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Chapter V Transport and Relaxation in Two Component Media

The calculation of effective macroscopic physical properties from a geometrical characterization of the microstructure is the second subproblem of the central question discussed in the introduction. The present chapter will focus on single phase fluid transport and dielectric relaxation in porous media as representative examples for this general problem. The third subproblem of passage between microscopic and macroscopic length scales which has played an important role in section III.A.5 will become more prominent in this and the following chapter. The upscaling problem appears in the present chapter as the need to find effective macroscopic equations of motion from averaging the underlying microscopic equations of motion. Successive spatial averaging allows passage to larger and larger length scales, and it can be carried out using systematic expansions in the ratio of length scales or selfconsistent effective medium theories. The idea of a selfconsistently determined homogeneous reference medium is central to the definition of an effective macroscopic physical property. Asymptotic expansions in the ratio of a microscopic length scale to a macroscopic scale are known as homogenization theory, and will be discussed in sections V.C.3 and V.C.4. Their purpose is to provide a systematic method of identifying useful macroscopic reference properties. Once a useful macroscopic desription is identified a generalized form of effective medium theory can be employed to calculate the effective macroscopic properties.