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0.3. Fluid Flow in Porous Media 7

is the saturation of the phase α. Here we suppose that the solid phase is
stable and immobile. Thus
α
ω α = φS α ω α
for a fluid phase α and

S α =1 . (0.10)
α:fluid
So if the fluid phases are immiscible on the micro scale, they may be miscible
on the macro scale, and the immiscibility on the macro scale is an additional
assumption for the model.
As in other disciplines the differential equation models are derived here
from conservation laws for the extensive quantities mass, impulse, and en-
ergy, supplemented by constitutive relationships, where we want to focus
on the mass.

0.3 Fluid Flow in Porous Media

Consider a liquid phase α on the micro scale. In this chapter, for clarity, we
d
write “short” vectors in R also in bold with the exception of the coordinate

3
vector x.Let ˜ α [kg/m ] be the (microscopic) density, ˜ q := η ˜ η ˜ v η ˜ α
α
[m/s]the mass average mixture velocity basedonthe particle velocity ˜ v η of
3
acomponent η and its concentration in solution ˜ η [kg/m ]. The transport
theorem of Reynolds (see, for example, [10]) leads to the mass conservation
law
˜
∂ t ˜ α + ∇· (˜ α ˜ q )= f α (0.11)
α
˜
with a distributed mass source density f α . By averaging we obtain from
here the mass conservation law
∂ t (φS α α )+ ∇· ( α q )= f α (0.12)
α
with α , the density of phase α, as the intrinsic phase average of ˜ α and
q ,the volumetric fluid velocity or Darcy velocity of the phase α,as the
α
extrinsic phase average of ˜ q . Correspondingly, f α is an average mass source
α
density.
Before we proceed in the general discussion, we want to consider some
specific situations: The area between the groundwater table and the imper-
meable body of an aquifer is characterized by the fact that the whole pore
space is occupied by a fluid phase, the soil water. The corresponding satu-
ration thus equals 1 everywhere, and with omission of the index equation
(0.12) takes the form
∂ t (φ )+ ∇· ( q)= f. (0.13)