# A Model for Simulating the Upward Flow of a Viscous Fluid in a Fracture Network
## 5 Discussion and Conclusions  
Fluid migration plays an important role in the oil accumulation mechanism and hence is key to oil exploration. It has been widely accepted that fractures are oil migration pathways and the main controlling factor for oil accumulation in faulted basins [40,41]. The main contribution of this work is establishing a validated model by combining 1D Navier-Stokes equations and linear elastic equations to simulate the upward fluid flow in the structural fractures. Considering the relationship between the temperature and the oil viscosity, we also coupled the model with the energy equations describing the thermal process in the fractures. Using the model, we have investigated the fluid migration and thermal process layer-by-layer for a single fracture, a bifurcated fracture, and a fracture network. Our results suggest that the parameters of both the fluid (Figures 5 and 9) and the surrounding rock matrix (Figures 6 and 8) affect the upward flow in the fractures. The outlet boundary condition is also very important. In good agreement with the experimental study by [PERSON] et al. [26], we found that the upward flow in an open fracture is mainly driven by the pressure difference between the inlet and outlet (Figure 4), so the fluid flow becomes a Darcy flow with a much lower speed in a closed fracture (Figure 7). As a result, the permeability is a more important factor affecting the profiles of the pressure and speed in a closed fracture (Figure 8(c1,c2)), compared with the profile in an open one (Figure 8(b1,b2)). Moreover, the liquid temperature in the closed fracture becomes closer to the rock matrix temperature as it gets closer to the outlet (Figure 7d), while the liquid temperature in the open fracture is basically maintained at the inlet temperature (Figure 4d). For the bifurcated fracture, we focused on the influences of fracture division, and the heterogeneity from the main channel to the branches on the upward flow. We found that the fracture division reduces the pressure in the main channel connected to the entrance of the whole fracture (Figure 18b), but increases the pressure gradient and the fluid speed there (Figure 18c), which agrees well with the findings of [PERSON] et al. [29]. The heterogeneity of the parameters between the main channel and the branches and between the branches has similar influences on the main channel but may affect the branches in opposite ways. As an example, for the flow in the main channel, decreasing the rock stiffness, either from the main channel to the branches or from one branch to the other, reduces its interior pressure but increases its speed. However, decreasing the stiffness from the main channel to the branches increases the outlet speed of both branches, while decreasing the stiffness from one branch to the other reduces the outlet speed of the branch with the higher stiffness. (Figure 12). In addition, we analyzed the influence of a closed branch on the upward flow in the whole bifurcated fractures (Figure 16), and found that closing a branch leads to a decrease in the internal pressure of the other branch channel as well as the main channel, an increase in the velocity of the main channel. Our result agrees well with those of [PERSON] et al. [42], who pointed out that  
Figure 20: (**a**,**b**), respectively, show the pressure and speed profiles in the channels on level 2 when the outlet of channel 1-2-1 is closed but other channels are open. For comparison, we plot the profiles of a homogeneous and open fracture network as light lines. the blockage of small fractures may have a significant impact on the local velocity field, increasing the flow velocity in the main channel, but it may also have the opposite effect on the channel velocity on the same line. Applying our model to the fracture network, we found that, with the increasing level of the network, the main contribution of the pressure difference shifts toward the entrance (Figure 18b), and the closer it is to the entrance, the greater is the speed of the fluid (Figure 18c).