## 3 Materials and Methods
### Dynamic Acoustic Rock Mechanical Parameter Test  
The elastic parameters in rock mechanics (elastic modulus, Poisson's ratio) are essential for the study of reservoir stress, inversion simulation calculations, and dynamic sealing evaluation in underground storage in depleted gas reserves. In order to accurately describe the evolution of reservoir stress, it is necessary to use acoustic and density logging data to measure the propagation velocity of ultrasonic waves in rock core samples to obtain dynamic mechanical parameters of rocks. The specific experimental steps are as follows:  
1. The preparation method of the rock core is consistent with static triaxial rock mechanical experiments.
2. Installation of rock core: Place the small rock core samples, which have been processed to the qualified size, into the rock ultrasonic parameter tester. When measuring the longitudinal wave velocity, vaseline is used as the coupling agent between the small rock core and the transducer, while when measuring the transverse wave velocity, the coupling agent between the small rock core and the transducer should be replaced with aluminum foil solid material to avoid errors caused by the rapid release of strain energy and the formation of stress waves. Apply a pressure of 0.05 MPa to fix the small rock core sample between the transducer and receiver.
3. Sonic wave testing: After the installation of the small rock core is completed, the sonic wave generator and oscilloscope are turned on. Considering the comparability of the results between static triaxial rock mechanics tests and dynamic sonic wave rock mechanics tests, the testing conditions for the small rock core sample are set to be consistent with the static triaxial rock mechanical test. The emission frequency of the transducer can be determined as: \[f\geq 40V_{p}\] (3)  
After setting the testing conditions, the sonic wave generator is used to emit longitudinal and transverse waves through the transducer to the fixed small rock core sample. After the sonic waves penetrate the small rock core, the transducer collects the sonic wave information, and the time difference between the transverse and longitudinal waves of the three small rock core samples in each well is recorded at the terminal. Additionally, after the testing is completed, the emission transducer and receiving transducer need to be connected to each other to measure the delay time between the transverse and longitudinal wave transducers and the instrument testing module. Then, the dynamic elastic modulus and dynamic Poisson's ratio are calculated separately using Equation (4).  
\[\begin{split} E_{d}&=2\rho V_{s}^{2}(1+\mu_{d})\\ \mu_{d}&=\frac{V_{p}^{2}-2V_{s}^{2}}{2\left(V_{p}^{2 }-V_{s}^{2}\right)}\\ V_{p}&=\frac{l}{\Delta t_{p}},V_{s}=\frac{l}{\Delta t _{s}}\end{split} \tag{4}\]  
where \(E_{d}\) is the dynamic elastic modulus of the rock, Pa; \(\mu_{d}\) is the dynamic Poisson's ratio of the rock; \(\rho\) is the density of the rock at the point of sound wave movement, kg/m\({}^{3}\); \(V_{p}\) is the longitudinal wave velocity, m/s; \(V_{s}\) is the transverse wave velocity, m/s; \(l\) is the length of the core sample of the reservoir rock to be tested in the direction of acoustic wave penetration, m.