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Convergence of a CVFE finite volume scheme for nonisothermal immiscible incompressible two-phase flow in porous media
Brahim Amaziane1; Mustapha El Ossmani2; El Houssaine Quenjel3; Youssef Zahraoui4
1 Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France and Inria, Paris research Center, Paris, France
2 University of Moulay Ismaïl, L2M3S–ENSAM, Meknès, Morocco and Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France
3 Chair of Biotechnology LGPM, CentraleSupélec, CEBB, Pomacle, France
4 University of Moulay Ismaïl, L2M3S–ENSAM, Meknès, Morocco
The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 263-304.

We consider a two-phase Darcy flow in a heterogeneous porous medium while taking into account the effects of temperature. The set of governing equations consists of the usual equations derived from the mass conservation of both fluids along with the Darcy–Muskat, the capillary pressure laws, and the energy balance equation. The problem is written in terms of the fractional flow formulation, i.e. the saturation of one phase, the nonisothermal global pressure and the temperature are primary unknowns. The spatial and temporal discretizations are carried out applying a vertex-centered finite volume scheme and the implicit Euler scheme, respectively, resulting in the final system of fully coupled nonlinear equations. Under some realistic assumptions on the data, we establish a sufficient condition on the mesh to demonstrate the discrete maximum principle for saturation and temperature. Various a priori estimates are derived that yield an existence result for discrete solutions. Based on preliminary estimates and compactness arguments, we prove the convergence of the numerical scheme to a weak solution of the continuous model. The open source platform DuMu X has been used to implement the resulting algorithm. Two numerical experiments are presented to illustrate the effectiveness and the robustness of the scheme.

Published online:
DOI: 10.5802/smai-jcm.113
Classification: 35K65, 35Q35, 65N12, 76S05, 76M12, 80A19
Keywords: Nonlinear degenerate system, Finite volume, Two-phase flow, Nonisothermal, Heterogeneous porous media, DuMu$^X$.

Brahim Amaziane 1; Mustapha El Ossmani 2; El Houssaine Quenjel 3; Youssef Zahraoui 4

1 Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France and Inria, Paris research Center, Paris, France
2 University of Moulay Ismaïl, L2M3S–ENSAM, Meknès, Morocco and Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France
3 Chair of Biotechnology LGPM, CentraleSupélec, CEBB, Pomacle, France
4 University of Moulay Ismaïl, L2M3S–ENSAM, Meknès, Morocco
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Convergence of a {CVFE} finite volume scheme for nonisothermal immiscible incompressible two-phase flow in porous media},
     journal = {The SMAI Journal of computational mathematics},
     pages = {263--304},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Brahim Amaziane; Mustapha El Ossmani; El Houssaine Quenjel; Youssef Zahraoui. Convergence of a CVFE finite volume scheme for nonisothermal immiscible incompressible two-phase flow in porous media. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 263-304. doi : 10.5802/smai-jcm.113. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.113/

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