Dispersion analysis of generalized wave equations under the single-parameter second-order strain gradient theory

Abstract

The construction of wave equations describing the wave propagation in real medium is a critical problem to be solved for oil and gas seismic exploration. Researches show that the microstructural interactions by different scales will trigger the heterogeneous response of the medium, which in turn has an impact on the mechanical behavior of macro-scales. The heterogeneous response triggered by microstructural interactions in the medium is portrayed by introducing higher-order spatial derivatives of the strain and additional parameters of the characteristic scale of the medium under the framework of the generalized continuum mechanics theory (GCMT). When considering the heterogeneous responses triggered by microstructural interactions of the medium, seismic waves propagate in a dispersion manner. In this paper, we introduce the generalized wave equations under the single-parameter second-order strain gradient theory (SSSGT) by considering the nonlocal effects, give the decoupled generalized wave equations using the Helmholtz decomposition theorem, and derive the expression of the phase-velocity of the P- and S-wave to analyze the theoretical dispersion. Combined with numerical modeling, we further investigate the dispersion in the propagation of P- and S-wave triggered by the microstructural interactions. Numerical modeling and dispersion analysis indicate that the wave propagation is affected by the characteristic scale of the micropore of the medium and the frequency, and the dispersion of P- and S-wave is prominent increasingly with the frequency and the characteristic scale of the micropore increasing.