# Enhanced Steady-State Solution of the Infinite Moving Line Source Model for the Thermal Design of Grouted Borehole Heat Exchangers with Groundwater Advection

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}-emission-reducing alternative for heating and cooling of residential and commercial buildings compared to conventional systems [1,2,3]. A typical GSHP system consists of a heat pump coupled with horizontal or, more commonly, vertical borehole heat exchanger (BHEs). According to [2,4], the increased use of these systems in the last decades can be ascribed mainly to the refocusing of energy policy in Europe and the significant economic growth in China. Compared to air source heat pump systems, the efficiency of GSHP systems is better and their environmental impact is lower [5,6]. However, installation costs are significantly higher, which affects their economic competitiveness [1,7]. For the design of cost-optimized systems, it is important to consider all relevant heat transfer effects of the ground source and to include as many geological characteristics for site-specific system tuning as possible.

^{light}[15] are widely used for the dimensioning process of BHEs [9]. Operating temperatures of BHEs in these programs are calculated by superposing the temperature change at the borehole wall and within the borehole itself, which are both due to the thermal loads (extraction or injection of heat), on undisturbed ground temperature. These two different temperature changes are commonly derived by established procedures: the temperature change at the borehole wall or further away in the subsurface is computed by extensions of the instantaneous point source model [16]. A resistance model (the borehole resistance R

_{b}, partially extended by thermal capacities) is applied to the temperature change in the borehole [12,13,14,15,16,17].

_{s}, which is defined as the ratio of the considered time t to the steady-state time t

_{s}as determined by Eskilson [21], and the dimensionless geometry r

_{b}/H [21]. If a borehole field of several interacting BHEs is considered, additional parameters such as the number and arrangement of the boreholes, and the distance between them, have an impact on the g-function as well. Furthermore, Eskilson [21] introduced the analytical solution of the finite line source for steady-state conditions including an isothermal boundary condition at the surface. Using the latter for dimensioning purposes, Esklison [21] recommended the application of the volume conservation principle in order to define the mean temperature at the borehole wall at steady-state conditions. In contrast, Zeng et al. [23] suggested using the integral mean temperature, which leads to nearly the same equation. Both the numerically determined transient g-functions of Eskilson [21] and the analytical finite line source model for steady-state conditions account for heat conduction in a homogeneous subsurface, thereby ignoring heterogeneity or advection associated with groundwater flow.

_{b}, determined by the infinite MLS, with the temperature occurring at the grouted borehole. The analytical model assumes that the groundwater flows horizontally directly around the vertical heat source, i.e., that heat conduction and advection occur within the borehole, which differs strongly from the real situation. For the case without grouting, the numerical and analytical model agree. In addition, horizontal groundwater flow induces a non-radial temperature distribution around the borehole; it must be examined whether this affects the borehole resistance and to what extent existing analytical methods to calculate the borehole resistance can still be used. An extra focus is placed on the validity of the infinite implementation of the MLS. Clearly, BHEs are better represented by finite lines and by including the three-dimensional effects from the ground surface and the borehole toe. The infinite MLS does not account for these but is a popular simplification and is easier to implement for BHE simulation than the finite MLS. As horizontal advection due to groundwater flow is expected to reduce the relative impact of the axial heat flow components at the top and bottom of a BHE, the infinite MLS may serve as a sufficient estimate, especially for dimensioning longer BHEs.

## 2. Materials and Methods

#### 2.1. Infinite Moving Line Source

- an initial temperature of the porous medium of zero (IC),
- a continuous source of constant strength $\dot{q}\left(t\right)$ generated at the point $P\left({x}^{\prime},{y}^{\prime}\right)=\left(0,0\right)$ from t = 0 onwards (BC),
- a surface temperature of zero located at infinity (BC).

#### 2.2. Finite Moving Line Source

_{,}and the borehole length H of the steady-state solution of the finite MLS with an isothermal boundary condition at the surface is deduced from the steady-state solution of the moving point source [41]:

#### 2.3. Numerical Model

^{®}[45]. Equivalent to the assumptions of the analytical model, a two-dimensional numerical model is set up, similar to that of Lazzari et al. [46]. In this way, the focus is set on the role of the grout, while keeping calculation time low. The error caused by ignoring the three-dimensional effects of a finite borehole is subsequently evaluated by comparison with the finite MLS (Equation (11)).

_{b}and a centered heat source in the grouting material is embedded in the simulated ground and is treated independently, particularly with regard to heat transfer processes and its boundary conditions towards the subsurface. Since the grouting of boreholes is applied to seal the borehole against the adjacent subsurface, the grouting material is modeled using heat conduction only, so that no groundwater flow occurs within the borehole itself. The model allows for different thermal properties of the grouting material in the borehole and the porous medium around the borehole. As is common in related works (e.g., [13]), the heat exchanger pipes are not modeled explicitly but represented by a concentric heat source in the grouted borehole.

#### 2.4. Correction of Infinite MLS and Application to BHE Design

## 3. Results and Discussion

#### 3.1. Compatibility of the Thermal Borehole Resistance Model for BHEs with Groundwater Advection

#### 3.2. Applicability of the Infinite MLS Model to Finite Boreholes

_{b}by H in Equation (5)). Translating the findings of Molina-Giraldo et al. [32] for a BHE with a length of 50 m, the discrepancy between the finite and the infinite MLS becomes irrelevant for Pe ≥ 0.015. The investigated range here does not cover such small Péclet numbers as by Molina-Giraldo [32], but the results are consistent. Furthermore, the infinite MLS model calculates (slightly) larger g-functions and, as a result, a greater temperature change than the finite MLS model. Hence, the infinite MLS model is the more conservative model and therefore favorable for designing BHEs of 30 m and more.

#### 3.3. Steady-State Thermal Conditions at the Wall of a Grouted Borehole

#### 3.4. Correction Function

^{light}for the planning of BHEs with groundwater advection, without changing further calculation methods and without the need for a complex numerical simulation. The correction function overcomes the deficiency of neglecting the effect of the non-permeable, grouted borehole, and thus prevents a temperature change prediction that is too low during the design phase.

#### 3.5. Demonstration Example

^{∙}K)/W, an undisturbed subsurface temperature of 12 °C, and a maximum temperature increase of 10 K, the required borehole depths and the resulting specific heat injection rates are determined by transforming Equation (14). Furthermore, the percentage deviation of the aforementioned properties calculated with and without the proposed correction in Equation (17) are listed as well. The percentage deviation expresses the discrepancy between both calculations compared with the calculation using the uncorrected g-function.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c_{p} | J/kg·K | Heat capacity at constant pressure |

f_{cor} | - | Correction function |

Fo | - | Fourier number (dimensionless time) |

g | - | g-function, dimensionless temperature response |

H | m | Borehole length |

I_{0} | - | Modified Bessel function of the first kind of order 0 |

K_{0} | - | Modified Bessel function of the second kind of order 0 |

k_{f} | m/s | Hydraulic conductivity |

Pe | - | Péclet number, the ratio of advective to diffusive heat transport |

$\dot{q}$ | W/m | Specific heat injection (positive) or extraction (negative) |

r | m | Radius |

R_{b} | (m·K)/W | Thermal borehole resistance |

t | s | Time |

U | m/s | Velocity |

x, y, z | m | Space coordinates, where the temperature is evaluated |

x´, y´, z´ | m | Space coordinates, where the heat source is located |

Greek symbols | ||

α | m^{2}/s | Thermal diffusivity |

Γ | - | The generalized incomplete gamma function |

$\Delta \vartheta $ | K | Temperature difference |

$\Delta \overline{\vartheta}$ | K | Mean temperature change |

ϑ | °C | Temperature |

${\overline{\vartheta}}_{E,0}$ | °C | Mean undisturbed subsurface temperature |

λ | W/(m·K) | Thermal conductivity |

ρ | kg/m^{3} | Density |

τ | s | Time at which the heat source is switched on |

φ | - | Porosity of the subsurface |

φ | - | Angle around the heat source, with φ = 0 corresponding to the direction of the groundwater flow in the plane perpendicular to the heat source and located behind the heat source in the groundwater flow direction. |

Subscripts: | ||

b | Referring to the borehole wall | |

Darcy | Referring to Darcy´s law | |

eff | Referring to the effective physical properties of the subsurface, which are i.e., volume-weighted unless otherwise specified | |

End | Referring to the state-state condition | |

grout | Referring to the grouting material | |

IMLS | Infinite moving line source | |

num | Referring to the numerical simulation(s) | |

f | Referring to the physical properties of the fluid | |

FMLS | Finite moving line source | |

s | Referring to the physical properties of the solid phase (rock matrix) | |

Abbreviations: | ||

BHE | Borehole heat exchanger | |

FLS | Finite line source | |

GSHP | Ground source heat pump | |

GW | Groundwater | |

MLS | Moving line source | |

UPS | Uninterrupted power supply |

## References

- Blum, P.; Campillo, G.; Münch, W.; Kölbel, T. CO
_{2}savings of ground source heat pump systems—A regional analysis. Renew. Energy**2010**, 35, 122–127. [Google Scholar] [CrossRef] - Bayer, P.; Saner, D.; Bolay, S.; Rybach, L.; Blum, P. Greenhouse gas emission savings of ground source heat pump systems in Europe: A review. Renew. Sustain. Energy Rev.
**2012**, 16, 1256–1267. [Google Scholar] [CrossRef] - Aditya, G.R.; Narsilio, G.A. Environmental assessment of hybrid ground source heat pump systems. Geothermics
**2020**, 87, 101868. [Google Scholar] [CrossRef] - Rees, S.J. An introduction to ground-source heat pump technology. In Advances in Ground-Source Heat Pump Systems; Rees, S., Ed.; Elsevier Reference Monographs; Elsevier: Amsterdam, The Netherlands, 2016; pp. 1–25. [Google Scholar]
- Yu, X.; Zhang, Y.; Deng, N.; Wang, J.; Zhang, D.; Wang, J. Thermal response test and numerical analysis based on two models for ground-source heat pump system. Energy Build.
**2013**, 66, 657–666. [Google Scholar] [CrossRef] - Saner, D.; Juraske, R.; Kübert, M.; Blum, P.; Hellweg, S.; Bayer, P. Is it only CO
_{2}that matters? A life cycle perspective on shallow geothermal systems. Renew. Sustain. Energy Rev.**2010**, 14, 1798–1813. [Google Scholar] [CrossRef] - Diao, N.; Li, Q.; Fang, Z. Heat transfer in ground heat exchangers with groundwater advection. Int. J. Therm. Sci.
**2004**, 43, 1203–1211. [Google Scholar] [CrossRef] - Ingenieure, V.D. Thermische Nutzung des Untergrunds: Erdgekoppelte Wärmepumpenanlagen; Beuth Verlag GmbH: Berlin, Germany, 2019. [Google Scholar]
- Reuss, M.; Karrer, H.; Gehlin, S.; Andersson, O.; Bjorn, H.; Nagano, K.; Katsura, T.; Metzner, M. IEA ECES ANNEX 27: Quality Management in Design, Construction and Operation of Borehole Systems. Final Report. 2020. Available online: https://iea-eces.org/wp-content/uploads/public/IEA-ECES-ANNEX-27-Final-Report-20201118.pdf (accessed on 7 December 2020).
- Rivera, J.A.; Blum, P.; Bayer, P. Increased ground temperatures in urban areas: Estimation of the technical geothermal potential. Renew. Energy
**2017**, 103, 388–400. [Google Scholar] [CrossRef][Green Version] - FascÌ, M.L.; Lazzarotto, A.; Acuna, J.; Claesson, J. Analysis of the thermal interference between ground source heat pump systems in dense neighborhoods. Sci. Technol. Built Environ.
**2019**, 25, 1069–1080. [Google Scholar] [CrossRef][Green Version] - EED Version 4—Earth Energy Designer: Update Manual. 2020. Available online: https://www.buildingphysics.com/manuals/EED4.pdf (accessed on 1 March 2021).
- Spitler, J.D.; Marshall, C.L.; Manickam, A.; Dharapuram, M.; Delahoussaye, R.D.; Yeung, K.W.D.; Young, R.; Bhargava, M.; Mokashi, S.; Yavuzturk, C.; et al. GLHEPro 5.0 For Windows: Users’ Guide. 2016. Available online: https://hvac.okstate.edu/sites/default/files/pubs/glhepro/GLHEPRO_5.0_Manual.pdf (accessed on 1 March 2021).
- Huber, A. Bedienungsanleitung zum Programm EWS; Huber Energietechnik AG: Zürich, Switzerland, 2016. [Google Scholar]
- Koenigsdorff, R. Oberflächennahe Geothermie für Gebäude: Grundlagen und Anwendungen Zukunftsfähiger Heizung und Kühlung; Fraunhofer IRB-Verl.: Stuttgart, Germany, 2011. [Google Scholar]
- Erol, S.; François, B. Multilayer analytical model for vertical ground heat exchanger with groundwater flow. Geothermics
**2018**, 71, 294–305. [Google Scholar] [CrossRef] - De Carli, M.; Tonon, M.; Zarrella, A.; Zecchin, R. A computational capacity resistance model (CaRM) for vertical ground-coupled heat exchangers. Renew. Energy
**2010**, 35, 1537–1550. [Google Scholar] [CrossRef] - Bennet, J.; Claesson, J.; Hellström, G. Multipole Method to Compute the Conductive Heat Flows to and Between Pipes in a Composite Cylinder; Notes on Heat Transfer; University of Lund: Lund, Sweden, 1987. [Google Scholar]
- Javed, S.; Spitler, J.D. Calculation of Borehole Thermal Resistance; Elsevier: Amsterdam, The Netherlands, 2016. [Google Scholar] [CrossRef]
- Claesson, J.; Hellström, G. Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Res.
**2011**, 17, 895–911. [Google Scholar] [CrossRef] - Eskilson, P. Thermal Analysis of Heat Extraction. Boreholes. Dissertation, University of Lund, Lund, Sweden, 1987. [Google Scholar]
- Javed, S.; Fahlén, P.; Claesson, J. Vertical ground heat exchangers: A review of heat flow models. In Proceedings of the Effstock 2009, Stockholm, Sweden, 14–17 June 2009. [Google Scholar]
- Zeng, H.Y.; Diao, N.R.; Fang, Z.H. A finite line-source model for boreholes in geothermal heat exchangers. Heat Trans. Asian Res.
**2002**, 31, 558–567. [Google Scholar] [CrossRef] - Angelotti, A.; Ly, F.; Zille, A. On the applicability of the moving line source theory to thermal response test under groundwater flow: Considerations from real case studies. Geotherm Energy
**2018**, 6, 12. [Google Scholar] [CrossRef][Green Version] - Antelmi, M.; Alberti, L.; Angelotti, A.; Curnis, S.; Zille, A.; Colombo, L. Thermal and hydrogeological aquifers characterization by coupling depth-resolved thermal response test with moving line source analysis. Energy Convers. Manag.
**2020**, 225, 113400. [Google Scholar] [CrossRef] - Chiasson, A.; O´Connell, A. New analytical solution for sizing vertical borehole ground heat exchangers in environments with significant groundwater flow: Parameter estimation from thermal response test data. HVAC&R Res.
**2011**, 17, 1000–1011. [Google Scholar] [CrossRef] - Claesson, J.; Hellström, G. Analytical Studies of the Influence of Regional Groundwater Flow on the Performance of Borehole Heat Exchangers. In Proceeding of the 8th International Conference on Thermal Energy Storage, Stuttgart, Germany, 20 August 2000. [Google Scholar]
- Hecht-Méndez, J.; de Paly, M.; Beck, M.; Bayer, P. Optimization of energy extraction for vertical closed-loop geothermal systems considering groundwater flow. Energy Convers. Manag.
**2013**, 66, 1–10. [Google Scholar] [CrossRef] - Katsura, T.; Nagano, K.; Takeda, S.; Shimakura, K. Heat Transfer Experiment in the Ground with Ground Water Advection. In Proceeding of the 10th International Conference on Thermal Energy Storage, Galloway, New Jersey, USA, 31 May–2 June 2006. [Google Scholar]
- Kölbel, T. Grundwassereinfluss auf Erdwärmesonden: Geländeuntersuchungen und Modellrechnungen. Ph.D. Thesis, Karlsruher Institut für Technologie, Karlsruhe, Germany, 2010. [Google Scholar]
- Mohammadzadeh Bina, S.; Fujii, H.; Kosukegawa, H.; Farabi-Asl, H. Evaluation of ground source heat pump system’s enhancement by extracting groundwater and making artificial groundwater velocity. Energy Convers. Manag.
**2020**, 223, 113298. [Google Scholar] [CrossRef] - Molina-Giraldo, N.; Blum, P.; Zhu, K.; Bayer, P.; Fang, Z. A moving finite line source model to simulate borehole heat exchangers with groundwater advection. Int. J. Therm. Sci.
**2011**, 50, 2506–2513. [Google Scholar] [CrossRef] - Stauffer, F.; Bayer, P.; Blum, P.; Molina Giraldo, N.A.; Kinzelbach, W. Thermal Use of Shallow Groundwater; First issued in paperback 2017; Environmental engineering; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2017. [Google Scholar]
- Sutton, M.G.; Nutter, D.W.; Couvillion, R.J. A Ground Resistance for Vertical Bore Heat Exchangers With Groundwater Flow. J. Energy Resour. Technol.
**2003**, 125, 183–189. [Google Scholar] [CrossRef] - Zhang, L.; Shi, Z.; Yuan, T. Study on the Coupled Heat Transfer Model Based on Groundwater Advection and Axial Heat Conduction for the Double U-Tube Vertical Borehole Heat Exchanger. Sustainability
**2020**, 12, 7345. [Google Scholar] [CrossRef] - Tye-Gingras, M.; Gosselin, L. Generic ground response functions for ground exchangers in the presence of groundwater flow. Renew. Energy
**2014**, 72, 354–366. [Google Scholar] [CrossRef] - Hellström, G. Ground Heat Storage: Thermal Analyses of Duct Storage Systems Theory. Ph.D. Thesis, University of Lund, Lund, Sweden, 1991. [Google Scholar]
- Rivera, J.A.; Blum, P.; Bayer, P. Analytical simulation of groundwater flow and land surface effects on thermal plumes of borehole heat exchangers. Appl. Energy
**2015**, 146, 421–433. [Google Scholar] [CrossRef] - Guo, Y.; Hu, X.; Banks, J.; Liu, W.V. Considering buried depth in the moving finite line source model for vertical borehole heat exchangers—A new solution. Energy Build.
**2020**, 214, 109859. [Google Scholar] [CrossRef] - Wagner, V.; Blum, P.; Kübert, M.; Bayer, P. Analytical approach to groundwater-influenced thermal response tests of grouted borehole heat exchangers. Geothermics
**2013**, 46, 22–31. [Google Scholar] [CrossRef] - Carslaw, H.S.; Jaeger, J.C. Conduction of Heat in Solids, 2nd ed.; Oxford Science Publications; Clarendon Press: Oxford, UK, 1959. [Google Scholar]
- Bear, J. Dynamics of Fluids in Porous Media; Dover Books on Physics and Chemistry; Dover: New York, NY, USA, 1988. [Google Scholar]
- Chaudhry, M.A.; Zubair, S.M. On a Class of Incomplete Gamma Functions with Applications: Chapman and Hall/CRC; Chapman and Hall/CRC: Boca Raton, FL, USA, 2001. [Google Scholar]
- Rivera, J.A.; Blum, P.; Bayer, P. Influence of spatially variable ground heat flux on closed-loop geothermal systems: Line source model with nonhomogeneous Cauchy-type top boundary conditions. Appl. Energy
**2016**, 180, 572–585. [Google Scholar] [CrossRef][Green Version] - COMSOL Multiphysics, version 5.6; COMSOL AB: Stockholm, Sweden, 2020.
- Lazzari, S.; Priarone, A.; Zanchini, E. Long-Term Performance of Borehole Heat Exchanger Fluids with Groundwater Movement: Excerpt from the Proceedings of the COMSOL Conference 2010 Paris. In Proceedings of the COMSOL Conference 2010, Paris, France, 17–19 November 2010. [Google Scholar]
- Gossler, M.A.; Bayer, P.; Zosseder, K. Experimental investigation of thermal retardation and local thermal non-equilibrium effects on heat transport in highly permeable, porous aquifers. J. Hydrol.
**2019**, 578, 124097. [Google Scholar] [CrossRef] - 48. COMSOL Multiphysics
^{®}. Subsurface Flow Module: User´s Guide. 2021. Available online: https://doc.comsol.com/5.4/doc/com.comsol.help.ssf/SubsurfaceFlowModuleUsersGuide.pdf (accessed on 24 August 2021). - Claesson, J.; Dunand, A. Heat Extraction from the Ground by Horizontal Pipes: A Mathematical Analysis; Swedish Council for Building Research: Stockholm, Sweden, 1983. [Google Scholar]
- Gu, Y.; O´Neal, D.L. Development of an Equivalent Diameter Expression for Vertical U-Tubes Used in Ground-Coupled Heat Pumps. Transactions-Am. Soc. Heat. Refrig. Air Cond. Eng.
**1998**, 104, 347–355. [Google Scholar] - Gu, Y.; O’Neal, D.L. An Analytical Solution to Transient Heat Conduction in a Composite Region with a Cylindrical Heat Source. J. Sol. Energy Eng.
**1995**, 242–248. [Google Scholar] [CrossRef] - Kavanaugh, S.P. Simulation and Experimental Verification of Vertical Ground-coupled Heat Pump Systems. Ph.D. Thesis, Oklahoma State University, Stillwater, OK, USA, 1985. [Google Scholar]
- Lazzarotto, A.; Pallard, W.M. Thermal Response Test Performance Evaluation with Drifting Heat Rate and Noisy Measurements. In Proceedings of the European Geothermal Congress, Hague, The Netherlands, 11–14 June 2019; pp. 1–9. [Google Scholar]
- Javed, S.; Claesson, J. New analytical and numerical solutions for the short-term analysis of vertical ground heat exchangers. ASHRAE Trans.
**2011**, 117, 3–12. [Google Scholar] - Lamarche, L.; Beauchamp, B. New solutions for the short-time analysis of geothermal vertical boreholes. Int. J. Heat Mass Transf.
**2007**, 50, 1408–1419. [Google Scholar] [CrossRef] - Van de Ven, A.; Koenigsdorff, R.; Hofmann, S. Entwicklung konsistenter Auslegungsmodelle für oberflächennahe geothermische Quellensysteme. In Proceedings of the BauSIM 2018. 7. Deutsch-Österreichische IBPSA-Konferenz, Karlsruhe, Germany, 26–28 September 2018; Karlsruher Institut für Technologie: Karlsruhe, Germany, 2018; pp. 508–515. [Google Scholar]
- Wagner, V.; Bayer, P.; Bisch, G.; Kübert, M.; Blum, P. Hydraulic characterization of aquifers by thermal response testing: Validation by large-scale tank and field experiments. Water Resour. Res.
**2014**, 50, 71–85. [Google Scholar] [CrossRef]

**Figure 3.**Percentage deviation of the g-functions calculated with the infinite and finite MLS with an isothermal boundary condition at the ground surface (

**a**) for a borehole radius of rb = 0.075 m and (

**b**) a borehole length of H = 30 m.

**Figure 4.**Deviation of the g-functions between the numerical model with a grouted borehole and the analytical infinite MLS model.

**Figure 5.**Percentage deviation of the g-functions depending on the effective thermal conductivity for different Darcy velocities.

**Figure 6.**Percentage deviation of the g-functions depending on the Péclet number for different effective thermal conductivities.

**Figure 7.**Second-degree polynomial correction function for the infinite MLS for steady-state conditions.

**Figure 10.**Deviation in borehole length depending on the Péclet number for the thermal properties as listed in Table 3 of (

**a**) Karst Limestone and (

**b**) Gravel and Sand (coarse).

**Table 1.**Examined parameter ranges for the comparison between the numerical model and the infinite MLS.

Parameter Description | Unit | Value Range | |||
---|---|---|---|---|---|

Darcy velocity υ_{Darcy} | cm/day | 2.94 | up to | 881.28 | |

Thermal conductivity of the solid phase λ_{s} | W/(m∙K) | 1.0 | 2.0 | 3.0 | 4.0 |

Porosity of the subsurface Φ | - | 0.1 | 0.3 | 0.5 | |

Thermal conductivity of the grouting material λ_{grout} | W/(m∙K) | 1.0 | 2.0 | ||

Borehole radius r_{b} | m | 0.06 | 0.075 | 0.09 | 0.105 |

Resulting Péclet numbers Pe | - | 0.1 | up to | 10 | |

Initial and boundary temperature ${\vartheta}_{E,0}$ | °C | 10 | |||

heat extraction rate $\dot{q}$ | W/m | −20 |

Borehole Radius r _{b} | Thermal Conductivity of the Grout λ _{grout} | Darcy Velocity υ _{Darcy} | Numerical Simulation with Advection | Numerical Simulation without Advection | Analytical Solution | ||
---|---|---|---|---|---|---|---|

Thermal Borehole Resistance R _{b} | Thermal Borehole Resistance R_{b} | Percentage Deviation | Thermal Borehole Resistance R_{b} | Percentage Deviation | |||

m | W/(m·K) | m/s | (m·K)/W | (m·K)/W | % | (m·K)/W | % |

0.060 | 1.0 | 3.40 × 10^{−}^{6} | 0.2102 | 0.2096 | −0.26% | 0.2104 | 0.09% |

2.0 | 3.40 × 10^{−6} | 0.1051 | 0.1048 | −0.26% | 0.1052 | 0.09% | |

1.0 | 1.70 × 10^{−}^{5} | 0.2102 | 0.2096 | −0.26% | 0.2104 | 0.09% | |

2.0 | 1.70 × 10^{−5} | 0.1051 | 0.1048 | −0.26% | 0.1052 | 0.09% | |

1.0 | 3.40 × 10^{−}^{5} | 0.2102 | 0.2096 | −0.26% | 0.2104 | 0.09% | |

2.0 | 3.40 × 10^{−5} | 0.1051 | 0.1048 | −0.26% | 0.1052 | 0.09% | |

1.0 | 1.70 × 10^{−}^{4} | 0.2102 | 0.2096 | −0.26% | 0.2104 | 0.09% | |

2.0 | 1.70 × 10^{−4} | 0.1051 | 0.1048 | −0.26% | 0.1052 | 0.09% | |

0.105 | 1.0 | 3.40 × 10^{−6} | 0.2991 | 0.2973 | −0.61% | 0.2994 | 0.10% |

2.0 | 3.40 × 10^{−6} | 0.1496 | 0.1487 | −0.59% | 0.1497 | 0.10% | |

1.0 | 1.70 × 10^{−5} | 0.2991 | 0.2973 | −0.61% | 0.2994 | 0.10% | |

2.0 | 1.70 × 10^{−5} | 0.1496 | 0.1487 | −0.59% | 0.1497 | 0.10% | |

1.0 | 3.40 × 10^{−5} | 0.2991 | 0.2973 | −0.61% | 0.2994 | 0.10% | |

2.0 | 3.40 × 10^{−5} | 0.1496 | 0.1487 | −0.59% | 0.1497 | 0.10% | |

1.0 | 1.70 × 10^{−4} | 0.2991 | 0.2973 | −0.61% | 0.2994 | 0.10% | |

2.0 | 1.70 × 10^{−4} | 0.1496 | 0.1487 | −0.59% | 0.1497 | 0.10% |

**Table 3.**Thermal and hydraulic properties of the subsurface used for the setup of demonstration examples [34].

Material | Thermal Conductivity λ | Vol. Heat Capacity c_{v} | Porosity Φ | Darcy Velocity υ_{Darcy} | Borehole Radius r_{b} | Péclet Number Pe |
---|---|---|---|---|---|---|

W/(m∙K) | MJ/(m^{3}∙K) | - | m/yr | - | ||

Groundwater * | 0.60 | 4.18 | ||||

Karst limestone * | 3.40 | 13.40 | 0.275 | 31.63 | 0.054 | 0.09 |

Sand (coarse) * | 0.8 | 1.40 | 0.385 | 23.14 | 0.054 | 0.23 |

Gravel * | 0.8 | 1.40 | 0.310 | 945.50 | 0.054 | 9.17 |

Gravel (modified) ** | 0.8 | 1.40 | 0.310 | 74.25 | 0.075 | 1.00 |

**Table 4.**Calculation example for the four cases listed in Table 3.

Heat Injection Rate | 8.00 | kW | |||||
---|---|---|---|---|---|---|---|

Borehole Resistance | 0.08 | (m·K)/W | |||||

Maximum Temperature Change of the Subsurface | 10 | K | |||||

Péclet Number | Model Selection | g- Function | Borehole Length | Percentage Deviation of the Borehole Length | Specific Heat Injection Rate | Percentage Deviation of the Heat Injection Rate | |

- | - | - | m | % | W/m | % | |

Karst limestone * | 0.09 | FLS | 6.6 | 383.52 | 74.39% | 20.86 | 42.66% |

IMLS without correction | 3.22 | 219.93 | 0.00% | 36.38 | 0.00% | ||

IMLS with correction | 3.32 | 224.60 | 2.13% | 35.62 | 2.08% | ||

Sand (coarse) * | 0.23 | FLS | 6.6 | 1226.29 | 161.34% | 6.52 | 61.74% |

IMLS without correction | 2.30 | 469.24 | 0.00% | 17.05 | 0.00% | ||

IMLS with correction | 2.49 | 501.66 | 6.91% | 15.95 | 6.46% | ||

Gravel * | 9.17 | FLS | 6.6 | 1202.67 | 1350.07% | 6.65 | 93.10% |

IMLS without correction | 0.11 | 82.94 | 0.00% | 96.46 | 0.00% | ||

IMLS with correction | 0.42 | 137.10 | 65.31% | 58.35 | 39.51% | ||

Gravel (modified) ** | 1.00 | FLS | 6.6 | 1202.67 | 414.82% | 6.65 | 80.58% |

IMLS without correction | 0.98 | 233.61 | 0.00% | 34.25 | 0.00% | ||

IMLS with correction | 1.34 | 2954.67 | 26.14% | 27.15 | 20.72% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Van de Ven, A.; Koenigsdorff, R.; Bayer, P.
Enhanced Steady-State Solution of the Infinite Moving Line Source Model for the Thermal Design of Grouted Borehole Heat Exchangers with Groundwater Advection. *Geosciences* **2021**, *11*, 410.
https://doi.org/10.3390/geosciences11100410

**AMA Style**

Van de Ven A, Koenigsdorff R, Bayer P.
Enhanced Steady-State Solution of the Infinite Moving Line Source Model for the Thermal Design of Grouted Borehole Heat Exchangers with Groundwater Advection. *Geosciences*. 2021; 11(10):410.
https://doi.org/10.3390/geosciences11100410

**Chicago/Turabian Style**

Van de Ven, Adinda, Roland Koenigsdorff, and Peter Bayer.
2021. "Enhanced Steady-State Solution of the Infinite Moving Line Source Model for the Thermal Design of Grouted Borehole Heat Exchangers with Groundwater Advection" *Geosciences* 11, no. 10: 410.
https://doi.org/10.3390/geosciences11100410