# Ensemble Kalman Filter Assimilation of ERT Data for Numerical Modeling of Seawater Intrusion in a Laboratory Experiment

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## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Saltwater Intrusion Experiment

#### 2.2. Data Assimilation Approaches

#### 2.3. Groundwater Flow and Transport Modeling

#### 2.4. Electrical Modeling

#### 2.4.1. Forward Modeling

#### 2.4.2. Inverse Modeling

#### 2.5. Ensemble Kalman Filter with Nonlinear Observations and Dual-Step Update

## 3. Modeling and Data Assimilation Setup

#### 3.1. Fine- and Coarse-Resolution Groundwater Model

#### 3.2. Setup and Calibration of the Electrical Model

#### 3.3. Setup of the Data Assimilation Scenarios

#### 3.3.1. Joint Assimilation Method

#### 3.3.2. Sequential Assimilation Method

#### 3.3.3. Evaluation Metrics

## 4. Results and Discussion

#### 4.1. Data Assimilation Performance for System State

#### 4.1.1. Unbiased Initial Ensemble (Scenario 1): Joint Assimilation

#### 4.1.2. Unbiased Initial Ensemble (Scenario 1): Sequential Assimilation

#### 4.2. Parameter Estimation: Joint Versus Sequential Assimilation

#### 4.2.1. Unbiased Initial Ensemble: Scenario 1

#### 4.2.2. Biased Initial Ensemble (Scenario 2)

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**(

**a**–

**c**) Simulated salt concentrations and (

**d**–

**f**) observed saltwater wedge at (

**a**,

**d**) 14 h, (

**b**,

**e**) 18 h, and (

**c**,

**f**) 22 h from the beginning of the experiment. The colorbar indicates simulated concentrations in kg of solute per kg of solution.

**Figure 4.**Setup of the electrical model for the saltwater intrusion experiment, including very large electrical resistivity padding to reproduce the geometry of the sandbox and tanks.

**Figure 5.**ERT pseudo-sections 14 h after the beginning of the experiment for the (

**a**–

**c**) reverse and (

**d**–

**f**) forward pole–dipole configurations: (

**a**,

**d**) apparent resistivities modeled with the fine groundwater grid; (

**b**,

**e**) apparent resistivities modeled with the coarse groundwater grid; (

**c**,

**f**) observed apparent resistivities. Colorbars indicate resistivity in $\mathrm{\Omega}\xb7$m.

**Figure 6.**Observed versus modeled resistances 14 h after the beginning of the experiment for a pole–dipole separation distance of (

**a**) 0.81 m, (

**b**) 1.08 m, and (

**c**) 1.50 m.

**Figure 7.**Inverted electrical resistivity distributions at (

**a**) 14 h, (

**b**) 18 h, and (

**c**) 22 h from the beginning of the experiment. Colorbars indicate bulk electrical resistivity in $\mathrm{\Omega}\xb7$m.

**Figure 8.**Scenario 1: initial ensemble of (

**a**) $lo{g}_{10}\kappa $ and (

**b**) $lo{g}_{10}{\alpha}_{L}$. (

**c**) Cross-plot of the prior joint distribution.

**Figure 9.**Scenario 2: initial ensemble of (

**a**) $lo{g}_{10}\kappa $ and (

**b**) $lo{g}_{10}{\alpha}_{L}$. (

**c**) Cross-plot of the prior joint distribution.

**Figure 10.**Box plots of the proportionality factor ensembles: (

**a**) before assimilation of measurements with 0.81 m electrode spacing; (

**b**) before assimilation of measurements with 1.08 m electrode spacing; (

**c**) before assimilation of measurements with 1.50 m electrode spacing; (

**d**) after assimilation of measurements with 0.81 m electrode spacing; (

**e**) after assimilation of measurements with 1.08 m electrode spacing; (

**f**) after assimilation of measurements with 1.50 m electrode spacing.

**Figure 11.**Evaluation metrics for concentration estimations by assimilating electrical resistance measurements: (

**a**) average absolute bias and (

**b**) ensemble spread. The first, second, and third assimilation substeps correspond to assimilations of data collected with 0.81 m, 1.08 m, and 1.50 m electrode spacing, respectively.

**Figure 12.**Concentration distributions at the (

**a**,

**c**,

**e**,

**g**,

**i**) first and (

**b**,

**d**,

**f**,

**h**,

**j**) last assimilation steps: (

**a**,

**b**) reference values; (

**c**,

**d**) before assimilation; (

**e**,

**f**) after assimilation of measurements with 0.81 m electrode spacing; (

**g**,

**h**) after assimilation of measurements with 1.08 m electrode spacing; (

**i**,

**j**) after assimilation of measurements with 1.50 m electrode spacing. The colorbar indicates concentration in kg of solute per kg of solution.

**Figure 13.**Additive factor box plots for the sequential approach in Scenario 1 (

**a**) before assimilation and (

**b**) after assimilation of inverted resistivities.

**Figure 14.**Evaluation metrics for concentration estimations by assimilating inverted electrical resistivity data: (

**a**) average absolute bias and (

**b**) ensemble spread.

**Figure 15.**Reference concentration distributions at times (

**a**) 14 h, (

**b**) 16 h, and (

**c**) 22 h, corresponding to the first, second, and last assimilation steps, respectively. Concentration distributions in Scenario 1 with assimilation of inverted resistivities: (

**d**) before and (

**g**) after the first assimilation step; (

**e**) before and (

**h**) after the second assimilation step; (

**f**) before and (

**i**) after the last assimilation step.

**Figure 16.**Scenario 1 (unbiased prior permeability): final distribution after last assimilation step of (

**a**) permeability and (

**b**) longitudinal dispersivity with the (

**a**,

**b**) joint approach and (

**d**,

**e**) sequential approach. Panels (

**c**,

**f**) show the corresponding scatter plots.

**Figure 17.**Scenario 2 (biased prior permeability): final distribution after last assimilation step of (

**a**) permeability and (

**b**) longitudinal dispersivity with the (

**a**,

**b**) joint approach and (

**d**,

**e**) sequential approach. Panels (

**c**,

**f**) show the corresponding scatter plots.

**Figure 18.**(

**a**) Predicted permeability ensemble, (

**b**) reference concentration distribution, (

**c**) predicted concentration distribution, and (

**d**) updated concentration distribution at the second assimilation step in Scenario 2 for the sequential approach.

**Figure 19.**Evaluation metrics for the estimation of the permeability: (

**a**) average absolute bias and (

**b**) ensemble spread.

**Figure 20.**Evaluation metrics for the estimation of the dispersivity: (

**a**) average absolute bias and (

**b**) ensemble spread.

Update Step | X | HX | D |
---|---|---|---|

1 | Predicted concentrations | Predicted resistances (or electrical resistivities) | Perturbed observed resistances (or inverted electrical resistivities) |

2 | Soil hydraulic parameters ($\kappa $ and ${\alpha}_{L}$) | Predicted concentrations | Updated concentrations |

Granular Material Parameters | Value |

Porosity (n) | 0.367 |

Bulk porous matrix compressibility ($\alpha $) | $1\times {10}^{-8}$ [kg/(m · s${}^{2}$)]${}^{-1}$ |

Permeability ($\kappa $) | $1.30\times {10}^{-10}$ m${}^{2}$ |

Longitudinal dispersivity (${\alpha}_{L}$) | 1$\times {10}^{-3}$ m |

Transversal dispersivity (${\alpha}_{T}$) | 1$\times {10}^{-4}$ m |

Fluid Parameters | Value |

Fluid viscosity ($\mu $) | 0.001 kg/(m · s) |

Freshwater salt concentration (${C}_{0}$) | 0.0 kg/kg |

Seawater salt concentration (${C}_{s}$) | 0.0446 kg/kg |

Freshwater density (${\rho}_{0}$) | 1000 kg/m${}^{3}$ |

Linear coefficient of density to concentration ($\partial \rho /\partial C$) | 722.15 (kg · kg)/(kg · m${}^{3}$) |

Fluid compressibility ($\beta $) | $4.47\times {10}^{-10}$ [kg/(m · s${}^{2}$)]${}^{-1}$ |

Molecular diffusivity (${D}_{m}$) | $1\times {10}^{-8}$ m${}^{2}$/s |

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

Cell size in sand tank (length × height) | 0.03 m × 0.03 m |

Cell size in saltwater tank (length × height) | 0.06 m × 0.03 m |

Cell size in freshwater tank (length × height) | 0.192 m × 0.03 m |

Freshwater resistivity | 20.92 $\mathrm{\Omega}\xb7$m |

Saltwater resistivity | 0.1486 $\mathrm{\Omega}\xb7$m |

Formation factor (Humble formula) | 5.33 |

Minimum dipole spacing | 0.03 m |

Maximum dipole spacing | 0.18 m |

Spacing factor between pole and dipole | 1 to 12 |

Beginning of electrode line (from upstream boundary) | 2.83 m |

End of electrode line (from upstream boundary) | 4.96 m |

Infinity pole (from upstream boundary) | 0.06 m |

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## Share and Cite

**MDPI and ACS Style**

Bouzaglou, V.; Crestani, E.; Salandin, P.; Gloaguen, E.; Camporese, M.
Ensemble Kalman Filter Assimilation of ERT Data for Numerical Modeling of Seawater Intrusion in a Laboratory Experiment. *Water* **2018**, *10*, 397.
https://doi.org/10.3390/w10040397

**AMA Style**

Bouzaglou V, Crestani E, Salandin P, Gloaguen E, Camporese M.
Ensemble Kalman Filter Assimilation of ERT Data for Numerical Modeling of Seawater Intrusion in a Laboratory Experiment. *Water*. 2018; 10(4):397.
https://doi.org/10.3390/w10040397

**Chicago/Turabian Style**

Bouzaglou, Véronique, Elena Crestani, Paolo Salandin, Erwan Gloaguen, and Matteo Camporese.
2018. "Ensemble Kalman Filter Assimilation of ERT Data for Numerical Modeling of Seawater Intrusion in a Laboratory Experiment" *Water* 10, no. 4: 397.
https://doi.org/10.3390/w10040397