The problem of determining the change of the volume and pressure of a gas cavity with time in a mildly sloping seam of the dome type, used as an underground gas storage vault, is solved by the method of combined asymptotic expansions; the gas storage vault has an arbitrary law of variation of discharge with time. A comparison is carried out with the existing exact solution for the case of a seam with constant thickness; good agreement is revealed. Calculations are carried out on the dynamics of a gas cavity in an actual seam. The case of a seam with a steep dome and a strong anisotropic permeability is also considered.