# A Model for Simulating the Upward Flow of a Viscous Fluid in a Fracture Network
## 2 The Upward Flow of a Viscous Liquid in a Single Fracture
### Result  
After validating our model, we employ it to investigate the upward flow of the viscous fluid in a single fracture. All the input parameters of the fracture, liquid, and rock matrix used in the computation are listed in Table 1, which are proposed in [18] and validated by the field works in [36; 37]. We solve the governing equation on 100 grids in the vertical coordinate, so that the grid size is 0.5 m.  
Figure 4 shows the profiles of (a) the liquid pressure, (b) the fracture aperture, (c) the liquid velocity, and (d) the liquid temperature in an open fracture in the steady state. The steady values of the liquid pressure and fracture aperture decrease along the fracture length. As the flux is constant through the fracture based on mass conservation, the liquid speed increases from the inlet to the outlet, which leads to the increasing pressure gradient as shown in (a). According to the positive correlation between the pressure and the leakage speed (Equation (6)), the leakage speed decreases in the vertical direction, along with the pressure. In addition, the leakage speed is around \(10^{-9}\) m/s in this case, approximately eight orders of magnitude lower than the liquid speed. The large difference between the liquid speed and the leakage speed causes a large value of the length scale, \(L_{1}\), in Equation (13), making the difference between the liquid temperature and inlet temperature small (according to Equation (12)). Figure 4d shows that the liquid temperature is very close to the inlet temperature, although it slightly increases along the fracture length, suggesting that the temperature of the rock matrix is not important in an open fracture. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Value** \\ \hline Fracture length & \(L\) & m & 50 \\ Original fracture aperture & \(h_{o}\) & m & 0.001 \\ Original pressure & \(p_{o}\) & Pa & \(2\times 10^{7}\) \\ Inlet pressure & \(p_{\text{in}}\) & Pa & \(3\times 10^{7}\) \\ Fracture stiffness & \(k_{n}\) & Pa/m & \(2\times 10^{10}\) \\ Permeability per thickness & \(\hat{k}_{p}\) & m & \(1\times 10^{-16}\) \\ Liquid density & \(\rho\) & kg/m\({}^{3}\) & 1000 \\ Liquid specific heat & \(C\) & J/(kg\(\cdot\)K) & 4199.2 \\ Reference viscosity module & \(\mu_{\text{ref}}\) & Pa s & 0.2 \\ Rock density & \(\rho_{m}\) & kg/m\({}^{3}\) & 2650 \\ Rock specific heat & \(C_{m}\) & J/(kg\(\cdot\)K) & 1003.8 \\ Rock thermal conductivity & \(K_{m}\) & W/(m\(\cdot\)K) & 5.23 \\ Inlet temperature & \(T_{\text{in}}\) & \({}^{\circ}\)C & 40 \\ Rock matrix temperature & \(T_{m}\) & \({}^{\circ}\)C & 100 \\ Reference temperature & \(T_{\text{ref}}\) & \({}^{\circ}\)C & 20 \\ Arrhenius activation energy & \(\Delta E_{T}\) & J/(g\(\cdot\)mol) & 20,091.84 \\ Reservoir porosity & \(\phi\) & - & 0.1 \\ Universal gas constant & \(R\) & J/(g\(\cdot\)mol\(\cdot\)K) & 8.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic input parameters for the model. The inlet temperature decides the liquid temperature, hence also decides the liquid viscosity in the fracture. Figure 5(a1,a2) shows that, although the inlet temperature does not obviously affect the pressure profile when it increases from 40 \({}^{\circ}\)C to 90 \({}^{\circ}\)C (a1), it increases the liquid speed due to the liquid viscosity decreases (a2). However, the ratio of the outlet speed to the inlet speed approximately keeps constant as 1.5, because the speed ratio relates to the aperture ratio and the inlet temperature never alters the fracture aperture at the inlet and outlet. The fracture length also does not change the inlet and outlet apertures, but it does alter the pressure profile in the fracture. Figure 5(b1) shows that the pressure gradient is lower in the longer fracture. Therefore, the liquid speed in the longer fracture is lower, as shown in Figure 5(b2). Figure 4: (**a**) The pressure profile, (**b**) the fracture aperture profile, (**c**) the velocity profile, and (**d**) the temperature profile in an open fracture.