The power-law characteristics in the microcrack system of a rock: inversion and analysis of laboratory velocity–pressure data

SUMMARY The pore-crack network, characterized by the distribution of pore aspect ratios and porosity, controls the seismic properties of a rock. The increase in laboratory ultrasonic velocity with effective pressure reflects the closure of microcracks, thereby allowing the pore-aspect-ratio spectrum to be inverted through rock-physics modelling. Previous studies on sandstone samples have suggested that the pore-aspect-ratio spectrum (porosity distribution versus pore-aspect-ratio distribution) follows a power-law distribution; however, its robustness across different lithologies, model dependence and origin remain poorly understood. In this study, these questions are further investigated. We compile laboratory ultrasonic velocity–pressure data from 50 rock samples spanning a wide range of lithologies and porosities. Then we invert the power-law exponent (the slope of the power-law distribution) from these data by integrating the pore-aspect-ratio spectrum into two independent rock-physics models: the existing Kuster–Toksöz (KT) model and the multicrack wave theory (MCWT) that incorporates the squirt-flow mechanism. The inversion results show that both models obtain power-law pore-aspect-ratio spectra across lithologies, with the MCWT showing improved modelling of the squirt-flow effect in saturated P-wave velocities, leading to smaller fitting errors. Theoretical modelling explains the differences between the two models in their fitting behaviour and inverted power-law exponent values. Furthermore, we demonstrate that the power-law pore-aspect-ratio spectrum can be derived from widely observed power-law fracture length and aperture distributions of the crack system. The result also establishes a relationship between the power-law exponent and porosity that well explains the global data trend from inverting the velocity–pressure data of rock samples. The significance of the power-law exponent and the applicability of the E-porosity relationship are also discussed. Our findings support a robust power-law pore-aspect-ratio spectrum across different lithologies and highlight its simplicity and practical applicability for analysing velocity–pressure data in cracked-porous rocks.