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Tytuł artykułu

On convergence of explicit finite volume scheme for one-dimensional three-component two-phase flow model in porous media

Wybrane pełne teksty z tego czasopisma
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this work, we develop and analyze an explicit finite volume scheme for a one-dimensional nonlinear, degenerate, convection–diffusion equation having application in petroleum reservoir. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze a numerical scheme corresponding to explicit discretization of the diffusion term and a Godunov scheme for the advection term. L∞ stability under appropriate CFL conditions and BV estimates are obtained. It is shown that the scheme satisfies a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem, and we mount convergence results to a weak solution of the problem in L1 . Results of numerical experiments are presented to validate the theoretical analysis.
Wydawca
Rocznik
Strony
510--526
Opis fizyczny
Bibliogr. 21 poz., rys.
Twórcy
  • Department of Mathematics, Laboratory of Nonlinear Partial Differential Equations, ENS Kouba, Algiers, Algeria
  • Department of Mathematics, Laboratory of Operator Theory and PDEs: Foundations and Applications, Faculty of Exact Sciences, University of El Oued, Algeria
autor
  • Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia
Bibliografia
  • [1] H. Wilhelm Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), 311–341, DOI: https://doi.org/10.1007/bf01176474.
  • [2] T. Arbogast, The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal. 19 (1992), no. 11, 1009–1031, DOI: https://doi.org/10.1016/0362-546X(92)90121-T.
  • [3] S. Boulaaras, A. Choucha, D. Ouchenane, and B. Cherif, Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms, Adv. Differ. Equ. 2020 (2020), 310, DOI: https://doi.org/10.1186/s13662-020-02772-0.
  • [4] G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986.
  • [5] Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, ESAIM, Philadelphia, USA, 2006.
  • [6] A. Choucha, S. Boulaaras, and D. Ouchenane, Exponential decay of solutions for a viscoelastic coupled Lame system with logarithmic source and distributed delay terms, Math. Meth. Appl. Sci. 44 (2021), no. 6, 4858–4880, DOI: https://doi.org/10.1002/mma.7073.
  • [7] A. Choucha, S. Boulaaras, D. Ouchenane, and S. Beloul, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, logarithmic nonlinearity and distributed delay terms, Math. Meth. Appl. Sci. 44 (2021), no. 7, 5436–5457, DOI: https://doi.org/10.1002/mma.7121.
  • [8] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modèles non Linéaires de laIngénierie Pétrolière, Mathématiques and Applications 22, Springer-Verlag, Berlin Heidelberg, 1996.
  • [9] Z. Chen and R. E. Ewing, Fully discrete finite element analysis of multiphase flow in groundwater hydrology, SIAM J. Numer. Anal. 34 (1997), no. 6, 2228–2253, DOI: https://doi.org/10.1137/S0036142995290063.
  • [10] M. Afif and B. Amaziane, On convergence of finite volume schemes for one-dimensional two-phase flow in porous media, J. Comput. Appl. Math. 145 (2002), no. 1, 31–48, DOI: https://doi.org/10.1016/S0377-0427(01)00534-9.
  • [11] R. Eymard and T. Gallouët, Convergence d’un schéma de type éléments finis-volumes finis pour un système formé d’une équation elliptique et d’une équation hyperbolique, ESAIM Math. Model. Numer. Anal. 27 (1993), no. 7, 843–861, http://www.numdam.org/item/M2AN_1993__27_7_843_0/.
  • [12] T. Gallouët and J. P. Vila, Finite volume schemes for conservation laws of mixed type, SIAM J. Numer. Anal. 28 (1991), no. 6, 1548–1573, DOI: https://doi.org/10.1137/0728079.
  • [13] R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, Solution of Equation in N , Techniques of Scientific Computing (Part 3), 2000, pp. 713–1018, DOI: https://doi.org/10.1016/S1570-8659(00)07005-8.
  • [14] K. W. Morton, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London, 1996.
  • [15] R. D. Lazarov, I. D. Mishev, and P. S. Vassilevski, Finite volume methods for convection–diffusion problems, SIAM J. Numer. Anal. 33 (1996), no. 1, 31–55, DOI: https://doi.org/10.1137/0733003.
  • [16] C. M. Marle, Multiphase Flow in Porous Media, Editions TECHNIP, Paris, 1981.
  • [17] Y. Jingxue, On the uniqueness and stability of BV solutions for nonlinear diffusion equations, Comm. Partial Differ. Equ. 15 (1990), no. 12, 54–67, DOI: https://doi.org/10.1080/03605309908820743.
  • [18] P. Daripa and S. Dutta, Modeling and simulation of surfactant-polymer flooding using a new hybrid method, J. Comput. Phys. 335 (2017), 249–282, DOI: https://doi.org/10.1016/j.jcp.2017.01.038.
  • [19] P. Daripa and S. Dutta, On the convergence analysis of a hybrid numerical method for multicomponent transport in porous media, Appl. Numer. Math. 146 (2019), 199–220, DOI: https://doi.org/10.1016/j.apnum.2019.07.009.
  • [20] P. Daripa, J. Glimm, B. Lindquist, and O. McBryan, Polymer floods: A case study of nonlinear wave analysis and of instability control in tertiary oil recovery, SIAM J. Appl. Math. 48 (1988), no. 2, 353–373, DOI: https://doi.org/10.1137/0148018.
  • [21] D. Kröner, Numerical Schemes for Conservation Laws, Wiley & Teubner, New York, 1997.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-a6db6c98-4493-4595-8060-4a46b67931b5
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