The geological resources of salt cavern gas storage in China are mostly complex layered salt beds with many thin inter-layers and high insoluble matter content. In the process of cavity-building by water-solution method, the insoluble matters in salt layers and inter-layers are peeled off and deposited at the bottom of salt cavern, occupying more than one-third of the whole cavity volume. These sediments have a large pore volume and strong compressibility, and they are filled with brine; as a result, they have great potential for gas storage. The research on the flow law of brine and sediment particles in gas-driven brine drainage in the sediments of salt caverns is the basis of utilizing the void space of sediments for gas storage. In this paper, salt cores of salt cavern gas storage wells in the Jintan District were selected, and the physical characteristics of insoluble sediments of the salt cores were analyzed. Then, a laboratory simulation device and experimental method of the gas-driven brine drainage were presented. Using artificial composite sediments in the experimental device, the following was tested: (i) the flow rates of brine and particles in the vicinity of the brine drain pipe in the sediments under different nitrogen displacement pressures, (ii) the relationship between the sand extraction amount and nitrogen displacement pressure of different brine drain pipes, (iii) the sand extraction amount of different sizes of particles with brine drain time, (iv) the cumulative sand extraction amount of different brine drain pipes, and (v) the effect of brine flow rate on the sand extraction amount. The results show that quartz, plagioclase, and ankerite account for 45–94% and clay accounts for 3.3–14.4% of the insoluble minerals of the salt cores from the Jintan District. The particle size distribution of the sediments ranges from 0.04 mm to 6 mm and can be divided into four ranges: <0.5 mm, 0.5 mm~2 mm, 2 mm~4 mm, and >4 mm. The mass percentage of each range is 37.9%, 36.5%, 17%, and 8.6%, respectively. There is a threshold pressure of the gas-driven brine drainage, where the larger the diameter of the sieve hole, the lower the threshold pressure, and the easier the pipe is to sand out. The diameter of the sieve hole has a great influence on the flow rate of the sediment particles near the brine drain pipe. The increase in nitrogen displacement pressure has a positive correlation with the flow rate of sediments near the pipe with 5 mm diameter sieve holes, but has little effect on the flow rate of sediments near pipes with 1.5 mm or 0.5 mm diameter sieve holes. The sand extraction amount is affected by factors such as the nitrogen displacement pressure, diameter of sieve hole, brine drain time, and brine flow rate in the process of gas-driven brine drainage. A higher nitrogen displacement pressure and brine flow rate lead to more sand extraction. A screen pipe with 1 mm diameter sieve holes is suggested to be used for sand control, the sieve holes are recommended to be machined in the shape of a trumpet with a small inlet section (i.e., 1 mm) and a large outlet end (i.e., 1.5 mm), and the brine flow rate is suggested to be about 30 m3/h when the brine removal is carried out in the sediments of salt cavern, which depends on the actual operation on site.