Abstract
We consider a model for two-phase flow in a porous medium posed in a domain consisting of two adjacent regions. The model includes dynamic capillarity and hysteresis. At the interface between adjacent subdomains, the continuity of the normal fluxes and pressures is assumed. For finding the semi-discrete solutions after temporal discretization by the θ-scheme, we proposed an iterative scheme. It combines a (fixed-point) linearization scheme and a non-overlapping domain decomposition method. This article describes the scheme, its convergence and a numerical study confirming this result. The convergence of the iteration towards the solution of the semi-discrete equations is proved independently of the initial guesses and of the spatial discretization, and under some mild constraints on the time step. Hence, this scheme is robust and can be easily implemented for realistic applications.
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References
Ahmed, E.: Splitting-based domain decomposition methods for two-phase flow with different rock types. Adv. Water Resour. 134, 103431 (2019)
Beliaev, A.Y., Hassanizadeh, S.M.: A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media. Transp. Porous Media 43, 487–510 (2001)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer New York (1991)
Caetano, F., et al.: Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Netw. Heterog. Media 5, 487–505 (2010)
Calugaru, D.G., Tromeur-Dervout, D.: Non-overlapping DDMs to solve flow in heterogeneous porous media. In: Domain Decomposition Methods in Science and Engineering, pp. 529–536. Springer-Verlag (2005)
Cao, X., Pop, I.S.: Two-phase porous media flows with dynamic capillary effects and hysteresis: Uniqueness of weak solutions. Comput. Math. Appl. 69, 688–695 (2015)
Gander, M.J., Dubois, O.: Optimized schwarz methods for a diffusion problem with discontinuous coefficient. Numer. Algor. 69, 109–144 (2014)
Gander, M.J., Halpern, L., Nataf, F.: Optimal schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41, 1643–1681 (2003)
Gander, M.J., Rohde, C.: Overlapping schwarz waveform relaxation for convection-dominated nonlinear conservation laws. SIAM J. Sci. Comput. 27, 415–439 (2005)
Karpinski, S., Pop, I.S., Radu, F.A.: Analysis of a linearization scheme for an interior penalty discontinuous galerkin method for two-phase flow in porous media with dynamic capillarity effects. Int. J. Numer. Meth. Engng. 112, 553–577 (2017)
Koch, J., Rätz, A., Schweizer, B.: Two-phase flow equations with a dynamic capillary pressure. Eur. J. Appl. Math. 24, 49–75 (2012)
Lions, P.L.: On the Schwarz alternating method III: A variant for nonoverlapping subdomains. In: Third International Symposium on Domain Decomposition Methods for Partial Dierential Equations, pp. 202–223. SIAM (1990)
List, F., Radu, F.A.: A study on iterative methods for solving richards’ equation. Comput. Geosci. 20, 341–353 (2016)
Lunowa, S.B.: Linearization and domain decomposition methods for two-phase flow in porous media. Master’s thesis, Eindhoven University of Technology (2018). URL https://research.tue.nl/en/studentTheses/linearization-and-domain-decomposition-methods
Lunowa, S.B., Rohde, C., Gander, M.J.: Non-overlapping Schwarz waveform-relaxation for quasi-linear convection-diffusion equations (2020). In preparation.
Mikelić, A.: A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure. J. Differ. Equ. 248, 1561–1577 (2010)
Mitra, K., van Duijn, C.J.: Wetting fronts in unsaturated porous media: The combined case of hysteresis and dynamic capillary pressure. Nonlinear Anal. Real World Appl. 50, 316–341 (2019)
Pop, I.S., Radu, F., Knabner, P.: Mixed finite elements for the richards’ equation: Linearization procedure. J. Comput. Appl. Math. 168, 365–373 (2004)
Radu, F.A., Pop, I.S., Knabner, P.: Newton-type methods for the mixed finite element discretization of some degenerate parabolic equations. In: Numerical Mathematics and Advanced Applications ENUMATH 2005, pp. 1192–1200. Springer Berlin Heidelberg (2006)
Schweizer, B.: Instability of gravity wetting fronts for Richards equations with hysteresis. Interfaces Free Bound. 14, 37–64 (2012)
Seus, D., Radu, F.A., Rohde, C.: A linear domain decomposition method for two-phase flow in porous media. In: Numerical Mathematics and Advanced Applications ENUMATH 2017, pp. 603–614. Springer International Publishing (2019)
Seus, D., et al.: A linear domain decomposition method for partially saturated flow in porous media. Comput. Methods Appl. Mech. Eng. 333, 331–355 (2018)
Acknowledgements
This work was supported by Eindhoven University of Technology, Hasselt University (Project BOF17NI01) and the Research Foundation Flanders (FWO, Project G051418N).
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Lunowa, S.B., Pop, I.S., Koren, B. (2021). A Linear Domain Decomposition Method for Non-equilibrium Two-Phase Flow Models. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_13
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