Abstract
In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized gradient flow structure, identifying the dissipative character of the physical system. The considered mixed finite element discretization is compatible with this structure, which gives access to a simple proof for the existence, uniqueness, and stability of discrete approximations. Moreover, still within the framework, the discretization allows for the development of finite volume type discretizations by lumping or numerical quadrature, reducing the computational cost of the numerical solution.
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References
Ambartsumyan, I., Khattatov, E., Nordbotten, J.M., Yotov, I.: A multipoint stress mixed finite element method for elasticity on simplicial grids. arXiv:1805.09920, accepted for SINUM
Atik, Y., Kabir, H., Vallet, G.: On a nonlinear system of BiotForchheimer type. Complex Var. Elliptic 1–23 (2019). https://doi.org/10.1080/17476933.2019.1695786
Baranger, J., Maitre, J.F., Oudin, F.: Connection between finite volume and mixed finite element methods. ESAIM-Math. Model Num. 30(4), 445–465 (1996)
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications, vol. 44. Springer (2013)
Both, J.W., Borregales, M., Nordbotten, J.M., Kumar, K., Radu, F.A.: Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017)
Both, J.W., Kumar, K., Nordbotten, J.M., Radu, F.A.: The gradient flow structures of thermo-poro-visco-elastic processes in porous media. arXiv e-prints arXiv:1907.03134 (2019)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, vol. 28. SIAM (1999)
Forchheimer, P.: Wasserbewegung durch boden. Z. Ver. Deutsch, Ing. 45, 1782–1788 (1901)
Girault, V., Wheeler, M.F.: Numerical discretization of a Darcy-Forchheimer model. Numer. Math. 110(2), 161–198 (2008)
Hu, X., Rodrigo, C., Gaspar, F.J., Zikatanov, L.T.: A nonconforming finite element method for the Biot’s consolidation model in poroelasticity. J. Comput. Appl. Math. 310, 143–154 (2017)
Klausen, R.A., Radu, F.A., Eigestad, G.T.: Convergence of MPFA on triangulations and for Richards’ equation. Int. J. Numer. Meth. Fl. 58(12), 1327–1351 (2008). https://doi.org/10.1002/fld.1787
Klausen, R.A., Winther, R.: Convergence of multipoint flux approximations on quadrilateral grids. Numer. Meth. Part D E 22(6), 1438–1454 (2006)
Markert, B.: A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua. Transport Porous Med. 70(3), 427 (2007)
Park, E.J.: Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Meth. Part D E 21(2), 213–228 (2005). https://doi.org/10.1002/num.20035
Peletier, M.A.: Variational Modelling: Energies, Gradient Flows, and Large Deviations. arXiv preprint arXiv:1402.1990 (2014)
Rui, H., Pan, H.: A block-centered finite difference method for the Darcy-Forchheimer model. SIAM J. Numer. Anal. 50(5), 2612–2631 (2012). https://doi.org/10.1137/110858239
Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44(5), 2082–2106 (2006)
Acknowledgements
This work is supported in part by the Research Council of Norway Project 250223, as well as the FracFlow project funded by Equinor through Akademiaavtalen.
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Both, J.W., Nordbotten, J.M., Radu, F.A. (2020). Free Energy Diminishing Discretization of Darcy-Forchheimer Flow in Poroelastic 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_17
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DOI: https://doi.org/10.1007/978-3-030-43651-3_17
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