## 3 Field Test Study on Liquid CO2 Phase-Transition Rock Fracturing
### Vibration Monitoring
#### 3.4.2 Energy Analysis  
As can be seen in Figures 12 and 13, the vibration signals produced by the liquid CO\({}_{2}\) phase-transition fracturing technology method were non-stationary random signals, which were similar to the blasting seismic waves. The Fourier transform could not accurately analyze the vibration signals. Therefore, in the current study, an HHT (Hilbert-Huang Transform) energy analysis principle was adopted to reflect the spectral characteristics and energy distribution concerns of the vibration signals from the perspective of the vibration signal energy [33; 34], for the purpose of evaluating the safety of the liquid CO\({}_{2}\) phase-transition fracturing technology method. It has been determined that HHT methods are suitable for analyzing non-stationary and non-linear signals. The methods consist of two steps: (1) The empirical mode decomposition of the vibration signals in order to obtain a series of intrinsic mode functions representing the time scale of the signal characteristics; (2) Hilbert transforms for each IMF (intrinsic mode function) component. The EMD (empirical mode decomposition) uses the mean of the upper and lower envelopes of the time series to determine the \"instantaneous equilibrium position\", and then extracts the IMF. An IMF is one of the oscillating modes of data recorded at a given time. The Hilbert spectrum can be obtained by the Hilbert transform of a signal as follows [33]:  
\[H(\omega,t)=\text{Re}\sum_{i=1}^{n}a_{i}(t)e^{\int\omega_{i}(t)dt} \tag{13}\]  
The Hilbert marginal spectrum can be obtained by the time integration of the Hilbert spectrum. \[h(\omega)=\int_{0}^{T}H(\omega,t)dt \tag{14}\]  
The marginal spectrum is the integral of the amplitude of each frequency component in time. The instantaneous energy spectrum of a signal can be obtained by the frequency integration of the square of the Hilbert spectrum. \[IE(t)=\int_{\omega}H^{2}(\omega,t)d\omega \tag{15}\]  
Therefore, it can be used to accurately describe the process of signal energy changing with time. In the current study, the above-mentioned method and MATLAB programming were adopted, and EMD decomposition was carried out by taking the vibration signal in the Z direction as an example. A total of 12 IMF components with different frequencies, and one monotonous residual component, were obtained, as shown in Figure 15. It was found that the frequencies of the IMF components had ranged from large to small from top to bottom. The frequencies of the IMF1, IMF2, and IMF3 components were found to be too high, and had not been in accordance with the characteristics of the vibration waves; therefore, those components were considered to be noise and were eliminated in subsequent analyses. The IMF4 to IMF7 components were considered to be the main components of the measured wave due to their high vibration intensities. The IMF10 to IMF12 components were found to have low intensities and low frequencies, which indicated that the measured waveform was mainly concentrated within a narrow frequency band. The EMD decomposition of the vibration signals in the X and Y directions was carried out using the same method described previously. The IMF components were obtained and brought into a Hilbert transform, which was obtained using the Hilbert energy spectrum. Figure 16 displays the Hilbert energy spectrum of the signals in the different vibration directions, which accurately shows the energy distribution of the IMF components. Then, by comparing the different Hilbert diagrams, it was found that the overall form of the Hilbert spectrum was basically the same in the three directions. The duration of the low frequency was longer, and the low-frequency components in the three directions had lasted until the end of the vibration wave. Meanwhile, the vibration energy in the three directions had displayed the largest energy in the first 0.2 s. Among those, the high frequency in the x-direction was the highest, followed by the z-direction, while the y axis was observed to be the lowest.