## 2 Materials and Methods
### Setting Process and Mechanical Analysis of DBS Slip
#### 2.1.2 Mechanical Analysis of the Slip During Setting Process  
The slip assembly serves as the primary anchoring mechanism during DBS setting and fracturing operations, securely locking the tool to the casing wall. This critical function prevents the axial displacement of the DBS while maintaining effective pressure-sealing integrity. However, during service, potential slip failure or excessive casing wall damage may occur, which could compromise well integrity and lead to serious safety incidents, ultimately threatening the safety of hydrocarbon production operations. Consequently, the mechanical analysis of the DBS mounting slips is essential. The DBS slip incorporates a multi-flap design. For analytical purposes, we isolate a single flap segment and conduct a stress distribution analysis. Figure 3 illustrates both the structural configuration and force distribution of this slip segment. Figure 2: Tight oil and gas volumetric fracturing process and soluble ball holder tool. During DBS setting operations, the slip assembly experiences multiple loading conditions including radial casing wall reaction forces, compressive loading from the sliding body, frictional forces at the sliding body/tailstock interface, and setting forces transmitted through individual cavity elements from the tailstock. Based on static equilibrium principles, the force balance on the slip must satisfy the following equations [47]:  
\[Q=F_{N}\cos\gamma-F_{\mathrm{f}}\sin\gamma \tag{1}\]  
\[F_{z}-F_{q}=F_{N}\sin\gamma+F_{f}\cos\gamma \tag{2}\]  
\[F_{f}=F_{N}\tan\varphi \tag{3}\]  
\[F_{Z}=W_{Z}/n \tag{4}\]  
\[F_{\mathrm{q}}=Q\tan\delta \tag{5}\]  
where \(Q\) is the radial load at the interface between the single-flap slip segment and casing inner wall, \(N\); \(F_{N}\) is the normal contact forces transmitted from the sliding body to single-flap slip, \(N\); \(F_{f}\) is the friction between the sliding body and the single-flap slip, \(N\); \(F_{Z}\) is the sealing force acting on single-valve slip, \(N\); \(F_{q}\) is the friction between single-flap slip and casing, \(N\); \(W_{Z}\) is the DBS seating load, \(N\); \(\gamma\) is the tilt angle of contact surface between the slip and sliding body, \({}^{\circ}\); \(\varphi\) is the friction angle between slip and sliding body, \({}^{\circ}\); \(n\) is the number of slip pieces. Combining Equations (1) through (5), we derive the following:  
\[Q=\frac{W_{Z}/n-Q\tan\delta}{\tan(\gamma+\varphi)} \tag{6}\]  
where \(\delta\) is the friction angle between the cavity and the casing wall, \({}^{\circ}\). Equation (6) can be transformed to yield the following:  
\[Q=\frac{W_{Z}}{n(\tan(\gamma+\varphi)+\tan\delta)} \tag{7}\]  
with the contact pressure distribution at the slip tooth-casing interface during occlusion as follows:  
\[\sigma=\frac{Q}{A} \tag{8}\]  
where: \(Q\) is the contact pressure between the slip teeth and the inner wall of the casing, \(\mathrm{N/mm^{2}}\); \(A\) is the contact area between the slip teeth and the inner wall of the casing, \(\mathrm{mm^{2}}\)  
Figure 3: Slip structure and force analysis. The structural geometry demonstrates that the slip-casing contact surface area is characterized by Equation (9):  
\[Q=F_{N}\cos\gamma-F_{\mathrm{f}}\sin\gamma \tag{1}\]0  
where \(m\) is the number of single-flap slip teeth; \(a\) is the slip teeth bite into the casing; \(\theta\) is the single-flap slip with rounded angle; \(D_{t}\) is the diameter of casing inner wall; \(\alpha\) is the slip tooth apex angle; \(\beta\) is the slip tooth inclination angle. Combining Equations (8) and (9), we obtain the following. \[Q=F_{N}\cos\gamma-F_{\mathrm{f}}\sin\gamma \tag{1}\]1  
The preceding mechanical analysis establishes the computational model for determining interfacial contact pressure between the casing and slip assembly during setting and sealing operations. As demonstrated by Equation (10), both the slip tooth apex angle and inclination angle significantly influence the slip-casing contact pressure distribution.