Generic modelling to develop thermal yield nomograms for coaxial deep borehole heat exchangers (DBHEs)

At present, there is little design guidance for coaxial DBHEs. Some guidance, in the form of a heat yield nomogram, has been published by Holmberg et al. (2016) and Banks (2023a), although the former was restricted to depths of less than 1 km and the latter was based on a simple line source analytical model (see equation 1 later in this paper). It is the authors’ experience that prospective developers sometimes have unrealistically high expectations of the sustainable thermal output of DBHEs. In the absence of simple models or design guidance, advanced and potentially time-consuming 3D numerical models may be applied. The authors argue that simpler, more transparent, analytical models may be adequate for the evaluation of single DBHEs (especially at the early design/feasibility stage), and that these can be used to derive a set of nomograms (look-up charts) that practitioners can use to gain a first-pass estimate of the thermal yield of a single DBHE. This approach has been adopted by the industry for shallow BHEs (MCS 2021; see above).

The model used for this study was developed by Beier (2020) on MATLAB software. It forms a 2D analytical solution and was originally designed for thermal response tests but has been used in studies for model benchmarking and a comparison of systems’ operation over their lifetime (e.g. Cai et al. 2022; Brown et al. 2023d; Kolo et al. 2023a). The model, which can simulate a constant heat load with a geothermal gradient implemented in the subsurface, solves the governing equations for heat transfer and boundary conditions using Laplace transformation, and it accounts for thermal storage in the circulating fluid. However, the model does have limitations including: (i) it can only operate with a fixed thermal power boundary; (ii) it does not fully account for thermal storage in the pipe, which can lead to errors (that are likely to be most significant during short-term simulations) if the pipe wall thickness is large; and (iii) it neglects axial heat transfer in the BHE grout and the surrounding ground (a limitation that may cause some discrepancies over an operational lifetime but which is generally not significant for short-term simulations, e.g. the several days’ duration of a thermal response test). Beier (2020) states that the ‘model can calculate transient temperatures at much larger time periods such as 10 or 20 years, corresponding to the design period of a borehole’.

The model has been validated by Beier (2020) against recorded measurements from a distributed thermal response test from Acuña and Palm (2010, 2013), and also Holmberg et al. (2016), providing an excellent fit between modelled and real data. The test data of Acuña and Palm (2010, 2013) have also been compared by Brown et al. (2023b) to other numerical models developed in OGS (e.g. Shao et al. 2016; Chen et al. 2019) and MATLAB (Brown et al. 2021), showing little discrepancy.

The Beier (2020) model has been applied in this study to a generic coaxial DBHE of varying depth (D) and rock thermal conductivity (⁠$λ$⁠) using the parameterization provided in Table 1. These two parameters only have been systematically varied, in order to keep output nomograms simple, and also because equation (1) suggests that these are the main controlling factors that are not specific to internal borehole design. For simplicity, other parameters have been kept constant, as previous sensitivity analyses of BHE (Banks 2012a, b; Le Lous et al. 2015) suggest that parameters such as rock volumetric heat capacity (⁠$ρ$c) and borehole diameter (r_b) have relatively little direct impact on the thermal output (these parameters are contained in the logarithmic portion of equation 1 rather than in the linear portion). The rock thermal conductivity (⁠$λ$⁠) is to be understood as a depth-weighted average over the length of the borehole: previous analytical and modelling studies confirm that stratal layering has only a very minor influence on thermal output (Banks 2022; Kolo et al. 2023a).

In the application of the Beier (2020) model in this paper, an initial temperature distribution with depth was calculated based on a surface temperature of 10°C (characteristic of annual average soil temperature across much of the UK: Met Office 2023) and a geothermal gradient of 0.025 K m⁻¹. Both the surface temperature and the geothermal gradient are reasonable, conservative, values for the UK. These parameters will have an impact on the rock temperature and thus on the thermal output of a DBHE but have not been systematically varied in this paper. The potential impact of these is considered in the Discussion. The thermal outputs of the DBHE described in this paper are thus characteristic of a reasonably efficient DBHE design (reasonably low R_b,eff) in temperate, tectonically stable, non-radiogenic terrains (e.g. much of northern Europe): the authors acknowledge that the derived nomograms will have a decreasing validity beyond those regions.

The thermal conductivity of the rock has been systematically varied between 1.6 and 3.6 W m⁻¹ K⁻¹ in 0.4 W m⁻¹ K⁻¹ increments. These values are regarded as being characteristic of the lithologies commonly found in the UK (Rollin 1987). The DBHE depths considered are 200 and 500 m, and then increasing in 250 m increments to 3000 m. At the maximum depth, the bottom hole temperature is 85°C using our parameterization.

Aspects relating to the choice of borehole construction material and (especially) circulating fluid flow rate will affect the borehole thermal output via the lumped parameter R_b,eff in equation (1). Moreover, it is acknowledged that the borehole and pipe diameters may exert a significant indirect effect on the borehole thermal output by affecting fluid flow velocity, turbulence, internal heat transfer and thus R_b,eff. These aspects will be highly borehole specific, and have been reviewed by several authors including Iry and Rafee (2019), Liu et al. (2019), Pan et al. (2020), Chen and Tomac (2023) and Chen et al. (2023). In this study, borehole construction and fluid flow have been parameterized generically and moderately conservatively but have not been systematically varied.

Two variants of the borehole have been studied: a borehole with grouted steel casing and an open-hole borehole (Fig. 3). The borehole parameterization broadly reflects the parameters applied to previous simulations of the proposed Newcastle Science City DBHE in the NE of England (Brown et al. 2023b, c, d; Kolo et al. 2023a), although the dimensions should not be construed as a recommendation for DBHE construction and some parameters have been adjusted to reflect more generically applicable values in the literature. The central pipe has been assumed to be made of HDPE with a thermal conductivity of 0.45 W m⁻¹ K⁻¹, although it is acknowledged that, in very deep hot environments, the thermotolerance and mechanical properties of HDPE may not make it a preferred choice. An alternative option would be steel, although its cost, weight and high thermal conductivity are not necessarily favourable (a high thermal conductivity increases thermal short-circuiting between the upward and downward coaxial flows). Another possibility might be some form of glass-reinforced thermotolerant plastic, which has a lower thermal conductivity than HDPE of 0.2–0.3 W m⁻¹ K⁻¹ (Dmitrevsky et al. 1987; East Coast Fibreglass Supplies 2023; Professional Plastics 2023). Therefore, the selection of HDPE in the model can be regarded as conservative.

The other parameter in the model that has a major influence on the results is the flow rate of the heat transfer fluid, which is assumed to be water in the model. The flow rate will often be selected so that it is able to yield:

• a transient turbulent flow regime in the annulus;

• acceptably low hydraulic head losses (MCS 2021 recommends that the electrical input power for the circulation pump should be less than 2.5% of the thermal output of the heat pump attached to a BHE); and

• a reasonable temperature differential at the surface heat exchanger (which might be in the range of 3–8°C for a water–water heat pump).

In order to simulate the unlined borehole in the Beier (2020) model, the thermal conductivity and volumetric heat capacity of the grout and casing have been set to the properties of the rock (the model requires a finite casing and grout thickness, so they cannot be omitted). This effectively results in an unlined borehole with a diameter of 161.7 mm (Fig. 3).

In this study, a single borehole cross-section has been selected for the entire borehole (Fig. 3) for each of the two cases. In reality, it is acknowledged that deep boreholes will often be constructed with several strings of ‘telescoped’ concentric, cement-grouted casing to different depths – the Beier (2020) model does not accommodate complex borehole designs. It is also acknowledged that Figure 3 may underestimate the average borehole diameter and thickness of cement-grouted annulus present (which may be subject to regulation or best-practice guidance) in a real deep borehole.

Simulations have been run for a 25 year period for each depth and thermal conductivity value by adjusting the constant heat extraction rate, in increments of 0.1 W m⁻¹, in the model until the temperature of the fluid entering the borehole reaches 5°C. This criterion is selected to prevent the practical risk of ice-crystal formation within the surface heat exchanger (Chen et al. 2019 used a similar temperature of 4°C). If an antifreeze additive were used, of course, lower temperatures (and greater heat yields) could be achieved.

For the Beier (2020) model, the flow (T_flow) and return (T_return) temperatures to and from the DBHE were averaged to yield an apparent average fluid temperature (T_ave) as ‘seen’ at the surface. The simulated data were then compared to the log-linear straight-line temperature evolution predicted by the line source heat equation (equation 1).

In an Excel spreadsheet environment, a best-fit log-linear trend line was fitted to data between 24 and 500 h, and equation (1) was solved to derive an estimate of the rock thermal conductivity and effective borehole thermal resistance from the slope and intercept, respectively. This procedure was repeated, using alternative subsets of data, between 24 and 3166 h and between 50 and 3166 h.

The OGS software (Kolditz et al. 2012), which was used for model benchmarking, employs the ‘Dual-Continuum’ method for the wellbore and surrounding formation, with finite-element spatial discretization (Shao et al. 2016; Chen et al. 2019). The method treats the wellbore as a series of 1D elements embedded in a 3D geological domain, saving computational time. It is a proven approach for deep coaxial systems with good comparisons to other numerical and analytical solutions (e.g. Cai et al. 2022; Kolo et al. 2022; Brown et al. 2023a). It has also been tested and validated against real data for a range of BHE configurations, exhibiting minimal discrepancy (e.g. Shao et al. 2016; Cai et al. 2021; Chen et al. 2021; Brown et al. 2023b).

In the OGS modelling environment, the same parameterization has been applied as in the Beier model. A constant heat-flow boundary is, however, applied to the base of the model, which results in a geothermal gradient of 0.025 K m⁻¹ for the rock thermal conductivity in the specific simulation. For example, for a thermal conductivity of 2.0 W m⁻¹ K⁻¹, the heat flow applied would be 2.0 W m⁻¹ K⁻¹  ×  0.025 K m⁻¹  =  0.05 W m⁻².

Earth Energy Designer (EED) is a commonly used industry analytical model for the design of borehole heat exchangers up to a maximum depth of 300 m (Hellström and Sanner 2000; BLOCON 2019). The analytical model employs the principle of superposition of heat extraction step functions, coupled to a library of mathematical g-functions for a wide variety of borehole array shapes, ultimately derived from numerical modelling. Much of the background to the model is described by Eskilson (1987).

EED version 4.2 (BLOCON 2019; Blomberg et al. 2019) has been used to simulate the 200 m-deep cased and unlined cased borehole scenarios, for rock thermal conductivities of 1.6 and 3.6 W m⁻¹ K⁻¹ using the constant heat extraction rates found using the Beier (2020) model. Parameterization was exactly as described in Table 1 and the simulation was run for 25 years, commencing in the month of January. A value of effective borehole thermal resistance (R_b,eff) was calculated by the software from the borehole geometry (Fig. 3), accounting for the internal heat transfer between the upward flow and the downward flow of fluid, and with no additional imposed grout contact resistance.

The results of the 25 year Beier (2020) simulations, which represent the data for further interpretation and discussion in this paper, are presented as Supplementary material A. The nomograms resulting from the simulations are presented as Figure 4 (cased borehole) and Figure 5 (unlined, open borehole). Continuous heat outputs of 2.4–5.1 kW (12.0–25.7 W m⁻¹) can be obtained from the 200 m borehole, ranging up to 165–281 kW (54.9 to 93.7 W m⁻¹) from a 3000 m borehole. Thermal output increases non-linearly with depth; it also increases with thermal conductivity, although the rate of increase is less than directly proportional. The underlying data are presented in Supplementary material A.

The nomograms compare well with findings from other modelling studies based on real DBHE scenarios: Kolo et al. (2023a) modelled a thermal yield of c. 50 kW for a 922 m DBHE in rocks with a thermal conductivity of 2.55 W m⁻¹ K⁻¹. This is slightly greater than predicted by Figure 4 and is explained by the lower fluid temperatures allowed and the higher geothermal gradient of 0.0334 K m⁻¹. Kolo et al. (2023b) modelled a thermal yield of around 200 kW in the context of a pre-feasibility study of a 2776 m DBHE in a mixed sedimentary sequence (c. 2.5 W m⁻¹ K⁻¹) of the Magnus offshore field (Petrofac 2022). Arguably, a comparison with modelling studies is of limited significance, given the similar software environments applied. A comparison with real DBHE trials is problematic, however, as these have typically been tested or operated for shorter periods and/or at higher fluid temperatures than are assumed in this paper's nomograms. For example (Chen and Tomac 2023):

• Hawaii geothermal project: 876.5 m DBHE, thermal output of 76 kW for a 7 day test;

• Weggis (Switzerland): 2300 m DBHE, thermal output of 42–50 kW (annual average);

• Weissbad (Switzerland): 1213 m DBHE, thermal output of 80 kW;

• Qingdao (China): 2605 m DBHE, thermal output of 448 kW for a 138 day test;

• Beijing (China): 1800 m DBHE, thermal output of 237–257 kW after 40–97 h tests; and

• Xi'an (China): 2500 m DBHE, thermal output c. 300 kW (Chen et al. 2023).

From Figures 4 and 5, the thermal outputs from the uncased borehole were slightly higher than for the cased borehole, and this can be ascribed to the thermal resistance represented by the low conductivity of the grout in the annulus behind the casing. The output from the unlined borehole is 50 W higher than the cased borehole for the ‘low-conductivity, 200 m’ scenario, increasing to 10.2 kW for the ‘high-conductivity, 3000 m’ scenario. The discrepancy is greatest for high rock conductivities (where the borehole thermal resistance is a more significant limiting factor) than for low rock conductivities (where the rock thermal conductivity is a more significant limiting factor).

In the models described here, a perfect thermal contact has been assumed between the grout and the rock wall of the borehole, and between the grout and the casing. In reality, the thermal resistance might be significantly larger than that represented in these idealized models, depending on the quality of the grouting job, the thickness of the grout and (arguably most importantly) the number of telescoped grouted casing strings employed when advancing the borehole.

To test the validity of the nomograms, a series of benchmarking scenarios were modelled and compared to the solutions from OGS (Fig. 8; see also Supplementary material B). Results indicate that there is generally a good match between the different modelling software; however, the solution from the model of Beier (2020) provides somewhat lower estimates of circulation fluid temperatures. One possible reason for this is that the Beier (2020) model does not explicitly model axial (vertical) heat transfer in the grout or ground (which OGS does).

For the range of depths modelled, the maximum difference in circulation fluid temperature was recorded for the higher thermal conductivity scenarios at 200 and 1000 m depth, while at 3000 m depth the lower thermal conductivity scenario showed more disagreement between model results. The recorded average fluid temperatures at the end of the solution showed a maximum discrepancy of (i) 0.6°C for 200 m, (ii) 0.9°C for 1000 m and (iii) 1.7°C for 3000 m. Nevertheless, the difference in solutions is relatively minor over the lifetime of a DBHE operating at a constant base load. Because of this discrepancy, the solutions from the model of Beier (2020) yield somewhat conservative (i.e. pessimistic) thermal power outputs in comparison to the solution from OGS.

The Beier model generally compares very well with results from EED, although the advised maximum depth limit of validity for EED is 300 m (BLOCON 2021), so deeper simulations were not compared. After 25 years of simulation, the Beier average fluid temperatures are typically around 0.2°C cooler than the EED simulation (Fig. 9). In the EED simulation, the fluid temperature evolves slightly less steeply than in Beier (2020). The observation may, at least in the latest part of the time series, be related to the EED model explicitly accounting for interactions in the vertical dimension with the atmosphere and underlying rocks (i.e. non-radial flow at a late time), which will tend to result in a flattening of the temperature trend towards steady state. However, the Beier model can be regarded as conservative relative to EED, as it delivers slightly lower average fluid temperatures.

From Figure 7, it can be seen that the fluid temperature evolution predicted by Beier (2020) closely approximates a log-linear trend. When examined closely, the evolution is very slightly concave upwards, indicating that the rate of temperature evolution with the logarithm of time decreases slowly during the simulation.

Inverse analysis of the best-fit log-linear trend using the line source heat equation (equation 1) yielded estimates of thermal conductivity of rock (⁠$λ$⁠) and borehole thermal resistance (R_b,eff). Due to the decelerating trend of temperature evolution with time, the analyses using data up to 500 h yielded slightly lower estimates of thermal conductivity than the analyses using data up to 3166 h (Table 2). In all cases, the estimates of thermal conductivity were lower than the ‘true’ value (i.e. input value to simulation), a finding that was also noted by Beier (2020). This indicates that the behaviour of the coaxial DBHE does not perfectly follow the line source heat equation; nor would one expect it to, given that the line source equation treats borehole thermal resistance as a constant value and neglects geothermal gradient, thermal storage effects in the borehole, borehole end effects and vertical heat migration. In a DBHE, internal heat transfer between the uphole and downhole fluid flows represents a significant component of borehole thermal resistance and, as this may progressively change, one should arguably not expect the ‘constant single value’ approach used in the line source heat equation to be perfectly valid.

Table 2 compares estimates of $λ$ and R_b,eff for different sections of data for each of three cases. The section of data between 50 and 3166 h gives the closest approximations to the imposed model value of $λ$ and also the highest (and most conservative) values of R_b,eff. Thus, for the spread of model scenarios, from 200 to 3000 m, and in cased and uncased boreholes, this section of data has been used to estimate values of effective borehole thermal resistance. The results are shown in Figure 10. The borehole resistances from this study broadly correspond with, although are a little lower than, values cited by Thomasson and Abdurafikov (2022), which is most likely due in part to the higher fluid flow rate applied in this study.

As one would expect, rock thermal conductivity has almost no influence on derived value of R_b,eff, as the value should theoretically depend solely on the borehole construction and flow rate. It transpires that depth is a major influence on R_b,eff, increasing significantly at depths greater than 1000 m. From Figure 10, it is inferred that internal heat transfer between the upflow and downflow fluid streams is not significant at a flow rate of 5 l s⁻¹ at depths shallower than 1000 m. At greater depths, internal heat transfer rapidly becomes the dominant component of borehole thermal resistance.

In this paper, we have not explored the effect of varying flow rate (nor the effect of varying borehole or pipe diameters, which would also alter the fluid flow velocity) but we know from previous studies (Kolo et al. 2023a) that lowering the flow rate will increase the internal heat transfer between the upflow and downflow fluid streams, which increases R_b,eff and imposes a finite borehole thermal resistance at shallower depths. In this study, we have assumed an HDPE central coaxial pipe; a more conductive (steel) pipe would allow more internal heat transfer and result in a higher R_b,eff. Some researchers have explored the possibility of insulated or vacuum-filled coaxial pipes in order to reduce the internal heat transfer (Rybach and Hopkirk 1995; Śliwa et al. 2017; Wirtén 2018).

Earth Energy Designer calculated values of effective borehole thermal resistance of 0.002 K m W⁻¹ for the unlined 200 m borehole, and 0.032 K m W⁻¹ for the cased borehole. These are marginally higher than the values derived from the slope and intercept of the Beier (2020) model (Fig. 10). EED has an advised depth limitation of 300 m (BLOCON 2021).

Figures 4 and 5 indicate that the relationship between depth and heat extraction (H) is non-linear with thermal output increasing disproportionately with depth. With reference to equation (2), this because the heat extraction (H) is:

• approximately proportionate to the available length (depth) of the borehole heat exchanger (D) but is also

• related to the difference between the average undisturbed ground temperature along the borehole length (T₀) and the average fluid temperature (T_ave). As borehole depth increases, T₀ also increases with the geothermal gradient.

• The geothermal gradient (which here is fixed at 0.025 K m⁻¹) and the surface temperature (fixed at 10°C), both of which will control T₀. As the geothermal gradient increases, T₀ and thus the thermal output H will both increase (equation 2). Additional nomograms could be generated for a range of geothermal gradients using the Beier (2020) code, as required.

• The acceptable minimum fluid inlet temperature during operation, which will be related to T_ave, depending on the flow rate. This minimum will depend on the fluid used but will typically be a few degrees above 0°C for water, and around or just below 0°C for an antifreeze-based solution.

• The effective borehole thermal resistance (R_b,eff), which will depend on the borehole construction and depth, and on the fluid flow rate. This paper suggests that, for DBHEs, the fluid flow rate (which controls the internal heat transfer between the upflow and downflow fluid) becomes increasingly important with increasing depth (Fig. 10). A high fluid flow rate is advantageous for the thermal performance of a DBHE (Kolo et al. 2023a) but, if it becomes excessive, it will create a high hydraulic resistance and consume large amounts of parasitic pumping power, to the overall detriment of the energetic efficiency of the DBHE.

• The effective borehole thermal resistance (R_b,eff) will also depend on the properties of the casing and (especially) the grout, if present. The modest difference between Figures 4 and 5 can be ascribed to the presence of a low thermal conductivity annular thickness (19 mm in our simulation) of cement grout behind the casing. Provided that the thermal conductivity of the grout is lower than that of the rock, R_b,eff would be expected to increase as the thickness of the grout increases (especially if multiple casing strings and grouted annuli are present).

The three models tested in this paper, OGS, EED and Beier (2020), yield similar trajectories for the evolution of borehole fluid temperature over a 25 year simulation. The Beier (2020) model yields the lowest fluid temperatures and the lowest thermal outputs, and can be regarded as the most conservative model for preliminary design purposes. The Beier (2020) model reproduces the results of EED closely, although the EED model is advised to be valid to depths of 300 m only. There is a relatively minor discrepancy between the Beier (2020) model and OGS, with OGS yielding somewhat higher fluid temperatures and higher thermal outputs; the relative discrepancy decreases as depth increases, from 7% at 200 m to 2–4% at 3000 m (Table 3). One possible reason for the discrepancies is that both EED and OGS take account of vertical heat transfer and borehole end effects, while the Beier (2020) model neglects axial (vertical) heat transfer in the grout and ground. If this were the dominant effect, one would expect the discrepancy to be larger for shallow boreholes, which is the case. Another source of discrepancy is how the various models account for the complex nature of heat transfer between solid materials and the circulating fluid. We can conclude, however, that for a first-pass study of a DBHE design, the Beier model yields acceptable and conservative results. For shallower boreholes, the discrepancy is larger but alternative models, such as EED, are available.

The Beier (2020) modelling indicates that the line source heat equation offers an acceptably close (but not perfect) approximation to the behaviour of a DBHE, even to depths of 3000 m, at least for the purposes of an early-stage feasibility study. This implies that the performance of the DBHE can be predicted using a spreadsheet or relatively straightforward analytical algorithm. Complex heat loads can be simulated by the principle of superposition and a set of heat loads of varying magnitude and duration applied as Heaviside-type step functions (Claesson and Eskilson 1988; Banks 2012a). It was, in fact, this approach that was used by Banks (2023a) to produce nomograms for partial loads. The main challenge to this approach is assigning a realistic value of effective borehole thermal resistance to equations (1) and (2). The borehole thermal resistance will depend strongly on fluid flow, and it is to this aspect that future research could usefully be dedicated.

Earlier analytical modelling by Banks (2022) indicated that the following factors could affect the application of a line heat source approach to modelling DBHE:

1. The polarity of flow: in this paper, we have assumed that for heat extraction, fluid flow down the annulus and up the central pipe (so-called CXA configuration) allows more efficient heat exchange than the opposite flow polarity. Banks (2022) explains the background to this assertion. The opposite is also true: that for heat rejection to a DBHE, flow down the central pipe and up along the annulus (CXC configuration) offers improved heat exchange. It is argued, however, that the polarity of flow would merely impact the value of R_b,eff and not the validity of the line heat source approach.

2. Heterogeneity of thermal properties of the rock: in this paper, the thermal properties of the rock are assumed to be homogeneous and isotropic. For the modelling of a shallow BHE, it is common practice to assume that a simple thickness-weighted average of thermal conductivity and volumetric heat capacity can be applied (Banks 2012a), yet systematic heterogeneity in rock thermal properties can have some impact on the performance of a DBHE (Banks 2022). For example, a large thickness of low-conductivity rock overlying a large thickness of high-conductivity rock at depth (where the temperature is higher) will result in better DBHE performance than where high-conductivity rock overlies low-conductivity rock (albeit with the same thickness-weighted average). Signorelli et al. (2007) carried out numerical modelling studies that reached similar conclusions. In both cases, however, the discrepancies were rather minor (c. 3%) and do not invalidate the line heat source method as an initial reconnaissance analytical modelling approach for DBHE. Kolo et al. (2023a) also found that the heterogeneity has relatively little impact on the DBHE thermal output.

Simulations, using the Beier (2020) analytical model, of 200–3000 m deep coaxial borehole heat exchangers of two generic designs, in rocks with an average thermal conductivity of 1.6–3.6 W m⁻¹ K⁻¹, indicate that the following continuous heat outputs can be obtained:

• 2.4–5.1 kW (12.0–25.7 W m⁻¹) from the 200 m borehole;

• 27–55 kW (27.3–54.8 W m⁻¹) from the 1000 m borehole; and

• 165–281 kW (54.9–93.7 W m⁻¹) from the 3000 m borehole.

The modelling work has allowed the construction of Figures 4 and 5, nomograms that can be applied by geoscientists, engineers and developers of DBHE at an early stage of feasibility study to gain an approximate idea of the thermal outputs that might be expected. This study has some limitations inasmuch as it has focused solely on varying borehole depth and thermal conductivity; it has not explored dependence on heat capacity (to which thermal output has previously been demonstrated to be relatively insensitive), borehole, casing or pipe diameter or flow rate (which are highly borehole specific). In this paper, reasonably generic values of these latter parameters have been selected. The modelling approach used in this paper could, however, be extended to generate further suites of nomograms for different geothermal gradients and fluid flow rates, which would further extend their applicability as a design tool.

The evolution of fluid temperature with the logarithm of time is approximately linear, implying that the behaviour of the temperature field in the rock or sediment surrounding the DBHE approximately conforms to the line source heat equation. This in turn implies that, provided an appropriate value of effective borehole thermal resistance can be assigned to the borehole, the line source heat equation (equations 1 and 2), in conjunction with superimposed Heaviside step functions, can be used to approximately simulate the behaviour of the DBHE under variable, rather than constant, thermal loads in a manner proposed by Eskilson (1987) and Banks (2012a).

The authors recognize that numerical models will continue to have an application in the detailed design of DBHEs, especially in circumstances where the geology or borehole construction is strongly inhomogeneous, anisotropic or asymmetrical: for example, where:

• the geology is not radially symmetrical (e.g. deviated or subhorizontal DBHE, faults in the vicinity of borehole or steeply dipping strata);

• there is a strong depth-dependent inhomogeneity in the geology (e.g. highly conductive deep rocks underlying poorly conductive shallow rocks);

• the borehole construction is complex (deviated boreholes, multiple grouted telescoped casing strings, partial grouting or variable materials).

• there are variable fluid flow conditions along the borehole length (e.g. where variable diameter causes varying fluid flow velocity down the borehole);

• groundwater flow is important for the subsurface advection of heat; or

• multiple DBHEs are envisaged.

The authors would like to thank the reviewers and editors of this paper for their diligence and useful critique and suggestions. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) license to any Author Accepted Manuscript version arising from this submission.

DB: conceptualization (lead), formal analysis (lead), methodology (lead), writing – original draft (lead); CSB: conceptualization (supporting), formal analysis (supporting), methodology (supporting), validation (equal), writing – original draft (supporting), writing – review & editing (equal); IK: conceptualization (supporting), formal analysis (supporting), methodology (supporting), validation (equal), writing – review & editing (equal); GF: conceptualization (supporting), funding acquisition (lead), methodology (supporting), project administration (lead), writing – review & editing (equal).

This work was funded by the UK Engineering and Physical Sciences Research Council (EPSRC) grants EP/T022825/1 and EP/T023112/1. The funding source is for the NetZero GeoRDIE (Net Zero Geothermal Research for District Infrastructure Engineering) and INTEGRATE (Integrating seasoNal Thermal storagE with multiple energy souRces to decArobonise Thermal Energy) projects, respectively.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The Beier model used in this study is available as a MATLAB code as supplementary material D to Beier's (2020) paper at https://ars.els-cdn.com/content/image/1-s2.0-S1359431120306992-mmc1.zip. The results from the modelling work are provided as Supplementary material A to this paper. A more complex version of Figure 8, showing fluid flow and return temperatures, is provided as Supplementary Material B.