# 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

- Difonzo, F.V.; Masciopinto, C.; Vurro, M.; Berardi, M. Shooting the Numerical Solution of Moisture Flow Equation with Root Water Uptake Models: A Python Tool. Water Resour. Manag.
**2021**, 35, 1–15. [Google Scholar] [CrossRef] - Broadbridge, P.; Daly, E.; Goard, J. Exact solutions of the Richards equation with nonlinear plant-root extraction. Water Resour. Res.
**2017**, 53, 9679–9691. [Google Scholar] [CrossRef] - Albrieu, J.L.B.; Reginato, J.C.; Tarzia, D.A. Modeling water uptake by a root system growing in a fixed soil volume. Appl. Math. Model.
**2015**, 39, 3434–3447. [Google Scholar] [CrossRef] - Bai, M.; Roegiers, J.-C. Fluid flow and heat flow in deformable fractured porous media. Int. J. Eng. Sci.
**1994**, 32, 1615–1633. [Google Scholar] [CrossRef] - Masters, I.; Pao, W.K.S.; Lewis, R.W. Coupling temperature to a double—Porosity model of deformable porous media. Int. J. Numer. Methods Eng.
**2000**, 49, 421–438. [Google Scholar] [CrossRef] - Stepanov, S.; Grigoriev, A.; Afanasyeva, N. Simulation of the process of infiltration into fractured porous soil in permafrost. Math. Notes NEFU
**2020**, 27, 105–117. [Google Scholar] - Helmig, R. Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems; Springer: Berlin/Heidelberg, Germany, 1997. [Google Scholar]
- Celia, M.A.; Binning, P. A mass conservative numerical solution for twophase flow in porous media with application to unsaturated flow. Water Resour. Res.
**1992**, 28, 2819–2828. [Google Scholar] [CrossRef] - Rathfelder, L.M. Abriola Mass conservative numerical solutions of the head-based Richards equation. Water Resour. Res.
**1994**, 30, 2579–2586. [Google Scholar] [CrossRef] - Ross, P.J. Efficient numerical methods for infiltration using Richards equation. Water Resour. Res.
**1990**, 26, 279–290. [Google Scholar] [CrossRef] - Berardi, M.; Difonzo, F.; Vurro, M.; Lopez, L. The 1D Richards’ equation in two layered soils: A Filippov approach to treat discontinuities. Adv. Water Resour.
**2018**, 115, 264–272. [Google Scholar] [CrossRef] - Berardi, M.; Difonzo, F.; Lopez, L. A mixed MoL–TMoL for the numerical solution of the 2D Richards’ equation in layered soils. Comput. Math. Appl.
**2020**, 79, 1990–2001. [Google Scholar] [CrossRef] - Berardi, M.; Difonzo, F.V. Strong solutions for Richards’ equation with Cauchy conditions and constant pressure gradient. Environ. Fluid Mech.
**2020**, 20, 165–174. [Google Scholar] [CrossRef] - Li, H.; Farthing, M.W.; Miller, C.T. Adaptive local discontinuous Galerkin approximation to Richards’ equation. Adv. Water Resour.
**2007**, 30, 1883–1901. [Google Scholar] [CrossRef] - Kumar, K.; List, F.; Pop, I.S.; Radu, F.A. Formal upscaling and numerical validation of unsaturated flow models in fractured porous media. J. Comput. Phys.
**2020**, 407, 109138. [Google Scholar] [CrossRef] - Vasil’ev, V.; Vasilyeva, M. An Accurate Approximation of the Two-Phase Stefan Problem with Coefficient Smoothing. Mathematics
**2020**, 8, 1924. [Google Scholar] [CrossRef] - Tubini, N.; Gruber, S.; Rigon, R. A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation. Cryosphere
**2021**, 15, 2541–2568. [Google Scholar] [CrossRef] - Efendiev, Y.; Hou, T.Y. Multiscale Finite Element Methods: Theory and Applications; Springer Science & Business Media: New York, NY, USA, 2009; Volume 4. [Google Scholar]
- Efendiev, Y.; Lee, S.; Li, G.; Yao, J.; Zhang, N. Hierarchical multiscale modeling for flows in fractured media using generalized multiscale finite element method. GEM-Int. J. Geomathematics
**2015**, 6, 141–162. [Google Scholar] [CrossRef] [Green Version] - Spiridonov, D.; Vasilyeva, M. Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Heterogeneous Medium. In Proceedings of the International Conference on Finite Difference Methods, Lozenetz, Bulgaria, 11–16 June 2018; Springer: Cham, Switzerland, 2018; pp. 517–524. [Google Scholar]
- Efendiev, Y.; Galvis, J.; Hou, T.Y. Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys.
**2013**, 251, 116–135. [Google Scholar] [CrossRef] [Green Version] - Spiridonov, D.; Vasilyeva, M.; Chung, E.T.; Efendiev, Y.; Jana, R. Multiscale Model Reduction of the Unsaturated Flow Problem in Heterogeneous Porous Media with Rough Surface Topography. Mathematics
**2020**, 8, 904. [Google Scholar] [CrossRef] - Stepanov, S.; Grigorev, A.; Vasilyeva, M.; Nikiforov, D.; Spiridonov, D. Multiscale model reduction of fluid flow based on the dual porosity model. J. Phys. Conf. Ser.
**2019**, 1158, 042025. [Google Scholar] [CrossRef] - Vasilyeva, M.; Vasil’ev, V.; Stepanov, S. Generalized multiscale discontinuous Galerkin method for solving the heat problem with phase change. J. Comput. Appl. Math.
**2018**, 340, 645–652. [Google Scholar] - Vasilyeva, M.; Stepanov, S.; Spiridonov, D.; Vasil’ev, V.; Finite, M. Element Method for heat transfer problem during artificial ground freezing. J. Comput. Appl. Math.
**2020**, 371, 112605. [Google Scholar] [CrossRef] - Weinan, E.; Engquist, B.; Huang, Z. Heterogeneous multiscale method: A general methodology for multiscale modeling. Phys. Rev. B
**2003**, 67, 092101. [Google Scholar] - Abdulle, A.; Weinan, E.; Engquist, B.; Vanden-Eijnden, E. The heterogeneous multiscale method. Acta Numer.
**2012**, 21, 1–87. [Google Scholar] [CrossRef] [Green Version] - Hajibeygi, H.; Bonfigli, G.; Hesse, M.A.; Jenny, P. Iterative multiscale finite-volume method. J. Comput. Phys.
**2008**, 227, 8604–8621. [Google Scholar] [CrossRef] - Lunati, I.; Jenny, P. Multiscale finite-volume method for compressible multiphase flow in porous media. J. Comput. Phys.
**2006**, 216, 616–636. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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