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.
