There is quite a bit of life that dies and gets buried every day. And this has been true non-stop for billions of years. It's not like the process ceased when the dinos went extinct, right?

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Correct, but the pressures and temperatures required to convert kerogen require the organic matter to be buried miles below the surface. Common source rocks for the oil and gas basins along the Atlantic margin are Cretaceous or Jurassic in age (so greater than 65 mm years old). This is coincidentally around the time the dinosaurs went extinct, but that is irrelevant since kerogen is derived mostly from plant life.

The real question is what rates active source rocks are expelling oil and how long it takes that oil to recharge reservoirs. Keep in mind that all of this is taking place on the scale of thousands to 100s of thousands of years and typical oil and gas fields are typically depleted on a scale of decades to a century. It’s very unlikely any source rock is actively expelling enough oil to keep up with the rate of depletion.