I heard of it as abiotic instead of abiogenic petroleum, but I think it is the same thing.
The proponents of abiotic oil that I have heard state that petroleum is produced deep in the earths crust or mantle by some unexplained process (the Keebler Elves maybe?) and slowly works its way up towards the surface. So there is no reason to worry about depleting oil reserves.
The problem with this is, is that the we find petroleum in the areas where the biogenic model says we should. If oil was produced by an abiogenic process, why do we find it in sedimentary and metamorphic rock formations and not igneous ones?
It also goes against what we know about hydrocarbon chemistry. When hydrocarbons are exposed to high enough amounts of heat and pressure, they break into smaller chains. This is the process called cracking, and is commonly used to make usable fuels out of heavier crude oil. If enough heat and pressure is applied, the carbon chains will break down to the simplest hydrocarbon, methane, also known as natueal gas.
Since heat and pressure increase the deeper we go into the earth, petroleum would break down, not form.
Biogenic pertoleum formation has a whole lot more evidence to back it up.