# A Generalized Semi-Analytical Solution for the Dispersive Henry Problem: Effect of Stratification and Anisotropy on Seawater Intrusion

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## Abstract

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## 1. Introduction

## 2. The Mathematical Model and Boundary Conditions

## 3. The Semi-Analytical Solution

- The length of the toe (${L}_{toe}$): The dimensionless distance between the seaside boundary and the point where the 50% isochlor intersects the aquifer bottom.
- The average vertical width of the mixing zone (${W}_{MZ}$): Defined as the average of the vertical dimensionless distances between the 10% and 90% isochlors. The mixing zone is defined by the interval $0.3\times {L}_{toe}$ to $0.7\times {L}_{toe}$.
- The total dimensionless salt flux (${Q}_{s}$): Represents the advective, diffusive and dispersive salt flux that enters the domain from the seaside boundary, normalized by the freshwater inland flux.
- The depth of the zone of groundwater discharge to the sea (${d}_{disch}$): Equal to the distance from the aquifer top surface to the point separating the discharge zone and seawater inland flow zone at the sea boundary.

## 4. The New Technique for Solving the System of Equations in the Spectral Space

## 5. Verification of the Fourier Series Solution: Stability and Comparison against Numerical Solution

#### 5.1. The Pure Diffusive Henry Problem

#### 5.2. The Dispersive Henry Problem

^{-4}) [57]. The anisotropy ratio (${r}_{k}$) and the rate of stratification ($\mathsf{\Upsilon}$) are kept the same as for the pure diffusive cases. Non-dimensional and corresponding physical parameters are given in Table 1 and Table 2. For homogenous and heterogeneous cases, oscillation-free solutions have been obtained using 4725 ($Nm=15$, $Nn=90$, $Nr=20$ and $Ns=160$) and 6,405 ($Nm=15$, $Nn=90$, $Nr=20$ and $Ns=240$) Fourier modes, respectively. This confirms that only the concentration Fourier modes in the x-direction are mainly affected by heterogeneity. The SA and numerical isochlors are depicted in Figure 5. SWI metrics are given in Table 3. As for the homogenous case, Figure 5 and Table 3 highlight the excellent agreement between the analytical and numerical solutions. The same remark, as in the homogenous case, can be made regarding the agreement between SA and numerical values of ${Q}_{s}$. Table 3 provides quantitative indicators that can be useful for benchmarking DDF codes in realistic configurations of anisotropy and heterogeneity. First an observation from the comparison between the Figure 4 and Figure 5 indicates that the effect of heterogeneity on SWI is more important in the dispersive case. Figure 5 shows that the solution of the dispersive HP is sharper and more realistic than for the diffusive case. This sharpness explains why the dispersive test case requires more Fourier modes than the diffusive one.

## 6. Effects of Anisotropy and Heterogeneity on SWI

#### 6.1. Effect of Anisotropy on SWI in a Homogenous Aquifer

#### 6.2. Coupled Effect of Anisotropy and Stratified Heterogeneity on SWI

## 7. Conclusions

- It derives the first SA solution of SWI with the DDF model in an anisotropic and heterogeneous domain with velocity-dependent dispersion. The SA solution is useful for testing and validating DDF numerical models in realistic configurations of anisotropy and stratification. In this context, we derived, analytically (using the Fourier series), quantitative indicators (i.e., seawater intrusion metrics ${L}_{toe}$ ,${W}_{MZ}$, ${d}_{disch}$ and ${Q}_{s}$) that can be effectively used for code verification.
- From a numerical point of view, an efficient technique is presented for solving the HP in the spectral space. With this technique, we showed that the governing equations in the spectral space can be solved with only the concentration as a primary unknown. The spectral velocity field can be analytically expressed in terms of concentration. This technique improves the practicality of the Henry problem’s SA solution and renders it more suitable for further studies requiring repetitive evaluations as in inverse modeling or sensitivity analysis.
- The developed SA solution is used to investigate the effects of anisotropy and stratification on SWI. This is the first time that these effects have been investigated analytically with the DDF model. In previous works, analytical studies on this issue have been limited to the sharp interface model. While in most of the existing studies, the effects of anisotropy and heterogeneity have been mainly discussed in regard to the position of the saltwater wedge, we provided here a deeper understanding of these effects on several metrics characterizing SWI.
- Taking advantage of the SA solution, we explained the contradictory results in regard to the effect of anisotropy on the position of the saltwater wedge. We showed that at a constant gravity number, the decrease in the anisotropy ratio leads to landward migration of the saltwater wedge. Contradictions observed in the previous studies are related to the way in which the anisotropy ratio is changed (whether by varying horizontal or vertical hydraulic conductivity). The SA solution shows also that anisotropy leads to a wider mixing zone and intensifies the saltwater flux to the aquifer. It leads to a shallower zone of groundwater discharge to the sea.
- The combined effects of anisotropy and stratification on SWI have been investigated. We showed that the width of the mixing zone is slightly sensitive to the rate of stratification. This sensitivity is more significant in highly anisotropic aquifers. Complementary effects of anisotropy and heterogeneity are observed on the saltwater wedge and toe position as well as on the saltwater flux, while opposite effects are observed on the depth of the groundwater discharge zone.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

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**Figure 2.**Variation of the horizontal and vertical hydraulic conductivities with respect to depth for $\mathsf{\Upsilon}=1.5$ and $\overline{{K}_{z}}=8.213\times {10}^{-3}\text{}\mathrm{m}/\mathrm{s}$. Depth datum is at the aquifer bottom surface.

**Figure 3.**Non-physical oscillations related to the Gibbs phenomenon: the heterogeneous pure diffusive case evaluated with insufficient number of Fourier modes.

**Figure 4.**Semi-analytical and numerical isochlors (10%, 50% and 90%) for the pure diffusive cases: (

**a**) homogenous and (

**b**) heterogonous aquifer.

**Figure 5.**Semi-analytical and numerical isochlors (10%, 50%, 90%) for the dispersive cases: (

**a**) homogenous and (

**b**) heterogonous aquifer.

**Figure 6.**Effect of anisotropy on the isochlors’ positions (90%, 50%, 10%): Results obtained for the dispersive homogenous case (see Table 1 for all parameters, except ${r}_{K}$, which is given in the figure).

**Figure 7.**Effect of anisotropy ratio on the 50% isochlor’s position when ${r}_{K}$ is changed based on the vertical hydraulic conductivity. The case ${r}_{K}=0.66$ corresponds to the homogenous dispersive case (Table 2). The case ${r}_{K}=0.16$ is obtained with reduced ${K}_{z}$ (divided by 4.125).

**Figure 8.**Variation of the SWI metrics ((

**a**) ${L}_{toe}$, (

**b**) ${W}_{MZ}$, (

**c**) ${Q}_{s}$ and (

**d**) ${d}_{disch}$) versus ${r}_{k}$. Results obtained for the dispersive case (see Table 1 for all parameters, except ${r}_{K}$ and $\mathsf{\Upsilon}$, which are given in the figure).

**Figure 9.**Coupled influence of anisotropy and heterogeneity on the main isochlors (10%, 50%, 90%). Results obtained for a fixed large-scale gravity number $\overline{NG}=3.11$. All other parameters are similar to the dispersive case in Table 1.

**Figure 10.**Effect of the heterogeneity for a varying large-scale gravity number. The 50% isochlor in the case of the homogenous aquifer with $\overline{NG}=3.11$ (solid line), heterogeneous aquifer with $\overline{NG}=3.11$ (dashed line) and heterogeneous aquifer with $\overline{NG}=0.84$ (dash-dotted line). Heterogeneous cases are obtained with $\mathsf{\Upsilon}=1.5$.

**Table 1.**Non-dimensional parameters used in the semi-analytical solution for the verification test cases.

Dimensionless Parameters | Value | Cases |
---|---|---|

$\overline{NG}=\frac{\overline{{K}_{z}}d\Delta \rho}{{\rho}_{0}{q}_{d}}.$ | 3.11 | All cases |

${b}_{m}=\epsilon {D}_{m}/{q}_{d}$ | 0.1 5 × 10 ^{−4} | Diffusive cases Dispersive cases |

Non-dimensional longitudinal dispersion ${b}_{L}={\alpha}_{L}/d$ | 0 0.1 | Diffusive cases Dispersive cases |

Transverse to longitudinal dispersion coefficients ratio ${r}_{\alpha}={\alpha}_{T}/{\alpha}_{L}$ | 0 0.1 | Diffusive cases Dispersive cases |

${r}_{K}={K}_{z}/{K}_{x}$ | 0.66 | All cases |

The rate of heterogeneity $\mathsf{\Upsilon}$ | 0 1.5 | Homogenous cases Heterogeneous cases |

Parameters | Value | Cases |
---|---|---|

$\Delta \rho $ [kg/m^{3}] | 25 | All cases |

${\rho}_{0}$ [kg/m^{3}] | 1000 | All cases |

${q}_{d}$ [m^{2}/s] | 6.6 × 10^{−5} | All cases |

$d$ [m] | 1 | All cases |

$\ell $ [m] | 4 | All cases |

$\overline{{K}_{z}}$ [m/s] | 8.213 × 10^{−3} | All cases |

${r}_{k}$ [-] | 0.66 | All cases |

$\epsilon $ [-] | 0.35 | All cases |

${D}_{m}$ [m^{2}/s] | 53.88 × 10^{−6}3.300 × 10 ^{−8} | Diffusive cases Dispersive cases |

${\alpha}_{L}$ [m] | 0 0.1 | Diffusive cases Dispersive cases |

${\alpha}_{T}$ [m] | 0 0.01 | Diffusive cases Dispersive cases |

$\mathsf{\Upsilon}$ [-] | 0 1.5 | Homogenous cases Heterogeneous cases |

**Table 3.**Physical parameters used in the numerical model for the verification test cases (L

_{toe}: length of the toe, W

_{MZ}: average vertical width of the mixing zone, Q

_{S}: total dimensionless salt flux, d

_{disch}: depth of the zone of groundwater discharge to the sea).

Semi-Analytical Solution | Numerical Solution | |||||||
---|---|---|---|---|---|---|---|---|

Metrics | ${L}_{toe}$ | ${W}_{MZ}$ | ${Q}_{s}$ | ${d}_{disch}$ | ${L}_{toe}$ | ${W}_{MZ}$ | ${Q}_{s}$ | ${d}_{disch}$ |

Diffusive homogenous | 0.74 | 0.78 | 1.06 | 0.57 | 0.74 | 0.79 | 1.09 | 0.56 |

Diffusive heterogeneous | 0.95 | 0.83 | 1.09 | 0.35 | 0.95 | 0.84 | 1.01 | 0.36 |

Dispersive homogenous | 1.54 | 0.29 | 1.07 | 0.46 | 1.53 | 0.29 | 1.09 | 0.47 |

Dispersive heterogeneous | 2.30 | 0.59 | 1.09 | 0.31 | 2.29 | 0.59 | 1.12 | 0.31 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fahs, M.; Koohbor, B.; Belfort, B.; Ataie-Ashtiani, B.; Simmons, C.T.; Younes, A.; Ackerer, P.
A Generalized Semi-Analytical Solution for the Dispersive Henry Problem: Effect of Stratification and Anisotropy on Seawater Intrusion. *Water* **2018**, *10*, 230.
https://doi.org/10.3390/w10020230

**AMA Style**

Fahs M, Koohbor B, Belfort B, Ataie-Ashtiani B, Simmons CT, Younes A, Ackerer P.
A Generalized Semi-Analytical Solution for the Dispersive Henry Problem: Effect of Stratification and Anisotropy on Seawater Intrusion. *Water*. 2018; 10(2):230.
https://doi.org/10.3390/w10020230

**Chicago/Turabian Style**

Fahs, Marwan, Behshad Koohbor, Benjamin Belfort, Behzad Ataie-Ashtiani, Craig T. Simmons, Anis Younes, and Philippe Ackerer.
2018. "A Generalized Semi-Analytical Solution for the Dispersive Henry Problem: Effect of Stratification and Anisotropy on Seawater Intrusion" *Water* 10, no. 2: 230.
https://doi.org/10.3390/w10020230