Model for nonlinear wave propagation derived from rock hysteresis measurements

We develop a method for modeling nonlinear wave propagation in rock at intermediate strain levels, that is, strain levels great enough that nonlinearity cannot be neglected but low enough that the rock does not incur macroscopic damage. The constitutive model is formulated using a singular-kernel endochronic formalism which satisfies a number of observational constraints, including producing a power law dependence of attenuation (Q(-1)) on strain amplitude. One free parameter represents cubic anharmonicity, and we set it to agree with laboratory observations of harmonic distortion. Another free parameter controls the amount of hysteresis; it is set to approximate laboratory stress-strain curves. The resulting phenomenological model provides a convenient means to parameterize laboratory observations in a form suitable for efficient wave propagation calculations. We solve one-dimensional wave propagation problems for this constitutive model using both finite difference and pseudospectral methods. Quasi-harmonic wave propagation in the Berea sandstone model shows several departures from results obtained with nonlinear elasticity: (1) more rapid decay with distance of the fundamental frequency component due to nonlinear, amplitude-dependent attenuation; (2) enhanced excitation of the third-order harmonic, in agreement with laboratory observations, and saturation, with propagation distance, of the harmonics. This behavior reflects competing effects of harmonic amplitude growth via nonlinear energy transfer from the source frequency and amplitude-dependent energy dissipation due to hysteresis. We also find that a two-frequency source function generates harmonics with frequencies which can be expressed as linear combinations of integer multiples of the source frequencies, in agreement with published laboratory results.