Simulating Subsurface Temperature under Variable

Journal of Porous Media 10(8), 769–782 (2007)

Simulating Subsurface Temperature under Variable Recharge Ashok K. Keshari1 and Min-Ho Koo2 1 2

Department of Civil Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India

Department of Geoenvironmental Sciences, Kongju National University, Kongju City, Chungnam 314-701, Korea E-mail: [email protected]

ABSTRACT A distributed numerical model utilizing MacCormack finite difference method is developed to simulate the transient subsurface temperature profiles under time varying groundwater recharge. It is not restricted to constant percolation rate and sinusoidal temperature variations as is the case in most models. The model is validated with the data available for the Nagaoka plain, Japan showing good agreement for the computed subsurface temperatures having correlation coefficient (as R2 ) and root mean square error (RMSE) as 0.89 and 0.42, respectively. The model is employed to investigate the alteration in the subsurface thermal regime due to variable recharge resulting from precipitation in South Korea. Results reveal that the subsurface thermal regime is influenced significantly by the temporal variability in groundwater recharge and temperature at the surface, and subsurface temperatures are not necessarily sinusoidal at all depths. Seasonal variations and the effect of short term hydrological phenomena are clearly depicted in simulated results. The temporal variations of the subsurface temperature are highly nonlinear near the ground surface and become almost linear beyond 10 m. The subsurface temperature profiles vary significantly from one cycle to other and tend to be vertical beyond 15 m. Its variability also depends upon the Peclet number.

769 Received April 10, 2006; Accepted October 21, 2006 c 2007 Begell House, Inc. Copyright °

Begell House Inc., http://begellhouse.com Downloaded 2007-9-6 from IP 210.106.80.162 by Prof. Min-Ho Koo (koo4446)

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NOMENCLATURE c co CN DN f (.) k O(.) q q∗

(RMSE∆T)cal t T z ZL Zo

specific heat of subsurface (J/Kg/◦ C) specific heat of water (J/Kg/◦ C) convection number diffusion number a dependent variable thermal conductivity of subsurface (W/m/◦ C) error order vertical groundwater flux (m/s) a variable dependent on vertical groundwater recharge and thermal properties of water and subsurface (m/s) calculated RMS error in the simulated subsurface temperature time (s) temperature (◦ C) vertical depth from the ground surface (m) depth at which the lowermost subsurface temperature is measured (m) depth at which the uppermost subsurface temperature is measured (m)

1. INTRODUCTION Understanding the interaction between groundwater recharge and the thermal regime of the subsurface has become an increasingly important concern for studies involving the quantification of groundwater recharge and thermal properties (Stallman, 1965; Bredehoeft and Papadopulos, 1965; Cartwright, 1970; Sorey, 1971; Boyle and Saleem, 1979; Taniguchi, 1993; Dapaah-Siakwan and Kayane, 1995; Tabbagh et al., 1999; Reiter, 2001; Keshari and Koo, 2002, 2007), infiltration and groundwater movement (Suzuki, 1960; Wierenga et al., 1970; Keys and Brown, 1978; Cartwright, 1979; Ronan et al., 1998; Constantz et al., 2003), water budgets (Keshari and Koo, 2002),

Greek Symbols α thermal diffusivity of the subsurface (m2/s) ∆t time interval (s) ∆z grid size (m) εT permissible error in subsurface temperature ζ1 depth-dependent temperature function ζ2 time-varying temperature function ρ density of subsurface (Kg/m3 ) ρo density of water (Kg/m3 ) Subscripts i space index t time derivative tt second-order time derivative z depth derivative Superscripts j time index p predictor

groundwater–surface water interactions (Silliman and Booth, 1993; Silliman et al., 1995), and quantifying disturbances in the thermal regime due to global warming and urbanization (Chisholm and Chapman, 1992; Mareschal and Beltrami, 1992; Harris and Chapman, 1995; Sakura et al., 1996; Taniguchi et al., 1999a,b). Wide potential uses of subsurface thermal data have recently generated great interest among researchers. The effect of instantaneous and/or seasonal changes in soil temperature at the surface is reflected in the soil temperature profile and is governed by the heat transport in the subsurface. The water during the precipitation infiltrates into the soil and percolates through the subsurface, resulting in an increase of soil moisture in the unsaturated zone and


Simulating Subsurface Temperature recharge to aquifers. During the movement of water through the subsurface, the heat is also transported by the advection process, and thus the redistribution of heat takes place within the subsurface. This process makes the convection velocity play a significant role in controlling the thermal stability of the subsurface. Thus it is interesting to investigate the response of the subsurface thermal regime to the transient groundwater recharge resulting from the precipitation, a time-variant hydrologic phenomenon. Numerous attempts have been made to analyze the simultaneous flow of heat and fluid through the porous medium (Stallman, 1965; Andrews and Anderson, 1979; Smith and Chapman, 1983; Vafai, 1984; Woodbury and Smith, 1985; Kumaran, 1988; Lapham, 1989; Vafai and S¨ozen, 1990). Analytical approaches for solving the simultaneous flow of heat and fluid through the subsurface as developed by Stallman (1965), Bredehoeft and Papadopulos (1965), Taniguchi (1993), and Tabbagh et al. (1999) have been used by several researchers (Sorey, 1971; Sakura, 1977; Boyle and Saleem, 1979; Cartwright, 1979; Jessop and Vigrass, 1989; Taniguchi and Sharma, 1993; Tabbagh et al., 1999) for estimating groundwater recharge and studying the interaction between the subsurface thermal regime and groundwater recharge. However, these approaches are applicable to only simplified initial and boundary conditions and simplified temperature distributions. Solutions and applications, as evident from the literature, are restricted to either steady state conditions with simplified Dirichlet-type boundary conditions or unsteady state conditions with an assumption of sinusoidal temperature fluctuations in the borehole as well as at the surface. The assumption of sinusoidal or harmonic temperature fluctuations can best be determined in real-life situations only with diurnal and annual periods of temperature oscillations, and thus these approaches cannot simulate the short-time recharge events resulting from the precipitation storms. Furthermore, the land surface heterogeneity in terms of topography, vegetation, soil cover, land use, geol-

771 ogy, and hydrological conditions may result in nonsinusoidal temperature variations even at larger time scales, and thus seasonal and secular variations need to be approximated in currently available approaches; this in turn may underestimate or overestimate simulations, depending on the specific field conditions. The secular variations indicate long-term changes with coarser time scales, for example, annual variations in several year cycles. The processes taking place within the subsurface at short time scales can also not be simulated. The other critical limitation in most of these approaches is that they consider a steady state water flow, which is not going to happen under the real-life conditions of hydrological problems. Because of temporal variability of precipitation, groundwater recharge is also a time-variant phenomenon, and thus the approaches or models must be capable of accounting for varying recharge with time. Analytical studies also deal with the simultaneous transient heat flow and steady state water flow (Cartwright, 1971, 1974; Ziagos and Blackwell, 1981, 1986). However, these studies are applicable for the horizontal water flow, which is shallow enough to interact with the surface. Silliman et al. (1995) presented a mathematical formulation for quantifying the flux across the sediment for conditions of onedimensional downflow with a constant flux over periods of days to weeks using temperature time series measurements in the water column and sediments. The solution is based on the widely available analytical solution for the chemical transport in terms of complementary error function and is applicable to simplified initial and boundary conditions. Taniguchi et al. (1999a,b) developed a series of type curves for the one-dimensional nonisothermal flow of an incompressible fluid through a homogeneous porous medium to estimate vertical groundwater fluxes under the conditions of linear and step increases in surface temperature. These type curves were applied to the subsurface thermal regime of the Tokyo metropolitan area to estimate the vertical groundwater flux under the condition of the surface warming caused by global


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warming and urbanization, and to the subsurface thermal regime of southwest western Australia to estimate the vertical groundwater flux under the condition of surface warming caused by forest clearance. A few numerical approaches (Andrews and Anderson, 1979; Lapham, 1989) were also reported on the simultaneous movement of heat and fluid to investigate the alteration in the subsurface thermal regime due to vertical groundwater movement. It is evident from the above discussion that most approaches reported in the literature, including the analytical and numerical approaches, simulate the alteration in the subsurface thermal regime due to steady state water flow and under simplified initial and boundary conditions. As discussed earlier, the groundwater recharge resulting from the precipitation is a time-variant phenomenon, and thus it is desirable to investigate the effect of transient phenomenon of groundwater recharge on the subsurface thermal regime. Furthermore, the land surface heterogeneity together with the hydrologic processes taking place at shorter time scales call for a generalized mathematical solution under generalized initial and boundary conditions. The objective of this article is to present a numerical model to simulate the subsurface temperature under time-varying groundwater recharge and for generalized initial and boundary conditions. The study also aims to investigate the alteration in the subsurface thermal regime due to variable recharge resulting from the precipitation in South Korea using precipitation time series of shorter intervals to gain better insight into the hydrogeothermal processes at shorter time scales. 2. MODEL DEVELOPMENT Temperature at the land surface is controlled principally by an energy balance between incoming solar radiation and outgoing long wavelength thermal radiation. Temperature below the land surface is affected by the flux of heat being conducted below from the land surface as well as the heat flux conducted out from the earth’s interior and the heat transfers result-

ing from other external or internal physical, chemical, and biological processes. However, in the subsurface between the ground surface and the top surface of the saturated layer, the heat conduction and the heat transfer resulting from the physical process, such as groundwater recharge, are of greater concern and are major processes that control the subsurface temperature distributions in real-life situations. Changes in the land surface temperature with time occur at several temporal scales; the largest of these changes are the daily and seasonal variations, both of which can have amplitudes of 10◦ C or more. The subsurface acts as a filter and attenuates these thermal waves with depth. The high-frequency or short-term variations die out more rapidly than long-term variations. When the precipitation occurs on the land surface, a part of the precipitation infiltrates down through the unsaturated zone as groundwater recharge, and the heat is transported below the land surface as mass transfer. The movement of water thus results in the transfer of heat through a convection process as well, which leads to the redistribution of heat within the subsurface and the alteration of the subsurface temperature distribution. The relative importance of these mechanisms in controlling the subsurface thermal regime is site-specific and may vary in the space and time domains. The partial differential equation (PDE) describing the one-dimensional simultaneous nonisothermal flow of heat and time-varying vertical groundwater recharge through the subsurface, a heterogeneous porous medium, can be expressed as

∂T ∂ = ∂t ∂z

µ ¶ ∂T ∂ ∗ α − (q T ) ∂z ∂z

(1)

where T is temperature, t is time, α is the thermal diffusivity of the subsurface, z is the vertical depth from the ground surface taken as positive downward, and q ∗ is a variable dependent on vertical groundwater recharge and thermal properties of the water and subsurface. The thermal diffusivity α and variable q ∗ are given by


Simulating Subsurface Temperature k cρ qco ρo q∗ = cρ

α=

773

(2)

which the lowermost subsurface temperature is measured.

(3)

Equation (1) is discretized using the MacCormack scheme based finite difference method, which is based on the second-order forward-time Taylor series. The Taylor series expansion for any generic dependent variable f (z, t) in terms of time can be expressed as

where k is the thermal conductivity of the aquifer, q is the vertical groundwater flux, co is the specific heat of water, ρo is the density of water, c is the specific heat of the subsurface, and ρ is the density of the subsurface. Equation (1) is a second-order nonlinear partial differential equation as the thermal diffusivity and the vertical groundwater recharge are variable in the space and time domains. For the numerical solution of the PDE described by Eq. (1), the space coordinate z must be a closed physical domain; however, the time coordinate can be an open final value. To simulate the transient temperature variation within the subsurface along a vertical profile, Eq. (1) is numerically solved with appropriate boundary conditions along with the initial temperature-depth profile as specific to an investigating area. The boundary conditions are expressed as the Dirichlet type (specified values of T ), the Neumann type (specified values of ∂T /∂z), or the mixed type (specified combinations of T and ∂T /∂z), as appropriate to the study area under investigation. In the present study, the uppermost subsurface is taken as the variable temperature boundary with known function with time, and the lowermost subsurface is taken as the no flux boundary. Thus the initial and boundary conditions considered in this study can be expressed as T (z, 0) = ζ1 (z)

(4)

T (Zo , t) = ζ2 (t)

(5)

∂T (z, t) |ZL = 0 ∂z

(6)

where ζ1 is a known depth-dependent temperature function and ζ2 is a known time-varying temperature function, Zo is the depth at which the uppermost subsurface temperature is measured (usually, this depth is taken as the ground surface), and ZL is the depth at

fij+1 = fij + ft |ji ∆t + 12 ftt |ji ∆t2 + O(∆t3 ) (7) where i and j are space and time indices, respectively, ∆t is the time interval, the subscript t with variable f denotes its time derivative, and O(∆t3 ) denotes the error of an order of ∆t3 . In the MacCormack scheme, ft |ji is determined directly from the PDE, and ftt |ji is determined from a first-order forward-time Taylor series expansion for ft |j+1 about grid point (i, j). Thus we get i ft |j+1 = ft |ji + ftt |ji ∆t + O(∆t2 ) i

(8)

Solving Eq. (8) for ftt |ji and substituting in Eq. (7), we get Eq. (9), which is solved for fij+1 by dropping the truncation error terms and replacing ft |ji and ft |j+1 in terms of spatial derivatives from the PDE: i fij+1 = fij +

j 1 2 [ft |i

+ ft |j+1 ]∆t + O(∆t3 ) (9) i

The MacCormack scheme employs a two-step predictor-corrector approach. In this approach, the subsurface temperature in the space and time domains, Tij+1 , is explicitly determined from the PDE described by Eq. (1) for the convection-diffusion heat transport within the subsurface. In the first step, the first-order forward difference finite difference approximation (FDA) is used for time and first-order space derivatives, whereas the second-order centered difference FDA is used for second-order space derivatives at grid point (i, j). In the second step, Eq. (9) is solved by evaluating Tz |ji following the same procedure as carried out in the first step and evaluating Tz |j+1 using the first-order backward difference FDA i based on the temperature values obtained in the first


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step at j + 1 time level, Tipj+1 . The resulting finite difference equations for the first and second steps can be expressed as j Tip j+1 = Tij − [CNji+1 Ti+1 − CNji Tij ] j +[DNji+1 Ti+1

−(DNji+1

+

DNji )Tij

+

(10)

j DNji Ti−1 ]

Tij+1 = 21 [Tij + Tip j+1 − CNj+1 Tip j+1 i

(11)

p j+1 p j+1 + CNj+1 + DNj+1 Ti+1 i−1 Ti−1 i p j+1 p j+1 − (DNj+1 + DNj+1 + DNj+1 i i−1 )Ti i−1 Ti−1 ]

where CN and DN are the cell convection number and cell diffusion number, respectively, and they are given as ∗j CNji+1 = qi+1

CNji = qi∗j

∆t ∆z

∆t ∆z

DNji+1 = αji+1 DNji = αji

(12) (13)

∆t ∆z 2

∆t ∆z 2

(14) (15)

CNj+1 = qi∗j+1 i

∆t ∆z

(16)

∗j+1 CNj+1 i−1 = qi−1

∆t ∆z

(17)

DNj+1 = αj+1 i i

∆t ∆z 2

(18)

j+1 DNj+1 i−1 = αi−1

∆t ∆z 2

(19)

where ∆z is grid size in the finite difference network. 3. SOLUTION METHODOLOGY Equations (10)–(19), together with Eqs. (4)–(6), constitute the basic governing equations for simulating the subsurface temperature under time-varying groundwater recharge. This set of equations in the proposed numerical model is capable of handling the Dirichlet-type boundary conditions having either constant temperature values or known temperature func-

tions of time at the top subsurface boundary and no flow boundary at the bottom subsurface. The other type of boundary conditions, as discussed earlier, can be incorporated into the proposed numerical model by suitably modifying Eqs. (5)–(6) to facilitate appropriate accounting of boundary conditions of the groundwater system under consideration. Equations (10)–(11) are in fact the basic equations in the developed MacCormack scheme–based numerical model for determining the subsurface temperature in space and time domains. The study area under consideration is first approximated as a block-centered or mesh-centered finite difference grid network, and the thermal and hydraulic properties of the study area are evaluated for each block or node. Equation (10) is then used, along with the initial and boundary conditions described by Eqs. (4)–(6), to determine the subsurface temperature values at various nodes of the finite difference grid for a specified time interval. The obtained values are known as the predicted or provisional values for this time step. Results obtained from the first step (predictor) are modified in the second step, known as the corrector step, to get the final results using Eq. (11) together with Eqs. (4)–(6) for this time step. The whole process is repeated for different time steps, and solutions are marched in time up to the desired time. The grid size and time scale for the considered application are chosen such that the cell convection and diffusion numbers are within the limits of CNji ≤ 0.9 and DNji ≤ 0.5. These limits have been taken based on the analysis carried out by Hoffman (1992) for the MacCormack scheme finite difference approximation of linear convection-diffusion equation using the von Neumann method, as the theoretical analyses of accuracy, order, stability, and consistency of the two-step MacCormack approximation of the nonlinear convection-diffusion equation are not available. This observation was examined while carrying out the analysis for the considered application in this study area and was found satisfactory to yield stable and convergent solutions from the developed numerical


Simulating Subsurface Temperature

775

model. The model has an error of the order O(∆t2 ) + O(∆x2 ). The model is calibrated and validated with the time series data of temperature measured at selected depths along the vertical profiles in the study area under consideration. During the calibration process, the model parameters are refined to yield the best results, and the optimized values of parameters are obtained for the study area. The model parameters are updated during the calibration and/or validation period to establish the reliability of simulation results based on the following criterion: (RMSE∆T )cal ≤ εT

(20)

where (RMSE∆T )cal is the calculated RMS error for the simulated temperature in space and time, and εT is the permissible error in the prediction of subsurface temperature for practical applications. 4. RESULTS AND DISCUSSION To validate the developed numerical model, it was evaluated with the measured groundwater temperature-depth profile for a study area located in the Nagaoka plain, Japan. For the details on the description of the study area and temperaturedepth profiles for various wells located in this study area, readers may refer to Kayane et al. (1985) and Taniguchi (1993). The groundwater temperature-depth profiles utilized in the present study were taken from these articles for depths of 3, 6, 9, 12, and 15 m below the ground surface in a well located in a recharge area at 2-month time intervals from August 1982 to August 1983. Temperature measurements were taken in a well of 10 cm diameter and 20 m depth by means of thermistors. The thermal conductivity and thermal capacity were taken as 1.58 W/◦ C/m and 2.73 × 106 J/m3 /◦ C, respectively, from the published literature (Taniguchi, 1993). Results were obtained from the developed numerical model with these thermal properties and taking the temperature measurements at 3 and 15 m below the ground surface as the top and bottom

subsurface Dirichlet boundaries with known temperature values varying with time. The mean hydraulic gradient was computed using observed groundwater level data, and the vertical hydraulic conductivity of the aquifer was estimated from borehole records and aquifer test data. These values were used to calculate groundwater flux from Darcy’s law. The computed value of mean hydraulic gradient is 1.87 × 10−2 for the period from August 1982 to August 1983, and the estimated value of vertical hydraulic conductivity is 8.91 × 10−6 m/s. Therefore the calculated groundwater flux from the hydraulic data is 1.67 × 10−7 m/s. The subsurface temperature variation is simulated for a constant groundwater recharge of 1.67 × 10−7 m/s, which has been obtained from the hydraulic data as the average groundwater recharge for this study area. The computed subsurface temperature values are shown in Fig. 1 with the measured temperature values as a scatterplot. It is evident from this figure that computed results are in good agreement with the measured ones, except for a few, and the deviation of these results may be attributed to the average value of groundwater recharge considered while simulating the subsurface temperature profile as there was no point information available for this well corresponding to the considered time frame. The statistical parameters, the correlation coefficient (R) between the computed and observed values of subsurface temperature and the root mean square error (RMSE), were calculated to test the goodness of fit of computed values with observed values. The calculated values of R2 and RMSE are 0.89 and 0.42, respectively, thus validating the developed numerical model for its applicability. The developed numerical model was applied to a study area located in South Korea (Fig. 2). South Korea is located in eastern Asia, on the southern half of the Korean Peninsula bordering the Sea of Japan and the Yellow Sea, within the geographic coordinates of 37◦ 000 N, 127◦ 300 E. The total surface area of South Korea is 98,480 sq km, with 99.70% as land surface and 0.30% as water surface. The terrain is mostly hills and mountains, and there are wide coastal plains


Keshari and Koo

Computed temperature (Celsius)

776

15 1:1 Line

R2 = 0.89 14 13 12 11 10 10

11

12 13 Observed temperature (Celsius)

14

15

Figure 1. Validation of the developed numerical model

Figure 2. Location of the study area in the west and the south, the lowest point being the Sea of Japan, 0 m, and the highest point being Hallasan, 1950 m. As per 1993 estimates, the land use pattern comprises 19% permanent arable land, 2% permanent crops, 1% permanent pastures, 65% forests and woodland, and 13% other. The total irrigated land is 13,350 sq km. The climate is temperate, with a

higher magnitude of rainfall in summer than in winter. The average annual precipitation in South Korea is 1300 mm; annual precipitation is a little higher in the central region, with a magnitude of about 1500 mm. More than half of the total rainfall amount is concentrated in summer, and less than 10% of the total precipitation is in winter.



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The surface temperature and precipitation measured at the Seoul meteorological station are utilized to investigate the temporal variation of subsurface temperature under the time-varying groundwater recharge. The ground surface is considered as the uppermost boundary, and thus the measured surface temperature is taken as the specified temperature boundary. On the basis of the literature, the depth 20 m below the ground surface is taken as the no flow boundary for temperature. Daily temperature data of 1998 are taken for the study. Figure 3 shows the temperature variation in a year on a daily scale. The temperature

varies between –11.9◦ C and 27.7◦ C, with an average value of 13.84◦ C. The precipitation is measured four times in a day at the Seoul meteorological station, and thus the daily precipitation for 1998 is obtained for the study. A time-varying groundwater recharge being equal to 15% of total precipitation with no time lag in its temporal distribution in a year on a daily scale is considered for the analysis. The temporal variations of precipitation and groundwater recharge in a year are shown in Figs. 4 and 5. The thermal conductivity and thermal capacity of the subsurface are taken to be equal to 1.60 W/m/◦ C and 2.80 × 106 J/m3 /◦ C.

30 25

Temperature (Celsius)

20 15 10 5 0 1/1/98 -5

3/2/98

5/1/98

6/30/98

8/29/98

10/28/98

12/27/98

-10 -15 Date

Figure 3. Daily variation of temperature at Seoul meteorological station

350 300

Precipitation (mm)

250 200 150 100 50 0 1/1/98

3/2/98

5/1/98

6/30/98

8/29/98

10/28/98

Date

Figure 4. Daily variation of precipitation at Seoul


12/27/98

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Keshari and Koo 50

Groundwater recharge (mm/day)

40

30

20

10

0 1/1/98

2/20/98

4/11/98

5/31/98

7/20/98

9/8/98

10/28/98

12/17/98

Date

Figure 5. Temporal variation of groundwater recharge in Seoul

Results are obtained from the developed numerical model with a finite difference grid of 1 m on a daily time scale. The temperature corresponding to 1 January is taken as the initial condition, and its distribution along the vertical depth up to 20 m is obtained by assuming an increasing gradient of 20 ◦ C/km. The subsurface temperature variation is simulated for a year on a daily time scale utilizing the hydrological, meteorological, and hydrogeological characteristics of the study area with a dynamic initial condition by simulating the subsurface thermal regime for several year cycles. Figure 6 shows the computed subsurface temperature variation in a year on a daily scale at 2, 5, 10, and 15 m depth below the ground surface. Results are shown for cycles 1, 5, and 10. These cycles indicate that solutions have been obtained using a dynamic initial condition with several cycles. It is evident from these figures that more gaps exist between the temporal variations at larger depth below the ground surface. The seasonal changes in subsurface temperature are clearly depicted in this figure. These variations show that subsurface temperatures are not necessarily sinusoidal at all depths, which was the assumption in most currently available models. It is also evident that curvatures in temperature variations diminish at larger depths. The temporal variations of the subsurface tem-

perature are highly nonlinear near the ground surface (less than 10 m) and become almost linear from 10 m onward from the ground surface. Figure 7 shows the computed temperature-depth profiles at various times with intervals of 3 months for the study area under consideration. The subsurface temperature profiles are shown for t = 90, 180, 270, and 365 days. The cycles in these figures are having the same meaning as described above. It is evident from these figures that temperature-depth profiles are nonlinear and are dependent on the characteristics of temperature change taking place at the land surface and the temporal variations in groundwater flux. The obtained subsurface temperature profiles vary significantly from one cycle to another and also at different times of the year. These profiles tend to be vertical beyond 15 m depth from the land surface and thus tend to adapt the natural thermal boundary condition at 20 m depth from the ground surface. Results obtained further show that the model is capable of simulating the boundary conditions adequately, and profiles are sensitive to the rate of groundwater recharge taking place within the subsurface. The variability in profiles depends on the Peclet number, which determines the dominancy of the advection process. The Peclet number varies between 0.00 and 1.51 for the simulated subsurface temperature-depth profiles.



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25

25

Cycle 1

Cycle 1

Cycle 10 15 10 5 0

(b)

Cycle 5

20

Cycle 5 Temperature (Celsius)


20

(a)

Cycle 10 15 10 5 0

1

61

121

181

241

301

361

1

61

121

Time (days)

25

241

301

361

25

Cycle 1

Cycle 1

(c)

Cycle 5 Cycle 10

15 10 5

(d)

Cycle 5

20


20


181 Time (days)

Cycle 10 15 10 5 0

0 1

61

121

181

241

301

361

1

61

121

181

241

301

361

Time (days)

Time (days)

Figure 6. Simulated subsurface temperature variation on daily scale: a) z = 2 m, b) z = 5 m, c) z = 10 m, and d) z = 15 m

5. CONCLUSIONS A distributed numerical model utilizing a MacCormack scheme for finite difference approximation of the partial differential equation describing the simultaneous flow of heat and water through the subsurface medium is developed to simulate the subsurface temperature profiles under time-varying groundwater recharge and generalized initial and boundary conditions. The model developed can simulate the transient behavior of groundwater recharge resulting from the precipitation storms and any temperature variations either at the ground surface or at a depth below the ground surface and could be used to elucidate the hydrogeothermal processes at a shorter time scale as well as to investigate the effects of seasonal and secular changes in temperature, precipitation, and groundwater recharge on the subsurface thermal regime.

These aspects make the model more attractive for real-life applications as the currently available models are restricted to constant percolation and sinusoidal temperature variations. The model is validated with the data available for the Nagaoka plain, located in Japan. The computed values of subsurface temperature are found to be in good agreement with the measured ones, having an R2 and RMSE of 0.89 and 0.42, respectively. The model is further applied to a study area located in South Korea to investigate temporal variations in subsurface temperature and subsurface temperature profiles under time-varying groundwater recharge resulting from precipitation. Results obtained for daily variations in subsurface temperature and subsurface temperature-depth profiles up to 20 m for several annual cycles reveal that the subsurface thermal regime


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Temperature (Celsius) 0

5

10


15

20

0

0

5

10

15

20

25

(a)

(b)

5

Depth (m)

Depth (m)

5 10 Cycle 1

15

10 Cycle 1

15

Cycle 5

Cycle 5 20

20

Cycle 10

25

Cycle 10

25 Temperature (Celsius)

Temperature (Celsius) 0

5

10

15

20

-5

25

5

10

20

5

Cycle 1

Depth (m)

10

10 Cycle 1

15

Cycle 5

Cycle 5 20

15

(d)

(c)

5

Depth (m)

0 0

0

15

30

0

Cycle 10

25

20

Cycle 10

25

Figure 7. Simulated subsurface temperature profile: a) t = 90 days, b) t = 180 days, c) t = 270 days, and d) t = 365 days

is influenced significantly by the thermal properties of the subsurface, hydrogeological setting, recharge duration, and temporal variability in groundwater recharge. The redistribution of the heat takes place due to groundwater recharge, even at shorter time scales. The temperature-depth profiles obtained for the study area are nonlinear and significantly depend on the temperature changes taking place at the ground surface. Seasonal variations and the effect of short-term hydrological phenomena are clearly depicted in simulated results. It is observed that subsurface temperatures are not necessarily sinusoidal at all depths. This finding limits the applicability of most currently available models as they are based on the assumption of sinusoidal temperature variation. The temporal variations of the subsurface temperature are highly nonlinear near the ground surface (less than 10 m) and become

almost linear from 10 m onward from the ground surface. The curvature in temporal variation of subsurface temperature diminishes at larger depths. The subsurface temperature profiles vary significantly from one cycle to other and also at different times of the year. These profiles tend to be vertical beyond 15 m, adapting to the natural thermal boundary condition at 20 m depth from the ground surface. The variability in profiles also depends on the Peclet number, which determines the dominancy of the advection process. ACKNOWLEDGMENT This research was partly supported by a grant (code 3-2-1) from Sustainable Water Resources Research Center of the 21st Century Frontier Research Program of the Korean government.



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REFERENCES

Kayane, I., Taniguchi, M., and Sanjo, K., Alteration of Groundwater Thermal Regime Caused by Advection, Hydrol. Sci. J., vol. 30, pp. 343–360, 1985.

Andrews, C. B., and Anderson, M. P., Thermal Alteration of Groundwater Caused by Seepage from a Cooling Lake, Water Resour. Res., vol. 15, pp. 595–602, 1979. Boyle, J. M., and Saleem, Z. A., Determination of Recharge Rates Using Temperature Depth Profiles in Wells, Water Resour. Res., vol. 15, pp. 1616–1622, 1979. Bredehoeft, J. D., and Papadopulos, I. S., Rates of Vertical Groundwater Movement Estimated from Earth’s Thermal Profile, Water Resour. Res., vol. 1, pp. 325–328, 1965. Cartwright, K., Groundwater Discharge in the Illinois Basin As Suggested by Temperature Anomalies, Water Resour. Res., vol. 6, pp. 912–918, 1970. Cartwright, K., Redistribution of Geothermal Heat by a Shallow Aquifer, Geol. Soc. Am. Bull., vol. 82, pp. 3197–3200, 1971. Cartwright, K., Tracing Shallow Groundwater Systems by Soil Temperatures, Water Resour. Res., vol. 10, pp. 847–855, 1974. Cartwright, K., Measurement of Fluid Velocity Using Temperature Profiles: Experimental Verification, J. Hydrol., vol. 43, pp. 185–194, 1979. Chisholm, T. J., and Chapman, D. S., Climate Change Inferred from Analysis of Borehole Temperatures: An Example from Western Utah, J. Geophys. Res., vol. 97, pp. 14,155–14,175, 1992. Constantz, J., Tyler, S. W., and Kwicklis, E., TemperatureProfile Methods for Estimating Percolation Rates in Arid Environments, Vadose Zone Hydrol., vol. 2, pp. 12–24, 2003. Dapaah-Siakwan, S., and Kayane, I., Estimation of Vertical Water and Heat Fluxes in the Semi-confined Aquifers in Tokyo Metropolitan Area, Japan, Hydrol. Processes, vol. 9, pp. 143–160, 1995. Harris, R. N., and Chapman, D. S., Climate Change on the Colorado Plateau of Eastern Utah Inferred from Borehole Temperatures, J. Geophys. Res., vol. 100, pp. 6367–6381, 1995. Hoffman, J. D., Numerical Methods for Engineers and Scientists, McGraw-Hill, New York, 1992. Jessop, A. M., and Vigrass, I. W., Geothermal Measurements in a Deep Well at Regina, Saskatchewan, J. Volcanol. Geotherm. Res., vol. 37, pp. 151–166, 1989.

Keshari, A. K., and Koo, M. H., Estimation of Groundwater Recharge from Ground Temperature Measurements along Vertical Profiles, Technologies for Analyzing the Groundwater Flow System, Conf. Proc., pp. 77–99, 2002. Keshari, A. K., and Koo, M. H., A Numerical Model for Estimating Groundwater Flux from Subsurface Temperature Profiles, Hydrol. Processes, 21, 2007, in press. Keys, W. S., and Brown, R. F., The Use of Temperature Logs to Trace the Movement of Injected Water, Ground Water, vol. 16, pp. 32–48, 1978. Kumaran, M. K., Comparison of Simultaneous Heat and Moisture Transport through Glass-Fiber and SprayCellulose Insulation, J. Thermal Insulation, vol. 11, pp. 6–16, 1988. Lapham, W. W., Use of Temperature Profiles beneath Streams to Determine Rates of Vertical Groundwater Flow and Vertical Hydraulic Conductivity, U.S. Geol. Surv. Water Supply Pap., 2337, 1989. Mareschal, J.-C., and Beltrami, H., Evidence for Recent Warming from Perturbed Geothermal Gradients: Examples from Eastern Canada, Clim. Dyn., vol. 6, pp. 135–143, 1992. Reiter, M., Using Precision Temperature Logs to Estimate Horizontal and Vertical Groundwater Flow Components, Water Resour. Res., vol. 37, pp. 663–674, 2001. Ronan, A. D., Prudic, D. E., Thodal, C. E., and Constantz, J., Field Study and Simulation of Diurnal Temperature Effects on Infiltration and Variably Saturated Flow beneath an Ephemeral Stream, Water Resour. Res., vol. 34, pp. 2137–2153, 1998. Sakura, Y., The Method for the Estimations of the Groundwater Velocities from Temperature Distributions: On the Groundwater around Sapporo, Japan, Stud. Water Temp., vol. 21, pp. 2–14, 1977. Sakura, Y., Uchida, Y., and Anderson, M. A., Effect of Groundwater Flow and Surface Warming on the Subsurface Thermal Regime in the Nobi Plain around Nagoya, Central Japan, Eos Trans. AGU, vol. 77, Spring Meet. Suppl., S97, 1996. Silliman, S. E., and Booth, D. F., Analysis of Time Series Measurements of Sediment Temperature for Identification of Gaining vs. Losing Portions of Juday Creek,


782

Indiana, J. Hydrol., vol. 146, pp. 131–148, 1993. Silliman, S. E., Ramirez, J., and McCabe, R. L., Quantifying Downflow through Creek Sediments Using Temperature Time Series: One-Dimensional Solution Incorporating Measured Surface Temperature, J. Hydrol., vol. 167, pp. 99–119, 1995. Smith, L., and Chapman, D. S., On the Thermal Effects of Groundwater Flow: 1. Regional Scale Systems, J. Geophys. Res., vol. 88, pp. 593–608, 1983. Sorey, M. L., Measurement of Vertical Groundwater Velocity from Temperature Profiles in Wells, Water Resour. Res., vol. 7, pp. 963–970, 1971. Stallman, R. W., Steady One-Dimensional Fluid Flow in a Semi-infinite Porous Medium with Sinusoidal Surface Temperature, J. Geophys. Res., vol. 70, pp. 2821–2827, 1965. Suzuki, S., Percolation Measurements Based on Heat Flow through Soil with Special Reference to Paddy Fields, J. Geophys. Res., vol. 65, pp. 2883–2885, 1960. Tabbagh, A., Bendjoudi, H., and Benderitter, Y., Determination of Recharge in Unsaturated Soils Using Temperature Monitoring, Water Resour. Res., vol. 35, pp. 2439–2446, 1999. Taniguchi, M., Evaluation of Vertical Groundwater Fluxes and Thermal Properties of Aquifers Based on Transient Temperature-Depth Profiles, Water Resour. Res., vol. 29, pp. 2021–2026, 1993. Taniguchi, M., and Sharma, M. L., Determination of Groundwater Recharge Using the Change in Soil Temperature, J. Hydrol., vol. 148, pp. 219–229, 1993. Taniguchi, M., Shimada, J., Tanaka, T., Kayane, I., Sakura, Y., Shimano, Y., Dapaah-Siakwan, S., and Kawashima, S., Disturbances of Temperature-Depth

Keshari and Koo

Profiles Due to Surface Climate Change and Subsurface Water Flow: 1. An Effect of Linear Increase in Surface Temperature Caused by Global Warming and Urbanization in the Tokyo Metropolitan Area, Japan, Water Resour. Res., vol. 35, pp. 1507–1517, 1999a. Taniguchi, M., Williamson, D. R., and Peck, A. J., Disturbances of Temperature-Depth Profiles Due to Surface Climate Change and Subsurface Water Flow: 2. An Effect of Step Increase in Surface Temperature Caused by Forest Clearing in Southwest Western Australia, Water Resour. Res., vol. 35, pp. 1519–1529, 1999b. Vafai, K., Convective Flow and Heat Transfer in Variable Porosity Media, J. Fluid Mech., vol. 147, pp. 233–259, 1984. Vafai, K., and S¨ozen, M., A Comparative Analysis of Multiphase Transport Models in Porous Media, Annu. Rev. Heat Transfer, vol. 3, pp. 145–162, 1990. Wierenga, P. J., Hagan, R. M., and Nielsen, D. R., Soil Temperature Profiles during Infiltration and Redistribution of Cool and Warm Irrigation Water, Water Resour. Res., vol. 6, pp. 230–238, 1970. Woodbury, A. D., and Smith, L., On the Thermal Effects of Three Dimensional Groundwater Flow, J. Geophys. Res., vol. 90, pp. 759–767, 1985. Ziagos, J., and Blackwell, D. D., A Model for the Effect of Horizontal Fluid Flow in a Thin Aquifer on Temperature-Depth Profiles, Trans. Geotherm. Resour. Counc., vol. 5, pp. 221–223, 1981. Ziagos, J., and Blackwell, D. D., A Model for the Transient Temperature Effects of Horizontal Fluid Flow in Geothermal Systems, J. Volcanol. Geotherm. Res., vol. 27, pp. 371–397, 1986.
