Why? Just give it back equally, and finance amelioration out of general revenue. (Not saying that's a good idea. You could finance amelioration out of a specific percentage of that tax.)
I suspect that would cause problems for the urban poor. Possibly also anyone for whom food makes up a large proportion of their annual budget, given both how reliant growing and transporting that is on oil and how reliant farmers are on internal-combustion vehicles.
Unfortunately, I'm not finding the numbers I need to get a good estimate. The important question is what portion of petroleum use winds up servicing those poor people versus the rest of the economy, because while it is certainly true that a small percentage increase in the costs of necessities hurts these people disproportionately, the money returned is a much greater portion of their income, and could very well dwarf the increased costs (indeed, this is what I would expect).
The very worst case, of course, is any who find themselves paying higher prices but unable for whatever reason to access the stipend.
