## 2 Coupling Calculation of Temperature and Seepage Field
### Coupling Analysis of Temperature Field and Seepage Field  
According to the analysis above, the control functions for the seepage field and temperature field coupling problem are:  
\[\left\{ \begin{matrix}a\
abla^{2}T+kb\big{\{}\
abla H\
abla T+T\
abla^{2}H\big{\}} =0\\ \
abla(K\
abla H)+\
abla(D_{T}\
abla T)=S_{s}\frac{\partial H}{\partial t} \end{matrix} \right. \tag{15}\]  
The temperature field and seepage field interact and influence each other in embankment engineering. Theoretically, there is an analytical solution that can simultaneously satisfy Equation (15), but it is difficult to obtain in most cases. To obtain a numerical solution, the one-dimensional problem is analyzed and discussed first. Assuming that the permeability coefficient \(K\) and water diffusivity \(D_{T}\) are constants, thermal conductivity \(\lambda\) is also a constant and does not change with temperature. The source and sink terms are not considered, and then Equation (15) can be rewritten as:  
\[\left\{ \begin{array}{ll}K\frac{d^{2}H}{dx^{2}}+D_{T}\frac{d^{2}T}{dx^{2}}=0 \\ a\frac{d^{2}T}{dx^{2}}+kb(\frac{dH}{dx}\frac{dT}{dx}+T\frac{d^{2}H}{dx^{2}})= 0\end{array} \right. \tag{16}\]  
The boundary conditions are:  
\[\left\{ \begin{matrix}H(0)=H_{1},H(l)=H_{2}&x\in[0,l]\\ T(0)=T_{1},T(l)=T_{2}&x\in[0,l]\end{matrix} \right.\]  
First of all, neglecting the influence of the seepage field on the temperature field sets \(k=0\) in Equation (16). The temperature distribution calculated by Equation (16) is:  
\[T_{0}=\frac{T_{2}-T_{1}}{l}x+T_{1} \tag{17}\]  
Substituting Equation (17) into Equation (16):  
\[H_{0}=\frac{H_{2}-H_{1}}{l}x+H_{1} \tag{18}\]  
\[a\frac{d^{2}T}{dx^{2}}+kb\bigg{(}\frac{H_{2}-H_{1}}{l}\frac{dT}{dx}\bigg{)}=0 \tag{19}\]  
Solving Equation (19), we can get:  
\[T_{1}(x)=\frac{1}{e^{\text{i}d}-1}\Big{[}(T_{2}-T_{1})e^{xx}+\Big{(}T_{1}e^{ \text{i}d}-T_{2}\Big{)}\Big{]} \tag{20}\]  
where \(\alpha=-[kb(H_{2}-H_{1})]/al\). Resubstituting \(T_{1}(x)\) into Equations (2)-(16):  
\[K\frac{d^{2}H}{dx^{2}}+D_{T}\frac{(T_{2}-T_{1})a^{2}}{e^{\text{i}d}-1}e^{xx}=0 \tag{21}\]With the boundary conditions, the solution of \(H_{1}(x)\) is:  
\[H_{1}(x)=\beta\frac{T_{2}-T_{1}}{e^{\mu l}-1}(1-e^{ax})+\frac{1}{l}[(H_{2}-H_{1})+ \beta(T_{2}-T_{1})]x+H_{1} \tag{22}\]  
where \(\beta=D_{T}/K\). Though the solutions of Equations (20) and (22) are not the analytical solutions of Equation (15), they satisfy the boundary conditions and could partially reflect the coupling effect. Thus, the approximate solution of Equation (15) is:  
\[T_{1}(x)=\frac{T_{2}-T_{1}}{l}x+T_{1}+(T_{2}-T_{1})\bigg{(}\frac{e^{ax}-1}{e^{ \mu l}-1}-\frac{x}{l}\bigg{)}=T_{0}(x)+T_{H}(x) \tag{23}\]  
\[H_{1}(x)=\frac{H_{2}-H_{1}}{l}x+H_{1}+\beta\frac{T_{2}-T_{1}}{e^{\mu l}-1}(1-e^ {ax})+\frac{1}{l}\beta(T_{2}-T_{1})x=H_{0}(x)+H_{T}(x) \tag{24}\]  
where \(T_{0}(x)\) is the temperature distribution caused by the gradient (\({}^{\circ}\)C); \(T_{H}(x)\) is the temperature distribution caused by the water gravity (\({}^{\circ}\)C); \(H_{0}(x)\) is the water head distribution caused by the gravity potential (m); and \(H_{T}(x)\) is the water head distribution caused by the temperature potential difference (m). For simplification, choosing embankment length and water head of upstream and downstream as \(l=100\) m, \(T_{2}=0\)\({}^{\circ}\)C, \(T_{1}=100\)\({}^{\circ}\)C, \(H_{1}=100\) m, \(H_{2}=0\), the physical and thermal parameters of the water and the porous medium are chosen as standard numbers: \(c_{w}=4.19\) KJ/(kg\(\cdot\)\({}^{\circ}\)C), \(\rho_{w}=10^{3}\) kg/m\({}^{3}\), \(D_{T}=1.01\times 10^{-12}\) m\({}^{2}\)/(s\(\cdot\)\({}^{\circ}\)C), \(\lambda=2.575\times 10^{-3}\) KJ/(m\(\cdot\)s\(\cdot\)\({}^{\circ}\)C). The permeability coefficients are chosen as follows: \(K=10^{-9}\) m/s, \(K=10^{-8}\) m/s, \(K=10^{-7}\) m/s and \(K=10^{-6}\) m/s for four different conditions to calculate \(\alpha\) and \(\beta\), and then calculate \(T_{0}(x)\), \(T_{1}(x)\), \(H_{0}(x)\) and \(H_{1}(x)\) relatively.