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

As $\omega\rightarrow 0$, one can represent the local fields as asymptotic series in the small parameter $-i\omega$

$$\begin{eqnarray} p_{fi} = p_{fi}^{(0)} -i\omega p_{fi}^{(1)} + O(\omega^2), \\ p_{ci} = p_{ci}^{(0)} -i \omega p_{ci}^{(1)} + O(\omega^2),\end{eqnarray}$$ (60) (61)

and equivalently for ${\bf u}_i$. The zeroth-order response corresponds to uniform fluid pressure in the pores and is therfore given by $p_{c1}^{(0)} = p_{c2}^{(0)} = \Delta P$ and

$$\frac{\overline{p}_{fi}^{(0)}}{\Delta P} = B_o = - \frac{a_... ...{13}}{a_{22} + 2 a_{23} + a_{33}} = \frac{1}{v_1/B_1 + v_2/B_2}$$ (62)

where the patchy-saturation a_ij have been employed. The fact that the quasi-static Skempton's coefficient in the patchy-saturation model is exactly the harmonic average of the constituents B_i is equivalent to saying that at low frequencies, the fluid bulk modulus is given by 1/K_f = v₁ /K_f1 + v₂ /K_f2. The quasi-static response is thus completely independent of the spatial geometry of the fluid patches; it depends only on the volume fractions occupied by the patches.

The leading order correction to uniform fluid pressure is then controlled by the boundary-value problem

$$\begin{eqnarray} \frac{K k}{\alpha \eta_1} \nabla^2 p_{f2}^{(1)} & = & \frac{\et... ...a p_{fi}^{(1)} &=& 0 \mbox{\hskip3mm on \hskip1mm} \partial E_{i}.\end{eqnarray}$$ (63) (64) (65) (66) (67)

It is now assumed that for patchy-saturation cases of interest (air/water or water/oil), the ratio $\eta_2/\eta_1$ can be considered small. To leading order in $\eta_2/\eta_1$, equations (63), (66), and (67) require that $p_{f2}^{(1)}({\bf r}) = \overline{p}_{f2}^{(1)}$ (a spatial constant). The fluid pressure in phase 1 is now rewritten as  

$$p^{(1)}_{f1}({\bf r}) = \overline{p}_{f2}^{(1)} - \frac{\eta... ...{k K} \left(1- \frac{B_o}{B_1}\right) \Delta P \Phi_1({\bf r})$$ (68)

where, from equations (64), (65) and (67) and to leading order in $\eta_2/\eta_1$, the potential $\Phi_1$ is the solution of the same elliptic boundary-value problem (25)-(27) given earlier.

Upon averaging (68) over all of $\Omega_1$, the leading order in $-i\omega$ difference in the average fluid pressures can be written  

$$\frac{\overline{p}_{f1} - \overline{p}_{f2}}{\Delta P} = -i... ...\frac{\eta_1 \alpha}{k K} \left(1- \frac{B_o}{B_1}\right) L_1^2$$ (69)

where L₁ is again the length defined by equation (24).

To connect this fluid-pressure difference to the increment $\dot{\zeta}_{\rm int}$ we use the divergence theorem and the no-flow boundary condition on $\partial E_i$ to write equation (53) as

$$-i\omega \zeta_{\rm int} = \frac{i\omega}{V} \frac{k}{\eta} ... ...v_1 \frac{\alpha}{K}\left(1- \frac{B_o}{B_1} \right) \Delta P.$$ (70)

Replacing $\Delta P$ with $\overline{p}_{f1} - \overline{p}_{f2}$ using equation (69) then gives the desired law $-i\omega \zeta_{\rm int} = \gamma_p (\overline{p}_{f1} - \overline{p}_{f2})$ with  

$$\gamma_p = \frac{v_1 k}{\eta_1 L_1^2} \left[1 + O\left(\frac{\eta_2}{\eta_1}\right)\right].$$ (71)

being the low-frequency limit of interest.