Transport - Environmental Modeling - page 59

10 9 m 2 /s at 4 C(H 1.23 € afner et al. 1992 ). For the same gas the same reference gives

10 5 m 2 /s at 0 C for the molecular diffusivity in air.
In order to formulate Fick's Law in multi-phase systems, such as in porous
media, two modifications have to be made. At first it has to be taken into account
that the area through which diffusive flux may occur is only a part of the total area
(see control volume in Fig. 2.4). Usually it is assumed that the area is reduced by the
same factor as the volume. The volumetric share of the pore space, porosity, is thus
taken as the factor that measures the share of the active area. For that reason a factor
y appears on the right side of ( 3.5 ) if applied in porous media. In unsaturated porous
media y represents volumetric water content.
The second correction is necessary to take into account that diffusion
pathlengths are necessarily longer if several phases are present. The situation is
illustrated in Fig. 3.1 . While in single phase systems the shortest path is available
for diffusive fluxes of particles, in a multi-phase environment such direct connec-
tion is impeded by obstacles. As pathlengths are longer in multi-phases, the
diffusive flux in those systems is smaller than in a single phase case. One may
also say that pathlenghts are prolonged, which yields a factor

a value of 1.98

greater than 1 in the

ϑ

denominator of the concentration gradient.
Pathlength prolongation also has to be considered in the calculation of flux j. The
flux in normal direction is smaller than the flux following the generally non-normal
pathline. The combined effect of both corrections with the length prolongation
factor ϑ

leads to the equation:

1
#

j ¼

2 D mol rc

(3.7)

In sedimentological and geochemical science (Boudreau 1996 ; Drewer 1997 )
Fick's Law is formulated with the correction factor as given in ( 3.7 ), which goes
back to Carman ( 1937 ). Later Carman ( 1956 ) used the term tortuosity factor for ,

which is misleading in ( 3.7 ), where it appears in the denominator. We prefer here to
adopt a notation that is often found in groundwater literature:

ϑ

j ¼tD mol rc (3.8)

Fig. 3.1 Comparison of
diffusion pathlengths in
single and multi-phase
systems