Overcoming Uncertainties Associated with Local Thermal Response Functions in Vertical Ground Heat Exchangers

Abstract

The short-term performance of ground heat exchangers (GHEs) is crucial for the optimal design of ground-source heat pumps (GSHPs), enhancing their contribution to sustainable energy solutions. Local short-time thermal response functions, or short-time g-functions (STGFs) derived from thermal response tests (TRTs), are of great interest for predicting the heat exchange due to their fast and simple applicability. The aim of this work is to perform a sensitivity analysis to assess the impact of thermal parameter variability and TRT operating conditions on the accuracy of the average fluid temperature (T_f) predictions obtained through a local STGF. First, the uncertainties associated with the borehole thermal resistance (R_b), transmitted from the soil volumetric heat capacity (C_S) or some models dependent on GHE characteristics, such as the Zeng model, were found to have a low impact in T_f resulting in long-term deviations of ±0.2 K. Second, several TRTs were carried out on the same borehole, changing input parameters such as the volumetric flow rate and heat injection rate, in order to obtain their corresponding STGF. Validation results showed that each T_f profile consistently aligned well with experimental data when applying intermittent heat rate pulses (being the most unfavorable scenario), implying deviations of ±0.2 K, despite the variabilities in soil conductivity (λ_S), soil volumetric heat capacity (C_S), and borehole thermal resistance (R_b).

1. Introduction

The increasing global demand for energy efficiency and sustainability has intensified interest in reducing the reliance on fossil fuels, particularly in the building sector, which accounts for approximately 40% of total energy consumption and a significant share of carbon emissions [1,2,3]. Among several renewable energy solutions, low-temperature geothermal energy has emerged as a valuable resource, widely utilized in ground-source heat pump (GSHP) systems. These systems integrate a conventional heat pump with a ground heat exchanger (GHE), where water or a water–antifreeze mixture facilitates heat exchange with the ground [4,5].

The efficiency of GSHPs is strongly influenced by the accurate design of the GHE, as borehole length estimation directly impacts system performance and initial installation costs. An overestimation of borehole depth can lead to unnecessary expenses, while an underestimation may compromise the system’s efficiency [6,7].

The thermal response test (TRT) is a widely adopted method to determine the thermal properties of the ground, which are essential for the design of an efficient GHE. This procedure involves circulating a heated fluid through the ground heat exchanger (GHE) while monitoring the inlet and outlet temperatures over time [8]. Subsequently, the infinite line source (ILS) model is widely employed due to its simplicity and fast computational performance, which takes the TRT data as input. This model incorporates three unknown thermal parameters: soil thermal conductivity (λ_S), soil volumetric heat capacity (C_s), and borehole resistance (R_b). It provides reliable estimates of λ_S, but the R_b estimation requires a previous determination of the soil volumetric heat capacity (C_s) [9,10]. The latter is frequently obtained from the literature [11] or laboratory tests, introducing uncertainties into R_b determination. Furthermore, the estimated R_b value, while valid for long-term operation, does not accurately represent transient short-term conditions within the borehole. During the initial stages of heat injection, transient thermal effects significantly influence the borehole response, impacting the mean fluid temperature (T_f) and potentially affecting the overall system performance [12,13,14]. Consequently, assuming a constant R_b in short-term scenarios may lead to inaccurate heat exchange predictions, emphasizing the need for refined modeling techniques.

To address these challenges, the finite line source (FLS) model offers a more comprehensive approach by integrating both transient and steady-state heat transfer behaviors. This model employs temperature response factors, known as g-functions, which account for variations in borehole geometry, time, and soil properties [15]. The g-function framework distinguishes between short-time g-functions (STGFs), which consider transient effects within the borehole materials [16,17], and long-time g-functions (LTGF), where the influence of the surrounding soil predominates [14,18]. This differentiation ensures a more accurate representation of the heat transfer process across different time scales.

For long-term heat transfer predictions, the LTGF applied in the FLS model can be derived using analytical methods under steady-state conditions [19], providing reasonable approximations of GHE performance despite inherent simplifications. However, these approaches are unsuitable for short-term predictions, needing the development of more advanced STGF determination techniques.

Several approaches have been proposed for STGF estimation: (1) numerical models derived from the finite element method (FEM) and finite volume method (FVM), which provide high accuracy in heat transfer simulations but at a high computational cost [20,21,22]; (2) lumped-element models (LEM) [16,23], which simplify the heat exchanger into discrete thermal resistance and capacitance entities, reducing computational time at the expense of slight accuracy loss; and (3) artificial intelligence techniques, e.g., artificial neural networks (ANNs) [24,25], which generate synthetic data for temperature response prediction, though requiring extensive pre-computation and large datasets. A key advantage of these numerical approaches is their ability to generate smooth response functions for both the ground behavior (STGF) and the time-dependent borehole resistance (R_b). These functions help to identify characteristic thermal profiles, which can be further refined through alternative experimental approaches. Recently, methods based on experimental data have emerged as a promising alternative, directly deriving STGFs from TRT measurements [26]. These methods achieve comparable accuracy with substantially reduced computational costs, making them a favorable choice for practical geothermal system modeling.

A relevant development in local short-time g-function (STGF_M) involves the segregation of borehole thermal behavior from the overall thermal response. The STGF initially derived from the TRT data compensates for the ILS model’s assumption of a constant R_b by exhibiting negative values at early times, thus deviating from an idealized smooth thermal profile. To mitigate this, a refined approach introduces an artificial fluid resistance (R_F) and approximates the segregated borehole resistance (R_bP), effectively filtering initial inconsistencies [27]. This procedure, while not significantly altering the calculated average fluid temperature (T_f), provides a more accurate representation of the borehole’s characteristic equations.

Consequently, the subsequent analysis focuses on evaluating uncertainties in STGF estimation without segregating R_b, ensuring a robust thermal parameter characterization for GHE performance assessment. Validating this fast and accurate procedure simplifies the design of GSHP systems, positively contributing to sustainability by promoting the adoption of low-temperature geothermal energy for building climatization and reducing dependence on fossil fuels. In this sense, this work conducted a sensitivity analysis to evaluate the influence of thermal parameter variability and TRT operating conditions on the consistency of the STGF_M. The analysis first focuses on uncertainties associated with R_b, transmitted from C_S or models dependent on GHE characteristics. Afterwards, multiple STGF_M of the same GHE were generated to model T_f under intermittent operation scenarios, assessing the influence of the variabilities in soil conductivity, volumetric heat capacity, and borehole thermal resistance.

2. Materials and Methods

2.1. Experimental Setup

The experimental setup consists of a single U-pipe ground heat exchanger (GHE) installed at the University of Jaén. The U-pipe was made of polyethylene (PE 100 SNR11 PN-16) with an external diameter of 40 mm and a wall thickness of 2 mm. The borehole had a depth of 130 m and a diameter of 150 mm, filled with bentonite to ensure proper thermal contact with the surrounding soil. Water was used as the heat transfer fluid circulating through the system.

A specialized thermal response test (TRT) device was connected to the U-pipe to conduct multiple tests under controlled conditions. The system operates as a closed-loop circuit, as shown in Figure 1, driven by a centrifugal hydraulic pump that ensures continuous fluid circulation. Heat is applied to the fluid using an electrical resistance immersed in a small tank. A T-valve configuration allows the fluid to either bypass or pass through the heating tank depending on the test requirements.

The setup was equipped with an ultrasonic volumetric flowmeter (range 0.25–25 L/min, ±3% accuracy) and piezo-resistive pressure sensors (range 0–1 MPa, ±0.003 MPa accuracy). Temperature measurements were obtained using PT-100 sensors with a range of 273 $\mathrm{K}$ to 323 $\mathrm{K}$ and an accuracy of ±0.5 $\mathrm{K}$, positioned at both the inlet and outlet ports. Additional components ensured system stability and safety, including an expansion tank, impurity filter, check valve, overpressure valve, and flow and temperature switches [28].

The electronic control module (ECM) regulates both the hydraulic pump and the electrical resistance, allowing precise control of volumetric flow and heating power. A variable frequency drive (VFD) controls the pump speed, while a solid-state relay modulates the electrical resistance via pulse-width modulation (PWM). Both actuators are managed by PID (proportional–integral–derivative) control algorithms, ensuring flexibility in conducting conventional TRTs, GTTs (ground temperature tests), and intermittent heat pulse cycles.

Before conducting the TRT, a GTT was performed to determine the undisturbed ground temperature (T_S). During this test, the circuit was filled with water and left to stabilize for several days. Once thermal balance is reached, the pump circulates the fluid at a low flow rate to maintain laminar flow and minimize heat losses [29]. These tests were conducted without heat injection; thus, the heating resistance was inactive at all times [30]. The outlet temperature of the fluid, which corresponds to the ground temperature at different depths, was recorded to establish the temperature profile along the borehole.

Following the GTT, the TRT was carried out by maintaining a constant volumetric flow in turbulent conditions while applying a uniform heat pulse (q = Q/H) for three to four days [31]. Throughout the test, inlet and outlet temperatures (T_in, T_out) were continuously recorded to calculate T_f, which was essential for determining the thermal properties of the ground. The reliability of the experimental results was ensured by evaluating measurement uncertainties, which can arise from sensor accuracy, data acquisition systems, calibration errors, and external power supply fluctuations.

2.2. Coupling ILS-FLS Models. Determining a Local STGF (STGF_M)

The finite line source (FLS) model is a well-established analytical method used to determine the average fluid temperature (T_f) over time within a ground heat exchanger (GHE), even when dealing with variable heat loads. This model operates by the superposition of the GHE’s outcomes due to the application of several unit heat pulses during a specific time period [14]. Essentially, it simulates the thermal interaction between the circulating fluid, borehole, and surrounding ground. In order to apply the FLS model, shown in Equation (1), a series of key thermal parameters are required as inputs, such as soil thermal conductivity (λ_S), undisturbed soil temperature (T_S), and borehole thermal resistance (R_b).

$$T_f=T_S+q{ R}_b+{\displaystyle \frac{q}{2 \pi {\lambda }_S}} g\left({\tau }_{dim},r_{dim}\right)$$

(1)

The thermal response factor, or g-function, represented as g (${\mathit{\tau }}_{\mathit{d}\mathit{i}\mathit{m}}$, ${\mathit{r}}_{\mathit{d}\mathit{i}\mathit{m}}$), is a dimensionless function that depends on the borehole geometry, time, and ground properties. Both the dimensionless time ${\mathit{\tau }}_{\mathit{d}\mathit{i}\mathit{m}}=\tau /{\tau }_S$ and radius ${\mathit{r}}_{\mathit{d}\mathit{i}\mathit{m}}=r_b/H$ are defined relative to the borehole depth H (m) and the steady-state time τ_S, which is given by H²/α_S [15,32]. The thermal diffusivity of the soil α_S (m²;/s) is determined by dividing the thermal conductivity λ_S (W/(m·K)) by the volumetric heat capacity C_S (J/(m³·K)). In a localized approach where the borehole radius (r_b) is predefined, the g-function can be expressed directly as a function of time τ, avoiding the need for dimensionless parameters such as ${\mathit{\tau }}_{\mathit{d}\mathit{i}\mathit{m}}$ and ${\mathit{r}}_{\mathit{d}\mathit{i}\mathit{m}}$ [26].

The thermal characterization of the GHE is commonly performed through the infinite line source (ILS) model, following Equation (2) [31]. This model, despite assuming certain simplifications (e.g., the R_b value is considered steady, referred to as R_b0), provides highly accurate estimations of the thermal parameter’s values. It was derived from the heat diffusion equation in one dimension, which models the borehole as an infinite line where the heat flow is propagated radially through the surrounding ground [33].

$$T_f=T_S+q{ R}_{b0}+{\displaystyle \frac{q}{4 \pi {\lambda }_S}} E_i\approx T_S+q{ R}_{b0}+{\displaystyle \frac{q}{4 \pi {\lambda }_S}} \left[\ln \left({\displaystyle \frac{4 {\alpha }_s \tau }{r_b^2}}\right)-\gamma \right]$$

(2)

$$T_f=a+m\ln \left(\tau \right) ; m={\displaystyle \frac{q}{4 \pi {\lambda }_s}} ; a=T_S+q{ R}_{b0}+{\displaystyle \frac{q}{4 \pi {\lambda }_S}} \left[\ln \left({\displaystyle \frac{4 {\alpha }_s }{r_b^2}}\right)-\gamma \right]$$

(3)

In Equation (2), the exponential integral function E_i [34] is developed to a term that contains Euler’s constant γ. To apply the ILS model, the temperature outcomes (T_f and T_S) from a TRT and GTT are required, whose measurement procedures are explained in Section 2.1. By plotting T_f against the natural logarithm of time, a linear trend emerges towards the end of the test period, as denoted in Equation (3). The slope ‘m’ of this line allows the determination of the soil thermal conductivity (λ_S), while the intercept ‘a’ on the y-axis enables the calculation of the borehole thermal resistance (R_b) once the soil volumetric heat capacity (C_S) is known.

The borehole initially exhibits a transient thermal regime when heat is injected into the ground, due to the non-stationary thermal behavior of the filling material and its interaction with the circulating fluid. This phenomenon is reflected in the time-dependent borehole thermal resistance, denoted as R_b(τ). At this stage, heat is both absorbed and conducted towards the surrounding soil, resulting in a dynamic variation of R_b. Over time, as heat flow stabilizes, R_b converges to its steady-state value R_b0, typically achieved during long-term system operation or by the end of a TRT [26,27]. At this point, heat transfer can be considered steady-state, involving convective heat transfer from the fluid to the inner pipe walls, followed by conductive heat transfer through the pipe and the filling material towards the outer borehole surface [35], with thermal capacity effects becoming negligible. While R_b0 is often determined using the ILS model, an accurate estimation of the transient resistance R_b(τ) requires more advanced models [36,37,38]. Consequently, the analysis of the GHE short-term performance is often limited by the computational resources and modeling tools available to the engineering team.

Furthermore, the heat transfer process from the borehole to the surrounding soil, extending to a distance where the undisturbed ground temperature (T_S) remains constant, is characterized by the g-function. It is composed of two distinct components: a short-time g-function (STGF), which captures relevant transitory thermal effects from the borehole, and a long-term g-function (LTGF), where heat transfer occurs primarily through radial conduction into the surrounding soil. The STGF and LTGF share a transitional region where both functions converge, between 12 and 200 h of heat injection [13,14,17,39], corresponding with the final stage of a TRT. This overlapping region provides an interval where transient and steady-state thermal behaviors balance, meaning that the transient R_b (R_b(τ)) tends to its steady-state value R_b0.

Cruz-Peragón et al. [26] proposed an efficient method for obtaining the STGF using experimental data from a TRT, referred to as local STGF (STGF_M), as a result of developing the initial approach from synthetic data carried out by Yavuzturk and Spitler [14]. This function characterizes the transient heat transfer dynamics between the circulating fluid and the surrounding soil, enabling accurate predictions even under varying thermal load conditions. In this approach, the STGF is derived from the FLS model, leading to Equation (4), after determining key ground thermal parameters from the ILS model and TRT temperature measurements, including soil thermal conductivity (λ_S), static borehole thermal resistance (R_b0), and the undisturbed ground temperature (T_S):

$${STGF}_M=g\left(\tau \right)={\displaystyle \frac{2 \pi {\lambda }_S}{q}} \left(T_f-T_S-q R_{b0}\right)$$

(4)

A slight limitation of this method is the assumption of a constant R_b0 value, as defined by the ILS model, which does not fully capture the transient nature of thermal exchange. To mitigate this issue, the STGF_M compensates by generating negative values, which can distort the expected thermal profile and introduce discrepancies in simulations of complex thermal scenarios. Through a segregation procedure of the short-term transient behavior, reference [27] demonstrated that the use of a constant R_b0 does not significantly affect the accuracy of the modeled thermal response. Additionally, the local STGF remains a fundamental tool for modeling the thermal response of ground heat exchangers (GHEs), offering valuable insights for both experimental research and practical applications [26,29].

The LTGF profile is determined through complex analytical expressions that model the heat transfer over extended time scales in vertical GHEs. These expressions incorporate error functions and geometric parameters of the borehole to accurately estimate heat distribution in the surrounding soil. One of the most recognized formulations for the LTGF was proposed by Diao et al. [19], presenting an integral expression that includes two complementary error functions (erfc). This approach enables faster computation compared to numerical methods, making it highly efficient for practical applications. This expression depends on key geometric and thermal parameters, such as borehole depth (H), axial radius (r), and soil thermal diffusivity (α_S), which in turn depend on λ_S and C_S. All these values are required before obtaining the transient evolution of that g-function.

2.3. Uncertainties Associated with the Thermal Parameters of the GHE

2.3.1. Uncertainty Associated with R_b0 Determination

The accurate characterization of GHEs depends on several key thermal parameters, each subject to inherent uncertainties. As explained in Section 2.2, the R_b value when the borehole has reached its steady-state behavior (R_b0) can be estimated through the ILS model. This model is extensively used because of its fast computation and simplicity. Nevertheless, the mentioned approach frequently requires the determination of C_S, which has a large uncertainty according to numerous authors, meaning a deviation in the range of 10% to 25% [10,40,41,42,43]. To evaluate the impact of this uncertainty, it is essential to analyze how different values of C_S affect the estimation of R_b0.

An alternative method to R_b0 determination involves laboratory analysis of filling material samples, which facilitate approximate values for thermal conductivity (λ_b) and volumetric heat capacity (C_b), both of which are essential for its estimation. These properties are obtained through experimental testing methods that measure the rate of heat transfer (e.g., following the ISO 8302 standard [44] using a guarded heat flow meter FOX-50) and the material’s ability to store heat (e.g., according to ASTM D4611 standard [45]). Consequently, conducting experimental work could provide more reliable and accurate property values. However, due to the inherent heterogeneity of the filling material compositions along borehole depth, an accurate value for these parameters cannot be defined. Instead, a range of values is established to account for variations in material properties and ensure more reliable thermal modeling [46,47], which introduces additional uncertainties. As a result, laboratory tests will not be considered hereon.

Another alternative method to estimate the constant R_b0 without relying on C_S was proposed by Zeng [48]. This approximation requires two primary input parameters: the thermal conductivity of the filling (λ_b) and the spacing between the centers of the U-tube pipes (S_S). In any case, it must be noted that these approximations do not consider transitory effects of the filling. However, determining λ_b can introduce uncertainty, as laboratory measurements may lack precision, and numerical simulations frequently imply high computational costs [49]. Additionally, the spacing S_S is not always uniform along borehole depth. Variations in pipe positioning due to installation factors or pipe deformation over long borehole lengths can lead to inconsistencies, further complicating the accuracy of R_b0 estimation. Despite these challenges, this model remains a valuable tool for borehole thermal resistance assessment, particularly in cases where the direct measurement of C_S is impractical.

2.3.2. Uncertainty Associated with the TRT-ILS Method for a Local STGF (STGF_M)

The TRT itself involves several uncontrolled variables due to the complexity of the system, being an outdoor test and taking 3 or 4 days to be carried out [50,51]: (i) External disturbances, such as environmental temperature, wind speed, pipe insulation, radiation absorption rate in the pipe surface, and so on, have relevant influence on the thermal response curve. These factors can cause variations in the temperature outcomes (T_in and T_out), and consequently in T_f, introducing additional uncontrolled heat exchange during the TRT. Similarly, the undisturbed soil temperature (T_S) measured during the GTT can also be affected, especially due to the low flow velocity, which increases sensitivity to external thermal inputs. Although various models have been proposed to quantify these effects [52,53], they are frequently complex and involve high computational cost. (ii) During the TRT, slight instabilities in heating power are likely to appear, which significantly influences the estimation of thermal response parameters obtained through the TRT [51]. (iii) The application of a different constant heating power significantly affects the heat transfer capacity (q (W/m)). Zhou et al. [54] carried out several TRTs over the same borehole, showing variations in q (W/m) up to 17.1% when the heating power (Q (W)) was increased by 15.7%. This occurs because heat injection creates a temperature gradient around the borehole, which induces variations in water density. As a result, convective heat transfer within the groundwater is triggered, leading to an overestimation of λ_S, whose deviations were up to 4.17%. (iv) The fluid velocity also affects the heat transfer capacity, mainly due to the influence on the heat transfer coefficient between the fluid and the wall pipe. Thus, the determination of λ_S is slightly affected by the selected volumetric flow rate (V_f) [54].

Another challenge is the lack of a universal evaluation method for determining these parameters, leading to inconsistencies in results depending on the approach used. The mentioned uncertainties impact the measurement of TRT and GTT outputs (T_f and T_S) and the subsequent parameter estimation process, such as the ILS model (λ_S and R_b0), which may affect the reliability of the ILS-FLS coupling. In this context, this study aims to validate whether the determination of the local short-time g-function (STGF_M) is independent of the specific TRT conducted. To achieve this, first, a STGF_M was derived from a TRT, trying different values of C_S to evaluate its influence. Later, several STGF_M derived from three different TRTs were compared, analyzing their convergence with the LTGF. This assessment helps ensure that the STGF_M remains a reliable tool for geothermal system modeling, regardless of uncontrolled variabilities in testing conditions and uncertainties in some parameter characterizations.

3. Results

Several TRTs (shown in

Table 1. Test conditions and derived thermal parameters from TRT experiments.

Vf (L/min)	Q (W)	λS (W/(m·K))	CS (J/(m3·K))10−6	Rb0 (m·K/W)	τloop (s)	εend (K)·10−2	dT1 (K/W) 10−3	dT2 (K/W) ·10−3
15 (ref)	4000	1.84	1.39	0.1115	873	7.02	2.4925	2.963
10	3510	1.93 (+5%)	1.45	0.1129(+1.3%)	1308	6.096	2.4227	2.822
20	4750	1.87 (+1.5%)	1.4	0.1018(−10%)	655	5.98	2.3906	2.905

) were conducted following ASHRAE standards to ensure precise control over heat injection and flow rate. The dispersion of these parameters, along with their peak values, remained within the recommended limits. The three selected sets of tests were chosen to cover a representative operational range commonly used in geothermal field applications. These variations allowed us to introduce controlled differences in the Reynolds number and heat injection rates, which are relevant factors influencing borehole behavior in the short term. These tests aimed to assess the accuracy of the local STGF in predicting the mean fluid temperature (T_f).

3.1. Reliability in Modeling T_f for Different Property Values from the Same TRT

Proper coupling between the STGF_M and LTGF, as demonstrated in Figure 2, ensures accurate determination of the local STGF. The TRT with a flow rate of V_f = 15 L/min and a heat rate of Q = 4000 W was used as the reference. The latter test conditions were used to generate the g-functions in Figure 2. Following the acquisition of the T_f profile and T_S value from respective tests (GTT for T_S), the ILS model was applied according to Equation (2)–(3), obtaining λ_S and R_b0, assuming a predefined C_S in the latter. However, a more precise characterization of the GHE is associated with the determination of the thermal parameters of the filling material (λ_b and C_b) and T_S using numerical methods or highly accurate experimental tests (with minimal dispersion). In this sense, the derived thermal parameters were refined using a 3D numerical model and a Design of Experiments methodology (3D-DoE approach) to enhance their reliability, as explained in reference [49], whose values are detailed in Table 1. Subsequently, once the thermal properties of the materials were determined, the STGF_M could be derived using the FLS model, as shown in Equation (4). As a result, T_f can be accurately reproduced across a wide range of operating conditions.

In this section, the uncertainty associated with the estimation of C_S is examined, assessing its impact on R_b0 when applying the ILS model and the generated T_f profile after the STGF_M determination. Differences greater than 100% in the value of the mentioned variable are compared. Thus, a reference value of C_S = 1.39·10⁶ J/(m³·K), which gives rise to R_b0 = 0.1115 m·K/W, is compared with higher values up to C_S = 3·10⁶ J/(m³·K) (R_b0 = 0.1440 m·K/W) and lower values down to C_S = 0.7·10⁶ J/(m³·K) (R_b0 = 0.08206 m·K/W). In these scenarios, the differences in the estimated R_b0 are less than 30% (ranging from 29.7% to 13.5%). Furthermore, the variation in the estimated T_f is reported to be less than 0.01 $\mathrm{K}$ (from 0.006 $\mathrm{K}$ to 0.003 $\mathrm{K}$) at the end of the test period (3 days), and less than 1 $\mathrm{K}$ (from 0.35 $\mathrm{K}$ to 0.55 $\mathrm{K}$) at 50 years (with LTGF), for a single borehole.

To ensure consistency in thermal modeling, it is essential to use all the parameters derived from the coupled ILS-FLS approach (λ_S, C_S, R_b0, and STGF_M). In the case that R_b0 calculated from other scenarios (such as 0.144 m·K/W and 0.08206 m·K/W) were used with the STGF_M determined from the reference scenario (from R_b0 = 0.1115 m·K/W), there would be a relevant deviation in the estimated T_f. For instance, a 50% discrepancy in R_b0 results in a constant deviation of approximately 2 $\mathrm{K}$ throughout the entire time period, affecting both short- and long-term simulations. While this difference may not be critical in the long term, it significantly impacts short-term heat rate pulse simulations, leading to inaccurate thermal predictions. Therefore, it is crucial to follow a consistent estimation process that maintains the coupling between the ILS and FLS models, ensuring reliable results in both transient and steady-state conditions.

A comparison between STGF_M and T_f was performed using thermal parameters derived from the ILS model [26] and the 3D-DoE approach [49] employed previously. The ILS model provided the following values: λ_S = 1.88 W/(m·K), and R_b0 = 0.141 m·K/W, from C_S = 2·10⁶ J/(m³·K), and T_S = 293.3 $\mathrm{K}$. In contrast, the 3D-DoE approach produced λ_S = 1.84 W/(m·K), R_b0 = 0.1115 m·K/W, C_S = 1.25·10⁶ J/(m³·K), and T_S = 293.6 $\mathrm{K}$. The latter also allowed us to obtain the filling parameters values: λ_b = 1.709 W/(m·K) and C_b = 0.99·10⁶ J/(m³·K). Despite initial deviations in the g-functions, as shown in Figure 3a, the resulting T_f predictions from both approaches revealed high consistency (see Figure 3b). Temperature errors were less than 0.4$\mathrm{K}$ during the first minute, primarily attributed to differences in T_S estimation, being negligible in the short term, and fluctuated within ±0.2 $\mathrm{K}$ over extended time periods. The T_f deviation between both procedures at the end of the TRT was only ε_end = 0.0702 $\mathrm{K}$. These findings highlight the robustness of the STGF_M approach and validate its reliability in generating T_f across different modeling methods.

The final selection of C_S is relegated to a secondary stage, as its influence is considerably smaller compared to λ_S: once λ_S has been estimated, the coupling of the ILS-FLS models determines R_b0 once T_S and C_S are known (the latter allows the calculation of α_S). Any uncertainties in these parameters propagate into the borehole thermal resistance estimation. Nevertheless, these errors are effectively mitigated when modeling T_f according to Equations (2)–(4), ensuring consistency between the estimated characteristics and the resulting thermal response functions. The process of generating STGF_M inherently incorporates the values of T_S, C_S, and R_b0, along with their associated uncertainties. Despite these variations, the STGF and LTGF curves consistently converge at the end of the test period (τ_end), reinforcing the accuracy of the modeling approach. In this context, the ILS model has been adopted as the primary method for determining R_b0.

The R_b0 was also estimated using the Zeng model [48], as a function of different values of borehole conductivity λ_b and tube spacing S_S. As shown in Figure 4, the model accurately adjusts to λ_b, but discrepancies are observed concerning S_S. Specifically, when R_b0 = 0.1115 m·K/W, the corresponding spacing should be 75 mm instead of 81.5 mm. If S_S = 75 mm is assumed, the resulting R_b0 is 0.1160 m·K/W. Despite these variations, the obtained R_b0 values remain within the same range derived from varying C_S earlier in this section. This consistency demonstrates that the uncertainty in R_b0 using this approach has a negligible impact on the modeling of T_f.

3.2. Reliability in Modeling T_f Across Multiple Tests in a Single Borehole

In Section 3.1, uncertainties in the STGF_M determination were evaluated for a single test. To further assess the reliability of the coupled ILS-FLS model, two additional TRTs were conducted on the same borehole. These tests corresponded to the flow rates and heating powers detailed in Table 1: the second test with V_f = 10 L/min and Q = 3510 W, and the third test with V_f = 20 L/min and Q = 4750 W. Deviations in parameter estimations (λ_S, C_S, and R_b0) derived from measurement uncertainties and differing calculation procedures.

This study compared the three TRTs performed under different conditions, aiming to identify consistent trends and validate the methodology. The results from all three tests under different heat injection rates (Q) are presented in Figure 5, illustrating variations in the STGF_M curves. A primary objective was to determine whether the STGF consistently aligns with the LTGF at τ_end when applying a logarithmic time approximation across all three curves. The observed consistency between these functions supports the robustness of the proposed approach.

The corresponding T_f profiles were obtained using the corresponding STGF_M presented in Figure 5, along with their respective thermal parameters and TRT conditions, in Equation (4). Afterwards, the three temperature profiles were compared using relative temperature deviations, with the first TRT (V_f = 15 L/min) serving as a reference. The performance indicators dT₁ = (T_end $-$ T_S)/Q for short-term performance and dT₂= (T_50years $-$ T_end)/Q for long-term were employed, which are normalized temperature differences used to evaluate the coupled ILS-FLS model across different time scales, with values ranging from [−4.12% to −2.6%] to [−4.9% to +2.28%], respectively. The estimation errors for temperature at the end of the TRT ranged between [−0.24 $\mathrm{K}$ and −0.39 $\mathrm{K}$], while at a 50-year projection, they fluctuated between [−0.78 $\mathrm{K}$ and −0.13 $\mathrm{K}$]. While these discrepancies do not present significant issues in steady-state conditions, their potential impact on short-term scenarios involving intermittent heat rate pulses must be assessed.

Thus, the GHE behavior was analyzed under dynamic conditions, conducting intermittent heat pulse simulations of 1 h pulses (Figure 6) and 4 h pulses (Figure 7). The recommended durations for these pulses, which improve the efficiency of ground-source heat pump (GSHP) systems, range from 2 to 6 h.

These figures illustrate that differences in thermal responses among the three function groups and thermophysical properties remain minimal. Notably, the discrepancies observed up to τ_loop contribute only marginally to the overall response, with a maximum deviation of approximately 1 $\mathrm{K}$ (see detail in Figure 7). Smoother initial segments of the STGF_M functions within this time interval would mitigate these discrepancies.

Throughout each pulse cycle, differences between the response models gradually decrease, with observed deviations around ±0.2 $\mathrm{K}$ at the end of each pulse. These minor differences, originally noted in Figure 5 during the estimation of the STGF_M curve, become more evident as pulse time progresses. Once a pulse concludes, the model accurately simulates ground cooling before repeating the cycle. This behavior highlights the importance of properly modeling intermittent pulses to ensure accurate thermal performance predictions for GSHP systems operating under variable load conditions.

To further validate the reliability of the modeling approach, the results were compared against those of an experimental 12 h pulse test conducted in a previous study [26]. The comparison, shown in Figure 8, examines the agreement between modeled and experimental data under intermittent heating conditions.

Direct curve comparisons are impractical during periods without heat injection, as the measured temperatures do not accurately represent T_f, presenting a notable challenge to this comparative analysis in those intervals. Furthermore, the ground temperature distribution is no longer radially uniform due to prior heat injection cycles, requiring long-term adjustments in the equivalent mean heat flow pulses.

Despite these complexities, the responses during heat injection periods show strong agreement between the model and experimental results. However, discrepancies increase towards the end of heat-free intervals, as small deviations in the estimation of STGF_M functions become more pronounced over time. These differences, initially observed in Figure 5, accumulate as the thermal pulses progress. Nonetheless, these discrepancies diminish towards the end of the TRT trials and remain negligible in the long term, reinforcing the model’s validity for long-duration heat pulse simulations.

3.3. Discussion of Results

Section 3.1 addressed the sensitivity of R_b0 determination to variations in the volumetric heat capacity of the soil (C_S), revealing that even significant changes in C_S (±100%) had a limited impact on the final value of R_b0 (less than ±30%) and resulted in negligible T_f deviations (under 0.01 $\mathrm{K}$ at 3 days and under 1 $\mathrm{K}$ at 50 years). Furthermore, the comparison between the ILS model and a 3D-DoE approach validated the experimental method, showing temperature deviations below ±0.2 $\mathrm{K}$ in almost the whole profile, reinforcing the credibility of the STGF_M derived using the proposed method.

It is important to employ all parameters obtained from the same coupled ILS-FLS model; otherwise, deviations in T_f generation of up to 2  $\mathrm{K}$ might occur. Lastly, to further support the robustness of the approach, the Zeng model was also applied. The resulting R_b0 values remained within the same range derived from varying C_S earlier in this section.

Section 3.2 extended the validation by testing the consistency of the STGF_M under three different TRT scenarios. Despite variations in flow rate and heating power, all tests resulted in STGF_M that aligned well with their corresponding LTGFs at τ_end. When applied to model T_f, relative temperature deviations remained below ±0.4 $\mathrm{K}$ in the short term and ±0.8 $\mathrm{K}$ over a 50-year period.

To assess the methodology under common operational conditions, intermittent heating pulses (1 h and 4 h durations) were simulated using the different STGF_M. The results showed only minor deviations between profiles, with differences around ±0.2 $\mathrm{K}$ at the end of each pulse. These findings were further validated by comparison with a 12 h experimental pulse from previous work [26], confirming that the STGF_M can accurately predict transient borehole thermal behavior under varying test and operational conditions.

4. Conclusions

The accurate determination of the short-time g-function (STGF) frequently requires time-consuming and complex numerical procedures, leading to significant computational overhead. This study demonstrates the reliability of thermal parameter values obtained by combining ILS-FLS models to calculate an experiment-based STGF (STGF_M), without the need of relying on numerical models. The following can be concluded:

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The consistency of this approach is supported by the agreement between STGF_M and LTGF curves across various experimental conditions.

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The error introduced in the determination of the steady borehole thermal resistance (R_b0) due to uncertainties in volumetric heat capacity (C_S), or by estimation methods such as Zeng’s model, was found to be negligible in the estimated T_f profile. Parameter variations within realistic ranges implied deviations in T_f modeling lower than 0.01 $\mathrm{K}$ at the end of the test period (3 days), and 1 $\mathrm{K}$ at 50 years (with LTGF).

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Parameters must be derived from the specific test. In particular, the same R_b0 value must be employed in conjunction with its corresponding STGF_M. Otherwise, deviations up to 2 $\mathrm{K}$ in the generated T_f profile can lead to discrepancies in the predicted thermal behavior, especially in intermittent heating (and subsequently, cooling) scenarios.

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The calculated STGF_M effectively captured the transient thermal behavior of the borehole heat exchanger, even under intermittent operational conditions. STGF_M determined from different cyclic heat injection experiments on the same borehole were used to generate the T_f profile, with observed deviations around ±0.2 $\mathrm{K}$ at the end of each pulse.

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Numerical models are not required, which makes the method highly suitable for practical applications. Thus, the proposed method offers a simplified and efficient alternative without compromising accuracy.

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The three selected TRT conditions reflect a representative operational range found in real-world geothermal applications.

Overall, this study confirms that ILS-FLS coupling is a reliable and accurate method for modeling borehole thermal response. The results emphasize the robustness of this approach across different test conditions and reinforce the need to maintain parameter consistency when applying the STGF_M methodology in geothermal system simulations. By enhancing the accuracy and efficiency of geothermal energy modeling, these findings contribute to more sustainable energy solutions, reducing energy losses and promoting long-term environmental benefits. Future developments will focus on adapting this methodology into more practical tools, enhancing its accessibility for professionals involved in GSHP system planning and implementation.