# Taking into Account both Explicit Conduits and the Unsaturated Zone in Karst Reservoir Hybrid Models: Impact on the Outlet Hydrograph

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−5}m·s

^{−1}and tends to be isotropic due to alteration processes [26,27]. The transmission zone constitutes the relatively unaltered part of the unsaturated zone, where water mainly flows vertically towards the saturated zone. In the transmission and saturated zones, at the scale of the flow unit, the matrix porosity and hydraulic conductivity are usually less than 2% and 10

^{−4}m·s

^{−1}respectively [24]. Flow processes in the unsaturated zone (soil, epikarst and transmission zone) can vary greatly in time and space [28,29,30,31]. Variable connectivity inside the flow path network controls the infiltration processes [19,29,32,33,34]. Flows in the unsaturated zone can be either direct through conduits or delayed because they slowly circulate in the matrix [32]. The karst unsaturated zone may therefore act as a main storage reservoir [35,36], whose complex functioning largely affects the shape of hydrographs [26,36,37,38,39,40,41,42,43]. However, the unsaturated zone is rarely represented explicitly in models of karst hydrodynamics [17,44,45,46,47,48,49]. Most modelling studies only consider the saturated zone of the aquifer [18,20,50,51,52,53,54,55].

## 2. Materials and Methods

#### 2.1. Description of the Hybrid Model and the Considered Karst Specificities

^{2}and a uniform thickness (250 m for the reference model). The outlet elevation is at 120 m. The Figure 1 presents several views of the model.

^{−3}km

^{2}to 10

^{−1}km

^{2}. The mesh cells support the porous fractured matrix, while a selection of mesh edges supports the discrete features that represent large conductive karst conduits. Both are homogeneous.

#### 2.2. Flow Equations and Model Parameters

^{−1}) in the function of the pressure head (ψ) and U is the sink-source term (s

^{−1}).

_{r}and θ

_{s}are residual and saturated water contents (-), respectively, and α (cm

^{−1}), n and m are empirical parameters. The moisture content equals porosity multiplied by saturation. The relative hydraulic conductivity K

_{r}(-) in the unsaturated zone follows this relation:

_{e}is effective saturation, generally defined as [72]

_{EK}, Φ

_{EK}, K

_{EK}) and transmission zone (TZ), respectively, and the flow capacity of the conduits (KS). Petrophysical values for the saturated zone (SZ) are assumed to be equal to those of the transmission zone (Φ

_{TZ−SZ}, K

_{TZ−SZ}). Due to the variable flooding of the epiphreatic zone, the thicknesses of the transmission zone and the saturated zone vary while their sum remains constant. The boundary condition at the outlet constrains the initial thickness of the saturated zone. Thus, the initial thickness of the transmission zone (Thk

_{TZ}) is the only geometrical parameter of interest for the lower subsystems. According to the literature, hydraulic conductivity is isotropic only in the epikarst. In the other subsystems, the ratio between the horizontal hydraulic conductivity (K

_{TZ−SZ}) and the vertical hydraulic conductivity is equal to 10, as usually assumed.

^{3}·s

^{−1}. For comparison, this value is 1000 times greater than that of the conduit scale model of Dal Soglio et al. [25], while the area of the model is 100 times greater. Nevertheless, the preliminary results led us to retain only values larger than this reference among all the tested values. Indeed, the simulated groundwater level locally exceeds the ground level if considering smaller values for conduit flow capacity together with other reference parameters. This highlights the importance of this parameter and the difficulty of calibrating it.

#### 2.3. Simulations and Evaluation Criteria

- Peak flow (maximum discharge value); in some cases, several local extrema are identified;
- Time after the event until peak flow;
- Discharge duration;
- Third order moment (skewness), which describes the shape of distribution;
- Fourth order moment (kurtosis), which is a flattening coefficient.

_{99}, the time necessary to drain 99% of recharge event water to the spring. Several tests have concluded that water drained after t

_{99}does not affect the results.

## 3. Results and Discussion

#### 3.1. Overview of Simulation Results and Hydrograph Typology

^{−1}at the 68th day. The values of the quantities obtained in the reference model should not be considered in absolute terms but only by comparison with the simulations carried out for other parameter sets. Effectively, these values depend both on the structure of the hypothetical aquifer constructed for the simulation and on the values retained to quantify all the parameters of the model. The long duration of draining observed in the simulation can be also related to the flow processes in the unsaturated matrix under variably saturated conditions. Indeed, matrix flows last longer in variably saturated conditions. As drainage occurs, saturation and hydraulic conductivity decrease, slowing the flow accordingly. Moreover, conduits draining the surrounding medium may dry it locally and create less conductive zones around them. This thereby limits the area of influence of conduits in the unsaturated zone. In the hypothetical aquifer matrix, the distance to the nearest conduit is highly variable, with some areas being very distant from the karst network, notably in each corner of the model (Figure 1c), which reinforces such behaviours. Recharge that is not drained towards the karst network flows vertically through the transmission zone, which acts as a buffer zone spreading the temporal distribution of the recharge event. For instance, in a model representing only the vicinity of a vertical conduit, with the same vertical organization of the medium and comparable properties, the two-day recharge event at the top of the model spans several dozens of days at the bottom of the transmission zone [25]. This result highlights the importance of the karst network structure and the distribution of distances from matrix to the nearest conduits in the model response.

#### 3.1.1. The Role of Epikarst Parameters

^{−2}m·s

^{−1}and thickness equal to 10 m. Finally, in this configuration, the epikarst parameters that have the greatest individual effect on the hydrograph are porosity and thickness for values ranging between [0.01; 0.10] and [0; 10 m], respectively. However, varying several parameters at a time should produce combined effects that could eventually be more important.

#### 3.1.2. The Role of Transmission and Saturated Zones Parameters

^{−6}m·s

^{−1}) of the transmission and saturated zones.

## 4. Evaluation of Models

#### 4.1. Model Assumptions

#### 4.2. Scaling Issues

#### 4.3. Evaluation of Models Outputs

#### 4.3.1. The Need of Hydrographs Descriptors

#### 4.3.2. Matching Model Outputs with Field Measurements

^{2}and a uniform thickness of 250 m: the peak flow value varies between 597 and 1063 L·s

^{−1}, the peak flow time varies between 4 and 204 days and the discharge duration varies between 912 and 3464 days. We use skewness and kurtosis descriptors for the shape of the hydrographs. Figure 3 shows kurtosis as a function of skewness for all the simulations and for hydrographs from the well-known karst systems described in Section 4.3.2. The values from simulations are consistent with the values from field sites. They cover the same ranges, and the reference simulation is almost centred. The long discharge durations could possibly be questioned, but these can probably be related to the huge uncertainty related to the upscaling issue. This is likely accentuated here by the model structure, with a poorly karstified area far from the represented karst network. These results nevertheless highlight the important delaying effect of the unsaturated zone.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation and corresponding model of karst system: (

**a**) illustrations of flows towards a spring and hydrograph; (

**b**) mesh of the model with locations of the vertical conduits (red points) and the outlet (blue arrow); (

**c**) horizontal slice of the mesh with projection of the conduits and location of cross-section (

**d**); (

**d**) vertical cross-section of the mesh with location of the terminal conduit; (

**e**) karst conduits network of the model.

**Figure 4.**Hydrograph typology established on the basis of the simulation results and the literature.

**Figure 5.**Hydrograph characteristics for various conduit properties. Plots are coloured according to the hydrograph typology: orange for type 2 and green for type 3.

**Figure 6.**Hydrograph results for various parameter sets. (

**a**) Hydrographs for varied porosities in the epikarst ϕ

_{EK}; (

**b**) hydrographs for varied thicknesses of the transmission zone Thk

_{TZ}; (

**c**) hydrographs for varied hydraulic conductivities of the transmission and saturated zones K

_{TZ-SZ}. Plots are coloured according to the hydrograph typology: red for type 1, orange for type 2, green for type 3 and purple for type 5.

**Figure 7.**Hydrograph characteristics for various values of several properties in epikarst. Plots are coloured according to the hydrograph typology: orange for type 2 and green for type 3.

**Figure 8.**Hydrograph characteristics for various values of several properties in the transmission zone and the saturated zone. Plots are coloured according to the hydrograph typology: red for type 1, orange for type 2, green for type 3 and purple for type 5.

**Table 1.**Value ranges for the properties of karst systems and karst modelling reported in the literature and model parameters [25].

Subsystem | Property (Units) | Values and Ranges of Values ^{1} from Literature | Model’s Values and Range of Values Min–Ref–Max |
---|---|---|---|

Epikarst (EK) | Thickness Thk _{EK} (m) | (0; >30) [24] (few meters; 10 ^{−15}) [23](3; 10) [1] (8; 12) [73] | 0–20–35 |

Porosity ϕ _{EK} (-) | (0.05; 0.1) [24,74] (0.1; 0.3) [27] >0.2 [1] | 0.01–0.1–0.25 | |

Horizontal ^{2} hydraulic conductivityK _{EK} (m·s^{−1}) | (10^{−7}; 10^{−4}) [41]10 ^{−5} [39](5 × 10 ^{−5}; 10^{−3}) [75](2 × 10 ^{−4}; 2 × 10^{−3}) [76]10 ^{−3} [77]>1000 * K _{TZ-SZ} [78] | 10^{−5}–10^{−2}–10^{−1} | |

Transmission and saturated zones (TZ–SZ) | Thickness Thk _{TZ} (m) | depending on the field site, usually tens of meters, <20; <50 [77] up to 700 [32] | 30–80–130 |

Porosity ϕ _{TZ-SZ} (-) | (0.004; 0.01) [1] 0.005 [79] (0.01; 0.02) [80] (0.024; 0.3) [81] | 0.005–0.01–0.025 | |

Horizontal ^{2} hydraulic conductivityK _{TZ-SZ} (m·s^{−1}) | (10^{−10}; 7 × 10^{−5}) [81](10 ^{−7} [46]; 10^{−6} [1,47]) [75](5 × 10 ^{−7}; 5 × 10^{−6} [39]) [9](10 ^{−6} [1,47]; 10^{−4} [79]) [80](10 ^{−5}; 10^{3}) [17] | 10^{−7}–10^{−5}–10^{−3} | |

Conduit (C) | Diameter D (m) | (0.08; 15) [64] (2; 10) [60] | Flow Capacity A _{C} * K_{C}(m ^{3}·s^{−1})10 ^{−2}–10^{−1}–10^{1} |

Section A _{C} (m^{2}) | (<1; >100) [82] | ||

Hydraulic conductivity K _{C} (m·s^{−1}) | (6 × 10^{−5}; 4 × 10^{−1}) [81](10 ^{−1}; 10) [17,75](3; 10) [80] 10 [9,47] | ||

Van Genuchten Model | Coefficient α (m ^{−1}) | (3.28 × 10^{−3}; 6.23 × 10^{−1}) [44]3.65 × 10 ^{−2} [47,49]10 ^{−2} [17,46] | 3.65 × 10^{−2} |

Empirical parameter n (-) | (0.01; 3) [44] 1.83 [47,49] 2 [17,46] | 1.83 | |

Residual water content θ_{r} (-)or Residual water saturation S _{r} (-) | θ_{r} = S_{r} = 0 [46]θ _{r} ∈(0.01; 0.05) [44]S _{r} = 0.05 [47]θ _{r} = 0.171 [17] | S_{r} = 0.05 |

^{1}Ranges of values from the literature are shown in parentheses.

^{2}When anisotropy is considered, values concordant to the strata are presented.

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## Share and Cite

**MDPI and ACS Style**

Dal Soglio, L.; Danquigny, C.; Mazzilli, N.; Emblanch, C.; Massonnat, G.
Taking into Account both Explicit Conduits and the Unsaturated Zone in Karst Reservoir Hybrid Models: Impact on the Outlet Hydrograph. *Water* **2020**, *12*, 3221.
https://doi.org/10.3390/w12113221

**AMA Style**

Dal Soglio L, Danquigny C, Mazzilli N, Emblanch C, Massonnat G.
Taking into Account both Explicit Conduits and the Unsaturated Zone in Karst Reservoir Hybrid Models: Impact on the Outlet Hydrograph. *Water*. 2020; 12(11):3221.
https://doi.org/10.3390/w12113221

**Chicago/Turabian Style**

Dal Soglio, Lucie, Charles Danquigny, Naomi Mazzilli, Christophe Emblanch, and Gérard Massonnat.
2020. "Taking into Account both Explicit Conduits and the Unsaturated Zone in Karst Reservoir Hybrid Models: Impact on the Outlet Hydrograph" *Water* 12, no. 11: 3221.
https://doi.org/10.3390/w12113221