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

The fluid pressure and confining pressure in the grains can again be developed as asymptotic series in $-i\omega$ [as in equations (60)-(61)]. The zero-order response corresponds to the static limit in which the fluid pressure is everywhere the same and given by $p_{f2}^{(0)} = \overline{p}_{f1} = B_o \Delta P$ with B_o = -(a₁₂+a₁₃)/(a₂₂ + 2 a₂₃ + a₃₃) and with the a_ij as given by equations (90)-(95). The detailed result for B_o can be expressed

$$\begin{eqnarray} &&\frac{1/K-(1-\alpha_2)/K_2^d}{B_o} = \frac{1}{K} - \frac{(1-\... ...right] + v_2 \frac{\alpha_2}{K_2^d} \left[\frac{1}{B_2} - 1\right]\end{eqnarray}$$ (98)

which reduces to the standard Gassmann expression given in the appendix (with a total porosity given by $v_1 + \phi_2 v_2$) when B₂ and $\alpha_2$ are themselves given by the Gassmann expressions. In this same zero-order limit, the undrained bulk modulus is defined as 1/K_o^u = a₁₁ + (a₁₂ + a₁₃) B_o which also reduces to the standard Gassmann expression when B₂ and $\alpha_2$ are themselves given by Gassmann expressions.

The leading-order in $-i\omega$ correction to uniform fluid pressure is thus governed by the problem

$$\begin{eqnarray} \nabla^2 p_{f2}^{(1)} &=& \frac{\eta \alpha_2}{k_2 K_2^d} p_{c2... ...(1)} &=& 0 \mbox{\hskip 3mm on \hskip 1 mm} \partial \Omega_{12}. \end{eqnarray}$$ (99) (100) (101)

Here, p_c2⁽⁰⁾ is the local confining pressure in the grain space in the static limit that can be written $p_{c2}^{(0)}({\bf r}) = \overline{p}_{c2}^{(0)} + \delta P ({\bf r})$. The average static confining pressure throughout the grains is determined from equation (84) with $P_c = \Delta P$ and $p_{f2}=p_{f1}=B_o \Delta P$ to yield  

$$\overline{p}_{c2}^{(0)} = \frac{(1-v_1 B_o)}{v_2} \Delta P.$$ (102)

The deviations $\delta P({\bf r})$ thus volume integrate to zero $\overline{\delta P} = 0$ and are formally defined

$$\delta P({\bf r}) = -\left(\frac{1-(v_1 + v_2 \alpha_2) B_o}... ...P - \frac{K_2^d}{\alpha_2} \nabla \cdot {\bf u}^{(0)}({\bf r}).$$ (103)

The local perturbations $\delta P({\bf r})$ are thus highly sensitive to the detailed nature of the grain packing and grain geometry. Fortunately, these perturbations do not play an important role in the theory.

The fluid pressure in the grains is now written in the scaled form  

$$p_{f2}^{(1)}({\bf r}) = -\frac{\eta \alpha_2 (1-v_1 B_o)}{v_2 k_2 K_s^d} \Delta P \, \Phi({\bf r})$$ (104)

where the potential $\Phi({\bf r})$ is independent of $\Delta P$ and is a solution of the elliptic problem

$$\begin{eqnarray} \nabla^2 \Phi({\bf r}) &=& -1 - \frac{v_2}{1- v_1 B_o} \frac{\d... ... \Phi &=& 0 \mbox{\hskip 3mm on \hskip 1 mm} \partial \Omega_{12}.\end{eqnarray}$$ (105) (106) (107)

To leading-order in $-i\omega$, an average of equation (104) gives

$$\begin{eqnarray} \overline{p}_{f1} - \overline{p}_{f2} &=& i\omega \overline{p}_... ... \alpha_2 (1-v_1 B_o)}{v_2 k_2 K_s^d} L_2^2 \Delta P + O(\omega^2)\end{eqnarray}$$ (108) (109)

where the squared length L₂² is defined  

$$L_2^2 = \overline{\Phi} = \overline{\Phi}_o \left[ 1 + \frac... ...\overline{\Phi_o \delta P}}{\overline{\Phi}_o \Delta P} \right]$$ (110)

with overlines denoting volume averages over the grain space and with the potential $\Phi_o$ defined as the solution of

$$\begin{eqnarray} \nabla^2 \Phi_o &=& -1 \\ {\bf n} \cdot \nabla \Phi_o &=& 0 \mb... ...Phi_o &=& 0 \mbox{\hskip 3mm on \hskip 1 mm} \partial \Omega_{12}.\end{eqnarray}$$ (111) (112) (113)

Although it is not generally true that $\overline{\Phi_o \delta P} = 0$ for all grain geometries, we nonetheless expect this integral to be small because $\Phi_o$ is a smooth function and $\overline{\delta P} = 0$. The local perturbations in the static confining pressure $\delta P({\bf r})$ require a solution of the static displacements throughout the entire grain space--a daunting numerical task. Whenever the length L₂ needs to be estimated, such as in the numerical results that follow, our approach is to simply use the reasonable approximation that $L_2^2 = \overline{\Phi}_o$.

Last, from the definition $\dot{\zeta}_{\rm int}$ of the internal transfer we have that to leading order in $-i\omega$

$$\begin{eqnarray} -i\omega \zeta_{\rm int}&=& \frac{i \omega k_2}{V\eta} \int_{\p... ...frac{v_2 k_2}{\eta L_2^2} (\overline{p}_{f1} - \overline{p}_{f2}).\end{eqnarray}$$ (114) (115) (116)

The normal ${\bf n}$ in equation (114) is outward to phase 1 which accounts for the sign change in equation(115). Note as well that equation (115) is a volume average of equation (99) while equation (116) follows from equations (102) and (109). The desired limit is thus $\lim_{\omega\rightarrow 0} \gamma(\omega) = \gamma_{sq} = v_2 k_2/(\eta L_2^2)$.