In 1972, the radioactive remains of such a nuclear reactor was found in the state of Gabon in West Africa, in the Oklo mines. Uranium extracted from that mine was abnormally short in U-235 isotopes, and upon examination, French scientists found that the uranium isotope levels had an uncanny resemblance to those in spent nuclear fuel from modern nuclear power plants. The evidence was strong enough to suggest a natural reactor, and further exploration confirmed it.
At the time of discovery, scientists were uncertain exactly how the Oklo reactor had operated without exploding or melting down. For 150 million years, it ran like clockwork with a 30 minute reaction cycle, followed by a 2.5 hour cool-down cycle, putting out an average of 100 kilowatts of power. And it was always exactly 30 minutes per cycle, without significant variation, which was baffling. But recent studies have finally solved the mystery by discovering the regulating mechanism: Water.
Under normal conditions, radioactive atoms like U-235 cast off neutron particles at speeds so high that most of the neutrons skip off the surface of other atoms and fly away. But if you put enough of the radioactive material together, the cast-off neutrons bounce around inside the mass, some slowing down enough to be absorbed into another atom's nucleus. The extra neutron causes the nucleus to become unstable and immediately split, which releases a large amount of energy. If one has enough radioactive material in sufficient density that a lot of nuclei split very rapidly (critical mass), the reaction increases exponentially, and results in an atomic explosion. Any less than that (subcritical mass), and it causes a sustained fission reaction, giving off energy as heat and radiation.
Fifteen such natural reactors have been found in the Oklo area, and they are now collectively referred to as the "Oklo Fossil Reactors." These natural reactors are providing useful data on long-term storage of spent nuclear fuel, as well as some insights into possible improvements in man-made reactors.