## 3 Difference Solution  
Let  
\[V_{i}=N_{\sigma}-\overline{u}_{i} \tag{16}\]  
By substituting Equation (16) for Equation (10), the consolidation equation and the solution conditions can be further transformed as follows:  
\[\frac{\partial^{2}V_{i}}{\partial Z^{2}}=\frac{c_{i}}{V_{i}}\left(\frac{ \partial V_{i}}{\partial Z}\right)^{2}+\frac{c_{\rm v01}V_{i}^{c_{i}-1}}{c_{ \rm v0i}}\frac{\partial V_{i}}{\partial T_{\rm v}} \tag{17}\]  
The corresponding solution conditions are transformed as follows:  
\[V_{i}(Z,0)=1 \tag{18}\]  
\[V_{1}(0,T_{\rm v})=N_{\sigma}-(N_{\sigma}-1)e^{-\pi T_{\rm v}} \tag{19}\]  
\[V_{2}(1,T_{\rm v})=N_{\sigma}-(N_{\sigma}-1)e^{-\beta T_{\rm v}}\ (\rm Continuous \rm{drainagebottom}) \tag{20a}\] \[\left.\frac{\partial V_{2}}{\partial Z}\right|_{Z=1}=0\ (\rm{Undrainedbottom})\] (20b) \[V_{1}(Z_{\rm f},T_{\rm v})=V_{2}(Z_{\rm f},T_{\rm v}) \tag{21}\]  
\[(V_{1})^{-c_{1}}\left.\frac{\partial V_{1}}{\partial Z}\right|_{Z=Z_{\rm f}}= \left.a(V_{2})^{-c_{2}}\frac{\partial V_{2}}{\partial Z}\right|_{Z=Z_{\rm f}} \tag{22}\]  
The Crank-Nicolson difference scheme is used to discretize Equation (15) into:  
\[ \begin{array}{l}\frac{V_{i,k+1}^{j}-2V_{i,k}^{j}+V_{i,k-1}^{j}+V_{i,k-1}^{j -1}-2V_{i,k+1}^{j-1}+V_{i,k-1}^{j-1}}{2\Delta Z}\\ =\frac{c_{i}}{V_{i,k}^{j-1}}\left(\frac{V_{i,k+1}^{j-1}-V_{i,k-1}^{j-1}}{2 \Delta Z}\right)^{2}+\frac{c_{\rm v01}}{c_{\rm v0}}\left(V_{i,k}^{j-1}\right) ^{c_{i}-1}\frac{V_{i,k}^{j}-V_{i,k}^{j-1}}{\Delta T}\end{array}  \tag{23}\]  
Organized:  
\[ \begin{array}{l}V_{i,k-1}^{j}-2\left[1+\frac{c_{\rm v01}}{\Delta c_{\rm v0i }}\left(V_{i,k}^{j-1}\right)^{c_{i}-1}\right]V_{i,k}^{j}+V_{i,k+1}^{j}\\ =-V_{i,k+1}^{j-1}+2\left[1-\frac{c_{\rm v01}}{\Delta c_{\rm v0i}}\left(V_{i,k} ^{j-1}\right)^{c_{i}-1}\right]V_{i,k}^{j-1}\\ \qquad\quad-V_{i,k-1}^{j-1}+\frac{c_{i}}{2V_{i,k}^{j-1}}\left(V_{i,k+1}^{j-1}- V_{i,k-1}^{j-1}\right)^{2}\end{array}  \tag{24}\]  
where \(\Delta Z\) is the space step; \(\Delta T\) is the time step; \(\lambda=\frac{\Delta T}{(\Delta Z)^{2}}\); \(k=1,2,3,\cdots n\). represents the number of space nodes; \(j=1,2,3,\cdots N\). represents the number of time nodes. The solution conditions:  
\[V_{i,k}^{0}=1\ i=0,1,2,\cdots,n. \tag{25}\]  
\[V_{1,0}^{j}=N_{\sigma}-(N_{\sigma}-1)e^{-\alpha\Delta T(j-1)}\ j=1,2,\cdots,N. \tag{26}\]  
\[V_{2,n}^{j}=N_{\sigma}-(N_{\sigma}-1)e^{-\beta\Delta T(j-1)}\ (\text{Continuousdrainagebottom}) \tag{27a}\]  
\[V_{2,n+1}^{j}=V_{2,n-1}^{j}\ (\text{Undrainedbottom}) \tag{27b}\]  
\[V_{1,n_{1}}^{j}=V_{2,n_{1}}^{j} \tag{28}\]  
\[(V_{1,n_{1}}^{j-1})^{-c_{1}}(V_{1,n_{1}}^{j}-V_{1,n_{1}-1}^{j})=a(V_{2,n_{1}}^ {j-1})^{-c_{2}}(V_{2,n_{1}+1}^{j}-V_{2,n_{1}}^{j}) \tag{29}\]  
In combination with the solution conditions, Equation (20) is expressed in the form of the following matrix:  
\[A^{j-1}V^{j}=B^{j-1} \tag{30}\]  
For double-sided continuous drainage (Continuous drainage bottom):  
\[A_{l,l}=-2\bigg{[}1+\frac{c_{v01}}{\lambda c_{v0i}}(V_{i,k}^{j-1})^{c-1}\bigg{]} \tag{31}\]  
\[A_{l,l-1}=A_{l,l+1}=1 \tag{32}\]  
where \(l=1,2,\cdots,n_{1}-1,n_{1}+1,\cdots,n-1\). \[A_{n_{1},n_{1}}=-(aV_{n_{1}}^{-c_{2}}+V_{n_{1}}^{-c_{1}}) \tag{33}\]  
\[V_{1,0}^{j}=N_{\sigma}-(N_{\sigma}-1)e^{-\alpha\Delta T(j-1)}\ j=1,2,\cdots,N. \tag{26}\]0  
\[V_{1,0}^{j}=N_{\sigma}-(N_{\sigma}-1)e^{-\alpha\Delta T(j-1)}\ j=1,2,\cdots,N. \tag{26}\]1  
\[V_{1,0}^{j}=N_{\sigma}-(N_{\sigma}-1)e^{-\alpha\Delta T(j-1)}\ j=1,2,\cdots,N. \tag{26}\]2  
\[V_{1,0}^{j}=N_{\sigma}-(N_{\sigma}-1)e^{-\alpha\Delta T(j-1)}\ j=1,2,\cdots,N. \tag{26}\]3  
\[V_{1,0}^{j}=N_{\sigma}-(N_{\sigma}-1)e^{-\alpha\Delta T(j-1)}\ j=1,2,\cdots,N. \tag{26}\]4  
where \(l=2,\cdots,n_{1}-1,n_{1}+1,\cdots,n-1\). \[ \begin{split} B_{1}&=-V_{1,2}^{j-1}+2\bigg{[}1- \frac{1}{\lambda}(V_{1,1}^{j-1})^{c-1}\bigg{]}V_{1,1}^{j-1}-V_{1,0}^{j-1}\\ &\quad+\frac{c_{1}}{2V_{1,1}^{j-1}}(V_{1,2}^{j-1}-V_{1,0}^{j-1}) ^{2}-V_{1,0}^{j}\end{split}  \tag{39}\]  
\[B_{n_{1}}=0 \tag{40}\]\[B_{n}=V_{2,n}^{j} \tag{41}\]  
The expressions for other coefficients A and B are the same as those for double-sided drainage conditions.