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

摘要

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 mathematicaldescriptions of wave propagation outside a borehole. By listing the mathematicaldescriptions here we can show the equivalence and interrelationships of many treatmentspresented in the literature. Part one will conclude with an outline of the familiardiscrete wavenumber technique as applied to modelling radiation outside a boreholefrom a point source inside a borehole. Part two will attempt to isolate the effects ofthe fluid-filled borehole on the radiation of a borehole source by comparing radiationpatterns for three cases: a point source in an infinite medium, a stress applied to thewall of an empty borehole (Heelan's (1953) result) and a point source in a fluid-filledborehole (Lee and Balch, 1982). Heelan's results will also be analyzed and defendedagainst criticism of them brought by Jordan (1962) and Abo-Zena (1978).The literature review will be thorough and will include the work done directly onmodelling downhole seismic sources and the comparatively larger amount of work doneon modelling sources for acoustic logging purposes which is directly applicable.Different authors publishing work on seismic sources have made widely differentsymmetry assumptions ranging from no symmetry to axisymmetry to symmetry alongthe z axis. These differences hamper the ability to directly compare results betweenthe different treatments. Compounding the differences in symmetry are the use ofdifferent 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 asource and the synthetics are calculated using the discrete wavenumber method. Thesynthetic seismograms are measured along vertical arrays of point receivers placed outside the borehole and for lithologies of Pierre shale, Solenhofen limestone, and Bereasandstone. The method and the resulting seismograms are outlined in this paper alongwith a brief description of the discrete wavenumber technique.