Mesoscopic flow equations

To obtain the transport law $-i\omega \zeta_{\rm int} = \gamma(\omega) (\overline{p}_{f1} - \overline{p}_{f2})$, the mesoscopic flow is analyzed in the limits of low and high frequencies. These limits are then connected using a frequency function that respects causality constraints. The linear fluid response inside the patchy composite due to a seismic wave can always be resolved into two portions: (1) a vectorial response due to macroscopic fluid-pressure gradients across an averaging volume that generate a macroscopic Darcy flux ${\bf q}_i$ across each phase and that corresponds to the macroscopic conditions $\overline{p}_{fi}= 0$ and $\nabla \overline{p}_{fi} \neq 0$; and (2) a scalar response associated with internal fluid transfer and that corresponds to the macroscopic conditions $\overline{p}_{fi} \neq 0$ and $\nabla \overline{p}_{fi}=0$. The macroscopic isotropy of the composite guarantees that there is no cross-coupling between the vectorial transport ${\bf q}_i$ and the scalar transport $\dot{\zeta}_{\rm int}$ within each sample.

The mesoscopic flow problem that defines $\dot{\zeta}_{\rm int}$ is the internal equilibration of fluid pressure between the patches when a confining pressure $\Delta P$ has been applied to a sealed sample of the composite. Having the external surface sealed is equivalent to the required macroscopic constraint that $\nabla \overline{p}_{fi}=0$. Upon taking the divergence of (2) and using equation (3), the diffusion problem controling the mesoscopic flow becomes

$$\begin{eqnarray} &&\frac{k}{\eta_i} \nabla^2 p_{fi} + i \omega \frac{\alpha}{K B... ...t \nabla p_{fi} = 0 \mbox{ \hskip2mm on \hskip1mm} \partial E_i, \end{eqnarray}$$ (56) (57) (58)

where $\Omega_i$ is the region that each phase occupies within the averaging volume, $\partial E_i$ is that portion of the external surface of the averaging volume that is in contact with phase i, and the brackets in equation (57) again denote jumps across the interface. One also needs to insert equations (3) and (4) into (1) to obtain a second-order partial-differential equation for the displacements ${\bf u}_i$. In general, the local confining pressures p_ci are determined using

$$p_{ci} = - K \nabla \cdot {\bf u}_i + \alpha p_{fi}$$ (59)

once the displacements ${\bf u}_i$ are known.