Contaminant Fate and Reactive Transport in Groundwater - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 856

increasing volume of the flow domain, resulting in a contaminated zone broader
than expected from the effect of advective flow alone. Two processes contribute to
hydrodynamic dispersion: molecular diffusion and mechanical dispersion.
Aqueous molecular diffusion describes the mass transport of a contaminant due
to the random thermal motion of molecules and atoms, also known as Brownian
motion. The mass flux of contaminant F per unit cross-sectional area [M L − 2 T − 1 ]
depends on the concentration gradient and can be expressed by Fick's first law
which, for a one dimensional case, is written as:

D aq ∂

C
∂

F =− (19.6)

x

where D aq is the aqueous diffusion coefficient [L 2 T − 1 ].
In porous media aqueous diffusion is hindered by the tortuous nature of the pores,
the diminished cross sectional area available for diffusion and by the size of the
pores. Therefore, an effective pore diffusion coefficient can be defined as:

D aq n δ

D e =

(19.7)

τ f

where τ f is

the tortuosity defined as the square ratio of the effective path length in the pore l e
[L] and the shortest distance l [L]:

δ is a dimensionless factor ( ≤ 1) accounting for pore constrictivity and

l e
l

2

τ f = > 1 (19.8)

In most practical cases the pore size distribution and tortuosities are unknown
and only the porosity is known. Therefore, the effective diffusion coefficient is
often described, by empirical correlations, as a function of the aqueous diffusion
coefficient and the porosity (Grathwohl 1998 ):

D aq n m

D e = (19.9)

where m is an empirical exponent (mostly close to 2). Under transient transport
conditions D e is divided by a capacity factor which accounts for the storage of the
contaminant in the porous media (i.e. the porosity for tracers). Molecular diffusion,
acting in the direction of a decreasing concentration gradient, tends to equalize the
concentration differences along discrete flow paths and is essential for mixing of
solutes across different flow paths.
Mechanical dispersion can be defined as the contaminant spreading caused by
local variation of the flow velocity. At the microscopic scale, these velocity varia-
tions depend on the tortuosity, on the size of the pores and on the variable friction
within an individual pore, with faster flow close to pore axis and slower flow in
proximity of the solid particle surface (Fig. 19.3 ).