## 4 Discussion  
From the experimental results, the nano-silica additive has significant effects on the rock erosion characteristics of the SC-CO\({}_{2}\) jet. How the nano-silica additive influences the erosion ability of SC-CO\({}_{2}\) jet, is likely to depend on the mass fraction of the nano-silica additive and the operating conditions. This is most likely due to the fact that operating conditions can have large effects on the physical properties of the SC-CO\({}_{2}\) fluid [42], which in turn, directly affects the acceleration of the nano-silica particles, as well as the impact characteristics of the nano-silica SC-CO\({}_{2}\) jet. In order to further understand the mechanism of the effects of nano-silica on the erosion characteristics of the SC-CO\({}_{2}\) jet, a preliminary was performed, as follows. According to the theory of the erosion of materials by solid particle impact, it is obvious that, when the other conditions are kept constant, the erosion intensity of nano-silica particles on the rock specimens is largely dependent on the impact velocity, which determines the magnitude of the impact loading and the stress wave energy [37]. Moreover, the impact velocity is affected by the forces which participate in the particle motion in the fluid [43]. These forces include interfacial drag, virtual mass, mechanical diffusion, friction, and gravitational forces. Also, the interfacial drag and the virtual mass force are the most important forces that act on the nano-silica particles. The interfacial drag acts in the same direction as the particles accelerate. Furthermore, the direction of the virtual mass force is opposite to the acceleration direction of the particles. The expressions of the forces are as follows [43]:  
\[F_{d}=\frac{3}{4}C_{d}\frac{\rho_{f}}{D_{a}}(V_{f}-V_{p})\Big{|}V_{f}-V_{p} \Big{|} \tag{2}\]  
\[F_{v}=\frac{1}{2}\rho_{f}V_{p}\frac{\text{d}(V_{f}-V_{p})}{\text{d}x} \tag{3}\]  
According to [PERSON]'s second law, the acceleration of the particles can be expressed as:  
\[a=\frac{(F_{d}-F_{p})}{m}=\rho_{f}\Bigg{[}\frac{3}{4m}\frac{\text{C}_{d}}{D_{ a}}(V_{f}-V_{p})\Big{|}V_{f}-V_{p}\Big{|}-\frac{1}{2m}V_{p}\frac{\text{d}(V_{f}-V_{ p})}{\text{d}x}\Bigg{]} \tag{4}\]  
As expressed in the equations, the density of the SC-CO\({}_{2}\) fluid can have large effects on the acceleration process of the nano-silica particles, and then affects the impact velocity and erosion action of the nano-silica SC-CO\({}_{2}\) jet. More specifically, when the other conditions are constant, the increase of fluid density can enhance the acceleration effect of particles in the jet flow. So, for the experiments on the effect of the inlet pressure on the erosion ability of the nano-silica SC-CO\({}_{2}\) jet, the density of the SC-CO\({}_{2}\) fluid, calculated by the property routine REFPROP 9.1 [44] at the  
Figure 10: Rock erosion characteristics under various fluid temperatures. nozzle inlet, increases with the increase of inlet pressure, as shown in Figure 11. The properties of the SC-CO\({}_{2}\) were calculated using the default equations in REFPROP. The increase of density will enhance the acceleration effect of the nano-silica particles in the SC-CO\({}_{2}\) fluid, as well as the impingement of the particles on the rock specimens. This can also be verified by the trend of the curves in Figure 7. With the increase of the inlet pressure, the mass fraction of 6 wt % initially has the largest erosion capability. Then, the mass fractions of 9 wt % and 12 wt % take the dominant place, at an inlet pressure between 40 MPa and 55 MPa. Finally, the mass fraction of 15 wt % shows the best erosion performance, at a pressure of 60 MPa. Similarly, for the experiments on the effect of the standoff distance (Figure 6) and the ambient pressure at constant inlet pressure (Figure 8), the density of SC-CO\({}_{2}\) fluid at the nozzle inlet is constant.