## Tuesday, January 31, 2012

### Calculating Biot characteristic frequency

In the theory of poro-elastic wave propagation, Biot (1956) defined a characteristic frequency (fc) given by this equation

For the fact-checkers out there, this relationship can be found by combining Biot (1956) equations 7.4, 6.8 and 3.7. A PDF of the paper can be found here.

At frequencies below fc, the Biot slow wave is not really a wave at all, but a highly attenuated diffusion phenomena. Above fc, the slow wave propagates with less attenuation and therefore would be more easily observable in lab experiments and has more influence on the reflection process.

The units used in poro-elasticity can be tricky. For example, the brine sandstone parameters of Dutta and Ode (1983) are pore fluid viscosity (0.01 poise), porosity (0.3), permeability (1 darcy), and in-situ pore fluid density (1 g/cc). You can imagine it is a rather horrific exercise to convert all this to MKS units for calculation of fc. However, we can navigate this very nicely using using wolfram|Alpha. The answer in this case if fc = 48379 Hz. Click the link in the previous sentence to do the calculation yourself. From there, you can put in other numbers and find fc in any case of interest.

To be sure you are on the right track, Dutta and Ode (1983) give a gas sand example with the same porosity and permeability as the previous calculation, but pore fluid viscosity is 0.00015 poise and density is 0.1 g/cc. From this you should find a characteristic frequency of 7257 Hz.

Biot, M. A., 1956a, Theory of propagation of elastic waves in a fluid-saturated porous solid, I: Low frequency range: Journal of the Acoustical Society of America, 28, no. 2, 168–178, doi:10.1121/1.1908239.

Dutta, N. C., and H. Ode, 1983, Seismic reflections from a gas-water contact: Geophysics, 48, 148–162, doi:10.1190/1.1441454.