# Modeling the Matrix-Conduit Exchanges in Both the Epikarst and the Transmission Zone of Karst Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overview of the Usual Representation of Hydrodynamics in Karst Systems and Related Properties

#### 2.1.1. Karst Subsystems

^{−5}m/s and effective porosity ranges from 1% to 10% [46,47]. Due to its relatively high porosity and hydraulic conductivity compared to those of the underlying rock, the epikarst can store a large amount of water in locally perched aquifers that generally drain horizontally towards preferential flow paths [11,48,49]. Three factors affect water storage in the epikarst: (1) its thickness and lateral continuity, (2) its average porosity, and (3) the relative rates of inflow and outflow of water [45].

^{−7}m/s and 10

^{−4}m/s, and porosity is usually less than 2% [45]. The transmission zone may also play a storage role [50,51], releasing water during the low water period [52].

#### 2.1.2. Matrix vs. Conduit Properties

^{−12}to 10

^{−6}m/s (assuming that permeability of 1 Darcy is equivalent to a hydraulic conductivity of 10

^{−5}m/s) at a small scale [67,68], while the conduits hydraulic conductivity is greater than 10

^{−1}m/s [69].

**Figure 1.**Schematic representation of water flux and exchanges through karst as a function of the hydrological period and corresponding numerical model. (

**a**) High flow conceptual model: the conduits drain the upper continuum and supply the lower continuum; (

**b**) Low flow conceptual model: the conduits drain the surrounding continuum; (

**c**) subsystems of the numerical model; (

**d**) associated mesh with boundary conditions.

#### 2.2. Modeling Approach

#### 2.2.1. Model Description, Boundary Conditions, and Evaluation Criteria

^{2}square so that the lateral boundaries are far enough from the conduit to have little effect on near-conduit exchanges. Hence, the presented results focus on a smaller area, up to 250 m from the conduit (e.g., Figure 2). The model is 140 m thick (from z = −40 m to z = 100 m) including the epikarst, transmission zone, and the upper part of the saturated zone. The model grid has 1405 vertex per slice and 104 slices. From the top of the model to a depth of 100 m, the layers are 1 m thick. The three deeper layers that correspond to the permanently saturated interval are thicker. They constitute a buffer zone to avoid a significant impact of the boundary conditions. The horizontal mesh is refined all around the vertical conduit.

#### 2.2.2. Flow Equations and Model Parameters

^{−1}). Solving the Richards’ equation requires constitutive relationships for saturation as well as the relative hydraulic conductivity. Several models based on statistical pore-size distribution have been developed [82,83,84,85,86,87]. Their application and calibration mostly concern unconsolidated porous media [88]. The literature contains a few applications of the Van Genuchten model [85] to a carbonate matrix [15,18,19,21]. However, the application of this empirical model to a fractured or karstified carbonate matrix is not well documented. In fact, estimating the relationship between saturation and relative hydraulic conductivity for karst media remains a challenge [89]. Here the Van Genuchten model is therefore applied with constant and uniform parameters from the literature for all simulations (Table 1). For the unsaturated zone, water content is equal to:

_{r}and θ

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

^{−1}), n, and m are empirical parameters. Note that setting either residual water content θ

_{r}or residual saturation S

_{r}is equivalent because the two parameters are linked: moisture content θ equals porosity ϕ multiplied by saturation S. However, some confusion may exist in the literature with residual saturation values directly set as equal to residual water content values [19] (Table 1). The relative hydraulic conductivity K

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

_{e}is effective saturation, generally defined as [85]:

_{EK}, K

_{EK}) are distinguished from those for the transmission and saturated zone that are equal (Φ

_{TZ−SZ}, K

_{TZ−SZ}). Assuming that the discrete feature supports most of the vertical conductivity induced by the karst and considering the contrast of anisotropy between the epikarst and transmission zone, hydraulic conductivity is set isotropic in the epikarst whereas it is set anisotropic everywhere else, with a horizontal conductivity ten times greater than the vertical, as frequently considered [54]. Note that the thickness of the epikarst (Thk

_{EK}) does not vary during the simulation whereas the thicknesses of the transmission zone and saturated zone may vary because of flooding of the epiphreatic zone. We thus consider the initial thickness of the transmission zone as the parameter of interest (Thk

_{TZ}). These parameters are modified around a reference simulation. The parameter values are in the range defined in Table 1. Only one parameter is changed at a time.

^{−2}and 10

^{1}m

^{3}/s

^{−1}(Table 1 and Figure 4).

## 3. Results

#### 3.1. Spatio-Temporal Evolution of the Flows at the Conduit Scale

**Figure 2.**Profiles of the saturation (blue) and hydraulic head (green) around the conduit, with plots zooming on the water level at three time steps: (

**a**) at steady state before the recharge event, (

**b**) at the end of the recharge event, (

**c**) 77 days after the event.

^{−1}(Figure 3f). Here, water from the conduit wet the unsaturated medium surrounding the conduit. However, the total flux from the conduit to the transmission zone remains below 3% of the flux entering the transmission zone by the conduit. As observed on the profiles, the influence on flows in the transmission zone remains limited to the area close to the conduit.

**Figure 3.**Vertical flux through horizontal planes at different elevations and mass balances, in the matrix and conduit respectively, as functions of time. (

**a**) Recharge at the top of the model (

**b**) Flux through a horizontal matrix section at different elevations in both the epikarst and transmission zone. (

**c**) Flux through a horizontal matrix section at different elevations in the transmission zone only with an adapted scale. (

**d**) Flux through a horizontal conduit section at different elevations in both the epikarst and transmission zone. (

**e**) Flux balance of matrix in the transmission zone. (

**f**) Flux balance of a conduit in the transmission zone.

#### 3.2. Impact of Parameter Variation at the Conduit Scale

^{−2}m

^{3}/s

^{−1}) limits the preferential flow through the conduit (Figure 4). Conversely, increasing the flow capacity of the conduits above a threshold value (here, 1 m

^{3}/s

^{−1}) does not have a greater impact on the response of the model, as referred to by Kovács et al. [74] for saturated conditions. We have therefore selected an intermediate value (10

^{−1}m3/s

^{−1}) for the reference model, to simulate a highly conductive conduit that ensures preferential flow but whose effect differs from that of a fixed-head boundary condition.

**Figure 4.**Vertical flux through the horizontal plane at the base of the transmission zone as a function of time, in the matrix (

**a**) and conduit (

**b**) respectively, for four values of flow capacity in the conduit.

_{EK}(from 10

^{−5}to 10

^{−1}m/s) while K

_{TZ−SZ}(10

^{−5}m/s) and other parameters remain constant. Conversely, Figure 6 shows three profiles of the saturation and hydraulic head for different values of K

_{TZ−SZ}(from 10

^{−7}to 10

^{−3}m/s) while K

_{EK}and other parameters remain constant (10

^{−2}m/s). The contrast of hydraulic conductivity between the epikarst and transmission zone mainly controls the flow direction in the epikarst during the recharge event. Other things being equal, a ratio of hydraulic conductivity higher than ten (one hundred for the ratio of vertical conductivity due to the difference of anisotropy) ensures horizontal preferential flow through the epikarst toward the conduit while saturation monotonically increases as a function of depth (Figure 5a versus Figure 8b,c). A ratio greater than about 10

^{4}is necessary to distinguish clearly specific behaviors in the epikarst as compared to the transmission zone: local saturation is higher at the bottom of the epikarst, sharp variation of flow direction between both zones, and the conduit is supplying the matrix in the transmission zone (Figure 5d and Figure 6a).

**Figure 7.**Profiles of the saturation (blue) and hydraulic head (green) around the conduit for various values of porosity of the epikarst. Each profile is taken at the time step for which the flux at the bottom output of the model is maximum. (

**a**) The porosity is the same throughout the entire model and equal to 0.01. (

**b**) The epikarst porosity is equal to 0.2.

**Figure 8.**Profiles of saturation (blue) and hydraulic head (green) around the conduit for various thicknesses of the epikarst. Each profile is taken at the time step for which the flux at the bottom output of the model is maximum. (

**a**) Model without an epikarst. (

**b**) Model with a 1 m thick epikarst. (

**c**) Model with a 10 m thick epikarst.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Profiles of the saturation (blue) and hydraulic head (green) around the conduit for various values of hydraulic conductivity of the epikarst. Each profile is taken at the time step for which the flux at the bottom output of the model is maximum. (

**a**) The hydraulic conductivity is the same throughout the entire model and equal to 10

^{−5}m/s. (

**b**) The epikarst hydraulic conductivity is equal to 10

^{−4}m/s. (

**c**) The epikarst hydraulic conductivity is equal to 10

^{−3}m/s. (

**d**) The epikarst hydraulic conductivity is equal to 10

^{−1}m/s.

**Figure 6.**Profiles of the saturation (blue) and hydraulic head (green) around the conduit for various values of hydraulic conductivity of the transmission and saturated zones. Each profile is taken at the time step for which the flux at the bottom output of the model is maximum. (

**a**) K

_{TZ-SZ}= 10

^{−7}m/s. (

**b**) K

_{TZ-SZ}= 10

^{−5}m/s. (

**c**) K

_{TZ-SZ}= 10

^{−3}m/s.

**Table 1.**Value ranges for the properties of karst systems and karst modeling reported in the literature and model parameters.

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

Epikarst (EK) | Thickness Thk _{EK} (m) | (0; >30) [45] (few meters; 10 ^{−15}) [38](3; 10) [37] (8; 12) [44] | 0–20–35 |

Porosity ϕ _{EK} (-) | (0.05; 0.1) [45,55] (0.1; 0.3) [47] >0.2 [37] | 0.01–0.1–0.25 | |

Horizontal ^{2} hydraulic conductivityK _{EK} (m/s) | (10^{−7}; 10^{−4}) [13]10 ^{−5} [56](5 × 10 ^{−5}; 10^{−3}) [57](2 × 10 ^{−4}; 2 × 10^{−3}) [58]10 ^{−3} [59] >1000*K _{TZ-SZ} [60] | 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 [59] up to 700 [61] | 30–80–130 |

Porosity ϕ _{TZ-SZ} (-) | (0.004; 0.01) [37] 0.005 [62] (0.01; 0.02) [63] (0.024; 0.3) [64] | 0.005–0.01–0.025 | |

Horizontal ^{2} hydraulic conductivityK _{TZ-SZ} (m/s) | (10^{−10}; 7 × 10^{−5}) [64](10 ^{−7} [18]; 10^{−6} [19,37]) [57](5 × 10 ^{−7}; 5 × 10^{−6} [56]) [65](10 ^{−6} [19,37]; 10^{−4} [62]) [63](10 ^{−5}; 10^{−3}) [17] | 10^{−7}–10^{−5}–10^{−3} | |

Conduit (C) | Diameter D (m) | (0.08; 15) [29] (2; 10) [33] | Flow Capacity A_{C} * K_{C} (m^{3}/s^{−1})10 ^{−2}–10^{−1}–10^{1} |

Section A _{C} (m^{2}) | (<1; >100) [66] | ||

Hydraulic conductivity K _{C} (m/s) | (6 × 10^{−5}; 4 × 10^{−1}) [64](10 ^{−1}; 10) [17,57](3; 10) [63] 10 [19,65] | ||

Van Genuchten Model | Coefficient α (m ^{−1}) | (3.28 × 10^{−3}; 6.23 × 10^{−1}) [15]3.65 × 10 ^{−2} [19,21]10 ^{−2} [17,18] | 3.65 × 10^{−2} |

Empirical parameter n (-) | (0.01; 3) [15] 1.83 [19,21] 2 [17,18] | 1.83 | |

Residual water content θ_{r} (-)or Residual water saturation S _{r} (-) | θ_{r} = S_{r} = 0 [18]θ _{r} ∈(0.01; 0.05) [15]S _{r} = 0.05 [19]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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**MDPI and ACS Style**

Dal Soglio, L.; Danquigny, C.; Mazzilli, N.; Emblanch, C.; Massonnat, G.
Modeling the Matrix-Conduit Exchanges in Both the Epikarst and the Transmission Zone of Karst Systems. *Water* **2020**, *12*, 3219.
https://doi.org/10.3390/w12113219

**AMA Style**

Dal Soglio L, Danquigny C, Mazzilli N, Emblanch C, Massonnat G.
Modeling the Matrix-Conduit Exchanges in Both the Epikarst and the Transmission Zone of Karst Systems. *Water*. 2020; 12(11):3219.
https://doi.org/10.3390/w12113219

**Chicago/Turabian Style**

Dal Soglio, Lucie, Charles Danquigny, Naomi Mazzilli, Christophe Emblanch, and Gérard Massonnat.
2020. "Modeling the Matrix-Conduit Exchanges in Both the Epikarst and the Transmission Zone of Karst Systems" *Water* 12, no. 11: 3219.
https://doi.org/10.3390/w12113219