# Multiscale Multiphysics Modeling of the Infiltration Process in the Permafrost

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Fine Grid Finite Element Approximation and Picard Iteration for Linearization

## 4. Generalized Multiscale Finite Element Method

## 5. Numerical Results Two-Dimensional Problem

- •
- Problem parameters $\sigma =2.0$, $\gamma =1.0$, $\beta =14.0$;
- •
- Volumetric heat capacity $c\rho $—thawed zone $2397.6\times {10}^{3}$ [J/m${}^{3}\xb7$K]; frozen zone $1886.4\times {10}^{3}$ [J/m${}^{3}\xb7$K];
- •
- Thermal conductivity $\alpha $—thawed zone $1.37$ [W/m·K], frozen zone $1.72$ [W/m·K];
- •
- Phase transition ${\rho}^{+}L$—75,330 $\times {10}^{3}$ [J/m].

## 6. Numerical Results Three-Dimensional Problem

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Domain $\mathsf{\Omega}$ with boundaries ${\mathsf{\Gamma}}_{in},\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Gamma}}_{st},\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Gamma}}_{s},\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Gamma}}_{b}.$

**Figure 2.**Illustration of Multiscale basis functions that are used to construct coarse grid approximation. Multiscale basis functions are constructed: based on the spectral characteristics of the local problems multiplied by partition of unity functions (the top is 2D and the bottom is 3D).

**Figure 3.**Computational domain and heterogeneous coefficient ${K}_{s}\left(x\right)$ (two-dimensional problem).

**Figure 4.**Numerical results for pressure that correspond to time step: (

**a**) $\tau =128$ (

**b**) $\tau =150$ (

**c**) $\tau =200$ (

**d**) $\tau =365$. This results are coarse grid solutions using 8 basis functions ($DO{F}_{c}=3968$).

**Figure 5.**Numerical results for temperature (

**a**) $\tau =150$ (

**b**) $\tau =200$ (

**c**) $\tau =320$ (

**d**) $\tau =365$. Where the white line is the isocline of zero for saturated soils and the black line is the isocline of zero for non-saturated soils. This results are coarse grid solution using 8 basis functions ($DO{F}_{c}=3968$).

**Figure 6.**Computational domain and heterogeneous coefficient ${K}_{s}\left(x\right)$ (three-dimensional problem).

**Figure 7.**Numerical results for pressure that corresponds to time step: (

**a**) $\tau =128$ (

**b**) $\tau =150$ (

**c**) $\tau =200$ (

**d**) $\tau =365$. This results are coarse grid solution using 8 basis functions ($DO{F}_{c}=$ 31,752).

**Figure 8.**Numerical results for temperature (

**a**) $\tau =150$ (

**b**) $\tau =200$ (

**c**) $\tau =320$ (

**d**) $\tau =365$, where white line is isocline of zero for saturated soils. This results are a coarse grid solution using 8 basis functions ($DO{F}_{c}=$ 31,752).

Month | Temperature ${}^{\circ}$C |
---|---|

January | −36.0 |

February | −31.9 |

March | −17.7 |

April | −2.8 |

May | 7.7 |

June | 16.7 |

July | 19.8 |

August | 17.3 |

September | 6.6 |

October | −4.7 |

November | −25.2 |

December | −36.4 |

**Table 2.**Relative ${L}_{2}$ and energy errors (%) for different number of multiscale basis functions. ($DO{F}_{f}=$ 29,041).

M | DOF | ${\left|\right|\mathit{e}\left|\right|}_{{\mathit{L}}^{2}}$ | ${\left|\right|\mathit{e}\left|\right|}_{\mathit{a}}$ | M | DOF | ${\left|\right|\mathit{e}\left|\right|}_{{\mathit{L}}^{2}}$ | ${\left|\right|\mathit{e}\left|\right|}_{\mathit{a}}$ |
---|---|---|---|---|---|---|---|

$20\times 5$ coarse grid | |||||||

$t=150$ | |||||||

Temperature | Pressure | ||||||

1 | 496 | 3.97 | 21.96 | 1 | 496 | 2.28 | 29.78 |

2 | 992 | 2.06 | 15.29 | 2 | 992 | 1.14 | 21.3 |

4 | 1984 | 0.88 | 9.43 | 4 | 1984 | 0.65 | 16.05 |

8 | 3968 | 0.33 | 4.97 | 8 | 3968 | 0.28 | 10.02 |

16 | 7936 | 0.07 | 1.91 | 16 | 7936 | 0.09 | 4.89 |

$t=200$ | |||||||

Temperature | Pressure | ||||||

1 | 496 | 2.77 | 14.78 | 1 | 496 | 2.19 | 29.06 |

2 | 992 | 1.3 | 10.9 | 2 | 992 | 0.82 | 21.3 |

4 | 1984 | 0.62 | 7.35 | 4 | 1984 | 0.46 | 16.53 |

8 | 3968 | 0.23 | 4.26 | 8 | 3968 | 0.16 | 8.56 |

16 | 7936 | 0.03 | 1.18 | 16 | 7936 | 0.04 | 3.83 |

**Table 3.**Relative ${L}_{2}$ and energy errors (%) for different number of multiscale basis functions. ($DO{F}_{f}=$ 522,774).

M | DOF | ${\left|\right|\mathit{e}\left|\right|}_{{\mathit{L}}^{2}}$ | ${\left|\right|\mathit{e}\left|\right|}_{\mathit{a}}$ | M | DOF | ${\left|\right|\mathit{e}\left|\right|}_{{\mathit{L}}^{2}}$ | ${\left|\right|\mathit{e}\left|\right|}_{\mathit{a}}$ |
---|---|---|---|---|---|---|---|

$20\times 20\times 8$ coarse grid | |||||||

$t=150$ | |||||||

Temperature | Pressure | ||||||

1 | 3969 | 3.27 | 16.8 | 1 | 3969 | 9.17 | 36.26 |

2 | 7938 | 2.67 | 14.75 | 2 | 7938 | 4.34 | 24.83 |

4 | 15,876 | 0.97 | 8.87 | 4 | 15,876 | 2.31 | 19.6 |

8 | 31,752 | 0.6 | 6.96 | 8 | 31,752 | 1.22 | 16.17 |

16 | 63,504 | 0.3 | 4.27 | 16 | 63,504 | 0.67 | 12.92 |

$t=200$ | |||||||

Temperature | Pressure | ||||||

1 | 3969 | 2.71 | 13.87 | 1 | 3969 | 8.57 | 31.61 |

2 | 7938 | 1.5 | 11.1 | 2 | 7938 | 3.63 | 20.53 |

4 | 15,876 | 0.66 | 6.35 | 4 | 15,876 | 1.89 | 15.09 |

8 | 31,752 | 0.43 | 5.53 | 8 | 31,752 | 1.01 | 12.26 |

16 | 63,504 | 0.18 | 3.12 | 16 | 63,504 | 0.52 | 9.04 |

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**MDPI and ACS Style**

Stepanov, S.; Nikiforov, D.; Grigorev, A.
Multiscale Multiphysics Modeling of the Infiltration Process in the Permafrost. *Mathematics* **2021**, *9*, 2545.
https://doi.org/10.3390/math9202545

**AMA Style**

Stepanov S, Nikiforov D, Grigorev A.
Multiscale Multiphysics Modeling of the Infiltration Process in the Permafrost. *Mathematics*. 2021; 9(20):2545.
https://doi.org/10.3390/math9202545

**Chicago/Turabian Style**

Stepanov, Sergei, Djulustan Nikiforov, and Aleksandr Grigorev.
2021. "Multiscale Multiphysics Modeling of the Infiltration Process in the Permafrost" *Mathematics* 9, no. 20: 2545.
https://doi.org/10.3390/math9202545