Geoscience Reference
In-Depth Information
D
MULTI
-
DIMENSIONAL CONSOLIDATION
The specific solution of consolidation problems in three dimensions requires
advanced mathematics. For a first estimation of the order of magnitude of the
effects induced by consolidation, a simple method is suggested (Verruijt). Consider
a general porous medium with volume
V
and surface
S
=
A
1
+
A
2
(Fig 6.4), where
A
1
is a drained surface (pore pressure is constant
u
= 0) and
A
2
is closed surface (no
flow condition). Adopting a volumetric approach, i.e. volume strain
and mean
effective stress
'
are related through a bulk modulus
K
:
' = K
, the general
storage equation, expressed by
(
w
/k
)
'
q
=
'
u =
(
w
/k
)
/
t =
(
w
/k
)
'/
t =
(1/
c
v
)
(
u
)/
t
(6.10)
w
.
30
Equation (6.10) is integrated
Here, the consolidation coefficient reads
c
v
= kK/
over the entire volume
[
'
u =
(1/
c
v
)(
u/
t
/
t
)]
dV
[
'
u
]
dV =
(1/
c
v
) (
[
u/
t
]
dV
[
/
t
]
dV
)
(6.11)
which requires the solution of three specific integrals. The second integral yields
with the introduction of a volume average pore pressure
(
[
u/
t
]
dV =
[
u dV
] /
t = V
[
u dV/V
] /
t = V
P/
t
(6.12)
The third integral in (6.11) is assumed to be zero, by considering the cases where
at time
t
= 0 a sudden loading is applied, which is kept constant during the
consolidation process. Hence
t
= 0, and the third term vanishes. The first
integral in (6.11) expresses the groundwater flow, since
/
[
'
u
]
dV =
(
w
/k
)
[
'
q
]
dV
(6.13)
q
is the net outflow out of a unit volume. It can be understood that in
the absence of singularities (point sources and wells) the compatibility of unit
volumes composing the entire volume, requires the flow at the boundaries to be in
accordance with the local flow. Mathematically this is equivalent to Gauss'
divergence theorem
where
'
[
'
q
]
dV =
q
'
dA
where
q
represents the boundary flux (
q
is perpendicular to boundary element
dA
). This boundary integral can be approximated by assuming a pore pressure
gradient at the boundary, according to an arbitrary parabolic shape of the actual
30
The compressibility of the pore fluid
)
can be included by adopting
c
v
= kK/
(
w
+n
)
K
).
Note, that the bulk modulus is involved and not the laterally confined compressibility
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