## 4 Breakdown Pressure Prediction with a Multi-Layer Neural Network
### Relative Importance of Influencing Variables  
Following the method proposed by [PERSON] et al. (2018), the procedure for quantifying the relative importance of the eight influencing factors mentioned earlier involves the following steps:  
Step 1--Keep all other variables constant while increasing or decreasing the input variable under analysis by 10% from its original value in the training dataset, generating two new input datasets;  
Step 2--Apply the trained neural network model to compute the results for these two datasets and calculate the difference between them. The greater the difference, the more significant the influence of that variable on the output;  
Step 3--Repeat this process for each input variable, then compute the average difference across all variables. The mean difference provides a measure of each input variable's influence on the predicted outcome. As illustrated in Figure 12, the relative importance of the factors is ranked as follows: fracturing fluid type > tensile strength > flow rate > maximum horizontal stress > Poisson's ratio > wellbore diameter > minimum horizontal stress > initial fracturing fluid temperature. Specifically, fracturing fluid type and tensile strength are the most sensitive variables, contributing 28.31% and 21.39% of the influence, respectively. This is due to the significant role of fracturing fluid properties in establishing the pore pressure system within the rock, and the critical impact of effective stress on fracture initiation. Tensile strength directly determines the rock's failure resistance, as it defines the relationship between effective circumferential stress at the wellbore or characteristic distance and the rock's ability to resist failure ([PERSON] et al., 2018). The flow rate accounted for 12.34% of the influence on breakdown pressure, as different fracturing fluids exhibit varying compressibility, resulting in distinct effects on pressure buildup within the same injection time. The initial fracturing fluid temperature had a minimal impact, likely due to two reasons: (1) the dataset included multiple types of fracturing fluids, with temperature variation closely tied to the fluid type, and (2) for SC-CO\({}_{2}\), the initial temperature only reflects pre-injection conditions, without accounting for temperature changes during the fracturing process, which significantly affect fracturing performance. Figure 11: Comparison of actual values and predictive values. Although the multi-layer neural network model built in this study achieved high predictive accuracy, the experimental data used were limited. Moreover, many key parameters (e.g., fracture toughness, permeability) that are crucial in the fracturing process were not included in the available literature. Future studies should expand the dataset and comprehensively test the mechanical properties of rocks used in fracturing experiments. Predicting the effects of multiple factors on breakdown pressure using deep learning represents a notable advancement in understanding the complex relationships between variables in fracturing operations. By quantifying the relative importance of factors such as fluid type, tensile strength, and flow rate, the trained neural network model provides valuable insights into their interactions. This knowledge can guide the optimization of fracturing parameters for more efficient and effective operations. The findings of this study align with those from previous research, which indicates that key variables like fluid type and rock strength disproportionately affect breakdown pressure. This can be attributed to the distinct properties and interactions of SC-CO\({}_{2}\) with rock formations. SC-CO\({}_{2}\), with its low viscosity and high diffusivity, penetrates microfractures and rock fissures more efficiently than water-based fluids, thereby lowering the breakdown pressure required for fracture initiation.