Does Biot’s theory have predictive power?

Abstract

Biot’s theory of elastic waves in fluid-saturated porous solids has two free parameters: the tortuosity α, characterizing the dynamic coupling between the solid and the fluid, and the structural factor δ, representing the geometric properties of the porous space. The meaning and significance of these parameters have not been sufficiently understood. The tortuosity has the physical meaning of the normalized mean square of the velocity of the pore fluid relative to the solid wall; it has a low-frequency but no high-frequency limits. The analytical calculation of the tortuosity for Biot’s slit-like pore provides its range of variability from approximately 1–100 in the frequency range of practical interest. The tortuosity has a significant effect on the properties of the Biot waves of the second kind in the high-frequency range. On the other hand, in realistically complex pore geometries, the values of the tortuosity are virtually unpredictable. This limits the usefulness of the Biot theory in predicting the wave propagation at high frequencies. At all frequencies, the effect of the structural factor is insignificant relative to the effect of the tortuosity. The conventional compressional wave (the wave of the first kind) is insensitive to both parameters at all frequencies. The frequencies of interest to seismic exploration are also free of the uncertainty imposed by the lack of constraints on the tortuosity as the only free parameter in Biot’s theory.