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Multiscale Computations for Flow and Transport in Heterogeneous Media

Efendiev, Yalchin and Hou, Thomas Y. (2008) Multiscale Computations for Flow and Transport in Heterogeneous Media. In: Quantum Treansport: Modelling, Analysis and Asymptotics - Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 11–16, 2006. Lecture Notes in Mathematics. No.1946. Springer , New York, NY, pp. 169-248. ISBN 978-3-540-79573-5.

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Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is due to disparity of scales. From an engineering perspective, it is often sufficient to predict macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. The purpose of this lecture note is to review some recent advances in developing multiscale finite element (finite volume) methods for flow and transport in strongly heterogeneous porous media. Extra effort is made in developing a multiscale computational method that can be potentially used for practical multiscale for problems with a large range of nonseparable scales. Some recent theoretical and computational developments in designing global upscaling methods will be reviewed. The lectures can be roughly divided into four parts. In part 1, we review some homogenization theory for elliptic and hyperbolic equations. This homogenization theory provides a guideline for designing effective multiscale methods. In part 2, we review some recent developments of multiscale finite element (finite volume) methods. We also discuss the issue of upscaling one-phase, two-phase flows through heterogeneous porous media and the use of limited global information in multiscale finite element (volume) methods. In part 4, we will consider multiscale simulations of two-phase flow immiscible flows using a flow-based adaptive coordinate, and introduce a theoretical framework which enables us to perform global upscaling for heterogeneous media with long range connectivity.

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Efendiev, Yalchin0000-0001-9626-303X
Additional Information:© 2008 Springer. The authors gratefully acknowledge financial support from US DOE under grant DE-FG02-06ER25727. T. Hou would like also to acknowledge a partial support from NSF grant ITR Grant ACI-0204932.
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG02-06ER25727
Series Name:Lecture Notes in Mathematics
Issue or Number:1946
Record Number:CaltechAUTHORS:EFElnm08
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11984
Deposited By: Archive Administrator
Deposited On:15 Oct 2008 23:38
Last Modified:08 Nov 2021 22:23

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