# A Model for Simulating the Upward Flow of a Viscous Fluid in a Fracture Network
## 4 The Upward Flow in a Fracture Network
### Result  
In this study, we focus on the dynamics of the upward flow in a 3-level fracture network, as shown in Figure 18a. At first, we assume the fracture network to be homogeneous, where all seven channels have the same parameters as listed in Table 1. The total length of the fracture is 50 m and all channels have same length. All channels on level 2 have open outlets. Figure 18b compares the pressure profile of the 3-level fracture network to that of the single and bifurcated fractures, suggesting that the interior pressure decreases with an increasing number of levels. Moreover, the inlet pressure gradient is higher while the outlet pressure gradient is lower in the fracture with more levels. As a result, the inlet speed of the fracture increases with the number of levels, whereas the outlet speed decreases, as shown in Figure 18c.  
We also investigate the heterogeneous fracture network, as shown in Figure 19a, where a channel on level 2 (i.e., channel 1-2-1) has different parameters from the others. Figure 19(b\({}_{\text{1}}\)) shows that the reduced stiffness of channel 1-2-1 decreases the interior pressure in all other channels on level 2. The interior pressure decreases more in channel 1-2-2 than in channels 2-2-1 and 2-2-2. In agreement with the cases of the single and bifurcated fractures, the speed in channel 1-2-1 increases with the reduced stiffness, as shown in Figure 19(b\({}_{\text{2}}\)). The increased inlet speed of channel 1-2-1 leads to more liquid flowing in, thus decreasing the speed in the other channels on level 2. Similarly to the interior pressure, the speed decreases more in channel 1-2-2 than in channels 2-2-1 and 2-2-2. Figure 19(c\({}_{\text{1}}\),c\({}_{\text{2}}\)) shows the effect of the increased permeability of channel 1-2-1 on the pressure and speed profiles of the channels on level 2, suggesting that the increasedpermeability decreases both the interior pressure and speed. Again, the degree of the decrease is larger in channel 1-2-2 than in channels 2-2-1 and 2-2-2. The above results indicate that changing the parameters in a channel leads to a greater effect on the flow dynamics in the closer channels.  
Next, we switch the outlet of channel 1-2-1 from open to closed and set the propagating speed, \(u_{\text{out}}\), to be zero, to study the effect of the heterogeneous boundary condition on the flow dynamics. Figure 20a shows that the closed outlet of channel 1-2-1 increases the interior pressure of all channels on level 2, and the interior pressure of channel 1-2-2 increases more than the pressures of channels 2-2-1 and 2-2-2. Figure 20b shows that the speed in closed channel 1-2-1 almost decreases to zero, and thus more liquid flows into other channels, increasing the speed there. Similarly to the interior pressure, the speed increases more in channel 1-2-2 than in channels 2-2-1 and 2-2-2.  
Figure 19: (**a**) illustrates the 3-level fracture network in which the parameters of channel 1-2-1 (red line) are different from those of the other channels. (**b**,**b2**), respectively, show the pressure and speed profiles in the channels on level 2 when the stiffness of channel 1-2-1 is at a lower value than that of other channels (4 \(\times\) 10\({}^{9}\) Pa/m). (**c1**,**c2**), respectively, show the pressure and speed profiles in the channels on level 2 when the permeability of channel 1-2-1 is at a higher value than that of other channels (1 \(\times\) 10\({}^{-13}\) m). For comparison, we plot the profiles of the homogeneous fracture network as light lines.  
Figure 18: (**a**) illustrates the 3-level fracture network with homogeneous parameters and open outlets. (**b**,**c**) show the profiles of the pressure and speed in the fracture network, respectively. For comparison, the profiles of the pressure and speed in the single (dashed-0 dotted lines) and bifurcated (dashed lines) fracture are also plotted in (**b**,**c**).