# Coupling SKS and SWMM to Solve the Inverse Problem Based on Artificial Tracer Tests in Karstic Aquifers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study site and Field Data

^{2}and a median elevation of about 940 m above sea level (a.s.l.). It belongs to the carbonate belt bordering the North of the French Pyrenees (Figure 1). The area was highly deformed during late Cenomanian to Tertiary Pyrenean orogeny induced by the transpressional strike-slip motion of the Iberic and European plates along the North Pyrenean fault [39,40]. The karstified part of the basin is characterized by metamorphic Jurassic to Cretaceous dolomites, limestones and calcareous marls. As a consequence of the metamorphism, matrix porosity was reduced to less than 1 percent [41]. These formations deep 70° to 90° southwards, under the slaty Albian-Cenomanian Ballongue flysch. The southern part of the area consists of a Cretaceous pull-apart basin opened during strike-slip motion along the Pyrenean margin [40,42]. A secondary fault from the North Pyrenean fault, the Alas fault, and the Balagué polje constitute the northern limit of the Baget drainage basin. The contact between the karstified calcareous formations and the impermeable flysch gives the valley orientation in the west-east direction. In the upstream part of the watershed, the Lachein river flows during periods of the high-water level; otherwise, the river flows downstream the perennial spring of the Baget watershed, Las Hountas. The downstream part of the watershed is characterized by the presence of loss and temporary and permanent resurgences on a spatially restrained area of around 1 km

^{2}(Figure 1). According to Mangin [41], voids consist of dissolution caves and in open fractures and joints. Several caves have been recognized and mapped, such as St Catherine, La Peyrère [43] and part of the system between La Peyrère, P2 Loss, and Moulo de Jaur [44]. Several approaches have been performed to attempt to establish voids geometry [45,46] and their influence in solute transport [10,13,41].

^{3}/s at the beginning down to 0.3 m

^{3}/s at the end of the campaign (Figure 2). A field fluorimeter GGUNFL-30 [48] was installed near the station B1, where the discharge has been measured since the late 1960s [41]. The fluorescence measurement is done on a 15-min sampling rate and the water sampling is performed at an hourly sampling rate. Water samples were analyzed by the CETRAHE laboratory (Orléans, France) using a spectrofluorimeter Hitachi F2500 and F7000. The injection point P2 Loss was chosen so the tracer reached the subterranean drainage system rapidly, getting close from an instantaneous injection (Dirac function).

#### 2.2. Modeling Approach

#### 2.2.1. Conduit Network Simulation

- Building a geological model of the studied area. It covered about 1 km
^{2}. Only the two main geological formations were considered: the slaty flysch in the southern part of the area and the Jurassic to Cretaceous metamorphic limestones in the northern part of the area. The contact between these two formations is oriented in the west-east direction. The calcareous formation was numerically considered as a homogeneous formation affected by structural heterogeneities (faults and fractures). Then, bedding planes, inception horizons, or even foliation were not considered for implementation of geological constraints in conduit networks simulation with SKS. This constitutes a strong hypothesis but seems to be acceptable regarding the small extension of the simulation area. Moreover, slaty formations constituted a boundary condition for the development of the conduit networks. - The structural heterogeneities (faults and fractures) over the area were considered in the SKS model. The main discontinuity direction was recognized from the satellite image, running 170° N to 10° N orientations [57,58,59] as well as the faults and fractures reported by the French geological survey (BRGM) in the BD_CHARM database [54]. The fracture model includes the main structures identified in the area and a set of stochastic fractures that is different for every simulation and generated following the statistical distributions derived from the field data. Besides, the observations made through speleological investigations [44] have been considered as conditional data in the conduit network simulations; thus, SKS reproduces this known conduit.
- The inlets and outlets can be identified and imposed in SKS. The inlets are composed of La Peyrère, P2 Loss, La Hillière and Moulo de Jaur. Moreover, some additional potential inlets can be randomly added over the area to ensure more physical realism and to allow potential feeding branches along with the solute transport to be considered. Then, Las Hountas, which is the perennial outlet of the Baget system, constitutes the only outlet of the synthetic conduits networks.

#### 2.2.2. Flow Simulation

^{−1}), $x$ is the longitudinal distance (L), $t$ is the time (t), $g$ is the gravitational acceleration (L·t

^{−2}), ${S}_{0}$ is the channel slope, ${S}_{f}$ is the friction slope, $A$ is the area of the flow cross-section and is a function of y upon the geometry of the conduit (L

^{2}) and $Q$ is the discharge with $Q=A\times V$ (L

^{3}·t

^{−1}).

^{−1/3}), $R$ is the hydraulic radius (L), $Q$ is the discharge (L

^{3}·t

^{−1}) and $A$ is the area of the flow cross-section (L

^{2}).

#### 2.2.3. Solute Transport Modeling

^{−3}), V is the volume (L

^{3}), t is the time (t), ${Q}_{i}$ and ${Q}_{o}$ are inflow (i) and outflow (o) rate (L

^{3}·t

^{−1}), ${c}_{i}$ and ${c}_{o}$ are the concentration of the influent and effluent (m·L

^{−3}), k is the decay constant (t

^{−1}), $s$ is the source (or sink) (m·t

^{−1}).

## 3. Results

#### 3.1. Model Setup

^{2}mesh grid. The temporal resolution in SWMM is a routing step of 15 s and a report step of 15 min. This provided a good compromise between model resolution and time of computation. Also, it allowed us to avoid numerical instabilities during dynamic wave routing for both flow and solute transport simulation [65]. A complete simulation, from the simulation of a synthetic conduit network to the simulation of the corresponding RTD, lasted about 5 min on a computer equipped with 8 Go RAM. The simulation was run over 1000 simulations and required approximately 84 h of computation.

#### 3.2. Statistical Analysis

^{2}, the mean flow velocity is 0.16 m/s and the mean length of the conduits between injection and recovery point was about 905 m. Dividing the length of the total conduits by the apparent distance between injection and recovery point gives and estimation of tortuosity. In karst systems, the tortuosity generally varies from 1.10 to 1.40 depending on the morphology of the conduit networks [70]. Considering an apparent distance of 850 m between injection and recovery sites [13], the mean tortuosity here is about 1.06, getting closer to the range of value for an angular maze conduit network morphology [70]. The variability of the mean flow section along the transport and the length of transport appears to be lower than for the mean flow velocity. Moreover, both present a unimodal distribution contrary to the mean flow velocity showing 3 modes (Figure 6). Nonetheless, the two modes showing higher flow velocity values count a low number of simulations in contrast with the mode showing values from 0.1 to 0.2 m/s. The mean flow velocity is considered both in temporal (during tracer recovery) and spatial (along with the entire tracer transport) terms. Considering spatial variations of the conduit section along with the tracer transport pathway may lead to significant variations in flow velocity.

#### 3.3. Conduit Geometry and Spatial Distribution of Flow

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location of the Baget karstic system, a geological map of the area (modified from [54]) and limits of the watershed (red dashed line). The study area concerns the downstream part of the watershed (black dashed line), in the mid mountains Lachein valley, characterized by altitude from 450 m a.s.l. to 750 m a.s.l. [55]. The pictures show remarkable features across the area: (

**A**) La Peyrère is a cave connected to the main drainage system, (

**B**) P2 Loss is the point of injection for artificial tracer tests, (

**C**) La Hillère is a temporary resurgence, (

**D**) Moulo de Jaur is a temporary resurgence and an intermediate observation point of tracer transport, (

**E**) Las Hountas is the perennial spring of the watershed and the tracer recovery point and (

**F**) B1 is the gauging station (discharge, water sampling, and fluorimeter), modified from [13].

**Figure 2.**Rainfall-discharge time series during artificial tracer tests and injection recovery time series. The artificial tracer tests have been performed in April 2018 (modified from [13]).

**Figure 3.**Coupling the Stochastic Karst Simulator (SKS) and Storm Water Management Model (SWMM) to solve the inverse problem based on artificial tracer tests. SKS is used to simulate the conduit network geometry and SWMM is used to simulate flow and solute transport, assuming an instantaneous complete mixing hypothesis.

**Figure 4.**Simulation of a karst conduit network over the downstream part of the Baget watershed using the SKS algorithm.

**Figure 5.**Tracer injection (green bars), experimental residence time distribution (RTD; black solid line) and simulated RTD curves: grey lines represent simulated RTD for all scenarios, green lines represent simulated RTD with an Nash-Sutcliff Efficiency coefficient (NSE) superior to 0.8 and the red line is the simulation providing the best NSE (0.89).

**Figure 6.**Histograms calculated over 1000 simulations on (

**a**) NSE coefficient, (

**b**) section area, (

**c**) the mean flow velocity and (

**d**) the conduit length. Grey bars are for all data and green bars represent the simulations providing an NSE superior to 0.8 between simulated RTD and observed RTD.

**Figure 7.**2D representation of the simulated tracer pathways, extracted from the synthetic conduit networks simulated with SKS.

**Figure 8.**(

**a**) Flow velocity along with the tracer transport and (

**b**) flow section along with the tracer transport. Green lines correspond to simulations providing an NSE superior to 0.8 and red lines correspond to simulation providing the best NSE (0.89).

Descriptive Statistics | Mean Flow Section Area (m^{2}) | Mean Flow Velocity (m/s) | Transport Length (m) |
---|---|---|---|

Min | 8.42 | 0.09 | 772.7 |

Max | 9.45 | 0.41 | 1007.0 |

Mean | 9.01 | 0.16 | 905.5 |

Standard Deviation | 0.20 | 0.06 | 43.1 |

Variation Coefficient | 0.02 | 0.14 | 0.05 |

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**MDPI and ACS Style**

Sivelle, V.; Renard, P.; Labat, D.
Coupling SKS and SWMM to Solve the Inverse Problem Based on Artificial Tracer Tests in Karstic Aquifers. *Water* **2020**, *12*, 1139.
https://doi.org/10.3390/w12041139

**AMA Style**

Sivelle V, Renard P, Labat D.
Coupling SKS and SWMM to Solve the Inverse Problem Based on Artificial Tracer Tests in Karstic Aquifers. *Water*. 2020; 12(4):1139.
https://doi.org/10.3390/w12041139

**Chicago/Turabian Style**

Sivelle, Vianney, Philippe Renard, and David Labat.
2020. "Coupling SKS and SWMM to Solve the Inverse Problem Based on Artificial Tracer Tests in Karstic Aquifers" *Water* 12, no. 4: 1139.
https://doi.org/10.3390/w12041139