fuel injector(s) would leak into the combustion chamber, so inside the engine. The rough running is the engine laboring to burn off the leaked fuel. Once the leaked fuel burns off then it runs fine. At a cold start, the leaked fuel has plenty of time to evaporate (overnight). If 5-10 minutes (say going inside a store) it's not enough time for much fuel to leak. But if the car is left for 20 mintues up to a few hours, the fuel can leak and then it sits there, unevaporated, until you start the car when it has to burn off.