## 3 Theoretical Models for Calculating Breakdown Pressure  
### Theoretical Model Based on Elasticity Mechanics  
Theoretical models used to predict hydraulic fracturing breakdown pressure are generally based on various assumptions. Common models include those based on linear elasticity, poroelasticity, characteristic distance, shear failure, energy release, and stress intensity factors [27; 28]. Several factors influence the breakdown pressure, such as the viscosity of the fracturing fluid [29], the stress state of the rock surrounding the borehole [6], the size and orientation of the borehole [30; 31], the pressurization rate [32; 33], the formation pore pressure [8], rock mechanical properties [7], and the reservoir temperature [19]. Assuming the reservoir behaves as a homogeneous, isotropic, linear elastic porous medium, and the rock surrounding the central borehole (wellbore) is subjected to plane strain conditions, the stresses in the system can be categorized as tensile (positive) and compressive (negative). As illustrated in Appendix A, the total circumferential stress around the borehole consists of three components. Upon the injection of high-pressure fluid, the total stress distribution around the borehole is altered. This stress alteration can be evaluated through the principle of superposition, where the original in situ stresses combine with the additional stresses induced by the fluid injection. This superimposed stress field ultimately governs the breakdown pressure and fracture propagation in the surrounding rock mass. When the circumferential effective stress \(\sigma_{\theta}\) at the wellbore wall reaches the tensile strength of the rock (\(\sigma_{t}\)), failure occurs, in accordance with the Hubbert-Willis criterion (i.e., H-W model) [6]. This criterion is predicated on the assumption that the rock is impermeable. Consequently, the circumferential stress component resulting from pore pressure can be disregarded, leading to breakdown pressure primarily governed by in situ stress and injection pressure. Furthermore, the criterion posits that fractures initiate perpendicular to the minimum horizontal stress (i.e., \(\theta=0\) or \(\pi\) in Equation (A1)). The H-W breakdown pressure \(P_{b}\) can be calculated with or without accounting for the initial pore pressure \(P_{0}\), as illustrated in Equations (1) and (2). \[P_{b}=-\sigma_{H}+3\sigma_{h}+\sigma_{t}-P_{0} \tag{1}\]  
\[P_{b}=-\sigma_{H}+3\sigma_{h}+\sigma_{t} \tag{2}\]  
In dense reservoirs, characterized by nanometer-scale permeability and the near incompressibility of water, hydraulic fracturing occurs rapidly, making fluid penetration negligible at the moment of injection and fracturing. As a result, the breakdown pressure for water as a fracturing fluid can be directly calculated using Equations (1) and (2). Conversely, SC-CO\({}_{2}\), with its low viscosity, absence of surface tension, and strong diffusion capabilities, demonstrates significant filtration effects. Consequently, the influence of pore pressure on breakdown pressure becomes significant and cannot be overlooked. This filtration effect introduces an additional circumferential stress component (\(\sigma_{\theta}^{\alpha}\)). Assuming that SCCO\({}_{2}\) penetrates the rock completely, the breakdown pressure can be determined using the Haimson-Fairhurst formula (i.e., H-F model) [7]. The H-F breakdown pressure \(P_{b}\), which can be calculated with or without accounting for the initial pore pressure \(P_{0}\), is expressed in Equations (3) and (4). \[P_{b}=\frac{-\sigma_{H}+3\sigma_{h}+\sigma_{t}-\alpha\frac{1-2\
u}{1-\
u}P_{0} }{2-\alpha\frac{1-2\
u}{1-\
u}} \tag{3}\]  
\[P_{b}=\frac{-\sigma_{H}+3\sigma_{h}+\sigma_{t}}{2-\alpha\frac{1-2\
u}{1-\
u}} \tag{4}\]  
The breakdown pressure models discussed above assume that rock failure occurs at a specific point on the wellbore wall, and do not consider the actual fluid pressure distribution within the rock. Numerical simulations by [PERSON] et al.