# A Review of the Modelling of Thermally Interacting Multiple Boreholes

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## Abstract

**:**

## 1. Introduction

## 2. Objectives of GHE Modeling

#### 2.1. Environmental Impacts

#### 2.2. Sustainability

#### 2.3. Thermal Interaction

## 3. Modeling Ground Heat Exchangers

**Figure 1.**Cross-section of a vertical ground heat exchanger (GHE). The fluid is ascending in one pipe and descending in the other.

#### 3.1. Analytical Approach

#### 3.1.1. Heat Transfer inside the Borehole

_{f}the circulating fluid temperature, T

_{b}the borehole wall temperature, and T

_{f}’ the temperature of the fluid entering the U-tube. Also, R

_{11}and R

_{22}are the thermal resistances between inlet and outlet legs of the U-tube and the borehole wall, respectively, and R

_{12}is the thermal resistance between the inlet and outlet legs of the U-tube (Figure 2). These thermal resistances can be calculated analytically using the Multipole Method [18,21]. It is seen in Equation (1) that the Quasi-3-D model is able to reflect the variation of the temperature of the circulating fluid (T

_{f}) along the tube (Z). Quasi-3-D models are preferred for design and analysis of GHEs, as they provide more accurate information on the heat flows inside the borehole.

1D (Equivalent diameter) [14] | 1D (Shape factor) [17] | 2D [18] | Quasi 3D [20] | |
---|---|---|---|---|

U-tube disposal | N | Y | Y | Y |

Quantitative expressions of the thermal resistance in the cross-section | N | N | Y | Y |

Thermal interference | N | N | N | Y |

Extinction between the entering and exiting pipes | N | N | N | Y |

Axial convection by fluid flow | N | N | N | Y |

Axial conduction in grout | N | N | N | N |

#### 3.1.2. Heat Transfer outside the Borehole

- -
- The ground is homogeneous in its thermal properties and initial temperature.
- -
- Moisture migration is negligible.
- -
- Thermal contact resistance is negligible between the pipe and grout and between the grout and soil.
- -
- The effect of ground surface is negligible for the initial 5–10 years (depending on the borehole depth).

_{0}is the initial temperature of the ground, and q’ is the heat flow rate per unit length of the borehole. Note that in Equation (4), the heat flow rate per unit length of the borehole is assumed constant along the borehole and steady throughout the operation time. Using this solution as a basic step pulse allows the calculation of temperature response to any load varying with time by considering piecewise constant heat extractions/rejections and superposing them in time as a series of load steps [36]. However, for long time periods, this process becomes computationally intensive. Some authors have focused on introducing efficient algorithms that lower the number of time steps [34,39], while others have used the convolution theorem in the frequency domain using fast Fourier transform to lower the computational burden of hourly temperature evaluations related to a time-varying heat load [28].

#### 3.2. Numerical Models

^{−}

^{2}m, while the size of the solution domain, which depends on the duration of system operation and its heating/cooling load, is approximately on an order of 10 m, making the domain extremely disproportionate. As a result, a large number of mesh elements is required for simulation of a single borehole and its surrounding soil. To achieve an inaccuracy of 2% or less for the steady state heat transfer analysis of boreholes, a minimum number of approximately 18 elements describing any circular shape of a horizontal cross section is needed [51]. In modelling the soil surrounding the borehole, a domain of a certain size can work well for one model, while it can be too small for another model requiring more boreholes, longer system performance durations or higher heating injection/removal rates. At the outer edge of the domain, a constant far-field temperature condition equal to the initial temperature is often applied. The sensitivity of the solution results to this boundary should always be examined and avoided by increasing the size of the domain. In three-dimensional modeling of a borehole system with typical flow velocities, a vertical element size of 2 m or less should often be applied to avoid inaccuracies of greater than 2% [51].

#### 3.3. Some Modeling Limitations

#### 3.4. Other Modeling Aspects

## 4. Conclusions

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Koohi-Fayegh, S.; Rosen, M.A.
A Review of the Modelling of Thermally Interacting Multiple Boreholes. *Sustainability* **2013**, *5*, 2519-2536.
https://doi.org/10.3390/su5062519

**AMA Style**

Koohi-Fayegh S, Rosen MA.
A Review of the Modelling of Thermally Interacting Multiple Boreholes. *Sustainability*. 2013; 5(6):2519-2536.
https://doi.org/10.3390/su5062519

**Chicago/Turabian Style**

Koohi-Fayegh, Seama, and Marc A. Rosen.
2013. "A Review of the Modelling of Thermally Interacting Multiple Boreholes" *Sustainability* 5, no. 6: 2519-2536.
https://doi.org/10.3390/su5062519