Compressional and Shear Waves

Compressional and shear waves have almost the same asymptotic behavior at low frequencies, but the analysis for shear waves is much shorter, so we will present only the shear wave analysis here.

The wavenumber k_s for shear wave propagation is determined by (14), and when $\omega \to 0$ we have $q \to i\rho_f \eta/\kappa \omega$, so

$$\begin{eqnarray} k_s^2 = {{\omega^2\rho}\over{\mu_d}}\left[1 + i{{\rho_f\kappa\omega}\over{\rho \eta}}\right]. \end{eqnarray}$$ (20)

Thus, when the loss tangent is a small number, we find the shear wave quality factor is

$$\begin{eqnarray} 1/Q_s\simeq {{\rho_f\kappa\omega}\over{\eta \rho}}. \end{eqnarray}$$ (21)

Total attenuation along the path of a shear wave is then determined by the integral $\int {{\rho_f\kappa\omega^2}\over{2\eta(\rho\mu)^{1/2}}}d\ell$along the path of the wave. We assume for the sake of argument that the fluid is the same throughout the reservoir. So all fluid factors as well as frequency are constant. The solid material parameters $\mu_d$ and $\rho_m$ and also the porosity $\phi$ (which is hidden in $\rho$) may vary in the reservoir, but these variations will be treated here as negligible compared the variations in the permeability $\kappa$. Thus, we find that the total attenuation along a path of length $L = \int d\ell$ is approximately proportional to $\int \kappa d\ell$. The average attenuation per unit length of the travel path is therefore proportional to $\int \kappa d\ell/L$, which is just the mean of the permeability along the wave's path. This result is also true for the compressional waves, but the other multiplicative factors are a bit more complicated in that case.