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INTERNATIONAL JOURNAL OF
NUMERICAL ANALYSIS AND MODELING, SERIES B
c 2013 Institute for Scientific
Computing and Information
Volume 4, Number 4, Pages 413–424
NUMERICAL MODELING AND ANALYTICAL VALIDATION
FOR THE MOVEMENT OF THERMAL FRONT IN A
HETEROGENEOUS AQUIFER THERMAL ENERGY STORAGE
SYSTEM
SAYANTAN GANGULY, N. SEETHA AND M. S. MOHAN KUMAR
Abstract. The aquifer thermal energy storage system (ATES) owing to growing demands for
sustainable energy has become a popular technology in last few decades for long term storing of
excess thermal energy. The efficiency of such a system depends entirely on the capacity of the
aquifer to retain heat and hence modeling the heat transfer and transient temperature distribution
in the aquifer due to hot water injection is essential. The present study is concerned about
presenting a two-dimensional numerical model for such an ATES system with block heterogeneities
where hot water is injected through an injection well into the heterogeneous porous aquifer with
lesser initial temperature. The transient temperature distribution in the porous aquifer or the
advancement of the hot water thermal front generated due to hot water injection into the porous
aquifer is modeled here using the multiphysics numerical code COMSOL. First the model is
developed for the general case including the convective and the conductive heat transport. Since
modeling of such complex systems are associated with numerical errors which may lead to possible
errors in estimation of temperature distribution in the aquifer, the results derived numerically
afterwards are compared and validated with an analytical model derived by the authors.
Key words. ATES, Heat transport, Hot water injection,Thermal front,Heterogeneous porous
media and Numerical and analytical modeling.
1. Introduction
Production of renewable and sustainable energy is one of the most important
topics of this century since the reserve of the fossil fuels on earth is going to be
exhausted in not so far future. The modern research thus concentrates a lot on
the exploration and exploitation of alternative sources of energy such as solar,
wind power and geothermal etc. With the production of energy the storage of the
excess thermal energy is also becoming important to make use of it in future. The
Aquifer Thermal Energy Storage (ATES) systems have become popular worldwide
for serving this purpose as a practically and economically feasible large scale open
energy storage system for heating as well as cooling buildings. Injecting the hot or
cold water into an aquifer for long term storage and extracting that in the time of
demand is the basic theory of this system. An ATES system operates in a full cycle
consisting of mainly four stages. The injection of hot/cold water into the aquifer,
storage of the hot/cold water, production and the heating/cooling of buildings (Sila
Dharma [1]). Practical feasibility and low cost of implementation and maintenance
have been behind the popularity of the system. Moreover it reduces the expense of
heating and cooling greatly and has got almost no adverse environmental effects.
Received by the editors October 4, 2012 and, in revised form, August 26, 2013.
2000 Mathematics Subject Classification. 35Q79.
413
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S. GANGULY, N. SEETHA AND M. KUMAR
The hot water byproduct of thermal power plant or the excess heat collected in
the solar energy system during periods of bright sunshine thus can be injected and
stored underground and extracted in the time of demand. Direct use of groundwater
with relatively high volumetric heat capacity makes ATES system more efficient
than any other system (e.g. closed system like borehole thermal energy system or
BTES) that can be used for this purpose (Kim et al. [2]). The effectiveness of
storing the thermal energy in an aquifer depends on the efficiency of the aquifer
to retain heat. Loss of heat from the aquifer to the surrounding media affects the
efficiency of the aquifer to be used as an ATES system and hence understanding the
heat transfer in the porous aquifer is crucial. Continuous injection of hot or cold
water into the natural groundwater in the aquifer generates an interface zone known
as the thermal front, across which the aquifer temperature varies from the injection
water temperature to the initial aquifer temperature. The thermal front with the
passage of time propagates through the porous aquifer medium. Thus modeling the
advancement of the thermal front or the transient temperature distribution due to
the thermal injection into the aquifer is essential for the effective design of the
injection production well system and for fixing the injection and extraction rates
into or from the aquifer, respectively.
The use of an aquifer as storage of hot water was first suggested in 1971 (Tsang [3]). Robbimov et al. [4] and Meyer and Todd [5] did some early studies
on this topic which mainly present analytical and semi-analytical solutions and economic considerations. Other analytical models and solutions on this topic were
developed by Sauty et al. [6], Chen and Reddel [7], Voigt and Haefner [8], Ziagos
and Blackwell [9], Li et al. [10] and Yang and Yeh [11], which present solutions
for idealistic systems under simplifying assumptions or qualitative estimations. A
three-dimensional numerical model for fluid flow and heat transport in an aquifer
was presented by Tsang et al. [12]. They found a heat recovery to storage ratio
of about 80%. An experimental study was performed by Molz et al. [13] to test
the concept of underground heat storage and to collect data for calibration of numerical models. Larson et al. [14] used the data collected by them for numerical
simulation of water and heat flow in a heat storage aquifer. Another analytical
solution was developed by Bodvarsson and Tsang [15] to investigate the movement
of the cold water thermal front through subsurface media with equally spaced horizontal fractures. Numerical simulations were also conducted by them to assess the
influence of the assumptions applied in analytical modeling. Another experimental
study was performed by Palmer et al. [16] on thermal injection and storage in
a shallow unconfined aquifer. The study provides three-dimensional temperature
distribution data which allows validating complex numerical models with experimental results. The experimental data also provide the quantification of physical
processes like advection, dispersion, retardation, buoyancy and boundary heat loss.
Molson et al. [17] developed a three-dimensional finite element model for simulating coupled density dependent groundwater flow and thermal energy transport.
They used the experimental data obtained by Palmer et al. [16] to validate their
model which showed the model agrees very well with experiment. Mongelli and
Pagliarulo [18] derived a model for temperature distribution in an unconfined semiinfinite aquifer. The results derived by them lead to the evaluation of the zone
influenced by recharge. Nagano et al. [19] did experimental study and performed
numerical simulations to investigate the importance of natural convection on forced
horizontal flow through saturated porous medium in an ATES system. Stopa and
NUMERICAL MODELING AND ANALYTICAL VALIDATION FOR ATES SYSTEM
415
Wojnarowski [20] developed an analytical model to investigate the movement of
the thermal front due to cold water injection in a geothermal reservoir considering
the heat capacity and density of rock and water to be functions of temperature.
Dickinson et al. [21] elaborated the theory of ATES and performed a numerical
study using software package HSTWin. They compared the numerical results with
operational data collected over 12 month period, which showed a good agreement
with each other.
In all the above mentioned studies the heterogeneity of an aquifer is not considered and study heat transport in a heterogeneous aquifer is rare in literature. The
objective of the present study is to develop a two-dimensional numerical model for
the transient temperature distribution and the propagation of the thermal front
generated due to thermal injection in a heterogeneous ATES system. As stated
earlier the numerical modeling has been performed using a multiphysics software
code COMSOL which is developed using finite element technique. A simple onedimensional model of the same system has also been presented afterwards. The
heat transfer processes taken into account were the advection and conduction. The
aquifer considered here is consisted of three sections with different thermo-geological
properties which take into account the heterogeneity of the medium. As the injection rate also may vary with time practically a pulse type time varying injection rate
has been considered in the model. Lastly the numerical results for transient temperature distribution in a one-dimensional homogeneous aquifer were compared with
a one-dimensional analytical model with an aim to validate the numerical model.
2. Mathematical and Numerical Modeling
The mathematical model of heat transfer in porous media is given by a secondorder partial differential equation for energy conservation in the model domain.
The equation is well known in literature presented in studies by Gringarten [22],
Wangen [23], Pao et al. [24]. The two- dimensional heat transport equation for the
porous aquifer for single phase fluid flow is given by
∂
∂t {(1 − φ)ρr cr + φρw cw }
2
2
(x,y,t)
(x,y,t)
λx ∂ T∂x
+ λy ∂ T∂y
2
2
+
∂
∂x {uw ρw cw T (x, y, t)}
+
∂
∂y {uw ρw cw T (x, y, t)}
=
(1)
where T is the temperature (in K); cr and cw are the specific heats of rock and
water, respectively (in J/Kg·K); φ is the porosity of the aquifer; ρr and ρw are
the densities of the rock and water, respectively (in kg/m3 ); uw is the velocity of
groundwater (in m/s); λx and λy are the thermal conductivities of the aquifer in
longitudinal and vertical directions, respectively (in W/m·K); t is injection time (in
seconds); x and y represents the distances in longitudinal and vertical direction,
respectively (in m). The above equation assumes local thermal equilibrium which
states that the temperature of each phase present in a representative elementary
volume (REV) equals to the average temperature of the REV. The assumption is
valid practically when no or very small phase change occurs in REV (Tao and Gray
[25]) and the fluid velocity is sufficiently small (Stopa and Wojnarowski [20]).
416
S. GANGULY, N. SEETHA AND M. KUMAR
Often aquifers are encountered which are consisted of block heterogeneities, i.e.
the aquifer is composed of blocks which have different thermo-geological properties.
The porous aquifer here is consisted of block heterogeneities having three of such
blocks of width 40 m, 30 m and 50 m, respectively with different thermo-geological
properties. The depth of the aquifer is considered to be constant and equal to
20 m. Hot water is considered being injected throughout the depth of a single
injection well fully penetrating into the aquifer at one end of the domain. The
temperature of the hot water is assumed to be 700 C (Tin ) which is kept constant
throughout the whole injection period. The initial temperature of the aquifer water
prior to the injection is assumed to be 200 C (T0 ). The aquifer is underlain and
overlain by rock layers of large thickness and high thermal conductivity. The whole
system is schematically presented in Fig.1. The numerical simulations of the present
study have been performed using multiphysics software COMSOL which solves
fluid flow and heat transport problems in porous media using the finite element
technique. The domain of total 120 m length and 20 m depth was discretized using
33280 triangular elements. Fixed injection water temperature of 700 C (353 K) is
considered as boundary condition at injection end and initial aquifer temperature
of 200 C (293 K) is taken as the second boundary condition at a long distance
away from the injection point. The second boundary condition is valid till the
thermal front reaches that distance. After that the temperature of that point starts
increasing. The top and bottom boundaries are considered as fixed temperature
boundaries with the temperature of the top boundary 200 C and linearly increasing
due to the geothermal gradient of 0.060 C/m to 21.20 C at the bottom. Hot water
is injected at a rate of 0.15 m3 /s for first 500 days and then the injection flux is
increased to 0.3 m3 /s. A small regional groundwater flow occurs through the aquifer
equal to 0.5 m3 /s. A simpler two-dimensional model with constant injection rate is
developed afterwards with an aim to compare the results with time varying injection
rate to judge the importance of the injection flux in the heat transfer phenomenon
in porous media. Results for a one-dimensional model (where the depth of the
domain is considered to be zero) of the same is also presented here for both time
varying and constant injection rate. Lastly the one-dimensional numerical model
is validated with an analytical solution derived by Ganguly et al. [26] for transient
temperature distribution in a homogeneous aquifer.
Figure 1. Schematic diagram of the heterogeneous porous medium with injection well installed
NUMERICAL MODELING AND ANALYTICAL VALIDATION FOR ATES SYSTEM
417
3. Analytical Solutions
The analytical solution for the transient temperature distribution in a homogeneous aquifer due to thermal injection is given by (Ganguly et al. [26])
T = T0 −
√2 (T0
Π
− Tin )
R∞
l1
exp(−ζ 2 −
U 2 x2
16λ2 ζ 2 )
(2)
where U = ρw cw uw . The lower limit of the integral equation is given by
1
C 2
l1 = x2 ( λt
)
(3)
where C = (1 − φ)ρr cr + φρw cw . Notice that the variation in the temperature
field in the aquifer arises due to the second term in the integral equation which in
turn arises due to the difference in temperature of the injected water (Tin ) and the
aquifer temperature prior to the injection (T0 ).
4. Results and Discussion
The thermo-geological properties of the three blocks of the heterogeneous porous
aquifer and that of the fluid are enlisted in Table 1. The results are presented first
for the general transient case (GTC) for the two-dimensional heterogeneous porous
aquifer with time varying injection rate. The temperature versus distance plot at
different injection times (i.e. the thermal front plots) for GTC with injection rate
0.15 m3 /s for first 500 days and 0.3 m3 /s afterwards, have been shown in Fig. 2.
The two-dimensional plots show that owing to continuous injection of hot water
into cold aquifer environment a thermal interface or the thermal front is set up
which propagates through the aquifer with time. The temperature of the aquifer
also increases gradually with the passage of injection time due to the advancement
of the thermal front. As the thermal front enters the second and third section (i.e.
at 40 m and 70 m, respectively) in the heterogeneous aquifer, a change in trend
in the temperature distribution is observed and sharp points are noticed on the
temperature distribution plot.
The transient temperature distribution plots have been shown in Fig. 3 for case
where the injection rate is kept constant and equal to 0.15 m3 /s. The plots show
that due to the increased injection rate after 500 days the advancement of the
thermal front has been more in GTC than the constant injection rate case. E.g. at
injection time of 1000 days the thermal front has penetrated 36 m in GTC whereas
the penetration has been 29 m in constant injection rate case. This evidently proves
that the system here is dominated by convective flux and as the convective flux is
directly proportional to the injection rate, due to the increase in injection rate the
thermal front advances more in GTC.
As mentioned earlier the efficiency of an ATES system depends on its capacity to retain heat and thus heat loss from the aquifer plays an important role in
the transient heat transport phenomenon. The effect of heat loss here is judged
by comparing the transient solution (with constant injection rate) with an ATES
system without heat loss for which the top and bottom boundaries are considered
418
S. GANGULY, N. SEETHA AND M. KUMAR
Table 1. Rock and fluid properties of three sections of the aquifer
Parameter name
Specific heats of rocks
(cr )(J/kg.K)
Density of the porous aquifer
(ρr )(kg/m3 )
Thermal conductivity of
the aquifer (longitudinal)
(λx )(W/m.K)
Thermal
conductivity
of the aquifer (vertical)
(λy )(W/m.K)
Porosity of the aquifer φ
Density
of
the
fluid
(ρr )(kg/m3 )
Specific heats of fluid
(ρw )(J/kg.K)
Section 1
850
Section 2
900
Section 3
1000
2650
2400
2700
2.0
0
1.5
1.0
0
0.7
0.15
985
0.20
985
0.10
985
4180
4180
4180
to be thermally insulated. The temperature distribution plots for the aquifer without heat loss have been shown in Fig. 4 at injection times of 100 and 1000 days.
The plots show that that the advancement of the thermal front for aquifer with
heat loss is always lesser than the aquifer without it. The thermal fronts in aquifer
without heat loss have reached 8 m and 32 m in 100 and 1000 days, respectively
whereas the advancement of that for aquifer with heat loss have been 6 m and 29
m, respectively. As the heat conductance over the boundaries is irreversible, for
an aquifer with large heat loss makes it inefficient to be used for thermal energy
storage.
Results for a one-dimensional study for the same heterogeneous aquifer with
the same time varying injection rate has been presented in Fig. 5(a) which shows
the temperature versus distance curves at different injection times. The curves
follow a nonlinear falling trend with the temperature of the injection water at one
end of the domain and approaching the initial aquifer temperature at the other.
Here also we observe the aquifer temperature gradually rises as the thermal front
proceeds through the aquifer with the passage of injection time and as the thermal
front enters into the second and third block of the domain we observe some change
in trend of the curve and a sharp point (at 40 m and 70 m, respectively) in the
temperature distribution curve which is triggered by the change of the thermogeological properties of the different block as in the heterogeneous porous media.
The temperature versus distance plots in case of uniform injection rate of the hot
water is shown in Fig. 5(b) which also shows that until 500 days the temperature
distribution is identical with GTC whereas after 500 days in GTC the thermal front
has advanced more into the aquifer due to the increase in injection rate since the
injection rate is directly proportional to the advective flux of heat transport in the
aquifer.
The one-dimensional numerical model used in the study is validated with the
analytical solution presented in section (3) by plotting the transient temperature
NUMERICAL MODELING AND ANALYTICAL VALIDATION FOR ATES SYSTEM
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2. Temperature distribution plots in the aquifer domain
for GTC at (a) 10 days, (b) 100 days, (c) 1000 days, (d) 2000 days,
(e) 5000 days and (f) 10000 days
419
420
S. GANGULY, N. SEETHA AND M. KUMAR
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3. Temperature distribution plots in the aquifer domain
for constant injection rate case at (a) 10 days, (b) 100 days, (c)
1000 days, (d) 2000 days, (e) 5000 days and (f) 10000 days
NUMERICAL MODELING AND ANALYTICAL VALIDATION FOR ATES SYSTEM
Figure 4. Temperature distribution plots in the aquifer domain
without heat loss at (a) 100 days and (b) 1000 days.
Figure 5. Temperature versus distance plots at different injection
times for (a) GTC and (b) constant injection rate case
421
422
S. GANGULY, N. SEETHA AND M. KUMAR
distribution at different injection times calculated by the numerical model and the
analytical solution in Fig. 6. The analytical solution given by equation (2) is an
integral solution the integral in which is solved in MATLAB by applying GaussKronrod quadrature technique. It is to be noted that although the upper limit of
the integral is infinity, the numerically effective range is much smaller than that.
To solve this problem the integral was tested for different upper limits and the limit
was fixed when almost no variation of result was noticed by varying the upper limit.
The temperature distribution plots derived numerically show very good agreement
with the analytical model at early injection times, but with passage of time it tends
to underestimate the propagation of the thermal front which is due to numerical
error (numerical dispersion).
Figure 6. Temperature versus distance plots at different injection
times for (a) GTC and (b) constant injection rate case
5. Conclusions
A two-dimensional numerical model for heat transport in a heterogeneous porous
aquifer thermal energy storage system is presented here, considering time varying
and constant injection rate. The primary target was to model the movement of
the thermal front with time, which is generated in the aquifer due to the thermal
injection. The numerical code used here has been validated with an analytical
solution for the heat transfer in a simple homogeneous aquifer. The conclusions
which can be drawn from study are
1. With continuous injection of hot water a thermal front is set up in the aquifer
and proceeds with passage of injection time. The aquifer temperature thus increases
gradually with time at a fixed distance and decreases with increasing longitudinal
distance at a fixed injection time.
2. The advective flux of heat transfer is proportional to the volumetric injection
rate of hot water to in an advection dominated system like the present one, since
the advective velocity of groundwater flow enhances due to increased injection rate.
NUMERICAL MODELING AND ANALYTICAL VALIDATION FOR ATES SYSTEM
423
Thus increase in hot water injection rate consequences in faster movement of the
thermal front through the aquifer.
3. The heat loss from the aquifer plays a crucial role in the transient heat
transport phenomenon in an ATES system. Due to heat loss the advancement of
the thermal front is always lesser in an aquifer with heat loss than that without it.
Large heat loss from an aquifer makes it inefficient for usage of long term thermal
energy storage.
4. The results for heat transfer in homogeneous aquifer due to thermal injection
derived numerically agrees with the results derived from the analytical solution
very well at early times, whereas the large times the numerical model tends to
underestimate the advancement of the hot water thermal front.
5. The thermo-geological properties in a heterogeneous porous aquifer influence
the transient heat transfer phenomenon. The influence is evident from the change
in trend in the temperature distribution plots when the thermal front enters from
one medium to another with different thermo-geological properties.
Lastly the present two-dimensional model gives some insight into the problem
of transient heat transport phenomenon in a heterogeneous porous aquifer due to
hot water injection into it. The results presented here can be effectively used in
design of the injection-production well scheme in a heterogeneous ATES system.
The model can also serve as a reference solution to more complex numerical models.
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8th International Symposium on Lowland Technology, Bali, Indonesia, 2012.
Department of Civil Engineering, Indian Institute of Science Bangalore, 560 012, India
E-mail : [email protected]
E-mail : [email protected]
E-mail : [email protected]