Processes and apparatus are disclosed for separating and purifying aqueous solutions such as seawater by causing a substantially impermeable mat of gas hydrate to form on a porous restraint. Once the mat of gas hydrate has formed on the porous restraint, the portion of the mat of gas hydrate adjacent to the restraint is caused to dissociate and flow through the restraint, e.g., by lowering the pressure in a collection region on the opposite side of the restraint. The purified or desalinated water may then be recovered from the collection region. The process may be used for marine desalination as well as for drying wet gas and hydrocarbon solutions. If conditions in the solution are not conductive to forming hydrate, a heated or refrigerated porous restraint may be used to create hydrate-forming conditions near the restraint, thereby causing gas hydrates to form directly on the surface of the restraint.