Slow Waves

In contrast, the slow compressional wave can have two very different types of behavior at low frequency depending on the magnitude of the permeability. The wavenumber k_- for slow wave propagation is determined by (15). To simplify this equation, we note that it is an excellent approximation to take

$$\begin{eqnarray} k_-^2\simeq b + f = {{\omega^2}\over{\Delta}} \left[qH - 2\rho_f C+\rho M\right]. \end{eqnarray}$$ (22)

So, at low frequencies, k_-² is proportional to q, whereas k_s² was inversely proportional to q. Then, for small frequencies but large values of the permeability, $q \to \rho_f[\alpha/\phi + i\eta/\kappa\omega]$. Substituting this into (22), we find that

$$\begin{eqnarray} k_-^2 = {{\omega^2}\over{\Delta}} \left[\alpha\rho_f H/\phi - 2\rho_f C + \rho M + i\eta\rho_f H/\kappa\omega\right]. \end{eqnarray}$$ (23)

So as $\omega \to 0$ for large $\kappa$, there will be an intermediate frequency regime in which the slow wave has a well-defined quality factor

$$\begin{eqnarray} 1/Q_- \simeq \eta\rho_f H/\kappa\omega(\alpha\rho_f H/\phi - 2\rho_f C + \rho M), \end{eqnarray}$$ (24)

which for strong frame materials reduces to

$$\begin{eqnarray} 1/Q_- \simeq \eta\phi/\alpha\kappa\omega. \end{eqnarray}$$ (25)

Except for some factors of density, porosity, and tortuosity, this expression is essentially the inverse of the corresponding expression for 1/Q_s. Obviously both factors cannot be small simultaneously except for a very limited range of frequencies, which is determined by the factor $\alpha\rho/\phi\rho_f$. Although the tortuosity $\alpha \ge 1$ in general it can have a wide range of values, for granular media it is typical to find $\alpha \simeq 2$ or 3. In addition, $\alpha$ is also scale invariant, i.e., it does not depend on the size of the particles composing the granular medium. So, the presence of $\alpha$ multiplying $\kappa$ in (25) does not change the fact that the slow-wave attenuation is strongly influenced by fluctuations in the permeability $\kappa$. Being proportional to the square of the typical particle sizes, the permeability is itself not scale invariant. There is nevertheless a fairly small range of frequencies in which the approximation in (25) is valid, say from about 20 kHz to a few MHz for $\kappa$'s on the order of 1 D ($\simeq 10^{-12}$m²). This is the range where a propagating slow wave might be expected to be seen, and in fact has been observed in laboratory experiments (Plona, 1980).

For still smaller permeabilities or smaller frequencies or both, the leading approximation for the slow wave dispersion is instead given by

$$\begin{eqnarray} k_-^2 \simeq i{{\omega \eta\rho_f H}\over{\kappa\Delta}}. \end{eqnarray}$$ (26)

This type of dispersion relation corresponds to a purely diffusive process having a diffusion coefficient ${\cal D} \simeq M\kappa/\eta\rho_f$.This result follows directly from the second equation in (2) when the porous frame is sufficiently rigid.

We reach the same conclusion about how fluctuating permeability affects the propagation or diffusion of increments of fluid content (i.e., masses of excess fluid particles) in both of these cases. For the wave propagation situation of (25), we clearly have, by simple analogy to the arguments given already, that the average attenuation per unit length along the wave's path is proportional to $\int \kappa^{-1}d\ell/L$. Similarly, in the limit of the diffusion process described by (26), then for a planar excitation diffusing through such a system in a direction perpendicular to the bedding planes, or for regions of isotropic random fluctuations  in permeability, we again expect the overall effective diffusion rate to depend on the same average quantity: $\int \kappa^{-1}d\ell/L$. Thus, measurements of slow waves or of fluid increment diffusion on the macroscale will measure an effective permeability that is largely controlled by the smallest permeability present in the system. Clearly, this is exactly the opposite dependence we found for the dependence of the shear wave and also for the fast compressional wave, and must cause difficulties for up-scaling in Biot's theory, where only one permeability parameter is available for the fitting of data.