American Institute of Mathematical Sciences

Advanced
May  2002, 2(2): 185-204. doi: 10.3934/dcdsb.2002.2.185

Analysis of upscaling absolute permeability

X.H. Wu 1, , Y. Efendiev 2, and  Thomas Y. Hou 3,

1. 

Applied & Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States
2. 

Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States
3. 

Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States

Received  January 2002 Published  February 2002

Flow based upscaling of absolute permeability has become an important step in practical simulations of flow through heterogeneous formations. The central idea is to compute upscaled, grid-block permeability from fine scale solutions of the flow equation. Such solutions can be either local in each grid-block or global in the whole domain. It is well-known that the grid-block permeability may be strongly influenced by the boundary conditions imposed on the flow equations and the size of the grid-blocks. We show that the upscaling errors due to both effects manifest as the resonance between the small physical scales of the media and the artificial size of the grid blocks. To obtain precise error estimates, we study the scale-up of single phase steady flows through media with periodic small scale heterogeneity. As demonstrated by our numerical experiments, these estimates are also useful for understanding the upscaling of general random media. It is further shown that the oversampling technique introduced in our previous work can be used to reduce the resonance error and obtain boundary-condition independent, grid-block permeability. Some misunderstandings in scale up studies are also clarified in this work.
Keywords: over-sampling., Upscaling, homogenization, resonance.
Mathematics Subject Classification: Primary: 65F10, 65F3.
Citation: X.H. Wu, Y. Efendiev, Thomas Y. Hou. Analysis of upscaling absolute permeability. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 185-204. doi: 10.3934/dcdsb.2002.2.185
[1]

Kundan Kumar, Tycho van Noorden, Iuliu Sorin Pop. Upscaling of reactive flows in domains with moving oscillating boundaries. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 95-111. doi: 10.3934/dcdss.2014.7.95
[2]

Zhiming Chen, Weibing Deng, Huang Ye. A new upscaling method for the solute transport equations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 941-960. doi: 10.3934/dcds.2005.13.941
[3]

Keaton Hamm, Longxiu Huang. Stability of sampling for CUR decompositions. Foundations of Data Science, 2020, 2 (2) : 83-99. doi: 10.3934/fods.2020006
[4]

Alexandre J. Chorin, Fei Lu, Robert N. Miller, Matthias Morzfeld, Xuemin Tu. Sampling, feasibility, and priors in data assimilation. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4227-4246. doi: 10.3934/dcds.2016.36.4227
[5]

Shixu Meng. A sampling type method in an electromagnetic waveguide. Inverse Problems and Imaging, 2021, 15 (4) : 745-762. doi: 10.3934/ipi.2021012
[6]

Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139
[7]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056
[8]

Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037
[9]

D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure and Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163
[10]

Philip Korman. Curves of equiharmonic solutions, and problems at resonance. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2847-2860. doi: 10.3934/dcds.2014.34.2847
[11]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems and Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709
[12]

Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems and Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033
[13]

T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[14]

Martin Hanke. Why linear sampling really seems to work. Inverse Problems and Imaging, 2008, 2 (3) : 373-395. doi: 10.3934/ipi.2008.2.373
[15]

Aku Kammonen, Jonas Kiessling, Petr Plecháč, Mattias Sandberg, Anders Szepessy. Adaptive random Fourier features with Metropolis sampling. Foundations of Data Science, 2020, 2 (3) : 309-332. doi: 10.3934/fods.2020014
[16]

Harun Karsli. On multidimensional Urysohn type generalized sampling operators. Mathematical Foundations of Computing, 2021, 4 (4) : 271-280. doi: 10.3934/mfc.2021015
[17]

Shixin Xu, Xingye Yue, Changrong Zhang. Homogenization: In mathematics or physics?. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1575-1590. doi: 10.3934/dcdss.2016064
[18]

Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1
[19]

Grégoire Allaire, Harsha Hutridurga. On the homogenization of multicomponent transport. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2527-2551. doi: 10.3934/dcdsb.2015.20.2527
[20]

Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (217)
  • HTML views (0)
  • Cited by (106)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy