# A Study on the Heat Transfer of Surrounding Rock-Supporting Structures in High-Geothermal Tunnels
## 2 Establishment of Finite Difference Model
### Algebraic Equations of Physical Quantities of Nodes  
#### 2.3.1 Discrete Equations for Internal Nodes  
(1) Discretization of diffusion terms  
Take the \(i\)-th time-layer diffusion term node (n, i) as an example, the difference expression is derived. Based on node (n + 1, i) and node (n - 1, i), the Taylor series expansion of function (t) to node (n, i) is obtained [27].  
\[t_{n+1}^{i}=t_{n}^{i}+\Delta r\frac{\partial t}{\partial r}\bigg{|}_{n,i}+ \frac{\Delta r^{2}}{2}\frac{\partial^{2}t}{\partial r^{2}}\Big{|}_{n,i}+\frac{ \Delta r^{3}}{6}\frac{\partial^{3}t}{\partial r^{3}}\Big{|}_{n,i}+\frac{\Delta r ^{4}}{24}\frac{\partial^{4}t}{\partial r^{4}}\Big{|}_{n,i}+\cdots \tag{6}\]  
\[t_{n-1}^{i}=t_{n}^{i}-\Delta r\frac{\partial t}{\partial r}\bigg{|}_{n,i}+ \frac{\Delta r^{2}}{2}\frac{\partial^{2}t}{\partial r^{2}}\Big{|}_{n,i}-\frac{ \Delta r^{3}}{6}\frac{\partial^{3}t}{\partial r^{3}}\Big{|}_{n,i}+\frac{ \Delta r^{4}}{24}\frac{\partial^{4}t}{\partial r^{4}}\Big{|}_{n,i}+\cdots \tag{7}\]  
The two equations are added and rewritten as an expression as follows:  
\[\frac{\partial^{2}t}{\partial r^{2}}\Big{|}_{n,i}=\frac{t_{n+1}^{i}-2t_{n}^{i }+t_{n-1}^{i}}{\partial r^{2}}+\mathcal{O}\Big{(}\Delta r^{2}\Big{)} \tag{8}\]  
where \(\mathcal{O}\Big{(}\Delta r^{2}\Big{)}\) represents the \(\Delta r\) whose minimum order is 2 in the remainder term. After omitting \(\mathcal{O}\Big{(}\Delta r^{2}\Big{)}\), the difference expression for the second derivative of the \(i\)-th time-layer diffusion node is obtained.  
\[\frac{\partial^{2}t}{\partial r^{2}}\Big{|}_{n,i}=\frac{t_{n+1}^{i}-2t_{n}^{i }+t_{n-1}^{i}}{\partial r^{2}} \tag{9}\]  
Similarly, we obtain the difference expression of the first derivative of the \(i\)-th time-layer diffusion term.  
\[\frac{\partial t}{\partial r}\Big{|}_{n,i}=\frac{t_{n+1}^{i}-t_{n-1}^{i}}{2 \Delta r} \tag{10}\]  
(2) Discretization of unsteady terms  
The Taylor series expansion of function (t) to node (n, i) is obtained.  
\[\frac{\partial t}{\partial t}\Big{|}_{n,i}=\frac{t_{n}^{i+1}-t_{n}^{i}}{\Delta r }+\mathcal{O}(\Delta\tau) \tag{11}\]  
Figure 1: Time–space regional discretization of one-dimensional unsteady heat conduction. Where \(r\) is the space coordinate and \(\Delta r\) is the space step. \(\tau\) represents the time coordinate and \(\Delta\tau\) is the time step. (n, i) is the position of a node on the time–space region, and the corresponding temperature is recorded as \(t_{n}^{i}\).  
where \(O(\Delta\tau)\) represents the \(\Delta\tau\) whose minimum order is in the remainder term. After omitting \(O(\Delta\tau)\), the difference expression for \(\frac{\partial t}{\partial\tau}\) is obtained.  
\[\frac{\partial t}{\partial\tau}\Big{|}_{n,i}=\frac{t_{n}^{i+1}-t_{n}^{i}}{ \Delta\tau} \tag{12}\]  
(3) Discrete equations of inner nodes  
Substituting Expressions (2), (3) and (4) into Expression (1), the discrete equation of inner nodes is obtained.  
\[t_{n}^{i+1}=\frac{F_{O}}{2}\left[\Big{(}2+\frac{\Delta\tau}{\tau_{n}}\Big{)}t_ {n+1}^{i}+\Big{(}2-\frac{\Delta\tau}{\tau_{n}}\Big{)}t_{n-1}^{i}+\left(\frac{2 }{F_{O}}-4\right)t_{n}^{i}\right] \tag{13}\]  
where \(F_{O}\) is the grid Fourier number \(F_{O}=\frac{\omega\Delta\tau}{\Delta\tau^{2}}\). \(\tau_{n}\) is the distance from the calculating point to the axis of the tunnel. \(t_{n}^{i}\) is the temperature of node \(n\) at time \(i\).