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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. 413 414 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. References [1] Sila Dharma,Modeling of aquifer thermal energy storage (ATES) using heat and solute transport in 3D (HST3D). Civil Engineering Dimension, 11(2), (2009) 119-125. [2] Kim, J., Lee, Y., Yoon, W.S., Jeon,J.S., M.H. Koo and Y. Keehm, Numerical modeling of aquifer thermal storage system, Energy, 35, (2010) 4955-4965. [3] Tsang, C.F., Aquifer Thermal Energy Storage, Presented at the Institute of Gas Technology, Symposium on Advanced Technologies for Storing Energy, Chicago, IL, 1978. 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Pao, W.K.S., Lewis, R.W. and Mastres, I., A fully coupled hydro-thermo-poro-mechanical model for black oil reservoir simulation, Int. J .Numer. Anal. Methods Geomech., 25, (2001) 1229-1256. Tao, Y.X. and Gray, D.M., Validation of local thermal equilibrium in unsaturated porous media with simultaneous flow and freezing, Int. Com. Heat Mass Transfer, 20, (1993) 323332. Ganguly, S., Seetha, N. and Mohan Kumar, M.S., Numerical simulation and analytical validation for transient temperature distribution in an aquifer thermal energy storage system, 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]