Geoscience Reference
In-Depth Information
Non-linear aspects
The viscosity, density and heat conduction are not constant with temperature, but
the specific heat capacity of water or soil between 0
C is rather a
constant. Related aspects, such as permeability, retardation and potental drop are
therefore influenced by the change of temperature and the development of heat
transfer creates heterogeneity and affects convection and the life cycle of a
geothermal system. For many fluids, viscosity is less sensitive to pressure (it is
usually ignored), but it shows a pronounced variation with temperature; for water
=
C and 100
=
2.5. The density of water is slightly dependent on temperature, and less
sensitive to pressure (water without air bubbles is usually considered relatively
incompressible); for water
20=C
/
70=C
1.03. Furthermore, density changes may
cause free convection. A test may clarify these aspects.
C
/
20
=
70
=
C
z
test box
z
pressure
temperature
1
T
2
l
2
2
p
2
l
1
k
2
p
t
L
v
k
1
l
t
p
1
p
T
T
1
T
2
T
1
Figure 14.8 Test on heat transport; non-linear effects
Consider a vertical tube filled with cold saturated sand with height
L
and with
initial temperature
T
2
, see Fig 14.8. From time
t
= 0, hot water with temperature
T
1
is infiltrated, driven by a constant pressure drop
p
1
-
p
2
=
2
l
2
. Note that the
volumetric weight of hot water and cold water differs. The heat is transported by
the induced pore water flow (retarded by the energy loss for heating up the grains)
and it spreads by conduction (and dispersion). Because of symmetry in the
spreading, a sharp heat front can be considered. It moves with velocity
v
=
Rw
=
Rq/n
, where
w
is the true pore water velocity,
R
the heat retardation factor (about
0.4),
q
the Darcy velocity and
n
the porosity. At time
t
, the heat front has
progressed over
l
t
= vt
. The heated area of the sand column has permeability
k
1
and
the cold part
k
2
. The permeability ratio is
k
2
/
k
1
= (
1
l
1
-
g
/
2
)/(
g
/
1
) =
1
/
2
, where
is
the intrinsic permeability (not affected by heat),
g
the gravity acceleration and
the
kinematic viscosity. Since the permeability changes due to the temperature change
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