Geoscience Reference
In-Depth Information
B =
sin(
D
(
2
/
2
c
v
1
))(sinh(
D
(
2
/
2
c
v
1
) +
?
cosh(
D
(
2
/
2
c
v
1
))
?
=
(
k
2
(
2
/
2
c
v
2
) / (
k
1
(
2
/
2
c
v
1
)
This particular example shows that, in a two-layered stratified porous medium,
maximum cyclic pore pressures take place at the interface and they may exceed the
total loading up to ~15%, and that pore fluid gradients at the interface in the finer
bed are relatively large. The latter implies that filter instability is more likely.
In practice, one commonly interprets cyclic pore pressures by applying the
standard formula for cyclic-damped response from the theory of heat conduction,
which states
u = u
0
exp(
G
) cos(
2
t
G
)
(16.12)
This is not correct for the examples discussed previously. It violates the principle
of effective stress and the fact that the total loading by waves, tides or rocking
submerged structures is transferred as a body force instead of a boundary force.
Travelling water waves on a seabed
The previous situation describes standing water waves. The solution for
travelling water waves on a seabed is extensively discussed in literature (see
Verruijt). The principle elastic solution for the pore pressures response in a seabed
subjected to long wave loading
u
=
A
cos(
i
2
t
x
) at
z
= 0 is
u = A
cos(
2
t
x
) (exp(
z
)
+ m
exp(
z
))/(1+
m
)
(16.13)
))
0.5
,
=
L
2
/(2
with
=
(1 +
i
(1+
+
m
= 2(1+
)
n
)
f
/E,
m
= 1/(1
2
),
c
v
T
),
= 2
/
L
, and
T
= 2
/
2
, where
L
and
T
are the wave length and wave period, and
E
and
is complex it will give rise to a phase
shift, which will rapidly attenuate with depth, since the real part of
are the elastic constants. Because
is large
compared to
. In case the pore water is incompressible,
=
0 holds and (16.13)
becomes
u
=
A
exp(
x
), showing that there is no phase lag and the
induced pore pressures attenuate with depth. For large waves, Verruijt also gives an
expression for the shear stress amplitude:
z
)cos(
2
t
^
=
A
z
exp(
z
), which for
z
< 1 is
^
/
linear with depth. Hence, the shear stress ratio
'
0
is a constant in the top of the
seabed. When this ratio approaches tan
, the seabed may become unstable.
Water waves on rigid structures
A semi-submerged coastal structure, founded on a filter bed and subjected to
wave loading, will rock on its foundation. These movements will cause
deformation in the seabed and hence induced pore pressures in the subsoil. The
situation at the edges of the foundation involves two phenomena; in fact, a
combination of the two situations presented in Fig 16.4a. For
= 1 (free-standing
wave loading) and
= 0 (effective drained loading under the structure), in a 2-D
setting, see Fig 16.6. Although the cyclic character of this induced phenomenon
will probably not create serious pore water flow, large local pore water gradients
Search WWH ::
Custom Search