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6 0. Modelling Processes in Porous Media with Differential Equations

that we have used so far is not suitable for processes at the laboratory or
technical scale, which take place in ranges of cm to m, or even for processes
in a catchment area with units of km. For those macroscales new models
have to be developed, which emerge from averaging procedures of the mod-
els on the microscale. There may also exist principal differences among the
various macroscales that let us expect different models, which arise from
each other by upscaling. But this aspect will not be investigated here fur-
ther. For the transition of micro to macro scales the engineering sciences
provide the heuristic method of volume averaging, and mathematics the
rigorous (but of only limited use) approach of homogenization (see [36] or
[19]). None of the two possibilities can be depicted here completely. Where
necessary we will refer to volume averaging for (heuristic) motivation.
d
Let Ω ⊂ R be the domain of interest. All subsequent considerations are
formal in the sense that the admissibility of the analytic manipulations is
supposed. This can be achieved by the assumption of sufficient smoothness
for the corresponding functions and domains.
Let V ⊂ Ω be an admissible representative elementary volume in the
sense of volume averaging around a point x ∈ Ω. Typically the shape and
the size of a representative elementary volume are selected in such a manner
that the averaged values of all geometric characteristics of the microstruc-
ture of the pore space are independent of the size of V but depend on
the location of the point x. Then we obtain for a given variable ω α in the
phase α (after continuation of ω α with 0 outside of α) the corresponding
macroscopic quantities, assigned to the location x,as the extrinsic phase
average

1
ω α := ω α
|V | V
or as the intrinsic phase average

1
α
ω α := ω α .
|V α |
V α
Here V α denotes the subset of V corresponding to α.Let t ∈ (0,T )be
thetimeat which theprocess is observed. Thenotation x ∈ Ωmeans the
vector in Cartesian coordinates, whose coordinates are referred to by x,
y,and z ∈ R. Despite this ambiguity the meaning can always be clearly
derived from the context.
Let the index “s” (for solid) stand for the solid phase; then

φ(x):= |V \ V s | |V | > 0
denotes the porosity, and for every liquid phase α,

S α (x, t):= |V α | |V \ V s |≥ 0