## 3 Materials and Methods
### Facilities  
Figure 2 is a schematic diagram of the experimental setup. The experiment was conducted with the use of a multifunctional SC-CO\({}_{2}\) jet test system developed by our research team independently, and had been used in some related research [13; 15]. The red arrows illustrate the flow direction of CO\({}_{2}\), while the blue ones show that of hot water. The experimental system consists of a series of units, including the CO\({}_{2}\) cartridge, the cooling unit converting gaseous CO\({}_{2}\) to liquid to facilitate pumping, the storage tank for temporary storage of liquid CO\({}_{2}\), the high pressure plunger pump applied to pressurize the liquid CO\({}_{2}\), the buffer tank employed to reduce the fluid fluctuation, the chamber applied to establish the ambient pressure, the counterbalance valve used to regulate the ambient pressure, and the cyclone de-sander utilized to purify the working SC-CO\({}_{2}\). Meanwhile, a water-bath device was used to heat the buffer tank and chamber to control the temperature of CO\({}_{2}\). The inlet pressure and fluid temperature can be regulated continuously through the control table from 0 to 75 MPa and 300 to 373 K, respectively. The ambient pressure in the chamber could be increased to 45 MPa. The SC-CO\({}_{2}\) was provided through the process of pressurization and heating and then flowed through the nozzle to form a SC-CO\({}_{2}\) jet. The jet discharged into the rock erosion chamber and then impinged on the specimen surface perpendicularly, as shown in Figure 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ratio**, \(R_{c}\) & **Density**, \(\rho_{a}\) & **Compressive Strength**, \(\sigma_{a}\) & **Modulus of Elasticity**, \(E_{a}\) & **Poisson\({}^{\prime}\)s Ratio**, \(v_{a}\) \\
**Cement/Sand/Water** & **(g/cm\({}^{3}\))** & **(MPa)** & **(GPa)** & **Poisson\({}^{\prime}\)s Ratio**, \(v_{a}\) \\ \hline
1/2/0.5 & 2.2 & 32.8 & 19.1 & 0.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical properties of the specimens. The detailed structural parameters of the nozzles used in the experiment are shown in Figure 3. Nozzle A was a commonly used convergent nozzle consisting of a contraction section and a straight pipe section. The convergent nozzle had an outlet diameter of \(d_{a}=2\) mm, a convergent angle of \(a_{1}=13.5^{\circ}\), and a straight pipe section length of \(l_{a}=2d_{a}\). Nozzle B was a Laval nozzle (Laval-1) which had three sections, namely, the convergent section, throat section and divergent section. The throat diameter of the Laval-1 nozzle was \(d_{b2}=2.0\) mm and the outlet diameter was \(d_{b1}=2.66\) mm. The divergent angle of the divergent section was \(a_{b1}=10^{\circ}\), the convergent angle of the convergent section was \(a_{b2}=13.5^{\circ}\), and the length of the divergent section and the throat section are \(l_{b1}=3.8\) and \(l_{b2}=1.0\) mm, respectively. Nozzle C was a typical Laval nozzle (Laval-2) with a smooth streamlined inner profile. The nozzle had a throat diameter of \(d_{c2}=2.0\) mm, an outlet diameter of \(d_{c1}=2.66\) mm, and an inlet diameter of \(d_{c3}=5\)mm. The lengths of the divergent section, the throat and the convergent section were \(l_{c1}=3.71\), \(l_{c2}=0.7\) and \(l_{c3}=1.8\) mm, respectively. Moreover, the divergent section \(l_{c1}\) was designed according to the characteristic line theory [40], and the convergent section \(l_{c3}\) was designed using the commonly used empirical Witoszynski formula [41]. The two Laval nozzles had the same designed Mach number of 2. In addition, the outside dimensions of the nozzle were the same, which were \(D_{a}=D_{b}=D_{c}=12\) mm and \(L_{a}=L_{b}=L_{c}=19\) mm. Figure 2: Schematic diagram of the rock erosion test system. Inlet pressure, \(P_{i}\), was defined as the pressure of the flow entering into the nozzle. Ambient pressure, \(P_{a}\), was defined as the pressure of the carbon dioxide in the chamber. Fluid temperature, \(T\), was defined as the temperature of the fluid near the nozzle inlet.