There are two ways to arrive at a borehole field length, and the best designs use both, in sequence. A sizing equation returns a fast, transparent estimate and shows why the field is the size it is; a full simulation then reports what the system actually does, hour by hour, over the whole design horizon. The two are often treated as rivals. They are better understood as the same model at two levels of detail — and used in that spirit, the first becomes the natural starting point for the second.
The sizing equation is the simulation equation, simplified
A simulation computes the fluid temperature from the entire load history by temporal superposition. In its common g-function form, the mean fluid temperature is
$$ T_m(t) = T_g + \frac{q\,R_b^{\ast}}{L} + \frac{1}{2\pi \lambda L}\sum_{i=1}^{n}\bigl(q(t_i)-q(t_{i-1})\bigr)\,g\!\left(t-t_{i-1},\,L\right) $$
where $T_m$ is the mean fluid temperature, $T_g$ the undisturbed ground temperature, $q$ the ground thermal load (W; reported in kW in Figure 1, and counted positive for heat injected into the ground), $L$ the total length of all boreholes in the field (m), $R_b^{\ast}$ the effective borehole thermal resistance (m·K/W), $\lambda$ the ground thermal conductivity (W/(m·K)), and $g$ the dimensionless g-function. The sum superimposes every step change in load, each weighted by the ground's response over the time elapsed since it occurred.
The g-function is not a single curve: it is the ground's thermal response for a given field configuration — the number of boreholes, their spacing and depth, and the ground's thermal conductivity and diffusivity. The borehole internals and the flow through them do not enter here; their effect is carried separately by the effective borehole resistance $R_b^{\ast}$.
Design does not ask for the temperature; it asks for the length that holds a chosen temperature limit. Rearranging the same expression isolates $L$ (Dion and Pasquier 2025):
$$ L = \frac{q\,R_b^{\ast} + \dfrac{1}{2\pi \lambda}\sum_{i=1}^{n}\bigl(q(t_i)-q(t_{i-1})\bigr)\,g\!\left(t-t_{i-1},\,L\right)}{T_m(t) - T_g} $$
The length still appears on both sides through the response, so it is found by iteration, but the question has been inverted: not what temperature results from this length? but what length holds this temperature?
The sizing equation is what this becomes when the full hourly history is reduced to three thermal pulses — a yearly average $q_y$, the monthly average $q_m$ in the month the peak occurs, and the hourly peak $q_h$ — acting over $t_y$ (typically ten years), $t_m$ (one month) and $t_h$ (four to six hours):
$$ L = \frac{q_y R_{gy} + q_m R_{gm} + q_h R_{gh} + q_h R_b^{\ast}}{T_m - T_g} $$
where $R_{gy}$, $R_{gm}$ and $R_{gh}$ are the response-factor resistances the g-function assigns to the three timescales. Nothing new has been introduced: the sizing equation is the simulation equation evaluated at three representative instants and solved for length rather than for temperature.
One step is worth making explicit. The equation is solved by setting the mean fluid temperature $T_m$ equal to the heat pump's operating limit $T_{\text{Lim}}$ — the minimum allowable temperature in heating, the maximum in cooling. It is therefore solved twice and returns two lengths: one governed by the heating limit, one by the cooling limit. They are rarely equal, and the longer of the two governs — it is the length carried forward. Selecting the shorter, or averaging the two, quietly under-designs the season that drives the field.
Why the sizing equation endures
The three-pulse form has anchored ground-loop design since the 1980s, when the first closed-loop sizing procedures were set down (Bose, Parker, and McQuiston 1985). In the standard grading of sizing algorithms — from L0 rules of thumb to L4 hourly simulation — it is the L2, three-pulse level (Spitler and Bernier 2016; Ahmadfard and Bernier 2019). It endures partly by inheritance — it predates fast hourly simulation and remains embedded in standards and practice (Kavanaugh and Rafferty 1997; ASHRAE 2023) — but mostly because it earns its place. It returns a single number, the total length $L$, that a designer can act on immediately, and its structure is legible: each term of the numerator is a distinct, physical contribution to the length.
- $q_y R_{gy}$ — the yearly term, large only when the ground loads are unbalanced;
- $q_m R_{gm}$ — the monthly term, from the average load of the peak month;
- $q_h R_{gh}$ — the hourly peak term, the brief extreme that sets the most demanding fluid temperature;
- $q_h R_b^{\ast}$ — the term that isolates the influence of the borehole resistance $R_b^{\ast}$.
Because each parameter enters the equation explicitly, the effect of changing any one of them on $L$ can be read directly — a transparency an hourly simulation, for all its realism, does not offer as readily: matching it would require a dedicated sensitivity analysis, re-running the simulation for each parameter in turn.
Read together, the four terms show why a field is the size it is, and therefore where to act. A tall yearly term points to an unbalanced load that hybridisation or load-side measures could correct (thermal load quality); a tall resistance term points to a borehole worth improving before adding length (grout, spacing and flow regime); a field dominated by the hourly term faces short extremes, often better answered by auxiliary capacity than by drilling.
What a simulation adds
The sizing equation buys its speed with an assumption: that the fluid temperature just reaches the limit and holds there. A real system is not pinned to its limit — its temperature moves with the weather, the load and the slow drift of the ground. For many load profiles the assumption is nonetheless a sound one: carrying the sized length into a full simulation confirms that the temperature limits are respected across the same horizon the sizing assumed, and the fast estimate lands where the detailed calculation agrees it should. Benchmarked against an independent design tool, sized lengths agree to within a few percent — diverging by only 3.7% and 5.2% for the least regular (L-shaped and rectangular) field geometries, most of that reflecting differences in modelling approach rather than the sizing equation itself (Ahmadfard and Bernier 2018).
Confirming the limit, though, is the least of what a simulation offers. Because it follows the system hour by hour over the whole design life, it also reports the quantities a length alone cannot: the annual energy consumption, the peak electrical demand, the seasonal coefficient of performance (sCOP), and the operating cost of the system — the measures on which a design is ultimately judged. And it resolves what a few representative instants cannot: the slow, multi-year drift of the ground temperature that only a long run reveals. A field that holds its limits comfortably in its first winter can drift a degree or more across two decades as the ground slowly warms or cools.
A linear workflow, two complementary tools
Seen this way, sizing and simulation are not competitors but complementary tools used in sequence. For a trial field geometry — a chosen number of boreholes at a given spacing — the path is linear:
1. Size. Solve the three-pulse equation for that geometry to obtain the total length $L$; with the number of boreholes $N$ fixed, the active length per borehole is $H = L/N$. Read the four-term breakdown to see what drives the result.
2. Simulate. Take that field as the starting point of a full simulation, follow the fluid temperature across the same design horizon used for sizing, and confirm the field holds its limits — while reading off the energy, peak-demand and cost metrics. Here the geometry is no longer fixed: boreholes can be added, the spacing or depth adjusted, and the design iterated interactively until performance and cost are optimized.
Because it resolves the full behaviour and the broader performance metrics, the simulation is the more complete tool, and the better suited to large projects, where the cost of over- or under-sizing is greatest. Each parameter in the sizing equation is a lever treated elsewhere in this series: the loads that drive the whole, the heat-pump performance that sets the temperature limits, the ground properties and layout that shape the response, and the borehole resistance that the fourth term flags directly. The sizing equation identifies which lever is worth pulling; the simulation confirms that pulling it produced the intended result.
Conclusion
A sizing equation and a simulation are not two methods but one, at two levels of detail. The first reduces the load history to three pulses and solves for length: fast, transparent and diagnostic. The second keeps the full history and reports what the system will actually do — its temperatures, its energy use, its cost — across the design horizon. Used in order, for a given geometry, they reinforce one another, and a design that uses both arrives at a field that is neither timid nor extravagant.
References
- Ahmadfard, M., and M. Bernier. 2018. Modifications to ASHRAE's sizing method for vertical ground heat exchangers. Science and Technology for the Built Environment 24 (7): 803–817.
- Ahmadfard, M., and M. Bernier. 2019. A review of vertical ground heat exchanger sizing tools including an inter-model comparison. Renewable and Sustainable Energy Reviews 110: 247–265.
- ASHRAE. 2023. ASHRAE Handbook — HVAC Applications. Atlanta: ASHRAE.
- Bose, J. E., J. D. Parker, and F. C. McQuiston. 1985. Design/Data Manual for Closed-Loop Ground-Coupled Heat Pump Systems. Atlanta: ASHRAE.
- Dion, G., and P. Pasquier. 2025. Ground heat exchanger sizing using borehole outlet transfer function. Science and Technology for the Built Environment 31 (10): 1198–1210.
- Kavanaugh, S. P., and K. Rafferty. 1997. Ground-Source Heat Pumps: Design of Geothermal Systems for Commercial and Institutional Buildings. Atlanta: ASHRAE.
- Spitler, J. D., and M. Bernier. 2016. Vertical borehole ground heat exchanger design methods. In Advances in Ground-Source Heat Pump Systems, ed. S. J. Rees, 29–61. Cambridge: Woodhead Publishing.
Part of the GSHP in Practice series.
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