On the applicability of the moving line source theory to thermal response test under groundwater flow: considerations from real case studies
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The classical methodology to perform and analyze thermal response test (TRT) is unsuccessful when advection contributes to heat transfer in the ground, due to the presence of a groundwater flow. In this study, the applicability, the advantages, and the limitations of the moving line source model to interpret TRT data are discussed. Two real TRT case studies from the Italian Alpine area are reported and analyzed, with both the standard infinite line source approach and the moving line source one. It is shown that the inverse heat transfer problem is ill-posed, leading to multiple solutions. However, besides minimization of the error between measurements and modeling, physical considerations help to discriminate among solutions the most plausible ones. In this regard, the MLS approach proves to be effective in the advection-dominated case. The original time criterion proposed here to disregard initial data from the fitting, based on a resistance–capacitance model of the borehole embedded in a groundwater flow, is validated in terms of convergence of the solution. In turn, in the case when advection and conduction are competitive, the MLS approach results more sensitive to ground thermal conductivity than to Darcy velocity. Although in this case a limited impact of the uncertainty in the groundwater velocity on the boreholes sizing is expected, future studies should focus on the development of a successful TRT methodology for this condition.
KeywordsThermal response test Moving line source Ground Thermal conductivity Groundwater Darcy velocity Inverse problem Ground-source heat pump
List of symbols
heat capacity (J/K)
heat capacity per unit mass (J/(kg K))
exponential integral function
borehole depth (m)
convective coefficient (W/(m2 K))
infinite line source
moving line source
Nusselt number (–)
Péclet number (–)
Prandtl number (–)
heat rate per unit length (W/m)
thermal resistance (m K/W)
Reynolds number (–)
Root Mean Squared Error (°C)
thermal response test
effective velocity (m/s)
thermal diffusivity (m2/s)
Euler constant (–)
polar coordinate (rad)
thermal conductivity (W/(m K))
dynamic viscosity (kg/(m s))
characteristic time (s)
Thermal response test (TRT) is a well-known experimental procedure allowing to derive in situ fundamental thermal–physical properties of the ground and of the borehole. Since its first developments in the mid-90s, it has spread rapidly, being now available in about 40 countries worldwide (Nordell 2011; Spitler and Gehlin 2015). Typically, it consists in forcing a thermal-carrier fluid at constant flow rate in a test borehole and in regulating the fluid inlet temperature in order to inject into the ground (or extract from it) a constant heat rate for 2–3 days. By choosing a physical model describing the TRT and by solving the associated inverse heat transfer problem, the average ground thermal conductivity, the borehole thermal resistance, and the ground undisturbed temperature can be assessed. Such parameters are the essential inputs for the design of ground heat exchangers (Zhang et al. 2014).
Different analytical models for interpreting the TRT data are available (Philippe et al. 2009), namely the infinite line source (ILS), the infinite cylindrical source (ICS), the finite line source (FLS), where the ILS is the most frequently adopted, due to its simplicity. Besides analytical models, numerical models can be used, requiring more modeling and computational effort, but offering the possibility to analyze heterogeneous geological conditions, groundwater flow, or non-conventional TRT procedures (Signorelli et al. 2007; Raymond et al. 2011).
When ground layers are saturated and significant groundwater flow occurs, the standard interpretation of the TRT, based on the assumption of pure conduction in the ground, fails and it becomes impossible to derive the ground thermal conductivity (Witte 2001; Sanner et al. 2013). Indeed, when groundwater flow is present, heat transfer in the ground occurs not only by conduction but also by advection. Consequently, in order to properly design boreholes operating in the presence of groundwater flow, Darcy velocity should be known, besides ground thermal conductivity. To this purpose, some authors (Wagner et al. 2013) recently proposed to adopt another analytical model, i.e., the Moving Line Source (MLS) to interpret groundwater-influenced TRTs and to derive both ground thermal conductivity and groundwater flow Darcy velocity. The MLS problem was firstly presented and discussed by Chiasson et al. (2000) and by Diao et al. (2004). By applying the MLS model to the results of several numerically simulated TRTs, Wagner et al. (2013) show that the true value of the Darcy velocity is generally underestimated. They attribute the discrepancy mainly to the difference between the hydraulic conductivity of the grouting material, which is almost null, and the aquifer. Therefore, they calculate, by numerical simulations, a correction factor to be applied to the velocity value derived from TRT fitting in order to obtain the true velocity. Such correction factor is derived for fitting thermal conductivity in the range of 1.5–4.5 W/(m K) and for fitting Darcy velocity up to 1 m/day. Finally, Wagner et al. (2013) apply their approach to three real TRTs. For the conduction-dominated case study, they conclude that the MLS approach does not provide any advantage, since although the thermal conductivity is estimated in a very narrow range, the Darcy velocity cannot be derived accurately. For the advection-dominated case study, they find several pairs of thermal conductivity and Darcy velocity values, negatively correlated, and conclude that in such conditions the MLS approach can only be used to derive a plausible range for the hydro-geological and thermal–physical properties of the ground. Finally, in the case where conduction and advection compete, the authors show that the possible pairs lie within a 10% range and thus the MLS approach provides a good estimate of the ground parameters.
In a subsequent study, Wagner et al. (2014) apply the MLS model to interpret both large-scale tank and field TRT experiments. They demonstrate that the test can also be used for hydro-geological characterization of the subsoil, since in both cases the evaluations of both experiments resulted in similar hydraulic conductivity ranges as determined by standard hydraulic investigation methods such as pumping tests and sieve analysis. It has to be pointed out that, since in the mentioned study the authors are not interested to use the TRT to determine the ground thermal conductivity, single parameter fits are carried out.
Chiasson and O’Connell (2011) adopt a parameter estimation technique to a TRT affected by significant groundwater flow and compare three different analytical solutions, namely the MLS, the one based on the groundwater g-function (Claesson and Hellstrom 2000), and a mass transport solution adapted by the authors using a mass-heat transport analogy, that can account for thermal dispersion phenomena. They find that only the mass-heat transport analogy yields a favorable comparison to field test data, while the other solutions do not produce a realistic comparison, implying that thermal dispersion is an important parameter, at least in situations with relatively high groundwater velocities. They also discuss the parameter estimation procedure remarking that multiple local minima are observed and recommending realistic constraints of the parameters for this kind of optimization. The role of thermal dispersion is also highlighted in a sensitivity analysis by Wagner et al. (2012), showing that disregarding dispersion can lead to overestimate the effective thermal conductivity by a factor up to 190%.
A completely different approach to TRT in the presence of a groundwater flow is recently proposed by Rouleau and Gosselin (2016), whose conceptual TRT is designed explicitly to derive information on groundwater flow velocity and direction. The authors propose to place a heating cable in the borehole before backfilling with grout and to place some temperature probes in different horizontal positions at the borehole wall. They assess the performance of the methodology by numerical simulations and find that it is more sensitive to ground thermal conductivity than to groundwater velocity.
Therefore, further efforts are necessary to identify the best methodology to perform and/or to analyze TRT data when groundwater flow is relevant. This study aims to investigate the applicability, the advantages, and the limitations of the MLS approach to TRT analysis. Since a criterion to assess the time from the TRT start since when the MLS model can be applied is lacking, an original time criterion is here proposed, based on physical considerations. The paper reports and deeply analyzes two case studies regarding TRTs performed in the Italian Alpine region. In each case, the ILS and the MLS approaches are applied and compared. The issue of the non-uniqueness of the solutions, namely the existence of several possible pairs of thermal conductivity and Darcy velocity values, is addressed in practice through physical considerations. Finally on the basis of a previous study by the authors (Angelotti et al. 2014), the impact of the uncertainty in the Darcy velocity is discussed, in terms of variation in the expected borehole energy performance.
The TRTs reported and discussed in this paper were performed by means of the mobile equipment GEOGert 2.0. The apparatus is equipped with three electrical resistances allowing to obtain a maximum thermal power of 8 kW. The monitoring module comprises three PT 100 thermal probes to measure inlet and outlet fluid temperatures and the ambient air temperature, and an electromagnetic flowmeter for measuring the mass flow rate. The minimum sampling time is 2 s.
The TRT data were interpreted firstly with the ILS model and then with the MLS model. The relevant equations for each model and the corresponding fitting methodology are outlined in the following.
Infinite line source analysis
Moving line source analysis
Results and discussion
Clavière case study
Description of the TRT case studies
Claviére (TO) (Delmastro 2014)
Gardolo (TN) (Zille 2013)
Dolostones with shallow clayish interpositions (0–150 m), compact crystalline limestone (150–170 m), aquifers levels at 2, 22, and 150 m
Dry gravel, saturated gravel, sand, and silt with saturated gravel (0–71.5 m), calcareous bedrock (71.5–115 m), aquifer level at 30 ÷ 32 m
Double U pipe, Φext = 40 mm, D = 152 mm, H = 170 m, Termoplast Plus grouting (λgr = 2.0 W/(m K))
Double U pipe, Φext = 32 mm, D = 130 mm, H = 115 m, Termoplast Plus grouting (λgr = 2.0 W/(m K))
Undisturbed ground temperature T0
Expected ground properties
λ = 2.7 W/(m K)
λ = 2.23 W/(m K)
α = 1.17·10−6 m2/s
α = 9.37·10−7 m2/s
MLS analysis: initial values for the parameter estimation technique
Rbh (m K/W)
It has to be mentioned that in this case the correction term proposed by Wagner et al. (2013) cannot be applied since the fitting Darcy velocity 6.6 m/day is beyond the range considered by the authors, suggesting that a broader range is worthy of investigation. Yet, on the basis of their work, it can be argued that the true Darcy velocity in this case may be more than 10 times higher than the fitting one.
Gardolo case study
The second thermal conductivity value (2.18 W/(m K)) corresponds to a group of solutions with the same borehole thermal resistance (0.149 m K/W), but a large range of flow velocities, namely 2·10−11–1.1·10−6 m/s. Actually this group of solutions results in a thermal conductivity quite close to the expected value, but the estimate of the groundwater flow velocity is not unique. Although this output can be overwhelming, it has to be noticed that the highest Darcy velocity taken from this family of solutions Péclet number, calculated by taking the borehole radius as the characteristic length, is about 0.1. According to the study by Angelotti et al. (2014), reporting also similar results from literature, when Pé = 0.1 groundwater influence on the borehole annual energy performance is limited to a maximum of 20%. Therefore, it can be argued that an inaccurate estimation of the Darcy velocity in this range does not affect significantly the borehole heat exchanger design, although further studies are necessary to clarify this issue.
The real groundwater velocity and the real ground thermal conductivity for the two analyzed TRT are actually unknown. Therefore, the considerations on the applicability of the MLS analysis to the case studies analyzed in this paper cannot be based on the capability of such model to estimate ground and groundwater properties correctly. Indeed performing quantitative hydro-geological investigations in situ like pumping tests (Wagner et al. 2014) is an expensive procedure, often requiring the drilling of more than one test borehole. The average ground thermal conductivity in situ can in principle be derived from laboratory measurements of homogeneous portions of the core sample extracted. However, especially in case of non-consolidated soils, preserving the sample density and humidity content in laboratory is not straightforward. In turn, a comprehensive validation of the MLS approach to TRT analysis can only come from laboratory tests. To this purpose, the authors are developing a physical model at reduced scale to study TRT procedure and interpretation in the presence of groundwater flow, under controlled and parametric conditions, namely ground composition, Darcy velocity, and source heat rate.
The analysis of the two real TRT cases presented in this paper clearly shows that finding both the ground thermal conductivity and the groundwater flow velocity from the time profile of the thermal-carrier fluid temperature measured in the standard TRT is an ill-posed inverse problem leading to multiple solutions. In turn, the determination of the borehole thermal resistance appears less critical, since multiple solutions lie in a very narrow range. To overcome the problem of multiple solutions, further efforts are necessary to develop new TRT procedures, where more quantities are monitored in order to determine a well-posed inverse problem. To this extent, the conceptual test recently proposed by Rouleau and Gosselin (2016) represents an interesting suggestion.
When applying the MLS approach, in order to discriminate among multiple (λ, vd) solutions, looking for the absolute minimum RMSE solution proves to be effective in the advection-dominated case (Clavière), but not when advection is small yet not irrelevant (Gardolo case). Physical considerations, given by the general knowledge of the hydro-geological conditions in situ, can lead to identify the most plausible solution, possibly not the best-fit one. In case conduction and advection compete, the MLS approach is not successful in determining the Darcy velocity, for which only a large range can be identified. According to literature, this advection regime may have a modest impact on the long-term energy performance of borehole heat exchangers. At the same time in this case, the standard ILS approach converges to an effective thermal conductivity, whose value larger than expected possibly includes the effects of groundwater. Further investigations are then necessary to understand if using such effective thermal conductivity for ground heat exchanger design leads to acceptable sizing.
Finally, the minimum time criterion developed in this study, although based on a simplified resistance–capacitance analogy for the borehole volume, proves to be effective in providing a basis for identifying the time since when the MLS model can be applied.
AA designed the study, developed the methodology, supervised the analysis, and drafted the manuscript. FL performed the analyses and interpreted the results. AZ provided the case studies and performed the geological analyses. All authors read and approved the final manuscript.
The authors warmly thank Riccardo Delmastro for providing the TRT data of the Clavière case study.
The authors declare that they have no competing interests.
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