Statistics of Boulder Encounters during Shaft Excavation

We study the problem of boulder encounter during shaft excavation and the bias that results from directly using statistics from borehole samples without correction for probe diameter. We specifically focus on two quantities, the rate of boulder encounters λ1 and the total length of obstructed shaft drilling L1. Assuming that boulders have spherical shape or cylindrical shape of a particular type, we evaluate how λ1 and the mean and variance of L1 depend on the shaft diameter, the distribution of the boulder diameter, and the minimum boulder diameter that constitutes an obstruction to drilling. The statistics on L1 assume that the boulder centers are distributed according to a homogeneous Poisson process in space. Finally, we briefly discuss the problem of inferring the spatial density of boulders and their diameter distribution from borehole data. Both λ1 and L1 increase significantly with increasing shaft diameter; hence the uncorrected statistics from borehole samples severely under-predict these two quantities during large-shaft drilling. Moreover, for typical parameter values, the mean value of L1 may be a significant fraction of the total shaft length.