# A New Normalized Groundwater Age-Based Index for Quantitative Evaluation of the Vulnerability to Seawater Intrusion in Coastal Aquifers: Implications for Management and Risk Assessments

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background and Analysis of the Vulnerability Methods

## 3. Method Design and Development

#### 3.1. Simulation of Coupled Variable-Density Flow and Transport

^{−1}]; $\rho $ is the mixture fluid density [ML

^{−3}]; $q$ is the specific discharge [LT

^{−1}]; $C$ is the dissolved salt concentration [ML

^{−3}]; $\varphi $ is the effective porosity [-]; $D$ is the hydrodynamic dispersion tensor [L

^{2}T

^{−1}] whose pore velocity-dependent entries are calculated following Bear and Cheng [28]; $K={k}_{s}{k}_{r}{\rho}_{f}g/\mu $ is the hydraulic conductivity tensor [LT

^{−1}]; ${k}_{s}$ is the intrinsic permeability tensor of the saturated porous medium [L

^{2}]; ${k}_{r}$ is the head-dependent relative hydraulic conductivity [-]; $g$ is the gravitational acceleration [LT

^{−2}]; $\mu $ is the dynamic viscosity of the fluid [ML

^{−1}T

^{−1}]; $H$ is the equivalent freshwater head [L]; ${\rho}_{r}=\left(\rho -{\rho}_{f}\right)/{\rho}_{f}$ is the relative density [-] with ${\rho}_{f}\approx 1000$ Kg/m

^{3}being the reference freshwater density; $\beta =\partial \rho /\partial C\approx 0.7$ is the rate of density variation relative to the salt concentration [-], and ${C}_{f}$ is the reference freshwater concentration [ML

^{−3}].

#### 3.2. Simulation of Groundwater Age and the Vulnerability Index Development

^{−1}]; and ${\mathsf{\Gamma}}_{in}$ is the union of all inflowing domain boundaries including the sea boundary ${\mathsf{\Gamma}}_{s}$. Notably, a unit growth of the age mass in the forward direction is added as a source term.

**x**). For large-scale problems, a computationally efficient way to solve the transport Equations (2) and (5) is to factor the global sparse transport matrix only once during the preconditioning step of the selected non-symmetric iterative sparse solver [56]. Overall, it is expected that the computational burden required by the introduced method is minimal even for large-scale applications.

## 4. Results

#### 4.1. Example 1: The Henry Problem (Homogeneous and Heterogeneous Domains)

#### 4.1.1. The Diffusive Henry Problem

**x**) = 0). In the following, this will be identified as the zero-vulnerability line for a 2D space (or surface in three-dimensional space) and denoted thereafter by ZVL (or ZVS) depending on the space dimension. Owing to salt diffusion, a mixing zone develops at intermediate positions between the saltwater and freshwater zones. The landward transport of the salt is greater with the aquifer depth due to the density stratification. The same phenomenon is observed for the calculated vulnerability index as shown in Figure 2c where the vulnerability to SWI is higher as the aquifer depth increases close to the coast. However, the vulnerability plume is less extensive than that for the concentration distribution. This is because only some, but not all, cells in the mixing zone are located behind the ZVL.

_{f}, was adopted.

_{p}= 4 × 10

^{−5}m

^{3}·s

^{−1}are visualized in Figure 3. Notably, the steady state concentration isochlors and vulnerability index plumes both advance landward because of pumping. Furthermore, their increased extension with the aquifer depth is noticed here again and is processed almost in the same way. However, as the pumping rate Q

_{p}reaches 10

^{−4}m

^{3}·s

^{−1}for the last test case as shown in Figure 4, saltwater upconing beneath the well becomes more pronounced implying significant vertical displacement of saltwater to the upper part of the aquifer. This is in agreement with the simulated steady state concentration isochlors and the vulnerability index profiles shown in Figure 3a,c, respectively. Hence, when the pumping rate in a well exceeds a critical value, yet to be determined, all the three-dimensional aquifer space separating this well to the coast becomes highly vulnerable to salinization. Although the selected example involves a confined aquifer, extrapolation to shallow unconfined aquifers suggests possible salinization of agricultural soils as the water table rises in winter periods and/or due to sea-level rise.

_{s}) at the observation points P

_{1,…,4}shown in Figure 1. Hence, an additional two transient simulations of the VDFT problem are performed for the standard Henry problem and the variant with the highest pumping rate. Additionally, the salinization rate (i.e., dC/dt) time series are equally monitored at these locations. A constant time step size of 5 min was selected. However, as the chosen numerical solution scheme was the total variation diminishing scheme smaller internal time steps were automatically selected to enforce the numerical stability to satisfy the Courant number criterion [29].

#### 4.1.2. The Heterogeneous Diffusive Henry Problems

_{b}= 0.1604 × 10

^{−9}m

^{2}; k′

_{b}= 3.2216 × 10

^{−9}m

^{2}; $\mp \delta $ is the decay/increase rate, and $z$ [L] is the aquifer depth. All the remaining physical and spatial discretization parameters remained identical as in the reference diffusive Henry problem. The effective diffusion coefficient was maintained as equal to that for the standard Henry problem (i.e., 1.886 × 10

^{−5}m

^{2}·s

^{−1}) for comparative purposes with the homogeneous simulations. Therefore, the results are expected not to be identical to those presented in [65] as they selected a smaller effective diffusion coefficient (i.e., 3.76 × 10

^{−6}m

^{2}·s

^{−1}) as suggested by other authors [67].

#### 4.1.3. The Dispersive Henry Problems

_{z}/K

_{x}) is smaller than for the ideal case (i.e., 1) as shown in Table 1.

#### 4.2. Example 2: The Akkar Coastal Aquifer (Lebanon)

^{−1}[68] to an average of 655 mg·L

^{−1}in 2013 [69]. Groundwater salinization is expected to extend further due to anticipated climate change impacts on water resources, especially in a semi-arid set up. Hence, in this work we investigate the vulnerability of this aquifer system to SWI. The performance of different management strategies is evaluated based on their distributed vulnerability indices.

^{−7}m·s

^{−1}) is set as a fixed flux on the top surface boundary. At the landward boundary, we assume a constant mountain-block recharge flux estimated to 6.4 × 10

^{−9}m·s

^{−1}from the east. For salt transport, we use a fixed salinity (i.e., 35 g·L

^{−1}) boundary condition at the seaside. A null concentration is fixed landward and on the top boundaries representing freshwater influx. A zero age Dirichlet boundary condition was prescribed on the top, left and right boundaries. As shown in Figure 7, two pumping wells are considered for groundwater abstraction. The first pumping well, P1, is located near the sea (i.e., at 1500 m from the coast) at 45 m depth while the second pumping well, P2, is located further inland (i.e., at 3000 m from the coast) at 30 m depth. Actual pumping rates at P1 and P2 amount to 579.25 L·h

^{−1}and 604.25 L·h

^{−1}, respectively.

^{−1}confirming that it remarkably delimits the areas more affected by SWI.

^{−1}, which amounts to 80% of the total pumping rate. The NSAVI isolines are drastically shifted seaward indicating a high potential for the aquifer remediation when using this management scenario. Indeed, the ZVL distance from the coast does not exceed 760 m leading to a reduced vulnerable area by almost one-third of the space that was initially under salinization threat.

## 5. Discussions

#### 5.1. Implications of the Zero-Vulnerability Line/Surface on Coastal Aquifers Management

#### 5.2. Implications for Optimal Management of Coastal Aquifers

#### 5.3. Monitoring Network Design and Implementation

## 6. Conclusions

- The normalized saltwater vulnerability index was obtained from steady state distributions of the normalized concentration and a restriction of the mean groundwater age to a mean saltwater age distribution. The approach provides a novel way to shift from the concentration space into a vulnerability assessment space to evaluate the threats to coastal aquifers.
- The vulnerability to seawater intrusion is higher as the aquifer depth increases in homogeneous and heterogeneous stratified aquifers. This behavior is not reproducible when using the data-driven indexing methods giving further evidence of their conceptual limitations.
- The zero-vulnerability line/surface delineates the coastal area for which there is a probability of seawater intrusion to occur. This novel concept has important implications for the optimal management of coastal aquifers and establishing related risk assessment methodologies.
- The zero-vulnerability line position differs from that of the 50% isochlor. As the strength of anthropogenic processes increase, the ZVL fits with increasing isochlor levels. Hence, the 50% isochlor is not a suitable metric to evaluate the vulnerability of coastal aquifers to seawater intrusion.
- The provided theoretical and field case study problems demonstrate the suitability of this approach to rank, compare and validate different scenarios for coastal water resources management.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of the Henry problem showing flow, transport, and direct age boundary conditions on the upstream freshwater and downstream seawater boundaries, respectively. Observation points P

_{1,…,4}are indicated in red color.

**Figure 2.**Spatial distribution patterns of (

**a**) the steady-state concentration isochlors, (

**b**) the direct mean groundwater age, and (

**c**) the vulnerability index to seawater intrusion for the standard Henry problem. The zero-vulnerability line (ZVL) is plotted with red color.

**Figure 3.**Spatial distribution patterns of (

**a**) the steady-state concentration isochlors, (

**b**) the direct mean groundwater age, and (

**c**) the vulnerability index to seawater intrusion for the modified Henry problem with a partially penetrating pumping well (blue line) discharging 4 × 10

^{−5}m

^{3}·s

^{−1}. The zero-vulnerability line (ZVL) is plotted with red color.

**Figure 4.**Spatial distribution patterns of (

**a**) the steady-state concentration isochlors, (

**b**) the direct mean groundwater age, and (

**c**) the vulnerability index to seawater intrusion for the modified Henry problem with a partially penetrating pumping well (blue line) discharging 10

^{−4}m

^{3}·s

^{−1}. The zero-vulnerability line (ZVL) is plotted with red color.

**Figure 5.**Time series of the normalized concentration (C/C

_{s}) and the salinization rate (dC/dt) at observation points P

_{1,…,4}for (

**a**) the standard diffusive Henry problem and (

**b**) its variant with the highest pumping rate.

**Figure 6.**Spatial distribution patterns of (

**a**) the steady-state concentration isochlors, (

**b**) the direct mean groundwater age, and (

**c**) the vulnerability index to seawater intrusion for the heterogeneous Henry problems with an exponentially decaying permeability with depth ($\mathsf{\delta}=3$). The zero-vulnerability line (ZVL) is plotted with red color.

**Figure 7.**Spatial distribution patterns of (

**a**) the steady-state concentration isochlors, (

**b**) the direct mean groundwater age, and (

**c**) the vulnerability index to seawater intrusion for the heterogeneous Henry problems with an exponentially increasing permeability with depth ($\mathsf{\delta}=-3$). The zero-vulnerability line (ZVL) is plotted with red color.

**Figure 8.**Spatial distribution patterns of (

**a**) the steady-state concentration isochlors, (

**b**) direct groundwater age, and (

**c**) the vulnerability index to seawater intrusion for the dispersive Henry problem. The zero-vulnerability line (ZVL) is plotted with red color.

**Figure 9.**Situation map of the Akkar plain in Northern Lebanon showing the lithology of outcropping formations, domain boundaries, the selected vertical X-Z transect for vulnerability assessments to SWI, and the high vulnerability line determined by GALDIT [70].

**Figure 10.**View of the vertical cross-section highlighted in Figure 9 composed from six aquifer units. The unconfined aquifer is subject to mountain-block recharge from the right (q), infiltrating recharge from the top (R), and seawater intrusion from the left side.

**Figure 11.**Spatial distribution patterns of the NSAVI vulnerability index and the zero-vulnerability line (ZVL) in the Akkar coastal aquifer (Lebanon) for (

**a**) the actual conditions and (

**b**) a newly proposed management scenario involving simultaneous relocation of the wells, decreasing the pumping density, and 20% decrease of the total flow rate.

Parameter | Diffusive Henry Problem | Dispersive Henry Problem |
---|---|---|

Freshwater density ${\rho}_{r}$ (Kg·m^{−3}) | 1000 | 1000 |

Seawater density ${\rho}_{s}$ (Kg·m^{−3}) | 1025 | 1025 |

Seawater salinity ${C}_{s}$ (Kg·m^{−3}) | 35 | 35 |

Vertical hydraulic conductivity ${K}_{z}$ (m·s^{−1}) | 0.01 | 0.01 |

Anisotropy ratio (${K}_{z}/{K}_{x})$ | 1 | 0.66 |

Effective porosity $\varphi $ | 0.35 | 0.35 |

Molecular diffusion of salt ${D}_{m}$ (m^{2}·s^{−1}) | 1.886 × 10^{−5} | 0 |

Longitudinal dispersivity ${\alpha}_{L}$ (m) | 0 | 0.1 |

Transverse dispersivity ${\alpha}_{T}$ (m) | 0 | 0.01 |

Freshwater flux q_{f} (m·s^{−1}) | 3.3 × 10^{−5} | 3.3 × 10^{−5} |

**Table 2.**Physical parameters used for the specific vulnerability assessments in the Akkar aquifer case study (Lebanon).

Parameter | Value |
---|---|

Freshwater density ${\rho}_{r}$ (Kg·m^{−3}) | 1000 |

Seawater density ${\rho}_{s}$ (Kg·m^{−3}) | 1025 |

Seawater salinity ${C}_{s}$ (Kg·m^{−3}) | 35 |

Dynamic viscosity $\mu $ (Pa.s) | 1.002 × 10^{−3} |

Unit 1: 1.01 × 10^{−12} | |

Unit 2: 1.01 × 10^{−12} | |

Vertical permeability ${k}_{z}$ (m^{2}) | Unit 3: 8.15 × 10^{−12} |

Unit 4: 5.09 × 10^{−13} | |

Unit 5: 1.01 × 10^{−14} | |

Unit 6: 3.05 × 10^{−13} | |

Anisotropy ratio (${K}_{z}/{K}_{x})$ | 0.1 |

Effective porosity $\varphi $ | Units 2, 3, 4: 0.3 |

Units 1, 5, 6: 0.2 | |

Molecular diffusion of salt ${D}_{m}$ (m^{2}·s^{−1}) | 10^{−9} |

Longitudinal dispersivity ${\alpha}_{L}$ (m) | 50 |

Transverse dispersivity ${\alpha}_{T}$ (m) | 5 |

Average recharge R (m·s^{−1}) | 3.86 × 10^{−7} |

Mountain-block recharge q (m·s^{−1}) | 6.40 × 10^{−9} |

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## Share and Cite

**MDPI and ACS Style**

Sbai, M.A.; Larabi, A.; Fahs, M.; Doummar, J. A New Normalized Groundwater Age-Based Index for Quantitative Evaluation of the Vulnerability to Seawater Intrusion in Coastal Aquifers: Implications for Management and Risk Assessments. *Water* **2021**, *13*, 2496.
https://doi.org/10.3390/w13182496

**AMA Style**

Sbai MA, Larabi A, Fahs M, Doummar J. A New Normalized Groundwater Age-Based Index for Quantitative Evaluation of the Vulnerability to Seawater Intrusion in Coastal Aquifers: Implications for Management and Risk Assessments. *Water*. 2021; 13(18):2496.
https://doi.org/10.3390/w13182496

**Chicago/Turabian Style**

Sbai, Mohammed Adil, Abdelkader Larabi, Marwan Fahs, and Joanna Doummar. 2021. "A New Normalized Groundwater Age-Based Index for Quantitative Evaluation of the Vulnerability to Seawater Intrusion in Coastal Aquifers: Implications for Management and Risk Assessments" *Water* 13, no. 18: 2496.
https://doi.org/10.3390/w13182496