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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 10^{4} 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 from one side, must exit the other side so that
().
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;
where is the surface
tension. During wave passage, one
can demonstrate (Pride and Flekkoy, 1999) that the mean curvature
changes as where
*H*_{1} is of the same order
as *H*_{o} and where is a dimensionless number called
the capillary number. The capillary number
is defined where is some estimate of the wave-induced Darcy flux
and that is thus bounded as the wave strain times phase velocity; i.e., m/s. For typical interfaces (like air and water),
we have Pa m
and Pa s. Thus, for linear wave problems,
and can be considered a very small number.

Writing the fluid
pressures as and
using the fact that is continuous, allows
the conservation
of energy at the interface to be expressed

| |
(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 to be continuous at the interface and since the assumption of
the grains being welded together [or having an overburden effective pressure
acting on them that is greater than the wave stress] requires that
is continuous,
we obtain that to leading order in
| |
(55) |

along the interface . This means that the fluid pressure equilibration
can be modeled using the standard displacement-stress continuity conditions along
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 ``'' in front of them.

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** Up:** Patchy-Saturation Transport
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Stanford Exploration Project

10/14/2003