Strain dependent attenuation: Observations and a proposed mechanism

Abstract

The measured attenuation (Q-•) of a rock is a function of a number of parameters, one of those being the applied strain amplitude. It is important to understand the effect that strain amplitude has on Q-• for several reasons: different measurement techniques use differing strain amplitudes and may measure a dissimilar Q-l, near source (large strain) wave propagation may behave highly non-linearly, and the strain amplitude dependence can provide insight into the attenuatiøn mechanism. A physical model based on the contact friction between crack surfaces in the rock has been developed to describe rock deformation and dissipation under large applied strain. The three'dimensional crack surfaces are characterized by a statistical distribution of asperity heights. The sliding contact of these sphericallytipped asperities dissipates frictional energy. HertzJan theory is applied to the average asperity contact and predicts that the large strain attenuation is given by Q-• = k[F/P 4/3, where k is a constantconsisting of the matrix elastic parameters, [ is the crack density, F is the strain amplitude, and P is the confining pressure. The total attenuation measured appears to be the sum of this strain dependent term and a strain independent term. The results of ultrasonic pulse transmission experiments are compared with the model's prediction. Both P and S waves with strain amplitudes from 10 -8 to 10 -5 were employed. Frequencies from 0.4 to 1.5 MHz were used in conjunction with rock confining pressures of 2 to 580 bars on dry Berea sandstone and lucite samples. The spectral ratio method and rise time technique were applied to deduce the Q-1 values. The observed data and other observations from the literature compare well with the model's prediction for the dependence of Q-• on large strain amplitude, crack density, and pressure. ston et al., 1979; Toksoz et al., 1979; Kovach and Nur, 1980; Spencer, 1981]. The effect of the amplitude of the propagating strain on Q-1 has been the subject of increasing research [Peselnick and Outerbridge, 1961; Gordon and Davis, 1968; Gordon and Rader, 1971; Mavko, 1979; Winkler et al., 1979; Johnston and Toksoz, 1980; Tittmann et al., 1981]. Attenuation in rocks generally increases with increasing strains once a certain threshold strain value has been reached (Figure 1). Similarly, strain amplitudes have been recognized by engineers as exerting a dominant influence on the dynamic moduli of soils [Richart et al., 1970; lmai et al., 1979; $toll, 1979].In this paper we propose a model of rock deformation and dissipation that predicts the effect that large strain amplitudes, pressure, and crack density have on attenuation. The model is based on Hertzian contact friction theory (after Mavko [1979]) and statistically defined crack surfaces [Walsh and Grosenbaugh, 1979] that are used to develop a mechanism for large-strain wave attenuation.Also presented are the results of experiments designed to elucidate the strain effect. All measurements were made by

A new model for analyzing the effect of fractures on triaxial deformation

“…The estimate inevitably lacks precision because it relies on observations of both the pressure dependence of Young's modulus and the dependence of the quality factor on pressure and strain amplitude. Further, measurements of quality factor show that Q -1 is proportional to strain amplitude only at amplitudes greater than 10 -5 (see [22] for reference to their measurements). We ) / ( dp dQ dp…”

“…Measurements of Q -1 by Stewart et al [22] provide another way to estimate the value of λα j . The estimate inevitably lacks precision because it relies on observations of both the pressure dependence of Young's modulus and the dependence of the quality factor on pressure and strain amplitude.…”

“…We find that λα j can be used in this way. Stewart et al [22] carried out a series of experiments on Berea sandstone samples, measuring P-and S-wave velocities and attenuation as a function of strain amplitude and confining pressure. We have taken their velocity data and converted them to moduli following standard procedure.…”

Temporal variations of coda Q: An attenuation-coefficient view

“…Many authors reported local-earthquake coda attenuation (Q c ) varying over time scales from several weeks to a few years, such as Chouet (1976Chouet ( , 1979, Aki (1980), Stewart et al (1983), Rhea (1984), Novelo-Casanova et al (1985), , Jin and Aki (1986), Peng et al (1986), ), Fehler et al (1988, Londono (1996), Gusev (1997), Del Pezzo et al (2004, Moncayo et al (2004), Novelo-Casanova et al (2006). The results were invariably interpreted by using the single-and multiple-scattering coda models, i.e.…”

Experimental and Analytic Characterization of Nonlinear Seismic Attenuation

“…In a similar fashion, Stewart et al,20 have proposed a physical attenuation model based on complex crack structure and friction which, like Mavko, provides a Q-proportional to peak strain and which is also proportional to the crack density and the -4/3 power of the confining pressure.…”