Depending on the physical and geometrical properties of a given porous medium, fluid flow can behave differently, going from a slow Darcian regime to more complicated Brinkman or even Forchheimer regimes for high velocity. The main problem is to determine where in the medium one regime is more adequate than others. In order to determine the low-speed and high-speed regions, this work proposes an adaptive strategy which is based on selecting the appropriate constitutive law linking velocity and pressure according to a threshold criterion on the magnitude of the fluid velocity itself. Both theoretical and numerical aspects are considered and investigated, showing the potentiality of the proposed approach. From the analytical viewpoint, we show existence of weak solutions to such model under reasonable hypotheses on the constitutive laws. To this end, we use a variational approach identifying solutions with minimizers of an underlying energy functional. From the numerical viewpoint, we propose a one-dimensional algorithm which tracks the transition zone between the low- and high-speed regions. By running numerical experiments using this algorithm, we illustrate some interesting behaviors of our adaptive model on academic cases and on small networks of intersecting fractures.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2022016
Keywords: Adaptive constitutive law, elliptic equation, fractured porous media, non-linearity, variational formulation
@article{M2AN_2022__56_2_565_0,
author = {Fumagalli, Alessio and Patacchini, Francesco Saverio},
title = {Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {565--592},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {2},
doi = {10.1051/m2an/2022016},
mrnumber = {4386475},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022016/}
}
TY - JOUR AU - Fumagalli, Alessio AU - Patacchini, Francesco Saverio TI - Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 565 EP - 592 VL - 56 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022016/ DO - 10.1051/m2an/2022016 LA - en ID - M2AN_2022__56_2_565_0 ER -
%0 Journal Article %A Fumagalli, Alessio %A Patacchini, Francesco Saverio %T Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 565-592 %V 56 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022016/ %R 10.1051/m2an/2022016 %G en %F M2AN_2022__56_2_565_0
Fumagalli, Alessio; Patacchini, Francesco Saverio. Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 2, pp. 565-592. doi: 10.1051/m2an/2022016
[1] , and , A reduced fracture model for two-phase flow with different rock types. Math. Comput. Simul. 137 (2017) 49–70. | MR | DOI
[2] , , , and , Domain decomposition for some transmission problems in flow in porous media. In: Numerical Treatment of Multiphase Flows in Porous Media (Beijing, 1999). Vol. 552 of Lecture Notes in Phys. Springer, Berlin (2000) 22–34. | MR | Zbl | DOI
[3] , , and , Décomposition de domaine et préconditionnement pour un modèle 3D en milieu poreux fracturé In: Proceeding of JANO 8, 8t Conference on Numerical Analysis and Optimization (2005).
[4] , Well-posed Stokes/Brinkman and Stokes/Darcy coupling revisited with new jump interface conditions. ESAIM: M2AN 52 (2018) 1875–1911. | MR | Zbl | Numdam | DOI
[5] and , A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. Theory Methods App. 60 (2005) 515–545. | MR | Zbl | DOI
[6] , and , On the well-posedness of generalized Darcy-Forchheimer equation. Boundary Value Prob. 2018 (2018) 1–17. | MR
[7] , and , Coupling Darcy and Stokes equations for porous media with cracks. ESAIM: M2AN 39 (2005) 7–35. | MR | Zbl | Numdam | DOI
[8] , and , A PDE-constrained optimization formulation for discrete fracture network flows. SIAM J. Sci. Comput. 35 (2013) B487–B510. | MR | Zbl | DOI
[9] , and , On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. 35 (2013) 908–935. | MR | Zbl | DOI
[10] , and , Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg, 2013. | MR | Zbl | DOI
[11] , and , Free energy diminishing discretization of darcy-forchheimer flow in poroelastic media. In: Finite Volumes for Complex Applications IX – Methods, Theoretical Aspects, Examples, edited by , , and . Springer International Publishing, Cham, (2020) 203–211. | MR | DOI
[12] , and , A generalization of the Darcy-Forchheimer equation involving an implicit, pressure-dependent relation between the drag force and the velocity. J. Math. Anal. App. 424 (2015) 785–801. | MR | Zbl | DOI
[13] , Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities. ESAIM: M2AN 43 (2009) 973–1001. | MR | Zbl | Numdam | DOI
[14] , , , , , , and , Benchmarks for single-phase flow in fractured porous media. Adv. Water Res. 111 (2018) 239–258. | DOI
[15] , , and , A reduced model for Darcy’s problem in networks of fractures. ESAIM: M2AN 48 (2014) 1089–1116. | MR | Zbl | Numdam | DOI
[16] , and , Modeling fractures as interfaces: a model for Forchheimer fractures. Comput. Geosci. 12 (2008) 91–104. | MR | Zbl | DOI
[17] and , Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79 (2009) 1309–1331. | MR | Zbl | DOI
[18] and , Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability. Trans. Am. Math. Soc. 372 (2019) 2255–2286. | MR | DOI
[19] , , and , Effective equations for two-phase flow in porous media: the effect of trapping on the microscale. Transp. Porous Media 69 (2007) 411–428. | MR | DOI
[20] , , , , , and , Porepy: an open-source software for simulation of multiphysics processes in fractured porous media. Comput. Geosci. 25 (2021) 243–265. | MR | DOI
[21] and , Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy-Forchheimer flow in the fracture. ESAIM: M2AN 48 (2014) 1451–1472. | MR | Zbl | Numdam | DOI
[22] , , and , Formal upscaling and numerical validation of unsaturated flow models in fractured porous media. J. Comput. Phys. 407 (2020) 109138. | MR | DOI
[23] and , A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20 (2016) 341–353. | MR | DOI
[24] , and , Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667–1691. | MR | Zbl | DOI
[25] and , A Darcy-Brinkman model of fractures in porous media. J. Math. Anal. App. 452 (2017) 1332–1358. | MR | DOI
[26] and , Variational Methods in Theoretical Mechanics. Springer, Berlin Heidelberg (1976). | MR | Zbl | DOI
[27] , Variational modelling: energies, gradient flows, and large deviations. Preprint (2014). | arXiv
[28] , and , Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168 (2004) 365–373. | MR | Zbl | DOI
[29] , , and , A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities. IMA J. Numer. Anal. 38 (2017) 884–920. | MR | DOI
[30] and , A mixed finite element method for second order elliptic problems. Lecture Notes Math. 606 (1977) 292–315. | MR | Zbl | DOI
[31] and , Mixed and hybrid methods. In: Handbook of Numerical Analysis. Vol. II, Handb. Numer. Anal., II. North-Holland, Amsterdam (1991) 523–639. | MR | Zbl
[32] and , A dimensionally reduced stokes-Darcy model for fluid flow in fractured porous media. App. Math. Comput. 384 (2020) 125260. | MR | DOI
[33] , and , An analysis of a mixed finite element method for a Darcy-Forchheimer model. Math. Comput. Modell. 57 (2013) 2325–2338. | MR | Zbl | DOI
[34] and , A potential formulation of non-linear models of flow through anisotropic porous media. Transp. Porous Media 45 (2001) 405–421. | MR | DOI
[35] , The Porous Medium Equation. Oxford University Press (2006). | MR | Zbl | DOI
[36] and , A criterion for non-Darcy flow in porous media. Transp. Porous Media 63 (2006) 57–69. | DOI
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