Word for someone who finds oil reservoirs

Petroleum geologist.

A petroleum geologist is a earth scientist who works in the field of petroleum geology, which involves all aspects of oil discovery and production. Petroleum geologists are usually linked to the actual discovery of oil and the identification of possible oil deposits or leads. [Wikipedia]

Petrogeologist is used also but it is less common.

Both my father and his father made their livings at precisely this job. The term used for it within the petroleum industry (in the United States, anyway) is exploration geologist.

A person who conducts seismic mapping of the strata of rock and other deposits in a particular area in order to help assess their potential as a source for oil or natural gas is called a stratigrapher.

Prospector.

Today prospectors are almost always (and when hired by serious interests, always) exploration geologists. Since their field is geology they are more likely to refer to themselves as geologists than prospectors.

But prospector covers both them and also those in earlier times whose methods were informed by less scientific knowledge than today.

Conversely, geologist covers geologists who apply their training to other ends.

If you're just guessing there might be oil down there and drill a well, you're a wildcatter. The days of wildcatters is long gone. All the crude that is ridiculously easy to find was found (and extracted) long ago.

If before drilling that expensive well, you drill a number of holes, sink small amounts of explosives into those holes, and apply some hairy mathematics on the seismograph readings (and other readings) that result to "see" what's down there, you're more likely to be a petroleum geologist (or more specifically, an exploration geologist).

The petroleum industry distinguishes between upstream, midstream, and downstream, and within upstream, it distinguishes between exploration and production. This is a question about exploration.