J. E. White

Signals in a Borehole Due to Plane Waves in the Solid

This paper presents a theory describing the acoustic waves set up in a fluid‐filled borehole by the passage of plane elastic waves in the surrounding solid. It consists of a mathematical development of the following process. A stress in the solid around the borehole will in general cause the hole to contract or expand. A contraction compresses the fluid and causes pulses to radiate in both directions as tube waves. At some observation point, one of these pulses is observed after a delay due to the propagation as a tube wave. A complex wave in the solid is expressed as a combination of stresses distributed along elementary lengths of the borehole, and the total acoustic wave received at a point is expressed as the summation of elementary pulses arriving after suitable delays. The expressions derived give pressure or fluid particle velocity in terms of motion or stresses in the solid for plane compressional waves and plane shear waves arriving at any angle. The method can be extended to the case of a spherical wave cutting the borehole.

ELASTIC WAVES ALONG A CYLINDRICAL BORE

This paper concerns axially symmetric solutions for waves propagating along a cylinder in an infinite elastic solid. Solutions are presented describing unattenuated propagation along the axis at phase velocities higher than shear and compressional speeds in the solid, in contradiction to earlier publications. Special attention is given to the limiting case of phase velocity equal to compressional speed in the solid, which at low frequencies very closely approximates the coupling of a fluid‐filled borehole to a plane compressional wave in the surrounding solid. Comparison with some experiments in a uniform section of Pierre shale show excellent agreement at low frequencies. In the low‐frequency limit, these solutions reduce to an approximate expression for borehole coupling published earlier by the author.

Reply by authors to discussion by Ellis Strick

A short quotation from the paper under discussion may help provide the background for this rebuttal: “The intended message of this paper is quite simple: Attenuation very nearly proportional to frequency over a wide frequency range does not demand substantial velocity variation within this frequency band and does not require any unusual velocity variation at low frequencies. The approach is to present one counter‐example to the published allegations that such velocity variation is essential, in view of casuality and mathematical considerations.” The counter‐example, a lumped‐element model, is full and complete and the reader is invited to refer to the paper. The authors did not propose this model as the ultimate description of seismic wave propagation in rocks. In fact, the paper contained this encouragement for further research.

Elastic Waves along a Cylindrical Bore

This paper discusses axially symmetric solutions for elastic waves propagating along an infinite cylinder, namely those which can be expressed as F(r) eikz e−iwt. M. A. Biot [J. Appl. Phys. 23, 997–1005 (1952)] presented results for phase velo|cities along the axis less than shear velocity in the solid and indicated that for higher phase velocities, the waves cannot propagate without attenuation. The present paper gives solutions for higher phase velocities, showing that no attenuation in the direction of the axis is required. Special attention is given to the limiting case of phase velocity equal to the speed of compressional waves in the solid, which yields the coupling of a fluid‐filled borehole to a plane compressional wave in the surrounding solid. In the low‐frequency limit, this coupling agrees with an approximate expression published earlier [J. Acoust. Soc. Am. 25, 906–915 (1953)].