Journal "Computational Technologies"

Article information

2020 , Volume 25, № 5, p.42-54

Shurina E.P., Itkina N.B., Trofimova S.A.

Mathematical modelling of the filtration process in a porous medium based on non-conformal mixed finite element formulation

The paper addresses mathematical modelling modelling for the process of fluid filtration in porous geological media under pressure. A computational scheme based on a mixed non-conformal variational formulation for solving the Darcy problem with a tensor coefficient of permeability of the medium was developed, implemented and verified. Moreover, the discontinuous Galerkin method for obtaining the finite element approximation was used.

A specialized hierarchical basis system for velocity in the 𝐻𝑑𝑖𝑣 space and a basis system with discontinuous functions on boundaries of finite elements for pressure in the 𝐿2 space were constructed.

Computational experiments on a class of problems close to real ones have shown that for a computational domain with multidirectional rectangular inclusions of arbitrary size and concentration, the numerical fields of pressure and velocity are determined with a relative error of 1e−2 even on a coarse grid. An increase in the contrast of the permeability coefficient of the medium with respect to the permeability coefficient of inclusions does not change the relative error in determining the numerical fields. To this end, we can conclude that the constructed computational scheme is stable to significant variation of the coefficient.

The authors have developed and verified a software package that is able to export a ready-made solution to the problem in the “.dat” format file for further graphical display and analysis in the Tecplot software package.

[full text]
Keywords: fluid filtration, Darcy problem, mixed formulation, finite element method, discontinuous Galerkin method

doi: 10.25743/ICT.2020.25.5.004

Author(s):
Shurina Ella Petrovna
Dr. , Professor
Position: General Scientist
Office: Novosibirsk State Technical University, Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
Address: 630073, Russia, Novosibirsk, Karl Marx Ave. 20
Phone Office: (343) 223-72-95
E-mail: shurina@online.sinor.ru

Itkina Natalya Borisovna
PhD. , Associate Professor
Position: Associate Professor
Office: Novosibirsk State Technical University
Address: 630073, Russia, Novosibirsk, Karl Marx Ave. 20
Phone Office: (383) 346-27-76
E-mail: itkina.nat@yandex.ru

Trofimova Svetlana Alekseevna
Position: Student
Office: Novosibirsk State Technical University, Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
Address: 630073, Russia, Novosibirsk, 20, Prospekt K. Marksa
E-mail: TrofimovaSA@ipgg.sbras.ru

References:

1. Kouznetsova V.G. Computational homogenization for the multi-scale analysis of multi-phase materials. Eindhoven: Technische Universiteit Eindvohen; 2002: 134.

2. Leont’ev N.E. Osnovy teorii fil’tratsii. Uchebnoe posobie [Fundamentals of the filtration theory: Study guide]. Moscow: Izd-vo Tsentra prikladnykh issledovaniy pri mekhaniko-matematicheskom fakul’tete MGU; 2009: 88. (In Russ.)

3. Leybenzon L.S. Podzemnaya gidrogazodinamika [Underground hydrodynamics]. Moscow: AN SSSR; 1953: 544. (In Russ.)

4. Belyaev A.Yu. Usrednenie v zadachakh teorii fil’tratsii [Averaging in filtration theory problems]. Moscow: Nauka; 2004: 198. ISBN:5-02-032909-6. (In Russ.)

5. Wang Y., Wang S., Xue Sh., Adhikary D. Numerical modeling of porous flow in fractured rock and its applications in geothermal energy extraction. Journal of Earth Science. 2015; 26(1):20–27.

6. Vafai K. Handbook of porous media. Second edition. LLC USA: Taylor Francis Group; 2005: 727.

7. Badea L., Discacciati M., Quarteroni A. Numerical analysis of the Navier — Stokes/Darcy coupling. Numerische Mathematik. 2010; 115(2):195–227.

8. Arnold D.N. Mixed finite element methods for elliptic problems. Computer Methods in Applied Mechanics and Engineering. 1990; (82):281–300.

9. Brezzi F. On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Analyse Num´erique. 1974; (2):129–151. Available at: https://doi. org/10.1051/m2an/197408R201291

10. Arnold D.N., Brezzi F. Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. Math. Modelling and Numer. Anal. 1985; (19):7–32.

11. Ladyzhenskaya O.A. Matematicheskie voprosy dinamiki vyazkoy neszhimaemoy zhidkosti [Mathematical problems of viscous incompressible fluid dynamics]. 2nd ed. Moscow: Nauka; 1970: 288. (In Russ.)

12. Brezzi F., Fortin M. Mixed and Hybrid Finite Element Methods. N.Y.: Springer-Verlag; 1991: 362.

13. Ainsworth M., Coggins P. The stability of mixed hp-finite element methods for Stokes flow on high aspect ratio elements. Siam J. Numer. Anal. 2000; 38(5):1721–1761.

14. Masud A., Hughes T.J.R. A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 2002; (191):4341–4370.

15. Brezzi F., Hughes T.J.R., Marini L.D., Masud A. Mixed discontinuous Galerkin methods for Darcy flow. Journal of Scientific Computing. 2005; 22(1):119–145.

16. Duran O., Devloo P.R.B., Gomes S.M., Valentin F. A multiscale hybrid method for Darcy‘s problems using mixed finite element local solvers. Comput. Methods Appl. Mech. Engrg. 2019; (354):213–244.

17. Xu W., Liang D., Rui H. A multipoint flux mixed finite element method for the compressible Darcy — Forchheimer models. Applied Mathematics and Computation. 2017; (315):259–277.

18. Carvalho P.G.S., Devloo P.R.B., Gomes S.M. On the use of divergence balanced H(div)-L2 pair of approximation spaces for divergence-free and robust simulations of Stokes, coupled Stokes — Darcy and Brinkman problems. Mathematics and Computers in Simulation. 2020; (170):51–78.

19. Armentano M.G., Stockdale M.L. A unified mixed finite element approximations of the Stokes — Darcy coupled problem. Computers and Mathematics with Applications. 2019; (77):2568–2584.

20. Arnold D.N., Brezzi F., Marini L.D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 2002; 39(5):1749;;1779.

21. Brezzi F., Marini D. A survey on mixed finite element approximations. IEEE Transactions on Magnetics. 1994; 30(5):3547–3551.

22. Sobolev S.L. Embedding theorems. Proc. 4th All-Union Math. Congr. 1963; (1):227–242. (In Russ.)

23. Solin P., Segeth K., Doleˇzel I.ˇ High-order finite element methods. Chapmanand & Hall, CRC; 2004: 388.

24. Shokin Yu.I., Shurina E.P., Itkina N.B. Sovremennye mnogosetochnye metody. Chast’ I. Mnogomasshtabnye metody: ucheb. posobie [Modern multigrid methods: Part I. Multiscale methods: Study Guide]. Novosibirsk: Izd-vo NGTU; 2010: 68. (In Russ.)

25. Bergamaschi L., Mantica S., Saleri F. Mixed finite element approximation of Darcy‘s law in porous media. Report CRS4 AppMath-94-20, CRS4. Cagliari, Italy; 1994: 21.

Bibliography link:
Shurina E.P., Itkina N.B., Trofimova S.A. Mathematical modelling of the filtration process in a porous medium based on non-conformal mixed finite element formulation // Computational technologies. 2020. V. 25. № 5. P. 42-54