#
Evaluation of Subsurface Heat Capacity through Oscillatory Thermal Response Tests^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

^{−1}. The results of the proposed methodology were compared 3-D numerical simulations and to a TRT with a constant heat injection rate with temperature response monitored from a nearby observation well. Results show that the OTRT succeeded to infer the expected subsurface heat capacity, but uncertainty is about 15% and the radial depth of penetration is only 12 cm. The parameters having most impact on the results are the subsurface thermal conductivity and the borehole thermal resistance. The OTRT performed and analyzed in this study also allowed to evaluate the thermal conductivity with similar accuracy compared to conventional TRTs (3%). On the other hand, it returned borehole thermal resistance with high uncertainty (15%), in particular due to the duration of the test. The final range of heat capacity is wide, highlighting challenges to currently use OTRT in the scope of ground-coupled heat pump system design. OTRT appears a promising tool to evaluate the heat capacity, but more field testing and mathematical interpretation of the sinusoidal response is needed to better isolate the subsurface from the BHE contribution and reduce the uncertainty.

## 1. Introduction

^{−1}during conventional TRTs [1,2]. Low-powered tests (10 to 25 W m

^{−1}) can also be conducted with a heating cable, which can even reduce the uncertainty associated to the subsurface TC because of the facility to evaluate the heat injection rate [3]. On the other hand, borehole thermal resistance cannot be properly assessed with a heating cable test. Other than being very compact and needing only 120 V power, the heating cable unit can provide a TC profile of the ground with several temperature sensors at depth, or with a fiber-optic cable [4,5]. Moreover, it does not require a BHE since it can be performed in open wells, provided that water is present to ensure thermal contact with the subsurface. Electric cable with heating and non-heating sections were also tested, but significant free convection occurs in the pipe (or well) according to the Rayleigh number stability criterion, thus allowing only 15% accuracy in TC estimation [4]. TRTs are commonly carried out for 48 to 72 h with a constant power ([2] and references therein). However, heat injection can be changed over time for different purposes. For instance, TRTs with step heat injection have been proposed to determine the optimal heat rejection/extraction rates for GCHP [6]. Constant-temperature TRTs have also been studied in order to reduce the testing time, increase the accuracy of the TC estimation via the slope method, and avoid the need of fixing the subsurface heat capacity (HC) to evaluate the borehole thermal resistance ([7] and references therein). To this regard, even though the range of variability of HC among geologic media is quite limited [8,9,10], a change from 1.5 to 3.2 MJ m

^{−3}K

^{−1}influences the thermal diffusivity (TD; ±40%) and thus the evaluation of the borehole thermal resistance (±10–23%) via conventional TRTs. In turn, we approximated that this could affect the evaluation of the total drilling length of BHEs by ±6–7%, with an impact of 3–4% on the total cost of the system. Moreover, HC is a crucial parameter in the design of underground thermal energy storage (UTES) systems, because it defines the amount of thermal energy that can be stored into the subsurface ([11,12] and references therein).

## 2. Materials and Methods

#### 2.1. Research Hypothesis

^{−1}) and diffusivity (m

^{2}s

^{−1}) ([14,15] and references therein).

- To perform a constant-heat-injection TRT while recording the thermal response in a nearby observation well in order to have a first estimate of the subsurface HC;
- To perform 3-D numerical simulation to validate the proposed methodology
- To carry out an OTRT in a BHE and compare the HC results to those obtained in steps 1 and 2.

#### 2.2. Oscillatory Heat Injection Theory and Analytical Approach for the Evaluation of Thermal Diffusivity

^{−1}) is the heat injected per unit length, ${q}_{\mathrm{p}}$ (W m

^{−1}) is the offset of the sinusoidal function, P (h) is the period of the oscillation and t (h) is time. This induces an oscillatory thermal response in the same well ($r={r}_{\mathrm{b}}$, with ${r}_{\mathrm{b}}$ borehole radius) which is described by the following equation given by Eskilson [21]:

^{−1}) denotes the resistance opposed by the surrounding medium, and is therefore called oscillatory resistance, and ${\mathsf{\u0278}}_{\mathrm{p}}$ (-) is the phase shift of the thermal response expressed as a fraction of P (0 < ${\mathsf{\u0278}}_{\mathrm{p}}$ < 1). ${R}_{\mathrm{p}}$ and ${\mathsf{\u0278}}_{\mathrm{p}}$ can be evaluated by comparing the heat injection and thermal response as described by the following Equations:

^{−1}) are the amplitudes of the thermal response and heat injection, respectively. Equation (2) is valid only if the system is linear time invariant, i.e., the oscillation frequencies of the heat injection and thermal response are the same, as described and demonstrated by [17]. If the heat source can be simplified to a heated line, there exists an analytical solution and Eskilson [21] derived the expressions for ${R}_{\mathrm{p}}$ (Equation (5)) and ${\mathsf{\u0278}}_{\mathrm{p}}$ (Equation (6)) as a function of ${r}_{\mathrm{pb}}$, a dimensionless factor described in Equation (7) which depends on the depth of investigation (${d}_{\mathrm{p}}$, assumed as the radial distance from the heated line, Equation (8)), which in turn varies according to the subsurface TD. Therefore, we have:

^{−1}K

^{−1}) is the subsurface TC, $\gamma $ is the Euler-Mascheroni constant and $\alpha $ (m

^{2}s

^{−1}) is the subsurface TD. Eskilson’s expressions are approximated to the first terms of the Taylor series expansion of the Kelvin function. The validity of this approximation is further discussed in Appendix A. It is finally important to highlight that the oscillatory resistance ${R}_{\mathrm{p}}$ depends on both the TC and TD of the subsurface (Equations (5), (7) and (8)), while ${\mathsf{\u0278}}_{\mathrm{p}}$ is only function of the diffusivity (Equations (6)–(8)).

- Subtraction of the linear component function of TC
_{heating}and ${R}_{\mathrm{bt}}^{*}$ in order to obtain the oscillatory component of the OTRT response (Figure 2B); - Comparison of the oscillatory heat injection and the oscillatory thermal response to evaluate ${R}_{\mathrm{p}}$ and ${\mathsf{\u0278}}_{\mathrm{p}}$. Evaluation of TD by means of the Equations (5)–(8) (Figure 2C);
- Evaluation of HC via the ratio TC
_{cooling}/TD.

#### 2.3. Description of the Test Site

^{−1}K

^{−1}and hydraulic conductivity of the highlighted fracture zones is 10

^{−8}m s

^{−1}. Darcy flux was estimated to 1 to 4 × 10

^{−9}m s

^{−1}, and Darcy velocity to 10

^{−10}m s

^{−1}[25,26,27]. Koubikana Pambou et al. [27] showed that GW does affect the subsurface TC inferred from TRT in correspondance of the fracture zones, with effective TC reaching maximum values of 2.0 W m

^{−1}K

^{−1}. However, this influence appears to be local and limited to fracture zones with thickness of few meters. Therefore, the OTRT analysis is expected to be unaffected by GW in the chosen 1-U BHE at the study site. The volumetric HC of the shales is estimated to be in the order of 1.8–2.4 MJ m

^{−3}K

^{−1}from the literature [8,9,10]. The shallowest 10 m of the sequence are expected to have a lower TC and a higher HC due to the higher water-filled porosity of the deposits (see Figure 4).

^{−1}K

^{−1}. The backfill is a geothermal grout mixture with water, bentonite and silica sand, with nominal TC of 1.7 W m

^{−1}K

^{−1}(error 10%, [28]). The expected grout TD is in the order of 0.44 mm

^{2}s

^{−1}. The fluid circulating in the BHE is water, with density of 1.0 × 10

^{3}kg m

^{−3}, dynamic viscosity of 1.37 × 10

^{−3}kg m

^{−1}s

^{−1}, and specific heat of 4.2 kJ kg K

^{−1}. The observation well OBS4 (diameter 0.051 m, depth 26 m) was drilled on purpose for the present study beside the 1-U BHE, at a distance of 1.2 m. Drilling a hole parallel to the BHE was challenging as expected, since the general deviation of the inclination of boreholes ranges from 1 to 10% of the depth. In order to have two parallel holes, it would have been necessary to drill them one after the other, with the same drilling machine and same operator. However, drilling another BHE was not possible due to field constraints. To partially reduce uncertainty, the inclination of OBS4 was measured to know the actual distance between the supposed linear heat source (1-U BHE, assumed vertical) and the observation well (OBS4). The assumption of the vertical BHE appears reasonable since the hole has a larger diameter (more than 2 times), a longer depth and was drilled with heavier machinery. The inclination analysis of OBS4 was carried out with the Gyro Master probe by SPT Semm Logging (inclination accuracy ± 0.05°). Results show that horizontal deviations with respect to the vertical are 0.05, 0.2, 0.9 and 2.1 m at depths of 5, 10, 15 and 21 m, respectively (Figure 5). The well is almost linear down to 10 m, with slight eastward inclination, then the deviation rate becomes bigger, and the inclination tends towards NNW. Distances to the BHE, assuming the latter to be vertical, are approximately 1.2, 1.3, 2.0, 3.0 and 4.4 m at depths of 5, 10, 15, 20 and 25 m (projected), respectively.

#### 2.4. Numerical Simulations

^{−3}K

^{−1}, named SC1, SC2 and SC3, respectively. TC was set to 1.7 W m

^{−1}K

^{−1}in all scenarios, so subsurface TD is 0.85, 0.71 and 0.61 mm

^{2}s

^{−1}, respectively. A test simulation demonstrated the negligible effect of the shallowest weathered unconsolidated sediments on the whole sequence. Therefore, only the shales were represented in the model in order to speed up the calculation.

^{−1}) is the flow rate, ${C}_{\mathrm{p}}$ (J kg

^{−1}K

^{−1}) is the specific heat of the working fluid, and ${R}_{\mathrm{bt}}$ and ${R}_{\mathrm{at}}$ are the borehole and internal resistances including the pipe resistance. With the BHE configuration having dimensions and characteristics described in Section 2.3 and a shank spacing ${x}_{\mathrm{c}}$ of 0.028 m, ${R}_{\mathrm{bt}}$ and ${R}_{\mathrm{at}}$ were evaluated to be 0.0798 and 0.3117 m K W

^{−1}, respectively. The equivalent resistance ${R}_{\mathrm{bt}}^{*}$ is finally 0.0894 m K W

^{−1}(error 10%), in agreement with that found with in-situ assessment [26,28]. In FEFLOW, ${R}_{\mathrm{bt}}$ was assigned to the two pipe-grout and the grout-soil resistances. The grout-grout resistance (${R}_{\mathrm{gg}}$) was calculated as follows [33]:

#### 2.5. Field Tests

#### 2.5.1. TRT with Constant Heat Injection

#### 2.5.2. OTRT

^{−1}K

^{−1}and ${R}_{\mathrm{bt}}^{*}$ 0.09 m K W

^{−1}, and it turned out to be 0.022 m. This value was also used for the analysis of the numerical simulations results.

## 3. Results

#### 3.1. TRT with Constant Heat Injection

^{−1}(Figure 6). The TC was evaluated via the recovery period in order to avoid the effect of the borehole thermal resistance and cable location, which can induce noise in the temperature signal of the heat injection period. This is because the heat source is placed in only one of the BHE pipes and thus its lateral position in the BHE slice remains unknown. The slope analysis of the recovery period outputs TC of 1.8–2.0 W m

^{−1}K

^{−1}from 5 to 20 m (Table 1). The error of the estimation δ (uncertainty) was calculated to be ca. 2.5% according to the methodology proposed by [39]. The results show a slight but clear difference in the first 10 m ($\lambda $ > 1.95 W m

^{−1}K

^{−1}) compared to deeper portions ($\lambda $ < 1.95 W m

^{−1}K

^{−1}), reflecting the local stratigraphy (see Figure 4). First and last sensors do not give reliable results due to violation of the ILS assumptions and the possible occurrence of convection cells. In particular, the last sensor shows a different behavior from the other sensors (see Figure 6). Therefore, they were not considered representative.

^{2}s

^{−1}) and higher in the deep portion (0.77–0.82 mm

^{2}s

^{−1}). The thermal response is therefore highly affected by the inclination of the observation well, i.e., the distance $r$. Using the TC obtained through the previous analysis of the recovery period in the BHE, HC is found at the depth of temperature sensors. Finally, two thermo-geological units can be distinguished: the unconsolidated deposits in the top 10–12.5 m showing higher HC (2.7–2.8 MJ m

^{3}K

^{−1}) and lower TD; the shale rock in its shallowest part showing lower HC (2.3–2.5 MJ m

^{3}K

^{−1}) and higher TD (Table 2). TC is however similar in the two units.

#### 3.2. Numerical Simulations

^{−1}K

^{−1}(SC3) to 1.71 W m

^{−1}K

^{−1}(SC1), while the borehole thermal resistance is 0.125 m K W

^{−1}. The overestimation of ${R}_{\mathrm{bt}}^{*}$ was expected due to the use of the analytical solution in FEFLOW. As shown by [30], the analytical approach overestimates the outlet BHE temperature in the first hours of simulation with respect to fully transient simulations, therefore raising the intercept of the linear regression.

#### 3.3. OTRT

^{−1}(19 to 39 W m

^{−1}, Figure 9). The median power injected was 29.5 W m

^{−1}. The flow rate was constant at 0.385 ± 0.002 L s

^{−1}(6.1 ± 0.03 GPM) throughout the entire test. Between 74 and 77 h, the monitoring system did not record any data, but the pump and heating element kept running as proved by back-up sensors along the circuit.

^{−3}K

^{−1}for ${R}_{\mathrm{p}}$ and ${\mathsf{\u0278}}_{\mathrm{p}}$, respectively, which means 10% and 5% error in the final HC estimation. However, the TD results closer to the expected value were found with p = −1, and the uncertainty related to the chosen value of p is within the final range of HC, as explained in the discussion.

^{−1}K

^{−1}(error 2.5%, Figure 10). By fixing the expected HC to 2.16 MJ m

^{−3}K

^{−1}, the borehole thermal resistance estimation returns 0.075 m K W

^{−1}. This was computed by considering a test duration from 10 to 142 h, representing a total of 5.5 days. The first 10 h of the test were ignored to avoid disturbances and heat transfer within the BHE [2]. In addition, in an OTRT it is also crucial to respect the period of oscillation and pick the same phases at the beginning and at the end to avoid influencing the linear regression. Therefore, in this case an interval of 132 h (11 periods of oscillation) was used to analyze the OTRT results.

^{−1}and ${\mathsf{\u0278}}_{p}$ is 0.1 (Table 6). The TD estimation through Equations (5) and (6) returns values of 0.96 mm

^{2}s

^{−1}(${d}_{\mathrm{p}}$ = 11 cm) and 0.48 mm

^{2}s

^{−1}(${d}_{\mathrm{p}}$ = 8 cm), respectively (Table 7). As a result, TD and HC are significantly different when estimated from ${R}_{\mathrm{p}}$ or ${\mathsf{\u0278}}_{\mathrm{p}}$. The TD and HC are closer to the expected subsurface values when evaluated via the oscillatory resistance, while they almost double when deduced from the phase shift (Table 8).

## 4. Discussion

^{−1}K

^{−1}, [26]) and temperature logs (1.79 W m

^{−1}K

^{−1}, [27]). This difference is most likely related to the shorter length investigated by this test (20 m), whereas previous tests surveyed the entire BHE length (154 m).

^{−1}K

^{−1}) and the observation well (TD of 0.8 mm

^{2}s

^{−1}and HC 2.16 MJ m

^{−3}K

^{−1}); and (2) thermal properties of the backfilling BHE grout (TC 1.7 W m

^{−1}K

^{−1}, TD of 0.44 mm

^{2}s

^{−1}and HC 3.9 MJ m

^{−3}K

^{−1}). Compared to the subsurface synthetic case, experimental ${R}_{\mathrm{p}}$ is 7% higher, while ${\mathsf{\u0278}}_{\mathrm{p}}$ difference is >15% (Table 9). Compared to the grout synthetic case, experimental ${R}_{\mathrm{p}}$ is higher by 17%, while ${\mathsf{\u0278}}_{\mathrm{p}}$ shows maximum deviation of 5% (Table 9). This situation is comparable to the numerical simulations, with both ${R}_{\mathrm{p}}$ and ${\mathsf{\u0278}}_{\mathrm{p}}$ closer to the experimental results. In turn, we can say that in both the experimental OTRT and the numerical simulations the evaluation of the TD by means of the oscillatory resistance returns values close to the subsurface, while the phase shift appears highly affected by that of the backfilling material (Table 10). As aforementioned, HC results in the numerical simulations have a maximum deviation in absolute value of 20% for the first scenario, with the others being rather close (<9%) to the specified HC imposed as model input (Table 11). A fully transient numerical simulation would likely achieve closer values, but the results show that the analytical scheme to implement the BHE in FEFLOW (less computationally heavy, faster and easily applicable) is comparable to the field case. In general, numerical simulations indicate that the OTRT mainly underestimates the subsurface HC, with a maximum deviation of 20% (Table 11).

^{−1}[26,28]). This uncertainty is higher than what can be found via conventional TRT [39], which makes the OTRT hardly trustworthy at present for the estimation of the borehole thermal resistance. Further tests are therefore necessary to investigate this aspect in detail. On the contrary, no major changes due to the duration of the test were highlighted on the oscillatory parameters. Possible differences were found being within the uncertainty range reported in Table 9.

^{−3}K

^{−1}with ${R}_{\mathrm{bt}}^{*}$ varying from 0.086 to 0.1 m K W

^{−1}. This means that a 10% error estimation in ${R}_{\mathrm{bt}}^{*}$ returns a 24% variation in the final HC estimation.

^{−1}, [26,28]), the subsurface heat capacity at the study site is estimated to be in the range of 1.9 MJ m

^{−3}K

^{−1}± 15%. Therefore, we can confirm that the heat capacity inferred by the OTRT is in agreement with the values expected after the heat diffusion analysis (2.1–2.2 MJ m

^{−3}K

^{−1}, Section 3.1) if we consider three aspects: the numerical simulations showed that the proposed methodology tends to underestimate HC; the depth of the observation well OBS4 only allowed to investigate the shallowest 10 m of shales, while the BHE extends to 154 m below ground; and the depth of investigation of the OTRT is limited to the close vicinity of the BHE (12 cm radius). However, the final uncertainty range is similar to that found in the literature and, therefore, it would unlikely justify the execution of an OTRT. In the authors’ opinion, more work has to be carried out in order to improve the mathematical description of the OTRT and the analytical formulation to analyze field data. In particular, work should be focused on the understanding of the delayed ${\mathsf{\u0278}}_{\mathrm{p}}$ found both in the field test and in the numerical simulations (0.1 vs. 0.085, see Table 9), as well as by [17]. It would be interesting to understand if this can be directly linked to the BHE configuration, and therefore to ${R}_{\mathrm{bt}}^{*}$, such to be taken into account in the TD estimation via Equation (6). This would permit the reduction of the uncertainty found in this work, since Equation (5) is affected by the estimation of the TC. This aspect goes beyond the scope of the present contribution.

## 5. Conclusions

^{−3}K

^{−1}± 15%) was likely underestimated. This fact was also confirmed by numerical simulations of OTRT. The result is affected by a range of variation similar to what can be found in the literature, therefore making the OTRT unlikely applicable at present. Higher-period tests (12–24 h) can be carried out within the conventional duration of TRTs. This would increase the penetration depth of the oscillatory signal, but the accuracy of the result is expected to decrease. The parameters having the greatest impact on the results are the subsurface TC and the borehole thermal resistance. While the former can be assessed with an OTRT, the latter cannot be defined with valid accuracy. This has direct effects on the calculation of the equivalent radius, i.e., the distance at which the oscillatory parameters ${R}_{\mathrm{p}}$ and ${\mathsf{\u0278}}_{\mathrm{p}}$ are calculated. Conversely to the Eskilson’s theory, ${R}_{\mathrm{p}}$ can be used to evaluate the subsurface heat capacity, while ${\mathsf{\u0278}}_{\mathrm{p}}$ is affected by a delay of 0.02. This outcome was found for both the experimental and numerical results, and it appears to be due to heat storage in the borehole, which is not considered in the borehole thermal resistance calculation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${A}_{\mathrm{t}}$ | amplitude of the thermal response [K] |

${A}_{\mathrm{h}}$ | amplitude of the heat injection [W m^{−1}] |

${d}_{\mathrm{p}}$ | depth of penetration [m] |

$D$ | deviation from the expected value [%] |

$H$ | borehole depth [m] |

$P$ | oscillation period [s] or [h] |

$q$ | heat injection/extraction rate [W m^{−1}] |

$r$ | distance [m] |

${r}_{\mathrm{b}}$ | borehole radius [m] |

${r}_{\mathrm{eq}}$ | equivalent borehole radius [m] |

${r}_{\mathrm{pb}}$ | dimensionless factor [-] |

${R}_{\mathrm{a}}$ | internal borehole thermal resistance [m K W^{−1}] |

${R}_{\mathrm{b}}$ | borehole thermal resistance [m K W^{−1}] |

${R}_{\mathrm{at}}$ | internal borehole thermal resistance including pipe resistance [m K W^{−1}] |

${R}_{\mathrm{bt}}$ | borehole thermal resistance including pipe resistance [m K W^{−1}] |

${R}_{\mathrm{bt}}^{*}$ | equivalent borehole thermal resistance including pipe resistance [m K W^{−1}] |

${R}_{\mathrm{gg}}$ | grout-to-grout resistance [m K W^{−1}] |

${R}_{\mathrm{p}}$ | oscillatory resistance [m K W^{−1}] |

$t$ | time [s] |

$T$ | temperature [K] or [°C] |

${x}_{\mathrm{c}}$ | shank spacing [m] |

Greek symbols | |

$\alpha $ | thermal diffusivity [m^{2} s^{−1}] |

$\gamma $ | Euler-Mascheroni constant 0.5772156649 [-] |

δ | relative error (uncertainty) of the estimation [%] |

${\lambda}_{\mathrm{g}}$ | thermal conductivity of the subsurface [W m^{−1} K^{−1}] |

${\lambda}_{\mathrm{gt}}$ | thermal conductivity of the grout [W m^{−1} K^{−1}] |

${\mathsf{\u0278}}_{\mathrm{p}}$ | phase shift [-] |

Abbreviations | |

1-U | single U |

BHE | borehole heat exchanger |

GW | groundwater |

GCHP | ground-coupled heat pump |

HC | heat capacity |

NNW | north/north-west |

NW | north-west |

OBS | observation well |

OPT | oscillatory pumping test |

OTRT | oscillatory thermal response test |

SDR | standard dimension ratio |

TC | thermal conductivity |

TD | thermal diffusivity |

TRCM | thermal resistance and capacity model |

TRT | thermal response test |

UTES | underground thermal energy storage |

## Appendix A. Analytical Validation of the Proposed Methodology

^{−1}K

^{−1}, TD of 0.7 mm

^{2}s

^{−1}and HC 2.43 MJ m

^{−3}K

^{−1}) and different BHE diameter (4.5 and 6 inches). A sinusoidal heat injection with amplitude of 10 W m

^{−1}and offset of 30 W m

^{−1}was used in the calculation of the oscillating temperature response. Target periods of oscillations were set to 12 and 24 h, so that the OTRT can last a maximum of 72 h. The equivalent borehole radius was calculated with grout TC equal to 1.7 W m

^{−1}K

^{−1}and ${R}_{\mathrm{bt}}^{*}$ equal to 0.09 m K W

^{−1}, resulting in 0.022 and 0.029 m for a 4.5-inch (0.114 m) and a 6-inch (0.152 m) BHE, respectively.

${\mathit{R}}_{\mathbf{p}}\text{}\left(\mathbf{m}\text{}\mathbf{K}\text{}{\mathbf{W}}^{-1}\right)$ | ${\mathbf{\u0278}}_{\mathbf{p}}$ (-) | ${\mathit{r}}_{\mathbf{p}\mathbf{b}}$ (-) | ${\mathit{d}}_{\mathbf{p}}\text{}\left(\mathbf{m}\right)$ | |
---|---|---|---|---|

4.5-inch/P 12 h | 0.140 | 0.089 | 0.316 | 0.097 |

6-inch/P 12 h | 0.118 | 0.106 | 0.420 | 0.099 |

4.5-inch/P 24 h | 0.165 | 0.071 | 0.230 | 0.141 |

6-inch/P 24 h | 0.142 | 0.081 | 0.305 | 0.149 |

$\mathsf{\alpha}\_{\mathit{R}}_{\mathbf{p}}\text{}\left({\mathbf{mm}}^{2}\text{}{\mathbf{s}}^{-1}\right)$ | ${\mathit{C}}_{\mathbf{v}}\_{\mathit{R}}_{\mathbf{p}}\text{}\left(\mathbf{MJ}\text{}{\mathbf{m}}^{-3}\text{}{\mathbf{K}}^{-1}\right)$ | $\mathsf{\alpha}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}\left({\mathbf{mm}}^{2}\text{}{\mathbf{s}}^{-1}\right)$ | ${\mathit{C}}_{\mathbf{v}}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}\left(\mathbf{MJ}\text{}{\mathbf{m}}^{-3}\text{}{\mathbf{K}}^{-1}\right)$ | |
---|---|---|---|---|

4.5-inch/P 12 h | 0.694 | 2.451 | 0.684 | 2.485 |

6-inch/P 12 h | 0.696 | 2.443 | 0.715 | 2.378 |

4.5-inch/P 24 h | 0.650 | 2.614 | 0.722 | 2.356 |

6-inch/P 24 h | 0.659 | 2.581 | 0.805 | 2.113 |

**Table A3.**Thermal diffusivity and heat capacity deviation D (%) from the expected 0.70 mm

^{2}s

^{−1}and 2.43 MJ m

^{−3}K

^{−1}, respectively.

$\mathit{D}\mathsf{\alpha}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | $\mathit{D}{\mathit{C}}_{\mathbf{v}}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | $\mathit{D}\mathsf{\alpha}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ | $\mathit{D}{\mathit{C}}_{\mathbf{v}}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ | |
---|---|---|---|---|

4.5-inch/P 12 h | −0.9 | 0.9 | −2.3 | 2.3 |

6-inch/P 12 h | −0.6 | 0.6 | 2.1 | −2.1 |

4.5-inch/P 24 h | −7.1 | 7.6 | 3.1 | −3.0 |

6-inch/P 24 h | −5.9 | 6.3 | 15.0 | −13.0 |

**Figure A1.**Comparison of the Taylor series expansion of the Kelvin funcion (blue line) with first (orange line) and second-order (green stars) approximations for the oscillatory resistance as a function of ${r}_{\mathrm{pb}}$. Equation (5) reported in the manuscript has been obtained by Eskilson with a first-order approximation.

**Figure A2.**Comparison of the Taylor series expansion of the Kelvin funcion (blue line) with first-order (orange line) and second-order (green stars) approximations for the phase shift as a function of ${r}_{\mathrm{pb}}$. Equation (6) reported in the manuscript has been obtained by Eskilson with a first-order approximation.

## Appendix B. Assumptions and Simplifications

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**Figure 1.**Comparison between conventional and oscillatory TRT (reprint with permission [17]).

**Figure 2.**Steps for the analysis of an OTRT: (

**A**) evaluation of λ

_{heating}and ${R}_{\mathrm{bt}}^{*}$ via the slope method; (

**B**) split of the linear and oscillatory components; (

**C**) comparison of the oscillatory heat injection and the oscillatory thermal response; (

**D**) analysis of the recovery period to estimate λ

_{cooling}.

**Figure 3.**Orthophoto view of the study site. BHEs and observation wells are indicated by red and blue dots, respectively. Numbers indicate the water level in meters above sea level (m a.s.l.), blue lines represent the local potentiometric field.

**Figure 5.**Inclination of the observation well OBS4 (red line) and comparison with the projection of the 1-U BHE (red dashed line and star).

**Figure 7.**Thermal response monitored in the observation well (T_obs) at 5 different depths, compared to temperature calculated with ICS and ILS response functions.

**Figure 9.**Observed temperature during the OTRT carried out in the field. Inlet (red) and outlet (blue) temperature curves are in absolute values. p-linear average (green) curves show increments with respect to the initial temperature.

**Figure 10.**Analysis of the OTRT carried out on the field: (

**A**) evaluation of λ

_{heating}and ${R}_{\mathrm{bt}}^{*}$ via the slope method; (

**B**) split of the linear and oscillatory components; (

**C**) comparison of the oscillatory heat injection and the oscillatory thermal response (the graph is cut at 75 h for simplicity); (

**D**) analysis of the recovery period to estimate λ

_{cooling}.

**Figure 11.**Numerical results (SC1, SC2, SC3) compared to the observations (obs) and the analytical synthetic case based on subsurface input parameters.

Depth (m) | λ (W m^{−1} K^{−1}) |
---|---|

2.5 | 2.26 |

5.0 | 1.97 |

7.5 | 2.00 |

10.0 | 1.97 |

12.5 | 1.93 |

15.0 | 1.86 |

17.5 | 1.91 |

20.0 | 1.86 |

22.5 | 3.58 |

Depth (m) | r (m) | λ (W m^{−1} K^{−1}) | α (mm^{2} s^{−1}) | C_{v} (MJ m^{−3} K^{−1}) |
---|---|---|---|---|

5.0 | 1.20 | 1.97 | 0.70 | 2.81 |

7.5 | 1.20 | 2.00 | 0.75 | 2.67 |

10.0 | 1.30 | 1.97 | 0.70 | 2.81 |

12.5 | 1.60 | 1.93 | 0.70 | 2.75 |

15.0 | 2.00 | 1.86 | 0.82 | 2.27 |

17.5 | 2.45 | 1.91 | 0.77 | 2.48 |

${\mathit{R}}_{\mathbf{p}}\text{}\left(\mathbf{m}\text{}\mathbf{K}\text{}{\mathbf{W}}^{-1}\right)$ | ${\mathbf{\u0278}}_{\mathbf{p}}$ (-) | |
---|---|---|

SC1 | 0.155 | 0.108 |

SC2 | 0.144 | 0.108 |

SC3 | 0.135 | 0.107 |

**Table 4.**Thermal diffusivity deduced from Equation (5) (${R}_{\mathrm{p}}$) and Equation (6) (${\mathsf{\u0278}}_{\mathrm{p}}$) with deviations from the input value.

α_INPUT | $\mathsf{\alpha}\_{\mathit{R}}_{\mathbf{p}}\text{}\left({\mathbf{mm}}^{2}\text{}{\mathbf{s}}^{-1}\right)$ | $\mathit{D}\mathsf{\alpha}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | $\mathsf{\alpha}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}\left({\mathbf{mm}}^{2}\text{}{\mathbf{s}}^{-1}\right)$ | $\mathit{D}\mathsf{\alpha}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ | |
---|---|---|---|---|---|

SC1 | 0.85 | 1.065 | 25.3 | 0.388 | −54.4 |

SC2 | 0.71 | 0.772 | 9.0 | 0.388 | −45.2 |

SC3 | 0.61 | 0.593 | −2.3 | 0.400 | −34.1 |

**Table 5.**Heat capacity deduced with TD from Table 4 and TC from the cooling period with deviations from the input value.

C_{v}_INPUT | ${\mathit{C}}_{\mathbf{v}}\_{\mathit{R}}_{\mathbf{p}}\text{}\left(\mathbf{MJ}\text{}{\mathbf{m}}^{-3}\text{}{\mathbf{K}}^{-1}\right)$ | ${\mathit{DC}}_{\mathbf{v}}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | ${\mathit{C}}_{\mathbf{v}}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}\left(\mathbf{MJ}\text{}{\mathbf{m}}^{-3}\text{}{\mathbf{K}}^{-1}\right)$ | ${\mathit{DC}}_{\mathbf{v}}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ | |
---|---|---|---|---|---|

SC1 | 2.0 | 1.610 | −19.5 | 4.417 | 120.9 |

SC2 | 2.4 | 2.194 | −8.6 | 4.363 | 81.8 |

SC3 | 2.8 | 2.827 | 1.0 | 4.192 | 49.7 |

${\mathit{R}}_{\mathbf{p}}\text{}\left(\mathbf{m}\text{}\mathbf{K}\text{}{\mathbf{W}}^{-1}\right)$ | $\mathsf{\delta}{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | ${\mathbf{\u0278}}_{\mathbf{p}}$ (-) | $\mathsf{\delta}{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ |
---|---|---|---|

0.152 | 4.10 | 0.100 | 3.47 |

**Table 7.**Thermal diffusivity deduced from Equation (5) (${R}_{\mathrm{p}}$) and Equation (6) (${\mathsf{\u0278}}_{\mathrm{p}}$) and related uncertainties δ (%).

$\mathsf{\alpha}\_{\mathit{R}}_{\mathbf{p}}\text{}\left({\mathbf{mm}}^{2}\text{}{\mathbf{s}}^{-1}\right)$ | $\mathsf{\delta}\mathsf{\alpha}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | $\mathsf{\alpha}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}\left({\mathbf{mm}}^{2}\text{}{\mathbf{s}}^{-1}\right)$ | $\mathsf{\delta}\mathsf{\alpha}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ |
---|---|---|---|

0.959 | 15.01 | 0.478 | 14.56 |

**Table 8.**Heat capacity value deduced with TD from Table 7 and TC from the cooling period, and related uncertainties δ (%).

${\mathit{C}}_{\mathbf{v}}\_{\mathit{R}}_{\mathbf{p}}\text{}\left(\mathbf{MJ}\text{}{\mathbf{m}}^{-3}\text{}{\mathbf{K}}^{-1}\right)$ | $\mathsf{\delta}{\mathit{C}}_{\mathbf{v}}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | ${\mathit{C}}_{\mathbf{v}}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}\left(\mathbf{MJ}\text{}{\mathbf{m}}^{-3}\text{}{\mathbf{K}}^{-1}\right)$ | $\mathsf{\delta}{\mathit{C}}_{\mathbf{v}}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ |
---|---|---|---|

1.775 | 14.85 | 3.562 | 14.85 |

**Table 9.**Comparison of oscillation parameters obtained from the analysis of field tests and numerical simulations. The two reference cases calculated with ground and grout properties are also reported.

${\mathit{R}}_{\mathbf{p}}\text{}\left(\mathbf{m}\text{}\mathbf{K}\text{}{\mathbf{W}}^{-1}\right)$ | $\mathsf{\delta}{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | ${\mathbf{\u0278}}_{\mathbf{p}}$ (-) | $\mathsf{\delta}{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ | ||
---|---|---|---|---|---|

Field | 0.152 | 4.10 | 0.100 | 3.47 | |

Numerical | SC1 | 0.155 | - | 0.108 | - |

SC2 | 0.144 | - | 0.108 | - | |

SC3 | 0.135 | - | 0.107 | - | |

Analytical | Ground | 0.142 | - | 0.085 | - |

Grout | 0.121 | - | 0.103 | - |

**Table 10.**Comparison of thermal diffusivity obtained from the analysis of field tests and numerical simulations. TD of ground and grout are also reported.

$\mathsf{\delta}\mathsf{\alpha}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | $\mathsf{\delta}\mathsf{\alpha}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ | ||||
---|---|---|---|---|---|

Field | 0.959 | 15.01 | 0.478 | 14.56 | |

Dα_${\mathit{R}}_{p}$(%) | Dα_${\mathsf{\u0278}}_{p}$(%) | ||||

Numerical | SC1 | 1.065 | 25.3 | 0.388 | −54.4 |

SC2 | 0.772 | 9.0 | 0.388 | −45.2 | |

SC3 | 0.593 | −2.3 | 0.400 | −34.1 | |

Analytical | Ground | 0.80 | - | 0.80 | - |

Grout | 0.44 | - | 0.44 | - |

**Table 11.**Comparison of heat capacity obtained from the analysis of field tests and numerical simulations. HC of subsurface and grout are also reported.

$\mathsf{\delta}{\mathit{C}}_{\mathbf{v}}\_{\mathit{R}}_{\mathbf{p}}\text{}(\%)$ | $\mathsf{\delta}{\mathit{C}}_{\mathbf{v}}\_{\mathbf{\u0278}}_{\mathbf{p}}\text{}(\%)$ | ||||
---|---|---|---|---|---|

Field | 1.775 | 14.85 | 3.562 | 14.85 | |

DC${\mathit{R}}_{p}$_{v}_(%) | DC${\mathsf{\u0278}}_{p}$_{v}_(%) | ||||

Numerical | SC1 | 1.610 | −19.5 | 4.417 | 120.9 |

SC2 | 2.194 | −8.6 | 4.363 | 81.8 | |

SC3 | 2.827 | 1.0 | 4.192 | 49.7 | |

Analytical | Ground | 2.16 | - | 2.16 | - |

Grout | 3.90 | - | 3.90 | - |

${x}_{c}$(m) | 0.030 | 0.028 | 0.025 | 0.023 |

${\mathit{R}}_{bt}^{*}$(m K W^{−1}) | 0.0858 | 0.0894 | 0.0952 | 0.0996 |

C${\mathit{R}}_{p}$_{v}_(MJ m^{−3} K^{−1}) | 1.645 | 1.775 | 2.011 | 2.209 |

D (%) | −7.3 | - | 13.3% | 24.4 |

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**MDPI and ACS Style**

Giordano, N.; Lamarche, L.; Raymond, J.
Evaluation of Subsurface Heat Capacity through Oscillatory Thermal Response Tests. *Energies* **2021**, *14*, 5791.
https://doi.org/10.3390/en14185791

**AMA Style**

Giordano N, Lamarche L, Raymond J.
Evaluation of Subsurface Heat Capacity through Oscillatory Thermal Response Tests. *Energies*. 2021; 14(18):5791.
https://doi.org/10.3390/en14185791

**Chicago/Turabian Style**

Giordano, Nicolò, Louis Lamarche, and Jasmin Raymond.
2021. "Evaluation of Subsurface Heat Capacity through Oscillatory Thermal Response Tests" *Energies* 14, no. 18: 5791.
https://doi.org/10.3390/en14185791