Model for nonlinear wave propagation derived from rock hysteresis measurements

We develop a method for modeling nonlinear wave propagation in rock at inte rmediate strain levels, that is, strain levels great enough that nonlineari ty cannot be neglected but low enough that the rock does not incur macrosco pic damage. The constitutive model is formulated using a singular-kernel en dochronic 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 i t to agree with laboratory observations of harmonic distortion. Another fre e parameter controls the amount of hysteresis; it is set to approximate lab oratory stress-strain curves. The resulting phenomenological model provides a convenient means to parameterize laboratory observations in a form suita ble for efficient wave propagation calculations. We solve one-dimensional w ave propagation problems for this constitutive model using both finite diff erence and pseudospectral methods. Quasi-harmonic wave propagation in the B erea sandstone model shows several departures from results obtained with no nlinear elasticity: (1) more rapid decay with distance of the fundamental f requency component due to nonlinear, amplitude-dependent attenuation; (2) e nhanced excitation of the third-order harmonic, in agreement with laborator y 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-dependen t energy dissipation due to hysteresis. We also find that a two-frequency s ource function generates harmonics with frequencies which can be expressed as linear combinations of integer multiples of the source frequencies, in a greement with published laboratory results.