Geoscience Reference
In-Depth Information
Linear model
The heat transport in a single layer (reservoir) with thickness
H
and in a uniform
steady groundwater flow at initial temperature
T
0
of 80 °C is considered (Fig
14.6a). At time
t
= 0, water of temperature
T
1
of 30 °C is injected at constant
volume rate
Q
. In this example, the lower boundary of the reservoir is sealed for
water and heat, and the upper boundary is sealed for water, but it can conduct heat
(thermal bleeding). If vertical temperature gradients in the reservoir are
disregarded, assuming the heat in the reservoir is uniform over the height, the
process is described by:
in the reservoir
T
T
T
T
w
cD
cD
cv
c
o
(14.3a)
L
T
x
x
y
y
x
t
H
T
'
T
'
in the overburden
(
c
)'
D
'
(
c
)'
(14.3b)
z
z
t
T
'
thermal bleeding (heat leakage
)
'
(14.3c)
w
o
z
z
0
0
at the interface
z =
0
T = T'
(14.3d)
Here, the parameters are: reservoir thermal longitudinal dispersion-diffusion
coefficient
D
L
=
c
+
a
L
v
[m
2
/s], reservoir thermal transversal dispersion-
diffusion coefficient
D
T
=
a
T
v
[m
2
/s], overburden thermal diffusion coefficient
D
' =
/
'/(
c
)', thermal convection velocity
v
=
q
(
c
)
f
/(
c
), soil heat capacity
c
=
n
(
c
)
f
c
)
s
[J/m
3
K], thermal conduction coefficient
+ (1
n
)(
[J/msK] in the reservoir and
' in the overburden, Darcy velocity
q
, density
, specific heat capacity
c
[J/kgK],
h
' = (
c
, longitudinal mechanical dispersivity
a
L
[m], transversal mechanical
dispersivity
a
T
[m] and porosity
n
. The dash characterises the overburden and the
suffix
f
refers to pore fluid and
s
to the soil matrix. The heat leakage at the interface
z
= 0, the thermal bleeding, is characterised by the heat flux
w
o
[J/m
2
s]. In this
approach, the parameters
c
)'/
,
c
,
D
L
,
D
T
,
D
',
a
L
,
h
',
q
and
H
are considered constant.
Usually, the diffusion effect is significantly smaller than the convection effect.
The reservoir temperature
T
becomes in this case (no vertical temperature
gradient in the reservoir)
h
'
x
D
'
T
T
(
T
T
)
erfc
(
(14.4)
0
1
0
H
4
v
(
vt
x
(
?
))
h
'
(
c
)'
2
with
?
D
L
D
'
(
)
< 0.2
and
h
'
vH
c
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