Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T13:39:48.651Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenized models for filtration and for acoustic wave propagation in thermo-elastic porous media

Published online by Cambridge University Press:  01 June 2008

A. MEIRMANOV
A. MEIRMANOV*
Affiliation:
Belgorod State University, 308015 Belgorod, Russia email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A system of differential equations describing the joint motion of thermo-elastic porous body and slightly compressible viscous thermofluid occupying pore space is considered. Although the problem is correct in an appropriate functional space, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification under various conditions imposed on physical parameters is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As a result, we derive Biot's system of equations of thermo-poroelasticity, a similar system, consisting of anisotropic Lamé equations for a thermoelastic solid coupled with acoustic equations for a thermofluid, Darcy's system of filtration, or acoustic equations for a thermofluid, according to ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 3 , June 2008 , pp. 259 - 284
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Acerbi, E., Chiado, Piat V., DalMaso, G. Maso, G. & Percivale, D. (1992) An extension theorem from connected sets and homogenization in general periodic domains. Nonlinear Anal. 18, 481–496.CrossRefGoogle Scholar
[2]Burridge, R. & Keller, J. B. (1981) Poroelasticity equations derived from microstructure. J. Acoustic Soc. Am. 70 (4), 1140–1146.CrossRefGoogle Scholar
[3]Biot, M. (1962) Generalized theory of acoustic propagation in porous dissipative media. J. Acoustic Soc. Am. 34, 1256–1264.CrossRefGoogle Scholar
[4]Clopeau, Th., Ferrin, J. L., Gilbert, R. P. & Mikelić, A. (2001) Homogenizing the acoustic properties of the seabed: Part II. Math. Comput. Model. 33, 821–841.CrossRefGoogle Scholar
[5]Gilbert, R. P. & Mikelić, A. (2000) Homogenizing the acoustic properties of the seabed: Part I. Nonlinear Anal. 40, 185–212.CrossRefGoogle Scholar
[6]Jikov, V. V., Kozlov, S. M. & Oleinik, O. A. (1994) Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, New York.CrossRefGoogle Scholar
[7]Lukkassen, D., Nguetseng, G. & Wall, P. (2002) Two-scale convergence. Int. J. Pure Appl. Math. 2 (1), 35–86.Google Scholar
[8]Meirmanov, A. (2007) Nguetseng's two-scale convergence method for filtration and seismic acoustic problems in elastic porous media. Siberian Math. J. 48 (3), 519–538.CrossRefGoogle Scholar
[9]Meirmanov, A. Darcy's law for a compressible thermofluid. Submitted to Asymptotic Analysis.Google Scholar
[10]Meirmanov, A. Acoustic and filtration properties of thermo-elastic porous media: Biot's equations of thermo-poroelasticity. Accepted for publication to Sbornik Mathematics.Google Scholar
[11]Meirmanov, A. M. & Sazhenkov, S. A. (2007) Generalized solutions to the linearized equations of thermoelastic solid and viscous thermofluid. Electron. J. Differential Equation. 41, 1–29.Google Scholar
[12]Nguetseng, G. (1989) A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623.CrossRefGoogle Scholar
[13]Nguetseng, G. (1990) Asymptotic analysis for a stiff variational problem arising in mechanics. SIAM J. Math. Anal. 21, 1394–1414.CrossRefGoogle Scholar
[14]Sanchez-Palencia, E. (1980) Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 129, Springer, Berlin.Google Scholar