## 2 Coupling Calculation of Temperature and Seepage Field
### Influence of Temperature on Seepage  
For embankment engineering, the temperature variation inside the system affects the physical and chemical parameters of water and porous media, thus affecting the distribution of the seepage field inside the dam body. The main parameters related to temperature distribution in seepage fields include porosity, specific heat capacity, thermal conductivity, and temperature conductivity. These parameters change little within the 10 \({}^{\circ}\)C range and can be regarded as constants [27; 28; 29]. The basic physical and chemical parameters related to temperature and seepage field are density \(\rho\), gravity \(\gamma\), dynamic viscosity \(\mu\), thermal conductivity \(\lambda\), thermal expansion coefficient \(\theta\), thermal diffusivity \(\alpha\), specific heat capacity \(c\), and kinematic viscosity \(v\). Research shows that within the range of temperature variation of 15 \({}^{\circ}\)C, the specific heat, density, thermal conductivity, and thermal diffusivity of water change little, and their effects can be ignored when analyzing the influence of temperature change on the physical and chemical properties of water. The changes in the kinematic viscosity coefficient, dynamic viscosity coefficient, and thermal expansion coefficient of water directly affect the seepage characteristics of water. Therefore, the changes should be taken into account during coupling calculation and analysis. The permeability coefficient of porous medium particles represents the degree of seepage, which is related to the type of porous medium, density, the dynamic viscosity coefficient, and the temperature of the permeable liquid. The Darcy permeability of the porous medium can be expressed as:  
\[k=Cd^{2}\frac{\gamma}{\mu}J \tag{1}\]  
where \(C\) is the factor determined by particle shape and size distribution; \(d\) is the particle diameter (m); \(Cd^{2}\) reflects the permeability of the medium which is only related to medium structure (particle size, shape, and distribution); \(\gamma\) is the water gravity (N\(\cdot\)m\({}^{-3}\)); \(\mu\) is the dynamic viscosity of water (Pa\(\cdot\)s); and \(J\) is the seepage slope. It can be noticed from Equation (1) that the permeability coefficient of the porous medium is inversely proportional to its dynamic viscosity coefficient, and the dynamic viscosity coefficient of the porous medium is affected by temperature. Therefore, the permeability coefficient of the porous medium is a function of temperature. The temperature field affects the distribution of the seepage field by affecting the permeability coefficient on one hand. On the other hand, the temperature potential gradient formed by temperature difference also affects fluid flows. As the temperature potential calculation is a complicated problem, the influence of temperature potential on water flow can only be expressed by an empirical formula of the temperature potential gradient. For one dimension case, it is:  
\[V_{T}=-D_{T}\frac{\partial T}{\partial x} \tag{2}\]  
where \(V_{T}\) is the fluid flux caused by temperature (m\(\cdot\)s\({}^{-1}\)); \(D_{T}\) is the diffusivity under temperature potential which contains the influence of the fluid, the heat expansion coefficient, and the physical and chemical coefficient change of the porous medium (m\(\cdot\)s\({}^{-1}\)); and \(\partial T/\partial x\) is the temperature gradient along x-direction. Thus, the seepage velocity along the x-direction is:  
\[V_{T_{x}}=-K(T)\frac{\partial H}{\partial x}-D_{T}\frac{\partial T}{\partial x} \tag{3}\]  
where \(H\) is the water head (m) and \(T\) is the temperature (\({}^{\circ}\)C).