We seek simple analytical solutions in a model of gas flow driven by a combination of buoyancy, viscous, and capillary forces. Traveling-wave solutions describe propagation of the top and bottom of the gas plume. The top of the plume has low gas saturation, but propagates much faster than the bottom. The theoretical maximum of the velocity of propagation of the top of the plume provides a simple conservative estimate of the time until plume evolution will dramatically slow down. A sequence of rarefaction and traveling-wave solutions characterizes the transition zones between the top and bottom stable regions. The analytical results are applied to studying carbon dioxide flow caused by leaks from deep geological formations used for CO₂ storage. The results are also applicable for modeling flow of natural gas leaking from seasonal gas storage, or for modeling of secondary hydrocarbon migration.