Phenomenological model of propagation of the elastic waves in a fluid-saturated porous solid with nonzero boundary slip velocity

It is known that a boundary slip velocity starts to play an important role when the length scale over which the fluid velocity changes approaches the slip length, i.e., when the fluid is highly confined, for example, fluid flow through porous rock or blood vessel capillaries. Zhu and Granick [Phys. Rev. Lett. 87, 096105 (2001)] have recently experimentally established the existence of a boundary slip in a Newtonian liquid. They reported typical values of the slip length of the order of few micrometers. In this light, the effect of introduction of the boundary slip into the theory of propagation of elastic waves in a fluid-saturated porous medium formulated by Biot [J. Acoust. Soc. Am. 28, 179-191 (1956)] is investigated. Namely, the effect of introduction of boundary slip upon the function F(kappa) that measures the deviation from Poiseuille flow friction as a function of frequency parameter kappa is studied. By postulating phenomenological dependence of the slip velocity upon frequency, notable deviations in the domain of intermediate frequencies in the behavior of F(kappa) are introduced with the incorporation of the boundary slip into the model. It is known that F(kappa) crucially enters Biot's equations, which describe dynamics of fluid-saturated porous solid. Thus, consequences of the nonzero boundary slip by calculating the phase velocities and attenuation coefficients of both rotational and dilatational waves with the variation of frequency are investigated. The new model should allow one to fit the experimental seismic data in circumstances when Biot's theory fails, as the introduction of phenomenological dependence of the slip velocity upon frequency, which is based on robust physical arguments, adds an additional degree of freedom to the model. In fact, it predicts higher than the Biot's theory values of attenuation coefficients of the both rotational and dilatational waves in the intermediate frequency domain, which is in qualitative agreement with the experimental data. Therefore, the introduction of the boundary slip yields threefold benefits: (1) better agreement of theory with experimental data since the parametric space of the model is larger (includes effects of boundary slip); (2) the possibility to identify types of porous medium and physical situations where boundary slip is important; and (3) constrain model parameters that are related to the boundary slip.