Method of Averaging the E ﬀ ective Thermal Conductivity Based on Thermal Response Tests of Borehole Heat Exchangers

: This work concerns borehole heat exchangers and their testing using apparatus for thermal response tests. In the theoretical part of the article, an equation was derived from the known equation of heat ﬂow, on which the interpretation of the thermal response test was based. The practical part presents the results of several measurements taken in the AGH Laboratory of Geoenergetics. They were aimed at examining the potential heat exchange capacity between the heat carrier and rock mass. Measurement results in the form of graphs are shown in relation to the examined, brieﬂy described wells. Result analysis made it possible to draw conclusions regarding the interpretation of the thermal response test. The method of averaging the measurement results was subjected to further study. The measuring apparatus recorded data at a frequency of one second, however such accuracy was too large to be analyzed e ﬃ ciently. Therefore, an average of every 1 min, every 10 min, and every 60 min was proposed. The conclusions stemming from the di ﬀ erences in the values of e ﬀ ective thermal conductivity in the borehole heat exchanger, resulting from di ﬀ erent data averaging, were described. In the case of three borehole heat exchangers, ground properties were identical. The e ﬀ ective thermal conductivity λ e ﬀ was shown to depend on various borehole heat exchanger (BHE) designs, heat carrier ﬂow geometry, and grout parameters. It is important to consider the position of the pipes relative to each other. As shown in the charts, the best (the highest) e ﬀ ective thermal conductivity λ e ﬀ occurred in BHE-1 with a coaxial construction. At the same time, this value was closest to the theoretical value of thermal conductivity of rocks λ , determined on the basis of literature. The standard deviation and the coe ﬃ cient of variation conﬁrmed that the e ﬀ ective thermal conductivity λ e ﬀ , calculated for di ﬀ erent time intervals, showed little variation in value. The values of e ﬀ ective thermal conductivity λ e ﬀ for each time interval for the same borehole exchanger were similar in value. The lowest values of e ﬀ ective thermal conductivity λ e ﬀ most often appeared for analysis with averaging every 60 min, and the highest—for analysis with averaging every 1 min. For safety reasons, when designing (number of BHEs), safer values should be taken for analysis, i.e., lower, averaging every 60 min.


Introduction
In the era of depleting conventional energy sources and increasingly pro-ecological policies, e.g., those of the European Union, it became natural to focus the attention on alternative energy sources.    TRT was developed mostly in Sweden [16]. A TRT is an indirect (in-situ) measurement method, which is the simplest and most exact way to determine precise thermal properties [17]. Thermal response tests were first suggested at an international conference in Stockholm [18]. A simple system was suggested, in which thermal energy at constant heating power was injected into a BHE while the BHE temperature was measured. Thermal response tests with mobile measurement equipment were first introduced in Sweden and USA in 1995. Since then, the method has developed and spread to several other countries in North America and Europe [17].
Until now, several studies regarding TRT have been conducted, e.g., [19][20][21][22][23]. Multiple modifications have been proposed, e.g., [24][25][26][27][28]. Traditional TRT applies a constant heating power to the pumped heat carrier [29]. The newer method, which holds the outflow heat carrier at a constant temperature, was shown to have many benefits, including shortening the test time and improving the operating constance Effective thermal conductivity λ eff in BHE is not the same value like weighted average thermal conductivity of geological layers λ. Difference between λ eff and λ is due to method imperfections. Basic TRT operational parameters are heating power and heat carrier flow rate. That values determine effective thermal conductivity λ eff in BHE.
Different values of λ eff are also observed in various BHE constructions when the geological condition is the same. Not only borehole thermal resistance R b changes with various BHE constructions. One of the most important reasons are heat exchange area and velocity of heat carrier.
In the paper, only values of λ eff in the function of measurement frequency are analyzed. Additional results are the comparison of values of λ eff in different BHEs of Geoenergetics Laboratory.
The paper includes mathematical background of TRT. The described method is a classical/traditional method of effective thermal conductivity λ eff in the BHE calculation. Next is the description of three different BHEs of Laboratory of Geoenergetics located at the Department of Drilling, Oil and Gas AGH UST. Next paragraph shows the results of TRT made on the BHEs. Interpretation of the results by different temperature measurement frequency during the TRT is described.
Many different commercial TRTs results are shown. On the basis of TRT data, the effect of frequency averaging of TRT results was determined. The paper finishes with conclusions and references.

Thermal Response Test
Thermal response test is based on Fourier's generally known differential heat conduction equation, which in cylindrical coordinates is as follows [37]: where: ρ-density of the medium, kg/m 3 , c p -specific heat of the ground, J/(kgK), λ-thermal conductivity of the ground, W/(mK), T-temperature, K, Using the method proposed for the Theis equation in hydrogeology [38] to solve this type of differential equation, the substitution was used: and After appropriate transformations, Equation (1) takes the form of: After a series of transformations, a mathematical relationship is obtained: where: q-unit heat transfer coefficient of the BHE, W/m, T 0 -initial temperature in the BHE, K.
Formula 6 is the basic equation on which the interpretation of the thermal response tests of the borehole heat exchangers is based. For a long time period further solution is possible. In the evaluations made of German TRTs, the minimum duration criterion as noted by [39] proved helpful: with t min -lower time limit of data to be used for analysis and r b -borehole radius.
Considering the integral of the exponential function further: it can be represented using the additive property of the integral relative to the integration interval and the integral function as follows: By replacing the values of the integral with its approximate value, a relation is eventually obtained: Energies 2020, 13, 3737 6 of 20 Test results show the relation between the heat carrier temperature and time. This allows for calculation of the effective thermal conductivity λ eff according to the formula [29]: (11) where: P-mean heating power during the TRT, W, H-depth of the borehole heat exchanger, m, k-the slope of the trend line-a straight line determined as the ratio of the mean temperature to the logarithm of the natural test time.
According to the mathematical model, effective thermal conductivity is identical to the thermal conductivity of the medium (rock mass). In reality, this value also depends on: • the design of the borehole heat exchanger, • TRT parameters, and • how the measurement results are averaged, as shown in this article. Its aim is to show the diversity of the final measurement results, which can be regarded as a manifestation of their uncertainty.

The Scope of the Conducted Research
Laboratory of Geoenergetics located at the Department of Drilling, Oil and Gas AGH UST contains borehole heat exchangers of different designs [4]. Three of them were subjected to the described research, short characteristics of which are given in Table 1. The location of aforementioned BHEs is shown in Figure 4. Similar studies for various BHE designs have already been described in literature [1,12,26]. • how the measurement results are averaged, as shown in this article. Its aim is to show the diversity of the final measurement results, which can be regarded as a manifestation of their uncertainty.

The Scope of the Conducted Research
Laboratory of Geoenergetics located at the Department of Drilling, Oil and Gas AGH UST contains borehole heat exchangers of different designs [4]. Three of them were subjected to the described research, short characteristics of which are given in Table 1. The location of aforementioned BHEs is shown in Figure 4. Similar studies for various BHE designs have already been described in literature [1,12,26,]. The study consisted of determining the effective thermal conductivity λeff in the borehole, which depended primarily on the thermal conductivity of the rocks in the lithological profile. Drilled lithological profile is presented in Table 2 [40]. • how the measurement results are averaged, as shown in this article. Its aim is to show the diversity of the final measurement results, which can be regarded as a manifestation of their uncertainty.

The Scope of the Conducted Research
Laboratory of Geoenergetics located at the Department of Drilling, Oil and Gas AGH UST contains borehole heat exchangers of different designs [4]. Three of them were subjected to the described research, short characteristics of which are given in Table 1. The location of aforementioned BHEs is shown in Figure 4. Similar studies for various BHE designs have already been described in literature [1,12,26,]. The study consisted of determining the effective thermal conductivity λeff in the borehole, which depended primarily on the thermal conductivity of the rocks in the lithological profile. Drilled lithological profile is presented in • how the measurement results are averaged, as shown in this article. Its aim is to show the diversity of the final measurement results, which can be regarded as a manifestation of their uncertainty.

The Scope of the Conducted Research
Laboratory of Geoenergetics located at the Department of Drilling, Oil and Gas AGH UST contains borehole heat exchangers of different designs [4]. Three of them were subjected to the described research, short characteristics of which are given in Table 1. The location of aforementioned BHEs is shown in Figure 4. Similar studies for various BHE designs have already been described in literature [1,12,26,]. The study consisted of determining the effective thermal conductivity λeff in the borehole, which depended primarily on the thermal conductivity of the rocks in the lithological profile. Drilled lithological profile is presented in Table 2 [40].
The study consisted of determining the effective thermal conductivity λ eff in the borehole, which depended primarily on the thermal conductivity of the rocks in the lithological profile. Drilled lithological profile is presented in Table 2 [40]. The study consisted of determining the effective thermal conductivity λeff in the borehole, which depended primarily on the thermal conductivity of the rocks in the lithological profile. Drilled lithological profile is presented in Table 2 [40].  Values of the thermal conductivity of rocks were determined based on literature data [12,41,42]. The measuring apparatus ( Figure 3) consisted of two modules together with a computer controlling and recording data ( Figure 5). It corresponds to the schematic of thermal response test devices and operation from Figure 1. The first module included electric heaters and a circulation pump, the second consisted of flow meters and thermometers. The computer program recorded all data necessary to calculate subsequent parameters. These data included the exact date and time from the start of the measurement, the temperature of the heating agent at the inflow and outflow of the borehole heat exchanger, the outside temperature, the difference in pressure at the inflow and outflow of the borehole heat exchanger, and the accumulated electricity. Values of the thermal conductivity of rocks were determined based on literature data [12,41,42]. The measuring apparatus ( Figure 3) consisted of two modules together with a computer controlling and recording data ( Figure 5). It corresponds to the schematic of thermal response test devices and operation from Figure 1. The first module included electric heaters and a circulation pump, the second consisted of flow meters and thermometers. The computer program recorded all data necessary to calculate subsequent parameters. These data included the exact date and time from the start of the measurement, the temperature of the heating agent at the inflow and outflow of the borehole heat exchanger, the outside temperature, the difference in pressure at the inflow and outflow of the borehole heat exchanger, and the accumulated electricity. Measurement results were recorded every second with time stabilization provided by the computer processor. The precision of the measurement time recording was high and equaled 0.2 milliseconds. Temperature measurements were performed with thermometers with an accuracy class of 1%. Flow meters had an average accuracy of 5%, up to 2% in an optimal operating range.
Data from TRT were obtained by summing all the records from every second and dividing by their number every 1 min, 10 min, and 1 h, which is a classic arithmetic mean. It was originally assumed that the median was an average value, but such averaging was more time consuming. Due to the very small differences between the median and the arithmetic mean, and a significantly shorter time of determining the latter estimator, it was decided to calculate the arithmetic mean. Data recording on the disc was carried out with precision to two decimal places. Measurement results were recorded every second with time stabilization provided by the computer processor. The precision of the measurement time recording was high and equaled 0.2 milliseconds. Temperature measurements were performed with thermometers with an accuracy class of 1%. Flow meters had an average accuracy of 5%, up to 2% in an optimal operating range.  Data from TRT were obtained by summing all the records from every second and dividing by their number every 1 min, 10 min, and 1 h, which is a classic arithmetic mean. It was originally assumed that the median was an average value, but such averaging was more time consuming. Due to the very small differences between the median and the arithmetic mean, and a significantly shorter time of determining the latter estimator, it was decided to calculate the arithmetic mean. Data recording on the disc was carried out with precision to two decimal places.

Measurement Results
Presented results include measurements for three boreholes of the Laboratory of Geoenergetics (number 1, 3, and 5). Each time, the test was preceded by a 24-h rinsing period without heating the heat carrier in order to effectively vent the system and equalize the temperature in the rock mass. In the next stage, heaters were started with a set power and worked for 100 h. This time ensured the stabilization of the effective thermal conductivity value, which is a basic parameter in the BHE measurements. On this basis, the ability of the rock mass to dissipate/store heat could be determined. The assumed heating power with which the heaters heated the working fluid was 2.8 kW. The heat carrier flow capacity was 20 dm 3 /min.
Effective thermal conductivity was calculated from Formula 10. The slope factor of the straight line k was determined with Microsoft Excel using the REGLINP function. It uses the least squares method.
The distribution of the average temperature in the graphs according to the time intervals of data averaging (every 1 min, 10 min, and 1 h) are shown in Figures 6-11. Figure 6 shows the measurement results for the borehole No. 1, with the measurement data averaging every 1 h. Figure 7 shows the same relationship but in a semi-logarithmic plot. Figure 8 shows the measurement results for borehole No. 3, with the measurement data averaging every 10 min. Figure 9 shows the same relationship but in a semi-logarithmic plot. Figure 10 shows the measurement results for borehole No. 5 with the measurement data averaging every 1 min. Figure 11 shows the same relationship but in a semi-logarithmic plot.
The curves are so wobbly (identically wobbly) because of no ideal thermal insulation. As shown in Figure 5, thermal insulation of pipes is used. Figure 2 (thermogram) shows the need for even better thermal insulation.
Every graph (Figures 6-11) was based on the same TRT. The precision of the temperature time recording with the use was the current computer science standard, equal to 0.2 milliseconds. Different results were based only on different averaging intervals.
Energies 2020, 13, x FOR PEER REVIEW 9 of 20 Effective thermal conductivity was calculated from Formula 10. The slope factor of the straight line k was determined with Microsoft Excel using the REGLINP function. It uses the least squares method.
The distribution of the average temperature in the graphs according to the time intervals of data averaging (every 1 min, 10 min, and 1 h) are shown in Figures 6-11. Figure 6 shows the measurement results for the borehole No. 1, with the measurement data averaging every 1 h. Figure 7 shows the same relationship but in a semi-logarithmic plot. Figure 8 shows the measurement results for borehole No. 3, with the measurement data averaging every 10 min. Figure 9 shows the same relationship but in a semilogarithmic plot. Figure 10 shows the measurement results for borehole No. 5 with the measurement data averaging every 1 min. Figure 11 shows the same relationship but in a semi-logarithmic plot.            The curves are so wobbly (identically wobbly) because of no ideal thermal insulation. As shown in Figure 5, thermal insulation of pipes is used. Figure 2 (thermogram) shows the need for even better thermal insulation.
Every graph (Figures 6-11) was based on the same TRT. The precision of the temperature time recording with the use was the current computer science standard, equal to 0.2 milliseconds. Different results were based only on different averaging intervals.

Interpretation of TRT Results
Values of the effective thermal conductivity λeff based on the conducted thermal response tests are presented in Table 3. Start time of calculation of λeff corresponded with Equation 7. End time corresponded with the end of heating time (end of TRT). The highest values are marked in red and the lowest in green. There is no correlation to be seen from the presented results.
The variability of the heat conduction value for borehole No. 1 is 0.028 Wm −1 K −1 , for borehole No. 3 it is 0.011 Wm −1 K −1 , and for borehole No. 5 it is 0.064 Wm −1 K −1 . The relative error resulting from the method of averaging equals 1.74%, 0.71%, and 4.05%, respectively.

Interpretation of TRT Results
Values of the effective thermal conductivity λ eff based on the conducted thermal response tests are presented in Table 3. Start time of calculation of λ eff corresponded with Equation (7). End time corresponded with the end of heating time (end of TRT). The highest values are marked in red and the lowest in green. There is no correlation to be seen from the presented results. The variability of the heat conduction value for borehole No. 1 is 0.028 Wm −1 K −1 , for borehole No. 3 it is 0.011 Wm −1 K −1 , and for borehole No. 5 it is 0.064 Wm −1 K −1 . The relative error resulting from the method of averaging equals 1.74%, 0.71%, and 4.05%, respectively. Table 4 shows different statistics. As presented, the volatility factor is less than 2.5%. With the same geology (the distance between the boreholes is shown in Figure 4), the value of this factor is quite high. This is due to the fact that the effective thermal conductivity λ eff is not the same as the average thermal conductivity of the rock mass λ, which, according to Table 2, equals 2.039 Wm −1 K −1 , which is the weighted average conductivity according to literature data. The accuracy results from averaging. Literature data is based mostly on the measuring dry samples. Is-situ measurements are not applicable in the presented research, because coring is too expensive. Real values of rocks heat conductivity in rock mass varies in high range. The reason is, for example, saturation, underground water filtration, and conductivity anisotropy.  Table 4 shows that the most advantageous design is the coaxial BHE, while single and double U-pipe exchangers are identical. The basic indicator that shows BHE quality is borehole thermal resistance R b . It will be described in a separate paper. The same influence on BHE quality also shows effective thermal conductivity λ eff . This value depends to some extent on the heat transfer from the carrier to the pipes. The heat penetration, however, also depends on the heat carrier's flow velocity. Heat penetration through the inner pipe and outer pipe is also of a different nature with U-pipe BHEs constructions.
For the latter case, the highest value of the coefficient of variation can be observed for different methods of the measurement results averaging. The value closest to the theoretical λ = 2.039 was measured in BHE-1, but it is not necessary that the effective thermal conductivity λ eff measured will match an average textbook rock mass thermal conductivity λ.
The same values are shown on below diagrams. For BHE-1 in Figure 12, for BHE-3 in Figure 13 and for BHE-5 in Figure 14. The number of significant digits results from averaging. One digit less gives the same averages in one case.
Energies 2020, 13, x FOR PEER REVIEW 12 of 20 high range. The reason is, for example, saturation, underground water filtration, and conductivity anisotropy. Table 4 shows that the most advantageous design is the coaxial BHE, while single and double Upipe exchangers are identical. The basic indicator that shows BHE quality is borehole thermal resistance Rb. It will be described in a separate paper. The same influence on BHE quality also shows effective thermal conductivity λeff. This value depends to some extent on the heat transfer from the carrier to the pipes. The heat penetration, however, also depends on the heat carrier's flow velocity. Heat penetration through the inner pipe and outer pipe is also of a different nature with U-pipe BHEs constructions.
For the latter case, the highest value of the coefficient of variation can be observed for different methods of the measurement results averaging. The value closest to the theoretical λ = 2.039 was measured in BHE-1, but it is not necessary that the effective thermal conductivity λeff measured will match an average textbook rock mass thermal conductivity λ. The same values are shown on below diagrams. For BHE-1 in Figure 12, for BHE-3 in Figure 13 and for BHE-5 in Figure 14. The number of significant digits results from averaging. One digit less gives the same averages in one case.    The diagrams of effective thermal conductivity λeff for individual heat exchangers depending on the time interval of measurement averaging, illustrate the differences between the values obtained for different boreholes with the same parameters, such as the flow rate of the heat carrier and the set heating power. For the sake of a more thorough statistical analysis on the impact of measurement averaging on TRT results, data from commercial thermal response tests were used [43]. Borehole and test data, as well as the interpretation results, are presented in Tables 5-9.  The diagrams of effective thermal conductivity λeff for individual heat exchangers depending on the time interval of measurement averaging, illustrate the differences between the values obtained for different boreholes with the same parameters, such as the flow rate of the heat carrier and the set heating power. For the sake of a more thorough statistical analysis on the impact of measurement averaging on TRT results, data from commercial thermal response tests were used [43]. Borehole and test data, as well as the interpretation results, are presented in Tables 5-9. The diagrams of effective thermal conductivity λ eff for individual heat exchangers depending on the time interval of measurement averaging, illustrate the differences between the values obtained for different boreholes with the same parameters, such as the flow rate of the heat carrier and the set heating power. For the sake of a more thorough statistical analysis on the impact of measurement averaging on TRT results, data from commercial thermal response tests were used [43]. Borehole and test data, as well as the interpretation results, are presented in Tables 5-9.   In Table 9, the lowest values are marked in green and the highest values in red. It can be seen that most often the lowest values occur for analysis with averaging every 60 min, and the highest for averaging every 1 min. For design safety reasons (designing the number of BHEs), averaging every 60 min should be used for analysis.
Due to the analysis of the commercial thermal response tests data, the effective thermal conductivity λ eff for borehole heat exchangers located throughout Poland was determined. Calculations were made for three time intervals of measurement results averaging: 1 min, 10 min, 60 min, in each individual case.
An important element for the analysis of the measurements was the rejection of the results recorded before and after conducting the test, as well as any anomalies, which could have been affected by external factors such as voltage drops, start-up and stabilization periods of the circulation pump operation, or the influence of atmospheric air temperature.
The output data of the conducted and analyzed tests were spreadsheets with a summary of all the parameters of the heat carrier flowing in the circulating system, recorded by the TRT device. For further analysis, additional calculations in the form of an average heat carrier temperature and the time from the beginning of the test and its natural logarithm were necessary.
By obtaining such data, it was possible to create appropriate graphs showing the relationship between the average temperature of the heat carrier, time, and the logarithm of time. A graph depicting the relationship between heating power, flow rate, and time was presented in order to determine their possible impact on the measurement process.
Slope factors were determined based on regression equations. The average heating power was calculated by summing the values recorded for each measurement and dividing it by the number of measurements. The depth of the exchanger, originating from the technical documentation, was given as additional information for each test.
The presented values of the effective thermal conductivity coefficient λ eff for each exchanger are similar and show no significant differences between each other. Only in four cases a discrepancy in results of more than 5% can be noticed, which may have been caused by external factors.
The calculated standard deviation and coefficient of variation values confirmed that the set heating power showed constancy and that any deviations from the average were small (less than 10%). The same is true for the effective thermal conductivity factor (the variability of this parameter is only a few percent), which indicates small differences in conductivity depending on the sampling rate of the test. The heating power shows a small dispersion of values over the measurement process.
It has been confirmed that the directional coefficient of the regression line affects the value of effective thermal conductivity, but these are not very large discrepancies. This is due to the number of results making up the graph of the average temperature dependency on the logarithm of time. More data allow for more accurate determination of the parameters, however it makes it is easier to disrupt the correctness of the measurement process, affecting the final result.

Conclusions
The thermal response tests were performed for 100 h in BHEs of different design and identical lithology. TRT data averaging intervals of every 1 min, 10 min, and 60 min were analyzed. Heat carrier temperature changes in time were recorded. The program used for the conversion of data from TRT apparatus were calculating the averages by an arithmetic mean method. It can be concluded that the most reliable (accurate) result would be reflected by a measurement with a time interval of every 1 min. Such data will be the largest in quantity. Rock mass properties were identical. The effective thermal conductivity λ eff is dependent on various BHEs designs, heat carrier flow geometry, as well as sealing slurry parameters. The position of the pipes in relation to each other also seems important to consider. As shown on the graphs, the best effective thermal conductivity λ eff has occurred in BHE-1 with a coaxial design. At the same time, this value is closest to the theoretical value of thermal conductivity of the drilled rocks λ.
By analyzing the results of the thermal response tests, the value of effective thermal conductivity can be easily determined. For this purpose, it is necessary to create a graph of the relation between the average temperature of the heat carrier and the logarithm of the working time of the TRT apparatus. Then, a regression line of this function and its directional factor should be determined.
The directional factor of the regression line shows different values, depending on the time interval for the considered BHE. As a result, the final value of the effective thermal conductivity coefficient λ eff for this BHE is slightly different. It is influenced by the method of averaging the obtained results during the thermal response test. In the majority of described cases, these differences are not very significant, and the values are comparable. In two cases, they are slightly larger, possibly because it is easier to miss the moment of the ground thermal saturation and the temperature graph smoothing at intervals of 60 min. It therefore appears that the interval of 1 min is more accurate for correct determination of the thermal parameters of the rock mass.
The results of the conducted analysis of the thermal response tests for the borehole heat exchangers located in Poland allowed to determine the directional coefficient of a regression line, and its influence on the effective thermal conductivity for individual exchangers, which was presented in Table 9.
Standard deviation and coefficient of variation values have confirmed that there were no significant deviations from the set value of heating power. They also showed that the effective thermal conductivity calculated for different time intervals shows little value variability.
Thermal conductivity values for each time interval, for the same exchanger, were similar in value. Investigated media have shown good potential for extensive installations acquiring heat from low temperature sources.
As shown in Table 9, the lowest values most commonly appear for analysis averaging every 60 min, and the highest-for analysis averaging every 1 min. For design safety reasons (designing the number of BHEs), safer values can be taken for analysis, i.e., averaging every 60 min. A better way is to use the most accurate estimate (1 min averaging) and then simply include a safety margin for design safety reasons. Adding safety should be a quantifiable adjustment, which can be based on the lowest (safer) value of averaging, sensors uncertainty, and the impact of ambient temperature.