It's a general misconception that a 'cavern' is left once an oil field is depleted. Actually, almost all oil pumped in the world comes from sandstone or limestone formations. This rock is reasonably porous and the oil, when discovered, is forced out of the formation to a the well zone by either natural gas pressure accompanying the oil or by pressurized water. The water usually used is highly salty water recovered from the oil producing strata. In the later stages of the life of the oil field, additional water, carbon dioxide or other chemicals are pumped under pressure into the strata to help in secondary or tertiary recovery.
By the way, when the oil well is first drilled and recoverable oil is discovered, the producing zone of rock is fractured, either chemically or hydraulically to assist in the flow of the oil droplets to the pipe stem...