# Dynamics of the Flow Exchanges between Matrix and Conduits in Karstified Watersheds at Multiple Temporal Scales

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}reacts with carbonate rock and (3) hydraulic gradient is sufficiently high to ensure the water renewal and to maintain the karstification potential [1]. This results in complex structures with a heterogeneous permeability field and non-linear hydraulic behavior. Drainage in karstic systems consists of an underground self-organized structure leading to preferential water pathways through a conduit network embedded in either a porous matrix which sometimes is highly permeable (i.e., Floridan Karst aquifer) or a low permeable fissured matrix [2,3]. This drainage organization might be divided between (1) a ‘main drainage system’ (MDS), associated to rapid transit time and (2) an ‘annexe drainage system’ (ADS), associated with a long period of water residence [4,5]. As a result, groundwater water flow in the karstic system could be divided into several compartments with contrasted hydrodynamic behavior.

^{2}.

## 2. Material and Methods

#### 2.1. Study Sites

^{2}and 13.2 km

^{2}[4]. They belong to the carbonated belt bordering the north of the French Pyrenees. The two systems share a common boundary oriented west-east. South of this boundary, the Baget area is affected by the Alas fault: Jurassic to Cretaceous metamorphic limestone dips 45° to 75° under slaty flysch on the southern border. The valley follows the contact between karstified calcareous formations and impervious flysch oriented in a west-east direction. Jurassic and Palaeozoic metamorphic dolomite and flysch constitute the northern border of the watershed. This corresponds to the Sainte-Suzanne marls, playing the role of a tight barrier. North of this barrier, the Aliou watershed is composed of Aptian limestone. The drainage is done in a north-south direction until the outlet located in the Aliou cave.

#### 2.2. Modeling Approach

#### 2.2.1. Conceptual Reservoir Modeling

_{AB}is the recession coefficient associated with the flow from reservoir A (either E, M, or C) to reservoir B (either M, C, or L) or to the outlet S and Q

_{AB}(L/T) is the discharge from A to B. Discharge in (L

^{3}/T) is computed by the product of Q

_{AB}with the total surface of the recharge area (R

_{A}).

_{EM}= a

_{MC}= 1, a

_{EC}= 2 and a

_{CS}= 4. Then we assume turbulent flow with non-linear law in conduit and capillary flow with linear law in the matrix. Both Aliou and Baget karstic watersheds are well karstified and conduit dominated with a short memory effect [31]. The emptying exponent between reservoir M and C is chosen to be equal to the emptying coefficient for E to M as the exchange between the conduit and the surrounding matrix is mainly determined by the hydraulic conductivity of the fissured system [37]. To give better physical realism to the model, it has been chosen to set all output discharge rate through reservoir C (Figure 2) Also, previous studies have shown that the rainfall-runoff relationship for both Aliou and Baget watersheds may be correctly described by making ET equal to zero [14,38,39].

#### 2.2.2. Model Calibration

_{sim}is the discretized time series simulated by the model, and q

_{obs}is the discretized experimental time series measured in the field. The NSE criterion is fluently used to judge the quality of a rainfall-runoff model [28,34,42,43]. Nonetheless, this criterion tends to favor the conformity of the model for the strong discharge values to the detriment of the weakest ones. Introducing a balance error estimation allows us to better describe the quality of the model in the case where overestimation of discharge during low water could introduce important mass errors between observed and simulated data. Also, the effects of bias introduced by some conditions such as large events can be reduced by data transformation [44] or by including the bias as an explicit component, as for the Kling-Gupta efficiency KGE [45]. Here the optimization function W

_{obj}is defined such as:

#### 2.2.3. Scale Effects

## 3. Results and Discussion

#### 3.1. Short-term Varibility of Internal Fluxes

_{MC}shows negative values, which means that the reservoir C feeds the reservoir M. Then, at the end of the rainfall event, the water level in all reservoir drops with different dynamics and after about two days without rainfall, the fluxes between reservoir M and C change direction. Then, the reservoir M feeds the reservoir C during recessional, until the next rainfall event (i.e., new water input in the system). These event scale dynamics are consistent with results obtained with a physically-based approach [58,59]. Nonetheless, these approaches need a correct knowledge of flow geometry to give a reliable discretization of the flow path in so-called “reach” were flow and transport parameters may be considered as homogeneous. The matrix-conduit dynamic had also been highlighted using conceptual modeling [60]. Duran [33] highlighted the dynamic fluxes between the conduit and a surrounding matrix over the Norville chalk aquifer using a KarstMod model coupled with turbidity and electrical conductivity analysis. In the study by Duran [33], the fluxes Q

_{MC}represent about 10% of the spring discharge. Over the Baget and Aliou watersheds, the contribution of the fluxes Q

_{MC}in the spring discharge is in the same order of magnitude. Zhang et al., [61] proposed a conceptual model coupled with dissolution rates estimations in “slow flow” and “fast flow” systems. They estimated the contribution of exchange from “slow flow” to “fast flow” systems in three catchments to vary between 64.1% and 87.5%.

#### 3.2. Long-Term Variability of Internal Dynamics

## 4. Conclusions

^{2}) in order to assess the dynamics of internal flow and water levels in the different compartments of the watershed (i.e., Epikarst, Matrix, and Conduit). The modeling has been conducted for two different time steps. Based on this modeling approach, hydrodynamics relationships between reservoirs M and C (representative of the matrix with slow flow and the conduit with fast flow and non-linear behavior, respectively) have been more preferentially explored on the short-term scale (i.e., a rainfall event) and on the long-term scale (i.e., inter-annual scale).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Localization of the Aliou (

**a**) and Baget (

**b**) karstic watersheds in the south of France with physiographic maps, modified from [30]. The circle indicates the position of the hydrometric station located at the outlet of the basin.

**Figure 2.**

**Top**: synthetic time series of rainfall, spring discharge (Q

_{spring}) and matrix-conduit flow dynamics (Q

_{MC});

**Bottom**: Structure of the reservoir model: epikarst (E), matrix (M) and conduit (C) are inter-connected from the top to the bottom of the structure. Q

_{MC}corresponds to the flow from reservoir M to reservoir C and depends on the difference between water levels in M and C. The model is considered on several time steps: (

**a**) the water level is higher in reservoir C, so Q

_{MC}< 0; (

**b**) the water level in reservoirs C and M are equal, so Q

_{MC}= 0 and (

**c**) the water level is higher in M so Q

_{MC}> 0.

**Figure 3.**Example of event base time series for KarstMod model on Aliou watershed on the hourly sampling rate: (

**a**) Rainfall time series with observed and simulated discharge; (

**b**) water level in KarstMod reservoirs E, M and C and (

**c**) fluxes between reservoir M and C. When Q

_{MC}> 0, the flow goes from reservoir M to C, otherwise the flow goes from reservoir C to M.

**Figure 4.**MASH of the daily rainfall (w = 30 days, Y = 15 years). The dates correspond to the starting year, noted Ys, for shifting horizon average, so the corresponding shifting horizon period is from the starting year Ys to Ys + 15 years. Due to the 15 years moving average window, the abscissa finished in 2003.

**Figure 6.**Contribution of the exchange between reservoir M and reservoir C to the total discharge on the annual scale (

**left**) and the monthly scale (

**right**).

[24 h] | [1 h] | |||||
---|---|---|---|---|---|---|

Period | Warm-Up | Calibration | Validation | Warm-Up | Calibration | Validation |

from | 1 January 1968 | 1 October1970 | 1 October 1980 | 1 October 2009 | 1 October 2010 | 1 October 2012 |

to | 30 September 1970 | 30 September 1980 | 31 December 2016 | 30 September 2010 | 30 September 2012 | 31 May 2017 |

Parameter | Calibration [24 h] | Calibration [1 h] | ||
---|---|---|---|---|

Aliou | R_{A} | Recharge area | 14.8 km^{2} | 14.2 km^{2} |

k_{EM} | Recession coefficient from E to M | 6.10 × 10^{−3} mm/day | 1.11 × 10^{−4} mm/h | |

k_{EC} | Recession coefficient from E to C | 9.83 × 10^{−3} mm/day | 2.69 × 10^{−3} mm/h | |

k_{CS} | Recession coefficient from C to S | 2.36 × 10^{−3} mm/day | 1.80 × 10^{−4} mm/h | |

k_{MC} | Recession coefficient from M to C | 3.17 × 10^{−1} mm/day | 2.41 × 10^{−2} mm/h | |

Baget | R_{A} | Recharge area | 15.8 km^{2} | 13.1 km^{2} |

k_{EM} | Recession coefficient from E to M | 3.36 × 10^{−3} mm/day | 2.09 × 10^{−4} mm/h | |

k_{EC} | Recession coefficient from E to C | 2.71 × 10^{−3} mm/day | 4.35 × 10^{−4} mm/h | |

k_{CS} | Recession coefficient from C to S | 1.21 × 10^{−3} mm/day | 7.21 × 10^{−4} mm/h | |

k_{MC} | Recession coefficient from M to C | 7.03 × 10^{−2} mm/day | 1.25 × 10^{−1} mm/h |

**Table 3.**Model performances for calibration and validation periods for daily and hourly KarstMod models (NSE: Nash Sutcliff Efficiency, BE: Balance Error and KGE: Kling Gupta Efficiency).

[24 h] | [1 h] | ||||
---|---|---|---|---|---|

Periods | Calibration | Validation | Calibration | Validation | |

(1970–1980) | (1980–2016) | (2010–2012) | (2012–2017) | ||

Aliou | NSE | 0.53 | 0.51 | 0.63 | 0.5 |

BE | 0.88 | 0.97 | 0.99 | 0.91 | |

KGE | 0.58 | 0.51 | 0.67 | 0.54 | |

Baget | NSE | 0.59 | 0.52 | 0.56 | 0.47 |

BE | 0.94 | 0.89 | 0.98 | 0.77 | |

KGE | 0.61 | 0.52 | 0.64 | 0.44 |

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**MDPI and ACS Style**

Sivelle, V.; Labat, D.; Mazzilli, N.; Massei, N.; Jourde, H.
Dynamics of the Flow Exchanges between Matrix and Conduits in Karstified Watersheds at Multiple Temporal Scales. *Water* **2019**, *11*, 569.
https://doi.org/10.3390/w11030569

**AMA Style**

Sivelle V, Labat D, Mazzilli N, Massei N, Jourde H.
Dynamics of the Flow Exchanges between Matrix and Conduits in Karstified Watersheds at Multiple Temporal Scales. *Water*. 2019; 11(3):569.
https://doi.org/10.3390/w11030569

**Chicago/Turabian Style**

Sivelle, Vianney, David Labat, Naomi Mazzilli, Nicolas Massei, and Hervé Jourde.
2019. "Dynamics of the Flow Exchanges between Matrix and Conduits in Karstified Watersheds at Multiple Temporal Scales" *Water* 11, no. 3: 569.
https://doi.org/10.3390/w11030569