High-frequency limit of $\gamma(\omega)$

In the extreme high-frequency limit, the fluid has no time to significantly escape from the porous grains (phase 2) and enter the main porespace (phase 1). As such, the fluid pressure distribution in each phase is reasonably modeled as

$$\begin{eqnarray} p_{f1}({\bf r}) &=& B_1^\infty \Delta P \\ p_{f2}({\bf r}) &=& ... ...infty \Delta P + C_2 \Delta P e^{-i^{3/2} \sqrt{\omega/D_2} \, x}\end{eqnarray}$$ (117) (118)

where x is again a local coordinate measuring distance normal to the interface $\partial \Omega_{12}$ and where D₂ is the fluid-pressure diffusivity within the porous grains that is given by $D_2 = k_2 K_2^d B_2/(\eta \alpha_2)$. In reality, the local confining pressure $p_{c2}(\bf r)$ throughout the grains has spatial fluctuations about the average value and we have made the approximation that $B_2 p_{c2}({\bf r}) \approx B_2^\infty \Delta P$ = the average fluid pressure throughout the grain space. It is easy to demonstrate that under undrained and unrelaxed conditions,

$$\begin{eqnarray} B_1^\infty &=& \frac{a_{13}a_{23} - a_{33} a_{12}}{a_{22} a_{33... ...=& \frac{a_{12}a_{23} - a_{22} a_{13}}{a_{22} a_{33} - a_{23}^2}. \end{eqnarray}$$ (119) (120)

Since these $B_i^\infty$ do not appear in the final result, we do not bother substituting in the a_ij constants from equations (90)-(95).

The continuity of fluid pressure p_f2 = p_f1 along $\partial \Omega_{12}$ (x=0) requires that $C_2=B_1^\infty-B_2^\infty$. The definition of $\dot{\zeta}_{\rm int}$ may now be used to write

$$\begin{eqnarray} -i\omega \zeta_{\rm int} &=& \frac{1}{V} \int_{\partial \Omega_... ...a B_2 K_2^d}} \frac{S}{V} (\overline{p}_{f1} - \overline{p}_{f2})\end{eqnarray}$$ (121) (122) (123)

where we have used, to leading order in the high-frequency limit, that $\overline{p}_{f1} - \overline{p}_{f2} = (B_1^\infty-B_2^\infty)\Delta P$. The desired limit is thus $\lim_{\omega\rightarrow \infty} \gamma(\omega) = \sqrt{-i\omega k_2 \alpha_2/(\eta B_2 K_s^d)} S/V$.