Numerical evaluation of thermal response tests

Geothermics 36 (2007) 141–166

Numerical evaluation of thermal response tests Sarah Signorelli a,∗,1 , Simone Bassetti a,1 , Daniel Pahud b , Thomas Kohl c a

Institute of Geophysics, ETH H¨onggerberg, CH-8093 Z¨urich, Switzerland b LEEE-DCT, SUPSI, 6952 Cannobio, Switzerland c GEOWATT AG, Dohlenweg 28, CH-8050 Z¨ urich, Switzerland Received 19 December 2003; accepted 31 October 2006 Available online 29 December 2006

Abstract Thermal conductivity is a key parameter in the design of borehole heat exchangers. Thermal response tests are becoming increasingly more popular for measuring in situ thermal conductivity but no theoretical investigations have been done so far that account for three-dimensional effects. We compared the results from a 3-D finite-element numerical model with those of a simple analytical linesource solution and tested their sensitivity to the duration of the tests. The effects of heterogeneous subsurface conditions, groundwater movement, and variable data quality are presented. Comparison with measured data emphasizes the importance of using more sophisticated numerical methodologies in interpreting thermal response test data. © 2006 CNR. Published by Elsevier Ltd. All rights reserved. Keywords: Thermal response test; 3-D numerical calculations; Borehole heat exchanger

1. Introduction Borehole heat exchanger (BHE) systems, mainly used for heating purposes, have become increasingly popular in Switzerland; to date (July 2006) nearly 30,000 have been installed. Switzerland is the world leader in this ecologically friendly technology in terms of area density (i.e. number of BHEs per km2 ; Rybach et al., 2000). Heat exchangers supply energy for heating and/or cooling small, decentralized buildings such as single-family houses, as well as ∗ 1

Corresponding author. Tel.: +41 44 242 14 54; fax: +41 44 242 14 58. E-mail address: [email protected] (S. Signorelli). Present address: GEOWATT AG, Dohlenweg 28, CH-8050 Z¨urich, Switzerland.

0375-6505/$30.00 © 2006 CNR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.geothermics.2006.10.006

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Nomenclature E1 k m N q r rb Rb t tc T T¯ Tcomp Tmeas

exponential integral constant in Eq. (5) [K] constant in Eq. (5) [K] number of measured data points heat injection rate per meter of borehole length [W m−1 ] radial distance [m] borehole radius [m] borehole resistance [K m W−1 ] time [s] time criterion [s] fluid temperature [◦ C] average fluid temperature of the circulation fluid [◦ C] computed fluid temperature [◦ C] measured fluid temperature [◦ C]

Greek symbols γ Euler’s constant [0.5772. . .] κ thermal diffusivity [m2 s−1 ] λ thermal conductivity [W m−1 K−1 ] λLS thermal conductivity estimated by the line-source model [W m−1 K−1 ] ρcp volumetric heat capacity [J m−3 K−1 ] Subscripts E end num assumed in the numerical model 0 initial, starting, or undisturbed

multi-family dwellings and commercial buildings. The most popular type of BHE consists of a closed circuit with double U-tubes in grouted boreholes, typically 100–300 m deep. The thermal properties of the subsurface geologic units largely determine the dimensions of the BHE installation. Estimated property values are generally sufficient for single-house applications and dimensioning of the systems is based on experience. For larger projects (>10 BHEs), however, a knowledge of the thermal conductivity of the subsurface will constrain the number of BHEs required to supply the energy needs of a given building, and consequently the costs involved. The thermal conductivity of the underground is often estimated by utilizing a thermal response test in a ready-to-operate BHE. This approach was first proposed by Morgensen (1983) and is based on an infinite line-source model (Carslaw and Jaeger, 1959). In 1995, the first mobile measurement devices were introduced in Sweden (Ekl¨of and Gehlin, 1996) and in the USA (Austin, 1998). Since then, the method has been improved and its use has spread to several other countries (Gehlin, 2002). This experimental methodology (e.g. Austin et al., 2000; Eugster and Laloui, 2002; Gehlin, 2002) attempts to constantly heat a fluid circulating through a ready-tooperate BHE, measuring the fluid temperature at its inlet and outlet. The set-up for this type of test is shown in Fig. 1.

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Fig. 1. Schematic illustration of the set-up for a thermal response test (Gehlin, 2002). T1 : temperature of produced fluid; T2 : temperature of injected fluid.

By evaluating the recorded temperatures versus time, we obtain an estimation of the average thermal conductivity in the vicinity of the BHE. In order to estimate the in situ thermal conductivity, the temperature changes during the recovery period (i.e. after the response test has been completed) are sometimes also measured and analyzed (Fujii et al., 2002). These data are less noisy than those recorded during the test itself but measuring the slowly recovering temperatures significantly increases the cost of the test. The advantage offered by response tests is that one can integrate the underground thermal properties along the entire length of a BHE, including groundwater and backfilling material, providing a so-called “effective” (not the “real”) thermal conductivity as defined under strict heat conduction assumptions. Laboratory measurements alone may lead to different values since they cannot correctly account for groundwater flow and water-filled cracks and pores. In some cases, however, it may prove dangerous to use this measured (or estimated) “effective” thermal conductivity as the representative value for dimensioning purposes, especially in advection-dominated areas. In the common representation of the heat transfer equation, advective and conductive processes are generally considered as two different physical terms. If the subsurface regime is dominated by advection these two processes cannot be described by one parameter only. Several mathematical models have been proposed to calculate the thermal conductivity from the measured temperature response, based on either analytical or numerical solutions of the heat

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conduction equation. Analytical models, such as the line-source or cylindrical model (Carslaw and Jaeger, 1959), are based on simple assumptions with regard to the geometry of the borehole and the heat exchanger, and are therefore generally straightforward. Recently, numerical techniques (e.g. Shonder and Beck, 1999; Yavuzturk et al., 1999) have become more popular because they can take into account specific borehole geometries and time-varying effects such as changes in heat injection rates. In this case, the additional model parameters involved in the analysis make interpretations more complex and time consuming. While the analytical line-source model is a simplification of the actual test, it is the most frequently used, and its accuracy in determining in situ thermal conductivities is generally accepted (Sch¨arli and Rybach, 2002). We will show that accurate results with the line-source model can only be obtained under specific conditions. The accuracy of the thermal response test depends largely on the length of the experiment, since the inlet and outlet temperatures measured at the beginning of the test reflect the thermal response of the BHE (i.e. its tubes, grout, etc.). The optimum duration is a topic of on-going debate (Austin, 1998; Gehlin, 1998; Smith and Perry, 1999), mainly because longer tests entail greater expense, but also because complex geological and hydrological conditions can affect the analysis of the collected data. In most evaluations of a thermal response test one assumes that the subsurface is homogenous and that there is a constant heat transfer rate along a BHE. In reality, the heat transfer from the fluid to the ground might be highest near the surface due to the natural temperature gradient, and, in the case of geologic heterogeneities, a particular layer may dominate the heat transfer process. Groundwater might also have an influence on the thermal response test. Although groundwater flow generally improves the efficiency of heating systems that are based on geothermal resources, it may lead to significant heat losses in underground thermal energy storage systems by carrying away the energy stored in the reservoir/aquifer. In order to investigate the different phenomena that could affect the analysis of thermal response test data, we have compared the results of analytical and 3-D numerical models. Our general approach was to use the numerical model to simulate various test cases and then utilize the computed temperature response as synthetic data in the more conventional analytical thermal response analysis. This should quantify the error introduced by the widely used line-source model. Finally, we present a numerical assessment of a real data set, along with a comparison to a line-source model. 2. Analytical methods In this section, we describe the analytical and numerical procedures used in the analysis of both synthetic and real data. 2.1. Line-source model The most widely used analytical procedure when interpreting BHE data is the line-source model. It is based on Kelvin’s line-source theory and has been applied to simulate the behavior of BHEs (Ingersoll and Plass, 1948; Sanner, 1992). Morgensen (1983) proposed this method to evaluate thermal response test data. The approach adopts the analytical solution for the response to an infinite constant-strength line-source within a homogeneous, isotropic, infinite medium. Assuming negligible vertical heat flow along the BHE and constant lateral heat flow, the temperature field around the BHE is only dependent on time, t, and radial distance, r, from the borehole

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axis. Using the so-called exponential integral E1 , the temperature field is given by (Carslaw and Jaeger, 1959):  2   ∞ r e−u q q du = E1 (1) T (r, t) − T0 = 2 4πλ r /4κt u 4πλ 4κt where T0 is the undisturbed ground temperature, q the heat injection rate per meter of borehole length, λ the thermal conductivity, and κ denotes the thermal diffusivity of the ground. The exponential integral can be approximated if the radius of the thermal front has penetrated the surrounding rock beyond the borehole wall and the effects of the BHE itself can be neglected. This simplified E1 is defined as:  2    2    n ∞ 2 r r 4κt n (r /4κt) ∼ (2) E1 = −γ − ln − (−1) = ln 2 − γ 4κt 4κt n n! r n=1

where γ is Euler’s constant [0.5772. . .]. The maximum error of this simplification is less than 10% for the time criterion tc ≥ 5r2 /κ. Thus, the temperature at the borehole wall (r = rb ) can be defined by:    5r2 4κt q − γ + T0 , with tc ≥ b ln (3) T (r = rb , t) = 2 4πλ κ rb Introducing a thermal borehole resistance between the fluid and the borehole wall, Rb , the average fluid temperature of the circulation fluid, T¯ , as a function of time can be written as:      4κ q 1 −γ + T0 T¯ (t) = T (r = rb , t) + qRb = ln(t) + q Rb + ln (4) 4πλ 4πλ rb2 If q is constant, the last two terms do not change with time and Eq. (4) becomes a simple linear relation: T¯ (t) = k ln(t) + m

(5)

where k is defined as k = q/4πλ, m being a constant. The line-source model assumes that T¯ corresponds to the average between the inlet (Tin ) and outlet temperature (Tout ) of the circulation fluid: Tin + Tout T¯ = 2

(6)

Now, if the change in the mean temperature, T¯ , is plotted versus the natural logarithm of time, the thermal conductivity based on the line-source model, λLS , is given by: λLS =

q ln(t2 ) − ln(t1 ) 4π T¯ (t2 ) − T¯ (t1 )

(7)

Here, it is not just the temperatures at times t1 and t2 that are used to calculate the regression line to estimate λLS in Eq. (7), but also all temperatures measured between t1 and t2 . In the field test the constant heat injection rate is generated by means of an electric heater of a few kW. The higher the energy loss in the subsurface, the higher the λLS . In the next section, the values of λLS are calculated from the temperature changes generated by a numerical model assuming a constant thermal conductivity, λnum .

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2.2. Finite-element code FRACTure Numerical models that simulate thermal response tests allow a more detailed representation of the borehole (i.e. of its geometry and thermal properties) and can account for variations in the heat injection rate. The most commonly used numerical models are based on parameter estimation techniques and on 1-D finite-difference borehole models (Shonder and Beck, 1999) or 2-D finite-volume models (Spitler et al., 1999; Yavuzturk et al., 1999; Austin et al., 2000). Because of the three-dimensional nature of the thermal field perturbed by an operating BHE system, we used the 3-D finite-element (FE) code FRACTure (Kohl and Hopkirk, 1995). This numerical program is suited to simulating coupled hydraulic, thermal, and even elastic processes under non-steady conditions. It was originally created to investigate the long-term behavior of hot dry rock reservoirs, but has been successfully applied to other types of geothermal system and other problems, including BHEs (e.g. Kohl et al., 2002). One particular feature of FRACTure is the combination of lower and higher dimensional elements, with the result that advective thermal transport in tube-like structures – as in the case of BHEs – can be accurately simulated using 1-D tube elements surrounded by 3-D matrix elements. The heat transfer through pipe walls from the circulating fluid to the ground is modeled as a thermal resistance whose value is a function of the fluid velocity-dependent thermal transfer coefficient (Kohl et al., 2002). In the FRACTure code, groundwater flow is computed using Darcy’s Law. The finite-element approach permits a flexible mesh generation. Tetrahedrons and prisms allow for a more accurate representation of the loop and borehole geometry of a BHE using different dimensionality for the pipe walls (<10−2 m) and the geological units (>102 m). The thermal and hydraulic properties of every element in the computational mesh can be changed. Dirichlet and Neumann boundary conditions can be set when simulating heat and mass transport. The transient behavior of selected parameters or boundary conditions, such as changes in heat injection rate or in surface temperature, can be simulated using time-dependent functions. A more detailed description of FRACTure is given in Kohl et al. (2002). This is the first analysis of thermal response tests using the FRACTure code (see Section 4.3). To this end, we applied parameter estimation to a single independent variable, the subsurface thermal conductivity, λ, which entails specifying all the other input parameters. In general the geometry of the BHE and the thermal properties of the grout are known, and the tube locations inside the borehole are not, so that the borehole geometry has to be assumed and may differ from the actual one. However, Austin et al. (2000) and Bassetti (2003) show that even significant variations in the shank spacing of the U-tubes (distance from center to center of the up- and down-tubes in each “U”) produce only small differences in the thermal conductivity estimations. The standard procedure is to minimize the difference between the measured and computed data by systematically varying the thermal conductivity, which would give a best estimate of the latter. The objective function for this optimization procedure, that is the sum of the squares of the errors (SSE) between the experimental and the simulated data, is given by: SSE =

N 

(Tmeas − Tcomp )2

(8)

n=1

where N is the total number of measured data points, and Tmeas and Tcomp are the measured and simulated temperatures, respectively. It is customary to minimize the square difference.

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3. Sensitivity analysis Before investigating an actual field experiment, we will discuss a sensitivity analysis based on synthetic data. The purpose of this exercise was to determine the sensitivity of the line-source model to test duration, borehole length and subsurface heterogeneity. First, we describe the computational mesh that was used and the method utilized to generate the synthetic data. Note that in Section 4 the same FE mesh will be used to evaluate the experimental data. 3.1. Borehole geometry and numerical discretization The FE mesh was based on a set of data obtained during a test in a 160-m deep borehole at Schweizerische Unfallversicherungsanstalt (SUVA) at Gisikon, Switzerland (see Table 1 and Section 4). In this test, the BHE was fitted with spacer-separated 40 mm diameter polyethylene pipes. Figure 2a illustrates the implementation of this borehole geometry with four tubes grouted in the center of the borehole. Different numbers of horizontal layers were used in the design of the mesh, depending on the total depth of the BHE system being modeled. Nodal spacing is fine around the borehole where the largest temperature gradients tend to occur. In the vertical direction the mesh generally has 20-m high elements, but their heights are reduced near the surface (to 1 m) and at the bottom of the borehole (height < 0.1 m) to reduce numerical instabilities. In the simulations discussed below, we assumed a constant surface temperature of 10 ◦ C and basal heat flow of 90 mW m−2 , resulting in a temperature gradient along the BHE. Note that the existence of such a gradient violates the line-source assumptions. Neumann-type boundaries are assumed laterally. To avoid boundary effects, the bottom and lateral boundaries of the mesh are placed at some distance from the BHE, which is located at the center of the mesh. In this case, the total dimensions of the 3-D mesh are 1500 m × 1500 m × 500 m (Fig. 2b). In the model, the assumed thermal conductivities, λ, and heat capacities, ρcp , for the tubing wall are typical for polyethylene (λpipe = 0.4 W m−1 K−1 , ρcp pipe = 1.62 MJ m−3 K−1 ) and for grout material (standard bentonite mixture; i.e. λgrout = 0.8 W m−1 K−1 ; ρcp grout = 2.0 MJ m−3 K−1 ). In the numerical model used in the sensitivity analyses given below, the thermal conductivity of the rock matrix, λnum , was set at 3 W m−1 K−1 , and the heat capacity, ρcp num , at 2.5 MJ m−3 K−1 . The changes in mean fluid temperature were computed using the FRACTure code and the assumed parameters and boundary conditions discussed above. A constant heating power of 9 kW Table 1 Characteristics of the SUVA borehole Depth Diameter Pipe material Outer pipe diameter Pipe thickness Spacer (shank spacing) Grouting material Heat carrier fluid Test flow rate

160 m 0.152 m Polyethylene 40 mm 3.7 mm 7.8 cm Quartz sand cement Water 810 L h−1

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Fig. 2. Computational mesh used to simulate a borehole heat exchanger (BHE). (a) Detailed view of the horizontal cross-section showing the four pipes in the centre of the BHE and the backfilling around them and (b) 3-D view.

was assumed throughout a 200-h test. From the synthetic temperature data given by the model, the thermal conductivity, λLS , was evaluated by applying the line-source model for different time intervals using Eq. (7). The value of λLS was determined either for a fixed end time, tE (=end of the considered time period), and a variable starting time, t0 (=start of the considered time period), or for fixed t0 and continuously increasing tE (i.e. increasing length of the analyzed data interval). With the exception of Fig. 3, all the results shown were obtained following the second

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method, which is the one generally used, with t0 = 10 h (∼tc ) and continuously increasing the end time from tE = 11 to 200 h. Thus, the first points on the left in Figs. 4 and 6–8 are the estimated conductivity computed from the 10–11 h data, and the last points are based on the temperatures collected during the 10–200 h period. 3.2. Numerical sensitivity Several studies have demonstrated that a fine computational mesh is required to properly determine the temperature changes associated with the operation of a BHE (e.g. Kohl et al., 2002). Finer discretization generally results in higher accuracy, but it is also more time consuming. To define the optimum computational mesh for our sensitivity analysis, we first set up several models using different spatial and temporal discretizations. Two factors in our general numerical model set-up, however, violate the line-source assumptions (see Section 2.1): (1) the temperature gradient along the BHE, and (2) the finite length of the BHE, both of which reflect real test conditions. All the model runs were therefore set up so as to fulfil the requirement of the line-source model by neglecting ground surface temperature boundary condition at the top and basal heat flow boundary condition at the bottom of the FE mesh. This leads to a uniform temperature along the BHE (no temperature gradient). By extending the FE mesh vertically over the depth range of the BHE only, we ensured that no thermal diffusion takes place below the BHE, whereby the effect of a finite source length was neglected. Four model runs were performed: • Run M1 : coarse spatial discretization (∼17,000 nodes)/coarse temporal discretization (t = 1 h). • Run M2 : fine spatial discretization (∼75,000 nodes)/coarse temporal discretization (t = 1 h). • Run M3 : fine spatial discretization (∼75,000 nodes)/fine temporal discretization (t = 5 min). • Run M4 : very fine spatial discretization (∼300,000 nodes)/coarse temporal discretization (t = 1 h). To illustrate the effects of discretization on the model results, the response test was simulated for a period of 200 h and λLS was estimated for a fixed tE = 200 h and a t0 that varied continuously from 1 to 199 h. Thus, the first point on the left in Fig. 3 is the estimated λLS for the 1–200 h data interval, while the last point to the right refers to that estimated for the 199–200 h interval. This evaluation method, in which the starting time of the time period considered, t0 , is progressively shifted to later times, reduces the influence of the borehole itself. Figure 3 shows that, for all these runs, the computed λLS values approach λnum for increasing t0 . However, the shape of the λLS curve for Run M1 is clearly different from the others. After reaching a maximum, the values of λLS in M1 decrease and are higher than for the other cases; this is caused by the very coarse discretization used. The λLS curves for the other model runs are nearly identical. The effect of the borehole itself is evident during the first hours, resulting in low values. After about 30 h, the computed λLS slowly approaches a constant value, but after 200 h the steady-state values have still not been reached. This analysis of synthetic data shows that, under ideal conditions and using appropriate computational meshes, the line-source approach can yield realistic thermal conductivities. The estimated λLS values for the interval 150–200 h for cases M2 , M3 , and M4 were 3.00, 2.98, and 2.99 W m−1 K−1 , respectively. These three runs gave almost identical values, all equal or close to λnum . The model runs required different computing times, ranging from 40 min to 10 h on a

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Fig. 3. Calculated thermal conductivities based on the line-source model, λLS , using numerical models with different spatial and temporal discretizations. The end time of the different evaluated data intervals is fixed at tE = 200 h. Starting time varies from t0 = 1 to 199 h. MI corresponds to “Run MI , where I: 1–4” (see text).

2 GHz PC. Run M2 is preferred because of the relatively short calculation time involved, and its discretization was used in the analyses that follow. 3.3. Length of measurement period The ideal duration of a thermal response test is the subject of on-going debate. In the literature there are recommendations for 60 h (Gehlin, 1998), 50 h (Austin et al., 2000), and 12–20 h (Smith and Perry, 1999). Since the calculation of thermal conductivities based on the line-source model is extremely sensitive to single data points, Gehlin and Hellstr¨om (2003) recommend using data evaluation periods longer than 30 h. It is difficult to determine the length of data interval needed for an accurate evaluation of conductivities based on experimental data sets. Austin (1998) and Witte et al. (2002) showed that small temperature variations, often caused by unstable power supplies or diurnal air temperature fluctuations, can lead to extremely variable results when data taken during different time periods of the same test are used in the evaluation, even when the test periods are long. This study will concentrate on the influence of starting time, t0 , on the line-source evaluation, and the impact of the required length of data interval on the calculation of thermal conductivity in the ground around the BHE. For this purpose, Run M2 (see previous section) was used to generate the synthetic thermal response. The error in the estimated thermal conductivity value should not exceed 10% after the critical time, tc , has been reached. This is the maximum error allowed for a thermal response test in practice (Eugster, 2002). The sensitivity analysis was performed under the line-source model requirements, as discussed earlier. However, other factors, such as borehole length, boundary conditions and unstable data, could further affect the error in the computed thermal conductivities. A different method was used in the evaluation of λLS based on the temperature response computed numerically than in Fig. 3. In this case, the simulated 200-h test was evaluated for

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Fig. 4. Results of the study to determine the duration of a thermal response test required to obtain a 10% accuracy using the line-source model. Plotted is the difference between the computed thermal conductivity, λLS , and that assumed in the numerical model, λnum . The starting point, t0 , for the different evaluated data intervals is set at 10, 20, 40, and 60 h. The end time, tE , is varied from t0 + 1 h up to 200 h (see text). Shading shows the region corresponding to the 95% confidence interval, when a 0.1 ◦ C white noise is superimposed on the synthetic thermal response.

increasing data interval lengths using different fixed starting times of 10 h (∼tc ), 20, 40, and 60 h, and variable end times of t0 < tE ≤ 200 h. The end time of the evaluated interval, tE , corresponds to the length of the test. Figure 4 shows the error corresponding to t0 = 10, 20, 40, and 60 h and increasing values of tE when the estimated thermal conductivities are compared with the assumed value (λnum = 3 W m−1 K−1 ) used in the numerical model to generate the synthetic thermal response test. It is evident that, at early t0 , λLS is lower than λnum , showing the influence of the borehole, caused by the lower thermal conductivity of the tubes and grouting material. At later starting times, t0 , and/or later end times, tE , the accuracy of the estimations based on the line-source model improves. The estimated λLS are generally lower than λnum , but in some cases, for late t0 and tE , they can be very slightly higher than the assumed value. Changes in t0 have a stronger effect on accuracy than changes in tE . For t0 = 10 h (∼tc ), the test must last at least 30 h for the error to drop below 10%, whereas at t0 = 60 h, the error is close to zero, irrespective of the evaluated interval length. Starting the evaluation at t0 ∼ 20 h (i.e. three times tc ), the error can always be expected to be less than 10%. These statements are true for synthetic data; that is, assuming constant heating and circulation rates and perfect measurements. When actual temperature measurement errors are considered, however, the accuracy of the estimations will be different. Witte et al. (2002) showed that a temperature disturbance of only ∼0.15 ◦ C affects the estimate of thermal conductivity significantly. To illustrate this effect, 50 new thermal response data sets were generated, superimposing random noise not exceeding 0.1 ◦ C (the general temperature measurement accuracy) onto the synthetic M2 thermal response data. The λLS was then evaluated from the new data sets. For the case of t0 = 40 h, the 95% confidence interval of these λLS evaluations is represented by the shaded area in

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Fig. 4, indicating that for small intervals (tE slightly higher than t0 ) the accuracy of the evaluation is compromised. The deviation from λnum ranges between 0 and >10%, compared to the <4% under ideal conditions, although the effect of the applied temperature disturbance vanishes at later tE . We also studied the effect of the disturbance for t0 = 10 h. In this case and because of the strong temperature increase at early times (low t0 ), the resulting confidence interval is smaller. At larger t0 , however, the influence of temperature disturbance on the accuracy of the λLS evaluation can become dominant. We should add at this point that a 0.1 ◦ C measurement accuracy is difficult to achieve in the field, and larger errors should be expected in actual thermal response tests. To summarize, a simple definition of optimal t0 and tE is not possible. Small values of t0 tend to underestimate the subsurface thermal conductivity, but show reduced sensitivity to temperature disturbances. In contrast, high values of t0 give good approximate values, but can be strongly affected by temperature variations. The thermal conductivities estimated on the basis of the line-source model, λLS , are generally lower than the actual values, but significant temperature disturbance may also result in higher values that could lead to an over-optimistic design of BHE fields. It must be remembered that the results discussed above were achieved with a model that was adapted to the requirements of the line-source model (assuming only a radial temperature field, no vertical temperature gradient). A different model set-up, in which both a temperature gradient and a finite length of the borehole are considered, could lead to different conclusions. 3.4. Effect of borehole length In this section, thermal response tests are simulated for “real” temperature fields. The effect of borehole length is investigated using the model assumptions for Run M2 (see Section 3.2). Now, however, we consider different boundary conditions; i.e. constant temperature at the surface and constant heat flow at the bottom of the model (located at 1 km in the deepest model run). Under actual field test conditions, a certain volume of the subsurface around the BHE is heated up and the resulting temperature field has the shape of an inverse funnel (see Fig. 5). The temperature distribution around the BHE can no longer be assumed radial as required by the line-source model. The steeper the sides of the “funnel”, the closer the temperature field will be to that assumed in the line-source model. Since the geometry of the funnel varies with depth, we would expect the deeper boreholes to fit the line-source requirements better. Borehole depths of 40, 80, 160, 320, and 400 m are considered here. The results are illustrated in Fig. 6 for fixed t0 = 10 h and increasing tE (analogous to one of the cases illustrated in Fig. 4). Comparing the 160 m model run to the t0 = 10 h results in Fig. 4, the estimated λLS is 2% higher because of the inclusion of more realistic boundary conditions. Generally, the estimated thermal conductivities are still lower than the values used in the numerical model (λnum = 3 W m−1 K−1 ). This seems to suggest that the temperature distribution in the subsurface around the 160-m deep BHE under “real” test conditions does not differ much from the radial temperature field. Figure 6 shows that, for tests lasting over 50 h, the computed λLS generally fits λnum for all model runs with different BHE lengths within 10%. Independent of the depths of the BHEs, accurate results can therefore be attained even though the temperature field developed around the BHE under “real” conditions does not strictly fulfil the line-source model requirements. Analysing the results of Fig. 6 further, it becomes apparent that other factors such as changes in transient behavior influence the evaluation. Even in an unusually long 500-h test, the 40-m deep borehole will not reach steady-state conditions in contrast to the deeper BHE models (Bassetti, 2003), and the evaluation of the temperature data computed by deeper BHE models does not

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Fig. 5. Computed temperature distribution around a 160-m deep borehole at the end of the thermal response test (i.e. at 200 h). The direction of the heat flux is schematically indicated by arrows. Note the different scales used on the x- and z-axis.

generate thermal conductivities that agree exactly with λnum . The λLS evaluations based on the 80 m model run overestimate λnum , whereas the estimates for 160, 320, and 400 m are slightly lower. The differences in the values given by the various models, as observed in Fig. 6, remain nearly constant (i.e. the differences do not change when the models simulate a 500-h test).

Fig. 6. Calculated thermal conductivities based on the line-source model, λLS , for different borehole lengths. The first point of the evaluated data interval is fixed at t0 = 10 h; the end time, tE , varies.

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Detailed investigations by Bassetti (2003) have shown that, if the boundary conditions correspond to those of the line-source model (as in Section 3.2), all model runs result in λnum with errors smaller than 0.5%. Therefore, the different deviations shown in the computed values (Fig. 6) cannot be the result of numerical errors but rather of diffusive thermal processes occurring next to an operating BHE. As mentioned earlier, the injection of heat during a thermal response test creates a temperature distribution in the subsurface that is similar in shape to an inverted funnel (see Fig. 5). Since these tests will cause a continuous heat flux from the borehole domain to the surface, the isotherms will be strongly curved near the surface. On the other hand, the temperature field is nearly radial at the bottom of the BHE and becomes more so as the depths of the exchangers increase. For example, a 40-m deep case will be more affected by surface conditions than one at 400 m depth. The mean temperature differences between the ascending and descending tubes of a BHE, T, will also show depth dependence. Deeper BHEs have higher T values and are therefore far more affected by thermal “short circuiting” between the tubes than shallower ones. This causes a non-radial 3-D thermal field in the vicinity of the borehole that is more significant in deeper boreholes. In other words, there is a complex interplay of the thermal field around a BHE caused by the radial “funnel” around the exchanger and the non-radial heat flow within the BHE itself (i.e. between its tubes). These effects are the main reason for the deviation of the λLS computed for the different BHE depths model from λnum when considering various BHE depths in the model. 3.5. Heterogeneous subsurface conditions and groundwater flow The amount of heat flowing between a BHE and the surrounding ground is determined by temperature gradients and thermal conductivities. This flow may vary along a BHE due to the particular distribution of subsurface temperatures and heterogeneities in the soil and rocks around the exchanger. A particular layer may have a dominant effect on the computed thermal conductivity λLS . Thermal response tests are generally performed by injecting hot fluids into the borehole. Most BHEs in Switzerland, however, operate under heat extraction conditions, although they are being used increasingly to inject heat into the subsurface during space-cooling of buildings. In heat injection tests, the largest temperature difference between the injectate and the ground occurs in the uppermost part of the BHE. When the cooled water returns to the subsurface during the operation of BHEs in the heating mode, the largest temperature differences will instead occur at the bottom of the borehole. The subsurface layers (or depth intervals) are thus stimulated differently under heat extraction and injection conditions and the estimated “effective” thermal conductivity may not correspond to the conductivity of the operational mode under investigation. We will now examine the influence of vertical heterogeneities along the length of a BHE for specific but simple cases. For this purpose, we have created two heterogeneous test cases in which the borehole length in the homogenous Run M2 (see Section 3.2) is divided into two 80-m thick units having different thermal conductivities. In Run M4-2 the upper layer has a thermal conductivity of 4 W m−1 K−1 and the lower one 2 W m−1 K−1 , while in Run M2-4 the conductivity distribution is reversed; it should be remembered that the thermal conductivity of the ground often increases with depth. The mean thermal conductivity (3 W m−1 K−1 ) corresponds to the homogenous 160-m deep case discussed in Section 3.4. The thermal response tests are simulated over a 200-h period, considering both operational modes (heat injection and heat extraction). Figure 7 summarizes the results of the assumed specific

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Fig. 7. Calculated thermal conductivities based on the line-source model, λLS , for the heterogeneous model Runs M2-4 and M4-2 compared to the homogeneous model Run M2 that uses the same average thermal conductivity. The first point of the evaluated data interval is set at t0 = 10 h; the end time varies.

model cases for a fixed t0 = 10 h. In the case of a homogenous subsurface, the computed λLS is nearly identical under both modes, showing a general underestimation. The heterogeneous (stratified) test cases generally give lower thermal conductivities than the homogenous cases (Fig. 7). The effects of the layer with the lower thermal conductivity seem to predominate slightly. In the stratified cases, the results depend on the test scenario considered. For example, the values of λLS for Run M2-4 are on average 3% higher for the heat extraction than for the heat injection response tests. In the heat extraction response tests, the largest temperature differences between the injected fluid and the surrounding ground occur in the deeper part of the BHE. The largest heat transfer between the fluid and the ground therefore occurs in that part of the exchanger and more energy can be dissipated into the deeper layer of the model. The effect is reversed in the heat injection tests. This might be the main reason for the observed differences in the results. For Run M4-2 , the results are inverted, with lower λLS for heat extraction than for heat injection. In the M2-4 case, when testing for heating applications (i.e. heat extraction), a standard heat injection test would give values that are ∼3% lower than the thermal conductivity value we would obtain in the heat extraction mode (see Fig. 7); were these values used, they could lead to a slight over-sizing of the BHE plant (i.e. boreholes would be drilled too deep and/or too many BHEs would be installed for a given project). For Run M4-2 , however, the one with the highly conductive upper layer, the effects would be reversed, with the result that the BHE plant might be slightly under-sized. Under actual field conditions, heat flow along the BHE will be much less uniform than in the simple proposed model, and even under these model conditions the effects of the different layers on the λLS evaluation will vary depending on the test scenario under investigation. However, the analytical evaluations of the heterogeneous test cases show only small differences with respect to the homogenous cases and all model runs achieve λnum within 10%.

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The presence of circulating groundwater generally increases the amount of heat that can be extracted from the ground by a BHE (Wagner and Clauser, 2002; Signorelli, 2004). Where groundwater flow is very slow, heat conduction and heat convection can be approximated by using an “equivalent” or “effective” thermal conductivity value, but this simplification is not valid where there is significant water movement (i.e. when the Peclet number, Pe, is greater than 1). It should be emphasized that, when evaluating the average ground thermal conductivity based on the line-source model, the method does not differentiate between thermal convection and conduction. In this section, we study the influence of groundwater flow on the estimated effective thermal conductivity. The effects of groundwater movement on the results of thermal response tests have been discussed by several authors (e.g. Gehlin, 1998; Sanner et al., 2000; Witte, 2002) and various theoretical studies have shown that these effects are important (Chiasson et al., 2000; Claesson and Hellstr¨om, 2000; Gehlin and Hellstr¨om, 2003). The impact of groundwater flow on thermal response tests was studied using a 3-D model and the FRACTure code. A 10-m thick permeable horizontal zone (i.e. an aquifer) was placed at 80 m depth into an otherwise homogeneous and impermeable 160-m thick 3-D system. The thermal conductivity of the aquifer was calculated assuming a 20% effective porosity and a rock matrix with a thermal conductivity of 3 W m−1 K−1 ; all other input parameters were equal to those given in Section 3.1. The aquifer was assumed to be saturated and the groundwater flow to be horizontal. The Darcy flow velocity, vD , given by the hydraulic conductivity and hydraulic pressure gradient in the aquifer, varies from 0.1 to 0.5, 1, and 2 m/day in model Runs MvD =0.1 , MvD =0.5 , MvD =1 , and MvD =2 , respectively. (The subscript vD indicates the velocity considered in a particular run.) Larger flow velocities could also be investigated, but we consider that the selected range represents the most relevant and interesting cases. In the context of our investigation, we can assume that groundwater temperatures below 20 m depth will remain constant throughout the year and that the operation of the BHE will change temperature only within a small area around the exchanger. For the sake of simplicity, we have therefore ignored the temperature dependence of the thermal properties of the ground in our calculations. In Fig. 8, the λLS obtained for the different simulation runs are compared with that of Run M2 , which only considers conduction (see Section 3.2). Here again, we set t0 at 10 h. Taking the diameter of the BHE as the characteristic flow length (L), the Peclet number is significantly lower than 1 for Darcy flow velocities of 0.1 m/day, or smaller. For these velocities, conduction will therefore dominate the thermal regime and the test results should not be affected by the moving groundwater, as shown by Chiasson et al. (2000). The effect of groundwater flow is already clearly visible in Run MvD =0.5 (Pe ∼ 1.7). For this run, the computed λLS for the 10–200 h data interval (i.e. 3.1 W m−1 K−1 ) is 7% higher than for Run M2 , at 2.9 W m−1 K−1 . This difference increases when flow velocity is increased. The computed thermal conductivity is 3.2 W m−1 K−1 for Run MvD =1 (Pe ∼ 3.4) and 3.4 W m−1 K−1 for MvD =2 (Pe ∼ 6.8). The impact of groundwater movement on the computed λLS not only depends on flow velocity, but also on flow volume (i.e. mass flow rate). Two additional models with aquifer thicknesses of 1 and 20 m and a constant flow velocity of 1 m/day were therefore considered, Runs MvD =1A and MvD =1B . As its aquifer thickness is 10 times smaller, the mass flow in Run MvD =1A is identical to that in MvD =0.1 , and the mass flow for Run MvD =1B corresponds to that of MvD =2 . For Run MvD =1A , the assumed groundwater flow does not have a significant impact on the computed λLS (Fig. 8); the results are not too different from those obtained for the zero-flow Runs M2 and MvD =0.1 . In Run MvD =1B , however, for the 10–200 h interval, the effects of groundwater

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Fig. 8. Calculated thermal conductivities based on the line-source model, λLS , for different groundwater flow velocities (see text), compared to the purely conductive model Run M2 , where the first point of the evaluated data interval is set at t0 = 10 h and the end time varies.

flow rate on the results are evident. In this case, the computed λLS is equal to 3.6 W m−1 K−1 , which is 24% higher than for Run M2 . Even though the flow rate is identical in Runs MvD =1B and MvD =2 , its impact on the results of MvD =1B is larger because of the thicker aquifer in contact with the BHE. The three-dimensionality of the problem, i.e. the disturbance to the temperature field by groundwater movement, is shown in Fig. 9 for Run MvD =1 . During a heat injection response test, the flowing groundwater removes heat from the ground around the BHE and reduces the temperature of the fluid circulating in the BHE. Figure 8 also indicates that the computed λLS becomes progressively larger with time for all simulations that include groundwater movement, as also observed by Witte (2002) in a field experiment. This author also showed that, under water-saturated subsurface conditions, heat injection and heat extraction tests yield different results. Thermal conductivities estimated from the injection tests were 10–15% too high and did not reflect actual subsurface conditions and, thus, should not be used directly in the design of BHE projects. In contrast, when there was no groundwater flow, Witte (2002) obtained similar conductivities from both types of test, similar to the results shown in Fig. 7. If there is significant groundwater flow, the subsurface conductivity cannot be correctly evaluated from the data of an actual thermal response test without further analysis. The movement of water needs to be incorporated into the interpretation of thermal response test data. Chiasson et al. (2000) showed that, as a first approximation, for the design of heating systems in areas where there is considerable groundwater movement, the values computed on the basis of the line-source model seem to be more reliable when the data were measured during short (50 h) tests than the data from longer tests. In any case, if a site has a non-uniform subsurface and significant groundwater flow, the thermal response tests will require appropriate field set-ups and data analyses in order to obtain reliable thermal conductivities before designing the heating or cooling (heat storage) projects.

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Fig. 9. Three-dimensional block showing the computed temperature field and the downstream thermal plume for Run MvD =1 , at the end of a simulated 200-h thermal response test run. The vertical BHE is at the center of the block. vD : Darcy flow velocity.

4. Evaluation of actual response test data 4.1. Data set Underground heat storage was integrated into the design of the SUVA building at Gisikon, Switzerland (Pahud, 2000), which entailed an evaluation of the thermal properties of the site subsurface. A thermal response test was carried out to determine in situ thermal conductivity using a 160-m deep BHE equipped with a double U-tube. Polyethylene spacers separated the four tubes and quartz sand cement was used as grouting material (thermal conductivity = 1.5 W m−1 K−1 , heat capacity = 2 MJ m−3 K−1 ). The characteristics of the BHE are summarized in Table 1. A borehole was drilled a short distance from the BHE for collecting soil samples. The geology at the site consists of unconsolidated sediments at the surface, overlying sandstones and conglomerates. The rocks, corresponding to the Upper Marine Molasse, have relatively high thermal conductivities. Laboratory measurements yielded a geometric average thermal conductivity over the 160-m BHE length of 3.6 W m−1 K−1 ; individual values vary between 2.9 and 4.75 W m−1 K−1 , with lower values found in the upper part of the borehole (Sch¨arli and Rybach, 1999); the average heat capacity is 2.3 MJ m−3 K−1 . Both boreholes were dry, suggesting little or no groundwater flow. Based on the results given in Section 3.5, one might expect the line-source approximation to provide a representative and reasonably reliable average thermal conductivity. The device used to monitor the thermal response was developed at Laboratoires de Syst`emes Energ´etiques at the Swiss Federal Institute of Technology in Lausanne (Laloui et al., 1998). The electric power of the device can be set to 3, 6, or 9 kW; its electric heater and the approximately 2-m long tubes connecting it to the BHE are insulated. The inlet and outlet fluid temperatures,

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Fig. 10. Data from the SUVA thermal response test.

air temperature inside and outside the device, fluid flow velocity and the electricity consumed by the heater and pump are measured every minute by a data logger and stored as 10 min averages. The fluid temperatures are measured with a precision of ±0.05 K, and are used to re-calculate the thermal power injected into the borehole based on the recorded flow rate, which is measured with a 1% precision. The fluid flow rate during the test was fairly small (Table 1), mainly because of the large pressure drop in the flowmeter and the relatively small (60 W) capacity pump. The measured temperature data (inlet, outlet, and air temperature) and the calculated heat injection rate are given in Fig. 10. The average ground temperature before starting the thermal response test was determined using a 17-h circulation test. Since the fluid flowing through the BHE initially did not have the same temperature as the ground, the test had to be run until the fluid temperature in the BHE had stabilized; the heat generated by the pump was assumed to be less than 55 W. Extrapolating the results from Bassetti (2003), the change in fluid temperature due to the heat generated by the pump should not be more than about 0.05 ◦ C. The average initial temperature of the ground was considered to be 12.4 ± 0.1 ◦ C. During the following 122-h thermal response test, an average of 6.2 kW of heat was injected. According to our earlier analysis (Fig. 4), the test should last about this length of time to produce good results. The heat injection rate varied within about 5% of the average (Fig. 10), the variations being attributed to changes in grid voltage. The measured fluid temperatures (Fig. 10) approached a nearly constant value at the end of the test, but these were perturbed by variations in the injection rate, which in turn affected the values of the estimated average ground thermal conductivities (see also Section 3.3). 4.2. Line-source model As shown in Fig. 4, the results of the line-source approach depend on the length of the data set used in the analysis, and are sensitive to small temperature disturbances (Witte et

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al., 2002). The data obtained from the thermal response test performed at the SUVA project were evaluated utilizing the usual methodology based on the line-source model (Pahud, 2000). Different measurement interval lengths were employed in the data evaluation, using a fixed starting time, t0 = 5 h (∼tc ). These lengths were increased in 10 min steps from tE = t0 up to 122 h. The estimated λLS values are given in Fig. 11, where the test durations correspond to the end time (tE ) of the evaluated data intervals. The estimated thermal conductivities, especially for short data intervals, are not stable (i.e. their values tend to oscillate). For evaluation intervals longer than 40 h, the instability decreases and the thermal conductivity approaches 3.0 W m−1 K−1 , a value that is about 20% smaller than the average measured in the laboratory (i.e. 3.6 W m−1 K−1 ). We will investigate whether this difference can be explained by the nonuniform heat injection rates that cannot be taken into account in the line-source model or by other effects. A closer inspection of the heat injection rate reveals variations on a daily scale that may be due to the unstable power supply. The line-source model assumes a constant rate of heat injection and cannot consider these variations. However, by assuming that the average power supply, and thus the average heat injection, are constant over an entire day, the effect of variable heat injection rates can be eliminated by evaluating 24-h data blocks (i.e. setting t0 = 5 h and tE = 29, 53, 77, and 101 h, respectively); the computed λLS values corresponding to these time blocks are shown in Fig. 11. As before, the four estimated λLS values decrease with time. The last day was a Sunday, when most factories and industries are closed. The average heat injection might therefore be slightly larger during the period t0 to tE 101 h because the grid voltage was higher then, resulting in the lowest value of all four computed λLS . However, ignoring this λLS , the average thermal conductivity estimate for the other three days (3.2 W m−1 K−1 ) is still about 15% lower than the laboratory measurements. Although the line-source model is highly sensitive to changes in heat injection rate, the variations in rate that were recorded cannot explain this difference in conductivity values.

Fig. 11. Calculated thermal conductivities [W m−1 K−1 ] based on the line-source model, λLS , and the SUVA data set. The first point of the evaluated data interval is set at t0 = 5 h; the end time varies.

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4.3. Numerical model The SUVA data set was analyzed using the FRACTure code, which can incorporate changes in heat injection rate in the parameter estimation procedure (see Section 2.2). The computation mesh used is described in Section 3.1, assuming a homogenous subsurface; the thermal properties of the grout and geometry of the BHE are given in Table 1, and the heat capacity of the subsurface as determined in the laboratory is given in Section 4.1. The relative position of the four tubes inside the borehole is the only other unknown parameter of the BHE system. Bassetti (2003) demonstrated that in thermal response tests the relative location of the tubes, which are separated by spacers, has only a minimal effect on the evaluation of the ground thermal conductivity. Several model runs were performed to identify the average thermal conductivity of the subsurface, which was then compared against the values based on the line-source model. All the numerical model runs began with the same steady-state temperature field, which was constrained by the ground surface temperature (10.5 ◦ C; Signorelli and Kohl, 2003) and by the average fluid temperature in the BHE (12.5 ◦ C), a value obtained from the circulation test performed prior to the thermal response test (Pahud, 2000). This average initial fluid temperature was assumed to correspond to that at half-borehole depth (i.e. at 80 m), so that a gradient of 2.5 ◦ C per 100 m was used. In model runs having different thermal conductivities, the heat flow at the bottom of the system was adjusted to achieve identical initial temperature fields. The simulation began with the 17-h circulation test so as to duplicate in the model the initial state of the SUVA data set. Afterwards, the thermal response test was modelled by varying the heat injection rate every 10 min based on the values measured during the field test. Simulations were performed using different average thermal conductivities (from 3.4 to 4.0 W m−1 K−1 ) in the model. The computed fluid temperature changes were compared against measured outlet temperatures. The thermal conductivity of the model showing the best fit with the field data was assumed to correspond to that of the subsurface. The results of the different model runs are given in Fig. 12; for clarity, this figure shows only the computed temperature histories when ground thermal conductivities of 3.5, 3.7, and 3.9 W m−1 K−1 are assumed in the model (λnum ), compared to field measurements; in reality, several other model runs were made. The figure illustrates how precisely the temperature variations can be modelled. The model run that best matches the field measurements is the one using λnum = 3.7 W m−1 K−1 , a value closer to the one obtained in the laboratory (i.e. 3.6 W m−1 K−1 ) than those based on the line-source model (see Section 4.2). 4.4. Discussion In the analysis of the SUVA test data, the most obvious reason for the roughly 15% difference between the average thermal conductivity evaluated by the line-source model and that computed by the numerical model is the unstable heat injection rate. However, a closer look at Fig. 12 reveals that, although the numerical simulations account for non-constant heat injection rates, timedependent effects can still be recognized, suggesting that other factors influence the results. The best-fit values every 10 min were therefore calculated and plotted as a smoothed curve (Fig. 13). These numerically computed best-fit thermal conductivities vary between 3.9 and 3.5 W m−1 K−1 and decrease with time. This trend includes shorter cycles where λnum seems to vary between locally lower and higher values on a daily scale; this cannot be explained by daily variations in power supply since the λnum evaluation considers the variation of heat injection rate (see Section 4.3).

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Fig. 12. SUVA test data analysis. Comparison of numerically calculated vs. measured fluid outlet temperatures for different assumed ground thermal conductivities (λnum ).

A plot of the difference (T) between air temperature and the average temperature of the inlet and outlet BHE fluids (mean temperature in Fig. 10) shows a clear (inverse) correlation with changes in λnum (Fig. 13). When the air temperature is higher than the mean fluid temperature (T > 0), heat seems to be added to the fluid. This increases the amount of heat injected into the subsurface, increasing the fluid temperature because of this higher rate, so that the analysis leads to lower thermal conductivity values. This effect is reversed when the air temperature is lower than that of the fluid (i.e. thermal energy losses yield higher average thermal conductivities). This seems to result from insufficient insulation of the test device and of the pipes connecting it to the BHE. Only if the air temperature is equal to that of the fluid (i.e. T = 0) will there be no external influence on the test. Figure 13 shows that the average value computed from the λnum values at these times is equal to the conductivity measured in the laboratory (i.e. 3.6 W m−1 K−1 ). The line-source model is a fast and easy method to evaluate thermal response test data, but estimates of the average ground thermal conductivity are sensitive to test conditions (mainly the stability of the heating power and the effects of weather conditions on heat injection rates). One advantage of the FRACTure code over the analytical line-source method is that various heat injection rates can be accurately simulated on any time scale. This allows a time-dependent evaluation of the temperatures where any data point can be fitted individually. Using the numerical code, we can also detect ambient temperature effects and achieve a more sophisticated data interpretation, as well as treating the borehole geometry and the surrounding geologic materials (horizontal and vertical heterogeneities) more accurately. Because the generation of the computational mesh needed for a FRACTure application is time consuming, this or other numerical codes might be more appropriate for large, complex BHE projects (Rohner et al., 2005), but the time and expense involved could be significantly reduced by using automatic mesh-generation software.

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Fig. 13. SUVA data set. Correlation between the calculated thermal conductivities based on the numerical model and the difference between the air temperature and the average temperature of the inlet and outlet BHE fluids (TAir–Fluid ).

5. Conclusions Thermal response models were studied using a 3-D finite-element numerical model. The temperatures given by the FRACTure code were evaluated by the analytical line-source model as if they were experimental data. This approach allowed us to estimate the accuracy of the general interpretation of the thermal response test data and to assess how long the tests would have to last to achieve a certain error level (i.e. 10%). The effects of borehole length, subsurface heterogeneity (when evaluating the test data using the line-source model), groundwater movement, and variable data quality on the line-source evaluation were discussed. The analysis based on the line-source model generally provides good results under ideal simulated test conditions, although the added effects of different errors can easily reach 10%. Our study on test duration clearly indicates that there is no straightforward definition on the clear starting or finishing point nor on how to estimate the length of the data interval to be evaluated. Early starting times are strongly influenced by the thermal response of the borehole heat exchanger itself, but temperature perturbations have a small impact on the calculated thermal conductivities. For late starting times, the opposite is true; perturbations can overwhelm the line-source evaluation. Under perfect simulated conductions, test durations of about 50 h generally provide good results. Since the subsurface is subjected to different thermal “stresses” during the operation of a BHE and during thermal response tests, i.e. cooling in the former case and heating during the latter, subsurface heterogeneity tends to affect the interpretation of the thermal response data. Although, for a stratified subsurface, analytically evaluated thermal conductivity is slightly different from that of a non-layered system, heterogeneity may become important if there is groundwater flow. We found that a heat extraction test likely provides more reliable data than an injection test when they are to be used in designing a BHE system for heating applications.

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Groundwater flow has a strong influence on the interpretation of thermal response test data when flow velocities are significantly higher than 0.1 m/day. In such cases, evaluations using the line-source model do not give stable values and the estimated thermal conductivities are questionable. Further investigations on this topic would help us to gain a better understanding of this effect. As part of this study, we analyzed the data from an actual thermal response test that was targeted at providing the data needed to design a BHE system for a building at Gisikon, Switzerland. The reported thermal conductivity values derived from the line-source model differ more than 20% from those measured in the laboratory, which leads one to question the reliability of the interpretation of the field test results. The FRACTure-based analysis explains this difference by providing a better understanding of the effects of disturbed data sets. The numerical code is more appropriate for a detailed evaluation of response tests since it can incorporate in the analysis borehole geometry and variations in heat injection rates, something the line-source model cannot do. Using FRACTure, we can resolve the thermal conductivity values on any time scale and thus identify time-dependent ambient temperature effects. The observed influence of air temperature on the results suggests that at Gisikon the test device and pipes connecting the device to the BHE might not have been properly insulated. Currently the device measures the BHE inlet and outlet fluid temperatures inside the device. Gehlin (2002) suggested fitting the sensors in the BHE tube below the subsurface to improve the reliability and usefulness of the thermal response test results. The possibility of identifying and incorporating into the analysis various effects on the collected data, such as time-dependent heat injection rates or the influence of air temperatures, should improve our interpretation of field data, but will require more time-consuming numerical modeling studies. We therefore recommend starting with a first-order estimation of the subsurface thermal conditions using the faster, but more conventional, line-source method. Later, and depending on the fluctuations in the estimated conductivities and their impact on the design of the heating (or cooling) system, one could consider re-evaluating the field results by means of numerical methods. A more reliable interpretation of the data sets should lead to a more accurate sizing of the BHE systems, which would in turn lead to a reduction in the overall cost of the project and its future operations. Acknowledgments This project was funded by “Projekt- und Studienfond der Elektrizit¨atswirtschaft” PSEL. Special thanks for their encouraging support go to E. Fischer and W. Rogg. The data on the SUVA response test were provided by the Swiss Federal Institute of Energy. Thanks are also extended to J.D. Spitler and R. Curtis for their useful reviews and suggestions. References Austin, W.A., 1998. Development of an in situ system for measuring ground thermal properties. M.S. Thesis. Oklahoma State University, Stillwater, OK, USA, 177 pp. Austin, W.A., Yavuzturk, C., Spitler, J.D., 2000. Development of an in situ system for measuring ground thermal properties. ASHRAE Trans. 106, 356–379. Bassetti, S., 2003. 3D Simulationen von geothermischen Response Tests. Diploma Thesis. Institute of Geophysics, Swiss Federal Institute of Technology, Z¨urich, Switzerland, 70 pp. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids, second ed. Oxford University Press, Oxford, UK, 510 pp.

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