| Summary: | 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
omission for our purposes, his work may nonetheless be seen as a starting point for the
modelling of downhole seismic sources. For instance, Lee and Balch (1982) developed
radiation patterns for fluid boreholes which were simple extensions of Heelan's results.
Additionally, one particular application of Heelan's theory is in the preliminary development of downhole seismic Sources that often require dry holes until the electronics can be properly shielded. For that reason, an exhaustive examination of the mathematics and physics that went into Heelan's first paper was undertaken to determine if his formulation was correct.
The fundamental basis of Heelan's work was a variant of the Sommerfeld integral,
an integral of cylindrical waves, in which he unfortunately did not specify the contour.
To overcome this obstacle of an unknown contour a parallel method suggested by
Brekhovskikh (1960, 1980) was implemented. Brekhovskikh used the Weyl integral,
an integral over plane waves, to duplicate Heelan's results for the radial and torsional
stresses. However he does no justification of the extensive algebra or analysis involved
and does not include the effects of axial stress. Thus in this paper, we have completed
and elucidated the work that Brekhovskikh initiated and moreover indirectly verified
that Heelan's results were correct.
Additionally, we found that Abo-Zena's and Heelan's initial formulations were
equivalent. The only difference was in a reversal of the separation of variables procedure necessary to replicate this work and also in Abo-Zena's USe of the Laplace
transform where Heelan used the Fourier transform. However, Abo-Zena's results do
extend Heelan's by allowing the source function to vary over the distance in which it
is applied. The far field results of Abo-Zena and Heelan are equivalent (White, 1983)
only if a 1/μ correction is applied to Abo-Zena's results.
The first half of this paper is very involved mathematically but much of the algebra
is relegated to Appendix A. Having verified that Heelan's results were correct we then
proceed to compare Heelan's results with well established point source representations
known in the literature (White, 1983) and also with radiation patterns from point
sources and stress sources in a fluid-filled borehole (Lee and Balch, 1982). These comparisons will help us isolate the propagation effects of the fluid and the geometrical
effect of the borehole. One unique aspect to our approach will be the consideration of
radiation from boreholes surrounded by varying lithologies instead of just the Poisson
solid as is commonly done. The lithologies to be considered include a soft sediment
(Pierre shale) and two more indurated sediments, Berea sandstone and Solenhofen
limestone. By following this approach we show that the effect on the radiation magnitude
can be substantial due to changes in lithology in addition to isolating the relative
effects of the borehole and the fluid.
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