# Time-Lapse 3D Electric Tomography for Short-time Monitoring of an Experimental Heat Storage System

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Direct Temperature Monitoring

#### 2.2. Numerical Simulations

^{®}[34]. Performed simulations follow basic state-of-the-art procedures that can be adopted in similar case studies and will offer spatial temperature maps to be compared with ERT results. For more details on the numerical simulations, including governing equations and constitutive laws, please refer to [34,35,36].

^{®}was adopted to solve the heat transport equation, where the temperature is subject to a set of Dirichlet, Neumann and Cauchy types’ boundary conditions. This tool makes use of the analytical solution proposed by [37] to solve the heat governing equations within the BHE. This approach, that assumes local steady state conditions and immediate thermal equilibrium between inlet and outlet pipes, was proved to output valuable results when the heat load is constant for times longer than 2–3 h [35].

^{−1}·K

^{−1}, [38]).

#### 2.3. Electrical Resistivity Tomography

_{i}is the estimated data variance, ρ

_{a,i,m}and ρ

_{a,i,c}are the measured and calculated apparent resistivities. The ill-posedness and non-linearity of the tomographic problem make it difficult to minimize the function in Equation (1). Indeed, apparent resistivities data are affected by random non-gaussian instrumental noise, difficult to model. The current approach is then based on a linearization procedure requiring an initial resistivity model, and on regularization according to the Tikhonov’s strategy [40]. With this regularization the condition of minimum roughness of the model is used as a stabilizing function [41]. Throughout the inversion iterations, the effect of non-gaussian noise is also appropriately managed using a robust data weighting algorithm [42,43]. With the adoption of the above described inversion strategies the minimizing functional is then transformed in:

_{i}are the elements of the weighting matrix, based on estimated data variances, ρ

_{i,c}and ρ

_{i,mo}are the calculated resistivities and the resistivities of an appropriate initial model m

_{o}(eventually null), L

_{i}are the elements of the regularizing matrix, which could account for specific constraints on the local data and, finally, α > 0 is the Lagrange multiplier, also called roughness coefficient or smoothness or damping.

_{o}has been adopted in order to correctly image time-lapse resistivity variations. The adopted initial model to be used as constrain, and as starting model for the inversions, was the one obtained after the inversion of the survey 1 (performed at 10 AM, see Table 1). The constrain to this initial model was obtained by means of the adoption of a Lagrange multiplier equal to 1 through all the iterations. The inversion is performed using an iterative Gauss-Newton scheme.

_{t}and T

_{0}are temperatures at respective and initial time steps and m is the fractional change in electrical resistivity (°C

^{−1}). A range of 0.018 ÷ 0.025 °C

^{−1}has been found by several authors for m [33,44,45,46] and it varies according to the type of fluid and sediments.

^{−1}was assumed after calibration tests at lab scale [27] and previous geophysical investigations at the field scale [32]. To obtain temperature differences images it is therefore possible to directly convert the resistivity difference (in percentage) of each voxel. Conversely, when the aim of the analysis is to obtain an absolute temperature value, a reference temperature (T

_{0}) should be assumed, as shown in Equation (3). The reference temperature T

_{0}could be adopted as homogenous within the whole investigated ground (undisturbed ground temperature of 14.2 °C) if the aim of the investigation is the long term TAZ of the system. Otherwise, in the short term, it could be assumed from the values registered by temperature sensors in the boreholes and in the monitoring hole at the beginning of the day of measurements if a direct comparison with local data is required.

## 3. Results

#### 3.1. Imaging TAZ

#### 3.2. Local Temperature Comparison

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Borehole thermal energy storage (BTES) system in Grugliasco with the disposition of the electrodes for the time-lapse 3D electrical resistivity tomography (ERT) acquisitions and plant connection; red pipes are inflow; blue pipes are outflow.

**Figure 2.**Meshing around the borehole heat exchangers (BHEs) in the FeFlow modelling environment with evidence (orange lines) of the connection between the BHEs related to the plant’s operation system; within BHEs red pipes are inflow, blue pipes are outflow.

**Figure 3.**Details on the area adopted to compare the monitoring approaches, with evidence of the BHEs location and plant connection; red pipes are inflow; blue pipes are outflow.

**Figure 4.**Temperature differences obtained from the interpretation of inverted resistivity data (Equation (3)) for four different time steps during the monitoring day. The location of BHEs is reported in the first panel.

**Figure 5.**Temperature differences obtained from the numerical simulations for four different time steps during the monitoring day. The location of BHEs is reported in the first panel.

**Figure 6.**Comparison of the two thermally affected zone (TAZ) (increase of 1 °C temperature from the surrounding) obtained from ERT data interpretation and numerical simulations.

**Figure 7.**Comparison of the temperatures obtained from the interpretation of inverted resistivity data by means of Equation (3) (dashed lines _rho) and temperatures from local direct measurements (continuous lines _T) near: The central double-U heat exchanger (DU) and two of the external single-U heat exchangers (A and B), for the whole monitoring day.

**Figure 8.**Comparison of the temperatures obtained from numerical simulations (bold lines inside the heat exchangers _BHE, dashed lines outside them _Soil) and temperatures from local direct measurements (continuous lines _T) near: The central double-U heat exchanger (DU) and two of the external single-U heat exchangers (A and B), for the whole monitoring day.

**Figure 10.**Comparison of the temperatures obtained from: Local direct measurements (continuous lines _T), numerical simulations (bold lines, inside the BHEs _BHE, dashed lines outside them _Soil) and inverted resistivity data (Equation (3), dashed lines _rho) near: (

**a**) The central double-U heat exchanger (DU) and two of the external single-U heat exchangers, A in (

**b**) and B in (

**c**), for the whole monitoring day.

Time | 3D ERT Surveys | Flow Rate [l h^{−1}] | Tin-Tout [°C] | Power [W] | Energy [kJ] |
---|---|---|---|---|---|

08:00 | 70 | 3.9 | 332 | 598 | |

08:30 | 70 | 8.4 | 676 | 1217 | |

09:00 | survey 0 | 200 | 7.5 | 2000 | 3600 |

09:30 | 190 | 9.0 | 2338 | 4208 | |

10:00 | survey 1 | 190 | 12.0 | 3127 | 5629 |

10:30 | survey 2 | 200 | 13.6 | 3473 | 6251 |

11:00 | survey 3 | 210 | 10.7 | 2618 | 4712 |

11:30 | survey 4 | 210 | 11.5 | 2801 | 5042 |

12:00 | survey 5 | 210 | 12.2 | 2984 | 5371 |

12.30 | survey 6 | 210 | 12.6 | 3097 | 5575 |

13:00 | survey 7 | 210 | 12.9 | 3173 | 5711 |

13:30 | survey 8 | 210 | 13.0 | 3195 | 5751 |

14:00 | survey 9 | 210 | 12.7 | 3131 | 5636 |

14:30 | survey 10 | 210 | 12.4 | 3060 | 5508 |

15:00 | survey 11 | 210 | 11.8 | 2911 | 5240 |

15:30 | survey 12 | 210 | 11.0 | 2718 | 4892 |

16:00 | survey 13 | 210 | 10.2 | 2508 | 4514 |

16:30 | survey 14 | 210 | 10.6 | 2801 | 5042 |

17:00 | survey 15 | 210 | 8.6 | 2320 | 4176 |

17:30 | 140 | 7.6 | 1263 | 2273 | |

18:00 | 80 | 6.9 | 639 | 1150 |

Parameter | Value |
---|---|

Porosity [-] | 0.30 |

Water content [%] | 50 |

Volumetric heat capacity of the solid phase [MJ·m^{−3}·K^{−1}] | 2.16 |

Thermal conductivity of the solid phase [W·m^{−1}·K^{−1}] | 4.50 |

Volumetric heat capacity of the fluid phase [MJ·m^{−3}·K^{−1}] | 2.10 |

Thermal conductivity of the fluid phase [W·m^{−1}·K^{−1}] | 0.30 |

Methodology | Temperature Increase [°C] |
---|---|

Direct Monitoring | 1.65 ± 0.03 |

Numerical Simulations | 1.50 ± 0.01 |

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**MDPI and ACS Style**

Comina, C.; Giordano, N.; Ghidone, G.; Fischanger, F. Time-Lapse 3D Electric Tomography for Short-time Monitoring of an Experimental Heat Storage System. *Geosciences* **2019**, *9*, 167.
https://doi.org/10.3390/geosciences9040167

**AMA Style**

Comina C, Giordano N, Ghidone G, Fischanger F. Time-Lapse 3D Electric Tomography for Short-time Monitoring of an Experimental Heat Storage System. *Geosciences*. 2019; 9(4):167.
https://doi.org/10.3390/geosciences9040167

**Chicago/Turabian Style**

Comina, Cesare, Nicolò Giordano, Giulia Ghidone, and Federico Fischanger. 2019. "Time-Lapse 3D Electric Tomography for Short-time Monitoring of an Experimental Heat Storage System" *Geosciences* 9, no. 4: 167.
https://doi.org/10.3390/geosciences9040167