# Uncertainty with Varying Subsurface Permeabilities Reduced Using Coupled Random Field and Extended Theory of Porous Media Contaminant Transport Models

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theory of Porous Media (TPM)

#### 2.2. Extended Theory of Porous Media (eTPM)

#### 2.2.1. Immiscible Constituents

#### 2.2.2. Miscible Concentration

#### 2.3. Field Equations and Constitutive Theory

#### 2.4. Evaluation of the Entropy Inequality

#### 2.5. Adaption of the Entropy Inequality

#### 2.6. The Chemical Potential as a Free Helmholz Energy Function for a Contaminant

#### 2.7. Stresses and Interaction Forces

#### 2.8. Numerical Treatment

- Balance of momentum for the mixture

- Balance of mass for the mixture

- Balance of mass for the contaminant

#### 2.9. Stabilized Boundary Conditions for Contaminant Transport

#### 2.10. Random Field Method

`R`with the package

`RandomFields`[42]. For the soil model with three layers, the Gaussian random field parameters are listed in Table 1.

#### 2.11. Physical Sandbox Experiment

#### 2.12. Field Scale Simulation

## 3. Results and Discussion

#### 3.1. Validation with the Physical Sandbox Experiment

#### 3.2. Transport of Contaminant in Groundwater through Three-Layer Heterogeneous Alluvial System

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BaRE | Bayesian recursive estimation |

CI | confidence interval |

eTPM | extended Theory of Porous Media |

GLUE | Generalized likelihood uncertainty estimation |

MCMC | Markov chain Monte Carlo |

MSE | mean square error |

RF | random fields |

REV | representative elementary volume |

TPM | Theory of Porous Media |

TDS | total dissolved solids |

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**Figure 4.**Density function of the Log-Gaussian distribution with $\widehat{\mu}=-17.26$ and $\widehat{\sigma}=0.38$. The probability (hatched area) of permeability values in the interval $\left[\mathrm{a},\mathrm{b}\right]=\left[{10}^{-8},{10}^{-7}\right]$ amounts to $0.99$.

**Figure 5.**Schematic representation of the physical tank test (

**a**), the 3 constructed layers and the sampling ports on tank (

**b**).

**Figure 6.**The comparison between the simulation (

**left**) and experiment (

**right**) for a three-layer physical sandbox. The color bar shows the permanganate concentration.

**Figure 7.**The comparison between measured and eTPM predicted permanganate (

**a**) concentration (

**b**) arrival time.

**Figure 8.**Permeability as realization of the three Log-Gaussian random fields for the three layers (here: a 50 meters section of the modelled soil). The colors represent the realized permeability values (cf. color legend).

**Figure 9.**The distribution of the contaminant concentration at different region of the aquifer at t = 128 days.

**Figure 10.**The distribution of the contaminant concentration in the area located between (85,21) and (90,25) at different times.

**Figure 11.**Histograms of (

**a**) the average contaminant concentration at $\mathrm{t}=140\phantom{\rule{4pt}{0ex}}\mathrm{days}$, (

**b**) contaminant arrival time for the region located between $(100,15)$ and $(150,100)$.

Soil Layer | Height | Value Interval $\left[\mathbf{a},\mathbf{b}\right]$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathbf{\Psi}$ |
---|---|---|---|---|---|

3 | 15 | $\left[{10}^{-7},{10}^{-6}\right]$ | $-14.97$ | $0.38$ | 8 |

2 | 20 | $\left[{10}^{-6},{10}^{-5}\right]$ | $-12.66$ | $0.38$ | 8 |

1 | 15 | $\left[{10}^{-8},{10}^{-7}\right]$ | $-17.27$ | $0.38$ | 8 |

Size (mm) | Hydraulic Conductivity (m/sec) | Porosity (%) | Hydraulic Gradient (-) | |
---|---|---|---|---|

Sand | 0.075–0.5 | $3.579\times {10}^{-6}$ | 21 | −0.0003 |

High permeability Sand | 0.5–2.36 | $2.754\times {10}^{-4}$ | 43 | −0.0005 |

Parameter | Simulation | Physical Sandbox |
---|---|---|

Number of layers | 3 | 3 |

Height of layers (m) | 50 | 50 |

Permeability (m/sec) | $3.3\times {10}^{-6}$–$2.6\times {10}^{-4}$ | $3.5\times {10}^{-6}$–$2.7\times {10}^{-4}$ |

Grain size distribution (mm) | - | 0.075–2.36 |

Porosity (%) | - | 21–43 |

Flow rate (lt/h) | 16 | 16 |

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**MDPI and ACS Style**

Seyedpour, S.M.; Henning, C.; Kirmizakis, P.; Herbrandt, S.; Ickstadt, K.; Doherty, R.; Ricken, T.
Uncertainty with Varying Subsurface Permeabilities Reduced Using Coupled Random Field and Extended Theory of Porous Media Contaminant Transport Models. *Water* **2023**, *15*, 159.
https://doi.org/10.3390/w15010159

**AMA Style**

Seyedpour SM, Henning C, Kirmizakis P, Herbrandt S, Ickstadt K, Doherty R, Ricken T.
Uncertainty with Varying Subsurface Permeabilities Reduced Using Coupled Random Field and Extended Theory of Porous Media Contaminant Transport Models. *Water*. 2023; 15(1):159.
https://doi.org/10.3390/w15010159

**Chicago/Turabian Style**

Seyedpour, S. M., C. Henning, P. Kirmizakis, S. Herbrandt, K. Ickstadt, R. Doherty, and T. Ricken.
2023. "Uncertainty with Varying Subsurface Permeabilities Reduced Using Coupled Random Field and Extended Theory of Porous Media Contaminant Transport Models" *Water* 15, no. 1: 159.
https://doi.org/10.3390/w15010159