Macroscopically microstructural effects on wave propagation in highly stressed fractured rocks

Stress-dependent seismic velocities in fractured rocks arise from the coupled deformation of a macroscopically continuous background matrix and stress-sensitive microstructures such as microcracks and aligned fractures. To capture this multi-source nonlinearity together with microstructural size effects, we develop a unified third-order strain-gradient acoustoelastic framework that embeds nonlocal strain-gradient micromechanics into classical acoustoelasticity based on third-order elastic constants, enabling micro–macro coupling through a total strain-energy function. 

We validate the theory using ultrasonic transmission measurements on two artificial sandstones sharing the same background matrix: an intact sample containing native microdefects and a cracked sample with uniformly implanted aligned penny-shaped cracks. Measurements were conducted under dry conditions at 500 kHz with hydrostatic pressure from 5 to 50 MPa, and anisotropic velocities were constrained using propagation directions normal and parallel to the bedding/crack plane. The proposed model reproduces the strongly nonlinear velocity–pressure trends in the low-pressure regime dominated by progressive crack closure, while remaining consistent with the near-linear regime at higher pressure.

A key outcome is a physically interpretable characteristic scale 𝑔 representing an evolving microstructural length associated with stress-driven changes in compliant pore space. We show that 𝑔 exhibits an asymptotic pressure dependence consistent with cumulative compliant-porosity evolution, and that these quantities are systematically correlated. Using effective-medium parameterizations for penny-shaped cracks (Hudson and Padé–Hudson), we further demonstrate that 𝑔² scales approximately linearly with fracture (crack) porosity across a range of crack aspect ratios and parameter ranges, supporting a robust micro–macro linkage.

These results provide a physics-guided route to connect stress-driven microstructural evolution with macroscopic wave observables, with implications for fracture characterization and seismic monitoring in stressed crustal systems.