Salt rock is considered as an ideal energy storage medium, and compressed air energy storage by a salt cavern can improve the utilisation efficiency of renewable energy. Salt rock in China mostly contains different interlayers, among which mudstone interlayers are the most common. At present, there are relatively few studies on the influence of mudstone interlayers on the long-term stable operation of gas storage. FLAC3D software was used to simulate the long-term operation of salt rock gas storage with different numbers of interlayers in the Yexian area of Pingdingshan. The results show that with the passage of time, the vertical displacement of the surrounding rock of the vertical single-cavity gas storage tank increases gradually. The maximum settlement value at the top of the surrounding rock is always greater than the maximum uplift value at the bottom. The horizontal displacement shows obvious symmetry with the vertical displacement at the top and bottom of the surrounding rock. The effect of the cyclic pressure interval on horizontal displacement is the same as that of vertical displacement. With the increase in the number of interlayers, the volume of the plastic zone gradually increases with the increase in the running time, and the increasing speed shows a growing trend.