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0.4. Reactive Solute Transport in Porous Media 11

0.4 Reactive Solute Transport in Porous Media

In this chapter we will discuss the transport of a single component in a
liquid phase and some selected reactions. We will always refer to water
as liquid phase explicitly. Although we treat inhomogeneous reactions in
terms of surface reactions with the solid phase, we want to ignore exchange
processes between the fluid phases. On the microscopic scale the mass con-
servation law for a single component η is, in the notation of (0.11) by
omitting the phase index w,
˜
∂ t ˜ η + ∇· (˜ η ˜ q)+ ∇· J η = Q η ,
where
2
J η := ˜ η (˜ v η − ˜ q)[kg/m /s] (0.26)
˜
3
represents the diffusive mass flux of the component η and Q η [kg/m /s]is
its volumetric production rate. For a description of reactions via the mass
action law it is appropriate to choose the mole as the unit of mass. The
diffusive mass flux requires a phenomenological description. The assump-
tion that solely binary molecular diffusion, described by Fick’s law,acts
between the component η and the solvent, means that
J η = −˜ D η ∇ (˜ η /˜ ) (0.27)
2
with a molecular diffusivity D η > 0[m /s]. The averaging procedure applied
on (0.26), (0.27) leads to
(1) (2) (1) (2)
∂ t (Θc η )+ ∇· (qc η )+ ∇· J + ∇· J = Q η + Q η
3
for the solute concentration of the component η, c η [kg/m ], as intrinsic
(1) (2)
phase average of ˜ η . Here, we have J as the average of J η and J ,
the mass flux due to mechanical dispersion, a newly emerging term at the
(1)
˜
macroscopic scale. Analogously, Q η is the intrinsic phase average of Q η ,
(2)
and Q η is a newly emerging term describing the exchange between the
liquid and solid phases.
The volumetric water content is given by Θ := φS w with the water
saturation S w . Experimentally, the following phenomenological descriptions
are suggested:
J (1) = −ΘτD η ∇c η

with a tortuosity factor τ ∈ (0, 1],
(2)
J = −ΘD mech ∇c η , (0.28)
and a symmetric positive definite matrix of mechanical dispersion D mech ,
which depends on q/Θ. Consequently, the resulting differential equation
reads

∂ t (Θc η )+ ∇· (qc η − ΘD∇c η )= Q η (0.29)