**6. Conclusions**

In this paper, we conducted a numerical study of the solution of the Darcy-Forchheimer model in high contrast heterogeneous media. To solve this problem, we used the mixed multiscale finite element method. The method showed good accuracy in two model problems. The obtained solutions were compared with the fine grid solution using the mixed finite element method. The study showed that the accuracy of this method depended on the number of multiscale basis functions and was almost independent of the influence of the nonlinear part of the equation. Mixed GMsFEM provided a good solution for any *β* values, and the accuracy of the method was improved by using more basis functions. The method showed good efficiency, since the number of Picard iterations with a large influence of the nonlinear part was much less than when solving a problem on a fine grid.

**Author Contributions:** The authors have contributed equally to the work.

**Funding:** D.S.'s and M.V.'s work was supported by the gran<sup>t</sup> of the Russian Scientific Found N17-71-20055 and the mega-gran<sup>t</sup> of the Russian Federation Government (N 14.Y26.31.0013). E.T.C.'s work was partially supported by the Hong Kong RGC General Research Fund (Projects 14304217 and 14302018) and the CUHK Direct Grant for Research 2018-19. The work of J.H. was supported by the China Postdoctoral Science Foundation Funded Project Grant No. BX20180266, 2018M642991, in part by the National Natural Science Foundation of China Grant No. 11901497, and in part by the Natural Science Foundation of Hunan Province Grant No. 2019JJ50607. The work of Y.H. was supported by the National Natural Science Foundation of China Grant No. 11971410 and in part by the Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018WK4006).

**Conflicts of Interest:** The authors declare no conflict of interest.
