## 2 Mathematical Modeling
### Governing Equations  
The governing equations of excess pore pressures in the unsaturated PVD foundation under the equal-strain assumption and time-dependent loading are concluded as the following matrix form:  
\[\mathbf{C}\overline{\mathbf{u}}_{i}=-\mathbf{C}_{\mathrm{v}}\Big{(}\mathbf{u }_{rr}+\frac{1}{r}\mathbf{u}_{r}\Big{)}+\mathbf{C}_{\mathrm{q}}\mathbf{q}_{i} \tag{1}\]  
where, \(\mathbf{C}=\left[\begin{array}{cc}1&C_{\mathrm{a}}\\ C_{\mathrm{w}}&1\end{array}\right]\), \(\mathbf{C}_{\mathrm{v}}=\left[\begin{array}{cc}C_{\mathrm{v}}^{\mathrm{a}}&0 \\ 0&C_{\mathrm{v}}^{\mathrm{w}}\end{array}\right]\), \(\mathbf{C}_{\mathrm{q}}=\left[\begin{array}{cc}C_{\mathrm{q}}^{\mathrm{a}}&0 \\ 0&C_{\mathrm{q}}^{\mathrm{w}}\end{array}\right]\), \(\mathbf{u}=\left[\begin{array}{cc}u_{\mathrm{a}}\\ u_{\mathrm{w}}\end{array}\right]\) and \(\mathbf{q}=\left[\begin{array}{cc}q\\ q\end{array}\right]\). \((\bullet)_{r}\) and \((\bullet)_{rr}\) are the first and second derivatives of a function upon \(r\), and \((\bullet)_{t}\) denotes the first derivative of a function upon \(t\). \(u_{\mathrm{a}}\) and \(u_{\mathrm{w}}\) are excess pore-air and water pressures respectively, \(q\) is the external loading, \(C_{\mathrm{a}}\) and \(C_{\mathrm{w}}\), respectively, are the interaction constants of air and water phases, \(C_{\mathrm{v}}^{\mathrm{a}}\) and \(C_{\mathrm{v}}^{\mathrm{w}}\) are the consolidation coefficients of air and water phases and \(\overline{u}_{\mathrm{a}}\) and \(\overline{u}_{\mathrm{w}}\), respectively, are average excess pore-air and water pressures, defined as:  
\[\overline{\mathbf{u}}=\frac{\int_{r_{\mathrm{w}}}^{r_{\mathrm{s}}}\mathbf{u}_ {\mathrm{s}}2\tau rdr+\int_{r_{\mathrm{s}}}^{r_{\mathrm{e}}}\mathbf{u}2\tau rdr} {\pi(r_{\mathrm{e}}^{2}-r_{\mathrm{w}}^{2})} \tag{2}\]  
where, \(\mathbf{u}_{\mathrm{s}}=\left[\begin{array}{cc}u_{\mathrm{a}_{\mathrm{s}}}&u _{\mathrm{w}_{\mathrm{s}}}\end{array}\right]^{\mathrm{T}}\), \(u_{\mathrm{a}_{\mathrm{s}}}\) denotes excess pore-air pressure in the smeared area and \(u_{\mathrm{w}_{\mathrm{s}}}\) is the excess pore-water pressure in the smeared area. The consolidation parameters are expressed as:  
\[C_{\mathrm{a}}=\frac{m_{2}^{\mathrm{a}}(u_{\mathrm{a}}^{0}+u_{ \mathrm{atm}})}{\big{(}m_{1}^{\mathrm{a}}-m_{2}^{\mathrm{a}}\big{)}\big{(}u_{ \mathrm{a}}^{0}+u_{\mathrm{atm}}\big{)}-n(1-S_{\mathrm{r}})} \tag{3a}\] \[C_{\mathrm{v}}^{\mathrm{a}}=\frac{k_{\mathrm{a}}RT}{gM\big{[}\big{(}m_{1}^{ \mathrm{a}}-m_{2}^{\mathrm{a}}\big{)}\big{(}u_{\mathrm{a}}^{0}+u_{\mathrm{atm} }\big{)}-n(1-S_{\mathrm{r}})\big{]}}\] (3b) \[C_{\mathrm{q}}^{\mathrm{a}}=\frac{m_{1}^{\mathrm{a}}\big{(}u_{ \mathrm{a}}^{0}+u_{\mathrm{atm}}\big{)}}{\big{(}m_{1}^{\mathrm{a}}-m_{2}^{ \mathrm{a}}\big{)}\big{(}u_{\mathrm{a}}^{0}+u_{\mathrm{atm}}\big{)}-n(1-S_{ \mathrm{r}})} \tag{3c}\]  
Figure 1: Consolidation modeling of unsaturated PVD foundation under time-dependent loading.  
\[C_{\rm w}=\frac{m_{1}^{\rm w}-m_{2}^{\rm w}}{m_{2}^{\rm w}} \tag{3d}\] \[C_{\rm v}^{\rm w}=\frac{k_{\rm w}}{\gamma_{\rm w}m_{2}^{\rm w}}\] (3e) \[C_{\rm q}^{\rm w}=\frac{m_{1}^{\rm w}}{m_{2}^{\rm w}} \tag{3f}\]  
where, \(m_{1}^{\rm a}\), \(m_{2}^{\rm a}\), \(m_{1}^{\rm w}\) and \(m_{2}^{\rm w}\), respectively, are the variations of net stress (\(q_{0}\)-\(u_{a}\)) and suction (\(u_{a}\)-\(u_{\rm w}\)), \(u_{\rm atm}\) is the atmospheric pressure, and \(u_{\rm atm}\) = 101.3 kPa, \(M\) is the air mass molecular, and \(M\) = 0.029 kg/mol, and \(n_{0}\) and \(S_{\rm r}\) are the initial porosity and saturation, respectively. \(R\) is the universal air constant, \(R\) = 8.314 J/(mol\(\cdot\)K), \(T\) is the absolute temperature, and \(T\) = 293 K, \(g\) is the gravitational acceleration, that is \(g\) = 9.8 m/s\({}^{2}\), and \(\gamma_{\rm w}\) is the unit weight of water, and \(\gamma_{\rm w}\) = 9.8 kN/m\({}^{3}\).