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Chapter 6: Conservation Equations
Darcy scale. The identification with Darcy’s law comes from the fact that the no-slip
condition is typically applied at the fluid-solid interface illustrated in Level III. The
system at Level III in Figure 6.2 is illustrated in Figure 6.3 without the biolfilm. There
the fluid is identified as the γ -phase and the solid as the impermeable κ-phase. The
development of local volume averaged equations requires that we define two types
of averages in terms of the averaging volume, V, illustrated in Figure 6.3. The first
of these is the superficial average of some function ψ γ defined according to
1
ψ γ = ψ γ dV (6.112)
V
V γ
while the second is the intrinsic average is defined by
1
γ
ψ γ = ψ γ dV (6.113)
V γ
V γ
These two averages are related according to

γ
ψ γ = ε γ ψ γ (6.114)
in which ε γ is the volume fraction of the γ -phase defined explicitly as

V γ
ε γ = (6.115)
V

L

k - phase
g - phase
r o

g
Averaging
volume, V

Figure 6.3. Two-phase system