This is most likely your EVAP system.
It could be a few things.
Some Audi A4/S4 owners complained about this specific issue and they might more prone to it, but so far I haven't experienced this. In all modern vehicles, there is a charcoal canister where the fumes in the fuel tank are stored as the tank vents when you are filling or when vapors are formed on a hot day. If it was saturated with liquid fuel (as happened with these folks), the vapor may not be able to be vented through this.
If line to the charcoal canister is plugged up, it would result in this as well.
If the canister purge valve has failed, it may not open when the engine is running to consume the excess fuel/vapor in the canister.
There may be a rollover valve on the fuel tank that is preventing the gasses from exiting the tank.
If you are in an area where the fuel pumps have a bellows to seal the filler, try to pull back slightly to allow fumes to escape out of the filler to see if this helps.
If that helps, diagnosing is the next step. You may be able to use a compressor to blow air through the evap line from the canister back to the fuel tank to clear an obstruction. With the evap line to the fuel tank disconnected, you could try to put a short piece of hose onto the canister and see if you can blow compressed air through it. If not, it's plugged and needs replacement.
If the purge valve fails, generally a DTC or fault code will be set.