Amp hour (Ah) is a unit of measurement that has nothing to do with power and it’s a totally different unit than Amp (Ampere). Amps are used to describe amp hour, but they are different units of measurement.
Amp hour describes the capable charge of a battery. Theoretically, we can draw 2 amps, continuously, for 60 minutes before our 2 amp hour battery is drained. Just think of amp hour as a way to know that one battery holds more of a charge than another. Just like the size of the gas tank in a car.
That said, if the drilling operation drew one amp continuously then yes a 2 amp hour battery would last two hours.