Modelling Of Downhole Seismic Sources I: Literature Review, Review Of Fundamentals, Impulsive Point Source In A Borehole

Abstract

This paper represents the first of a two paper sequence comprising a multi-faceted introduction to the numerical and analytical modelling of seismic sources in a borehole. Part one will present a literature review and a review of the fundamental mathematical descriptions of wave propagation outside a borehole. By listing the mathematical descriptions here we can show the equivalence and interrelationships of many treatments presented in the literature. Part one will conclude with an outline of the familiar discrete wavenumber technique as applied to modelling radiation outside a borehole from a point source inside a borehole. Part two will attempt to isolate the effects of the fluid-filled borehole on the radiation of a borehole source by comparing radiation patterns for three cases: a point source in an infinite medium, a stress applied to the wall of an empty borehole (Heelan's (1953) result) and a point source in a fluid-filled borehole (Lee and Balch, 1982). Heelan's results will also be analyzed and defended against criticism of them brought by Jordan (1962) and Abo-Zena (1978). The literature review will be thorough and will include the work done directly on modelling downhole seismic sources and the comparatively larger amount of work done on modelling sources for acoustic logging purposes which is directly applicable. Different authors publishing work on seismic sources have made widely different symmetry assumptions ranging from no symmetry to axisymmetry to symmetry along the z axis. These differences hamper the ability to directly compare results between the different treatments. Compounding the differences in symmetry are the use of different time dependencies ([superscript -iwl], [superscript iwl]) and the use of different Hankel function or modified Bessel function strategies. Therefore, the mathematical fundamentals of wave propagation in a borehole from the different symmetry systems are presented here in a consistent notation and are related to each other and treatments in the literature to help eliminate duplication of effort. Finally, wave propagation outside a borehole is examined using synthetic seismograms. For the synthetic seismograms, a point source inside the borehole is used as a source and the synthetics are calculated using the discrete wavenumber method. The synthetic seismograms are measured along vertical arrays of point receivers placed outside the borehole and for lithologies of Pierre shale, Solenhofen limestone, and Berea sandstone. The method and the resulting seismograms are outlined in this paper along with a brief description of the discrete wavenumber technique.