# Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

**Remark**

**1.**

## 3. Fine Grid Approximation

## 4. Coarse Grid Approximation

**Remark**

**2.**

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Girault, V.; Wheeler, M.F. Numerical discretization of a Darcy-Forchheimer model. Numer. Math.
**2008**, 110, 161–198. [Google Scholar] [CrossRef] - Park, E.J. Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Methods Partial Differ. Equ.
**2005**, 21, 213–228. [Google Scholar] [CrossRef] - Pan, H.; Rui, H. Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput.
**2012**, 52, 563–587. [Google Scholar] [CrossRef] - Rui, H.; Pan, H. A Block-Centered Finite Difference Method for the Darcy-Forchheimer Model. SIAM J. Numer. Anal.
**2012**, 50, 2612–2631. [Google Scholar] [CrossRef] - Rui, H.; Zhao, D.; Pan, H. A block-centered finite difference method for Darcy-Forchheimer model with variable Forchheimer number. Numer. Methods Partial Differ. Equ.
**2015**, 31, 1603–1622. [Google Scholar] [CrossRef] - Rui, H.; Liu, W. A Two-Grid Block-Centered Finite Difference Method For Darcy-Forchheimer Flow in Porous Media. SIAM J. Numer. Anal.
**2015**, 53, 1941–1962. [Google Scholar] [CrossRef] - Huang, J.; Chen, L.; Rui, H. Multigrid methods for a mixed finite element method of the Darcy-Forchheimer model. J. Sci. Comput.
**2018**, 74, 396–411. [Google Scholar] [CrossRef] [PubMed] - Chen, Z.; Hou, T. A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput.
**2003**, 72, 541–576. [Google Scholar] [CrossRef] [Green Version] - Aarnes, J.E.; Efendiev, Y.; Jiang, L. Mixed multiscale finite element methods using limited global information. Multiscale Model. Simul.
**2008**, 7, 655–676. [Google Scholar] [CrossRef] - Aarnes, J.E. On the use of a mixed multiscale finite element method for greaterflexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul.
**2004**, 2, 421–439. [Google Scholar] [CrossRef] - Aarnes, J.E.; Kippe, V.; Lie, K.A. Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Resour.
**2005**, 28, 257–271. [Google Scholar] [CrossRef] - Chung, E.T.; Efendiev, Y.; Lee, C. Mixed generalized multiscale finite element methods and applications. Multiscale Model. Simul.
**2015**, 13, 338–366. [Google Scholar] [CrossRef] [Green Version] - Chung, E.T.; Leung, W.T.; Vasilyeva, M. Mixed GMsFEM for second order elliptic problem in perforated domains. J. Comput. Appl. Math.
**2016**, 304, 84–99. [Google Scholar] [CrossRef] - Chung, E.T.; Leung, W.; Vasilyeva, M.; Wang, Y. Multiscale model reduction for transport and flow problems in perforated domains. J. Comput. Appl. Math.
**2018**, 330, 519–535. [Google Scholar] [CrossRef] - Zhang, J.; Xing, H. Numerical modeling of non-Darcy flow in near-well region of a geothermal reservoir. Geothermics
**2012**, 42, 78–86. [Google Scholar] [CrossRef] - Chang, J.; Nakshatrala, K.B.; Reddy, J.N. Modification to Darcy-Forchheimer model due to pressure-dependent viscosity: Consequences and numerical solutions. J. Porous Media
**2017**, 20. [Google Scholar] [CrossRef] - Ciarlet, P.G. The Finite Element Method for Elliptic Problems; SIAM: Philadelphia, PA, USA, 2002. [Google Scholar]
- Adams, R.; Fournier, J. Sobolev Spaces; Academic Press: New York, NY, USA; London, UK; Toronto, ON, Canada, 1975. [Google Scholar]
- Chan, H.Y.; Chung, E.; Efendiev, Y. Adaptive mixed GMsFEM for flows in heterogeneous media. Numer. Math. Theory Methods Appl.
**2016**, 9, 497–527. [Google Scholar] [CrossRef] [Green Version] - Li, D.; Engler, T.W. Literature review on correlations of the non-Darcy coefficient. In Proceedings of the SPE Permian Basin Oil and Gas Recovery Conference, Midland, TX, YSA, 15–17 May 2001. [Google Scholar]
- Muljadi, B.P.; Blunt, M.J.; Raeini, A.Q.; Bijeljic, B. The impact of porous media heterogeneity on non-Darcy flow behaviour from pore-scale simulation. Adv. Water Resour.
**2016**, 95, 329–340. [Google Scholar] [CrossRef]

**Figure 1.**Illustration of the heterogeneous property, coarse grid, and multiscale basis functions in the local domain.

**Figure 4.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 1 with $\beta =0$.

**Figure 5.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 2 with $C=34.93$.

**Figure 6.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 1 with $C=34.93$.

**Figure 7.**Fine grid solution (

**top**) and multiscale solution using eight multiscale basis functions (

**bottom**). Coefficient k from Test 2 with $\beta =0$.

**Table 1.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $\beta =0$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % |
---|---|---|---|---|---|---|---|

1 | 320 | 10.069 | 1.212 | 1 | 320 | 11.279 | 1.451 |

2 | 540 | 1.112 | 0.031 | 2 | 540 | 2.943 | 0.104 |

4 | 980 | 0.253 | 0.001 | 4 | 980 | 0.579 | 0.004 |

8 | 1860 | 0.061 | 0.001 | 8 | 1860 | 0.152 | 0.001 |

**Table 2.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $C=10.24$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.334 | 1.856 | 9 | 1 | 320 | 11.042 | 2.155 | 7 |

2 | 540 | 2.329 | 0.136 | 8 | 2 | 540 | 3.585 | 0.252 | 6 |

4 | 980 | 2.174 | 0.129 | 7 | 4 | 980 | 2.897 | 0.199 | 5 |

8 | 1860 | 2.054 | 0.102 | 6 | 8 | 1860 | 2.856 | 0.195 | 5 |

Iterations on the fine grid = 5 | Iterations on the fine grid = 5 |

**Table 3.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $C=34.93$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.389 | 2.052 | 24 | 1 | 320 | 11.021 | 2.374 | 22 |

2 | 540 | 2.766 | 0.171 | 23 | 2 | 540 | 4.059 | 0.319 | 20 |

4 | 980 | 2.567 | 0.151 | 21 | 4 | 980 | 3.469 | 0.248 | 19 |

8 | 1860 | 2.561 | 0.151 | 19 | 8 | 1860 | 3.446 | 0.244 | 17 |

Iterations on the fine grid = 21 | Iterations on the fine grid = 21 |

**Table 4.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with $C=1581.14$.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.415 | 2.181 | 387 | 1 | 320 | 11.033 | 2.518 | 399 |

2 | 540 | 2.998 | 0.197 | 403 | 2 | 540 | 4.364 | 0.364 | 388 |

4 | 980 | 2.814 | 0.177 | 389 | 4 | 980 | 3.868 | 0.294 | 371 |

8 | 1860 | 2.799 | 0.175 | 368 | 8 | 1860 | 3.843 | 0.291 | 348 |

Iterations on the fine grid = 1162 | Iterations on the fine grid = 1143 |

**Table 5.**Relative error in the ${L}_{2}$ norm for different numbers of multiscale basis functions. Coefficient k from Test 1 (left) and Test 2 (right) with C = 71,554.17.

M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter | M | ${\mathit{DOF}}_{\mathit{c}}$ | ${\mathit{e}}_{\mathit{u}}$, % | ${\mathit{e}}_{\mathit{p}}$, % | #iter |
---|---|---|---|---|---|---|---|---|---|

1 | 320 | 10.416 | 2.185 | 744 | 1 | 320 | 11.033 | 2.522 | 794 |

2 | 540 | 3.003 | 0.198 | 854 | 2 | 540 | 4.372 | 0.365 | 809 |

4 | 980 | 2.821 | 0.177 | 886 | 4 | 980 | 3.878 | 0.295 | 805 |

8 | 1860 | 2.805 | 0.176 | 864 | 8 | 1860 | 3.853 | 0.291 | 777 |

Iterations on the fine grid = 16,489 | Iterations on the fine grid = 14,466 |

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## Share and Cite

**MDPI and ACS Style**

Spiridonov, D.; Huang, J.; Vasilyeva, M.; Huang, Y.; Chung, E.T.
Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model. *Mathematics* **2019**, *7*, 1212.
https://doi.org/10.3390/math7121212

**AMA Style**

Spiridonov D, Huang J, Vasilyeva M, Huang Y, Chung ET.
Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model. *Mathematics*. 2019; 7(12):1212.
https://doi.org/10.3390/math7121212

**Chicago/Turabian Style**

Spiridonov, Denis, Jian Huang, Maria Vasilyeva, Yunqing Huang, and Eric T. Chung.
2019. "Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model" *Mathematics* 7, no. 12: 1212.
https://doi.org/10.3390/math7121212