Attenuation and dispersion of elastic waves in fluid-saturated rocks due to pore fluid viscosity is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies can be studied using an asymptotic analysis of Rytov's exact dispersion equations. Since the wavelength of the shear wave in the fluid (viscous skin depth) is much smaller than the wavelength of the shear or compressional waves in the solid, the presence of viscous fluid layers requires a consideration of higher-order terms in the low-frequency asymptotic expansions. This expansion leads to asymptotic low-frequency dispersion equations. For a shear wave with the directions of propagation and of particle motion in the bedding plane, the dispersion equation yields the low-frequency attenuation (inverse quality factor) as a sum of two terms which are both proportional to frequency ω but have different dependencies on viscosity η: one term is proportional to ω/η, the other to ωη. The low-frequency dispersion equation for compressional waves allows for the propagation of two waves corresponding to Biot's fast and slow waves. Attenuation of the fast wave has the same two-term structure as that of the shear wave. The slow wave is a rapidly attenuating diffusion-type wave, whose squared complex velocity again consists of two terms which scale with iω/η and iωη.
For all three waves, the terms proportional to η are responsible for the viscoelastc phenomena (viscous shear relaxation), whereas the terms proportional to η−1 account for the visco-inertial (poroelastic) mechanism of Biot's type. Furthermore, the characteristic frequencies of visco-elastic ωV and poroelastic ωB attenuation mechanisms obey the relation ωVωB = AωR2, where ωR is the resonant frequency of the layered system, and A is a dimensionless constant of order 1. This result explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic theories that imply ω ≪ ωR. The poroelastic mechanism dominates over the visco-elastic one when the frequency-indepenent parameter B = ωB/ωV = 12η2/μsρfhf2 ≪ 1, and vice versa, where hf is the fluid layer thickness, ρf the fluid density, and μs represents the shear modulus of the solid.