My understanding of the way it works is that the engines use X amount of fuel to operate at full power.
The speed the vessel or vessels move at is related to the power generated by the engines vs. the total mass the engines are attempting to move.
In your case, a tug moving from point A to B will always use the same amount of fuel, whether it weighs 40,600 (just the tug) or 12,540,600 (tug + OH). It will just take a lot longer, as you experienced, since the Power produced by the engines will move the combined mass much slower than only the tug.
3100 Power pushing 40,600 tons = ~13.1 tons per unit of engine power
vs.
3100 Power pushing 12,540,600 tons = ~4,045.4 tons per unit of engine power