Stress-Dependent Reflection and Transmission of Elastic Waves in Rocks With Elastic-Inelastic Deformations Under Confining, Uniaxial, and Pure Shear Prestresses

Insights into stress-dependent reflection and transmission (R/T) of elastic waves are crucial for geophysical applications involving stress- and angle-dependent seismic data. Stress-induced deformations in rocks typically exhibit an elastic-inelastic progressive transformation due to stress accumulation, crack closure, and crack growth prior to mechanical failure. Based on the truncated Taylor expansion of strain-energy function for small prestress deformations, the classical theory of acoustoelasticity with second- and third-order elastic constants (2oeCs and 3oeCs) is insufficient to handle inelastic deformations caused by the closure or opening of compliant microstructures. Referred to the statically deformed state, Fu and Fu [1] formulate the Padé Piola-Kirchhoff stress equations of motion for wave-induced small dynamic fields superposed on severely deformed rocks under large-magnitude prestress. The resulting Padé acoustoelastic equations of motion describe wave propagation in rocks with elastic-inelastic deformations in terms of 2oeCs, 3oeCs, Padé coefficients, and static stresses. The stress- and angle-dependent R/T of waves at the stressed interface with a large static bias remains largely unaddressed. We first compare conventional and Padé acoustoelastic plane-wave analyses for P- and S-wave velocities. Theoretical predictions by Padé acoustoelasticity agree well with laboratory measurements made on Portland and Massilon sandstones under confining and uniaxial prestresses, respectively. We formulate the Padé acoustoelastic R/T coefficients at an interface between rocks with locally uniform elastic-inelastic deformations, assuming negligible boundary dislocations, which holds under moderate inelastic deformation but may not apply to large plastic flow. We examine three typical loading modes (confining, uniaxial, and pure shear), with each considering the incidence of upgoing and downgoing P- and S-waves. The relevant Knott equations are derived for the estimation of stress-dependent R/T coefficients. We investigate the energy conservation at the interface to confirm the accuracy of stress- and angle-dependent R/T coefficients under hydrostatic stress conditions. Comparisons between different levels of prestresses show the important effect of inelastic strains due to the closure of compliant pores on the energy distribution of R/T waves. Stress-dependent intercepts, slopes, critical angles, and R/T energy ratios are closely related to the sensitivity of upper and lower media to pressure and the type and magnitude of prestress. Particularly, the stress-induced closure of compliant pores must be considered for amplitude-variation-with-offset analyses in geophysical applications.