# Mobile Sources Mixing Model Implementation for a Better Quantification of Hydrochemical Origins in Allogenic Karst Outlets: Application on the Ouysse Karst System

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

_{3}

^{−}values (median: 436 mg.L

^{−1}); (ii) Water coming from sinking rivers fed by spring coming from igneous rocks with low mineralization but relatively higher K

^{+}values (median: 4.2 mg.L

^{−1}); (iii) Highly mineralized water coming from deep evaporitic layers and feeding another sinking river with very high sulfate concentrations (median: 400 mg.L

^{−1}). Sliding window cross-correlation analyses and hydrochemical analyses during a flood event are performed to implement a mobile source mixing model approach. This approach shows significant differences with a simple fixed source mixing model and appears more reliable but requires more time and money to carry out. The results and conclusion of this study will be used for forecasting and managing operational actions for resource management.

## 1. Introduction

## 2. Context of the Study Area

#### 2.1. Geology and Hydrogeology

^{−1}and 135 m h

^{−1}[25].

#### 2.2. Hydrological and Hydrogeological Context

^{−1}to 5000 L.s

^{−1}, the Alzou river flowrate varies between 20 L.s

^{−1}and 8000 L.s

^{−1}, and finally, the Ouysse flowrate varies from 20 L.s

^{−1}to 10,000 L.s

^{−1}[17]. Several tracer tests have identified the connections between the losses of these three rivers and the springs of Cabouy, Saint Sauveur, and Fontbelle [25].

## 3. Materials and Methods

#### 3.1. Continuous Data Monitoring and Discrete Sampling

^{−1}(EC), and ±0.1 °C (T).

_{3−}70% pure) was applied for cation measurement. Alkalinity (HCO

_{3}

^{−}) was directly measured on-site by titration.

^{2+}, Mg

^{2+}, Na

^{+}, K

^{+}, HCO

_{3}

^{−}, Cl

^{−}, SO

_{4}

^{2−}, NO

_{3}

^{−}) was carried out by ICP-MS.

#### 3.2. Methods

#### 3.2.1. Correlation Analyses on Hydrological Time Series

#### 3.2.2. Source Mixing Calculation

_{f}) of two end members (V

_{1}and V

_{2}) in a stationary mass transfer is the sum of all end members (V

_{f}= V

_{1}+ V

_{2}). If two non-covariant variables of concentration A and B are present in each end member, the mixing equations are as follows (Equation (1)):

_{1}, …, K

_{n}are the concentrations of the variables (A to K) for every end member (1 to n). A

_{f}, …, K

_{f}are the values for each variable in the mix. Solving this matrix equation leads to know α, …, η, the contributions of every end member to the chemical balance of f.

#### 3.3. The Mobile Sources Mixing Model Approach

## 4. Results and Discussion

#### 4.1. Sliding Windows Cross-Correlation Analyses on Long-Term Monitoring Results

#### 4.1.1. Crosscorrelation between Sinks and Springs Waterlevel

- In most cases (apart from Theminettes and Cabouy), the delay between water elevation at sinks and water elevation at springs during flood events varies inside a hydrological annual cycle. The lag is shorter in Spring, when the rainfall is maximum (in frequency and intensity) and when the maximum storage capacity of the reservoir is reached. The flood response is thus shorter and more intense during Spring or Summer storms, longer and buffered during the Winter recharge period. Pressure waves have thus the fastest transfer velocity during these high-flow periods, with a minimal loss of amplitude.
- Seasonal lag value variability is much more pronounced in the oriental allogenic sub-catchment of the Ouysse system (mainly between the Themines and Theminettes sinks and the Cabouy spring, and at a minor degree with the Saint-Sauveur spring), than in the occidental autogenic sub-catchment (mainly drained by the Saint-Sauveur and Fontbelle spring). In this latter sub-catchment, the intensity and the reaction time of water level during flood events are quite homogeneous all year long, and much more decorrelated with the water level variations at Themines and Theminettes. This autogenic subsystem appears much more transmissive, with less storage capacity. The rapidity and the intensity of the flood response seem moderately dependent on both the rainfall intensity and the storage state of the reservoir. This subsystem may be relatively simply organized, with a clear major drainage axis with very few minor branching. Conversely, the oriental allogenic subsystem appears less transmissive, with a higher storage capacity, suggesting a bigger development of the karst network, and/or a complex organization including secondary tributaries/branching of unknown development.
- The hydrological behavior regarding the Alzou sink appears singular: water level lags are shorter between the Alzou sinks and the system outlets, probably because the distances between this sink and the springs are shorter (~10 km) than between the other sinks (Themines, Theminettes) and the system outlets (~20 km).
- Lags and amplitude correlation are remarkably close between Themines and Theminettes sinks, regarding the system outlets. The only significant difference is for the maximum lag between Themines and Cabouy (113 h) which is one order of magnitude higher than the other values calculated from cross-correlation analysis (typically lower than 28 h).
- The hydrological behavior of the downstream Ouysse river appears to be mainly correlated to the hydrological behavior of Cabouy and Saint-Sauveur springs, which brings the major volumetric contributions to the main flow of the system outlet.
- From a methodological point of view, the results exposed above show that the presence of rapid and intense variations of water level in karst systems induces a bias in any hydrological interpretation if cross-correlation analysis is conducted only considering a global approach (correlation calculations conducted throughout the whole chronicle) rather than considering a sliding windows cross-correlation method.

#### 4.1.2. Crosscorrelation between Sinks and Springs Electric Conductivity

- For the Cabouy spring, the maximum correlation estimated from the global method and the median correlation estimated from the sliding window method are consistent. The lags of the water mass transfer estimated by the global method vary from ~80 h (from the Themines and Theminettes sinks) to 128 h (from the Alzou sink). In that case, the solute transfer from the Alzou sink to the system outlets seems to be slower than the one from the Limargue sinks. Considering the sliding window method, the lags from the Alzou sink appear to have very limited variations (121–133 h), compared to the ones from Themines and Theminettes that varies on a larger range (respectively 15–130 h and 131–272 h).
- For the Fontbelle spring, the maximum correlation estimated with the global method and the median correlation estimated with the sliding window method are generally inconsistent. The lags of the conductivity estimated by the global method vary from 185 h (from the Théminettes sink) to about 500 h (Alzou and Thémines). In that case, there is a large difference in lag between each sink and the Fontbelle spring, conversely to the estimation made for the Cabouy spring. That is quite surprising, since the Themines and Theminettes sinks are both located at the Limargue inlet of the oriental allogenic sub-catchment. Considering the sliding window method, the median lags estimated appear globally one order of magnitude lower (from ~67 h from the Alzou and Themines sinks, to 91 h from the Theminettes sink), but with a variability that can be very high (notably in the case of the Themines sink: 0–480 h).
- Comparable observations can be made for the Saint-Sauveur spring and the downstream Ouysse river: maximum lags estimated from both methods can be fairly consistent, but can vary inside a large range (e.g., 0–353 h from Themines to Saint-Sauveur).

- The water from the rivers comes from the Limargue area and is fed by Segala igneous bedrock springs, with low mineralization, but relatively high concentrations of Cl
^{−}, Na^{+}, and K^{+}. - The highly mineralized water from the Alzou subsystem is characterized by a high conductivity due to its evaporitic origin.
- Water directly infiltrated on the autogenic karst area, with chemistry controlled by the dissolution of the Middle and Upper Jurassic limestones.

#### 4.2. Mixing Model Results

#### 4.2.1. Geochemical Decomposition of the Flood Chemiograph during High-Frequency Flood Monitoring

#### 4.2.2. Source Water Bodies Signatures Identification

_{3}

^{−}for the autogenous karst; (ii) SO

_{4}

^{2−}as the marker for the Alzou pole; (iii) K

^{+}for the Limargue sinks.

_{3}

^{−}(karst), and K

^{+}(Limargue) in Equation (2). The results are shown in Figure 5B.

_{3}

^{−}, K

^{+}, SO

_{4}

^{2−}in Equation (2), marking the waters of the autogenic karst, the Limargue sinks, and the Alzou respectively. Figure 5A illustrates these results. Proportions are stable before the flood’s arrival, with the autogenous karst as the main contributor (~60%) and with complementary contributions from the Alzou (~10%) and Limargue (~30%) sinks. During the two flood peaks, the karst water contributes up to 90% while the contributions of the Limargue sinks drop to less than 10%. The contribution of the Alzou sink becomes negligible (~0%). However, during the first peak, we observe a slight increase in the contribution of the Limargue (~50%) which could be linked to the fact that we observe the first flood after the low water level. This increase is not observed at the second peak nor on the decomposition of the 2016 flood because the karst and epikarst are filled by the last flood and the piston effect counterbalances the arrival of the loading of the sinks. After the flood, the proportions of water from each origin tend to return to the initial values (karst ~75%, Limargue ~20%, Alzou ~5%). The contributions are quite similar to what was observed in the 2016 flood deconvolution (Lorette et al., 2020) with, however, smaller contributions from the Alzou (30–10%) in low water and larger contributions for the Limargue sinks (30% in low water). In both cases, the karst/sink proportions are very similar.

_{3}

^{−}, Limargue sinks with K

^{+}, and Alzou with SO

_{4}

^{2−}).

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Sliding correlation analyses between the water level for the Cabouy station and the different stations at the sinks; the results of the sliding cross-correlation analyses are displayed in red dots (vertical axis: correlation, label: time lag between the two sites in hours).

**Figure A2.**Sliding correlation analyses between the water level for the Ouysse station and the different stations at the sinks; the results of the sliding cross-correlation analyses are displayed in red dots (vertical axis: correlation, label: time lag between the two sites in hours).

**Figure A3.**Sliding correlation analyses between the water level for the Saint Sauveur station and the different stations at the sinks; the results of the sliding cross-correlation analyses are displayed in red dots (vertical axis: correlation, label: time lag between the two sites in hours).

## Appendix B

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**Figure 2.**Illustration of a sliding window CCA between rainfall and outlet flow, adapted from Delbart.

**Figure 3.**Illustration of how the mobile source mixing model works: the arrows represent an example of time lag of 1.5 day at which the water mass might transfer from all endmembers to the mix.

**Figure 4.**Sliding correlation analyses between the water levels for the Fontbelle station and the different stations at the sinks; the results of the sliding cross-correlation analyses are displayed in red dots (vertical axis: correlation, label: the time lag between the two sites in hours).

**Figure 5.**Evolution of the contributions results of the mobile source mixing model for Cabouy (

**A**), Courtilles (

**B**) Saint Sauveur (

**C**), and Fontbelle (

**D**) alongside water level. The dotted lines correspond to a positive and negative 5% error for each point.

**Figure 7.**Contributions results of the mobile source mixing model for Cabouy compared with fixed source model.

**Figure 8.**Contributions results of the mobile source mixing model for Saint Sauveur compared with fixed source model.

**Figure 9.**Contributions results of the mobile source mixing model for Fontbelle compared with fixed source model.

**Table 1.**Summary of statistics on the flow/water level, temperature, and conductivity at each monitored station.

Flow (L/s) | Temperature (°C) | Conductivity (µS/cm) | Waterlevel (cm) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Min | Median | Max | Min | Median | Max | Min | Median | Max | Min | Median | Max | |

Theminettes | <10 | 247 | 14,000 | 4.1 | 12.9 | 22.9 | 78 | 387 | 662 | / | / | / |

Themines | <10 | 196 | 29,200 | 1.7 | 12.3 | 23.2 | 130 | 330 | 945 | / | / | / |

Alzou | 20 | 160 | 10,800 | 4.5 | 13.5 | 23 | 262 | 1311 | 2145 | / | / | / |

Cabouy | / | / | / | 10.5 | 13.3 | 16.5 | 218 | 563 | 847 | 30 cm | 72 cm | 196 cm |

Saint Sauveur | / | / | / | 10.6 | 13.3 | 19 | 489 | 609 | 788 | 54 cm | 110 cm | 227 cm |

Fontbelle | 0 | 137 | 1375 | 12.8 | 13.4 | 14 | 501 | 599 | 857 | / | / | / |

Ouysse Cales | 454 | 2218 | 94,200 | 9.5 | 13.2 | 18.8 | 351 | 571 | 984 | / | / | / |

**Table 2.**Compared results of the global CCA and the sliding window CCA between sinks waterlevel and springs waterlevel.

CABOUY | Global CCA | Median and [Min–Max] of Sliding Windows CCA | ||
---|---|---|---|---|

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 8 | 0.4 | 8 | 0.87 |

[5 to 111] | [0.72 to 0.94] | |||

THEMINES | 12 | 0.47 | 11 | 0.87 |

[9 to 113] | [0.69 to 0.96] | |||

THEMINETTES | 11 | 0.5 | 9.5 | 0.89 |

[8 to 10] | [0.84 to 0.93] | |||

FONTBELLE | Global CCA | Median and [Min–Max] of Sliding Windows CCA | ||

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 6 | 0.43 | 6 | 0.74 |

[0 to 14] | [0.06 to 0.89] | |||

THEMINES | 10 | 0.48 | 8 | 0.75 |

[0 to 12] | [0.14 to 0.93] | |||

THEMINETTES | 9 | 0.54 | 8 | 0.69 |

[0 to 11] | [0.028 to 0.93] | |||

ST SAUVEUR | Global CCA | Median and [Min–Max] of Sliding Windows CCA | ||

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 6 | 0.89 | 5.5 | 0.87 |

[0 to 113] | [0.47 to 0.96] | |||

THEMINES | 8 | 0.87 | 7.5 | 0.89 |

[1 to 11] | [0.24 to 0.96] | |||

THEMINETTES | 7 | 0.89 | 7 | 0.91 |

[5 to 19] | [0.61 to 0.97] | |||

OUYSSE | Global CCA | Median and [Min–Max] of Sliding Windows CCA | ||

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 8 | 0.85 | 6.5 | 0.88 |

[1 to 116] | [0.1 to 0.98] | |||

THEMINES | 9 | 0.85 | 9 | 0.9 |

[0 to 28] | [0.74 to 0.98] | |||

THEMINETTES | 10 | 0.84 | 9 | 0.85 |

[2 to 28] | [0.76 to 0.96] |

**Table 3.**Compared results of the global ACC and the sliding window ACC between sinks’ electrical conductivity and springs’ electrical conductivity.

CABOUY | Global CCA | Median and [Min–Max] of Sliding Windows CCA | ||
---|---|---|---|---|

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 128 | 0.81 | 127 | 0.74 |

[121 to 133] | [0.62 to 0.86] | |||

THEMINES | 76 | 0.74 | 97 | 0.77 |

[15 to 130] | [0.58 to 0.84] | |||

THEMINETTES | 84 | 0.64 | 188 | 0.63 |

[131 to 272] | [0.53 to 0.8] | |||

FONTBELLE | Global CCA | Median Sliding Windows CCA | ||

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 489 | 0.85 | 66 | 0.65 |

[0 to 81] | [0.55 to 0.67] | |||

THEMINES | 500 | 0.12 | 67.5 | 0.46 |

[0 to 480] | [0.02 to 0.74] | |||

THEMINETTES | 185 | 0.3 | 91 | 0.3 |

[15 to 158] | [0.1 to 0.78] | |||

ST SAUVEUR | Global CCA | Median Sliding Windows CCA | ||

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 131 | 0.74 | 5 | 0.57 |

[0 to 255] | [0.28 to 0.76] | |||

THEMINES | 273 | 0.68 | 249 | 0.6 |

[0 to 353] | [0.36 to 0.81] | |||

THEMINETTES | 264 | 0.65 | 278.5 | 0.49 |

[164 to 416] | [0.13 to 0.75] | |||

OUYSSE | Global CCA | Median Sliding Windows CCA | ||

Lagmax (h) | correlation (Rxy) | Lagmax (h) | correlation (Rxy) | |

ALZOU | 392 | 0.36 | 193 | 0.15 |

[132 to 288] | [0.04 to 0.45] | |||

THEMINES | 248 | 0.21 | 280 | 0.18 |

[129 to 472] | [0.02 to 0.86] | |||

THEMINETTES | 259 | 0.05 | 130.5 | 0.23 |

[68 to 320] | [0.16 to 0.85] |

**Table 4.**Summary of statistics on the flow/water level, temperature, and conductivity at each monitored station.

sinks | borehole | springs | |||||
---|---|---|---|---|---|---|---|

Themines | Alzou | Courtilles | Cabouy | Saint Sauveur | Fontbelle | ||

Ca (mg/L) | min | 29.72 | 125.65 | 58.61 | 82.77 | 99.95 | 116.00 |

median | 42.97 | 217.00 | 86.38 | 124.37 | 126.00 | 131.58 | |

max | 82.20 | 446.00 | 114.10 | 153.99 | 151.00 | 172.78 | |

variance | 5.96 | 268.51 | 10.41 | 12.54 | 8.31 | 4.54 | |

HCO_{3} (mg/L) | min | 106.14 | 200.08 | 173.24 | 302.56 | 300.00 | 306.22 |

median | 152.50 | 275.72 | 259.86 | 335.50 | 340.38 | 337.33 | |

max | 263.52 | 329.40 | 322.08 | 417.24 | 531.79 | 451.40 | |

variance | 28.55 | 18.65 | 19.62 | 17.24 | 25.54 | 8.42 | |

SO_{4} (mg/L) | min | 13.10 | 127.00 | 11.70 | 8.56 | 2.53 | 4.32 |

median | 20.00 | 400.00 | 15.50 | 16.00 | 8.58 | 7.25 | |

max | 71.70 | 921.00 | 44.30 | 73.70 | 70.60 | 64.40 | |

variance | 3.92 | 853.66 | 1.34 | 7.84 | 6.50 | 3.53 | |

K (mg/L) | min | 3.27 | 2.96 | 1.35 | 0.38 | 1.08 | 0.50 |

median | 4.22 | 5.41 | 2.70 | 1.99 | 2.11 | 1.23 | |

max | 11.63 | 8.63 | 4.02 | 3.77 | 5.01 | 2.76 | |

variance | 0.10 | 0.07 | 0.02 | 0.01 | 0.01 | 0.01 | |

Cl (mg/L) | min | 8.87 | 7.85 | 7.95 | 7.95 | 2.43 | 5.68 |

median | 10.40 | 12.20 | 8.83 | 8.84 | 7.99 | 8.38 | |

max | 14.90 | 18.00 | 9.87 | 14.70 | 12.20 | 10.40 | |

variance | 0.03 | 0.23 | 0.01 | 0.09 | 0.14 | 0.10 | |

Na (mg/L) | min | 4.41 | 2.42 | 2.39 | 0.46 | 2.78 | 0.65 |

median | 6.17 | 6.16 | 4.63 | 3.57 | 3.86 | 3.55 | |

max | 11.22 | 9.01 | 6.82 | 7.83 | 7.54 | 6.17 | |

variance | 0.10 | 0.12 | 0.04 | 0.15 | 0.09 | 0.06 | |

Mg (mg/L) | min | 8.26 | 11.50 | 6.89 | 1.94 | 2.55 | 0.81 |

median | 12.27 | 24.55 | 11.03 | 4.70 | 3.95 | 3.22 | |

max | 20.20 | 59.60 | 14.97 | 9.09 | 10.60 | 8.21 | |

variance | 0.67 | 9.82 | 0.41 | 0.25 | 0.29 | 0.15 | |

NO_{3} (mg/L) | min | 8.01 | 0.04 | 10.90 | 5.39 | 5.14 | 422.84 |

median | 12.20 | 22.30 | 16.10 | 19.55 | 11.90 | 13.40 | |

max | 13.80 | 52.20 | 17.80 | 26.20 | 30.80 | 30.90 | |

variance | 0.05 | 4.44 | 0.06 | 0.49 | 1.03 | 1.45 |

NASH | Correlation | |||||
---|---|---|---|---|---|---|

Limargue | Karst | Alzou | Limargue | Karst | Alzou | |

cabouy | 0.04 | 0.17 | 0.89 | 0.19 | 0.41 | 0.94 |

saint sauveur | 0.24 | 0.70 | 0.76 | 0.49 | 0.84 | 0.87 |

fontbelle | 0.44 | 0.52 | 0.21 | 0.66 | 0.72 | 0.45 |

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## Share and Cite

**MDPI and ACS Style**

Viennet, D.; Lorette, G.; Labat, D.; Fournier, M.; Sebilo, M.; Araspin, O.; Crançon, P.
Mobile Sources Mixing Model Implementation for a Better Quantification of Hydrochemical Origins in Allogenic Karst Outlets: Application on the Ouysse Karst System. *Water* **2023**, *15*, 397.
https://doi.org/10.3390/w15030397

**AMA Style**

Viennet D, Lorette G, Labat D, Fournier M, Sebilo M, Araspin O, Crançon P.
Mobile Sources Mixing Model Implementation for a Better Quantification of Hydrochemical Origins in Allogenic Karst Outlets: Application on the Ouysse Karst System. *Water*. 2023; 15(3):397.
https://doi.org/10.3390/w15030397

**Chicago/Turabian Style**

Viennet, David, Guillaume Lorette, David Labat, Matthieu Fournier, Mathieu Sebilo, Olivier Araspin, and Pierre Crançon.
2023. "Mobile Sources Mixing Model Implementation for a Better Quantification of Hydrochemical Origins in Allogenic Karst Outlets: Application on the Ouysse Karst System" *Water* 15, no. 3: 397.
https://doi.org/10.3390/w15030397