# American Institute of Mathematical Sciences

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. Google Scholar [2] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. 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, (1991), 109. 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), (1974), 213. Google Scholar [5] A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North-Holland, (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. 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), 62. 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., 13 (2005), 941. 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. 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, (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. doi: 10.1016/0020-7225(81)90054-9. Google Scholar [12] L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998). Google Scholar [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, $2^{nd}$ edition, (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. doi: 10.1002/cnm.852. Google Scholar [15] H. Haller, Verbundwerkstoffe mit Formgedächtnislegierung: Mikromechanische Modellierung und Homogenisierung,, Dissertation, (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. 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. 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, (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. 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). 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), 62. 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. 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. 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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##### 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. Google Scholar [2] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. 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, (1991), 109. 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), (1974), 213. Google Scholar [5] A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North-Holland, (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. 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), 62. 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., 13 (2005), 941. 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. 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, (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. doi: 10.1016/0020-7225(81)90054-9. Google Scholar [12] L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998). Google Scholar [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, $2^{nd}$ edition, (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. doi: 10.1002/cnm.852. Google Scholar [15] H. Haller, Verbundwerkstoffe mit Formgedächtnislegierung: Mikromechanische Modellierung und Homogenisierung,, Dissertation, (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. 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. 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, (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. 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). 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), 62. 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. 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. 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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