\widetilde{u}_{\mathrm{ w}}-\frac{k_{\mathrm{a}}}{\widetilde{C}_{\mathrm{v}}g}\left(u_{\mathrm{a}}^{0}+ \widetilde{C}_{\mathrm{a}}u_{\mathrm{w}}^{0}+\frac{\widetilde{D}_{\mathrm{a}} }{s+b}\right) \tag{7}\]  
where \(\widetilde{C}_{\mathrm{a}}=\widetilde{m}_{2}^{\mathrm{a}}/\left[\widetilde{ m}_{1\mathrm{k}}^{\mathrm{a}}-\widetilde{m}_{2}^{\mathrm{a}}-u_{\mathrm{ atm}}n_{0}(1-S_{\mathrm{r}0})/\left(\overline{u}_{\mathrm{a}}^{0}\right)^{2}\right]\);  
\[\widetilde{C}_{\mathrm{v}}^{\mathrm{a}}=k_{\mathrm{a}}RT/\left\{g\overline{ u}_{\mathrm{a}}^{0}M\left[\widetilde{m}_{1\mathrm{k}}^{\mathrm{a}}- \widetilde{m}_{2}^{\mathrm{a}}-u_{\mathrm{atm}}n_{0}(1-S_{\mathrm{r}0})/ \left(\overline{u}_{\mathrm{a}}^{0}\right)^{2}\right]\right\}\]  
\[\widetilde{D}_{\mathrm{a}}=m_{1\mathrm{k}}^{\mathrm{a}}q_{0}b/\left[\widetilde{ m}_{1\mathrm{k}}^{\mathrm{a}}-\widetilde{m}_{2}^{\mathrm{a}}-u_{\mathrm{atm}}n_{0}(1-S_{ \mathrm{r}0})/\left(\overline{u}_{\mathrm{a}}^{0}\right)^{2}\right]\]  
\[\mathrm{and}\ \widetilde{m}_{1\mathrm{k}}^{\mathrm{a}}=-1/E_{0\mathrm{n}}^{ \mathrm{a}}-1/(E_{\mathrm{n}}^{\mathrm{a}}+\eta_{\mathrm{n}}^{\mathrm{a}}s),\ \widetilde{m}_{2}^{\mathrm{a}}=-1/E_{0\mathrm{b}}^{\mathrm{a}}-1/(E_{1 \mathrm{s}}^{\mathrm{a}}+\eta_{\mathrm{s}}^{\mathrm{a}}s).\]  
In the above equations, \(\widetilde{C}_{\mathrm{a}}\) is the interaction constant related to the air phase in viscoelastic foundation; \(\widetilde{C}_{\mathrm{v}}^{\mathrm{a}}\) is the consolidation coefficient associated with air phase in viscoelastic foundation; \(\widetilde{m}_{1\text{k}}^{\text{a}}\) is the volume change coefficient of the air phase skeleton corresponding to the net normal stress (\(\sigma-u_{a}\)) when the viscoelastic foundation \(K_{0}\) is loaded; \(\widetilde{m}_{2}^{\text{a}}\) is the air phase volume change coefficient corresponding to matric suction (\(u_{\text{a}}-u_{\text{w}}\)) when loaded on the viscoelastic foundation \(K_{0}\); \(R\) represents the universal air constant, and \(R=8.314\)\(J/(\text{mol-K})\); \(u_{\text{atm}}\) represents the atmospheric pressure; \(n_{0}\) represents the incipient porosity and \(S_{\text{r0}}\) is the incipient degree of saturation; \(\eta_{\text{n}}^{\text{a}}\) and \(\eta_{\text{s}}^{\text{a}}\) represent the viscous coefficients of the air phase accounting for variations in net normal stress and matric suction; T represents the absolute temperature; \(E_{1j}^{\text{a}}/E_{0j}^{\text{a}}\) are the stiffness coefficients of air phase; and the subscript \(j\) (=\(n\) or \(s\)) represents the net normal stress or matric suction, respectively. In addition, the subscripts 0 and 1 of \(E_{1j}^{\text{a}}/E_{0j}^{\text{a}}\) represent the elastic body \(L_{0}\) and Kelvin body, respectively. Moreover, considering exponential loading, the governing equation of the water phase for 1D consolidation of unsaturated soils [38] can be formulated as detailed below:  
\[\frac{\partial u_{\text{w}}}{\partial t}=-C_{\text{w}}\frac{\partial u_{ \text{a}}}{\partial t}-C_{\text{v}}^{\text{w}}\frac{\partial^{2}u_{\text{w}}} {\partial z^{2}}+D_{\text{w}}\text{e}^{-bt} \tag{8}\]  
Implementing the Laplace transform and [PERSON]'s correspondence principle in Equation (8) leads to the following equation (the details are shown in [PERSON] et al.