Stochastic interpretation of thermal response test with TRT-SInterp
Introduction
A thermal response test (TRT) aims at measuring in situ the thermal properties of geological material and borehole heat exchanger (BHE) prior to the design of a ground-coupled heat pump system. In practice, a TRT consists to induce a temperature change in the fluid circulating in the pipes of a BHE and to find a set of thermal parameters allowing numerical reproduction of the temperatures measured in the BHE (see Fig. 1).
Interpretation of TRT is usually achieved with a simplification of the infinite line-source model (Ingersoll et al., 1954). The analysis consists to fit, on a semi-log plot, the first-order approximation of the exponential integral function to the experimental temperatures (Gehlin, 2002). The slope coefficient and y-intercept of the fitted model being connected to the equivalent ground thermal conductivity and borehole resistance (Witte, 2013), a solution to this inverse problem is obtained directly by regression. Parameter uncertainty can then be assessed easily through conventional statistical or geostatistical analysis (Focaccia et al., 2013). The so-called graphical slope method has the merit of being easy to use and understand but several limitations restrict its application to real field conditions.
The most stringent requirement of the graphical method is undoubtedly the need to keep the heating or cooling power (Witte et al., 2002) constant throughout the test duration, which may prove difficult in practice. Although alternate approaches are available to get around this limitation (e.g. Beier and Smith, 2003), the truncation error made by the linearization of the infinite line-source model and the assumption of thermal steady state in the borehole induce a significant model error preventing use of early temperature measurements.
Interpretation of TRT relies on a specific BHE model. As rightly pointed out by Witte (2013), the assumptions inherent to the interpretation model induce an error that in turn affects the outcome of the interpretation. Serious efforts have been made by the community to develop more complex, and hopefully, more accurate models (Al-Khoury et al., 2010, Zarrella et al., 2011, Bauer et al., 2011, Zarrella et al., 2013). Unfortunately, as soon as complex models are used, linearization of the problem is no longer possible and identification of the parameters has to be done through inverse modeling by coupling the interpretation model to an optimization algorithm. Although use of nonlinear estimation or sweeping techniques is now more common (Wagner and Clauser, 2005, Signorelli et al., 2007, Marcotte and Pasquier, 2008b, Wagner et al., 2012, Li and Lai, 2012), its usage remains limited apparently due to the complexity and computational burden associated to inverse problem solution.
In this article, a program designed to analyze TRTs by deterministic or stochastic inversion, TRT-SInterp, is presented. By giving more flexibility to the interpretation of TRTs, the program aims to foster investigation of new testing strategies (step test, thermal recovery, addition of temperature sensors in the grout and fluid, etc.) and maybe reduce tests duration and parameters uncertainty. This article provides sufficient insight on how to use TRT-SInterp, analyze its results and maybe modify it to fulfill the requirements of a specific research project. The latest version of the source code is available to the community for research activity on cageo as well as on the author's webpage.
Section snippets
Solution of the direct problem
The evolution of the fluid temperature during a TRT can be simulated with analytical solutions or simplified numerical models. Even if these models can directly predict the fluid temperature circulating in a BHE, it has become common in the field of geoexchange to use them only to generate a transfer function g describing the thermal behavior of a BHE and to solve the direct problem by convolving g with an input function f describing the temporal variation of the heating power.
The theoretical
Implementation
TRT-SInterp is implemented in Matlab (The MathWorks Inc., 2013) with a special attention paid to the code optimization and to a reduction of the computation time. Among the various strategies used to reduce the computation time, Matlab built-in functions have been widely used on multidimensional arrays and whenever possible, the code has been vectorized and use of for loop avoided. However, the program can be easily modified for parallel computing to distribute the inversions over the workers
Application to a real test case
To assess the heat transfer potential of a BHE made of highly conductive pipes, Pasquier and Groleau (2009) performed a long duration TRT on a BHE containing 20 sensors distributed at various depths in the fluid and in the sand filling a borehole. During the TRT, temperature measurements were recorded at 30-second interval during 12.1 days by 11 probes (accuracy of ±0.22 °C) installed in the fluid and by 9 additional probes (accuracy of ±0.5 °C) installed in the BHE along the pipes (see Fig. 1).
Conclusion
In this contribution, a program designed to analyze thermal response tests by deterministic or stochastic inversion was presented. The program simulates a borehole heat exchanger by a finite line-source model or a thermal resistance and capacity model and allows identification of six parameters. If the thermal resistance and capacity model are used as the interpretation model, it is possible to integrate to the inversion the temperature measurements made at various depths in the fluid and grout
Acknowledgments
The author wish to thank two anonymous referees for their constructive and helpful comments and suggestions. Some special thanks are also dedicated to Mr. Stéphane Gonthier from Versaprofiles Inc. who allowed the author to use the experimental dataset used in this work as well as Misters Louis Jacques and Denis Marcotte for suggesting useful additions to TRT-SInterp. This research was financed by a Natural Sciences and Engineering Research Council of Canada (NSERC) research grant.
References (30)
- et al.
Efficient finite element modeling of shallow geothermal systems
Comput. Geosci.
(2010) - et al.
Transient 3D analysis of borehole heat exchanger modeling
Geothermics
(2011) - et al.
A software tool for geostatistical analysis of thermal response test dataGA-TRT
Comput. Geosci.
(2013) - et al.
New solutions for the short-time analysis of geothermal vertical boreholes
Int. J. Heat Mass Transf.
(2007) - et al.
Parameter estimation of in-situ thermal response tests for borehole ground heat exchangers
Int. J. Heat Mass Transf.
(2012) - et al.
Fast fluid and ground temperature computation for geothermal ground-loop heat exchanger systems
Geothermics
(2008) - et al.
On the estimation of thermal resistance in borehole thermal conductivity test
Renew. Energy
(2008) - et al.
The importance of axial effects for borehole design of geothermal heat-pump systems
Renew. Energy
(2010) - et al.
Short-term simulation of ground heat exchanger with an improved TRCM
Renew. Energy
(2012) - et al.
Efficient computation of heat flux signals to ensure the reproduction of prescribed temperatures at several interacting heat sources
Appl. Therm. Eng.
(2013)
Joint use of quasi-3d response model and spectral method to simulate borehole heat exchanger
Geothermics
Numerical evaluation of thermal response tests
Geothermics
Evaluating thermal response tests using parameter estimation for thermal conductivity and thermal capacity
J. Geophys. Eng.
Numerical sensitivity study of thermal response tests
Renew. Energy
Error analysis of thermal response tests
Appl. Energy
Cited by (31)
A versatile method for estimating grout and ground thermal properties in a thermal response test
2023, Applied Thermal EngineeringThermal response tests: A biased parameter estimation procedure?
2021, GeothermicsDevelopment of chiller-attached apparatus for accurate initial ground temperature measurement: Insights from global sensitivity analysis of thermal response tests
2021, Energy and BuildingsCitation Excerpt :Recently, probabilistic estimation methods have been developed to overcome the limitations of deterministic estimation methods that provide only point estimates. Pasquier [18] transformed the deterministic estimation to a stochastic one through a Monte Carlo simulation in which the unknown parameter sets were sampled using uniform distributions. Li et al. [19] also conducted parameter estimation with a Monte Carlo simulation to evaluate the robustness of the stepwise estimation method developed to overcome the issue of correlation between the estimated parameters and to utilize the data interval during which the parameters show large sensitivity to changes in fluid temperature.