# Numerical solution to the problem of two-phase filtration with heterogeneous coefficients by the finite element method

### Abstract

We consider the process of ﬁltration of a two-phase ﬂuid in a porous, heterogeneous medium. This process is described by a coupled system of equations for saturation, ﬁltration rate, and pore pressure. We consider mathematical models with and without capillary forces, in the presence of which, for saturation, we have a nonstationary convection-diﬀusion equation. Since this process is characterized by a signiﬁcant predominance of the convective term in the equation for saturation, countercurrent approximations are used by adding non-uniform artiﬁcial diﬀusion. Speed and pressure are approximated using a mixed ﬁnite element method. The results of numerical calculations for a two-dimensional case with strongly heterogeneous permeability coeﬃcients of a porous medium are presented. Several cases of relative ﬂuid permeability associated with linear and nonlinear coeﬃcients and the presence of capillary forces are

considered.

### References

[1] | Aziz K., Settari A., Petroleum reservoir simulation, Applied Sci. Publ., London, 1979 |

[2] | Bear J., Dynamics of fluids in porous media, Dover Publ., New York, 1988 |

[3] | Chen Z., Huan G., Ma Y., Computational methods for multiphase flows in porous media, Southern Methodist University, Dallas, 2006 |

[4] | Vasil'ev V. I., Popov V. V., and Timofeeva T. S., Vychislitel'nye Metody v Razrabotke Mestorozhdenij Nefti i Gaza, Izdat. SO RAN, Novosibirsk, 2000 |

[5] | Vabishhevich P. N., “Explicit-implicit computational algorithms for multiphase filtration problems”, Math. Models Comput. Simul., 22:4 (2010), |

[6] | Vabishchevich P. N., Vasil'eva M. V., “Iterative solution of the pressure problem for the multiphase filtration”, Math. Modell. Anal., 17:4 (2012), |

[7] | Vasilyeva M. V., “Numerical modelling of filtration at multiprocessor systems”, Mat. Zametki YaGU, 17:2 (2010), |

[8] | Afanas'eva N. M., Vasil'eva M. V., and Zaharov P. E., “Parallel'noe chislennoe modelirovanie processa zavodnenija neftjanogo mestorozhdenija”, Mat. Zametki YaGU, 18:1 (2011), |

[9] | Chung E. T., Leung W. T., Vasilyeva M., “Mixed GMsFEM for second order elliptic problem in perforated domains”, J. Comput. Appl. Math., 304 (2016), |

[10] | Akkutlu I. Y., Efendiev Y., Vasilyeva M., Wang Y., “Multiscale model reduction for transport and flow problems in perforated domains. Multiscale model reduction for shale gas transport in poroelastic fractured media”, J. Comput. Physics, 353 (2018), |

[11] | Vasilyeva M. V., Vasilyev V. I., and Timofeeva T. S., “Numerical solution by the finite elements method to the problems of transfer by diffusion and convection in strongly heterogenuous media”, Uchen. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki, 158, no. 2, 2016, |

[12] | Alekseev V. N., Vasilyeva M. V., and Stepanov S. P., “Iterative methods for solving the problems of flow and transfer in perforated domains”, Vestn. Severo-Vostoch. Federal. Univ., 2016, no. 5, |

[13] | Vasil'ev V. I., Vasilyeva M. V., and Nikiforov D. Ya., “Solution to the problems of one-phase filtration by the finite elements method on the computation cluster”, Vestn. Severo-Vostoch. Federal. Univ., 2016, no. 6, |

[14] | Brezzi F., Fortin M., Mixed and hybrid finite element methods, Springer-Verl., Berlin, 1991 |

[15] | Raviart P. A., Thomas J. M., “A mixed finite element method for 2-nd order elliptic problems”, Mathematical aspects of finite element methods, Springer-Verl., Berlin; Heidelberg, 1977, |

[16] | Carstensen C., “A posteriori error estimate for the mixed finite element method”, Math. Comput, 66:218 (1997), |

[17] | Samarskij A. A. and Vabishhevich P. N., Methods for Convection-Diffusion Problems, Editorial URSS, Moscow, 2004 |

[18] | Donea J., Huerta A., Finite element methods for flow problems, John Wiley & Sons Ltd, Chichester, 2003 |

[19] | Brooks A. N., A Petrov-Galerkin finite element formulation for convection dominated flows, PhD thesis, California Institute of Technology, 1981 |

[20] | Christie M. A., Blunt M. J., “Tenth SPE comparative solution project: A comparison of upscaling techniques”, SPE Reservoir Simulation Symposium, Society of Petroleum Engineers, 2001 |

[21] | Logg A., Mardal K.-A., Wells G. N., Automated solution of differential equations by the finite element method. The FEniCS Book, Springer-Verl., Berlin, 2011 |

[22] | Geuzaine C., Remacle J.-F., Software GMSH, http://geuz.org/gmsh |

[23] | Software package PARAVIEW, http://www.paraview.org |

*Mathematical notes of NEFU*, 24(2), pp. 46-62. doi: https://doi.org/10.25587/SVFU.2017.2.9245.

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