Abstract
In many natural and technical applications in porous media fluid’s flow behavior is highly affected by fractures. Many approaches employ mixed-dimensional models that model thin features as dimension-reduced manifolds. Following this idea, we consider porous media where dominant heterogeneities are geometrically represented by sharp interfaces. We model incompressible two-phase flow in porous media both in the bulk porous medium and within the fractures. We present a reliable and geometrically flexible implementation of a fully conforming finite volume approach within the DUNE framework for two and three spatial dimensions. The implementation is based on the new dune-mmesh grid implementation that manages bulk and surface triangulation simultaneously. The model and the implementation are extended to handle fracture junctions. We apply our scheme to benchmark cases with complex fracture networks to show the reliability of the approach.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Blatt, M., Burchardt, A., Dedner, A., Engwer, C., Fahlke, J., Flemisch, B., Gersbacher, C., Gräser, C., Gruber, F., Grüninger, C., Kempf, D., Klöfkorn, R., Malkmus, T., Müthing, S., Nolte, M., Piatkowski, M., Sander, O.: The distributed and unified numerics environment, version 2.4. Arch. Numer. Softw. 4, 13–29 (2016)
Chalons, C., Rohde, C., Wiebe, M.: A finite volume method for undercompressive shock waves in two space dimensions. ESAIM Math. Mod. Num. Anal. 51, 1987–2015 (2017)
Dune-MMesh: A DUNE grid implementation based on CGAL Delaunay triangulations. Release 1.1. https://gitlab.dune-project.org/samuel.burbulla/dune-mmesh
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis VII (2000)
Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., Helmig, R.: DuMux: DUNE for multi-phase, component, scale, physics, ... flow and transport in porous media. Adv. Water Resour. 34, 1102–1112 (2011)
Fumagalli, A., Scotti, A.: A numerical method for two-phase flow in fractured porous media with non-matching grids. Adv. Water Resour. 62, 454–464 (2013)
Giovanardi B., Scotti A., Formaggia L.: A hybrid XFEM–Phase field (Xfield) method for crack propagation in brittle elastic materials. Comput. Methods Appl. Mech. Eng. 320 (2017)
Jaffré, J., Mnejja, M., Roberts, J.E.: A discrete fracture model for two-phase flow with matrix-fracture interaction. Procedia Comput. Sci. 4 (2011)
Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26, 1667–1691 (2005)
Miehe, C., Mauthe, S., Teichtmeister, S.: Minimization principles for the coupled problem of Darcy-Biot-type fluid transport in porous media linked to phase field modeling of fracture. J. Mech. Phys. Solids 82, 186–217 (2015)
Mishra, S., Jaffré, J.: On the upstream mobility scheme for two-phase flow in porous media. Comput. Geosci. 14, 105–124 (2010)
Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam (1977)
The CGAL Project: CGAL user and reference manual. Version 4.14. https://doc.cgal.org/4.14/Manual/packages.html (2019)
Acknowledgements
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project Number 327154368—SFB 1313.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Burbulla, S., Rohde, C. (2020). A Fully Conforming Finite Volume Approach to Two-Phase Flow in Fractured Porous Media. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_51
Download citation
DOI: https://doi.org/10.1007/978-3-030-43651-3_51
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43650-6
Online ISBN: 978-3-030-43651-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)