# A Model for Simulating the Upward Flow of a Viscous Fluid in a Fracture Network
## 3 The Upward Flow of Liquid in a Bifurcated Fracture
### Results  
We first evaluate the role of the rock stiffness on the flow in the open bifurcated fracture with the boundary condition at the outlets defined by Equation (28). We assume the stiffness of the branches (channels 1 and 2) to be the same as shown in Figure 12(a1-a3), which shows the effect of the changing stiffness from the main channel (channel 0) to the branches (channels 1 and 2) on the pressure and speed profiles. Compared with the single fracture (as shown in Figure 4), both the pressure gradient and speed vary sharply at the connection of the channels in the bifurcated fracture due to the flow diversion. If the stiffness of the branches is higher than that of the main channel, the pressure at the channel connection increases ((a2)), but the inlet and outlet speeds of the bifurcated fracture decrease ((a2)) because of the reduced pressure gradient at the inlet and outlet. Figure 12: (**a1**) illustrates the bifurcated fracture when channels 1 and 2 have the same parameters. (**a2**,**a3**), respectively, compare the profiles of the pressure and speed in an open bifurcated fracture with homogeneous stiffness (2 \(\times\) 10\({}^{10}\) Pa/m) with those in open bifurcation fractures, where the stiffness of channels 1 and 2 is higher (2 \(\times\) 10\({}^{11}\) Pa/m) or lower (4 \(\times\) 10\({}^{9}\) Pa/m) than that of channel 0 (2 \(\times\) 10\({}^{10}\) Pa/m). (**b1**) illustrates a bifurcated fracture in which channel 2 has different parameters from channels 0 and 1. (**b2**,**b3**), respectively, present the profiles of the pressure and speed in the open bifurcated fractures, where the stiffness of channel 2 is 4 \(\times\) 10\({}^{9}\) Pa/m but that of channels 0 and 1 is 2 \(\times\) 10\({}^{10}\) Pa/m. For comparison, we plot the profiles in the bifurcated fracture with the homogeneous stiffness as light lines. We also investigate the flow dynamics in the bifurcated fracture when the stiffness of channel 2 is lower than that of channels 0 and 1, as shown in Figure 12(b1). Figure 12(b2,b3) shows that the lower stiffness of channel 2 causes a higher interior pressure and higher speed than that of channel 1, which is in agreement with the case of a single channel (Figure 6(a1,a2)). Moreover, the lower stiffness in channel 2 reduces the interior pressure in channels 0 and 1. It also increases the pressure gradient at the inlet of channel 0 while decreasing the pressure gradient at the outlet of channel 1 ((b2)), hence increasing the inlet speed but decreasing the speed at the outlet ((b3)). The permeability also affects the flow in the open bifurcated fracture. Figure 13(a1) compares the pressure profiles between a fracture with homogeneous permeability and a fracture in which the permeabilities of channels 1 and 2 are equal and higher than the permeability of channel 0, and suggests that the high permeability of channels 1 and 2 reduces the interior pressure in the whole fracture. Due to the increased pressure gradient at the inlet, the inlet speed of the fracture is higher than that with homogeneous permeability, as shown in Figure 13(a2). The outlet speed, however, becomes lower, as a result of the increased leakage through the wall of the fracture. Figure 13(b1,b2) shows the profiles of the pressure and speed, respectively, when the permeability of channel 2 is higher than that of channels 0 and 1. The higher permeability of channel 2 reduces the interior pressure in all three channels ((b1)), and causes a higher speed in channel 0 than that with homogeneous permeability ((b2)) because of the increased pressure gradient. The inlet speed of channel 2 is higher than that of channel 1, suggesting that more liquid flows into channel 2. This reduces liquid speed in channel 1. The outlet speed of channel 2, however, is lower than that of channel 1, as a result of the higher leakage. The last parameter we investigate for the open bifurcated fracture is the channel length.