February  2015, 8(1): 55-76. doi: 10.3934/dcdss.2015.8.55

Homogenization of thermal-hydro-mass transfer processes

1. 

Department of Mathematics, Soochow University, Suzhou 215006, China, China

Received  December 2012 Revised  February 2013 Published  July 2014

In the repository, multi-physics processes are induced due to the long-time heat-emitting from the nuclear waste, which is modeled as a nonlinear system with oscillating coefficients. In this paper we first derive the homogenized system for the thermal-hydro-mass transfer processes by the technique of two-scale convergence, then present some error estimates for the first order expansions.
Citation: Shixin Xu, Xingye Yue. Homogenization of thermal-hydro-mass transfer processes. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 55-76. doi: 10.3934/dcdss.2015.8.55
References:
[1]

G. Allaire, Homogénéisation et convergence à deux échelles, application à un problème de convection-diffusion, C. R. Acad. Sci. Paris, 312 (1991), 581-586.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[3]

G. Allaire, Homogenization of the unsteady Stokes equations in porous media, in Progress in partial differential equations: Calculus of variations, applications (eds. Bandle C, et al.) (Pont-à-Mousson, 1991), Pitman Research Notes in Math. Ser., 267, Longman Higher Education, Harlow, 1992, 109-123.  Google Scholar

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I. Babuška, Solution of problem with interfaces and singularities, in Mathematical Aspects of Finite Elements in Partial Differential Equations (ed. C. de Boor), Academic Press, New York, 1974, 213-277. Google Scholar

[5]

A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.  Google Scholar

[6]

G. S. Bodvarsson, W. Boyle, R. Patterson and D. Williams, Overview of scientific investigations at Yucca Mountain-the potential repository for high-level nuclear waste, J. Contam. Hydro., 38 (1999), 3-24. doi: 10.1016/S0169-7722(99)00009-1.  Google Scholar

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T. A. Buscheck, N. D. Rosenberg, J. A. Blink, Y. Sun and J. Gansemer, Analysis of thermo-hydrologic behavior for above-boiling and below-boiling thermal-operating modes for a repository at Yucca Mountain, J. Contam. Hydro., 62-63 (2003), 441-457. Google Scholar

[8]

Z. M. Chen, W. B. Deng and H. Ye, A new upscaling method for the solute transport equations, Discrete Contin. Dyn. Syst., Ser. A, 13 (2005), 941-960. doi: 10.3934/dcds.2005.13.941.  Google Scholar

[9]

Z. M. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp., 72 (2003), 541-576. doi: 10.1090/S0025-5718-02-01441-2.  Google Scholar

[10]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford University Press, 1999. Google Scholar

[11]

H. I. Ene and E. Sanchez-Palencia, Some thermal problems in flow through a periodic model of porous media, Int. J. Eng. Sci., 19 (1981), 117-127. doi: 10.1016/0020-7225(81)90054-9.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Berlin Heidelberg New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[14]

L. N. Guimaraes, A. Gens, M. Sanchez and S. Olivella, THM and reactive transport analysis of expansive clay barrier in radioactive waste isolation, Commun. Numer. Meth. Engng., 22 (2006), 849-859. doi: 10.1002/cnm.852.  Google Scholar

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H. Haller, Verbundwerkstoffe mit Formgedächtnislegierung: Mikromechanische Modellierung und Homogenisierung, Dissertation, TU-München 1997. Google Scholar

[16]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar

[17]

E. Marusic-Paloka and A. Piatnitski, Homogenization of a nonlinear convection- diffusion equation with rapidly oscillating coefficients and strong convection, J. Lond. Math. Soc., 72 (2005), 391-409. doi: 10.1112/S0024610705006824.  Google Scholar

[18]

G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[19]

O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[20]

S. B. Otsmane, G. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231.  Google Scholar

[21]

D. Patriarche, E. Ledoux, J. L. Michelot, R. S. Coincon and S. Savoye, Diffusion as the main process for mass transport in very low water argillites: 2. Fluid flow and mass transport modeling, Water Resource Research, 40 (2004), W01516. doi: 10.1029/2003WR002700.  Google Scholar

[22]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods Averaging and Homogenization, Springer, 2008.  Google Scholar

[23]

J. Rutqvist and C. F. Tsang, Analysis of thermal-hydrologic-mechanical behavior near an emplacement drift at Yucca Mountial, J. Contam. Hydro., 62-63 (2003), 637-652. Google Scholar

[24]

H. R. Thomas, Y. He, M. R. Sansom and C. L. W. Li, On the development of a model of the thermo-mechanical-hydraulic behaviour of unsaturated soils, Engineering Geology, 41 (1996), 197-218. doi: 10.1016/0013-7952(95)00033-X.  Google Scholar

[25]

C. F. Tsang, O. Stephansson and J. A. Hudso, A discussion of thermo-hydro-mechanical (THM) process associated with nuclear waste repositories, Int. J. Rock Mech. Min. Sci., 37 (2000), 397-402. doi: 10.1016/S1365-1609(99)00114-8.  Google Scholar

[26]

V. V. Zhikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogénéisation et convergence à deux échelles, application à un problème de convection-diffusion, C. R. Acad. Sci. Paris, 312 (1991), 581-586.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[3]

G. Allaire, Homogenization of the unsteady Stokes equations in porous media, in Progress in partial differential equations: Calculus of variations, applications (eds. Bandle C, et al.) (Pont-à-Mousson, 1991), Pitman Research Notes in Math. Ser., 267, Longman Higher Education, Harlow, 1992, 109-123.  Google Scholar

[4]

I. Babuška, Solution of problem with interfaces and singularities, in Mathematical Aspects of Finite Elements in Partial Differential Equations (ed. C. de Boor), Academic Press, New York, 1974, 213-277. Google Scholar

[5]

A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.  Google Scholar

[6]

G. S. Bodvarsson, W. Boyle, R. Patterson and D. Williams, Overview of scientific investigations at Yucca Mountain-the potential repository for high-level nuclear waste, J. Contam. Hydro., 38 (1999), 3-24. doi: 10.1016/S0169-7722(99)00009-1.  Google Scholar

[7]

T. A. Buscheck, N. D. Rosenberg, J. A. Blink, Y. Sun and J. Gansemer, Analysis of thermo-hydrologic behavior for above-boiling and below-boiling thermal-operating modes for a repository at Yucca Mountain, J. Contam. Hydro., 62-63 (2003), 441-457. Google Scholar

[8]

Z. M. Chen, W. B. Deng and H. Ye, A new upscaling method for the solute transport equations, Discrete Contin. Dyn. Syst., Ser. A, 13 (2005), 941-960. doi: 10.3934/dcds.2005.13.941.  Google Scholar

[9]

Z. M. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp., 72 (2003), 541-576. doi: 10.1090/S0025-5718-02-01441-2.  Google Scholar

[10]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford University Press, 1999. Google Scholar

[11]

H. I. Ene and E. Sanchez-Palencia, Some thermal problems in flow through a periodic model of porous media, Int. J. Eng. Sci., 19 (1981), 117-127. doi: 10.1016/0020-7225(81)90054-9.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Berlin Heidelberg New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[14]

L. N. Guimaraes, A. Gens, M. Sanchez and S. Olivella, THM and reactive transport analysis of expansive clay barrier in radioactive waste isolation, Commun. Numer. Meth. Engng., 22 (2006), 849-859. doi: 10.1002/cnm.852.  Google Scholar

[15]

H. Haller, Verbundwerkstoffe mit Formgedächtnislegierung: Mikromechanische Modellierung und Homogenisierung, Dissertation, TU-München 1997. Google Scholar

[16]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar

[17]

E. Marusic-Paloka and A. Piatnitski, Homogenization of a nonlinear convection- diffusion equation with rapidly oscillating coefficients and strong convection, J. Lond. Math. Soc., 72 (2005), 391-409. doi: 10.1112/S0024610705006824.  Google Scholar

[18]

G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[19]

O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[20]

S. B. Otsmane, G. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231.  Google Scholar

[21]

D. Patriarche, E. Ledoux, J. L. Michelot, R. S. Coincon and S. Savoye, Diffusion as the main process for mass transport in very low water argillites: 2. Fluid flow and mass transport modeling, Water Resource Research, 40 (2004), W01516. doi: 10.1029/2003WR002700.  Google Scholar

[22]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods Averaging and Homogenization, Springer, 2008.  Google Scholar

[23]

J. Rutqvist and C. F. Tsang, Analysis of thermal-hydrologic-mechanical behavior near an emplacement drift at Yucca Mountial, J. Contam. Hydro., 62-63 (2003), 637-652. Google Scholar

[24]

H. R. Thomas, Y. He, M. R. Sansom and C. L. W. Li, On the development of a model of the thermo-mechanical-hydraulic behaviour of unsaturated soils, Engineering Geology, 41 (1996), 197-218. doi: 10.1016/0013-7952(95)00033-X.  Google Scholar

[25]

C. F. Tsang, O. Stephansson and J. A. Hudso, A discussion of thermo-hydro-mechanical (THM) process associated with nuclear waste repositories, Int. J. Rock Mech. Min. Sci., 37 (2000), 397-402. doi: 10.1016/S1365-1609(99)00114-8.  Google Scholar

[26]

V. V. Zhikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

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