## 2 Materials and Methods
### Facilities  
The experiments were conducted on a multifunction SC-CO\({}_{2}\) jet testing system, independently developed by our team, as shown in Figure 2. The red arrows illustrate the direction of the flow of CO\({}_{2}\), while the blue arrows show that of hot water. This system consisted of several parts, which are: the CO\({}_{2}\) cartridge; the cooling part, transforming gaseous CO\({}_{2}\) to liquid; the storage tank, storing CO\({}_{2}\) temporarily; the high pressure plunger pump, pressurizing the liquid CO\({}_{2}\); the surge tank, reducing the fluid fluctuation; the counterbalance valve, used to build the ambient pressure; and the cyclone de-sander, purifying the working SC-CO\({}_{2}\). Simultaneously, a water-bath heater was used to heat the surge tank and jet chamber. The continuous regulation of the ranges of the nozzle inlet pressure and fluid temperature, were 2 MPa to 75 MPa and 290 to 373 K, respectively. The ambient pressure in the tests could be set up to 45 MPa. The pure SC-CO\({}_{2}\) jet was provided through the pressurizing and heating processes, and then discharged into a rock erosion chamber and perpendicularly impinged on the specimen surface, as shown in Figure 2.  
The nano-silica SC-CO\({}_{2}\) jet was achieved in three steps. First, a certain mass fraction of the nano-silica additive was dispersed into liquid CO\({}_{2}\) through 25 min of mechanical agitation, and was then sonicated by an ultrasonic probe (Hielscher Up400S, Hielscher Ultrasonics, Teltow, Germany) for another 25 min in the storage tank. Then, after flowing through the pressurizing and heating units, the liquid CO\({}_{2}\) changed into a supercritical state. Finally, a high speed nano-silica SC-CO\({}_{2}\) jet was formed after having passed through a convergent nozzle, which was then impinged on the specimen surface. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & **Length** & **Diameter** & **Density** & **Compressive Strength** & **Modulus of Elasticity** & **Poisson’s** \\  & **(mm)** & **(mm)** & **(g/cm\({}^{3}\))** & **(MPa)** & **(GPa)** & **Ratios** \\ \hline
1 & 98.5 & 50 & 2.6 & 38.2 & 8.38 & 0.12 \\
2 & 100.5 & 50 & 2.5 & 38.5 & 8.32 & 0.11 \\
3 & 99.3 & 50 & 2.5 & 37.9 & 8.35 & 0.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical and mechanical properties of the rock specimens. Figure 2: Schematic diagram of the rock erosion test system of a SC-CO\({}_{2}\) jet. Pressure measurements were taken at the nozzle inlet and in the chamber at a 1 kHz sampling rate, using BD/SENSORS DMP334 transducers (BD | SENSORS, Therestein, Germany) with an accuracy of 0.175% FS. The temperatures in the chamber and at the nozzle inlet were measured using Omega T thermocouples, whose total accuracy was \(\pm\)1 K. The real-time data were obtained by a QuantumXMX840 data acquisition and monitoring system. During each test, erosion of the rock by the SC-CO\({}_{2}\) jet did not begin until the pressure and temperature obtained by the transducers was maintained at the wanted values. In this way, the practical application conditions for a SC-CO\({}_{2}\) jet in the bottom hole were simulated. As a convergent nozzle is a typical rock erosion nozzle of wide application, this kind of nozzle was used to investigate the effects of a nano-silica additive on the rock erosion characteristics of a SC-CO\({}_{2}\) jet. The structure of the convergent nozzle used in the experiments is shown in Figure 3. The exit diameter was \(d_{0}\) = 1 mm, the length of the exit was \(l_{1}\) = 2\(d_{0}\), the length of the conical section was \(l_{2}\) = 11\(d_{0}\), and the convergent angle was \(\alpha\) = 13.5\({}^{\circ}\). The rock erosion capacity of the SC-CO\({}_{2}\) jet was evaluated by the erosion volume, \(V\). The erosion volume was achieved in three steps, as shown in Figure 4. First, a certain amount of salt, which had a density of \(\rho_{s}\) = 2165 kg/m\({}^{3}\), was measured using an electronic balance with a resolution of 0.001 g, and the mass was \(m_{1}\).