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In borehole seismic experiments the presence of the borehole has a significant effect on observations. Unfortunately, including boreholes explicitly in modeling schemes excludes the use of some methods (e.g., frequency-wavenumber) and adds prohibitively to the cost of others (e.g., finite difference). To overcome this problem, the authors use the concept of an effective source/receiver array to replace the explicit representation of the borehole by a distributed seismic source/receiver. This method mimics the presence of the borehole at seismic frequencies under a wide variety of conditions without adding a significant computational cost. It includes the effects of dispersive and attenuative tube wave propagation, the generation of secondary sources at interfaces and caliper changes, and the generation of conical waves in low-velocity layers. Comparison with a finite-difference scheme with an explicit borehole representation validates the approach. The modeling method applied to a continuity logging geometry demonstrates that the presence of guided waves does not uniquely imply bed connectivity. Results for a single-well imaging geometry emphasize the dominance of the tube wave in the hydrophone synthetics and demonstrates the necessity of using clamped geophones for single-well experiments. The concept of an effective source/receiver array is an efficient way of including borehole phenomena inmore » seismic modeling methods at minimal extra computational cost.« less

Radiation of elastic waves from a point force or from a localized torque into a transversely isotropic medium has been formulated in terms of displacement potentials, and transient waveforms have been computed by numerical Fourier inversion. For isotropic sandstone, this procedure yields P‐ and S‐wave pulses whose arrival times and magnitudes agree with theory. For a range of anisotropic rocks, arrival times of quasi‐P‐waves and quasi‐S‐waves agree with asymptotic theory. For extreme anisotropy, some quasi‐S‐wave pulses arrive at times which are not predicted by asymptotic theory. Magnitudes have not been compared with results of asymptotic theory, but decrease with distance appears to be in agreement. This Fourier inversion method gives near‐source changes in waveform which are not obtainable from the asymptotic theory.

White (1983), among others, has shown that the propagation and attenuation of Stoneley waves in a fluid‐filled borehole contain information on the porosity and permeability of the surrounding formation. Prior to extracting permeability from attenuation, the correct viscous loss of Stoneley waves in a borehole in a nonporous medium must be completely understood. With this motivation in mind, the purpose of this paper is to derive the viscous loss due to drag at the wall of a borehole from the exactly solvable case of a perfectly cylindrical borehole in an elastic, nonporous formation filled with a fluid that is viscous in shear. Thus the fluid pressure and fluid motion in the small but critical region near the fluid‐solid interface can be calculated. We have obtained dispersion and attenuation of trapped waves and tube waves, but here we report only on tube waves.

Acoustic waveform measurements in boreholes have important applications in fracture hydrology and radioactive waste disposal, but ambiguities in existing interpretation techniques remain a problem. We have addressed the problem by using residue theory to predict the relative excitation of various modes contained in experimental waveforms. A plane‐geometry model involving a layer of fluid between two elastic half‐spaces is shown to provide velocity dispersion curves for propagating modes that are very similar to those for the fluid‐filled borehole. We use the plane‐geometry model to illustrate the effects of the confined borehole fluid on surface and body waves traveling along the borehole in the elastic solid. We also computed excitation functions for some of the lowest‐order symmetric modes, calculated the time‐domain response of the trapped modes following the shear head waves, and compared them to waveforms recorded in boreholes through several homogeneous formations. The insight into the mode composition of the experimental waveforms obtained in these formations is used to construct amplitude logs that should be especially sensitive to variations in the presence of fluid‐filled fractures in the borehole wall. Initial tests show the technique is most successful when the waveform is dominated by the fundamental tube wave, and yet frequencies remain relatively high. The model analysis indicates these conditions can only be obtained when the borehole diameter is not much larger than that of the logging tool.

Calculations for an unconsolidated sand with partial gas saturation show a 20 percent increase in compressional wave velocity between 1 and 100 hz and attenuation of 27 db/1000 ft at 31 hz and 82 db/1000 ft at 123 hz. Shear velocity and attenuation are not affected. Fluid‐flow waves are shown to be responsible for the dispersion and attenuation at low frequencies; relations are derived by extending Gassmann’s viewpoint to include coupling between fluid‐flow waves and seismic body waves. This appears to be an important loss mechanism for heterogeneous porous rocks.

Many measurements have been made on fluid‐saturated porous rods executing extensional, flexural, and torsional motion. Measurements for extensional and flexural motion yield a loss parameter for Young's modulus waves [Formula: see text], and the measurement for torsional motion yields [Formula: see text] for shear waves. [Formula: see text] has then been calculated for compressional waves in bulk rock, on the assumption that the fluid‐saturated rock is an isotropic solid. I point out the fallacy of computing [Formula: see text] from these measurements and also urge workers to recognize the losses due to simple fluid viscosity in interpreting their data on extensional waves in rods. By application of published theory, I show that peaks in attenuation of extensional waves are to be expected at frequencies of several hertz to several kilohertz, depending upon rod radius. Computed curves are compared with published measurements on Navajo sandstone saturated with water, ethanol, and n‐decane. In each case, computed peak frequency agrees with published measurements. Shift of the peak frequency with temperature from 4 °C to 25 °C is due to change of viscosity of the saturating fluid (water).

The model proposed describes the radiation of the primary seismic pulse from an explosive source. The model incorporates a viscous Voigt liquid in description of the nonelastic rock deformation that the explosion causes. The problem is solved in the frequency domain, followed by Fourier synthesis to obtain waveforms. Comparisons of the theoretical results with field data in three sedimentary environments of shale, sandstone, and marl indicate good quantitative agreements in shape and duration of transient pulses. Values of angular normal stress obtained at the deformed boundary suggest a dynamic tensile strength of rocks that decreased with duration of the tensile stress transient. The radius of the deformed zone (B) scales with the charge size Q approximately as [Formula: see text] where n is [Formula: see text] or larger.

A simple method is described for computing the waveform to be expected from a closely‐spaced pattern of shots or geophones when the individual signal is a pulse. Four strength distributions are compared, and the case of uniform strength along the pattern is discussed in detail. It is concluded that if the individual signal is a short pulse, such as a seismic reflection, then the performance of the pattern is quite comparable for the four source distributions considered.

This paper discusses a particular mechanism for energy loss in granular media, involving sticking and sliding between grains at areas of contact. Earlier publications have developed expressions for attenuation in regular packings of spheres, attributing all the loss to sliding friction. The present work extends the model by incorporating static friction and thereby derives an expression for attenuation which is in better agreement with measured values. The attenuation coefficient for both shear and compressional waves is found to be: 1. Independent of vibration amplitude at low amplitudes; 2. Directly proportional to frequency; 3. Dependent on the difference between static and sliding coefficients of friction; and 4. Dependent on the packing of the granules.

Starting with a simple word statement of the reciprocity which exists between forces and displacements in a general elastic solid, it is shown that low‐frequency radiation of shear and compressional waves from relatively complex sources can be obtained by solving relatively simple problems in static elasticity. Illustrative examples include radiation from radial and tangential pairs of forces acting on the wall of a cylinder, pressure in a finite cylinder, and a pair of radial forces in a hole in a plate. For the last case, measurements in a plexiglas plate compare favorably with computations.

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.

This paper describes a means of combining the voltages from a three‐component seismometer so as to suppress noise waves arriving randomly from all directions and to indicate the direction of arrival of any particular wave‐train, such as might be caused by an earthquake, explosion, or moving vehicle. Each of the horizontal components of motion is electronically multiplied by the vertical component, with or without phase shift, and the two resulting products are displayed as a vector pointing to the source of seismic waves. The equipment assembled to perform these operations is referred to as the Motion Product Detector. Mathematical analysis and small‐scale tests of the Motion Product Detector on the Research Center grounds look promising.

This experimental study of the generation of shear waves by explosive sources stemmed from Heelan’s theoretical result that pressure acting on the wall of a cylindrical hole in a solid should radiate shear waves quite as effectively as compressional waves. The measurements confirm this expectation, but good overall agreement was not achieved until expressions were derived which take into account radiation from strong water‐pulse waves in the shothole. Our results show that the ratio of shear‐to‐compressional amplitudes generated by an explosive source increases as the charge size decreases. At an angle of 45 degrees, the ratio is approximately unity for a charge consisting of 10 ft of Primacord. We found that the shot‐generated water pulse (tube wave) is a strong shear source, continuously generating shear energy in the formation as it travels in the borehole. This drastically affects the directivity of SV waves and in Pierre shale gives a pattern whose maximum is near‐vertical. This suggests the possibility of prospecting with shear waves, using a distributed charge detonated at shear velocity to generate substantial downward‐direction shear energy in the earth. However, the substantially larger attenuation of shear waves compared to compressional waves has discouraged us from pursuing this further.

Several papers in recent years have dealt with the causality‐imposed relation between attenuation and dispersion for waves in lossy solids, with emphasis on seismic waves. While the published formulas for dispersion within a particular frequency band are supported by experimental evidence within that band, the mathematical behavior of these expressions outside the band, particularly at low frequencies, is physically unacceptable. In the present paper, one‐dimensional seismic waves are modeled as propagation along a simple lumped‐element transmission line, leading to expressions for attenuation and velocity as functions of frequency which not only satisfy the experimental data available, but exhibit no objectionable behavior outside the range of available data. This is achieved by introducing a resistive element whose value is inversely proportional to frequency. Numerical application of the Hilbert transform shows the condition of causality to be satisfied by this model quite accurately.

Transient waveforms have been computed for a family of source and receiver types and for geometry corresponding to plausible acoustic logging tools in oil wells. The acoustic tool now in commercial use is one member of the family. The computations also treat flexural waves and torsional waves. All quantities are expressed in terms of appropriate vector and scalar potentials, and output waveforms are obtained by numerical evaluation of triple Fourier transforms. When loss parameters are included in the stress‐strain relations, the Fourier inversion formulas are free of singularities and numerical integration is straightforward. For the purely elastic case, it is necessary to cope with the singularities of the integrands in performing the numerical integrations. Dispersion curves and other features agree with earlier publications on wave propagation along concentric cylinders.

Mr. Savit has sent me a copy of his letter to the editor. I suggest that the “seemingly contradictory conclusions” mentioned by Mr. Savit might largely be cleared up by repeating in their entirety the two sentences which contain the portion of a sentence quoted by Mr. Savit: “It was anticipated that for large values of Δτ, either of the last two cases would result in smaller residual signal than was obtained with the uniform distribution. This turns out to be true by a small factor, but it appears that, for small values of Δτ, the details, of the strength distribution have very little effect on the resulting waveform.”