This work considers the management of an aquifer with stochastic recharge and finite boundaries. For a class of such models we show that there is a positive threshold of the water stock that the manager should aim at keeping. Thus, under the optimal extraction regime at a time when the water stock is lower than this threshold, there should be no pumping from the aquifer, while at times when the water stock is greater than the threshold, all water in excess of the threshold level should be pumped. The threshold level happens to coincide with the aquifer boundary for sufficiently high unit cost of pumping, in which case, only the runoff water of excess recharge should be collected.