Analytical and Preisach-based petrophysical models to describe the acoustic hysteresis of compressional and shear wave velocities

Summary Acoustic wave propagation under pressure is inherently nonlinear and often exhibits hysteresis in elastic properties. Understanding this behaviour is important in rock physics and exploration seismology because pressure-dependent elastic responses contain information on evolving rock microstructure. We present a petrophysical model that describes the relationship between acoustic P- and S-wave velocities and effective rock pressure during both loading and unloading phases. Instead of prescribing a single microscopic mechanism, the model introduces an effective extensive state variable that represents the cumulative effect of pressure-induced microstructural evolution. Depending on the dominant physical process, this variable may be associated with compliant pore and microcrack volume, grain-contact evolution, or other pressure-sensitive structural descriptors. The model distinguishes between open and closed microstructural states and describes their stress-dependent evolution through different sensitivities during loading and unloading, thereby reproducing irreversible velocity-pressure behaviour within a unified framework. Model parameters are estimated from laboratory-measured acoustic velocities using linearized joint inversion. The inversion further enables the reconstruction of an acoustically weighted effective descriptor that characterises the pressure-dependent contribution of compliant microstructural evolution to the observed velocity response. In parallel, we apply the Preisach theory of hysteresis to pressure-dependent acoustic velocities. In this framework, compliant microstructural features are represented by effective hysteretic units (hysteron) characterized by distributed closing and opening pressures. The resulting Preisach model reproduces the measured hysteresis with high accuracy and enables a model-dependent reconstruction of the Preisach density function, which provides an effective, non-unique representation of hysteretic switching contributions across closing and opening pressures. By combining the petrophysical and Preisach approaches, we introduce two diagnostic quantities, the Ratio of Hysteron Jumps (RHJ) and the Cumulative Hysteron Jumps (CHJ), which quantify the relative and cumulative contributions of hysteretic elements throughout loading and unloading. While both approaches reproduce the observed velocity–pressure hysteresis, they provide complementary information. The petrophysical model describes the macroscopic evolution of elastic properties via an effective state variable and its acoustically weighted representation, whereas the Preisach model resolves the distribution and activation of hysteretic switching contributions in Preisach space via the reconstructed density function. The results demonstrate that acoustic hysteresis measurements contain not only information on effective elastic properties but also recoverable signatures of pressure-dependent microstructural evolution relevant to seismic interpretation.