Environmental Engineering Reference
In-Depth Information
The discharge vector has the dimension [m
2
/s] and denotes the volume of
water flowing in the
x-y-
plane per unit space of
z
-direction. Discharge vector and
Darcy-velocity u [m/s] are related according to the formula:
H
ðx; yÞ
u
for the confined aquifer
ðx; yÞ¼
q
(14.9)
hðx; yÞ
u
ðx; yÞ
for the unconfined aquifer
with
H
thickness of the confined aquifer [m] and
h
height of watertable above the
base of the unconfined aquifer [m].
For the real interstitial velocity v [m/s] holds:
H y
v
ðx; yÞ
for the confined aquifer
q
ðx; yÞ¼
(14.10)
hðx; yÞy
v
ðx; yÞ
for the unconfined aquifer
with porosity
¼K rh
(with hydraulic conductivity
K
),
a formula for the calculation of the potential
y
. Using Darcy's Law u
'
from the piezometric head
h
can be
given:
K H hðx; yÞþC
c
for the confined aquifer
'ðx; yÞ¼
(14.11)
2
1
2
K hðx; yÞ
þ C
u
for the unconfined aquifer
h
[m] is the piezometric head above the base, both for the confined and the
unconfined case. In the unconfined aquifer,
h
corresponds to the position of the
groundwater table.
C
u
and
C
c
are constants that are irrelevant for the flow field:
when the potential is differentiated, the two constants vanish. However,
C
u
and
C
c
are relevant for the relation between
h
and
'.
Details are given below.
The condition that the head has no jump, where the aquifer changes from
confined to unconfined state, yields a condition for
C
u
und
C
c
. If both formulae
for the marginal condition
h ¼ H
are evaluated, both potential values are equal
under the condition:
1
2
KH
2
C
c
¼ C
u
(14.12)
, with which it is possible to describe
aquifers being partly confined and partly unconfined. Altogether one may thus
write:
One obtains a continuous potential
'ðx; yÞ
1
2
K H
2
K H hðx; yÞ
þ'
0
for the confined aquifer
'ðx; yÞ¼
(14.13)
2
1
2
K hðx; yÞ
þ'
0
for the unconfined aquifer
where the notation
'
0
is used instead of
C
u
. The transition between confined and
unconfined situation is given for the (critical) potential value:
1
2
KH
2
'
crit
¼
þ '
0
(14.14)