Modelling of Downhole Seismic Sources Ii: an Analysis of the Heelan/brekhovskikh Results and Comparison of Point Source Radiation to Radiation from Boreholes

The work of Heelan (1952, 1953a,b) was one of the first studies of wave propagation from a cylindrical boundary. Heelan attempted to model the radiation emanating from a cylindrical shot hole filled with dynamite. To do so he applied a constant stress to a finite length of an empty infinite cylindrical cavity embedded in an infinite elastic, homogeneous medium. The stresses he considered were axial, torsional, and radial stresses. The radial and axial stresses were required to be proportional to each other and of the same duration. To date Heelan's work has been referenced in over 100 articles and 15 different journals including recent works (Paulsson, 1988) . His results have also been compared with results from the reciprocity theorem (White, 1953, 1960) and played an integral part of important books including those by Brekhovskikh (1960, 1980) and White (1965, 1983). His fundamental contributions were the description of shear wave lobes, the famous four-leaved rose, generated from a radial source in a borehole and that the radiation patterns for an axial source and a torsional source in a borehole have the same geometries as the point axial and torsional sources in infinite media. Despite the importance of this work, Heelan's results have been criticized by Jordan (1962) who dismissed the work as mathematically unsound and Abo-Zena (1977) who devoted an appendix of his 1977 paper to criticizing Heelan's results. The main point of contention has been the use of contour analysis in his first paper (Heelan, 1953a). Although Heelan's work did not include a fluid-filled borehole which is a crucial