Phosphate Depletion

A TOD poster and Energy Bulletin writer put together an intriguing analysis of phosphate depletion. As farmers use the phosphorus from rock phosphates as fertilizer to replenish the amount used up by crops, modern agricultural practices practically demands an ongoing and unlimited supply to keep production up. Although not quite as non-renewable as fossil fuel and helium sources, easily accessible phosphates practically assumes a non-replenishable role in planning for future resource utilization.

The authors, Dery and Anderson, applied Hubbert Linearizations to depletions on several scales and came up with what I would consider a valid empirical model for the current state of phosphate depletion.

As a nit, the "World Rock Phosphate Production" red curve they fit to above shows significantly asymmetry about the peak, particularly in the tails. Does this fit to the sigmoid function in the logistic derivation? If they fit to a sigmoid, the curve actually should show perfect symmetry about the peak. See below for a mirror image on the curve superimposed to show the asymmetry issue (note that by eyeballing, it might actually improve their visual fit).

Just recently, some real nitpickers have raked global warming climate scientists over the coals for an error correction that you can barely even detect in the graphs, see here and here. So it pays to make these corrections before the Freepers start to take potshots.

Otherwise, the post brings up some valid issues. Looking at a survey produced by the USGS [PDF], the authors seem not to even consider the potential critical importance of phosphate depletion:

U.S. phosphate rock annual production capacity is expected to fall during the next year when one mine in Florida and one mine in Idaho are anticipated to close because of depleted reserves. Production rates will not be affected though, owing to the reopening of a mine in Idaho and expansion at others in Florida. Complicated permitting procedures and public opposition to new mines have slowed the development of new mines in Florida. More than one-half of the current production capacity was expected to be exhausted by 2015, which would necessitate new mines or imports to maintain current production rates and leading world supplier status.

If you read between the "lah-di-dah" lines, the USGS authors seem to assume some new discoveries will take the place of ongoing depletion. As everyone seems to agree on, all plants and therefore all high-yield crops require phosphorus ... so we really have no substitute, other than to find more.

Of course, we can probably better back up a real argument for phosphate depletion by using better discovery and production models than the Hubbert Linearization heuristic favored by many oil depletion analysts and by Dery and Anderson themselves. In particular, I would suggest that the dispersive discovery model and the shock model would work and provide a set of valid physical parameters to use as a basis for analysis that the HL techniques do not provide. For one thing, it seems intuitive that the phosphorus deposits need easy access for profitable extraction. Fourteen years ago phosphate rock went for around $20 per ton (the post-processed fertilizer about $240 per ton), while a barrel of oil is about $70 for 0.136 tons (or $500/ton). That must have some impact on how far and deep we will go to recover phosphate rock, so a discovery model like the dispersive model we use for oil makes some sense.