SEISMIC WAVE ATTENUATION CHARACTERISTICS FROM PULSE EXPERIMENTS

Abstract

Laboratory data on the transmission of stress waves in rocks indicate that the attenuation exponent (db/ft) for steady state sine waves is roughly proportional to frequency in dry cores at atmospheric pressure. Measurements of the effects of pressure and water saturation, however, show that much more work is required before laboratory results can be extrapolated to conditions in the earth. Field experiments avoid these problems, but not, of course, without introducing others. The field tests and theoretical studies of Ricker indicate that the attenuation exponent is proportional to the square of the frequency at low frequencies; these results have not, however, been accepted unreservedly. We propose further experiments of the type conducted by Ricker, but with emphasis on recording the wave at several distances so that the changes in shape can be observed as the wave travels through the medium. To use such data to determine the attenuation exponent and the wave propagation equation, we propose that the pulses be converted to equivalent steady state sine wave data by Fourier integral analysis. Geometric effects in the particular experiment can be eliminated by mathematical analysis and the attenuation exponent calculated from the steady state magnitudes and phases at different distances. The basis of the analysis method is the fact that the manner in which the attenuation exponent appears in the harmonic solution of the most general linear wave equation is independent of the manner in which it varies with frequency. Hence at each frequency the attenuation exponent can be calculated from the steady state data for that frequency. The method is applied to data obtained in a quarry sandstone. Although the results are not consistent for different transmission distances, it is believed that data from additional experiments of this kind can be used to determine the attenuation exponent and the wave propagation equation. In formulating the method, the Boltzmann superposition principle is used in which the general three‐dimensional stress‐strain relations for an isotropic material require two elastic constants and two memory functions. A preliminary study of the memory function for dilatational waves yields some interesting limitations on possible functions when we impose the restriction of elastic behavior with static stresses and strains. If the Laplace transform of this function is analytic at the origin, then the attenuation exponent increases with the square or some higher even power of the frequency at low frequencies. To obtain any other variation, the memory function transform must have a branch point at the origin (poles and essential singularities are ruled out). No memory function will yield an attenuation exponent precisely proportional to frequency over any frequency range, but a class of memory functions may exist yielding an attenuation exponent proportional to frequency raised to a power arbitrarily close to unity at low frequencies. This analysis is based on the assumption of linearity, which we do not wish to abandon until forced to by experimental data.