Percolation Theory and Fire Spread

Abstract

Percolation theory deals with the statistics of random arrays. Two-dimensional percolation theory has been used as a model of forest fires. The important result is that if burnable elements are randomly placed on a two-dimensional array with a packing ratio p, then there is a value p_c below which fires will not propagate, and above which a fire will propagate from one end of the array to the other.

At p = p_c,. the system exhibits greatest fluctuations. Furthermore, at p equals; p_c fire properties such as the mean position of the fire-front and the total number of burnt sites should vary as power-laws. The critical exponent for these power-laws, as determined from simple percolation theory, were compared with those experimentally delermined by burning random arrays of matchsticks (with ignitable heads) in a square lattice. The most notable discrepancy is that the theory predicts that at critical percolation a fire-front decelerates, whereas the experiments indicate acceleration

Although simple percolation theory yields qualitative insights into expected fire behaviour, a correct quantitative theory needs to allow for both pilot ignition of adjacent matches and radiant ignition of distant matches.