Mathematical Geosciences

, Volume 48, Issue 1, pp 3–23 | Cite as

Estimating Thermal Response Test Coefficients: Choosing Coordinate Space of The Random Function

  • Roberto Bruno
  • Francesco Tinti
  • Sara Focaccia
Special Issue


In shallow geothermal systems, the main equivalent underground thermal properties are commonly calculated with a thermal response test (TRT). This is a borehole heat exchanger production test where the temperature of a heat transfer fluid is recorded over time at constant power heat injection/extraction. The equivalent thermal parameters (thermal conductivity, heat capacity) are simply deduced from temperature data regression analysis that theoretically is a logarithmic function in the time domain, or else a linear function in the log-time domain. By interpreting the recorded temperatures as a regionalized variable whose drift is the regression function, in both cases the formal problem is a linear estimation of the mean. If the autocorrelation function (variogram, covariance) of residuals is known, coefficient variance can be directly deduced. Coefficient estimates are independent of the drift form adopted, and the residuals are the same in the same points. The random function is different in the time domain, however, and in the log-time domain. In fact, residual variograms are different due to the transformation of the coordinate space. This paper uses a TRT case study to examine the consequences of coordinate space transformation for a random function, namely its variogram. The specific question addressed is the choice of coordinate space and variogram.


Thermal response test Random function Geothermal energy Pseudo-variogram models Non-linear transformation 



The authors sincerely thank Dr. Markus Proell, Ph.D, the ZAE Bayern and all the IEA-ECES Annex 21 group for the invaluable help in understanding thermal response test issues and processes and for giving them the opportunity to work on the Ravensburg TRT dataset, used for the case study. The authors also sincerely thank the reviewers for their encouragement and the useful suggestions that have made this paper much more convincing and complete.


  1. Austin WA, Yavuzturk C (2000) Development of an in-situ system and analysis procedure for measuring ground thermal properties. ASHRAE Trans 106(1):365–379Google Scholar
  2. Beier RA, Smith MD (2003) Minimum duration of in-situ tests on vertical boreholes. ASHRAE Trans 109(2):475–486Google Scholar
  3. Boettcher S, Sibani P (2011) Aging in dense colloids as diffusion in the logarithm of time. J Phys Condens Matter 23:065103CrossRefGoogle Scholar
  4. Bruno R, Focaccia S, Tinti F (2011) Geostatistical modeling of a shallow geothermal reservoir for air conditioning of buildings. In: Proceeding of IAMG, Salzburg, pp 145–162Google Scholar
  5. Bruno R, Mercuri S, Tinti F, Witte H (2013) Probabilistic approach to TRT analysis: evaluation of groundwater flow effects and machine–borehole interaction. In: Proceedings of European geothermal congress, Pisa, ItalyGoogle Scholar
  6. Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New York (ISBN: 978-0-470-18315-1)CrossRefGoogle Scholar
  7. Dieterich JH (1978) Time-dependent friction and the mechanics of stick-slip. Pageoph 116(4–5):790–806CrossRefGoogle Scholar
  8. Edwards N (2005) Marine controlled source electromagnetics: principles, methodologies, future commercial applications. Surv Geophys 26:675–700CrossRefGoogle Scholar
  9. Eklof C and Gehlin S (1996) TED–a mobile equipment for thermal response tests. Master’s thesis: 198E, Lulea University of Technology, SwedenGoogle Scholar
  10. Focaccia S, Bruno R, Tinti F (2013) A software tool for geostatistical analysis of thermal response test data: GA-TRT. Comput Geosci 59:163–170CrossRefGoogle Scholar
  11. Gaddum JH (1945) Lognormal distributions. Nature 156:463–466CrossRefGoogle Scholar
  12. IGSHPA (2013) Closed-loop/geothermal heat pump systems: design and installation standards. In: International ground source heat pump association. Oklahoma State University, StillwaterGoogle Scholar
  13. Lloyd CD (2010) Local models for spatial analysis. CRC Press, Taylor & Francis Group, London, p 352 (ISBN: 9781439829196)Google Scholar
  14. Lutkephol H, Xu F (2012) The role of the log transformation in forecasting economic variables. Empir Econ 42:619–638CrossRefGoogle Scholar
  15. Marcotte D, Pasquier P (2008) On the estimation of thermal resistance in borehole thermal conductivity test. Renew Energ 33(11):2407–2415CrossRefGoogle Scholar
  16. Mogensen P (1983) Fluid to duct wall heat transfer in duct system heat storages. In: Proceedings of the international conference on subsurface heat storage in theory and practice, appendix, Part II, StockholmGoogle Scholar
  17. Mouri H (2013) Log-normal distribution from a process that is not multiplicative but is additive. Phys Rev E 88(4). doi: 10.1103/PhysRevE.88.042124
  18. Mozyrska D, Torres DFM (2009) The natural logarithm on time scales. JDSGT 7(1):41–48Google Scholar
  19. Sanner B, Hellström G, Spitler J, Gehlin S (2005) Thermal response test–current status and world-wide application. In: Proceedings of world geothermal congress, Antalya, TurkeyGoogle Scholar
  20. Theis CV (1935) The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. Trans Am Geophys Union 16(2):519–524CrossRefGoogle Scholar
  21. Tinti F, Focaccia S, Bruno R (2015) Thermal response test for shallow geothermal applications: a probabilistic analysis approach. Geotherm Energy 3(6). doi: 10.1186/s40517-015-0025-5
  22. Witte HJL, van Gelder GJ, Spitler JD (2002) In situ measurement of ground thermal conductivity: a Dutch perspective. ASHRAE Trans 108(1):263–272Google Scholar
  23. Zhang C, Guo Z, Liu Y, Cong X, Peng D (2014) A review on thermal response test of ground-coupled heat pump systems. Renew Sust Energ Rev 40:851–867CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2015

Authors and Affiliations

  • Roberto Bruno
    • 1
  • Francesco Tinti
    • 1
  • Sara Focaccia
    • 1
    • 2
  1. 1.Department of Civil, Chemical, Environmental and Materials EngineeringUniversity of BolognaBolognaItaly
  2. 2.CERENAInstituto Superior Técnico de LisboaLisbonPortugal

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