# Effects of Geometry on Artificial Tracer Dispersion in Synthetic Karst Conduit Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. COMSOL Multiphysics^{®}

#### 2.2. Physical Interfaces

#### 2.2.1. Turbulent Flow

^{−1}mostly characteristic of a low-flow period, a minimum Reynolds number of 10,000 can be estimated. Therefore, we obviously choose to operate in turbulent flow regimes [40,41]. The incorporation of turbulence equations aims to capture the intricate flow characteristics and complexities inherent in high-Reynolds-number scenarios, providing a more comprehensive understanding of the tracer test behavior in several karst conduits geometries. The CFD module within COMSOL Multiphysics incorporates the single-phase flow branch, which consists of various subbranches with physics interfaces. The turbulent flow k–ε interface appears as the more relevant, since it is designed to simulate single-phase flows at elevated Reynolds numbers. It allows a numerical resolution of the Reynolds-averaged Navier–Stokes (RANS) equations coupled to the continuity equation to ensure mass conservation. The model introduces two additional transport equations and two dependent variables: the turbulent kinetic energy, k, and the turbulent dissipation rate, ε. In turbulent flow, the Navier–Stokes equation maintains its identity with the laminar counterpart, yet an additional component emerges in the transport equation [42]. Equation (1) represents the transport equation for k, where Pk is the production term presented in Equation (2). The transport equation for ε is shown in Equation (3).

^{T}represents the turbulent viscosity shown in Equation (4) and C

_{μ}is a model constant.

#### 2.2.2. Transport of Diluted Species (TDS)

#### 2.3. Case Studies

^{−1}, which aligns with field velocities within a realistic field range. The concentration at the inflow is estimated by Equation (7)

_{0}is the initial concentration established at 1 mol/m

^{3}, and $\sigma $ and $\mathsf{\mu}$ represent the mean and standard deviation of the distribution.

#### 2.4. Residence Time Distribution

## 3. Results and Discussions

#### 3.1. Simulation Results Using COMSOL Multiphysics

#### 3.1.1. Case 1: Influence of the Distance of the Outlet from the Inlet

#### 3.1.2. Case 2: Influence of the Size of Pools

#### 3.1.3. Case 3: Influence of the Position of Pools

#### 3.1.4. Case 4: Influence of the Diameter of Conduits

^{−1}for D = 2 m to 0.471 s

^{−1}for D = 6 m. Then, the RTD maximum value slightly decreases. This increased flow velocity can result in faster solute transport through the system. As solutes spend less time within the conduits before exiting, the RTD maximum value tends to decrease.

#### 3.1.5. Case 5: Influence of the Angle of Connection

#### 3.1.6. Case 6: Influence of the Number of Branches

## 4. Discussion

## 5. Conclusions

^{−1}. Second, numerical results show that the pool size and position have no significant effect on the BTCs, consistent with previous studies. Third, conduit diameter variation depicts an effect on RTD values and on the arrival time of solute. The RTD maximum values tend to increase first and then decrease. Finally, we have examined the influence of the connection angle and the number of branches. Results show that the connection angle between two conduits has an essential impact on the transport process, with a significant impact on RTD maximum values, arrival time, and the occurrence of multi-peaked BTCs. Furthermore, the variation in the number of branches compares how the conduit network’s structural complexity affects solute transport within the karstic system.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**RTDs corresponding to various geometries involving the variation of distances between inlets and outlets, as well as symmetry alterations. Within this figure, (

**a**) corresponds to L = 20 m, (

**b**) represents L = 10 m up, and (

**c**) signifies L = L (symmetric) (The concentration field shown in the figures corresponds to the peak).

**Figure 2.**RTDs corresponding to various geometries involving the pool sizes. Within this figure, (

**a**) corresponds to LP = 2 m, (

**b**) represents LP = 4 m up, (

**c**) signifies LP = 6 m, (

**d**) indicates L = 8 m down, and (

**e**) corresponds to L = 10 m (the concentration field shown in the figures corresponds to the peak).

**Figure 3.**RTDs corresponding to various geometries involving the pool positions: (

**a**) corresponds to L-L-L, (

**b**) represents L-L-2L-3L, (

**c**) 2L-L-2L-3L, and (

**d**) 3L-L-2L-3L (the concentration field shown in the figures corresponds to the peak).

**Figure 4.**RTDs corresponding to various geometries involving conduit diameter. Within this figure, (

**a**) corresponds to D = 2 m, (

**b**) represents D = 4 m, (

**c**) signifies D = 6 m, (

**d**) indicates D = 8 m, and (

**e**) corresponds to D = 10 m (the concentration field shown in the figures corresponds to the peak).

**Figure 5.**RTDs corresponding to various geometries involving angle of connection (

**a**) corresponding to alpha = 10, (

**b**) alpha = 20, (

**c**) alpha = 30, (

**d**) alpha = 40, (

**e**) alpha = 50; (

**f**) alpha = 60, (

**g**) alpha = 70, (

**h**) alpha = 80, and (

**i**) alpha = 90 (the concentration field shown in each figures corresponds to the peak).

**Figure 6.**RTDs corresponding to various geometries involving number of branches. Alpha = 30 (

**a**) corresponds to n = 1, (

**b**) n = 2, and (

**c**) n = 3 (the concentration field shown in the figures corresponds to the peak).

**Figure 7.**RTDs corresponding to various geometries involving several branches for alpha = 90 within figure: (

**a**) corresponds to n = 1, (

**b**) illustrates n = 2 and (

**c**) shows n = 3 (the concentration field shown in the figures corresponds to the concentration peak).

**Table 1.**Values of the parameters of the k epsilon turbulence model, which is a widely used approach in computational fluid dynamics (CFDs) to simulate turbulent flows by solving transport equations for turbulence kinetic energy and its dissipation rate.

Constant | Value |
---|---|

C_{μ} | 0.09 |

C_{ε1} | 1.44 |

C_{ε2} | 1.92 |

σ_{k} | 1.0 |

σ_{ε} | 1.3 |

**Table 2.**Presentation of the different geometries considered in CFD simulation affecting BTCs within complex karst conduit systems. Each case investigates specific aspects: Case 1: variable inlet-outlet distance/Cases 2–3: impact of pool characteristics (size and position)/Case 4: conduit diameter/Case 5: connection angles between conduits/Case 6: number of branches from 1 up to 3. They correspond to the main geometries already identified in karst networks on a large scale [11] that can be of interest in artificial tracer dispersion discussion.

Cases | Conditions | Parameters | Illustration |
---|---|---|---|

Case 1 | Distance of the outlet from the inlet (D _{0} = D_{conduit} = 2 m) | L = L_{0}L = 10 m L = 20 m | |

Case 2 | Size of pools (D_{0} = D_{conduit} = 2 m and L_{PO} = 2 m) | L_{P} = 2×_{PO}L _{P} = 3×L_{PO}L _{P}= 4×L_{PO}L _{P}= 5×L_{PO} | |

Case 3 | Position of pools (Cubic Pools 6 × 6 m and L’O = 2 m) | L’ = 2 × L’_{O}L’ = 2 × L’ _{O}L’ = 3 × L’ _{O} | |

Case 4 | Diameter of conduits | D = 2 × D_{0}D = 3 × D _{0}D = 4 × D _{0}D = 5 × D _{0} | |

Case 5 | Angle of connection (D _{0} = D_{conduit} = 2 m) | Alpha α α= 10…90 | |

Case 6 | Numbers of branches | N = 1, 2, 3 |

Notations | Definitions | Units |
---|---|---|

RTD | Distribution of the time that solutes spend within the system | [s^{−1}] |

T_{peak} | Elapsed time to peak concentration | [s] |

T_{arrival} | Elapsed time to solute last detected | [s] |

Distance | Max Value of RTD (s^{−1}) | Peak Time (s) | Arrival Time (s) |
---|---|---|---|

L = 20 m (a) | 0.338 | 1600 | 2900 |

L = 10 m up (b) | 0.265 | 1400 | 2300 |

L = 0 m up (c) | 0.149 | 1300 | 2000 |

Lp | Max Value of RTD (s ^{−1}) | Peak Time (s) | Arrival Time (s) |
---|---|---|---|

Lp = 2 m (a) | 0.110 | 400 | 800 |

Lp = 4 m (b) | 0.113 | 450 | 820 |

Lp = 6 m (c) | 0.125 | 500 | 820 |

Lp = 8 m (d) | 0.117 | 500 | 820 |

Lp = 10 m (e) | 0.113 | 500 | 820 |

**Table 6.**Statistical parameters from RTDs for the third case (variation in the position of the pools).

Lp | Max Value of RTD (s ^{−1}) | Peak Time (s) | Arrival Time (s) |
---|---|---|---|

L-L-L-L (a) | 0.135 | 650 | 1200 |

L-L-2L-3L (b) | 0.143 | 600 | 1300 |

2L-L-2L-3L (c) | 0.187 | 500 | 1300 |

3L-L-2L-3L (c) | 0.255 | 250 | 1400 |

D | Max Value of RTD (s ^{−1}) | Peak Time (s) | Arrival Time (s) |
---|---|---|---|

D = 2 m (a) | 0.338 | 1300 | 2000 |

D = 4 m (b) | 0.424 | 1100 | 1800 |

D = 6 m (c) | 0.471 | 1100 | 1800 |

D = 8 m (d) | 0.447 | 1100 | 1600 |

D = 10 m (e) | 0.423 | 1100 | 1600 |

**Table 8.**Statistical parameters from RTDs for the fifth case (variation in the angle of connection).

Alpha | Max Value of RTD (s ^{−1}) | Peak Time (s) | Arrival Time (s) | |
---|---|---|---|---|

Alpha 10 (a) | First peak | 0.358 | 900 | 1900 |

Second peak | 0.066 | 1150 | 1900 | |

Alpha 20 (b) | First peak | 0.269 | 800 | 2000 |

Second peak | 0.081 | 1200 | 2000 | |

Alpha 30 (c) | First peak | 0.187 | 800 | 2200 |

Second peak | 0.053 | 1500 | 2200 | |

Alpha 40 (d) | First peak | 0.319 | 1100 | 2100 |

Second peak | 0.0359 | 1500 | 2100 | |

Alpha 50 (e) | First peak | 0.25 | 700 | 1400 |

Second peak | - | - | 1400 | |

Alpha 60 (f) | First peak | 0.567 | 600 | 900 |

Second peak | - | - | 900 | |

Alpha 70 (g) | First peak | 0.476 | 600 | 900 |

Second peak | - | - | 900 | |

Alpha 80 (h) | First peak | 0.427 | 600 | 900 |

Second peak | - | - | 900 | |

Alpha 90 (I) | First peak | 0.425 | 600 | 900 |

Second peak | - | - | 900 |

**Table 9.**Statistical parameters from RTDs for the final case (variation in the number of branches for alpha = 30).

Alpha 30 | Max Value of RTD (s ^{−1}) | Peak Time (s) | Arrival Time (s) | |
---|---|---|---|---|

a | First peak | 0.187 | 800 | 5200 |

Second peak | 0.52 | 1600 | 5200 | |

b | First peak | 0.616 | 1500 | 8000 |

Second peak | 0.709 | 2300 | 8000 | |

c | First peak | 0.474 | 2600 | 10,000 |

Second peak | 0.232 | 3800 | 10,000 |

**Table 10.**Statistical parameters from RTDs for the final case (variation in the number of branches for alpha = 90).

Alpha 90 | Max Value of RTD (s ^{−1}) | Peak Time (s) | Arrival Time (s) | |
---|---|---|---|---|

a | First peak | 0.485 | 600 | 7100 |

Second peak | 0.009 | 4400 | 7100 | |

b | First peak | 0.245 | 700 | 8000 |

Second peak | 0.025 | 4700 | 8000 | |

c | First peak | 0.157 | 900 | 10,000 |

Second peak | 0.022 | 5900 | 10,000 |

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**MDPI and ACS Style**

Rabah, A.; Marcoux, M.; Labat, D.
Effects of Geometry on Artificial Tracer Dispersion in Synthetic Karst Conduit Networks. *Water* **2023**, *15*, 3885.
https://doi.org/10.3390/w15223885

**AMA Style**

Rabah A, Marcoux M, Labat D.
Effects of Geometry on Artificial Tracer Dispersion in Synthetic Karst Conduit Networks. *Water*. 2023; 15(22):3885.
https://doi.org/10.3390/w15223885

**Chicago/Turabian Style**

Rabah, Amal, Manuel Marcoux, and David Labat.
2023. "Effects of Geometry on Artificial Tracer Dispersion in Synthetic Karst Conduit Networks" *Water* 15, no. 22: 3885.
https://doi.org/10.3390/w15223885