Characterization of heat transport processes
in geothermal systems
..............................................................................................................................................................
Ulvi Arslan1 and Heiko Huber2*
1
Institute of Materials and Mechanics in Civil Engineering (IWMB), Technical University
Darmstadt (TUD), Petersenstraße 12, 64287 Darmstadt, Germany; 2CDM Smith Consult
GmbH, Neue Bergstraße 13, 64665 Alsbach, Germany
.............................................................................................................................................
Abstract
Experimental field tests as well as laboratory tests have been conducted. In addition to field tests, a
geothermal laboratory device has been developed. With the help of this device, the heat transport
processes in different geological and hydrogeological conditions can be simulated. The different types of
heat transport mechanisms in geothermal systems can be characterized clearly. Furthermore, it is
possible to determine the increase of the effective thermal conductivity of a line source with rising
groundwater flow velocities. With the geothermal data gathered, common numerical programs are
verified and optimized. Therefore, all measured data are reconsidered by numerical back analysis.
Keywords: geothermal; energy; heat transport; laboratory
*Corresponding author.
heiko.huber@cdmsmith.
com
Received 13 November 2012; revised 1 March 2013; accepted 5 March 2013
................................................................................................................................................................................
1 INTRODUCTION
In times of global warming, renewable energies are getting
more important. Geothermal energy is the auspicious renewable energy in the field of geotechnical engineering.
Geothermal energy is thermal energy stored below earth’s
surface. The heat contained in deep geothermics (.400 m
below earth’s surface) originated mainly by radioactive decay of
persistent isotopes, while the heat contained in shallow geothermics (,400 m below earth’s surface) originated mainly from
solar irradiance upon the earth’s surface. On average, the temperature increases 0.038C m21 of depth, which is called geothermal gradient. Therefore, 99% of earth is hotter than 10008C,
while 99% of the remaining 1% is even hotter than 1008C. At
depths of about 1 km, temperatures of 35–408C can be achieved.
The geothermal heat transfer can be derived from the first
law of thermodynamics for a closed system, which can be
written as Equation (1).
dU ¼ dQ þ dW
ð1Þ
Between any two equilibrium states, the change of the inner
energy dU is equal to the sum of change of energy by a
heating process dQ and the change of work done dW at the
system. Derivation and transformation of Equation (1) leads to
Equation (2).
rc
@T
_
_ þW
¼ divðqÞ
@t
ð2Þ
where r is the density (kg m23); c is the spec. heat capacity
(W s kg21 K21); T is the temperature (K); t is the time (s); q_ is
the heat flux (W m22);
_ is the thermal source (W m22).
W
The change of the temperature T in time is heat flow
density plus thermal sources while heat flow always occurs
from a higher temperature object to a cooler temperature as
described by the second law of thermodynamics, indicated by
the negative first right-hand term. After all different types of
heat flow mechanisms can be distinguished, such as conduction, convection and radiation. The conductive heat transfer
bases on Fourier’s law and can be written as Equation (3).
q_ cond ¼ l gradðTÞ
ð3Þ
where l thermal conductivity (W m21 K21).
The convection (here transport of heat energy by groundwater movements) bases on the analogous Darcy’s law and can
be written as Equation (4).
q_ conv ¼ ðrcÞ divðvTÞ
ð4Þ
where v is the Darcy velocity (ms21).
Heat transport by thermal radiation is defined as Equation
(5). For temperatures (8 – 158C) of shallow geothermics, radiation is ,1% of the total heat transport and can therefore be
neglected.
q_ rad ¼ 1 s T 4
International Journal of Low-Carbon Technologies 2013, 8, 71–79
# The Author 2013. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
doi:10.1093/ijlct/ctt014 Advance Access Publication 14 April 2013
ð5Þ
71
U. Arslan and H. Huber
where s is the Stefan– Boltzmann constant (W m22 K24); 1 is
the relation of emissivity.
Each phase of the multiphase body soil has its own combination of heat flow mechanisms. If heat flow by radiation is
neglected, the heat transfer equation for the fluid phase (index
F) can be written as Equation (6).
ðrcÞF
@TF
¼ ðrcÞF vF divðTF Þ þ divðlF gradðTF ÞÞ
@t
_F
þ hAðTS TF Þ þ W
ð6Þ
where h is the heat transfer coefficient (W m22 K21); A is the
area (m2).
While the heat transfer equation for the solid phase (index S)
of a multiphase body can be written as
ðrcÞS
@TS
_S
¼ divðlS gradðTS ÞÞ þ hAðTF TS Þ þ W
@t
ð7Þ
Current numerical programs based on the finite element
method (FEM) or the finite difference method (FDM) describe
the geothermal energy transport with different assumptions.
These methods combine Equations (6) and (7) with the assumption that the temperature level of the fluid and the solid
phase is equal (local thermal equilibrium). Furthermore, the
thermal conductivity, density and heat capacity are uniformed
from separated values of each phase to total values of the multiphase body (index SF). With these assumptions, Equations (6)
and (7) can be written as Equation (8) introducing the porosity
n, which describes the ratio of pore volume to the total volume.
ðrcÞSF
@T
_ SF ð8Þ
¼ ðrcÞF nvF divðTÞ þ divðlSF gradðTÞÞ þ W
@t
where n is the porosity .
Objective of geothermal engineering is to use that energy
potential with suitable systems extracting or inserting heat
energy from or into the ground through heat transfer. Further
general information on geothermal basics and its background
are given in [1] and [2].
For the design of smaller geothermal systems, the geothermal, geological and hydrogeological values can be estimated
according to common literature. For the design of complex
geothermal systems, the subsoil has to be modeled with numerical approaches based on the FEM or the FDM using
values evaluated in laboratory or in situ. In saturated soils,
especially in areas of groundwater flow, the energy transport of
the fluid phase (conduction and convection) and the solid
phase (conduction) and their interaction has to be considered
separately for a proper numerical modeling of geothermal
systems. Even in case of low-groundwater flow velocity of
about 1027 ms21, the role of the convective heat transport
cannot be neglected, as described by [3] and [4].
Available monitored data of the ratio of heat energy transported by convection to the entire transported heat energy in
dependence of the groundwater flow velocity is insufficient
72
International Journal of Low-Carbon Technologies 2013, 8, 71– 79
[5]. For the optimization of the modeling of geothermal
systems, a sufficient database of groundwater flow-influenced
geothermal systems is essential.
For a better understanding of the geothermal heat transfer
processes like conduction, convection, radiation and their
interaction, a large-scale geothermal laboratory device has been
invented and developed at the Technical University Darmstadt.
2 LABORATORY DEVICE
2.1 Assembling
To investigate the different types of heat transport mechanisms
in detail a large-scale laboratory device has been developed at
the Technical University Darmstadt, [6] (Figure 1).
The laboratory device is constructed in a large scale with
outer dimensions of 312/76/90 cm (L/W/H) and inner dimensions of 297/64/71 cm (L/W/H). The construction is very
massive with an acryl glass wall of 1.5 cm supported by a steel
frame in a U 80 shape placed vertically every 50 cm and horizontally at the top and the bottom. The apparatus is covered by
an acryl glass plate. Every wall, the bottom and the cover plate
as well as its connections are waterproof up to a high pressure.
The device can be filled with different kind of watersaturated soil. With its massive construction confined, water
even of high pressure can be simulated. At the borders of the
device, a connection for the water inlet and water outlet is installed. At the outside of the device, the water inlet is connected to a water supply tank of adaptable height. The water
outlet is connected to a hydraulic pipeline, which is also variable in its installation height. Inside of the device, the water
inlet and outlet are enlarged to diffusers of a perforated metal
plated and a geotextile being installed 13.3 cm from each
border. The diffusers spread the water flow uniformly over the
whole section of the device. According to the chosen difference
in height DH between the hydraulic head of the water supply
tank and the pipelines at the water outlet, different flow velocities of the water running through the device can be regulated
according to the permeability of the installed soil in consideration of Darcy’s law.
Figure 1. Large-scale laboratory device.
Characterization of heat transport processes in geothermal systems
At about one-third of the length of the device (83.3 cm), a
line source is installed vertically. This line source is constructed
by a copper tube with an outer diameter of 1.8 cm, which is
filled with silicon oil surrounding a heating element. With help
of this line source, a thermal load can be applied to the installed soil steady or transient in time. The laboratory device is
thermally isolated with different layers of Styrofoam. A constantly tempered fluid is circulating in copper tubes between
these different layers of Styrofoam. Therefore, a chosen constant temperature can be applied to the device.
2.2 Measuring system
The laboratory device is equipped with an extensive measuring
system. The hydraulic head of the water supply tank, the water
outlet pipelines and different points of the inside of the laboratory device is monitored with 12 water standpipes. To be able
to compare the different hydraulic heads, the water standpipes
are gathered next to each other in a water harp.
At the water inlet and the water outlet, the mass flow is
measured constantly with flow meters. At the water outlet, the
whole section of the device is divided into nine chambers of
the same area with separated water outlets. The mass flow
through these nine chambers can be measured separately.
Therefore, the homogeneity of the water flow over the whole
section of the device can qualitatively be verified.
After all, 33 temperature sensors Pt 100 are installed in or at
the laboratory device to determine constantly the development
of temperature over time. Two Pt 100 are installed at the water
inlet and the water outlet to measure the temperature of the
flowing water. One sensor is installed inside of the line source
to guarantee an accurate thermal load. Thirty Pt 100 are
located in a horizontal section in the middle of the height of
the device. According to an expected symmetric temperature
plume, 28 of the sensors in the soil are installed at one side of
the symmetric axis. Two of them are installed mirrored to the
symmetry axis to verify the expected symmetry.
According to a performed sensitivity analysis, the sensors
are located next to the line source in direction of the temperature plum, beside and behind the line source. The influence of
the convective heat transport can be determined by comparing
the temperature development of the sensors before the line
source with the sensors behind the line source. A drawing of the
arrangement of the measuring system is given in (Figure 2).
2.3 Experimental results
After all 42 scenarios with different geological conditions like
coarse and fine sands as well as different hydrogeological conditions like confined and unconfined groundwater with different Darcy velocities were performed. Some of the results are
Figure 2. Arrangement of the measuring system.
International Journal of Low-Carbon Technologies 2013, 8, 71– 79 73
U. Arslan and H. Huber
given in the following. Temperature development of chosen
temperature sensors in the experiment numbered Sz 3a 4 are
shown in (Figure 3). In Sz 3a 4, a constant thermal load of
23.3 W m21 was set to the line source for 2 days, which was installed in uniform, loose coarse sand. The groundwater was
confined with an over pressure of about 0.3 m, and a Darcy
velocity of 0.45 m day21 was applied.
Starting from a tempered level of the whole device of 208C,
the temperature inside the line source (31) increases to about
28.58C after 2 days. Right beside the line source (11), an increase of the temperature to about 25.88C after 2 days can be
determined. The temperature sensors (1), (4) and (8), with a
distance of 10 cm to the line source show a temperature increase of about 28K after 2 days. The temperature sensors
numbered (2), (5) and (16) with a distance of 20 cm to the
line sources show an increase of the temperature of about
1.28K, while the sensors numbered (21) with a distance of
30 cm to the line source show an increase of 1.18K. The slope
of the temperature increase of the red-marked sensors in
groundwater flow directions (8), (16) and (21) is parallel to
each other with an amount of about 0.318K day21. The greenmarked temperature sensors (1) and (2), which are located
beside the line source, are parallel to each other, too, but with
a lower slope of about 0.288K day21. The blue-marked temperature sensors (4) and (5), being located before the line
source in groundwater flow direction, show the smallest slope
of about 0.258K day21. This difference in the slopes of the
sensors before, beside and behind the line source is caused due
to the convective heat transport.
In the scenarios Sz 3a 2– Sz 3a 7, the Darcy velocity was
varied between 0 and 1.12 m day21, while all other values were
kept constant. Therefore, the variation of the temperature
development inside the laboratory device depending on the
Darcy velocity was investigated. The temperature development
over time of the temperature sensors inside the line source
(31), right beside the line source (11) and in a distance of
10 cm to the line sourced in groundwater flow direction for
the scenarios Sz 3a 2– Sz 3a 7 is shown in (Figure 4). A strong
correlation between the Darcy velocity and the slope of the
temperature increase can be determined. The higher the Darcy
velocity, the smaller is the slope of the temperature increase,
and therefore, the smaller is the absolute temperature of the
sensors after 2 days of the test. While the temperature of
the sensor inside the line source (31) rises in Sz 3a 7
(v ¼ 0 m day21) to a value of about 29.18C after two days, the
temperature of the same sensor rises to a value of only 27.28C
in Sz 3a 2 (v ¼ 1.12 m day21). This trend can also be determined for the temperature sensors numbered (11) and (8). The
higher Darcy velocity causes a high dispersion of the applied
heat energy of the line source. This high thermal dispersion
leads to a larger spatial extension of the temperature plume by
lower temperatures inside the plume.
3 NUMERICAL BACK ANALYSIS
The gathered experimental geothermal laboratory data were
back-analyzed by numerical methods.
3.1 Model description
The numerical back analysis was performed with FEFlow, an
FEM-based code for combined transient heat and transient
flow transport simulation. The laboratory device was simulated
as a 2D, horizontal, water-saturated problem. The simulation
area was divided into .130,000 three-noded triangles with a
higher density of elements inside and right beside the line
source and at the borders of the simulation area. The duration of
Figure 3. Temperature development over time of chosen temperature sensors in Sz 3a 4.
74
International Journal of Low-Carbon Technologies 2013, 8, 71– 79
Characterization of heat transport processes in geothermal systems
the simulation was chosen according to the performed scenarios
to 2 days, each. The automatic time stepping control scheme was
chosen as the predictor–corrector Adams-Bashforth/Trapezoid
rule.
The modeling area was simulated with three different materials. Material 1 defines the water-saturated soil inside the laboratory apparatus, Material 2 and Material 3 describes the
line source simulated as a copper tube (Material 2), which is
filled with silicon oil (Material 3). As initial conditions, the
temperature of the whole laboratory device was set to 208C,
and the hydraulic head was set 0.3 m above the height of the
laboratory device. As boundary conditions, the hydraulic head
of the diffusers were defined constant, and the temperature of
the border and the diffusers were set constant. The properties
of the three materials were chosen according to performed
laboratory tests and where varied in bandwidths (Table 1).
Most of the values of the water-saturated sand (Material 1)
were determined in laboratory with help of standard geotechnical and geothermal testing according to national standards.
The thermal conductivity of the solid and the fluid phase of
the sand were determined with help of the fast thermal conductivity meter tk04 byTeKa, Berlin. The volumetric heat capacity of the solid and the fluid phase of the sand were
determined with help of the differential scanning calorimeter
Figure 4. Temperature development over time of chosen sensors in Sz 3a 2– Sz 3a 7.
Table 1. Properties of the Materials 1– 3
Water-saturated soil (Material 1)
Thermal conductivity of solid lS
Thermal conductivity of fluid lF
Vol. heat capacity of solid cS
Vol. heat capacity of fluid cF
Porosity n
Permeability k
Dispersivity aL/aT
Darcy velocity v
Copper tube (Material 2)
Thermal conductivity of solid lS
Vol. heat capacity of solid cS
Porosity n
Permeability k
Silicon oil (Material 3)
Thermal conductivity of fluid lF
Vol. heat capacity of fluid cF
Porosity n
Permeability k
Unit
Bandwidth
Standard value
Remark
W m21 K21
W m21 K21
MJ m23 K21
MJ m23 K21
–
ms21
m m21
m day21
2.5– 6.0
0.597
1.0– 2.5
4.18
0.4– 0.5
3.8.1023
0/0– 0.25/0.025
0– 0.46
3.85
0.597
1.73
4.18
0.46
3.8.1023
0/0
0.155
Determined in laboratory, tk04 method
Determined in laboratory, tk04 method
Determined in laboratory, DSC 200 F3
Determined in laboratory, DSC 200 F3
Determined in laboratory, weight measuring
Determined in laboratory, DIN 18300
According to common literature [7]
Determined in laboratory. Water mass flow
W m21 K21
MJ m23 K21
–
m s21
0.05–0.07
0.03–300
0
1.0.10220
0.056
3.45
0
1.0.10220
According to common literature [2]
According to common literature [2]
According to common literature [7]
According to common literature [7]
W m21 K21
MJ m23 K21
–
m s21
5– 100
1– 40
1
1
100
10
1
1
According to common literature [2]
According to common literature [2]
According to common literature [7]
According to common literature [7]
International Journal of Low-Carbon Technologies 2013, 8, 71– 79 75
U. Arslan and H. Huber
Figure 5. Sensitivity analysis: variation of lS, cS, n and v of Material 1 (sand).
DSC 200 F3 by NETZSCH GmbH. The porosity was determined by measuring the weight of the sand and the water
insight the laboratory device. The permeability was determined
according to DIN 18300, while the Darcy velocity was measured with help of the mass of the water running through the
laboratory device per square meter and time. The values of the
copper tube (Material 2) and the silicon oil (Material 3) were
chosen due to common literature, such as [2] and [7]. The
values of Material 1, determined in laboratory, correspond very
well to values for that material in accordance to standard
values from common literature. Nevertheless, the range of the
bandwidth for the numerical analysis of the values for that
material is chosen widely, to get insight into the influence of
these values. The thermal conductivity and the permeability of
the silicon oil were chosen very high to guarantee a fast distribution of the temperature inside the line source. The thermal
conductivity of the copper tube was chosen very small, to
simulate the heat transfer from the flowing silicon oil to the
copper tube and from the copper tube to the water-saturated
sand with flowing groundwater.
3.2 Sensitivity analysis
The influence of every single material property was investigated
with help of a numerical sensitivity analysis. While all other
76
International Journal of Low-Carbon Technologies 2013, 8, 71– 79
Figure 6. Sensitivity analysis: variation of aL and aT of Material 1 (sand).
properties were set to the standard value, one property was
varied between the chosen bandwidth according to (Table 1).
The results of the sensitivity analysis are summarized in
(Figures 5– 8). The continuous blue line is the temperature development of the temperature sensor inside the line source
(31) gathered in the experiment numbered Sz 3a 5.
A variation of the thermal conductivity l, the heat capacity
c and the porosity n of the water-saturated sand leads to a parallel displacement of the temperature, with only small variation
Characterization of heat transport processes in geothermal systems
Figure 7. Sensitivity analysis: variation of cS and lS of Material 2 (copper).
Figure 8. Sensitivity analysis: variation of cF and lF of Material 3 (oil).
Figure 9. Comparison of the experimental laboratory data with the data from the numerical back analysis of Sz 3a 5.
of the temperature slope. The slope of the temperature increase
without a parallel displacement changes depending on the
simulated Darcy velocity v (Figure 4).
A variation of the thermal dispersivity of the water-saturated
sand leads only to a small parallel displacement of the slope
of temperature increase. This is caused due to the small
International Journal of Low-Carbon Technologies 2013, 8, 71– 79 77
U. Arslan and H. Huber
Figure 10. Comparison of the experimental laboratory data and the numerical back analysis of Sz 3a 7– Sz 3a 10.
dimensions of the laboratory device. The thermal dispersivity
only affects the temperature development in field experiments
with larger dimensions (Figure 6).
A variation of the thermal conductivity l of the copper
tube and the silicon oil lead to a parallel displacement of the
temperature with only small variation of the temperature
slope. A variation of the heat capacity of the copper tube and
the silicon oil leads to a displacement of the temperature increase at the beginning of the thermal load. Owing to a higher
heat capacity, the temperature retains longer inside the line
source. This leads to higher temperatures at the beginning of
the experiment (Figures 7 and 8).
3.3 Results
With the help of the sensitivity analysis the best fitting parameters to the laboratory data of the scenario Sz 3a 5 were
found, whereby a Darcy velocity of 0.155 m day21 and a constant thermal load of 23.3 W m21 were applied. A comparison
of the experimental laboratory data (continuous line) and the
numerical back analysis (dashed line) for scenario Sz 3a 5
inside the line source (31), right beside the line source (11) as
well as 10 cm (8) and 30 cm away from the line source in
groundwater flow direction is given in (Figure 9).
After all, a good fit of the numerical back analysis to the experimental laboratory data was achieved, and the numerical
model was calibrated for the scenario Sz 3a 5. To validate the
numerical model, the scenario Sz 3a 7– Sz 3a 10 were backanalyzed. In the experimental laboratory scenarios of Sz 3a 7–
Sz 3a 10, all geological and hydrogeological values were kept
constant, while only the thermal load applied on the line
source was varied between 14.8 W m21 (Sz 3a 8) and
40.1 W m21 (Sz 3a 9). Owing to the variation of the constant
thermal load, the temperature inside the line source (31)
increases to values of 26 – 368C after 2 days.
A comparison of the experimental laboratory data (continuous lines) and the numerical back analysis (dashed lines) for
78
International Journal of Low-Carbon Technologies 2013, 8, 71– 79
the scenario Sz 3a 7– Sz 3a 10 inside the line source (31) is
shown in (Figure 10). After all, a good fit of the numerical
back analysis to the experimental laboratory data was achieved.
The numerical model was validated.
4 CONCLUSION
With the help of the developed geothermal laboratory device,
data of high accuracy were gathered. In .40 scenarios, different geological and hydrogeological values were varied.
A numerical model was developed and validated by the extensive experimental data. In future work, all the results of the
laboratory experiments will be compared with the performed
geothermal field test [8]. Therefore, the numerical model can
also be validated for in situ experiments. With the help of this
developed model, validated by extensive geothermal laboratory
as well as in situ data geothermal systems can be simulated and
therefore preoptimized. That means, especially for shallow geothermal closed-loop systems, such as borehole heat exchangers,
the length of borehole heat exchangers can be shortened, and
money can be saved.
REFERENCES
[1] Ingersoll LR, Plass HJ. Theory of the ground pipe heat source for the heat
pump. ASHVE Transactions 1948;47:339– 48.
[2] Carslaw HS, Jaeger JC. Conduction of Heat in Solids, 2nd edn. Oxford
University Press, 1959, 510.
[3] Pannike S, Kölling M, Panteleit B, et al. Auswirkung hydrogeologischer
Kenngrößen auf die Kältefahnen von Erdwärmesondenanlagen in
Lockersedimenten, Grundwasser – Zeitschrift der Fachsektion Hydrogeologie
1/2006, 2006.
[4] Katsura T, Nagano K, Hori S, et al. Investigation of groundwater flow
effect on the thermal response test result. In: Proceedings of 11th Energy
Characterization of heat transport processes in geothermal systems
Conservation Thermal Energy Storage Conference. Stockholm. 15 – 17 June
2009.
[5] Glück B. Simulationsmodell Erdwärmesonden’ zur wärmetechnischen
Beurteilung von Wärmequellen, Wärmesenken und Wärme-/Kältespeichern.
Rud. Otto Meyer-Umwelt-Stiftung, Hamburg, 2008.
[6] Huber H, Arslan U, Stegner J, et al. Experimental and numerical modelling
of geothermal energy transport. In: Proceedings 13th International
Conference of the International Association for Computer Methods and
Advances in Geomechanics (IACMAG 2011), Melbourne, Australia, 09 – 11
May 2011, Vol. 1, pp. 455 – 459, 2011.
[7] Bear J. Dynamics of Fluids in Porous Media, Volume 1, Environmental
science series, Dynamics of Fluids in Porous Media, Policy Sciences Book
Series. American Elsevier Pub. Co., 1972.
[8] Huber H, Arslan U. Geothermal Field tests with Groundwater Flow. In:
Proceedings Thirty-Seventh Workshop on Geothermal Reservoir Engineering.
Stanford University, Stanford, California, 30 January 2012– 1 February 2012.
International Journal of Low-Carbon Technologies 2013, 8, 71– 79 79