## 2 Theoretical Background  
### Working Principles of SPWJ  
As shown in Figure 2, the SPWJ is generated by a single-stage, single-acting super-charger with hydraulic oil as the driving medium. The hydraulic oil drives a piston to reciprocate back and forth so that the small-diameter end of the piston squeezes water in a pressurization cavity and intermittently sprays a PWJ through the nozzle. As shown in  
Figure 1: Mining coabled methane and geothermal resources through multiple rock layers. Figure 3a, when the piston presses the water body forward, the water in the pressurization chamber is squeezed and sprayed as a PWJ, and the one-way valve is closed at this time. When the piston moves backward, water is replenished in the chamber. The state of the nozzle is controlled by the flow rate of water replenishment, so the nozzle sprays a low-pressure continuous jet or intermittent water drops, and the one-way valve is in an open state, as shown in Figure 3b. The reciprocating motion of the piston depends on the reversing slide valve at the upper end of the piston, and automatic commutation can be performed. Given the characteristics of the SPWJ, the authors focused on the piston extrusion stroke. As shown in Figure 3, a typical SPWJ-generating device includes a squeezing plunger, an impact chamber, and a nozzle installed in the chamber. Compared to conventional high-pressure continuous water jets, the variable cross-section extrusion type  
Figure 3: Two working states of the SPWJ generator: (**a**) stroke high-pressure water jet; (**b**) return low-pressure water jet. Figure 2: Structure and working principle of the SPWJ generator. pulse water jet generator converts low-pressure fluid into high-pressure pulse water jets, as shown in Figure 4a. The occurrence of SPWJ is composed of two interrelated basic mechanisms, namely, the strong disturbance of the compression wave and the periodic change in volume. When the squeezing piston pushed into the impact chamber comes into contact with water, the generated pressure disturbance propagates to the bottom at the speed of sound and is reflected after propagating to the end. The pressure behind the compression wave is always higher than the pressure before, which also means that when the piston squeezes the water in the impact cavity, the pressure in the cavity will rise in a discontinuous manner. In other words, when the first compression wave continuously reaches its peak, the subsequent wave is the coupling wave of the compression wave and reflection wave, showing discontinuous rising and falling oscillation. When the piston retreats, the pressure in the cavity quickly drops to the initial input pressure, which is in good agreement with the actual test results (as shown in Figure 4b). To further clarify the mechanism of the SPWJ, the control body is shown in Figure 5. The mass conservation equation is applied to obtain the initial value equation involving the plunger and water mass, expressed in Equation (1) [38]:  
Figure 4: The occurrence mode and pressure curve of SPWJ and CWJ: (**a**) differences in modes of action; (**b**) differences in pressure curves. Table 1 presents the output pressure of the SPWJ and CWJ, and the erosion and crushing behavior of granite by the SPWJ was studied by using the obtained nozzle output peak pressure. \[\left\{ \begin{array}{l}\rho av+\frac{\phi}{\rho}A(h-x)-A\rho x=0\\ p=E_{wf}(\rho/\rho_{w}-1)\\ l_{c}v^{\frac{\bullet}{0}}+v^{2}/2=\int_{0}^{p}dp/\rho\\ \frac{\bullet}{x}=-(A/M)p\\ x(0)=v(0)=p(0)=0\\ \frac{\star}{\alpha}(0)=U_{0}\\ \rho(0)=\rho_{w}\end{array} \right.