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Asymptotic Behavior of a Solution of Relaxation System for Flow in Porous Media

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Theory, Numerics and Applications of Hyperbolic Problems I (HYP 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 236))

Abstract

We introduce a novel modeling of phase transitions in thermal flow in porous media by using hyperbolic system of balance laws, instead of system of conservation laws. We are interested in two different behaviors of the balance system: the long time behavior, in which we study the solution with fixed relaxation term and very large time; and the behavior of the solution when the relaxation term is taken to zero and the time is fixed. We also are interested in solving the question: “Does this balance system tend to the conservation system under equilibrium hypothesis?”. To answer this question, we introduce a projection technique for the wave groups appearing in the system of equations and we study the behavior of each group. For a particular Riemann datum, using the projection method, we show the existence of a decaying traveling profile supported by source terms and we analyze the behavior of this solution. We corroborate our analysis with numerical experiments.

E. Abreu thanks for financial support through grants FAPESP No. 2014/03204-9, CNPq No. 445758/2014-7 and UNICAMP/FAEPEX No. 519.292-0280/2014. W. Lambert was supported through FAPERJ grant No. E-26/110.241/2011. A. Bustos thanks FAPESP for a graduate fellowship through grant No. 2011/23628-0.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Campinas, Campinas, SP, 13083-970, Brazil

    E. Abreu

  2. Universidad Sergio Arboleda, Cra. 13, numbers 74-64, Bogotá, Colombia

    A. Bustos

  3. ICT - UNIFAL, Cidade Universitária - BR 267 Km 533 Rodovia José Aurélio Vilela, Poços de Caldas, Alfenas, MG, Brazil

    W. J. Lambert

Authors
  1. E. Abreu
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  2. A. Bustos
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  3. W. J. Lambert
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Correspondence to E. Abreu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Würzburg University, Würzburg, Germany

    Christian Klingenberg

  2. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Michael Westdickenberg

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Abreu, E., Bustos, A., Lambert, W.J. (2018). Asymptotic Behavior of a Solution of Relaxation System for Flow in Porous Media. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_2

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