Geoscience Reference
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pore pressure field in the entire volume (Fig 6.4). The maximum pore pressure is
caused by a sudden jump at
t
= 0 and decaying with time. This parabolic shape is
characterised by a maximum pressure
u
m
at an average distance
L
1
(drainage
length) from the drained border
A
1
. The relation between the average pressure
P
and the maximum pressure
u
m
for a 2D parabolic shape is
P
=
2
/
3
u
m
.
slope: 2u
m
/L
1
slope: 2u
m
/L
1
slope: 2u
m
/L
1
slope: 2u
m
/L
1
parabola
parabola
parabola
parabola
u
m
u
m
u
m
u
m
P
P
P
P
q
q
q
q
L
1
L
1
L
1
L
1
A
1
A
1
A
1
A
1
V
V
V
V
A
2
A
2
Figure 6.4
Introduction of Darcy's law yields at the border
A
1
for a parabola
q
= 2
(
k/
w
)
u
m
/
L
1
= 3 (
k/
w
)
P
/
L
1
. The integral (6.13) can now be evaluated
q
'
dA =
q
'
dA
1
=
[3(
k
/
w
)
P/L
1
]
dA
1
=
3(
k
/
w
)
P/L
1
dA
1
=
3(
k
/
w
)
P A
1
/L
1
Thus
[
'
u
]
dV =
3
P A
1
/
L
1
(6.14)
and equation (6.11) becomes with (6.12) and (6.14)
3
P A
1
/L
1
=
(
V/c
v
)
P/
t
(6.15)
With the introduction of a second length scale
L
2
=
V
/
A
1
called the hydraulic
radius (it is related to the drainage capacity of the considered volume) this equation
becomes
P =
P/
t
with
= L
1
L
2
/(3
c
v
)
(6.16)
The general solution, assuming
P
=
P
0
at
t
= 0, yields
31
P = P
0
exp[
t/
] =
P
0
exp[
3
c
v
t/L
1
L
2
]
(6.17)
31
For an axi-symmetric parabolic shape the factor
P
=
1
/
2
u
m
leading
P/P
0
= exp[4
ct/L
1
L
2
],
and for a spherical symmetry
P
=
2
/
5
u
m
giving
P/P
0
= exp[
5
c
v
t/L
1
L
2
]. But for a wick drain
it becomes
P/P
0
exp[ 0.25
c
v
t/L
1
L
2
]
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