Geoscience Reference
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with time, the specific discharge
q
increases with time, and so does the heat front
velocity
v
and position
l
t
.
Considering mass conservation between the hot and cold area at any moment,
leads to
k
p
k
p
1
1
2
2
k
p
p
k
p
p
l
L
l
1
1
t
2
t
2
1
t
2
t
q
q
q
p
1
2
t
k
k
l
L
l
1
2
1
t
2
t
l
(
L
l
)
1
t
2
t
k
2
(
1
l
l
)
1
2
L
q
2
(14.7)
l
1
(
1
1
1
t
L
2
2
One may recognise the density and viscosity effect. Next,
l
t
=
L
and
q
are
calculated.
Rq
Rk
l
t
v
t
t
(
(
1
1
1
)
2
(
1
l
l
)
t
1
2
2
n
nL
2
2
2
1
1
2
t
1
Rk
l
t
2
1
1
2
1
(
1
(
l
l
)
t
1
2
2
2
nL
L
2
2
2
1
1
(
1
2
2
k
2
1
(
l
l
)
1
2
L
2
q
(14.8)
1
2
t
Rk
1
1
2
1
with
(
1
(
l
l
)
1
2
2
nL
2
2
2
2
< 1, this result indeed shows that
q
increases with time when hot
water is injected. Obviously, when cold water is injected in a hot aquifer, the
discharge will decrease with time. For a temperature drop of 50
Since
1
/
1
2
=
C, this increase or
decrease can be in the order of 40% (significant!).
The viscosity and density effect in heat dispersion is shown in Fig 14.9. In
reality, the heat front is not uniform and preferential channels can be formed,
particularly in horizontal flow, where the density change will cause two-
dimensional free convection (warm water tends to move upward). Here, numerical
simulation is required. Practically, the density effect is marginal for geothermal
systems. The viscosity effect is however pronounced. If disregarded, it may lead to
underestimation of the life cycle of deep geothermal systems and for cyclic shallow
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