# One-Dimensional Consolidation of Multi-Layered Unsaturated Soil with Impeded Drainage Boundaries
## 3 Solution Formulations  
In the Laplace domain, the excess pore pressure vector can be defined as  
\[\overline{{\bf u}}^{(i)}(z,s)=\int_{0}^{\infty}{\bf u}^{(i)}(z,t)e^{-st}dt \tag{10}\]  
where \(s\) is the Laplace transform parameter. Using the initial conditions (4) to perform the Laplace transform of the system (1) leads to:  
\[{\bf W}^{(i)}\overline{{\bf u}}_{zz}^{(i)}-s\overline{{\bf u}}^{(i)}+\overline {{\bf F}}^{(i)}=0 \tag{11}\]where:  
\[\textbf{W}^{(i)}=-\Big{[}\textbf{C}^{(i)}\Big{]}^{-1}\textbf{K}^{(i)},\ \widetilde{\textbf{F}}^{(i)}=\Big{[}\textbf{C}^{(i)}\Big{]}^{-1}\Big{[}s \widetilde{\textbf{Q}}^{(i)}-\textbf{Q}_{0}^{(i)}\Big{]}+\textbf{u}_{0}^{(i)} \tag{12}\]  
Note that the matrix \(\textbf{W}^{(i)}\) has the following diagonalized form:  
\[\textbf{W}^{(i)}=\textbf{\theta}^{(i)}\boldsymbol{\Lambda}^{(i)}\Big{[} \textbf{\theta}^{(i)}\Big{]}^{-1} \tag{13}\]  
with:  
\[ \begin{array}{c}\textbf{\theta}^{(i)}=\left[\begin{array}{cc}1&\textbf{ \theta}_{12}^{(i)}\\ \textbf{\theta}_{21}^{(i)}&1\end{array}\right],\ \boldsymbol{\Lambda}^{(i)}=\left[ \begin{array}{cc}\Lambda_{1}^{(i)}&0\\ 0&\Lambda_{2}^{(i)}\end{array}\right],\\ \Lambda_{1,2}^{(i)}=\Big{\{}2\Big{[}C_{\text{a}}^{(i)}C_{\text{w}}^{(i)}-1 \Big{\}}\Big{]}^{-1}\Big{\{}c_{\text{v}}^{(i)}+c_{\text{v}}^{\text{w}(i)}\pm \Big{[}\psi^{(i)}\Big{]}^{\frac{1}{2}}\Big{\}}\\ \psi^{(i)}=\Big{[}c_{\text{v}}^{\text{a}(i)}-c_{\text{v}}^{\text{w}(i)}\Big{]} ^{2}+4c_{\text{a}}^{(i)}C_{\text{w}}^{(i)}c_{\text{v}}^{\text{a}(i)},\ \psi^{(i)},\ \theta_{12}^{ (i)}=C_{\text{a}}^{(i)}c_{\text{v}}^{\text{w}(i)}\Big{\{}\Big{[}C_{\text{a}}^ {(i)}C_{\text{w}}^{(i)}-1\Big{]}\Lambda_{2}^{(i)}-c_{\text{v}}^{\text{a}(i)} \Big{\}}^{-1}\\ \theta_{21}^{(i)}=C_{\text{w}}^{(i)}c_{\text{v}}^{\text{a}(i)}\Big{\{}\Big{[}C_ {\text{a}}^{(i)}C_{\text{w}}^{(i)}-1\Big{]}\Lambda_{1}^{(i)}-c_{\text{v}}^{ \text{w}(i)}\Big{\}}^{-1}\end{array}  \tag{14}\]  
Reformulate the system (11) using Equation (13) as  
\[\Big{[}\textbf{\theta}^{(i)}\Big{]}^{-1}\widetilde{\textbf{u}}^{(i)}_{zz}-s \Big{[}\boldsymbol{\Lambda}^{(i)}\Big{]}^{-1}\Big{[}\textbf{\theta}^{(i)} \Big{]}^{-1}\widetilde{\textbf{u}}^{(i)}+\Big{[}\boldsymbol{\Lambda}^{(i)} \Big{]}^{-1}\Big{[}\textbf{\theta}^{(i)}\Big{]}^{-1}\widetilde{\textbf{F}}^{( i)}=0 \tag{15}\]  
This system is diagonal for the vector of \(\Big{[}\textbf{\theta}^{(i)}\Big{]}^{-1}\widetilde{\textbf{u}}^{(i)}\) and has the general solution:  
\[\widetilde{\textbf{u}}^{(i)}=\textbf{\theta}^{(i)}\Big{[}\textbf{E}_{1}^{(i)} (z,s)\textbf{A}_{1}^{(i)}+\textbf{E}_{2}^{(i)}(z,s)\textbf{A}_{2}^{(i)}\Big{]} +s^{-1}\widetilde{\textbf{F}}^{(i)} \tag{16}\]  
with:  
\[\textbf{E}_{1}^{(i)}(z,s)=\Big{[}\textbf{E}_{2}^{(i)}(z,s)\Big{]}^{-1}=\left[ \begin{array}{cc}e^{\sqrt{s/\Lambda_{1}^{(i)}}(z-z_{i-1})}&0\\ 0&e^{\sqrt{s/\Lambda_{2}^{(i)}}(z-z_{i-1})}\end{array} \right] \tag{17}\]  
where \(\textbf{A}_{1}^{(i)}\) and \(\textbf{A}_{2}^{(i)}\) are the two coefficient vectors for the \(i\)th soil layer. The Laplace transforms for the boundary conditions (7) and continuity conditions (8) are:  
\[\widetilde{\textbf{u}}^{(1)}_{z}(0,s)-h^{-1}\textbf{R}_{\text{t}}\widetilde{ \textbf{u}}^{(1)}(0,s)=0,\ \widetilde{\textbf{u}}^{(n)}_{z}(h,s)+h^{-1}\textbf{R}_{\text{b}}\widetilde{ \textbf{u}}^{(n)}(h,s)=0 \tag{18}\]  
\[\widetilde{\textbf{u}}^{(\ell)}(z_{\ell},s)=\widetilde{\textbf{u}}^{(\ell+1)}( z_{\ell},s),\ \textbf{k}^{(\ell)}\widetilde{\textbf{u}}^{(\ell)}_{,z}(z_{\ell},s)=\textbf{k}^{( \ell+1)}\widetilde{\textbf{u}}^{(\ell+1)}_{,z}(z_{\ell},s) \tag{19}\]  
Substituting Equation (16) into these equations further produces:  
\[\left[ \begin{array}{cccccccc}\textbf{T}^{(1)}&&&&\\ \boldsymbol{\Sigma}^{(1)}&\boldsymbol{\Gamma}^{(2)}&&&&\\ &\ddots&\ddots&&&&\\ &&&\boldsymbol{\Xi}^{(i-1)}&\boldsymbol{\Gamma}^{(i)}&&\\ &&&\boldsymbol{\Xi}^{(i)}&\boldsymbol{\Gamma}^{(i+1)}&&\\ &&&&\ddots&\ddots&\\ &&&&\boldsymbol{\Xi}^{(n-1)}&\boldsymbol{\Gamma}^{(n)}&\boldsymbol{\mathbb{B}}^ {(n)}\end{array} \right]_{4t\times 4t\times 4t}\left\{