Page 33 - Geothermal Energy Renewable Energy and The Environment

16 Geothermal Energy: Renewable Energy and the Environment

Table 2.3
Thermal conductivity of some common materials, in w/m-k
material 25°c 100°c 150°c 200°c
Quartz a 6.5 5.01 4.38 4.01
Alkali feldspar b 2.34
Dry sand a 1.4
Limestone a 2.99 2.51 2.28 2.08
Basalt a 2.44 2.23 2.13 2.04
Granite a 2.79 2.43 2.25 2.11
Water c 0.61 0.68 0.68 0.66

a Clauser, C. and Huenges, E., Rock Physics and Phase Relations, Washington, DC:
American Geophysical Union, 105–26, 1995.
b Sass, J. H., Journal of Geophysical Research, 70, 4064–65, 1965.
c Weast, R. C., CRC Handbook of Chemistry and Physics, Boca Raton, FL: CRC Press,
Inc., 1985.

The relationship represented by Equation 2.4 applies directly to instances where temperature
measurements are made in a well or borehole, allowing projection to depth of the temperature of
a potential geothermal resource. In essence, the geometry represented by this problem consists of
a slab of material across which a temperature difference exists. By measuring the heat capacity
of the geological material and the temperature difference between two different depths in a well
it is possible to deduce how many J/s (i.e., watts) are flowing through an area, and thus project to
depth what the temperature may be. Suppose, for example that, at a potential geothermal site located
in basalt, an exploration well was drilled to a depth of 2000 m and the temperature measured
at the bottom of the well was 200°C. If we average the thermal conductivity of basalt between
25°C (the temperature at the ground surface) and 200°C (Table 2.3), which is reasonable given the
nearly linear change in k with temperature over this temperature interval, and assume that our
th
measurements are representative of each square meter of surface area at the site, the rate of heat
flow at this site is

2.2 W/m-K × 1 m × (473 K – 298 K)/(1000 m) = 0.193 W.
o
2
o
2
Given that the global average heat flow is 87 mW/m , the value at our hypothetical site is sugges-
tive of a significant heat source at depth that could warrant further investigation.
However, this approach is not suitable for calculating how much heat a geological body, such
as a magma chamber, will transfer to its surroundings over time since the geometry of the magma
system is not planar. If we assume the heat source can be conceptualized as a cylindrical body, then
the form of the equation becomes
dq /dt = k × 2π × r × l × dT/dr, (2.5)
th
th
where r is the radius of the cylinder and l is the length. By integrating this equation, heat conduction
as a function of radial distance from the body can be determined from

dq /dt = k × 2π × l × (T – T )/(ln (r /r )), (2.6)
1
2
2
1
th
th
where the subscripts 1 and 2 refer, respectively, to the inner and outer locations relative to the center
of our heat source.