Analysis of dispersive characteristics and scale effects of seismic waves under generalized wave equations

Considering the complex multi-scale microstructural interactions in the medium, observable scale effects emerge in seismic wave propagation. In the framework of the generalized continuum mechanics theory, the fundamental constituents of the medium are modeled as volumetric micro-elements, incorporating the effects of long-range forces between them. By introducing higher-order gradients of rotation/strain and scale parameters characterizing the spatial microstructure of the medium, the scale effects during seismic wave propagation can be better captured under current computational conditions. In this paper, the generalized wave equations and its decoupled expression are given under the unified framework with considering the influence of the couple stress effect and the strain gradient effect on wave propagation. Theoretical analysis and numerical experiments demonstrate that when accounting for the complex spatial microstructural interactions, even in an isotropic homogeneous medium, both P-wave and S-wave propagation exhibit dispersion, and the dispersive characteristics are closely related to scale parameters. In addition, the variation of wave energy distribution at the interface of the medium caused by scale effects is analyzed. It is found that the existence of the scale effect can enhance the wavefield information related to the interface in the seismic record, which can provide a new idea for the fine structure detection and physical property information interpretation of the reservoir.