A theory for acoustic propagation in a porous and elastic medium developed by M. A. Biot. Compressional and shear velocities can be calculated by standard elastic theory from the composite density, shear and bulk modulus of the total rock. The problem is how to determine these from the properties of the constituent parts. Biot showed that the composite properties could be determined from the porosity and the elastic properties (density and moduli) of the fluid, the solid material, and the empty rock skeleton, or framework. To account for different frequencies of propagation, it is also necessary to know the frequency, the permeability of the rock, the viscosity of the fluid and a coefficient for the inertial drag between skeleton and fluid.

Unlike the Gassmann model, the Biot theory takes into account frequency variations, and allows for relative motion between fluid and rock framework. As a result, it predicts some of the observed changes in velocity with frequency, as well as the critical frequency below which the Gassmann model is valid. It also predicts the existence of a so-called slow wave in addition to the shear wave and the compressional, or fast wave. The slow wave arises when the fluid and the skeleton move 180⁰ out-of-phase with each other. Its velocity is related to fluid mobility, but unfortunately has been observed only in the laboratory, not on logs. At logging frequencies, it degenerates into a diffusion phenomenon rather than a wave, and is apparently too highly attenuated to be observed in real rocks. However, in permeable formations, the Stoneley wave couples into the slow wave, causing the attenuation and dispersion that allow the measurement of Stoneley permeability.

The full Biot theory is used mainly to analyze laboratory data. For practical log interpretation, it is more common to use the simpler Gassmann model.


Reference:

See Biot MA: Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid: I Low Frequency Range, Journal of the Acoustical Society of America 28, (1956):168-178.

Biot MA: Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid: II Higher Frequency Range,Journal of the Acoustical Society of America 28, (1956): 179-191.