Modeling Seismic Wave Propagation Through Non-Uniform Fracture Stiffness: Theory and Implementation of Stress-Dependent Approaches

In this work, we examine three models for representing fractures with non-uniform stiffness due to stress. The first is an effective medium model that maps fracture properties onto a Transversely Isotropic medium. The second represents each fracture explicitly, modeling the two surfaces and the displacement discontinuity boundary conditions. The third is a localized effective medium (LEM), which models a thin zone around each fracture as an effective medium while leaving the rest of the material unchanged. We apply all three models to simulate a laboratory experiment where waves propagated through multiple parallel fractures under controlled conditions. To evaluate the models, we first conduct numerical experiments using the Bandis et al. (1983) stress-dependent stiffness approach, comparing the resulting waveforms. We find that the explicit and LEM models generally agree well. Next, we compare the waveforms for waves propagating parallel and perpendicular to the fractures with the experimental data. Both the explicit fracture model and the LEM reproduce many observed effects on wave propagation, including waveform changes due to fracture orientation and stiffness variation. These results demonstrate that both explicit and localized effective medium models reliably capture the impact of fractures on seismic waves. This validates their use in forward modeling of wave propagation in fractured media and supports their application in inversion workflows to estimate fracture properties from waveform data.