Most old oil fields already do this, it is called "enhanced recovery". You have injection wells where you pump down water and production wells where a water/oil mix comes up. The water you use is mostly contaminated stuff you want to get rid of any ways.

You do have pump the water in under pressure though, just letting it pour down the hole will not work since rock is more dense the water. Imagine you were standing on a sponge in the shower, the sponge will not absorb much water since you are squishing all the pore spaces closed.