## 2 Laboratory Tests on Nonlinear Flow of Fractures
### Experimental Results and Analysis of the Nonlinear Flow of Fractures  
When the flow velocity is small or at a low Reynolds number, the fluid flow conforms to the cubic law [63]:  
\[Q=-\frac{\text{we}^{3}}{12\mu}\
abla P \tag{3}\]  
where \(w\) is the width of the fracture, \(e\) is the aperture of the fracture, \(\mu\) is the dynamic viscosity coefficient of the fluid, \(Q\) is the volume flow, and \(\
abla P\) is the water pressure gradient. When the flow velocity is increasing, the inertial effects induced by the contact areas and surface roughness of the fracture should be considered, and the Forchheimer law is adopted to describe its nonlinear term [64]:  
\[-\
abla P=AQ+BQ^{2} \tag{4}\]  
where \(A=12\mu/\text{we}^{3}\) and \(B=\beta\rho/\text{w}^{2}e^{2}\) are the linear and nonlinear coefficients, respectively; \(\mu\) is the dynamic viscosity coefficient of the fluid; \(\beta\) is the Forchheimer coefficient; \(\rho\) is the fluid density; \(w\) is the width of the fracture; and \(e\) is the aperture of the fracture. The Reynolds number provides a method of comparison for the ratio of inertial force to viscous force; to quantify the relative importance of two forces under given flow conditions, the Reynolds number is defined as [65]:  
\[\text{Re}=\frac{\rho\text{ve}}{\mu}=\frac{\rho Q}{\mu w} \tag{5}\]  
where \(\rho\) is the fluid density, \(v\) is the flow velocity, \(w\) is the width of the fracture, \(e\) is the aperture of the fracture, and \(\mu\) is the dynamic viscosity coefficient of the fluid. Figure 5: (**a**) Diagram of the permeability measurement apparatus. (**b**) Real picture of the apparatus. The critical Reynolds number for the transition from linear flow to nonlinear flow can be calculated by follow equation [65; 66]:  
\[\text{Rec}=\frac{\text{A}\rho}{9\text{B}_{\text{i}\text{W}}} \tag{6}\]  
Flow experiments have been carried out for fracture samples with different filling conditions. All experimental results show that the relationship between the water pressure gradient \(\
abla\)P and volume flow Q is strongly nonlinear and the Forchheimer law could describe the results well. The correlation coefficient R\({}^{2}\) could reach more than 0.99. Figure 6 shows that an increase in flow velocity leads to a nonlinearity of flow. Non-Darcy flow is caused by non-negligible inertial damage due to a change in flow direction or velocity along the flow path, and a greater velocity results in more significant losses of inertia. A decreasing proportion in the water pressure gradient is greater than an increasing proportion of flow due to the inertial losses, and the relationship between the water pressure gradient and flow is no longer linear [1; 53]. Table 2 shows comparisons of the linear coefficient A and the nonlinear coefficient B. Coefficients A and B increase with a decrease in the particle size of the filling material, and the maximum coefficients are obtained under the condition without filling. Coefficient A is related to the viscous friction at the water and rock interface and represents the intrinsic permeability. It seems that a smaller particle of sand forms a denser pore structure, which makes the flow channel become narrower, and results in a lower flow capacity and  
Figure 6: Relationship of \(\
abla\)P and Q for a fracture: (**a**) filled with 50 mesh quartz sand, (**b**) filled with 100 mesh quartz sand, (**c**) filled with 200 mesh quartz sand, and (**d**) without filling. permeability. Coefficient B describes the inertia effect due to an irreversible kinetic energy loss caused by flow acceleration or deceleration. The contact area of the fracture surface increases with a decrease in the particle size of the filling material, thus leading to an increment in the curvature of the flow path and producing an increase in the inertia effect during the flow process thereby ultimately, increasing Coefficient B. As shown in Figure 7, the Reynolds number increases with an increase in water pressure.