TY - JOUR
AU - Clauser,, Christoph
AB - Abstract Thermal response tests (TRT) record the temperature variation of closed-loop shallow borehole heat exchangers (BHE) due to fluid circulation. The average change of fluid temperature is directly related to the rock thermal conductivity λ around the well. If environmental and experimental conditions satisfy the usual experimental standards, TRT can predict effective ground thermal conductivity within an error of approximately ±10%. This accuracy is generally accepted as sufficient for an appropriate prediction of the geothermal heat yield. However, the line source approach (on which the analysis of the TRT experiment is based) does not allow us to derive thermal capacity independently from thermal conductivity, and the soil thermal capacity ρc is usually assumed constant here. We calculate the response temperature of a synthetic TRT experiment as a reference for a subsequent joint estimation of rock thermal conductivity and thermal capacity. Within a reasonable computing time, a comprehensive parameter estimation is impossible, if coupled fluid flow and heat transport in the BHE tubes are explicitly simulated. Therefore, we substitute the BHE tube by a constant heat source with diffusive heat transport only. Although this simplification limits the application of the method to synthetic TRT data, we perform a systematic study of the method's accuracy to analyse thermal capacity with respect to data noise and test duration. Finding the minimum misfit with respect to the reference experiment, we obtain both thermal conductivity and thermal capacity, i.e. more information on ground thermal properties than the line source theory can provide. The effect of the additional information on the ground thermal capacity is demonstrated by a numerical simulation of a real TRT, where the fluid flow and the heat transport within the BHE tube are explicitly simulated. Nevertheless, thermal capacity is generally variable within ±20% for the same rock type. In our analysis, this uncertainty results in a variation of ±2% of the outlet temperature. thermal response test, geothermal energy, ground thermal properties, parameter estimation 1. Introduction The thermal power of a borehole heat exchanger (BHE) depends mainly on the thermal properties of the ground, in particular, on the thermal conductivity λ. In order to forecast the thermal power reliably and thus minimize borehole depth and the associated drilling cost, it is a common practice to perform thermal response tests (TRT) prior to installing large BHE systems or many shallow BHE (van Gelder 2001). A TRT experiment records the outlet temperature of a closed-loop shallow BHE due to constant heating or cooling by the fluid circulation. The change of outlet fluid temperature is directly related to the rock thermal conductivity around the well. The temperature time series is usually analysed using the line source theory (described briefly in section 2). The analysis is straightforward, but involves many assumptions. In particular, the TRT yields an ‘effective’ thermal conductivity due to the integration of the ground thermal properties along the entire length of the BHE, including near-borehole effects and convective heat transport by groundwater flow (Signorelli 2004). It is not possible to determine the rock thermal capacity ρc with the line source theory. Therefore, thermal capacity has often been neglected in the TRT analysis. Recently, progress has been made in analysing TRT data more precisely (Signorelli 2004) by means of numerical simulations based on finite-element codes (Kohl and Hopkirk 1995, Kohl et al2002). In principle, numerical simulation can overcome all restrictions of the line source approach. For instance, sensor noise, variation of surface temperature and BHE fluid circulation rate, borehole grout material and stratification of the ground can be considered. In contrast to the line source theory, numerical models require initial values of ground thermal properties to solve the forward problem (section 3). Therefore, unknown model parameters need to be estimated: the model with the optimum fit of measured and simulated outlet-temperature yields information on the realistic ground thermal properties, including the rock thermal capacity. The accurate simulation of coupled fluid flow and heat transport within small tubes requires a high-resolution grid and accordingly small time steps. Resulting computing times on a high-end workstation for simulating a single TRT easily exceed several hours. Therefore, a comprehensive parameter estimation cannot be performed in a reasonable time. To overcome this restriction, we do not simulate the flow and heat transport within the BHE tube simultaneously. Instead, we substitute the loop by a constant heat source with diffusive heat transport only (figure 1, diffusive model). Although this simplification limits the application of this method to synthetic TRT data, it permits us to perform a systematic study of the method's accuracy with respect to data noise and test duration for determining thermal capacity. The model parameters of this synthetic TRT are chosen with respect to a real TRT, performed by GroenHolland BV (Amsterdam) using a concentric tube. The GroenHolland test facility and its model parameters are briefly described in section 4. Figure 1. Open in new tabDownload slide Two different model approaches used in this study. Left: for parameter estimation, the BHE (3) is substituted by a homogeneous cylinder with constant heat extraction. Right: model for simulation of the coupled fluid flow and heat transport within the BHE tubes; (4) outer concentric tube; (5) inner concentric tube. (1) Ground surrounding the BHE; (2) backfilling grout; : temperature within the BHE due to heat extraction. Tin and Tout are the inlet and outlet temperature, respectively. Figure 1. Open in new tabDownload slide Two different model approaches used in this study. Left: for parameter estimation, the BHE (3) is substituted by a homogeneous cylinder with constant heat extraction. Right: model for simulation of the coupled fluid flow and heat transport within the BHE tubes; (4) outer concentric tube; (5) inner concentric tube. (1) Ground surrounding the BHE; (2) backfilling grout; : temperature within the BHE due to heat extraction. Tin and Tout are the inlet and outlet temperature, respectively. In the past, thermal capacity was often assumed constant in the TRT and BHE analysis. In section 6, we study the effect of an uncertainty in the ground thermal capacity of up to ±33% on the production temperature of a real TRT by keeping thermal capacity constant. For this purpose, the coupled flow and heat transport are simulated simultaneously within the concentric tube (convective model). In this paper, model geometry and model parameters for both diffusive and conductive simulations are adjusted with respect to the GroenHolland TRT parameters. 2. Line source theory The TRT analysis is generally based on Kelvin's line-source theory (Theis 1935, Carslaw and Jaeger 1959). Therefore, the closed-loop BHE is approximated by a line source in a homogeneous medium with constant thermal properties. The temperature variation around the line source in space and time can be described as (Gehlin 2002, Signorelli 2004) (1) where T0 is the initial undisturbed ground temperature, κ = λ/ρc the isobaric rock thermal diffusivity, Q the constant heat injection rate, z the length of the BHE and γ = 0.577… is Euler's constant. The error of the approximation in equation (1) is less than 2.5% for t > 20 r2/κ. For a line source of length z, the average BHE temperature caused by the specific radial heat flow q = Q/z through the BHE is (2) where RB is the thermal borehole resistance between the BHE fluid and the borehole wall. Using numerical simulations, (Signorelli 2004) shows that corresponds to the average of inlet and outlet temperatures of the BHE fluid: . Equation (2) specifies as a linear function of the logarithm of time t. Thus, the rock thermal conductivity can be derived from the slope a of this linear relation: (3) Considering random and systematic experimental errors and the approximations invoked by the analytical model, TRT can predict effective rock thermal conductivities within an error of approximately ±10% (Witte et al2002, Gehlin and Spitler 2002). This is commonly accepted as sufficient for an appropriate heat yield prediction. 3. Numerical simulation The accuracy of the TRT method is limited by (1) near-borehole effects due to the generally low grout thermal conductivity; (2) vertical heat flow variation due to the variations of the natural temperature gradient which is largest near to the surface (Signorelli 2004); (3) heat convection by the groundwater flow; and (4) duration of the test interval. The influence of these effects on the TRT accuracy is discussed in detail by Austin et al (2000), Gehlin (2002) and Signorelli (2004). As mentioned in section 1, these effects can be quantified if numerical simulations are used instead of the analytical line-source solution, equations (2) and (3). Generally, U-tubes or double U-tubes are used for TRT. This is quite challenging for numerical simulations because it requires a fine grid to map the BHE tubes accurately (Signorelli 2004). However, GroenHolland BV also used concentric tubes. We restrict our analysis to a cylindrical model, with the centre of the concentric tubes on the model axis. This means we can significantly reduce the model dimension, the grid complexity and accordingly the number of grid cells. However, simulation of the groundwater flow, which can have a significant effect on the BHE power (Wagner and Clauser 2002), cannot be considered using this model geometry. The equations for coupled flow and heat transport on a FD grid in a cylindrical coordinate system are (4) (5) where h is the hydraulic head, Ss the specific storage coefficient of the tubes and W is the fluid source/sink at the tube inlet/outlet. The hydraulic conductivity K of the tubes and the rock thermal diffusivity κ are constant scalars. The equations of coupled flow and heat transport are solved numerically for a real TRT in section 6. The thermal power P of a TRT or a BHE is related to the fluid heat capacity ρfcf, the difference between outlet and inlet temperatures, Tout - Tin, and the circulation rate Qf of the fluid: (6) where Qf = WΔV is the product of the fluid source/sink W in equation (4) and the volume ΔV of the injection cell of the model. For our comprehensive parameter estimation study of ground thermal properties, we approximate the BHE tube by a solid cylinder with constant heat extraction instead of solving the coupled flow and heat transport equations (4) and (5). Within this cylinder, heat extraction is calculated using a volume heat sink H corresponding to a heat flow through the cylindrical surface at r = rB equal to that of the BHE: (7) where VBHE is the total volume of the concentric BHE tube. The TRT experiment can then be simulated by solving the 2D heat conduction problem in cylindrical coordinates: (8) Using equation (8), we can perform comprehensive parameter estimation, because grid and time step size can significantly be reduced compared to the coupled problem, equations (4) and (5). Different model concepts are illustrated in figure 1. In this paper, we use the FD simulation code SHEMAT (Clauser 2003), which was customized to perform parameter estimation and to use load-time functions for time-dependent source/sink terms. However, any geothermal simulation code capable of simulating flow and heat transport on cylindrical FE or FD grids may be used for this kind of analysis as well. 4. GroenHolland TRT The main purpose of a TRT experimental apparatus is to provide a constant heat load to the earth. Groenholland developed a test method, based on the use of a heat pump. The heat load is applied by keeping the temperature difference ΔT = Tout - Tin constant between outlet and inlet. This heat flow is actively controlled. In contrast to common TRT, the GroenHolland test facility can also provide cooling in order to minimize groundwater convection around the well and thus provide a more reliable thermal conductivity. Heat injection or extraction is selected according to near-surface ground temperature. Minimizing the difference between the near-surface ground temperature and circulation fluid temperature helps us to reduce errors. A detailed description of the test facility is provided by Witte (2001) and Witte et al (2002). The two sensors which record the fluid temperature at the inlet and outlet are calibrated. Witte et al (2002) assume an instrumental error of ΔTin = ΔTout = ±0.05 K for each sensor, which added a total error of in quadrature yield. For the experiment described, the average flow rate and thus the heat extraction has a random measurement error of ±1.2% equivalent to ΔH = ±0.23 kW m-3. The data of the GroenHolland TRT experiment are summarized in table 1. Table 1. Properties and design of a thermal response test experiment performed by GroenHolland BV (Amsterdam). Property Unit Value Depth BHE z m  40 Borehole diameter rG m  0.5 Undisturbed ground temperature T0 °C  13.75 Average fluid circulation rate Qf m3 h-1  0.73 Circulation fluid heat capacity cf J kg-1 K-1  3876 Circulation fluid thermal W m-1 K-1  0.5016 conductivity λf Circulation fluid density ρf kg m-3  1026 Circulation fluid kinematic m2 s-1  2.01 × 10-6 viscosity νf Total experiment period t0 h  106.6667 Cooling power P W -959.61 Outer pipe diameter rB m  0.04 Outer pipe wall thickness m  0.0037 Inner pipe diameter di m  0.025 Inner pipe wall thickness m  0.0023 Pipe material thermal conductivity W m-1 K-1  0.42 Pipe material thermal capacity MJ m-3 K-1  2.19 Hydraulic conductivity K m s-1  168 Specific storage coefficient Ss m-1  10-10 Property Unit Value Depth BHE z m  40 Borehole diameter rG m  0.5 Undisturbed ground temperature T0 °C  13.75 Average fluid circulation rate Qf m3 h-1  0.73 Circulation fluid heat capacity cf J kg-1 K-1  3876 Circulation fluid thermal W m-1 K-1  0.5016 conductivity λf Circulation fluid density ρf kg m-3  1026 Circulation fluid kinematic m2 s-1  2.01 × 10-6 viscosity νf Total experiment period t0 h  106.6667 Cooling power P W -959.61 Outer pipe diameter rB m  0.04 Outer pipe wall thickness m  0.0037 Inner pipe diameter di m  0.025 Inner pipe wall thickness m  0.0023 Pipe material thermal conductivity W m-1 K-1  0.42 Pipe material thermal capacity MJ m-3 K-1  2.19 Hydraulic conductivity K m s-1  168 Specific storage coefficient Ss m-1  10-10 Open in new tab Table 1. Properties and design of a thermal response test experiment performed by GroenHolland BV (Amsterdam). Property Unit Value Depth BHE z m  40 Borehole diameter rG m  0.5 Undisturbed ground temperature T0 °C  13.75 Average fluid circulation rate Qf m3 h-1  0.73 Circulation fluid heat capacity cf J kg-1 K-1  3876 Circulation fluid thermal W m-1 K-1  0.5016 conductivity λf Circulation fluid density ρf kg m-3  1026 Circulation fluid kinematic m2 s-1  2.01 × 10-6 viscosity νf Total experiment period t0 h  106.6667 Cooling power P W -959.61 Outer pipe diameter rB m  0.04 Outer pipe wall thickness m  0.0037 Inner pipe diameter di m  0.025 Inner pipe wall thickness m  0.0023 Pipe material thermal conductivity W m-1 K-1  0.42 Pipe material thermal capacity MJ m-3 K-1  2.19 Hydraulic conductivity K m s-1  168 Specific storage coefficient Ss m-1  10-10 Property Unit Value Depth BHE z m  40 Borehole diameter rG m  0.5 Undisturbed ground temperature T0 °C  13.75 Average fluid circulation rate Qf m3 h-1  0.73 Circulation fluid heat capacity cf J kg-1 K-1  3876 Circulation fluid thermal W m-1 K-1  0.5016 conductivity λf Circulation fluid density ρf kg m-3  1026 Circulation fluid kinematic m2 s-1  2.01 × 10-6 viscosity νf Total experiment period t0 h  106.6667 Cooling power P W -959.61 Outer pipe diameter rB m  0.04 Outer pipe wall thickness m  0.0037 Inner pipe diameter di m  0.025 Inner pipe wall thickness m  0.0023 Pipe material thermal conductivity W m-1 K-1  0.42 Pipe material thermal capacity MJ m-3 K-1  2.19 Hydraulic conductivity K m s-1  168 Specific storage coefficient Ss m-1  10-10 Open in new tab Figure 2 shows the inlet and outlet temperatures over the entire period of 106 h. The sudden decrease of Tin and Tout after approximately 2 × 104 s is coherent with a decrease of the circulation flow rate Qf which may by due to a significant air temperature change. Unfortunately, air temperature was not monitored during this experiment. The difference between the inlet and outlet temperatures is kept constant during the entire period of 106 h, providing a constant cooling power. Figure 2. Open in new tabDownload slide Measured injection (Tin) and production (Tout) temperatures of the GroenHolland TRT (solid lines); long dashes: injection fluid circulation rate Qf; short dashes: linear regression of the average fluid temperature (equation (3)). Figure 2. Open in new tabDownload slide Measured injection (Tin) and production (Tout) temperatures of the GroenHolland TRT (solid lines); long dashes: injection fluid circulation rate Qf; short dashes: linear regression of the average fluid temperature (equation (3)). Applying the line source theory (equation (3)) to the average fluid temperature yields a thermal conductivity of approximately 1.96 W m-1 K-1. Therefore, we use a thermal conductivity of λ0 = 2.0 W m-1 K-1 as a reference value for the parameter estimation. 5. Parameter estimation The temperature within the BHE is simulated using a large number of models based on equation (8) with different values of ground thermal conductivity and ground thermal capacity. The model with the optimum fit between measured and simulated temperatures yields the most realistic ground thermal properties. The FD grid extends over a radial distance of 150 m and a depth of 100 m. The vertical resolution is kept constant at 10 m, while in the radial direction the grid size increases logarithmically by a factor of 1.2, starting from 0.02 m on the axis. The thermal properties of the four innermost nodes (r < 0.02 m) from z = 0 m to z = 40 m correspond to the values of the circulation fluid (table 1). The thermal conductivity of the reference model is set to the value derived from the line source theory, λ0 = 2.0 W m-1 K-1. The thermal capacity ρc0 of the reference model was set to 2.0 MJ m-3 K-1 due to lack of any other information. The initial temperature of the entire model on the model axis is set to 13.75 °C including the BHE (table 1). Temperatures at the top, bottom and lateral boundaries are fixed to the initial values. While the distance of the lateral and bottom boundary to the BHE is large enough to prevent any influence on the temperature at the BHE, the constant temperature at the model top is an approximation, which was accepted due to lack of any surface temperature monitoring data. The heat extraction rate within the cylinder which represents the BHE is set to H = P/V = -19.09 kW m-3 (table 1). Additionally, near-borehole effects are considered as the thermal properties of nodes between the outer BHE tube, and the borehole radius corresponds to the values of the grout material (λG = 0.8 W m-1 K-1, ρcG = 2.0 MJ m-3 K-1). The total time step number is n = 800 resulting in a time step size of Δt = 480 s. Figure 3 shows the temperature within the BHE over a total period of 106 h using the reference values λ0 and ρc0 for ground thermal properties. The temperature decrease deviates from the expected linear behaviour for very short times of less than 104 s. This effect is caused by the grout material, which has a far lower thermal conductivity than the surrounding ground. Additionally, the thermal conductivity λLS derived from the line source theory, equation (3), is shown for different analysis intervals t2 - t1, whereas t2 represents the total measurement time t0 (kept constant) and t1 corresponds to the time axis values t. If short times less than approximately 3 × 104 s are disregarded, λLS accounts for up to 95% of the expected value. Figure 3. Open in new tabDownload slide Average GHE temperature (solid line) of an artificial TRT based on the data from table 1. The dashed line shows the thermal conductivity λLQ derived from equation (3) for different values of t1 (corresponding to the time axis) but a constant value of t2 = 106 h. Figure 3. Open in new tabDownload slide Average GHE temperature (solid line) of an artificial TRT based on the data from table 1. The dashed line shows the thermal conductivity λLQ derived from equation (3) for different values of t1 (corresponding to the time axis) but a constant value of t2 = 106 h. The parameter estimation method was performed considering realistic data noise by load time functions. Therefore, a random error of 1.2% is added to the input heat source time series. The temperature sensor noise is considered by adding a random noise of the maximum amplitude of 0.07 K to the outlet temperature. Thermal conductivity and thermal capacity are systematically varied from 1.5 W m-1 K-1 ⩽ λi ⩽ 2.5 W m-1 K-1 and 1.0 MJ m-3 K-1 ⩽ ρcj ⩽ 3.0 MJ m-3 K-1 with increments of 0.05 W m-1 K-1 and 0.05 MJ m-3 K-1, respectively. In summary, the TRT experiment described above is simulated 800 times with different initial values for the ground thermal properties but with identical BHE and grout properties. In order to avoid more unknown ground properties, we disregarded ground porosity and thus pore fluid thermal conductivity and capacity. The squared differences ΔT2k of the BHE temperatures corresponding to the reference and modified conditions, (λ0, ρc0) and (λi, ρcj), respectively, are summed up for all n simulation time steps. Thus, the total misfit M between the reference model and simulation can be described as (9) Figure 4 shows the temperature misfit M over the inverse thermal diffusivity κ. Each solid line corresponds to a different ground thermal conductivity λi. There is no minimum misfit for λi < 1.85 W m-1 K-1 and λi > 2.25 W m-1 K-1. The absolute minimum corresponds to the real thermal conductivity of 2.0 W m-1 K-1. Figure 4. Open in new tabDownload slide Misfit M (equation (9)) with respect to the reference model (λ0, ρc0) versus 1/κ = ρc/λ. The curve parameters denote the rock thermal conductivity in W m-1 K-1. The data were analysed over a time interval of 106 h (=800 sampling values). Figure 4. Open in new tabDownload slide Misfit M (equation (9)) with respect to the reference model (λ0, ρc0) versus 1/κ = ρc/λ. The curve parameters denote the rock thermal conductivity in W m-1 K-1. The data were analysed over a time interval of 106 h (=800 sampling values). Thermal conductivity and inverse thermal diffusivity are within 1.90 W m-1 K-1 < λ < 2.10 W m-1 K-1 and 1.35 × 106 m-2 s <1/κ < 0.75 × 106 m-2 s, respectively. This yields 2.57 MJ m-3 K-1 > ρc > 1.58 MJ m-3 K-1 for thermal capacity. Therefore, the mean thermal capacity derived from figure 4 is ρc = 2.08 MJ m-3 K-1, which corresponds almost exactly to the expected value of ρc = 2.0 MJ m-3 K-1. For a TRT, it is most desirable to reduce the total testing time (here: t0 = 106 h) in order to minimize costs. Figure 5 shows the same experiment, but analysed for a period of t0 = 53 h. Obviously, the minimum misfit can be determined as well as in figure 4. However, finding a minimum is almost impossible for a testing period of t0 = 26 h (figure 6). Figure 5. Open in new tabDownload slide Misfit M (equation (9)) with respect to the reference model (λ0, ρc0) versus 1/κ = ρc/λ. The curve parameters denote the rock thermal conductivity in W m-1 K-1. The data were analysed over a time interval of 53 h (=400 sampling values). Figure 5. Open in new tabDownload slide Misfit M (equation (9)) with respect to the reference model (λ0, ρc0) versus 1/κ = ρc/λ. The curve parameters denote the rock thermal conductivity in W m-1 K-1. The data were analysed over a time interval of 53 h (=400 sampling values). Figure 6. Open in new tabDownload slide Misfit M (equation (9)) with respect to the reference model (λ0, ρc0) versus 1/κ = ρc/λ. The curve parameters denote the rock thermal conductivity in W m-1 K-1. The data were analysed over a time interval of 26 h (=200 sampling values). Figure 6. Open in new tabDownload slide Misfit M (equation (9)) with respect to the reference model (λ0, ρc0) versus 1/κ = ρc/λ. The curve parameters denote the rock thermal conductivity in W m-1 K-1. The data were analysed over a time interval of 26 h (=200 sampling values). In conclusion, thermal conductivity and thermal capacity can be well determined and with good precision by the parameter estimation. The great advantage of this method is that it permits us to allow all kinds of environmental influences. For example, if the ground temperature is monitored in the vicinity of the well near the surface, which was done by GroenHolland in another experiment (Witte et al2002), numerical simulation of TRT can improve the quality of the results. Additionally, the sensitivity of the method can be analysed with respect to different time periods. In this case, the real ground thermal properties can be determined even for a total measurement period of only half the original duration. Similar studies were performed by Shonder and Beck (1999) and Austin et al (2000), but focused mainly on grout and ground thermal conductivity. Although rock thermal capacity was not considered in parameter estimations, Austin et al (2000) forecast an uncertainty of up to 6.3% in the calculation of ground thermal conductivity, if the rock thermal capacity is changed by ±25%. 6. Ramification for BHE power We performed a similar analysis, but considering simulating explicitly coupled fluid flow and heat transport within the BHE tube. Load-time functions define the transient temperature Tin and circulation rate Qf at the tube inlet (corresponding to Tin and Qf in figure 2). The outlet temperature Tout is simulated with respect to the initial values of ground thermal properties. A random temperature error with a maximum amplitude of 0.07 K is added to the outlet temperature after the simulation. Due to the large computing time, we strongly reduced the parameter space and analysed only the outlet temperature for a small number of different ground thermal capacities. The simulated outlet temperature is compared with that in the GroenHolland experiment. In contrast to the analysis in section 5, a refined FD grid was used in order to accurately map the concentric tube geometry. The outer and inner tube walls consist of three FD columns, the pipes of five columns, respectively. This results in a minimum grid size of the pipe walls and the pipes of less than 1 mm and 2.5 mm, respectively. The vertical grid size grows logarithmically by a factor of 1.2 from 2.5 mm at the surface to a maximum value of 2.5 m. The grid size decreases again by the same factor towards the bottom of the BHE in order to obtain a high resolution where the fluid flows from the outer into the inner pipe. The grid size is increased again from the BHE bottom towards the lower boundary of the FD model at z = 100 m. Flow in the concentric tube is assumed to be linear and the flow velocity is calculated from Darcy's law. The Reynolds number Re for the flow regime in the inner pipe of the GroenHolland TRT is (Signorelli 2004) (10) which corresponds to fully developed turbulent flow. In reality, the flow velocity vz vanishes at the wall surface. Therefore, heat transfer from the ground to the circulation fluid is correctly calculated by adding a heat transfer coefficient to equation (5) (Kohl et al2002). Once a turbulent flow regime is established, the power production and thus the BHE production temperature becomes almost steady state, which means that the heat-transfer coefficient between the outer tube wall and BHE fluid can be neglected (Signorelli 2004). Under these conditions, we assume that the Darcy flow is appropriate for the simulation of the TRT response temperature. Prior to the BHE simulation, a stability analysis is performed in order to determine an appropriate time step size Δt. Accordingly, Δt is increased over seven decades from 3.84 × 10-4 s to 3.84 × 103 s. No temperature difference was observed for time step sizes up to 0.384 s. Therefore, Δt is set to 0.384 s, corresponding to a total of one million time steps. The model parameters (temperature, fluid flow rate Qf and injection temperature Tin) correspond to the transient values of the GroenHolland TRT. As in the previous parameter estimation, temperatures are kept constant at the top, bottom and lateral boundaries. The sum of squared residuals with respect to the GroenHolland data is shown in table 2. The bold letters label the sensor noise sensitivity. A thermal conductivity of 2.0 W m-1 K-1 yields the best agreement between simulated and measured temperatures which corresponds to the result from the line source theory (section 4). In contrast to the synthetical data in the previous section, no clear minimum misfit can be found with respect to thermal capacity. The analysis suggests that the thermal capacity of the ground is less than 1.56 MJ m-3 K-1 which is close to the lower limit of commonly occurring soil types (Austin et al2000). Table 2. Total difference in K2 between simulated and measured TRT production temperature, calculated from 1000 time monitoring points ([λ] = W m-1 K-1; [ρc] = MJ m-3 K-1). Bold letters are below the measurement error M of 0.072K21000 = 4.9K2. λ ρc 1.00 1.50 1.75 2.00 2.25 2.50 3.00 1.34 188.2 44.3 14.1  4.7 10.8 30.7  96.7 1.56 175.9 36.4 10.2  4.6 14.6 38.1 111.0 1.78 167.6 32.8  8.5  5.4 17.5 42.2 118.0 2.00 159.2 27.4  7.3  6.3 21.7 48.1 129.0 2.22 148.4 24.4  6.4  8.0 25.9 53.4 138.7 2.44 143.2 21.7  5.3  9.4 27.9 59.6 145.1 2.66 136.5 19.8  5.0 10.9 32.6 65.0 155.1 λ ρc 1.00 1.50 1.75 2.00 2.25 2.50 3.00 1.34 188.2 44.3 14.1  4.7 10.8 30.7  96.7 1.56 175.9 36.4 10.2  4.6 14.6 38.1 111.0 1.78 167.6 32.8  8.5  5.4 17.5 42.2 118.0 2.00 159.2 27.4  7.3  6.3 21.7 48.1 129.0 2.22 148.4 24.4  6.4  8.0 25.9 53.4 138.7 2.44 143.2 21.7  5.3  9.4 27.9 59.6 145.1 2.66 136.5 19.8  5.0 10.9 32.6 65.0 155.1 Open in new tab Table 2. Total difference in K2 between simulated and measured TRT production temperature, calculated from 1000 time monitoring points ([λ] = W m-1 K-1; [ρc] = MJ m-3 K-1). Bold letters are below the measurement error M of 0.072K21000 = 4.9K2. λ ρc 1.00 1.50 1.75 2.00 2.25 2.50 3.00 1.34 188.2 44.3 14.1  4.7 10.8 30.7  96.7 1.56 175.9 36.4 10.2  4.6 14.6 38.1 111.0 1.78 167.6 32.8  8.5  5.4 17.5 42.2 118.0 2.00 159.2 27.4  7.3  6.3 21.7 48.1 129.0 2.22 148.4 24.4  6.4  8.0 25.9 53.4 138.7 2.44 143.2 21.7  5.3  9.4 27.9 59.6 145.1 2.66 136.5 19.8  5.0 10.9 32.6 65.0 155.1 λ ρc 1.00 1.50 1.75 2.00 2.25 2.50 3.00 1.34 188.2 44.3 14.1  4.7 10.8 30.7  96.7 1.56 175.9 36.4 10.2  4.6 14.6 38.1 111.0 1.78 167.6 32.8  8.5  5.4 17.5 42.2 118.0 2.00 159.2 27.4  7.3  6.3 21.7 48.1 129.0 2.22 148.4 24.4  6.4  8.0 25.9 53.4 138.7 2.44 143.2 21.7  5.3  9.4 27.9 59.6 145.1 2.66 136.5 19.8  5.0 10.9 32.6 65.0 155.1 Open in new tab We suggest that the low resolution of this analysis with respect to the rock thermal capacity is caused by the low outlet temperature resolution of ±0.1 K of the data provided to us. Unfortunately, the data were already processed and the original, high accuracy time series was unavailable. The accuracy of only one digit is visible in figure 7 by the ±0.1 K fluctuations in inlet and outlet temperatures. Figure 7 also shows a small misfit between the measured and simulated outlet temperatures for λ = 2.0 W m-1 K-1 and ρc = 2.0 MJ m-3 K-1. This example demonstrates that the outlet temperature can be precisely simulated considering all known temporal variations and fluctuations of model parameters. Surprisingly, the convincing fit between the measured and simulated outlet temperatures is achieved disregarding temperature variations at the top boundary, although Signorelli (2004) finds a significant effect of surface temperature variation on the TRT results. Figure 7. Open in new tabDownload slide Difference between measured inlet (Tin) and outlet (Tout) temperatures (solid lines) and simulated outlet temperature (dashed line). Long dashes: injection fluid circulation rate Qf. The simulation is performed using λ = 2.0 W m-1 K-1 and ρc = 2.0 MJ m-3 K-1. Time-dependent fluid flow within the BHE tubes is simulated using Tin and Qf at the BHE inlet. Figure 7. Open in new tabDownload slide Difference between measured inlet (Tin) and outlet (Tout) temperatures (solid lines) and simulated outlet temperature (dashed line). Long dashes: injection fluid circulation rate Qf. The simulation is performed using λ = 2.0 W m-1 K-1 and ρc = 2.0 MJ m-3 K-1. Time-dependent fluid flow within the BHE tubes is simulated using Tin and Qf at the BHE inlet. In order to study the magnitude of the effect of the unknown rock thermal capacity ρc on the outlet temperature of this TRT, we compared the simulated outlet temperature for ground thermal capacities ρc varying from 1.33 MJ m-3 K-1– 2.66 MJ m-3 K-1 with respect to a model with ρc = 2.00 MJ m-3 K-1. The ground thermal conductivity λ = 2.0 W m-1 K-1 was kept constant. Different time series in figure 8 show the temperature differences with respect to the reference simulation without considering temperature sensor noise. Figure 8. Open in new tabDownload slide Influence of varying rock thermal capacity on outlet temperatures. ΔT is the total difference of simulated outlet temperatures for different rock thermal capacities with respect to a reference model with ρc = 2.0 MJ m-3 K-1. The curve parameters ΔTi, i = 1–6 correspond to ρc1 = 1.33 MJ m-3 K-1, ρc2 = 1.56 MJ m-3 K-1, ρc3 = 1.78 MJ m-3 K-1, ρc4 = 2.22 MJ m-3 K-1, ρc5 = 2.44 MJ m-3 K-1 and ρc6 = 2.66 MJ m-3 K-1. The rock thermal conductivity is set to λ = 2.0 W m-1 K-1. Figure 8. Open in new tabDownload slide Influence of varying rock thermal capacity on outlet temperatures. ΔT is the total difference of simulated outlet temperatures for different rock thermal capacities with respect to a reference model with ρc = 2.0 MJ m-3 K-1. The curve parameters ΔTi, i = 1–6 correspond to ρc1 = 1.33 MJ m-3 K-1, ρc2 = 1.56 MJ m-3 K-1, ρc3 = 1.78 MJ m-3 K-1, ρc4 = 2.22 MJ m-3 K-1, ρc5 = 2.44 MJ m-3 K-1 and ρc6 = 2.66 MJ m-3 K-1. The rock thermal conductivity is set to λ = 2.0 W m-1 K-1. The simulated average temperature difference with respect to the reference model varies from -0.048 K (ρc = 1.33 MJ m-3 K-1) to 0.036 K (ρc = 2.66 MJ m-3 K-1), which is small (≈4%) compared to the average measured temperature difference between the inlet and outlet of 1.188 K of the reference model. Because the temperature difference is related linearly to the thermal power of the TRT, changing the ground thermal capacity by ±33% (±20%) yields a change in the thermal power of approximately ±4% (±2%). However, the temperature differences are smaller than the measurement error of 0.07 K. This means that an uncertainty of ±33% in the thermal capacity cannot be resolved with this TRT. 7. Summary and conclusion This paper presents for the first time an approach to obtain both the rock thermal conductivity and rock thermal capacity from synthetical TRT data considering realistic data noise. The parameter estimation is applied to the TRT data by varying λ and ρc of the rock surrounding the BHE. The analysis was performed substituting the concentric BHE tube with a cylinder with constant heat extraction (diffusive model) as an alternative to simulating coupled fluid flow and heat transport within the BHE tubes (convective model), which consumes too much computing time. This way, we performed a systematic analysis of the accuracy of the method for thermal capacity determination with respect to data noise and test duration. The model with an optimum fit of measured and simulated data provides information about realistic ground thermal properties. The sensitivity of the parameter estimation method is sufficient to obtain the ground thermal properties of a synthetical model even if heat extraction power noise is considered. Additionally, the total required measurement period was analysed. It was found that the unknown ground thermal properties can be determined even if the original measurement period is reduced by half. To date, geothermal energy forecasts of BHE are generally based on the rock thermal conductivity only. Besides heat advection by groundwater flow, thermal conductivity is the most important ground parameter controlling the thermal power of a BHE. Therefore, TRT design and analysis is often restricted to thermal conductivity alone. First attempts to improve the line source analysis were recently made using heat transport simulation codes. In general, the ground thermal capacity ρc is disregarded in the TRT analysis, because ρc varies within 20% of 2.3 MJ m-3 K-1 within many rock types because of self-compensating factors (Beck 1988, Mottaghy et al2005). Nevertheless, if this average variation of ±20% in the ground thermal capacity is incorporated in the TRT analysis, the geothermal energy yield may change by ±2%. First attempts to improve the efficiency of a FE heat transport code have been undertaken recently by Al-Khoury et al (2004). In this work, two distinct model elements are combined in order to reduce the total number of FE elements to a great extent: a 3D soil element capable of simulating coupled heat and groundwater flow, and a line (1D) heat pipe element capable of simulating a pseudo 3D heat flow in a typical vertical BHE consisting of pipe-in, pipe-out and grout. In principle, the parameter estimation, based on this new heat transport code, can overcome all restrictions of the line source approach and the diffusive model of this parameter estimation. Then, the parameter estimation technique would also provide a precise prediction of the vertical variation of ground thermal properties and ground water flow in the vicinity of a BHE. Acknowledgments Henk Witte (GroenHolland BV, Amsterdam) is gratefully acknowledged for the TRT data and discussion. Jörn Bartels (Neubrandenburg) and Thomas Kohl (Zürich) are gratefully acknowledged for constructive and helpful comments. References Al-Khoury R , Leyens D , Kölbel T . , 2004 Efficient 3d finite-element analysis for geothermal heating and cooling systems Int. 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TI - Evaluating thermal response tests using parameter estimation for thermal conductivity and thermal capacity
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/2/4/S08
DA - 2005-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/evaluating-thermal-response-tests-using-parameter-estimation-for-gC20POX2Y0
SP - 349
VL - 2
IS - 4
DP - DeepDyve
ER -