# Characteristics of Deep Groundwater Flow and Temperature in the Tertiary Pohang Area, South Korea
## 3 Methods  
### Theoretical Background of Thermal Heat Transfer  
The geothermal flux inside the Earth variably affects the thermal gradient [25]. The transient heat transfer in homogeneous and isotropic saturated porous media can be expressed as follows [26]:  
\[\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+ \frac{\partial^{2}T}{\partial z^{2}}-\frac{c_{0}\rho_{0}}{\kappa}\Bigg{[} \frac{\partial(v_{x}T)}{\partial x}+\frac{\partial\big{(}v_{y}T\big{)}}{ \partial y}+\frac{\partial(v_{z}T)}{\partial z}\Bigg{]}=\frac{c\rho}{\kappa} \frac{\partial T}{\partial t} \tag{1}\]  
where \(T\) denotes the temperature at time \(t\); \(c_{0}\) and \(\rho_{0}\) denote the specific heat and density of the fluid, respectively; \(c\), \(\rho\), and \(\kappa\) denote the specific heat, density, and thermal conductivity of the rock matrix in fluid-filled rocks, respectively; and \(v_{x}\), \(v_{y}\), and \(v_{z}\) denote the fluid velocities in the \(x\)-, \(y\)-, and \(z\)-directions, respectively. In the case of 1D heat conduction in the vertical direction [11], the assumptions are as follows: the borehole is filled with fluid at a steady state and without the influence of drilling and pumping; thermal conductivity is constant at the object section; fluid flow occurs only in the vertical direction; temperature changes in the borehole occur owing to the vertical flow; and the fluid flow rate of the object section is uniform. From Equation (1), the 1D steady-state heat transfer equation for an inclined fracture zone can be expressed as follows [11]:  
\[\frac{\partial^{2}T}{\partial z^{2}}-\frac{c_{0}\rho_{0}}{\kappa}\frac{ \partial(vT)}{\partial z}=0 \tag{2}\]  
where \(v\) is the fluid velocity vector. Equation (1) can be simplified as Equation (4) based on [PERSON]'s assumptions [27], as follows:  
\[\frac{\partial(vT)}{\partial z}\approx\ v_{\xi}\ \gamma \tag{3}\]  
\[\frac{\mathrm{d}^{2}T}{\mathrm{d}z^{2}}-\frac{\alpha\gamma}{L}=0 \tag{4}\]  
where \(v_{\xi}\) denotes the fluid velocity vector in the fracture direction (the direction of the local coordinate \(\xi\)), \(\gamma\) denotes the average magnitude of the temperature gradient in the fracture zone, and \(\alpha\) denotes the Peclet number, equal to \(\frac{c_{0}\rho_{0}v_{z}L}{\kappa}\).  
When the upper and lower boundary temperatures of the aquifer or geological formation are \(T=T_{0}\) (at \(z=0\)) and \(T=T_{L}\) (at \(z=L\)), respectively (Figure 2), Equation (4) can be expressed as follows [27]:  
\[\frac{T_{z}-T_{0}}{T_{L}-T_{0}}=z_{D}+\frac{\beta}{2}z_{D}(z_{D}-1) \tag{5}\]  
where \(L\) denotes the thickness of the fractured aquifer, i.e., the vertical distance between \(z=0\) and \(z=L\). The parameter \(z_{D}\) (dimensionless) is equal to \(z/L\). The parameter \(\beta\) (dimensionless) can be expressed as follows:  
\[\beta=\alpha\sin\!\omega \tag{6}\]where \(\omega\) denotes the average angle of the fracture zone from the horizontal plane. The parameter \(\beta\) has a different sign depending on the direction of the flux: zero in the case of conduction, positive in the case of downward flow, and negative in the case of upward flow.