For as long as the writers need. May be four days.. a week, or exactly seventeen days, two hours and fifty three minutes, just enough time to get to star base insert name and get saved!
According to the TNG Tech Manual there are smaller fuel tanks on deck 9 of the Saucer Section that are used during seperated flight mode. If the Saucer was going to be seperated for a long period of time, then there are several ways for it to refuel. The impulse engines run on deuterium which is easily replicated. Or they could use the transporter to lock onto interstellar gas. Or they could use the shuttles ot visit nearby worlds and collect it. If for some reason, these options were not available, then they could make deuterium from regular water. It would not be a difficult thing to do as any first year chemistry student can do it. There are several wastewater recycling centers on a Galaxy-class ship that make drinking water from sewage. Instead, these recycling centers could take the sewage and make deuterium from it. Also, the ship would not have to use the engines except when accelerating or decelerating. The rest of the time, the engines would be used only for power generation which would decrease the amount of fuel needed. So, as long as the crew are well fed, then the ship would have an endless suplly of deuterium. Admiral William Thomas Riker 19:31, August 13, 2011 (UTC)
It is probably Starfleet policy that Galaxy-class starships operating in separated-flight mode is to be deemed a very extraordinary event, the duration of which is to be minimized. Nevertheless, in order to allow a separated-flight mode, there must be engineering and operational contingencies, as described above, built in ProfessorTrek 06:32, August 14, 2011 (UTC)