# Modeling the Potential of Submarine Groundwater Discharge to Facilitate Growth of Vibrio cholerae Bacteria

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Setup

_{S}) were particularly in focus as gradient-driven exchanges and small-scaled variations of flow directions took place mainly in that area. To maximize the representation of the hydrodynamic and solute transport processes right at the coast, the spatial resolution of the mesh was increased in this area using the implemented triangle generator (Delaunay triangulation). Based on model setups in [12], the sea side boundary featured a small vertical part at its lower end to avoid highly acute angles leading to distorted mesh elements and potential numerical issues.

#### 2.2. Governing Equations

^{−1}] is the specific storage coefficient of the fluid, h [m] is the hydraulic potential, t [T] is the time, v [L T

^{−1}] is the Darcy velocity vector, and q [L T

^{−1}] is the outflow/inflow rate (sink/source) of the model system [46].

^{−3}] is the mass concentration. D [L

^{2}T

^{−1}] represents the hydrodynamic dispersion tensor including effects of molecular diffusion D

_{D}and longitudinal and transverse dispersivities (α

_{L}, α

_{T}), respectively [47].

^{−1}] is the hydraulic conductivity tensor, f

_{µ}[–] is the fluid-specific viscosity ratio function, h [L] is the hydraulic potential, and $\tilde{\rho}=$ relative density $=\left(\frac{\rho -{\rho}_{0}}{{\rho}_{0}}\right)$. $\rho $ and ${\rho}_{0}$ [M L

^{−3}] describe the fluid density and reference fluid density. e quantifies the gravitational unit vector with respect to global coordinates [48].

_{D}is the effective molecular diffusion [L

^{2}T

^{−1}], α

_{L}and α

_{T}are longitudinal and transverse dispersivities [L], and I [-] is the unit tensor [49].

#### 2.3. Parametrization

_{Min}= 0°) and the maximum value (β

_{Max}= 76.1°) approached more natural (shallow) conditions. Besides six different slope angles (Table 1), model runs varied individual parameters to elucidate their impact on SGD and V. cholerae growth. Values of controlling factors of SGD were selected to represent a realistic spectrum based on literature (Table 1).

_{X}= 0.8 m d

^{−1}) and varying hydraulic conductivities (8 m d

^{−1}≤ K ≤ 85 m d

^{−1}).

#### 2.4. Model Evaluation

^{−5}s

^{−1}). SGD zone lengths are determined by a digital measuring tool implemented in FEFLOW and represent the sea-side boundary part at which streamlines outflow from the aquifer (Figure 2a). Potentially optimal conditions for V. cholerae growth were measured likewise, as their requirements were met along a restricted part of the SGD zone—referred to as V. cholerae habitat zone, represented by local salinity values C

_{S}≤ 1000 mg L

^{−1}at the direct interface between sea and aquifer (Figure 2b).

## 3. Results

_{X}) on both the SGD zone and the V. cholerae habitat zone. The habitat zones expand non-linearly with increasing inflow, while saltwater intrusion diminishes (Figure 3). V. cholerae habitat zones are generally smaller than the SGD area and disappear below q

_{X}= 0.07 m d

^{−1}. Also, specific SGD flux and hydraulic gradient correlate linearly with inflow (Figure 4). Altered hydraulic conductivities impact SGD and V. cholerae zones; the observed changes in SGD zone, V. cholerae habitat zone (Figure 5), and hydraulic gradient (Figure 6) decrease exponentially with increasing aquifer conductivity with higher rates of change at low K-values. Specific SGD fluxes show a weak positive correlation to changes of hydraulic conductivity (Figure 6).

_{X}and lowest K, respectively. For the considered K-values (Table 1), the V. cholerae habitats in this study range from 1 m to a maximum of 36 m, which corresponds to a possible habitat extension from 1.2% to 42.3% of the entire seaside boundary. With a constant inflow of q

_{X}= 1.3 m d

^{−1}and 85 m d

^{−1}≥ K ≥ 8 m d

^{−1}we note a range from 4 m to 36 m (4.7 % to 42.3 %). Furthermore, the model shows that there is no linear relation between V. cholerae habitat zones and volumes of SGD; while SGD rate increases with increasing permeability (Figure 6), habitat areas decrease (Figure 5).

_{L}, α

_{T}) on the hydraulic potential distribution in coastal aquifers is described in [13,44], which is in accordance with the results obtained here. Accordingly, SGD zones only show minor changes with varying dispersivity parameters (Figure 7). Different from previous results, V. cholerae habitats do not follow the SGD zone response. Higher mixing potential between groundwater and intruding seawater significantly decreases the preferred habitat of V. cholerae assuming a static concentration cap of C

_{max}= 1000 mg L

^{−1}(Figure 7). The maximum expansion of V. cholerae habitat zones at different dispersivities (75 m ≥ α

_{L}≥ 2.5 m) ranges between 15 m and 30 m (the difference corresponds to 17.6 % of the entire seaside boundary) for K = 8 m d

^{−1}.

^{−1}) but converge as the hydraulic conductivity decreases. Specific SGD fluxes correspond to the SGD zone growth behavior in much smaller dimensions (Figure 8). Highest potential changes in V. cholerae zone lengths caused by different coastal slope angles are 6.5 m at K = 17 m d

^{−1}.

## 4. Discussion

_{X}. This is also true for different values of influx q

_{X}. The exponential growth of SGD and V. cholerae habitat zone lengths as only the hydraulic conductivity changes is attributed to the FEFLOW-implemented Darcy equation

_{T}and α

_{L}—slight positive correlations can be attributed to the fact that higher dispersivities reduce density contrasts which weakens the SWI and favors a larger discharge zone for groundwater [12,59]. The strong influence of groundwater and seawater mixing on V. cholerae habitats in this study originates in their assumed dependence solely on salinity. As high mixing potential spreads the transition zone between groundwater and SWI, more discharging groundwater is enriched with salt and higher mass concentrations arise along the SGD zones.

_{L}and α

_{T}values.

_{S}≤ 1000 mg L

^{−1}). Regarding the global frequency of coastal slope angles, most relevant simulation results range between 54° < β < 76° [60]; within these bounds SGD and V. cholerae habitats increase as the sea-side boundary inclines, which could be partially owed to an increasing hydraulic gradient to overcome a growing hydrostatic potential on an expanding slope.

## 5. Conclusions

- Sufficient fresh groundwater inflow and hydraulic conductivity of a coastal aquifer are the most important controls that govern V. cholerae growth. Demanding a high freshwater supply (C
_{S}≤ 100 mg L^{−1}) to the coastal slope and thus a high hydraulic gradient between the hinterland and sea to displace saltwater intrusion, a preferably high groundwater inflow and low hydraulic conductivity within the bounds of a highly conductive material (e.g., sand) create the most favorable conditions for V. cholerae growth, which is not necessarily linked to the specific SGD flux. - Although not significantly affecting the SGD volume, dispersion facilitates non-halophilic bacterial expansion at decreasing values, keeping the fresh groundwater flow low in salt by preventing mixing processes between fresh and saline groundwaters.
- Coastal slope had substantially less impact on the estimated habitat areas than the other analyzed parameters.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch (not to scale) of the conceptual model build regarding geometrical and parametrical properties along the boundaries, where q

_{X}describes the horizontal and q

_{Y}the vertical groundwater inflow, C the solute concentration of the fluids and h corresponds to the static hydraulic potential of the sea. The depth of the aquifer is z and β shows the coastal slope angle.

**Figure 2.**(

**a**) Typical flow pattern, solute distribution and Darcy velocities within homogeneous aquifer models. The submarine groundwater discharge (SGD) zone length extends across the area of seaward directed flow arrows along the seaside boundary. (

**b**) Salinity distribution in the aquifer model and marine sediments (seaside BC), isolines indicate concentration limits such as optimized areas for V. cholerae growth (C

_{S}≤ 1000 mg L

^{−1}).

**Figure 3.**(

**a**) Length of SGD zones and (

**b**) length of V. cholerae zones plotted against the groundwater inflow q

_{X}. Each differently colored data series reflects a hydraulic conductivity value K [m d

^{−1}].

**Figure 4.**(

**a**) Specific SGD flux and (

**b**) hydraulic gradient plotted against the groundwater inflow q

_{X}. Each differently colored data series reflects a hydraulic conductivity value K [m d

^{−1}] as seen in Figure 3.

**Figure 5.**(

**a**) Length of SGD zones and (

**b**) length of V. cholerae zones plotted against the hydraulic conductivity K. Each differently colored data series reflects an inflow value q

_{X}[m d

^{−1}].

**Figure 6.**(

**a**) Specific SGD flux and (

**b**) hydraulic gradient plotted against the hydraulic conductivity K. Each differently colored data series reflects an inflow value q

_{X}[m d

^{−1}] as seen in Figure 5.

**Figure 7.**(

**a**) Length of SGD zones and (

**b**) V. cholerae zones plotted against the longitudinal dispersivity α

_{L}. Each differently colored data series reflects a hydraulic conductivity value K [m d

^{−1}].

**Figure 8.**(

**a**) Length of SGD zones and (

**b**) V. cholerae zones plotted against the coastal slope angle tan(β). Each differently colored data series reflects a hydraulic conductivity value K [m d

^{−1}].

**Table 1.**Parameters and related values that varied or remained constant throughout the simulations. The density ratio corresponds to assumptions made in [50].

Varying Parameter | Symbol | Unit | Value |

Groundwater inflow | q_{X} | m d^{−1} | 0.07; 0.17; 0.4; 0.8; 1.3 [12,51] |

Hydr. conductivity | K | m d^{−1} | 8; 17; 34.4; 43; 85 [36,38,52,53,54] |

Long. dispersivity | α_{L} | m | 2.5; 5; 10; 15; 25; 35; 50; 75 [12,55] |

Trans. dispersivity | α_{T} | m | α_{L} × 0.1 [12,55] |

Coastal slope angle | β | ° | 0; 11.3; 28.1; 53.8; 69.7; 76.1 |

Constant Parameter | Symbol | Unit | Value |

Porosity aquifer | ϕ_{S} | – | 0.3 [56,57] |

Salinity groundwater | C_{F} | mg L^{−1} | 100 [44] |

Salinity seawater | C_{S} | mg L^{−1} | 35000 [44] |

Density groundwater | ρ_{F} | kg m^{−}^{3} | 1000 [58] |

Density seawater | ρ_{S} | kg m^{−}^{3} | 1026 [58] |

Density ratio | d | – | 0.026 |

Molecular diffusion coefficient | D_{M} | m^{2} s^{−1} | 1 × 10^{−9} [58] |

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**MDPI and ACS Style**

Vollberg, F.; Walther, M.; Gärdes, A.; Moosdorf, N. Modeling the Potential of Submarine Groundwater Discharge to Facilitate Growth of *Vibrio cholerae* Bacteria. *Hydrology* **2019**, *6*, 39.
https://doi.org/10.3390/hydrology6020039

**AMA Style**

Vollberg F, Walther M, Gärdes A, Moosdorf N. Modeling the Potential of Submarine Groundwater Discharge to Facilitate Growth of *Vibrio cholerae* Bacteria. *Hydrology*. 2019; 6(2):39.
https://doi.org/10.3390/hydrology6020039

**Chicago/Turabian Style**

Vollberg, Felix, Marc Walther, Astrid Gärdes, and Nils Moosdorf. 2019. "Modeling the Potential of Submarine Groundwater Discharge to Facilitate Growth of *Vibrio cholerae* Bacteria" *Hydrology* 6, no. 2: 39.
https://doi.org/10.3390/hydrology6020039