# Quantification of Groundwater Vertical Flow from Temperature Profiles: Application to Agua Amarga Coastal Aquifer (SE Spain) Submitted to Artificial Recharge

^{*}

## Abstract

**:**

^{−9}and 7.5 × 10

^{−7}m/s has been found. These values decrease near the surface, where the flow is mainly horizontal.

## 1. Introduction

## 2. Study Area

## 3. Methodology

#### 3.1. Experimental Data Introduction and Processing

^{2}), whereas for a maximum irrigation value, only the area where a permanent sheet was observed is considered. Even though the irrigation pipe system surrounds the 8 ponds, not all of them were activated simultaneously, and seawater does not frequently cover the whole area.

#### 3.2. The Inverse Problem Protocol

## 4. Numerical Model

## 5. Inverse Problem Applied to Vertical Flow Analysis

## 6. Discussion

^{−7}to 7.3 × 10

^{−7}m/s by applying Bredehoeft and Papadopulos’ solution [10], and the second, using the seepage meters procedure, has an interval of 2.3 × 10

^{−9}–1.2 × 10

^{−6}m/s. Employing Bredehoeft and Papadopulos’ solution, we obtain an average of 3.5 × 10

^{−7}m/s. Other scenarios in which temperature data were used to estimate vertical velocities had similar results. Lapham [1] studied upward flow in a river–groundwater interaction scenario in two different locations in the US (Hardwick and New Braintree). The values ranged from 3.5 × 10

^{−8}m/s to 7 × 10

^{−7}m/s. According to Stallman [34], percolation rates of 2.3 × 10

^{−7}m/s can be detected with profiles resulting from diurnal temperature fluctuations. This rate can go down to 0.35 × 10

^{−7}m/s in low conductivity soils with a wide range of temperature variations and carefully taken measurements. Considering annual temperature fluctuations, detection can reach values of 0.11 × 10

^{−7}m/s.

_{bouyancy}:

_{Darcy}) of 8.6 × 10

^{−7}and 3.5 × 10

^{−8}m/s from the previous study of Alhama [32]. These values are the lowest attributed to the Pliocene–Quaternary materials in the upper aquifer, which limit the vertical flow (transversal to layers arranged horizontally). These are the standard values of μ (1 × 10

^{−3}kg m

^{−1}s

^{−1}), ρ

_{e,w}(10

^{3}kg m

^{−3}), g (9.81 m s

^{−2}), and β ([1,2] × 10

^{−4}°C

^{−1}).

## 7. Conclusions

^{−7}to 7 × 10

^{−7}m/s. The second (P-II) compared the TDPTS from a year in which the recharge was negligible (2011, scenario 5) with another in which the recharge was significant (2022, scenario 5). The results varied between 3.5 × 10

^{−8}and 7.5 × 10

^{−7}m/s. In all the scenarios, the velocity decreased near the surface (from 4.9 × 10

^{−7}m/s averaged at 12 m depth to 2.8 × 10

^{−7}m/s averaged at 3 m depth). It was negative (downward flow) at 3 m depth in scenario 6, in which intensive artificial recharge was taking place.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\mathrm{C}}_{\mathrm{c}}$ | capacitor |

${\mathrm{c}}_{\mathrm{e}}$ | volumetric heat capacity of the rock–fluid matrix (J m^{−3} k^{−1}) |

${\mathrm{c}}_{\mathrm{e},\mathrm{w}}$ | volumetric heat capacity of water (J m^{−3} k^{−1}) |

$\mathrm{EC}$ | electrical conductivity (mS/cm) |

$\mathrm{g}$ | gravity (m s^{−2}) |

${\mathrm{G}}_{\mathrm{cd}}$ | current generator relative to the heating circuit, placed in the lower half of the cell, parallel to the lower resistance |

${\mathrm{G}}_{\mathrm{cu}}$ | current generator relative to the heating circuit, placed in the upper half of the cell, parallel to the upper resistance |

$\mathrm{H}$ | vertical depth of the aquifer (m) |

J_{C} | current intensity flowing through a capacitor (A) |

$\mathrm{k}$ | permeability (m^{2}) |

${\mathrm{k}}_{\mathrm{m}}$ | heat conductivity of the rock–fluid matrix (J/(s m °C)) |

${\mathrm{K}}_{\mathrm{Darcy}}$ | hydraulic conductivity (m s^{−1}) |

$\mathrm{MCT}$ | Mancomunidad de los Canales del Taibilla |

${\mathrm{R}}_{\mathrm{cd}}$ | resistor placed in the lower half of the cell, relative to heat flow |

${\mathrm{R}}_{\mathrm{cu}}$ | resistor placed in the upper half of the cell, relative to heat flow |

$\mathrm{T}$ | temperature (°C) |

${\mathrm{T}}_{\mathrm{z}}$ | temperature at depth z (°C) |

${\mathrm{T}}_{\mathrm{z}\text{}\mathrm{simulated}}$ | temperature simulated at depth z (°C) |

${\mathrm{T}}_{\mathrm{bottom}}$ | temperature at the bottom of the domain (°C) |

$\mathrm{TDPTS}$ | temperature–depth profile time series |

TI | thermal inertia |

${\mathrm{T}}_{\mathrm{ini}}$ | initial temperature (°C) |

${\mathrm{T}}_{\mathrm{surface}}$ | mean surface soil temperature (°C) |

$\mathrm{t}$ | time (s) |

$\mathrm{V}$ | voltage (V) |

${\mathrm{v}}_{\mathrm{advection}}$ | advective component of velocity (m s^{−1}) |

${\mathrm{v}}_{\mathrm{bouyancy}}$ | velocity caused by buoyancy effects (m s^{−1}) |

${\mathrm{v}}_{\mathrm{z}}$ | vertical water flow velocity (m s^{−1}) |

${\mathrm{v}}_{\mathrm{z},\mathrm{upward}\text{}\mathrm{flow}}$ | vertical upward water flow velocity component (m s^{−1}) |

${\mathrm{v}}_{\mathrm{z},\mathrm{recharge}\text{}\mathrm{flwo}}$ | vertical downward water flow velocity component (m s^{−1}) |

${\mathrm{v}}_{\mathrm{z},\mathrm{o},\mathrm{i}}$ | inverse problem protocol velocity “i” |

${\mathrm{v}}_{\mathrm{z},\mathrm{o},1}$ | inverse problem protocol initial velocity |

$\mathrm{z}$ | vertical spatial coordinate (m) |

$\mathsf{\alpha}$ | thermal diffusivity (m^{2} s^{−1}) |

$\mathsf{\beta}$ | thermal expansion coefficient of water (°C^{−1}) |

$\Delta $ | gradient operator |

$\Delta $T | amplitude due to sinusoidal boundary condition at the surface (°C) |

$\Delta {\mathrm{T}}_{\mathrm{interval}}$ | temperature gradient for each depth interval (°C) |

${\left(\Delta \mathrm{T}\right)}_{\mathrm{z}=0}$ | amplitude of the temperature variation at $\mathrm{z}=0$ (°C) |

$\Delta {\mathrm{v}}_{\mathrm{z}}$ | groundwater velocity increase (ms^{−1}) |

$\mathsf{\mu}$ | water viscosity (kg m^{−1} s^{−1}) |

${\mathsf{\rho}}_{\mathrm{e}}$ | wet bulk density of the rock–fluid matrix (kg m^{−3}) |

${\mathsf{\rho}}_{\mathrm{e},\mathrm{w}}$ | water density (kg m^{−3}) |

$\mathsf{\tau}$ | period of the sinusoidal thermal wave (s) |

$\mathsf{\phi}$ | mathematical function |

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**Figure 1.**The Agua Amarga salt marsh and the seawater irrigation area. Geographic coordinates are in the corner (Base image from Google Earth [California, USA], Landsat Copernicus [USA], and Maxar Technologies [Colorado, USA]).

**Figure 5.**Total volume of seawater used for irrigation in July (

**left**) and August (

**right**) for the period 2010–2022.

**Figure 6.**TDPTS averaged for years with and without recharge (2010–2015 and 2016–2022, respectively) in the months of July (

**left**) and August (

**right**).

**Figure 7.**TDPTS for piezometer P-8 in 2011 (no artificial recharge,

**left**) and 2020 (intensive artificial recharge,

**right**).

**Figure 9.**Influence of velocity on function variations (Ψ) for each scenario at the selected depths.

Scenario | Time Interval | ${\mathbf{k}}_{\mathbf{m}}$ (cal °C ^{−1} s^{−1} m^{−1}) | ${\mathsf{\rho}}_{\mathbf{e}}{\mathbf{c}}_{\mathbf{e}}$ (cal °C ^{−1} m^{−3}) | ${\mathbf{T}}_{\mathbf{surface}}$ (°C) | ${\mathbf{T}}_{\mathbf{bottom}}$ (°C) |
---|---|---|---|---|---|

1 | July 2010–2015 | 0.45 | 0.75 × 10^{−6} | Average air temperature 2010–2015: 26.5 | Average temperature at 16 m depth in the piezometer 2010–2015: 20.5 |

2 | July 2016–2022 | 0.45 | 0.75 × 10^{−6} | Average seawater temperature 2016–2022: 26.8 | Average temperature at 16 m depth in the piezometer 2016–2022: 20.5 |

3 | August 2010–2015 | 0.45 | 0.75 × 10^{−6} | Average air temperature 2010–2015: 27.1 | Average temperature at 16 m depth in the piezometer 2010–2015: 20.6 |

4 | August 2016–2022 | 0.45 | 0.75 × 10^{−6} | Average seawater temperature 2016–2022: 27.3 | Average temperature at 16 m depth in the piezometer 2016–2022: 20.5 |

Scenario | Time Interval | ${\mathbf{k}}_{\mathbf{m}}$ (cal °C ^{−1} s^{−1} m^{−1}) | ${\mathsf{\rho}}_{\mathbf{e}}{\mathbf{c}}_{\mathbf{e}}$ (cal °C ^{−1} m^{−3}) | ${\mathbf{T}}_{\mathbf{surface}}$ (°C) | ${\mathbf{T}}_{\mathbf{bottom}}$ (°C) |
---|---|---|---|---|---|

5 | 2011 | 0.45 | 0.75 × 10^{−6} | Sinusoidal function based on monthly air temperature 2011 | Constant value calculated from mean temperature at 16 m depth 2011: 20.6 |

6 | 2020 | 0.45 | 0.75 × 10^{−6} | Sinusoidal function based on monthly air temperature 2020 | Constant value calculated from mean temperature at 16 m depth 2020: 20.3 |

**Table 3.**Selected velocities for every depth according to the protocol in P-I. The symbol [*] refers to the results of limited reliability.

Scenario | Velocity at 3 m Depth (m/s) | Velocity at 6 m Depth (m/s) | Velocity at 9 m Depth (m/s) | Velocity at 12 m Depth (m/s) |
---|---|---|---|---|

1 | 7 × 10^{−7} | 7 × 10^{−7} | 7 × 10^{−7} | 7 × 10^{−7} |

2 | 3 × 10^{−8} * | 4 × 10^{−7} | 4 × 10^{−7} | 4 × 10^{−7} |

3 | 2 × 10^{−7} | 4 × 10^{−7} | 4 × 10^{−7} | 4 × 10^{−7} |

4 | 2 × 10^{−9} * | 1 × 10^{−7} | 2 × 10^{−7} | 2 × 10^{−7} |

**Table 4.**Selected velocities for every depth according to the protocol in P-II for scenarios 5 (2011) and 6 (2020). The symbol [*] refers to results of limited reliability.

Month (Year) | Velocity at 3 m Depth (m/s) | Velocity at 6 m Depth (m/s) | Velocity at 9 m Depth (m/s) | Velocity at 12 m Depth (m/s) |
---|---|---|---|---|

January (2011) | 4.0 × 10^{−7} | 6.0 × 10^{−7} | 6.0 × 10^{−7} | 6.0 × 10^{−7} |

February (2011) | 1.5 × 10^{−7} | 7.5 × 10^{−7} | 7.5 × 10^{−7} | 7.5 × 10^{−7} |

March (2011) | 1.0 × 10^{−7} | 2.5 × 10^{−7} | 5.0 × 10^{−7} | 5.0 × 10^{−7} |

August (2011) | 2.5 × 10^{−7} | 7.5 × 10^{−7} | 7.5 × 10^{−7} | 7.5 × 10^{−7} |

September (2011) | 0.0 * | 7.0 × 10^{−7} | 7.0 × 10^{−7} | 7.0 × 10^{−7} |

October (2011) | 0.0 * | 7.0 × 10^{−7} | 7.0 × 10^{−7} | 7.0 × 10^{−7} |

November (2011) | 1.0 × 10^{−8} * | 2.0 × 10^{−7} | 5.0 × 10^{−7} | 5.0 × 10^{−7} |

January (2020) | 4.0 × 10^{−7} | 5.0 × 10^{−7} | 5.0 × 10^{−7} | 5.0 × 10^{−7} |

February (2020) | 1.5 × 10^{−7} | 7.0 × 10^{−7} | 7.0 × 10^{−7} | 7.0 × 10^{−7} |

March (2020) | 1.5 × 10^{−7} | 1.5 × 10^{−7} | 1.5 × 10^{−7} | 3.5 × 10^{−7} |

June (2020) | 5.0 × 10^{−8} * | 6.0 × 10^{−7} | 6.0 × 10^{−7} | 6.0 × 10^{−7} |

July (2020) | −3.0 × 10^{−8} * | 7.0 × 10^{−7} | 7.0 × 10^{−7} | 7.0 × 10^{−7} |

August (2020) | −2.0 × 10^{−7} | 8.0 × 10^{−8} | 2.0 × 10^{−7} | 2.0 × 10^{−7} |

September (2020) | 4.0 × 10^{−9} * | 8.0 × 10^{−8} | 1.5 × 10^{−7} | 4.0 × 10^{−7} |

October (2020) | 4.5 × 10^{−8} * | 8.0 × 10^{−8} | 1.0 × 10^{−7} | 4.0 × 10^{−7} |

November (2020) | 3.5 × 10^{−8} * | 3.5 × 10^{−8} | 6.0 × 10^{−8} | 1.0 × 10^{−7} |

V_{z,1} (m/s) | V_{z,2} (m/s) | Transient Time (days) |
---|---|---|

0.0 × 10^{−7} | 7.00 × 10^{−7} | 111 |

1.00 × 10^{−7} | 7.00 × 10^{−7} | 104 |

2.00 × 10^{−7} | 7.00 × 10^{−7} | 82 |

4.00 × 10^{−7} | 7.00 × 10^{−7} | 65 |

Depth Interval (m) | ΔT_{interval} (°C) | v_{advection} P1 (m/s) | v_{advection} P2(m/s) | v_{bouyancy}(m/s) |
---|---|---|---|---|

0–4.5 | 5.75 | [2 × 10^{−9}–7 × 10^{−7}] | [0–4 × 10^{−7}] | 9.79 × 10^{−9} |

4.5–7.5 | 2.25 | [1 × 10^{−7}–7 × 10^{−7}] | [3.5 × 10^{−8}–7.5 × 10^{−7}] | 3.83 × 10^{−9} |

7.5–10.5 | 1.05 | [2 × 10^{−7}–7 × 10^{−7}] | [6 × 10^{−8}–7.5 × 10^{−7}] | 1.79 × 10^{−9} |

10.5–16 | 0.35 | [2 × 10^{−7}–7 × 10^{−7}] | [1 × 10^{−7}–7.5 × 10^{−7}] | 5.96 × 10^{−10} |

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**MDPI and ACS Style**

Jiménez-Valera, J.A.; Alhama, I.; Trigueros, E.
Quantification of Groundwater Vertical Flow from Temperature Profiles: Application to Agua Amarga Coastal Aquifer (SE Spain) Submitted to Artificial Recharge. *Water* **2023**, *15*, 1093.
https://doi.org/10.3390/w15061093

**AMA Style**

Jiménez-Valera JA, Alhama I, Trigueros E.
Quantification of Groundwater Vertical Flow from Temperature Profiles: Application to Agua Amarga Coastal Aquifer (SE Spain) Submitted to Artificial Recharge. *Water*. 2023; 15(6):1093.
https://doi.org/10.3390/w15061093

**Chicago/Turabian Style**

Jiménez-Valera, José Antonio, Iván Alhama, and Emilio Trigueros.
2023. "Quantification of Groundwater Vertical Flow from Temperature Profiles: Application to Agua Amarga Coastal Aquifer (SE Spain) Submitted to Artificial Recharge" *Water* 15, no. 6: 1093.
https://doi.org/10.3390/w15061093