From the Wikipedia page on abiogenic petroleum origin:

Abiogenic petroleum origin is a term used to describe a number of different hypotheses which propose that petroleum and natural gas are formed by inorganic means rather than by the decomposition of organisms.

The abiogenic hypothesis regained support in 2009 when researchers at the Royal Institute of Technology (KTH) in Stockholm reported they believed they had proven that fossils from animals and plants are not necessary for crude oil and natural gas to be generated.

The weight of evidence currently shows that petroleum is derived from ancient biomass. However, it still has to be established conclusively, which means that abiogenic alternative theories of petroleum formation cannot be dismissed.

Abiogenic hypotheses generally reject the supposition that certain molecules found within petroleum, known as biomarkers, are indicative of the biological origin of petroleum. They contend that these molecules mostly come from microbes feeding on petroleum in its upward migration through the crust, that some of them are found in meteorites, which have presumably never contacted living material, and that some can be generated abiogenically by plausible reactions in petroleum.

Occurrences of abiotic petroleum in commercial amounts in the oil wells in offshore Vietnam are sometimes cited, as well as in the Eugene Island block 330 oil field, and the Dnieper-Donets Basin.

Modern geologists think that commercially profitable deposits of abiotic petroleum could be found, but no current deposit has convincing evidence that it originated from abiotic sources.

Panhandle-Hugoton field (Anadarko Basin) in the south-central United States is the most important gas field with commercial helium content. Some abiogenic proponents interpret this as evidence that both the helium and the natural gas came from the mantle.

The Bạch Hổ oil field in Vietnam has been proposed as an example of abiogenic oil because it is 4,000 m of fractured basement granite, at a depth of 5,000 m.

A major component of mantle-derived carbon is indicated in commercial gas reservoirs in the Pannonian and Vienna basins of Hungary and Austria. Natural gas pools interpreted as being mantle-derived are the Shengli Field and Songliao Basin, northeastern China.

The Chimaera gas seep, near Çıralı, Antalya (southwest Turkey), has been continuously active for millennia and it is known to be the source of the first Olympic fire in the Hellenistic period. On the basis of chemical composition and isotopic analysis, the Chimaera gas is said to be about half biogenic and half abiogenic gas, the largest emission of biogenic methane discovered; deep and pressurized gas accumulations necessary to sustain the gas flow for millennia, posited to be from an inorganic source, may be present.

Incidental arguments for abiogenic oil

Given the known occurrence of methane and the probable catalysis of methane into higher atomic weight hydrocarbon molecules, various abiogenic theories consider the following to be key observations in support of abiogenic hypotheses:

The serpentinite synthesis, graphite synthesis and spinel catalysation models prove the process is viable.

The likelihood that abiogenic oil seeping up from the mantle is trapped beneath sediments which effectively seal mantle-tapping faults.

Outdated mass-balance calculations for supergiant oilfields which argued that the calculated source rock could not have supplied the reservoir with the known accumulation of oil, implying deep recharge.

The presence of hydrocarbons encapsulated in diamonds.

The proponents of abiogenic oil also use several arguments which draw on a variety of natural phenomena in order to support the hypothesis:

The modeling of some researchers shows the Earth was accreted at relatively low temperature, thereby perhaps preserving primordial carbon deposits within the mantle, to drive abiogenic hydrocarbon production.

The presence of methane within the gases and fluids of mid-ocean ridge spreading centre hydrothermal fields.

The presence of diamond within kimberlites and lamproites which sample the mantle depths proposed as being the source region of mantle methane.

The presence of methane on Saturn's moon Titan and in the atmospheres of Jupiter, Saturn, Uranus and Neptune is cited as evidence of the formation of hydrocarbons without biological intervention.

According to Thomas Gold, a major proponent of this theory: “Hydrocarbons are not biology reworked by geology (as the traditional view would hold), but rather geology reworked by biology.”

Further reading:

Inorganic Origin of Petroleum

The Origin of Crude Oil or Petroleum: Biotic or Abiotic?

Are oil and water shortages manufactured in order to control the economy and human populations? A closer look at abiotic petroleum and primary water.

The Abiotic Oil Controversy