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)
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