Reflection and transmission of plane waves at an interface separating two poro-viscoelastic materials with continuity and elastic consistence
Authors: Liu Xu , Li Huijian , Al-Shuhail Abdullatif A , Liu Bo , Ren Zhengyong 
Published: January 2021

The seismic reflection/transmission (R/T) problems have been investigated for various complicated porous materials described by various poro-viscoelastic theories. However, it was also reported that even for a simple plane wave at a plane interface between two dissimilar isotropic viscoelastic solids, they were proved to be more complicated than expected and still required further study. The main difficulty is to correctly select the signs of the vertical components of the slowness vectors (or vertical slowness q) of the viscoelastic R/T waves to calculate R/T coefficients (including the relevant displacement amplitudes and phase delays) without unphysical discontinuity but with elastic consistence. Logically, if this problem had not been solved in the relatively simple case of waves in isotropic viscoelastic materials, it would not be expected to be solved in the complicate porous media. The existing criterion to select the sign of vertical slowness for poro-viscoelastic waves is the radiation condition that has been reported to cause unphysical discontinuity for the R/T coefficients of viscoelastic waves. In this paper, the radiation condition is illustrated to cause the unphysical discontinuities. Meanwhile, the continuity criterion is for the first time applied in the R/T coefficients calculation for poro-viscoelastic waves. The continuity criterion is simply suggested to be conducted in the complex vertical slowness square (q2) plane by swapping the sign of the q for the relevant wave whose q2 locus crosses the branch cut. Furthermore, the elastic consistence is also for the first time considered for poro-viscoelastic waves. We stress the importance of consistence between the R/T coefficient equations of poro-viscoelastic waves and those of pure elastic waves, which includes three consistent assumptions: time dependence of harmonic waves, the z-direction of the coordinate system and particle motion (polarization) vectors. With the numerical examples of poro-viscoelastic interfaces, the R/T coefficients of three types of incident waves, that is, homogeneous S waves, inhomogeneous S waves and homogeneous P waves, are investigated for frequencies 1000 and 0.001 Hz, representing highly and weakly dissipative poro-elastic waves, respectively. The calculated R/T coefficients are shown to be continuous without unphysical discontinuity under the continuity criterion. For the cases of homogeneous S- or P-wave incidence, the R/T coefficients of poro-viscoelastic waves with 0.001 Hz well match the corresponding coefficients of elastic waves in terms of amplitudes and phases, and thus the elastic consistence is well confirmed. For the highly dissipative poro-elastic waves (1000 Hz), the R/T coefficients significantly depend on the degrees of inhomogeneity of the incident waves. In a similar manner to the viscoelastic wave, a higher degree of inhomogeneity of an incident wave better tends to smooth the R/T coefficients of poro-viscoelastic waves. The methods suggested in this paper to solve the problems of R/T coefficient continuity and elastic consistence for the poro-viscoelastic waves can also be applied for viscoelastic waves as the special case.