# Transient Evolution of Inland Freshwater Lenses: Comparison of Numerical and Physical Experiments

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}= 0.89, 0.94, 0.85). Predicted lens degradation times corresponded to the observed lenses, which demonstrated the utility of numerical models and physical models to assess IFL geometry and position. Improved understanding of IFL dynamics provides water-resource exploration and development opportunities in drylands throughout the Arabian Peninsula and elsewhere with similar environmental settings.

## 1. Introduction

## 2. Materials and Methods

^{2}= 0.34, p = 0.1304). The study concluded that the differences in lens geometry between the physically modeled lenses and the analytical solution were due to the transient nature of IFLs, subsequently motivating the need for numerical studies that could address IFL changes in geometry through time.

^{−3}), K is hydraulic conductivity tensor (LT

^{−1}), h is the freshwater head (L), ρ

_{f}is the freshwater density (ML

^{−3}), z is elevation, S

_{s}is the freshwater specific storage (L

^{−1}), θ is porosity (1), and q

_{s}is a source or sink (T

^{−1}) of fluid with density ρ

_{s}.

^{−3}), D is the solute dispersion tensor, v is the advective velocity (LT

^{−1}), and C

_{s}is the source or sink concentration (ML

^{−3}). The linking of Equations (1) and (2) requires a function to relate concentration and fluid density.

#### 2.1. Numerical Model Design and Parameters

_{x}, K

_{y}, K

_{z}), were used from measurements taken during the previously conducted physical laboratory model study with a constant head permeameter during the laboratory experiments from sediments comprised of medium to fine sand (d

_{50}= 0.55 mm) to represent the highly permeable gravel sediments in the upper layer of the Kuwait Group Aquifer [36]. The porosity ($\theta $) was calculated during the previous study using the sum of solid and pore volumes. Additional representative properties of variable-density values through porous media were selected after Konikow, et al. [40] and others, as shown in Table 1.

#### 2.2. Boundary and Initial Conditions

_{1}) with a constant concentration of 35,700 mg/L for the full duration of the model (200 h). The right side of the domain included a general-head boundary with a general-head elevation at 0.405 m (h

_{2}) and boundary-head elevation at 0.40 m for a distance of 2 m, which generated a slope of 0.25% to represent the gentle gradient. The top layer of the domain included a recharge boundary with a width (w

_{r}) of 0.42 m. The initial condition of the freshwater head (h

_{f}) was calculated to 0.4075 m based on the change in water table elevation between the left and right boundaries of the domain. The initial saltwater concentration (C

_{s}) of 35,700 mg/L was selected for the brackish to saline water from the Raudhatain Depression (7000–50,000 mg/L) and used with the physical model simulations. Evaporation, similar to the previously conducted physical laboratory simulations, was considered negligible. A summary of the boundary conditions is shown in Table 2.

#### 2.3. Observation Wells and Geometry Data

#### 2.4. Numerical Settings

^{−5}m because of the small range of measurement values from the physical laboratory model. The flow solver used the incomplete Cholesky preconditioning method.

^{−8}. The lengths of transport time steps were calculated using a Courant number of 0.75. The time steps for the flow calculations for stress periods 1, 2, and 3 are 10, 50, and 50, respectively. The maximum number of transport steps was set at 3000 with a maximum step size set at 0.125 h. Simulation results were saved at every time step for both flow and transport calculations to use for analyses.

#### 2.5. Simulations

_{q}) were recorded and subtracted from the time observed for each simulated lens to reach a maximum thickness (T

_{max}). The initial recharge rates (R

_{i}) used in physical experiments were divided by the calculated time differences to equal the adjusted recharge rates (R

_{a}). The beginning of the second stress period was set to the observed time of initial flux, and the end of the second stress period was set at the time of the observed maximum thickness to match the maximum thickness times of the physical model to the numerical model. Three stress periods were assigned to each simulation to apply the recharge and represent (1) a period of no recharge, (2) a period of freshwater recharge applied across the recharge boundary, and (3) a period of no recharge for a total duration of 200 h. The duration of each stress period was set accordingly (Table 3). Freshwater was infused with uranine, which is a tracer dye that allowed for the measurements of IFL thicknesses and lengths for ten hours for each simulation before the tank wall interfered with the lens geometry, which was compared to the numerical model using concentration data exported from the software. Statistical model evaluation techniques were used to determine the accuracy of the simulated data [41].

## 3. Results

#### 3.1. Model Calibration and Results

#### 3.2. Water Table Elevation Results

^{2}= 0.89). The Nash–Sutcliffe efficiency (NSE) indicates a good fit between the observed and simulated data (NSE = 0.84). The root mean square error (RMSE) indicates a small error value appropriate for model evaluation (RMSE = 0.007). The percent bias (PBIAS) value suggests nearly zero model overestimation bias of water table elevation (PBIAS = −0.7).

#### 3.3. Lens Geometry Results

^{2}= 0.94) and (R

^{2}= 0.85). The NSE indicated a good to moderate fit between the observed and simulated data (NSE = 0.65) and (NSE = 0.82). The RMSE indicates an error value appropriate for model evaluation (RMSE = 0.03) and (RMSE = 0.06). The PBIAS value suggests a slight to moderate model overestimation bias of IFL thickness (PBIAS = −16.3) and (PBIAS = −0.4). For lengths, Simulations 2 and 3 correlates more linearly; however, Simulation 1 deviates from the linear trend likely from a decrease in velocity during the physical model simulation from hydraulic conductivity variations due to the packing of the sand (Figure 8). A summary of these results is presented in Table 5.

#### 3.4. Velocity Results

_{x}) from the center layer and column to the last column of the downgradient or right side of the domain (x = 1.0–2.0 m, z = 0.25 m) for each simulation before, during, and after the recharge stress period as shown in Table 6.

#### 3.5. Degradation Results

## 4. Discussion

^{−7}–2.85 × 10

^{−6}m/s. The numerical model showed an increase in average velocity during the recharge stress period by an order of magnitude (10

^{−5}m/s), which aligns with the range of recharge rates used in the physical and numerical model simulations 3.33 × 10

^{−6}–6.67 × 10

^{−6}m/s. Based on these values, freshwater recharge that entered the center of the Raudhatain depression is estimated to travel 4000 meters towards the periphery between 20 and 200 years, depending on the hydraulic conductivity. This estimate aligns with those by Yihdego and Al-Weshah [43] and Parsons Engineering and Construction Corporation [44] who reported an inferred flow velocity between 11 and 245 m/year based on

^{14}C and

^{3}H age dating, as well as an estimated groundwater velocity of 20–90 m/year based on a hydraulic conductivity range of 40–80 m/day. Estimates such as these inform geochemical studies that aim to approximate IFL water age, such as the study by Kuldzhayev [45], which determined the onset of freshwater accumulation of an IFL in the Karakum desert to be 3400 years. Investigations into the formation, geometry, and extent of freshwater lenses over long temporal scales are needed for inland and coastal environments to quantify the supply of drinking water and the effects of external changes such as precipitation and sea-level rise [13,30], but also geomorphological [46] and anthropogenic impacts [47,48].

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Conceptual diagram of an inland freshwater lens (IFL). Modified from Rotz and Milewski [1].

**Figure 2.**Map of Kuwait showing the location of the Raudhatain (top) and Umm Al-Aish (bottom) IFLs highlighted in red.

**Figure 3.**Diagram of observational data collected including water table elevation and IFL geometry (i.e., thickness, length) from previously conducted physical laboratory simulation.

**Figure 4.**Conceptual diagram of the numerical model with domain dimensions, boundary conditions, and observation well locations.

**Figure 6.**Numerical model Simulation 2 of IFL development and transient evolution. The dimensions of each cross-section match the numerical model domain of 2.0 m (L) × 0.5 m (H) × 0.10 m (W).

**Figure 10.**Observed vs. simulated thicknesses with best fit exponential regression lines to represent IFL degradation for Simulation 1 (red circles), Simulation 2 (green triangles), and Simulation 3 (purple squares).

**Figure 11.**A second recharge pulse is applied to Simulation 2 to show the potential applications for IFL models that consider IFL transient evolution.

Input Parameter | Value | Description |
---|---|---|

K_{x}, K_{y}, K_{z} | 0.0015 m/s | Hydraulic conductivity |

θ | 0.39 | Porosity |

S_{s} | 1 × 10^{−5} | Specific storage |

S_{y} | 0.34 | Specific yield |

D_{m} | 1 × 10^{−10} m^{2}/s | Molecular diffusion coefficient |

Δρ/ΔC | 0.7143 | Slope of fluid density to solute concentration |

δ_{L} | 1 × 10^{−5} m | Longitudinal dispersivity |

δv/Δl | 0.02 | Vertical/longitudinal dispersivity |

Boundary Condition | Position (x,y) | Values |
---|---|---|

Constant head | 0, 0.41 | 35,700 mg/L |

General head | 2, 0.405 | calculated |

Recharge | 0.79, 1.21 | 0 mg/L |

Evaporation | Entire domain | 0 m/s |

**Table 3.**The initial recharge rate (R

_{i}), time of recharge flux (T

_{q}), time of observed maximum thickness (T

_{max}), adjusted recharge (R

_{a}) rates, and stress period times for the numerical model simulations.

Simulation | R_{i} (m/s) | T_{q} (s) | T_{max} (s) | R_{a} (m/s) | Stress Periods Starts (1, 2, 3) (s) |
---|---|---|---|---|---|

1 | 4.72 × 10^{−5} | 1620 | 6480 | 3.33 × 10^{−5} | 1620, 6480, 720,000 |

2 | 6.48 × 10^{−5} | 1296 | 6120 | 4.72 × 10^{−5} | 1296, 6120, 720,000 |

3 | 7.86 × 10^{−5} | 1008 | 5400 | 6.67 × 10^{−5} | 1008, 5400, 720,000 |

**Table 4.**The objective function (Φ), hydraulic conductivity estimates (K

_{x}, K

_{z}), and confidence intervals for the numerical simulations using the parameter estimation tool (PEST) tool.

Run | Φ (m) | K_{x} (m/s) | K_{z} (m/s) | Confidence Interval for K_{x} (m/s) | Confidence Interval for K_{z} (m/s) |
---|---|---|---|---|---|

1 | 4.8 × 10^{−4} | 7.6 × 10^{−4} | 7.0 × 10^{−4} | 7.6 × 10^{−4}–1.2 × 10^{−3} | 2.5 × 10^{−5}–1.9 × 10^{−2} |

2 | 5.5 × 10^{−3} | 3.6 × 10^{−4} | 4.3 × 10^{−4} | 2.5 × 10^{−4}–5.3 × 10^{−4} | 2.1 × 10^{−5}–8.5 × 10^{−3} |

3 | 1.5 × 10^{−3} | 4.3 × 10^{−4} | 9.0 × 10^{−4} | 5.3 × 10^{−4}–1.0 × 10^{−3} | 5.8 × 10^{−5}–4.7 × 10^{−3} |

**Table 5.**Summary of model evaluation statistics for IFL water table elevation, thickness, and length for Simulations 1, 2, and 3. The evaluation statistics are defined as a measure of fit (R

^{2}), the Nash Sutcliffe Efficiency (NSE), root mean square error (RMSE), and percent bias (PBIAS).

Run | Slope-Intercept Form | R^{2} | NSE | RMSE | PBIAS |
---|---|---|---|---|---|

Water table elevation | y = 1.0477x – 0.0244 | 0.89 | 0.84 | 0.007 | −0.7 |

Thickness | y = 1.0856x + 0.0133 | 0.94 | 0.65 | 0.03 | −16.3 |

Length | y = 1.0134x – 0.0094 | 0.85 | 0.82 | 0.06 | −0.4 |

**Table 6.**Average groundwater velocity (v

_{x}) measurements of the numerical model simulations before, during, and after the recharge stress event.

Run | v_{x} Before Recharge Stress Period (m/s) | v_{x} During Recharge Stress Period (m/s) | v_{x} After Recharge Stress Period (m/s) |
---|---|---|---|

1 | 6.88 × 10^{−6} | 2.40 × 10^{−5} | 7.04 × 10^{−6} |

2 | 3.86 × 10^{−6} | 6.99 × 10^{−5} | 3.85 × 10^{−6} |

3 | 5.64 × 10^{−6} | 6.48 × 10^{−5} | 4.84 × 10^{−6} |

**Table 7.**Lens degradation results between the physical model and numerical model simulations. Units have been converted from seconds to hours.

Run | Physical Model Regression | Degradation Time (h) | Numerical Model Regression | Degradation Time (h) |
---|---|---|---|---|

1 | y = 1.699 × 10^{−1} e^{−0.069x} | 41 | y = 2.023 × 10^{−1} e^{−0.072x} | 42 |

2 | y = 2.697 × 10^{−1} e^{−0.047x} | 70 | y = 3.017 × 10^{−1} e^{−0.05x} | 68 |

3 | y = 2.597 × 10^{−1} e^{−0.1x} | 33 | y = 3.185 × 10^{−1} e^{−0.087x} | 40 |

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**MDPI and ACS Style**

Rotz, R.; Milewski, A.; Rasmussen, T.C.
Transient Evolution of Inland Freshwater Lenses: Comparison of Numerical and Physical Experiments. *Water* **2020**, *12*, 1154.
https://doi.org/10.3390/w12041154

**AMA Style**

Rotz R, Milewski A, Rasmussen TC.
Transient Evolution of Inland Freshwater Lenses: Comparison of Numerical and Physical Experiments. *Water*. 2020; 12(4):1154.
https://doi.org/10.3390/w12041154

**Chicago/Turabian Style**

Rotz, Rachel, Adam Milewski, and Todd C Rasmussen.
2020. "Transient Evolution of Inland Freshwater Lenses: Comparison of Numerical and Physical Experiments" *Water* 12, no. 4: 1154.
https://doi.org/10.3390/w12041154