# 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
In panel (**a**), the solid line represents the steady state while the dashed line is the initial profile. Figure 5: The profiles of pressure and speed with different inlet temperatures (**a1,a2**) and different fracture lengths (**b1,b2**) in an open fracture. The red and blue colors, respectively, represent the pressure and speed profiles. The properties of the rock matrix also play an important role in the flow dynamics of the fracture. Figure 6(a\({}_{1}\)) shows that the pressure profile is straighter and the interior pressure value is lower for a higher rock stiffness. Unlike the properties of the liquid, the stiffness of the rock can change the aperture of the fracture. According to Equation (17), the larger stiffness and lower interior pressure should lead to a smaller aperture, which agrees with the experimental study by [PERSON] et al. [24]. Figure 6(a\({}_{2}\)) shows that both the liquid speed and its gradient decrease with increasing stiffness, suggesting an increase in the resistance. Moreover, the ratio of the outlet speed to the inlet speed also decreases with increasing stiffness as a result of the increasing ratio between the outlet and inlet apertures. Figure 6(b\({}_{1}\)) shows the effect of the permeability on the pressure profile, and indicates that the pressure profile is straighter and the interior pressure is slightly lower at a higher permeability. The change in the speed profile with the permeability is more obvious than that of the pressure profile. As shown in Figure 6(b\({}_{2}\)), the liquid speed at a permeability of \(10^{-16}\) m is higher than that without any permeability. The reason for the higher liquid speed is that the viscosity of the liquid is higher with a finite permeability. According to Equations (12) and (13), the liquid temperature increases when the leakage velocity, \(u_{L}\), increases from zero to a finite value, which then increases the liquid viscosity. As the permeability increases from \(10^{-16}\) m to \(10^{-14}\) m, the inlet speed continuously increases but the outlet speed decreases, due to the increasing volume of liquid leaving through leakage. Figure 7 shows the steady profiles of (a) liquid pressure, (b) fracture aperture, (c) liquid speed, and (d) liquid temperature in a closed fracture with \(u_{\text{out}}=0\). The used parameters are listed in Table 1. Compared with the results shown in Figure 4, the liquid pressure and fracture aperture change slightly from the inlet to the outlet (a,b), and the liquid speed is more than two orders of magnitude lower than that in the open fracture (c). Moreover, the liquid temperature is higher (d) than that in the open fracture. The reason for the higher liquid temperature is the higher ratio of the leakage speed to the liquid speed. In a closed fracture, the speed ratio is more than two orders of magnitudes higher than that in the open fracture. The higher speed ratio leads to the lower value of length scale \(L_{1}\) in Equation (12), thus increasing the effect of the rock matrix temperature on the liquid temperature. Figure 6: The profiles of the pressure and speed with different rock stiffnesses (**a\({}_{1}\),a\({}_{2}\)**) and different rock permeabilities (**b\({}_{1}\),b\({}_{2}\)**) in an open fracture. The red and blue colors, respectively, represent the pressure and speed profiles. Figure 8 further presents the effects of rock properties, including the rock matrix temperature, stiffness, and permeability, on the dynamics and temperature of the liquid in a closed fracture. (a\({}_{1}\),a\({}_{2}\)) show that the pressure gradient and the liquid speed are lower at the lower rock matrix temperature, due to the higher viscosity of the liquid. The lower rock temperature causes a lower liquid temperature (a\({}_{3}\)), and hence a higher liquid viscosity. (b\({}_{1}\)) shows that the interior pressure is higher but the pressure gradient is lower at the lower rock stiffness. The lower pressure gradient leads to a lower liquid speed in the fracture, as shown in (b\({}_{2}\)).