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&
2
dy
r
S
W
(
r
,
t
)
W
(
u
)
exp(
y
)
,
u
(14.10)
y
4
tkH
u
with
S
the aquifer storativity. For
u
< 0.01 the formula simplifies to
W
(
u
) =
-0.5772 - ln(
u
). The transmissivity
kH
can be obtained from the gradient of
s
and
log(
t
). It is difficult to obtain a reliable value for the storativity
S
. One usually
adopts
S
= 0.001.
For a semi-confined aquifer the well-function is given by Hantush
55
&
2
)
dy
r
W
(
r
,
t
)
W
(
u
,
)
)
exp(
y
)
,
)
,
kHC
(14.11)
4
y
y
u
with
the leakage factor, and
C
=
H'/k'
the hydraulic resistance of the semi-
confining layer (
H'
the height and
k'
the permeability). It is commonly applied by
geohydrologists. Note that in equation (14.11) an instantaneous leakage without
consolidation is adopted from a (rigid) aquitard. In fact, such a system is quite
unrealistic for shallow systems.
Besides some modifications (well storage, skin effect, anisotropy, partial
penetration) these well-functions comprise the core of the state of the art for
pumping tests and water well systems. If one includes time-dependent
consolidation in the aquitard, the well function becomes, adopting a semi two-
dimensional process (Barends)
&
2
)
f
dy
i
W
(
r
,
t
)
W
(
u
,
)
,
)
exp(
y
)
(14.12)
4
y
y
u
2
u
H
'
2
with
f
f
(
;
),
;
/(
1
),
i
i
y
c
t
v
Here,
c
v
is the consolidation coefficient of the aquitard. (14.12) is based on an
approximate Laplace inverse technique; the error involved is less than 1%. The
function
f
i
depends on the condition of the adjacent aquitard. Three situations can
be considered
1 (infinite) thick aquitard
f
1
=
;
(14.13a)
2 finite aquitard drained at the outside
f
2
=
;
coth(
;
)
(14.13b)
3 internal aquitard
f
3
=
;
tanh(
;
/2)
(14.13c)
55
For large value of time
t
Hantush' function transforms to a Bessel function,
W
(0,
)
) =
2
K
0
(
)
). Other simplifications lead to
K
0
(
)
) =
(
/2
)
) exp(
)
), for
)
>> 1, and
K
0
(
)
) =
ln(
)
/1.123), for
)
<< 1.
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