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A Numerical Method for an Inverse Problem Arising in Two-Phase Fluid Flow Transport Through a Homogeneous Porous Medium

  • Aníbal CoronelEmail author
  • Richard Lagos
  • Pep Mulet
  • Mauricio Sepúlveda
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

In this paper we study the inverse problem arising in the model describing the transport of two-phase flow in porous media. We consider some physical assumptions so that the mathematical model (direct problem) is an initial boundary value problem for a parabolic degenerate equation. In the inverse problem we want to determine the coefficients (flux and diffusion functions) of the equation from a set of experimental data for the recovery response. We formulate the inverse problem as a minimization of a suitable cost function and we derive its numerical gradient by means of the sensitivity equation method. We start with the discrete formulation and, assuming that the direct problem is discretized by a finite volume scheme, we obtain the discrete sensitivity equation. Then, with the numerical solutions of the direct problem and the discrete sensitivity equation we calculate the numerical gradient. The conjugate gradient method allows us to find numerical values of the flux and diffusion parameters. Additionally, in order to demonstrate the effectiveness of our method, we present a numerical example for the parameter identification problem.

Notes

Acknowledgements

We thank the anonymous reviewer for their insightful comments and suggestions. AC thanks to DIUBB 172409 GI/C, DIUBB 183309 4/R, and FAPEI at U. del Bío-Bío, Chile. RL thanks to PY-F1-01MF16 at U. de Magallanes, Chile. PM thanks to Spanish MINECO projects MTM2014-54388-P and MTM2017-83942-P and Conicyt PAI-MEC folio 80150006. MS thanks to Fondecyt 1140676 and BASAL project CMM, U. de Chile and CI2MA, U. de Concepción, and by Conicyt project Anillo ACT1118 (ANANUM).

References

  1. 1.
    M. Afif, B. Amaziane, On convergence of finite volume schemes for one-dimensional two-phase flow in porous media. J. Comput. Appl. Math. 145(1), 31–48 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Bürger, A. Coronel, M. Sepúlveda, On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels. Appl. Numer. Math. 56(10), 1397–1417 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. Bürger, A. Coronel, M. Sepúlveda, A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modeling sedimentation-consolidation processes. Math. Comput. 75(253), 91–112 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Bürger, A. Coronel, M. Sepúlveda, A numerical descent method for an inverse problem of a scalar conservation law modelling sedimentation, in Proceedings of ENUMATH 2007, Graz, Austria, September 2007, ed. by K. Kunisch, G. Of, O. Steinbach, pp. 225–232 (Springer, Heidelberg, 2008)Google Scholar
  5. 5.
    R. Bürger, A. Coronel, M. Sepúlveda, Numerical solution of an inverse problem for a scalar conservation law modelling sedimentation, in Hyperbolic Problems: Theory, Numerics and Applications. Proceedings of Symposia in Applied Mathematics, vol. 67, Part 2 (American Mathematical Society, Providence, 2009), pp. 445–454Google Scholar
  6. 6.
    A. Coronel, F. James, M. Sepúlveda, Numerical identification of parameters for a model of sedimentation processes. Inverse Prob. 19(4), 951–972 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Donat, F. Guerrero, P. Mulet, Implicit-explicit methods for models for vertical equilibrium multiphase flow. Comput. Math. Appl. 68(3), 363–383 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Y. Jingxue, On the uniqueness and stability of BV solution for nonlinear diffusion equations. Commun. PDE 15, 1671–1683 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type (Translated from the Russian by S. Smith). Translations of Mathematical Monographs, vol. 23 (AMS, Providence, 1968)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Aníbal Coronel
    • 1
    Email author
  • Richard Lagos
    • 2
  • Pep Mulet
    • 3
  • Mauricio Sepúlveda
    • 4
  1. 1.Departamento de Ciencias BásicasUniversidad del Bío-BíoChillánChile
  2. 2.Departamento de Matemática y Física, Facultad de CienciasUniversidad de MagallanesPunta ArenasChile
  3. 3.Departament de MatemàtiquesUniversitat de ValènciaBurjassotSpain
  4. 4.CI2MA & Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile

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