If one were to model this phenomenon, a Gaussian plume would probably be the most straightforward method. So, what do we know? According to the site submitted by Duck Duck Goose:

On the average a fart is composed of about 59% nitrogen, 21% hydrogen, 9% carbon dioxide, 7% methane, and 4% oxygen.

We can use that info to find the expected density of the gas using the ideal gas law comparing 98.6F to STP. For the purposes of this equation, we’d assume the stinky parts are homogeneous with the bulk of the fart at least until the room is filled. We’d also assume that the plume would have a turbulent interface.

There would have to be two parts to the equation, one with a pure momentum flux (buoyancy), that would be used in the case of a person sitting down (with their sphincter pointing down, therefore not imparting any horizontal component into the plume’s velocity vector). The additional part, the mass flux, would therefore (and obviously) take into account the horizontal velocity imparted onto the plume by the bum. Finally, ambient wind speed must be taken into account.

Now, to use my least favorite textbook phrase: I’ll leave the solution to the reader.

Disclaimer: I took most of this info about plumes from the wiki page. Most of what I know about plumes is from reading about how a MEOI (maximally exposed offsite individual) would be effected in the event of a radionuclide spill. Most of the actual simulations are done via computer programs.