Elastic nonlinearity in rock: On the relative importance between higher-order elastic constants and hysteresis

Abstract

Rocks are extremely elastically nonlinear, even at strain as low as 10−7. Recent simulations of dynamic elastic pulsed wave experiments and comparison with static and resonance test predictions revealed that the physical mechanism for nonlinearity in rocks cannot be attributed to higher-order nonlinear coefficients alone. Static stress-strain tests and resonance measurements show in addition an undeniable hysteretic behavior of stress and modulus versus strain. Therefore, hysteresis has been introduced into the dynamic nonlinear wave equation by means of a discontinuous term in the modulus. The new theoretical model is based on four parameters: the first and second nonlinearity constants, attenuation, and hysteresis strength. In doing so, rich harmonic spectra and nonlinear waveforms observed in dynamic pulse mode experiments can be simulated using realistic values of higher-order elastic constants and hysteresis. Furthermore, the model provides characterization criteria for rock types depending on the relative importance of hysteresis and nonlinearity parameters. Chalk, for instance, can have large first and second nonlinearity parameters, because it shows a rich harmonic spectrum but no hysteresis. On the other hand, the nonlinear spectra of several sandstones can be attributed almost entirely to the first nonlinear coefficient and to hysteresis. [Work supported by DOE/OBES/UCal.]