# 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
### Model
#### 2.1.1 Governing Equations  
We describe the steady-state Newtonian flow in a single fracture of inclination \(\alpha\) from the horizontal coordinate using the following 1D Navier-Stokes equations along the horizontal coordinate,  
\[\frac{\partial^{2}u}{\partial x^{2}}=\frac{1}{\mu}(\frac{\partial p}{\partial z }-\rho g\text{sin}\alpha)\,,\hskip 14.226378 ptx\in[0,h]\,, \tag{1}\]  
where \(u\) represents the vertical speed, \(\mu\) is the dynamic viscosity, \(p\) is the pressure, \(\rho\) is the density, \(g\) is the gravitational acceleration, and \(h\) is the aperture of the fracture. In this study, we neglect the gravitational term because it is approximately three orders of magnitude lower than the pressure term. The boundary conditions on the two walls are no-slip, as follows,  
\[u(0)=u(h)=0\,, \tag{2}\]  
Integrating Equation (1), while accounting for the boundary condition of Equation (2), we obtain the relationship between the mean vertical speed, \(\bar{u}\), and the pressure gradient, \(\partial p(z,t)/\partial z\), as follows,  
\[\bar{u}(z,t)=-\frac{h^{2}(z,t)}{12\mu}\frac{\partial p(z,t)}{\partial z}\,, \tag{3}\]  
Our model is concerned with laminar flow between two infinite plates. Therefore, the following relationship should be satisfied,  
\[Re=-\frac{\rho\bar{u}(z,t)h(z,t)}{\mu}<2300\,, \tag{4}\]  
where Re represents the Reynolds number. Considering the impact of matrix leakage on the fracture walls [30], we apply the mass conservation law to a short length, \(\mathrm{d}z\), in the vertical coordinate to obtain  
\[\frac{\partial(\rho h\mathrm{d}z)}{\partial t}=(\rho\bar{u}h-\frac{1}{2}\frac {\partial(\rho\bar{u}h)}{\partial z}\mathrm{d}z)-(\rho\bar{u}h+\frac{1}{2} \frac{\partial(\rho\bar{u}h)}{\partial z}\mathrm{d}z)-2\rho u_{L}\mathrm{d}z\,, \tag{5}\]  
Figure 2: Model of a single fracture. For an opened fracture, the triangle surrounded by the dashed line is removed, while the triangle seals the outlet for a closed fracture. where \(u_{L}\) represents the leakage velocity from the natural fracture to the rock matrix. Based on [PERSON]'s law, the leakage rate is proportional to the pressure difference in the \(x\) direction across the fracture wall and can be computed as  
\[u_{L}(z,t)=\frac{\xi_{p}}{\mu}(p(z,t)-p_{b})\,, \tag{6}\]  
where \(\hat{k}_{p}\) is the ratio of the permeability of the rock foundation to the rock thickness, and \(p_{b}\) represents the back pressure. Simplifying Equation (5) gives a one-dimensional continuity equation as  
\[\frac{\partial(\rho h)}{\partial t}=-\frac{\partial(\rho hh)}{\partial z}-2 \rho u_{L}\,, \tag{7}\]  
We assume the density of the liquid is constant because of the very low compressibility, but consider the variation of the viscosity along the vertical coordinate. Therefore, Equation (7) transforms into  
\[\frac{\partial h(z,t)}{\partial t}=-h(z,t)\frac{\partial a(z,t)}{\partial z}- \bar{u}(z,t)\frac{\partial h(z,t)}{\partial z}-2u_{L}(z,t)\,, \tag{8}\]  
Substituting Equation (3) into Equation (8), we obtain the following nonlinear equation for the evolution of the aperture,  
\[\frac{\partial h}{\partial t}=\frac{h^{3}}{12\mu}\frac{\partial^{2}p}{ \partial z^{2}}-\frac{h^{3}}{12\mu^{2}}\frac{\partial\mu}{\partial z}\frac{ \partial p}{\partial z}+\frac{h^{2}}{4\mu}\frac{\partial p}{\partial z}\frac{ \partial h}{\partial z}-2u_{L}\,, \tag{9}\]  
In this study, we use the relationship given by [PERSON] and [PERSON] [30] to describe the variation of the viscosity caused by the temperature,  
\[u(0)=u(h)=0\,, \tag{2}\]0  
where \(\mu_{\text{ref}}\) represents the reference viscosity module and \([T]\) the temperature thinning module, defined as  
\[u(0)=u(h)=0\,, \tag{2}\]1  
where \(\Delta E_{T}\) represents the Arrhenius activation energy required to induce thermal thinning, \(R\) is the universal gas constant, \(T\) is the temperature of the fluid, and \(T_{\text{ref}}\) is the reference temperature. The temperature distribution of fractures during oil migration has been studied [31; 32; 33].