Phase Velocity and Attenuation

With all of the double-porosity coefficients now defined, the compressional phase velocity and attenuation may be determined by inserting a plane-wave solution into the effective single-porosity Biot equations [in the form (1)-(4)]. This gives the standard complex longtitudinal slowness s of Biot theory  

$$s^2 = b \mp \sqrt{b^2 - \frac{\rho \tilde{\rho} - \rho_f^2}{MH-C^2}},$$ (29)

where  

$$b = \frac{{\rho M + \tilde{\rho} H - 2 \rho_f C}} {2({MH-C^2})}$$ (30)

is simply an auxiliary parameter, and where H, C and M are the Biot (1962) poroelastic moduli defined in terms of the complex frequency-dependent parameters of equations (11)-(13) as

$$\begin{eqnarray} H &=& K_U + 4G/3 \\ C &=& B K_U \\ M &=& \frac{B^2}{1-K_D/K_U} K_U.\end{eqnarray}$$ (31) (32) (33)

The complex inertia $\tilde{\rho}$ corresponds to rewriting the relative flow resistance as an effective inertial effect

$$\tilde{\rho} = - {\eta}/[i\omega k(\omega)].$$ (34)

Taking the minus sign in equation (29) gives an s having an imaginary part much smaller than the real part and that thus corresponds to the normal P-wave. Taking the positive sign gives an s with real and imaginary parts of roughly the same amplitude and that thus corresponds to the slow P-wave (a pure fluid-pressure diffusion across the seismic band of frequencies). We are only interested here in the properties of the normal P-wave.

 

QandVdp
Figure 1 The attenuation and phase velocity of compressional waves in the double-porosity model of Pride and Berryman (2003a). The 5 cm embedded spheres of phase 2 have frame moduli (K^d₂ and G₂) modeled using the modified Walton theory given in the appendex in which both K^d₂ and G₂ vary strongly with the background effective pressure P_e (or overburden thickness). These spheres of porous continuum 2 were embedded into a phase 1 continuum modeled as a consolidated sandstone.

The P-wave phase velocity v_p and the attenuation measure Q_p^-1 are related to the complex slowness s as

$$\begin{eqnarray} v_p &=& 1/{\rm Re}\{s\} \\ Q_p^{-1} &=& {\rm Im}\{s\}/{\rm Re}\{s\}.\end{eqnarray}$$ (35) (36)