Yes mutt, it's an incomplete question, but yours is the best response so far.
From another perspective, even if not Field Limited on the 6000 ft runway with the "Increased" Takeoff speeds available (Which I assume to be the case as the original poster refers to Obstacle Limit Only), the 12000 ft runway still wins from the obstacle perspective.
Even if the aircraft was capable of the Structural Limit as the maximum field limit (even at the increased speeds) on the 6000 ft runway, then, at the same weight on the 12000 ft runway, the obstacles are a further 6000 ft distant, thus requiring lesser obstacle clearance gradient for their clearance. The only exception to this is a very UP sloping runway where the 1st segment obstacle may be the runway itself, with essentially level ground beyond and close-in (This happens on a few very long UP sloping runways that we use (Melbourne RWY 34 for example at +0.9% where the aircraft 'basic' 1st segment gradient might only be of the order of +0.5%, and weight must be reduced to achieve satisfactory 1st segment clearance)).
Too many variables, here's one more of them.
Happy New Year to all,
Old Smokey