Modelling Of Downhole Seismic Sources II: An Analysis Of The Heelan/Brekhovskikh Results And Comparison Of Point Source Radiation To Radiation From Boreholes

摘要

The work of Heelan (1952, 1953a,b) was one of the first studies of wave propagationfrom a cylindrical boundary. Heelan attempted to model the radiation emanating froma cylindrical shot hole filled with dynamite. To do so he applied a constant stress toa finite length of an empty infinite cylindrical cavity embedded in an infinite elastic,homogeneous medium. The stresses he considered were axial, torsional, and radialstresses. The radial and axial stresses were required to be proportional to each otherand of the same duration.To date Heelan's work has been referenced in over 100 articles and 15 differentjournals including recent works (Paulsson, 1988) . His results have also been comparedwith results from the reciprocity theorem (White, 1953, 1960) and played an integralpart 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 theradiation patterns for an axial source and a torsional source in a borehole have thesame 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) whodevoted an appendix of his 1977 paper to criticizing Heelan's results. The main pointof 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 crucialomission for our purposes, his work may nonetheless be seen as a starting point for themodelling of downhole seismic sources. For instance, Lee and Balch (1982) developedradiation 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 byBrekhovskikh (1960, 1980) was implemented. Brekhovskikh used the Weyl integral,an integral over plane waves, to duplicate Heelan's results for the radial and torsionalstresses. However he does no justification of the extensive algebra or analysis involvedand does not include the effects of axial stress. Thus in this paper, we have completedand elucidated the work that Brekhovskikh initiated and moreover indirectly verifiedthat Heelan's results were correct.Additionally, we found that Abo-Zena's and Heelan's initial formulations wereequivalent. 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 Laplacetransform where Heelan used the Fourier transform. However, Abo-Zena's results doextend Heelan's by allowing the source function to vary over the distance in which itis 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 algebrais relegated to Appendix A. Having verified that Heelan's results were correct we thenproceed to compare Heelan's results with well established point source representationsknown in the literature (White, 1983) and also with radiation patterns from pointsources 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 geometricaleffect of the borehole. One unique aspect to our approach will be the consideration ofradiation from boreholes surrounded by varying lithologies instead of just the Poissonsolid as is commonly done. The lithologies to be considered include a soft sediment(Pierre shale) and two more indurated sediments, Berea sandstone and Solenhofenlimestone. By following this approach we show that the effect on the radiation magnitudecan be substantial due to changes in lithology in addition to isolating the relativeeffects of the borehole and the fluid.