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Numerical and Analytical Modelling of Downhole Seismic Sources: The Near and Far Field

Jeffrey A. Meredith

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on October 31, 1990 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

The physical and mathematical description of seismic sources is well developed in the fields of earthquake seismology. In contrast, the description of radiation from seismic sources that have been placed in boreholes is poorly developed. Most of the work describing downhole seismic sources has consisted of the evaluation of integrals in the far field using the method of stationary phase. These far field descriptions, however, fail to describe long-standing experimental observations made in low velocity sediments surrounding fluid-filled boreholes. In particular, the far field descriptions fail to duplicate the relative increase in shear wave amplitude in the vertical direction for low velocity sediments. However, the far field representations do describe radiation into high velocity sediments fairly well.

Two approaches, one analytical and one numerical, were used in this thesis to enhance the description of radiation from downhole seismic sources. The numerical approach involved development of a Thomson-Haskell type algorithm to investigate radiation from axial, torsional, radial and volume point sources. The analytical approach was to verify, explain and most importantly extend the far field analysis for the empty and fluid-filled borehole.

Using the analytical and numerical approaches and comparison between the two techniques proved to be a powerful tool for the description of radiation from downhole seismic sources. The presence of fluid in the borehole, the presence of casing surrounding the borehole, and the velocities of the lithology surrounding the borehole were shown to be important elements governing the observed radiation. One exception is that in the case of axial and torsional sources, radiation was found to be independent of borehole effects.

For radial and point sources, P wave radiation is somewhat affected by the borehole but not significantly by the presence of fluid in the borehole. The P wave radiation is essentially spherical with a slight perturbation along the borehole axis. The effect of casing on the P wave radiation pattern is limited to reducing the amplitude.

Radiation of S waves from radial and point sources was most significantly affected by the borehole. Radiation of S waves from empty borehole and into high velocity sediments surrounding fluid-filled boreholes was found to be qualitively similar and well understood. Radiaition of S waves into low velocity sediemnts surrounding boreholes was found to be controlled by the tube wave travelling up the borehole. In fact, the tube wave generates conical wave fronts which produce part of the radiated S wave field. This has been hypothesized on the basis of experiemntal evidence and here the physics and mathematics of the phenomena is thoroughly explored. It is shown that these conical wavefronts are in fact Mach waves analagous to their counterparts in aerodynamics and seismology. Mathematically, Mach waves are governed by a tube wave pole and the proimity of the integration path to this pole governs the behavior of the radiated S wave field. The dependence differs according to the velocity range:

• ß > af the shear wave velocity is greater than the fluid velocity and the S wave radiation is essentially similar to that for an empty borehole. As ß approaches af a greater effect due to the tube wave pole is observed.
• af > ß > CT when the shear wave velocity is in between the fluid velocity and the tube wave velocity the existence of shear wave lobes is confirmed.
• af > CT > ß when the shear wave velocity is less than the tube wave velocity, a supershear condition exists and Mach waves are radiated away from the borehole. If the borehole is cased CT will increase to be closer to af and this velocity range is more likely to be encountered.

Calculation of the residue for the tube wave pole allows the contribution of the Mach wave to be isolated and calculated. Through this residue calculation, analogous to the calculation of the residue of the Rayleigh wave in seismology, the existence of the necessary phase delays for Mach wave radiation are derived and this phase delay was seen to be equivalent to that determined by geometric arguments. Additionally, a geometric decay of 1/r could be derived from the calculation of the residue, where r2 is the horizontal distance between the source and receiver not the total distance.

Comparison of seismograms generated with the numerical algorithm and those generated with the radiation pattern formulas showed good agreement when lithologies with high velocities surrounded the source borehole. Comparison of the algorithm results with two data sets from published experiments measuring radiation from point sources (dynamite) into low velocity sediments surrounding the borehole showed excellent agreement and helped resolve long standing differences with the theoretical predictions. A third data set showing radiation from a downhole axial source into a low velocity sediment also agreed with the algorithm results quite well.