# Determining the Relation between Groundwater Flow Velocities and Measured Temperature Differences Using Active Heating-Distributed Temperature Sensing

^{1}

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^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Laboratory Setup Flow Simulator

^{−3}m

^{3}(when filled) and kept at a constant water level using a pump and spillway. By changing the height of the inflow water tank and the height of the outflow hose on the downstream side of the tank, the hydraulic head difference between in- and outflow of the tank was controlled and so was the groundwater velocity within the simulator. Groundwater flow velocities (Darcy flux) between 0.1 and 8 m d

^{−1}were generated in this way. The background temperature inside the flow simulator was not constant during the experiments, due to shutting down the flow overnight (in the 2017 experiments) and changing inflow temperatures, despite the fact that the temperature of the hall, in which the simulator was placed, was kept constant at 20 °C. During the 2018 experiments, flow was continued overnight for a more stable background temperature.

^{−1}) of the sediment were experimentally determined after the flow and heat transport experiment.

#### 2.2. Measuring Temperature Using AH-DTS

^{®}Curve Fitting Toolbox) to determine the equilibrium temperature as being the asymptote of the heating curve using:

^{2}of 0.80 over all heating curve fits. To correct for the drift in the background temperature the base temperature after each heating run was determined by fitting an exponential function on the cooling curve using:

^{2}of all cooling curve fits was 0.79. The resulting base temperature (based on the asymptote $e$) was subtracted from the asymptote of the equilibrium temperature resulting in a more consistent $\u2206T$. We chose this method because shortening the heating periods can be a great advantage for field applications.

#### 2.3. Numerical Modeling

^{−1}), $h$ is the hydraulic head (m) and $x$ and $y$ are the distance (m). The heat transfer is represented by the heat transport equation [18]:

^{−3}) and the bulk specific heat capacity (J kg

^{−1}°C

^{−1}) of the saturated sediment (b) and that of water (w), $T$ is the temperature (°C), $t$ is time (seconds) and ${k}_{eff}$ is the effective thermal conductivity of water and solids, (W m

^{−1}°C

^{−1}). ${q}_{x}$ and ${q}_{y}$ are the components of the specific discharge vector:

^{−1}) is the input of heat to the heating cable and A (m

^{2}) is the cross-sectional area of the heating wire. The parameters used for the heat input and the sediment properties are given in Table 1.

^{−1}, 1.89 m d

^{−1}).

#### 2.4. Analytical Solution for Direct Groundwater Flow Velocity Estimation Using $\Delta T$

^{−1}),${r}_{h1}$ (m) is the radius of the heating part of the heating cable (m), ${r}_{h2}$ (m) is the total radius of the heating cable and ${r}_{tot}$ (m) is the sum of the radii of the heating cable, the distance through the sediment and the radius of the fiber-optic cable used to measure temperature change combined. ${k}_{tot}$ (W m

^{−1}K

^{−1}) is the combined thermal conductivity calculated (harmonic mean) as materials in series [26]:

^{−3}), $u$ is the flow velocity (in m s

^{−1}), $\mu $ is the dynamic viscosity of the water (kg m

^{−1}s

^{−1}), and $v$ is the kinematic viscosity of the water (in m

^{2}s

^{−1}), ${C}_{f}$ is the specific heat capacity of the fluid (in J kg

^{−1}°C

^{−1}). For L we used the diameter of the heating cable.

## 3. Results

#### 3.1. Experimental Results

^{−1}results in a 0.38 °C lower $\u2206T$ for all data points. However, as suggested by Equation (13), the $\u2206T$ versus groundwater flow velocity relationship is non-linear, which was also concluded in a study using higher flow rates [8].

#### 3.2. Analyzing Results with a Numerical Modeling Code (FlexPDE)

^{−1}K

^{−1}in steps of 0.5 W m

^{−1}K

^{−1}. Table 3 shows the mean absolute errors for the calibration location A4 and the validation location A3.

^{−1}K

^{−1}was used to determine the best fit solution by minimizing the error. The mean absolute errors for the variations of the second calibration round are provided in Table 4. A best fit between observations and model data was obtained for ${K}_{s}$ = 5.0 W m

^{−1}K

^{−1}, within ranges (2–8.4 W m

^{−1}K

^{−1}) found in literature [19,28,29].

#### 3.3. Applying Analytical Solution (Equation (13)) for Flow Velocity Estimation

^{2}= 0.85. $\u2206T$ ranged from 5 to 8.4 °C with flow velocities up to 8.8 m d

^{−1}. The calculated velocities are in line with the measured velocities.

## 4. Discussion

^{−1}were relatively unstable. The hydraulic head difference between the in-and outflow of the flow simulator for these flow velocities needed to be very low (<2.5 cm) which is difficult to control with the relative coarse sand used in the experiment. Small fluctuations in the in-and outflow system can have a significant effect.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Equation (13) Derived Based on Equation (6)

- $Q$ as heat input to the cable (W m
^{−1}), - ${r}_{h1}$ the radius of the heating part of the heating cable (m),
- ${r}_{h2}$ the total radius of the heating cable (m),
- ${r}_{tot}$ is the sum of the radii of the heating cable, the distance through the sediment and the radius of the fiber-optic cable used to measure temperature change combined (m),
- ∆T as the temperature change due to heating (°C), and
- ${k}_{tot}$ is the combined thermal conductivity calculated (harmonic mean) as materials in series [26]:

- ○
- ${k}_{hc}$ and ${f}_{hc}$ are the thermal conductivity and fraction of thickness in respect to the total radius ${r}_{tot}$ of the heating cable core,
- ○
- ${k}_{h}$ and ${f}_{h}$ are the thermal conductivity and fraction of thickness in respect to the total radius ${r}_{tot}$ of the heating cable protection,
- ○
- ${k}_{j}$ and ${f}_{j}$ are the thermal conductivity and fraction of thickness in respect to the total radius ${r}_{tot}$ of the DTS cable jacket,
- ○
- ${k}_{a}$ and ${f}_{a}$ are the thermal conductivity and fraction of thickness in respect to the total radius ${r}_{tot}$ of the DTS cable aramid protection,
- ○
- ${k}_{m}$ and ${f}_{m}$ are the thermal conductivity and fraction of thickness in respect to the total radius ${r}_{tot}$ of the DTS fiber mantle,
- ○
- ${k}_{f}$ and ${f}_{f}$ are the thermal conductivity and fraction of thickness in respect to the total radius ${r}_{tot}$ of the DTS fiber,
- ○
- ${k}_{eff}$ and ${f}_{sed}$ are the thermal conductivity and fraction of thickness in respect to the total radius ${r}_{tot}$ of the saturated sediment.

- Parameter $h$ is the heat transfer coefficient $\left({\mathrm{W}\mathrm{m}}^{-2}{\mathrm{K}}^{-1}\right)$ defined by [6]:

- ○
- L is the characteristic length of the flow (m),
- ○
- Nu the Nusselt number (-),
- ○
- ${k}_{eff}$ the combined thermal conductivity of sediment and water as given by [18]:

- ▪
- $n$ is the porosity
- ▪
- ${k}_{f}$ and ${k}_{s}$ the thermal conductivity of the water and the sediment solid material.

- The Nusselt number is given by

- ○
- $Re$ is the Reynolds number (-)

- ▪
- $\rho $ the density of the water (in kg m
^{−3}) - ▪
- $u$ the groundwater flow velocity (in m s
^{−1}) - ▪
- $\mu $ the dynamic viscosity of the water (kg m
^{−1}s^{−1}) - ▪
- $v$ the kinematic viscosity of the water (in m
^{2}s^{−1})

- ○
- $Pr$ is de Prandtl number (-)

- ▪
- ${C}_{f}$ the specific heat capacity of the fluid (in J kg
^{−1}°C^{−1}) - ▪
- ${k}_{f}$ is the thermal conductivity of the fluid (in W m
^{−1}°C^{−1})

- ○
- $C$ and $m$ are constants which are determined empirically and standard values for $C$ and $m$ are available depending on the values for $Re$ and $Pr$ (Hilpert, 1933; Bergman et al., 2011).

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**Figure 1.**(

**a**) Schematic side view of the flow simulator, in 2D vertical view. The frame contains the Distributed Temperature Sensing (DTS) cables and is placed inside the tank [15]; (

**b**) Sideview picture of the flow simulator showing the dimensions; (

**c**) schematic top view of the cable locations as placed in the flow simulator. The cable locations selected for the FlexPDE modeling are A3 and A4 (see Section 2.3).

**Figure 2.**(

**a**) Temperatures for the cable section placed in the cooler reference temperature bath based on internal calibration of the Silixa Ultima DTS (blue), Calibrated and smoothed temperature based on the separated calibration bath (green) and PT100 measurement data of the temperature in the calibration bath (red); (

**b**) Temperatures for the cable section placed in the warmer reference temperature bath.

**Figure 3.**Cross-section showing the setup of the heating cable and the fiber-optic cables. (

**a**) Cable setup for the 2017 experiments; (

**b**) Improved cable setup for the 2018 experiments.

**Figure 4.**Box-whisker plots of the measured data from the flow simulator showing the groundwater flow velocity (x-axes) versus $\u2206T$ (Y-axes) for the data measured in the 2018 experiments. The measurement data for all measurement values of all cables in the experiments for a specific experimental run are combined into one box. The orange line of the box shows the median of the data, and the bottom and top edges of the box respectively the 25th and 75th percentiles. The whiskers stretch to the most extreme datapoints not being outliers that are marked as open circle markers.

**Figure 5.**Temperature of the groundwater as modeled in FlexPDE. The direct surroundings of the Active Heating-Distributed Temperature Sensing (AH-DTS) cable at location A3 is shown, containing the duplex fiber-optic cable and the two heating cables. The flow is from left to right. Plotted area 20 mm × 20 mm of the 2 m × 1 m modeled area; (

**a**) temperature in and around the cable at time 2200 s; (

**b**) at time 2790 s; (

**c**) at time 9390 s; (

**d**) at time 9400 s.

**Figure 6.**Comparison of model and measurement data for the calibration location A4 in the flow simulator for three experimental runs. The blue line shows the temperature as measured in the experiments, the red line the modeled temperature for the best fit solution. The dotted green lines present the variations of ${K}_{s}$ that were calculated.

**Figure 7.**Comparison of model and measurement data for the validation location A3 in the flow simulator for three experimental runs. The blue line shows the temperature as measured in the experiments, the red line the modeled temperature for the best fit solution. The dotted green lines present the variations of ${K}_{s}$ that were calculated.

**Figure 8.**Measured $\u2206T$ groundwater flow velocity and the modeled groundwater flow velocity (Equation (13)) using measured $\u2206T$ as input shown for the 2018 experiments for the optimized $C$ and $m$ constants.

Parameter | Parameter | Reference |
---|---|---|

Heat input Q (W m^{−1}) | 33.33 | Based on measured voltage input and cable resistance |

Horizontal hydraulic conductivity (m d^{−1}) | 40 | Determined in laboratorial experiments |

Effective porosity (-) | 0.41 | Determined in laboratorial experiments |

Medium | Thermal Conductivity (W m^{−1} K^{−1}) | Specific Heat (J kg^{−1} K^{−1}) | Density (g cm^{−3}) | Reference |
---|---|---|---|---|

Sand | 5.1 ^{1} | 830 | 2.65 | [19,20,21] |

Water | 0.591 | 4186 | 0.99 | [19] |

Heating cable core | 390 | 385 | 8.96 | |

Heating cable protection material | 0.2 | 100 | 1.2 | [22] |

DTS protective jacket | 0.196 | 1565 | 1.121 | [23] |

DTS aramid protection | 0.04 | 1420 | 1.44 | [24] |

DTS fiber mantle | 0.196 | 1565 | 1.121 | [23] |

DTS fiber | 2 | 1430 | 2.2 | [25] |

^{1}The thermal conductivity of sand was used as the calibration parameter of the model.

**Table 3.**Mean absolute error between model estimation and measurement results of the first calibration round.

Thermal Conductivity (W m^{−1} K^{−1}) | Location A4 (°C) | Location A3 (°C) |
---|---|---|

4.0 | 0.670 | 0.596 |

4.5 | 0.372 | 0.323 |

5.0 | 0.149 | 0.172 |

5.5 | 0.322 | 0.382 |

6 | 0.523 | 0.581 |

6.5 | 0.700 | 0.758 |

7 | 0.857 | 0.914 |

**Table 4.**Mean absolute error between model estimation and measurement results for the second calibration round.

Thermal Conductivity (W m^{−1} K^{−1}) | Location A4 (°C) | Location A3 (°C) |
---|---|---|

4.85 | 0.192 | 0.164 |

4.9 | 0.172 | 0.159 |

4.95 | 0.156 | 0.162 |

5.0 | 0.149 | 0.172 |

5.05 | 0.152 | 0.188 |

5.1 | 0.163 | 0.207 |

5.15 | 0.178 | 0.227 |

5.2 | 0.195 | 0.249 |

5.25 | 0.215 | 0.271 |

5.3 | 0.236 | 0.294 |

5.35 | 0.257 | 0.316 |

5.4 | 0.279 | 0.338 |

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## Share and Cite

**MDPI and ACS Style**

Bakx, W.; Doornenbal, P.J.; van Weesep, R.J.; Bense, V.F.; Oude Essink, G.H.P.; Bierkens, M.F.P.
Determining the Relation between Groundwater Flow Velocities and Measured Temperature Differences Using Active Heating-Distributed Temperature Sensing. *Water* **2019**, *11*, 1619.
https://doi.org/10.3390/w11081619

**AMA Style**

Bakx W, Doornenbal PJ, van Weesep RJ, Bense VF, Oude Essink GHP, Bierkens MFP.
Determining the Relation between Groundwater Flow Velocities and Measured Temperature Differences Using Active Heating-Distributed Temperature Sensing. *Water*. 2019; 11(8):1619.
https://doi.org/10.3390/w11081619

**Chicago/Turabian Style**

Bakx, Wiecher, Pieter J. Doornenbal, Rebecca J. van Weesep, Victor F. Bense, Gualbert H. P. Oude Essink, and Marc F. P. Bierkens.
2019. "Determining the Relation between Groundwater Flow Velocities and Measured Temperature Differences Using Active Heating-Distributed Temperature Sensing" *Water* 11, no. 8: 1619.
https://doi.org/10.3390/w11081619