# Flow and transport in perforated and fractured domains with Robin boundary conditions

• Gavrilieva Uygulaana S., lanasemna@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677891, Russia
• Alekseev Valentin N., alekseev.valen@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677891, Russia
• Vasilyeva Maria V., vasilyevadotmdotv@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677891, Russia
Keywords: transport equation, Stokes problem, perforated domain, fractured domain, numerical modeling, Robin boundary condition, numerical stabilization, SUPG, ﬁnite element method

### Abstract

We consider transport and ﬂow problems in perforated and fractured domains. The system of equations is described by the Stokes equation for modeling ﬂuid ﬂow and the equation for the concentration transfer of a certain substance. Concentration is supplemented by inhomogeneous boundary conditions of the third type which simulate the occurring reaction on the faces of the modeled object. For the numerical solution of the problem, a ﬁnite-element approximation of the equation is constructed. To obtain a sustainable solution to the transport problem, the SUPG (streamline upwind Petrov–Galerkin) method is used to stabilize the classical Galerkin method. The computational implementation is based on the Fenics computational library. The results of solving the model problem in perforated and fractured domains are presented. Numerical studies of various regimes of heterogeneous boundary conditions were carried out.

### References

 [1] Chung E. T. et al., “Multiscale model reduction for transport and flow problems in perforated domains”, J. Comput. Appl. Math., 330 (2018), 519–535 [2] Chung E. T., Vasilyeva M., Wang Y., “A conservative local multiscale model reduction technique for Stokes flows in heterogeneous perforated domains”, J. Comput. Appl. Math., 321 (2017), 389–405 [3] Vasilyeva M. V. and Prokopyev G. A., “Numerical solution of the two-phase filtration problem with nonhomogenuous coefficients by the finite element method”, Mat. Zamet. SVFU, 24:2 (2017), 46–62 [4] Chung E. T., Iliev O., Vasilyeva M. V., “Generalized multiscale finite element method for non Newtonian fluid flow in perforated domain”, AIP Conference Proceedings. AIP Publishing, 1773:1 (2016), 100001 [5] Chung E. T. et al., “Generalized multiscale finite element methods for problems in perforated heterogeneous domains”, Appl. Anal., 95:10 (2016), 2254–2279 [6] Battiato I. et al., “Hybrid models of reactive transport in porous and fractured media”, Adv. Water Resources, 34:9 (2011), 140–1150 [7] Battiato I., Tartakovsky D. M., “Applicability regimes for macroscopic models of reactive transport in porous media”, J. contaminant hydrology, 120 (2011), 18–26 [8] Tomin P., Lunati I., “Hybrid Multiscale Finite Volume method for two-phase flow in porous media”, J. Comput. Physics, 250 (2013), 293–307 [9] Brezzi F., Fortin M., Mixed and hybrid finite element methods, Springer Ser. Comput. Math., 15, Springer Science & Business Media, New York, 2012 [10] 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, 292–315 [11] Donea J., Huerta A., Finite element methods for flow problems, John Wiley & Sons, Chichester, 2003 [12] Brooks A. N., A Petrov-Galerkin finite element formulation for convection dominated flows: diss., California Institute of Technology, 1981 [13] Vabishchevich P. N., Vasil'eva M. V., “Explicit-implicit schemes for convection-diffusion reaction problems”, Numer. Anal. Appl., 5:4 (2012), 297–306 [14] Afanas'eva N. M., Vabishchevich P. N., Vasil'eva M. V., “Unconditionally stable schemes for convection-diffusion problems”, Russian Mathematics, 57:3 (2013), 1–11 [15] Brooks A. N., Hughes T. J. R., “Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”, Comp. Methods Appl. Mechanics and Engineering, 32:1-3 (1982), 199–259 [16] Riviere B., Wheeler M. F. Discontinuous Galerkin methods for flow and transport problems in porous media, Intern. J. Numer. Methods in Biomedical Engineering, 18:1 (2002), 63–68 [17] Cockburn B., Shu C. W., “The local discontinuous Galerkin method for time-dependent convection-diffusion systems”, SIAM J. Numer. Anal., 35:6 (1998), 2440–2463 [18] Alekseev V. N., Vasilyeva M. V., and Stepanov S. P., “Iterative methods of solving the flow and transfer problem in perforated domains”, Vestn. SVFU, 2016, no. 5, 67–79 [19] Vasilyeva M. V., Vasilyev V. I., and Timofeeva T. S., “Numerical solution by the finite element method of the diffusion and convection problems in strongly heterogenuous media”, Uch. Zap. Kazansk. Univ., Ser. Fiz.-Mat. Nauki, 158:2 (2016), 243–261
How to Cite
Gavrilieva, U., Alekseev, V. and Vasilyeva, M. (&nbsp;) “Flow and transport in perforated and fractured domains with Robin boundary conditions”, Mathematical notes of NEFU, 24(3), pp. 65-77. doi: https://doi.org/10.25587/SVFU.2018.3.10890.
Issue
Section
Mathematical Modeling