# American Institute of Mathematical Sciences

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

## Analysis of upscaling absolute permeability

 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.
Citation: X.H. Wu, Y. Efendiev, Thomas Y. Hou. Analysis of upscaling absolute permeability. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 185-204. doi: 10.3934/dcdsb.2002.2.185
 [1] Zhiming Chen, Weibing Deng, Huang Ye. A new upscaling method for the solute transport equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 941-960. doi: 10.3934/dcds.2005.13.941 [2] Kundan Kumar, Tycho van Noorden, Iuliu Sorin Pop. Upscaling of reactive flows in domains with moving oscillating boundaries. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 95-111. doi: 10.3934/dcdss.2014.7.95 [3] Alexandre J. Chorin, Fei Lu, Robert N. Miller, Matthias Morzfeld, Xuemin Tu. Sampling, feasibility, and priors in data assimilation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4227-4246. doi: 10.3934/dcds.2016.36.4227 [4] Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709 [5] Martin Hanke. Why linear sampling really seems to work. Inverse Problems & Imaging, 2008, 2 (3) : 373-395. doi: 10.3934/ipi.2008.2.373 [6] T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125 [7] Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems & Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295 [8] Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 2014, 34 (7) : 2847-2860. doi: 10.3934/dcds.2014.34.2847 [11] Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139 [12] Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056 [13] Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547 [14] Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911 [15] Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757 [16] Reuven Cohen, Mira Gonen, Avishai Wool. Bounding the bias of tree-like sampling in IP topologies. Networks & Heterogeneous Media, 2008, 3 (2) : 323-332. doi: 10.3934/nhm.2008.3.323 [17] Valery Y. Glizer, Vladimir Turetsky, Emil Bashkansky. Statistical process control optimization with variable sampling interval and nonlinear expected loss. Journal of Industrial & Management Optimization, 2015, 11 (1) : 105-133. doi: 10.3934/jimo.2015.11.105 [18] Anupama N, Sudarson Jena. A novel approach using incremental under sampling for data stream mining. Big Data & Information Analytics, 2017, 2 (5) : 1-13. doi: 10.3934/bdia.2017017 [19] Lorenzo Audibert. The Generalized Linear Sampling and factorization methods only depends on the sign of contrast on the boundary. Inverse Problems & Imaging, 2017, 11 (6) : 1107-1119. doi: 10.3934/ipi.2017051 [20] Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1

2016 Impact Factor: 0.994