Analysis with first-order approximations (FOA)

The first-order approximation (FOA) models interpret the heating phase of the TRT through a simple linear regression, yielding the ground thermal conductivity λ_s and an experimental effective borehole resistance R_b^*, the value to be read against the theoretical one on the Test borehole panel. A recovery phase is not required for this: when the test includes one, its own regression can be run as an independent check on λ_s, not as a second measurement the interpretation depends on.

Heating phase analysis - The thermal response of a GHE during the heating phase can be modeled with reasonable accuracy by:

T(t) = T_g + q R_b^* + (q / 4πλ_s) E₁(r_b² / 4α_st)

where T(t) is the mean fluid temperature, T_g is the undisturbed ground temperature, q is the constant unit heating power, R_b^* is the steady-state borehole effective resistance, λ_s is the ground thermal conductivity, r_b is the borehole radius, α_s is the ground thermal diffusivity and E₁ is the exponential integral function.

This is the infinite line-source model, whose analytical solution goes back to Carslaw and Jaeger (1959) and which has been the basis for TRT interpretation since Mogensen (1983): the borehole is idealized as a constant-strength line of heat in an infinite, homogeneous, conduction-only medium.

Using the first-order approximation of E₁ allows expressing the mean fluid temperature by a regression model of the form T(t) = m ln(t) + b, with m = q/(4πλ_s). This equation has an approximation error of less than 10% for t ≥ t_cr = 5 r_b²/α_s.

The slope of the late-time fluid temperature against ln(t) gives the conductivity, and the intercept is what allows the borehole resistance to be identified, which is why the active length H and the late-time portion of the test must both be right. The slope method itself, and its assumptions, are reviewed in detail by Gehlin (2002) and Spitler and Gehlin (2015).

A dual-range slider is used to interactively select data and compute the regression coefficient m associated with these points. To refine your selection, click the slider, then press the Q (←) or E (→) keys to move the start of the window and the A (←) or D (→) keys to move its end. The critical time t_cr is drawn with a vertical red line and corresponds to the moment after which the approximation error is smaller than 10%; data located before the red line should not be used for the analysis. The t_cr reported by the console is the position of that red line. On the chart, the mean fluid temperature is plotted against ln(t), on which the line-source model predicts a straight line, and the fitted regression is overlaid on the selected points. A late-time portion that curves away from that line, rather than tracking it, is one of the few visual signs of groundwater advection or of a heating power that did not hold constant.

Recovery phase analysis - At the beginning of the recovery phase, heating is stopped and the fluid returns towards the undisturbed temperature. Using the FOA of E₁ and the principle of temporal superposition, the recovery response can be modeled by:

T(t) = T_g + (q / 4πλ_s) ln(t / (t − t_h))

where t is the time since the beginning of the heating phase and t_h is the heating phase duration. The FOA gives a regression model of the form T(t) = m ln(t/(t − t_h)) + b, with m = q/(4πλ_s). Further first-order forms of the recovery, built on the time derivative of the temperature rather than on the temperature itself, are developed by Pasquier (2018).

Again, a dual-range slider selects the data and computes m, plotted this time against the ratio t/(t − t_h) rather than against ln(t). The critical time t_cr is drawn with a vertical red line; because time runs backwards on this axis, the exclusion rule reverses, and data located after the red line should not be used for the analysis.

The unit power q follows your selection - The slope alone does not give λ_s: the model reads it from m = q/(4πλ_s), so the unit power q enters the result just as directly. It is worth knowing exactly which power each phase uses, because one of the two moves as you work.

• Heating phase — q is the mean heating power measured from the beginning of the heating phase to the last point selected, divided by the active length H. Moving the upper handle of the regression window therefore changes q as well as the slope, and both change λ_s. This is intended: it keeps the power consistent with the stretch of test the line is being asked to describe. It also means a window that stops just after a power interruption carries that interruption in its average.
• Recovery phase — q is the power measured over the whole heating phase, not over the recovery window, since it is the injection that created the thermal disturbance the recovery relaxes from. Moving the recovery window changes the slope but not q.

Both values are reported, in W/m, in the result files written alongside the project.

Averaging method for the fluid temperature - Two options are offered to estimate the mean fluid temperature during the heating phase. The first is the arithmetic mean of the borehole inlet and outlet fluid temperature (the only option for the recovery phase). The second is the so-called p-linear average, deemed to better represent the mean fluid temperature in the GHE.

The arithmetic mean is known to overestimate the mean fluid temperature, and therefore the borehole resistance, when the inlet-to-outlet temperature difference is large; the p-linear average was introduced by Marcotte and Pasquier (2008) to reduce this bias.

Ground volumetric capacity - The ground volumetric heat capacity C_s (in millions of J/m³/K, from 1.6 to 2.8) is used only to compute the critical time t_cr. This value must be provided because the FOA method cannot estimate it.

Assumptions and limitations - The FOA models assume a constant heating power during the heating phase, a thermal steady-state in the GHE, a constant circulation flow rate, the absence of groundwater flow around the GHE, the lack of axial effects and no heating during the recovery phase. If one or more of these assumptions are not met, the FOA model should not be used.

The pipe arrangement is not among them. The line is fitted to the measured mean fluid temperature and to the unit power q = Q/H, and neither depends on how many pipes the hole carries, so the method applies to any configuration of the test borehole: Single U-Loop, either Double U-Loop, Coaxial or Standing Column. The effective resistance R_b^* read from the intercept is simply the one that configuration produces, whatever it is.

The constant-power assumption is the one most often broken in the field. Where the injected power drifted or stopped, the choice is between accepting the bias and fitting the measured history itself, either by superposing line sources in time and estimating the parameters numerically (Mazzotti et al., 2018) or by the short-term g-function of the next node.

Of these, the no-groundwater assumption is the one most often violated in the field and the hardest to detect from the temperature record alone. Where advection is known to be significant, the conduction-only line source is the wrong model outright, and a moving line-source formulation such as that of Pasquier and Lamarche (2022) is required to recover the ground properties. It need not affect the whole borehole to matter: Voirand and Maragna (2025) report tests where a distributed measurement traced the enhanced heat transfer to one depth interval rather than to the whole borehole, something the inlet and outlet temperatures that feed the regression cannot localise.

Summary & charts: the regression residuals and the results in the console

Residual histogram - The panel to the right of the inputs shows the distribution of the regression residuals, the difference between each measured mean fluid temperature and the fitted line, over the window currently selected. The residuals of a least-squares fit always average zero, so what is worth looking at is their shape: a narrow, symmetric bell says the line describes the data; a skewed or double-peaked distribution says it does not, usually because the window still contains data from before t_cr or spans a change in the heat injection. The histogram is bounded on the percentiles of the residuals rather than on their extremes, so a handful of outliers cannot flatten it into a single bar.

Where the regression results are reported - The numbers themselves, λ_s, R_b^*, t_cr, and the mean, standard deviation and RMSE of the residuals, are reported on the second page of the console, in two columns, one for the heating phase and one for the recovery phase. Selecting this node brings that page up, and the figures update as the regression windows move.

TRT in practice: cross-checking the two regressions

When a recovery phase was recorded, treat the heating-phase and recovery-phase conductivities as an independent cross-check: they are derived from different data through different regressions, so close agreement is reassuring, whereas a marked discrepancy is a warning sign. The usual culprits are an unstable heat injection rate, a test cut short before t_cr, or groundwater advection, exactly the conditions under which the line-source assumptions fail. In those cases, turn to the short-term g-function, which uses the full transient and tolerates a varying power. Note also that the recovery phase returns only λ_s, not R_b^*, since the borehole resistance is carried in the heating-phase intercept.

The window the line is fitted over moves λ_s the most — more than measurement noise, and more than any of the properties entered on the other panels. Oh et al. (2022) rank the experimental factors that govern an analytical interpretation and find the start of the fit ahead of the test duration. Report the regression window alongside the conductivity, and place its start on t_cr rather than wherever the line happens to look straightest.

The ground volumetric capacity C_s is the one input here that need not be known precisely. It does not affect the conductivity λ_s directly: it only sets the critical time t_cr, the boundary before which the approximation is invalid. A rough value is therefore adequate: if it places t_cr slightly off, simply discard the data points near the line rather than agonising over C_s.