Direct Calibration of the Heelan Model Using Vibration Data

The model of waves emanating from a blast hole proposed by Heelan, and further developed by Blair and Minchinton, provides an approximation of particle motion in an isotropic, continuous and homogenous medium caused by a cylindrical pressure source. Previous work in the literature shows that the shapes of the peak strain and peak particle velocity contours generated by this model are influenced by priming location. This paper presents the results of experimental investigations into the effects of priming position in an explosive column and the resulting shape of the surrounding stress and particle velocity fields. Fifteen emulsion charges were detonated in a limestone quarry, of which eight were bottom primed and seven top primed. The charges were fully confined, with no breakout to any surface. Particle velocity was monitored using tri-axial sensors coupled to the rock using cement, and pressure sensors were placed in water filled boreholes for fourteen of the experiments. Statistically significant differences in peak pressure for bottom and top primed shots were observed. Blair and Minchinton developed two pressure source functions, one of which is scaled using the charge weight scaling law. New models of the explosive pressure source are developed and compared to those found in the literature. These models are directly calibrated by predicting the peak velocities achieved at the known locations of the velocity sensors, then minimizing the residual sum of squares between the predicted and observed peak vibration values. Due to the complexity of the model the residual sum of squares cannot be minimized using standard non-linear optimization techniques, so a stochastic, agent based global optimization method is employed. Models are then cross validated by comparing the relative predicted differences in stress between bottom and top priming position to the observed differences.