Local continuity conditions on $\partial \Omega_{12}$

At the pore scale, the interface separating one fluid patch from the next is a series of meniscii. Roughness on the grain surfaces keeps the contact lines of these meniscii pinned to the grain surfaces. Pride and Flekkoy (1999) argue that the contact lines of an air-water meniscus will remain pinned for fluid-pressure changes less than roughly 10⁴ Pa which correspond to the pressure range induced by linear seismic waves. So as a wave passes, the meniscii will bulge and change shape but will not migrate away. This makes the problem vastly more simple to analyze theoretically.

One porous-continuum boundary condition is that all fluid volume that locally enters the interface $\partial \Omega_{12}$ from one side, must exit the other side so that ${\bf n} \cdot {\bf Q}_1 = {\bf n} \cdot {\bf Q}_2$ ($={\bf n} \cdot {\bf Q}$). Another boundary condition is that the difference in the rate at which energy is entering and leaving the interface is entirely due to the work performed in changing the miniscii surface area. Before the wave arrives, each miniscus has an initial mean curvature H_o fixed by the static fluid pressures initially present; $p_{f1}^o - p_{f2}^o = \sigma H_o$ where $\sigma$ is the surface tension. During wave passage, one can demonstrate (Pride and Flekkoy, 1999) that the mean curvature changes as $H = H_o + \epsilon H_1 + O(\epsilon^2)$ where H₁ is of the same order as H_o and where $\epsilon$ is a dimensionless number called the capillary number. The capillary number is defined $\epsilon = \eta \vert{\bf Q}\vert/\sigma$where $\vert{\bf Q}\vert$ is some estimate of the wave-induced Darcy flux and that is thus bounded as the wave strain times phase velocity; i.e., $\vert{\bf Q}\vert < 10^{-3}$ m/s. For typical interfaces (like air and water), we have $\sigma\gt 10^{-2}$ Pa m and $\eta \approx 10^{-3}$ Pa s. Thus, for linear wave problems, $\epsilon \ll 10^{-4}$ and $\epsilon$can be considered a very small number.

Writing the fluid pressures as $p_{fi} = p_{fi}^o + \delta p_{fi}$ and using the fact that ${\bf n}\cdot {\bf Q}$ is continuous, allows the conservation of energy at the interface to be expressed

$$\left[{\bf n} \cdot \left\{ \mbox{\boldmath$\tau$}_i \cdot \... ...\right] = \sigma {\bf n} \cdot {\bf Q} H_o [1 + O(\epsilon)].$$ (54)

The brackets on the left-hand side deonte the jump in energy flux across the interface, while the right-hand side represents the rate at which work is performed in stretching the meniscii. Since conservation of momentum requires ${\bf n} \cdot \mbox{\boldmath$\tau$}$to be continuous at the interface and since the assumption of the grains being welded together [or having an overburden effective pressure $(1-\phi)(\rho_s-\rho_f)g h$ acting on them that is greater than the wave stress] requires that $\dot {\bf u}$ is continuous, we obtain that to leading order in $\epsilon$

$$\delta p_{f1} = \delta p_{f2}$$ (55)

along the interface $\partial \Omega_{12}$. This means that the fluid pressure equilibration can be modeled using the standard displacement-stress continuity conditions along $\partial \Omega_{12}$ that were also employed in the double-porosity analysis; i.e., capillary effects can be neglected. In what follows, the fluid pressures correspond to the changes induced by the wave and so we cease to explicitly write the ``$\delta$'' in front of them.