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With the development of salt cavern gas storage technology, the construction of large-scale salt cavern gas storage using sediment voids is expected to solve the problems of low effective volume formation rate and poor construction economy of high-impurity salt mines. At present, there are few studies on the long-term operational mechanical behavior of salt cavern gas storage under the influence of sediment accumulation. The present paper studies the influence of sediment height, particle gradation, and operating pressure on the stability of salt caverns by constructing a coupling model of sediment particle discontinuous medium and surrounding rock continuous medium. The continuous–discontinuous coupling algorithm is suitable for analyzing the influence of sediment height and particle gradation on the creep shrinkage of salt caverns. The increase in sediment height slows down the creep shrinkage of the cavern bottom, which strengthens the restraining effect on the surrounding rock of the cavern. As a result, the position of the maximum displacement of the surrounding rock moves to the upper part of the cavern. The sediment particle gradation has little effect on the cavern volume shrinkage rate. The greater the coarse particle content, the smaller the cavern volume shrinkage rate. The greater the operating pressure, the more conducive to maintaining the stability of the cavern. This situation slows down the upward movement of the sediment accumulation and increases the gas storage space in the upper part of the cavern. The obtained results can provide a reference for evaluating the long-term operational stability of sediment-type high-impurity salt cavern gas storage.