## 3 Theoretical Models for Calculating Breakdown Pressure
### Comparison of Theoretical and Experimental Results  
As confirmed in the literature, the H-W, H-F, and Ito models yield reasonable estimates for calculating breakdown pressure. However, the fundamental assumptions underlying these models often fail to fully reflect real conditions. Moreover, breakdown pressure models based solely on tensile strength face challenges in accurately predicting the complex fracture processes involved. Consequently, a notable discrepancy exists between theoretical predictions and experimental values [27; 28]. To calculate the breakdown pressure corresponding to the theoretical models discussed, we combined experimental measurements with previously published results [35; 36]. The parameters used for these calculations are detailed in Table 2, where the initial pore pressure was assumed to be 0 MPa.  
Figure 6 compares the breakdown pressure values obtained from laboratory experiments with those calculated using the aforementioned theoretical models. In Figure 6, the H-W model and H-F model are denoted as H-W and H-F, respectively. The H-W model assumes that the rock is either impermeable or possesses low permeability, while the H-F model assumes high permeability in the rock. It is evident from Figure 6 that the experimental data do not consistently fall within the theoretical ranges predicted by these models, indicating that classical breakdown pressure models fail to capture the anomalous phenomena observed during experiments. Several factors contribute to this discrepancy, as follows: (1) the assumptions underlying the theoretical models do not encompass all the influencing factors present in the experiments, and (2) these models, typically applied to hydraulic fracturing, may not be suitable for predicting breakdown pressure in SC-CO\({}_{2}\) fracturing.  
Various fracturing parameters such as the type of fracturing fluid, flow rate, temperature, and confining pressure significantly influence the experimental breakdown pressure values. For instance, an increase in the flow rate corresponds to a rise in breakdown pressure, as it relates to the rate of pressurization. Although the Ito model accounts for both pressurization rate and wellbore diameter, the upper and lower bounds of this model do not encompass the breakdown pressure for specimens 3 and 9 (Figure 6). This limitation may be attributed to the inherent heterogeneity of natural rock.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & \
\begin{tabular}{c} **Biot** \\ **Coefficient** \\ \end{tabular} & \(\sigma_{t}\) **(MPa)** & \(\
u\) & \(r_{w}\) **(m)** & \(d_{c}\) **(m)** \\ \hline Value & 0.9 & \
\begin{tabular}{c} 15.63 (1\#-5\#) \\ 18.18 (6\#-11\#) \\ \end{tabular} & 0.20 (1\#-5\#) & 0.01 &
\begin{tabular}{c} 0.0024 (1\#-5\#) \\ 0.0017 (6\#-11\#) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters used in the theoretical models.  
Figure 6: Comparison between theoretical and experimental results. The areas shaded in green, blue, yellow, and red represent the scheme of fracturing fluid type, flow rate, temperature, and confining pressure, respectively. Black circle means the value of the breakdown pressure.