Coda-wave interferometric and lock-in measurements of slow dynamic nonlinearity

We study the universal linear-in-log-time recoveries of material moduli in rock and cements after minor conditioning (by strains of order 10−6), recoveries that proceed in the laboratory over decades in time from msecs to minutes. Digital-lock-in and coda-wave-interferometricprocedures reveal changes in ultrasonic wave-speed with high precision and fine time resolution. These methods are particularly useful for separating the intrinsic material contributions to early time relaxation spectra from the obfuscating effects of conditioning ring-down. We apply these procedures to (1) examine very early and very late times, where recovery must deviate from log-linearity, (2) elucidate dependence on conditioning strains, and strain rates, (3) measure the effect of conditioning durations and thresholds, and (4) detect dependence on environment. We confirm reports of a conditioning threshold in Berea sandstone, at about one micro-strain, below which there is no slow dynamics and above which SD is linear in pump strain. Our measurements challenge reports of roll-offs in relaxation spectra at early times. They also fail to support speculations of dependence on conditioning strain rate—at least in our parameter regime with strains of order 10−6 and pump frequencies of order kHz. [Work supported by the US DOE Award No. DE-SC0021056.]