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0.4. Reactive Solute Transport in Porous Media 13

and s η is decoupled and extracted from (0.30). If the time scales of transport
and reaction differ greatly, and the limit case k η →∞ is reasonable, then
(0.30) is replaced by

f η (x, c η (x, t),s η (x, t)) = 0 .
If this equation is solvable for s η , i.e.,
s η (x, t)= ϕ η (x, c η (x, t)) ,

the following scalar equation for c η with a nonlinearity in the time
derivative emerges:

∂ t (Θc η + b ϕ η (·,c η )) + ∇· (qc η − ΘD∇c η )= Q (1) .
η
If the component η is in competition with other components in the sur-
face reaction, as, e.g., in ion exchange, then f η has to be replaced by a
nonlinearity that depends on the concentrations of all involved components
c 1 ,...,c N , s 1 ,...,s N . Thus we obtain a coupled system in these variables.
Finally, if we encounter homogeneous reactions that take place solely in the
(1)
fluid phase, an analogous statement is true for the source term Q η .

Exercises

0.1 Give a geometric interpretation for the matrix condition of k in (0.16)
and D mech in (0.28).

0.2 Consider the two-phase flow (with constant α ,α ∈{w, g})
∂ t (φS α )+ ∇· q = f α ,
α
q = −λ α k (∇p α + α ge z ) ,
α
= 1 ,
S w + S g
=
p g − p w p c
with coefficient functions

p c = p c (S w ) , λ α = λ α (S w ) , α ∈{w, g}.
Starting from equation (0.23), perform a transformation to the new
variables
q = q + q , “total flow,”
w
g
S
1 1 λ g − λ w dp c
p = (p w + p g )+ dξ , “global pressure,”
2 2 S c λ g + λ w dξ
and the water saturation S w . Derive a representation of the phase flows in
the new variables.