No.
To explain: an oil reservoir is not a 'cavern', not a balloon, but the small void spaces left between grains of porous rock. The reservoir itself can be envisioned by taking a sack of marbles and dumping them in a glass. If you look, there is space between the marbles, and those spaces are interconnected. The marbles dont flaot when the glass is filled with water, but stick a straw in there and you can suck the water out.
On a smaller scale, the spaces between sand grains work the same way. On a larger scale, so does the reservoir of an oil or gas field, with innumerable smaller pores connected, filled with oil and gas. Remove the oil and gas, and while some settling may occur, there is no cavern to collapse, the contact between the grains that supported the caprock will continue to support it, because it was never one big void. The gas dissolved in the oil expands as the pressure is lowered by producing oil and fills the pore space nicely, but the grains in the rock do the heavy lifting, the oil and gas just fill the leftover space.
Notice, as you suck the water out of the glass full of marbles, the marbles don't collapse. Neither will the rock in the gulf.
Thank you!!
An awesome analogy, much better than I’ve seen. I knew FReepers would blow this one out!! (pun intended)