Mathematical Geosciences

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

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

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.


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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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