Exact solutions in a model of vertical gas migration

This work is motivated by the growing interest in injecting carbon dioxide into deep geological formations as a means of avoiding atmospheric emissions of carbon dioxide and consequent global warming. One of the key questions regarding the feasibility of this technology is the potential rate of leakage out of the primary storage formation. We seek exact solutions in a model of gas flow driven by a combination of buoyancy, viscous and capillary forces. Different combinations of these forces and characteristic length scales of the processes lead to different time scaling and different types of solutions. In the case of a thin, tight seal, where the impact of gravity is negligible relative to capillary and viscous forces, a Ryzhik-type solution implies square-root of time scaling of plume propagation velocity. In the general case, a gas plume has two stable zones, which can be described by travelling-wave solutions. The theoretical maximum of the velocity of plume migration provides a conservative estimate for the time of vertical migration. Although the top of the plume has low gas saturation, it propagates with a velocity close to the theoretical maximum. The bottom of the plume flows significantly more slowly at a higher gas saturation. Due to local heterogeneities, the plume can break into parts. Individual plumes also can coalesce and from larger plumes. 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