## 5 Conclusions  
This study presents an integrated approach to predicting breakdown pressure in SC-CO\({}_{2}\) fracturing by combining experimental data with a multi-layer neural network model. The findings confirm that SC-CO\({}_{2}\) exhibits superior fracturing performance compared to traditional hydraulic fracturing fluids, primarily due to its high diffusivity and effective infiltration of microfractures, thereby lowering the breakdown pressure required for fracture initiation. Elevated confining pressures significantly increase breakdown pressure, highlighting the critical role of in situ stress. Additionally, the study found that higher flow rates also led to higher breakdown pressures, suggesting that increased flow rates enable the faster build-up of injection pressure, thus causing rock failure more rapidly. The effect of temperature was also evident, with breakdown pressure decreasing as temperature increased, indicating that higher temperatures facilitate fracturing by reducing the effective stress in the rock.  
A comparison of experimental results with theoretical predictions from the H-W and H-F models revealed discrepancies, indicating that classical breakdown pressure models fail to fully capture the complex phenomena observed during SC-CO\({}_{2}\) fracturing. This is partly because these models were originally designed for hydraulic fracturing and operate under simplifying assumptions, such as low or high rock permeability, which may not align with actual experimental conditions. Moreover, the Ito model, which incorporates parameters, such as pressurization rate and wellbore diameter, also failed to predict the breakdown pressures of certain specimens, likely due to the inherent heterogeneity of natural rock.  
Using a fully connected neural network, this study successfully captured the nonlinear relationships among multiple influencing factors (fracturing fluid type, flow rate, temperature, tensile strength, and confining pressure) and the breakdown pressure. The model outperformed traditional predictive models, achieving an \(R\) value of 0.9123 for the testing set, thereby demonstrating strong predictive capability. The sensitivity analysis identified fracturing fluid type and tensile strength as the most influential factors, contributing 28.31% and 21.39% to the overall breakdown pressure, respectively. These insights offer a data-driven foundation for optimizing fracturing design and improving operational efficiency.  
Despite promising results, this study acknowledges limitations, including the relatively small dataset size and the exclusion of critical parameters such as fracture toughness and rock permeability. Future research should expand the dataset and incorporate a more comprehensive range of rock properties to further enhance the model's predictive accuracy. Overall, the application of a multi-layer neural network model provides a robust, adaptable framework for predicting fracturing outcomes, particularly for unconventional fracturing techniques such as SC-CO\({}_{2}\), as traditional models often fail to account for complex variable interactions.  
Conceptualization, X.Z.; methodology, X.Z.; software, M.Z.; validation, S.L. and H.L.; formal analysis, X.Z. and M.Z.; investigation, H.L.; resources, X.Z.; data curation, M.Z.; writing--original draft preparation, X.Z.; writing--review and editing, M.Z. and S.L.; visualization, X.Z. and M.Z.; supervision, S.L.; project administration, X.Z.; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant number 42307208), the Shanghai Pujiang Programme (Grant number 23 PJD105), and the Postdoctoral Researcher Programme of China (Grant number GZB20230529).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.