Analysis with the short-term g-function

This model works from the short-term g-function of the test borehole: the temperature response the borehole produces for a unit step of heat, resolved from the very first minutes rather than only at late times. It accounts for the pipe and borehole geometry, the thermal properties (λ and α) of the fluid, pipes, grout and geological materials, and the fluid vertical velocity in the pipe loop. For a given set of input parameters it simulates the fluid temperature at the borehole inlet T(z=0 − ↓) and outlet T(z=0 − ↑). It is for now limited to a single U-loop borehole, whatever configuration the Test borehole panel carries.

That response is not solved from scratch at each move of a slider, which would be far too slow to be interactive. It is evaluated by a pre-trained artificial neural network that reproduces the output of a transient 3D finite-element model to within 1 × 10^-4 °C (Pasquier and Marcotte, 2020). The network is therefore a fast, deterministic stand-in for a detailed physical computation, not a statistical model fitted to your data: for the same inputs it returns the same response, every time.

Coupled with an efficient temporal superposition scheme (Marcotte and Pasquier, 2008; Pasquier et al., 2013), the model captures the temperature changes due to variations in heating power for a TRT shorter than 21 days. This allows the simultaneous analysis of heating and recovery phases, pulsed TRTs and interrupted TRTs.

Behind the sliders, the model does not simulate your test directly. For the geometry and properties you supply, it reconstructs the borehole's unit response, its g-function, which is then convolved with the actual, measured heat-injection history. Because the real power signal drives the convolution, the model follows every ramp, pause and shut-off in the test, which is precisely why it succeeds on the early hours, the recovery, and pulsed or interrupted tests where the line-source model cannot. Two papers anchor this approach. Pasquier (2018) established that the first hours of a test carry recoverable information about the ground, rather than merely noise to be discarded, recovering a thermal conductivity within 10% of a reference value from the first three hours alone. Tests deliberately made of successive pulses fall within the same framework (Fossa et al., 2018).

Excitation function method - The drop-down at the top of the panel selects how the temperature difference across the borehole, from which the excitation of the convolution is built, is obtained from the record:

• T(↓) − T(↑), z=0 — directly from the two measured temperatures. This is the default, and the right choice whenever both probes are trustworthy.
• Q/(V·C_f) — from an energy balance on the measured heating power and flow rate. The two are two routes to the same physical signal, so a marked disagreement between the fits they produce is itself informative: it means the power, the flow rate and the temperature difference are not mutually consistent, which usually points at a mis-scaled flow meter or at a fluid mixture other than the one declared.

The flow rate handed to the model is the mean over the heating phase alone, not over the whole record. On a test where the pump stops during the recovery, an average over everything would pull that input below the range the model was established over.

Interactive calibration - Six single-handle sliders are used to interactively select λ_s, C_s, λ_g, C_g, s and T_g. To refine a selection, first click a slider, then press the A (←) or D (→) keys; Q (←) and E (→) do the same here. Each move re-runs the simulation and updates the fit statistics and the chart.

The measured inlet and outlet temperatures are drawn as thick red and blue curves, and the temperatures simulated by the model as thinner curves in those same two colours, lightened rather than a third, unrelated one: the hue marks the channel, inlet or outlet, and the intensity marks the origin, measured or modelled. Calibration consists in moving the parameter sliders until the thin curves lie on the thick ones. Judge the fit over the entire record, heating and recovery alike: a set of parameters that reproduces the heating phase while missing the recovery has not identified the ground, it has merely matched a slope.

The ranges of the first four are those of the training envelope: λ_s from 0.5 to 6.0 W/m·K, λ_g from 0.5 to 4.0 W/m·K, and both volumetric capacities from 1.5 to 2.9 MJ/m³·K. T_g runs from −10 to 40 °C, wide enough to accept a cold subsurface or a warm site without clipping, and its handle is placed on the value determined by the Undisturbed temperature panel as soon as one is available. s is the shank spacing, the centre-to-centre distance between the two pipes, the same quantity entered on the Test borehole panel, so the two read directly against one another. Its range is not fixed but derived from the borehole itself, spanning nearly the full space available between the two pipes without letting either cross the borehole wall. Its step is one micrometre, fine enough that the response can really be tuned rather than jumped over. The end of this page gives a practical way to work through the six sliders.

Range of validity - Because the underlying network was trained over a finite range of conditions, it should be used as an interpolator, not an extrapolator. The training envelope covers the great majority of conventional tests: a ground conductivity from 0.25 to 8 W/m·K and a grout conductivity from 0.25 to 5 W/m·K, volumetric capacities from 1.4 to 3.0 MJ/m³·K, boreholes 50 to 250 m long with a radius up to 0.1 m, pipe inner radii from 12 to 27 mm and outer radii from 15 to 36 mm, a pipe conductivity from 0.3 to 1.0 W/m·K, a fluid volumetric capacity from 3.8 to 4.3 MJ/m³·K, and flow rates from 15 to 35 L/min. The sliders are deliberately narrower than that envelope, so that a normal interpretation stays inside it. What the sliders do not police are the quantities read from the other panels: a borehole geometry or a flow rate outside these bounds, or a configuration other than a single U-tube, falls outside what the model learned and its estimates become unreliable. The test must also fit within the 21-day window; a longer record is truncated to it. When inputs sit near these edges, lean on the first-order approximation as a check.

Assumptions and limitations - The assumptions underlying this model are a constant circulation flow rate and the absence of groundwater flow around the GHE. If one or more of these assumptions are not met, it should not be used to analyze the TRT. Two conditions sit outside the training envelope altogether rather than at its edge, and no slider position repairs them: a ground that is layered rather than uniform, and groundwater advection. Both are active research subjects, and a network trained on a heterogeneous medium with advection has since been demonstrated (Rose et al., 2024), but the network shipped here does not carry them.

One limitation is not an assumption to be checked but a boundary of the model itself: the network was trained on single U-loop boreholes only, and represents any other test borehole as one regardless of what the Test borehole panel shows. The first-order approximation carries no such restriction.

Summary & charts: the simulated curves, their residuals and the calibrated parameters

Residual histogram, and where the fit is reported - The panel to the right shows the distribution of the residuals, measured minus simulated. The numbers behind it, the current value of each of the six parameters, the ground and grout thermal diffusivities α = λ/C they imply, the mean, standard deviation and RMSE of the residuals, and an experimental effective borehole resistance R_b^*, computed by the same multipole method as the Test borehole panel but from these calibrated parameters instead of the entered ones, are reported on the third page of the console, which selecting this node brings up.

The model predicts both fluid temperatures, and both are drawn on the chart: the convolution yields the outlet temperature, and the inlet temperature follows by adding the temperature difference across the borehole.

TRT in practice: a good fit is not proof of the right parameters

Where to start. Set λ_s and T_g from the first-order approximation, then adjust the remaining four sliders to drive the RMSE down. Calibrate the parameters the test is most sensitive to first, typically λ_s and T_g, before fine-tuning the grout properties and the spacing, which influence mainly the early-time response. A good fit reproduces both the heating and the recovery temperatures, not just one of them.

Reading the fit while you calibrate. Unlike a least-squares regression, nothing forces these residuals to be centred, so their mean carries information: a mean away from zero says the simulated curve sits above or below the measurement throughout the test, a bias that no amount of scatter explains and that usually points at T_g or at the ground conductivity. The RMSE is the figure the sliders above are driving down, and the histogram beside it says how: a narrow symmetric bell centred on zero is a fit, a shifted one is a bias, and a broad or double-peaked one says the model follows part of the test only.

Calibrating six parameters against one temperature record is an inverse problem, and inverse problems rarely have a single answer. Several combinations can reproduce the same measurements about equally well: a slightly lower conductivity offset by a warmer T_g, or a poorer grout offset by a wider pipe spacing, can land within a few hundredths of a degree of one another in RMSE. Driving the misfit down therefore proves that a combination is compatible with the data, not that it is the one the ground actually holds.

Two more habits keep this in check. Constrain what you can measure independently rather than fitting it: take T_g from the Undisturbed temperature panel instead of tuning it freely, and hold the geometry to what the completion record says. Then try to falsify the fit rather than to admire it: move a parameter away from its calibrated value and see whether the RMSE truly objects. A parameter the test does not constrain will barely move it, and that is worth knowing before quoting it in a report. Where the answer stays ambiguous, report a range and say which parameters were fixed rather than a single set of six figures the data cannot separate.