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

Abstract

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.

Appendix: Pseudo-variogram Models for the Transformation in the Space of Logarithms

See Appendix Tables 4, 5 and 6. The pseudo-variogram is always included in the standard interval [0, C]. Given a data field of fixed dimension in ${\{}t{\}}$ where the variogram is stationary, the pseudo-range decreases when the field translates towards increasing coordinate values (Fig. 6a). On the contrary, in the case of variogram stationary in ${\{}\tau {\}}$, by keeping the data field of fixed dimension in ${\{}t{\}}$, the pseudo-range increases when the field translates towards increasing coordinate values (Fig. 6b).

Table 4 Pseudo-variogram equations for the spherical model

Table 5 Pseudo-variogram equations for the exponential model

Table 6 Pseudo-variogram equations for the Gaussian model