Transport - Environmental Modeling - page 60

where the tortuosity t , with values

between 0 and 1 in Fick's Law for multi-phase systems, a formulation which was
already proposed by Bear ( 1972 ). 3 The factor

is defined as another factor, aside from factor y

t and the prolongation factor are thus

2 . The coefficient of the gradient, consisting of
three factors, can be termed effective diffusivity .

connected by the formula t ¼ 1 =#

D eff ¼ ytD mol

(3.9)

Care is advisable with the term 'effective,' because it is not used in the same way
in scientific literature. Sometimes the product of diffusivity and tortuosity, without
porosity, gets the predicate 'effective.' Sometimes the term effective is omitted at
all. In contrast to effective diffusivity the single phase diffusivity is often referred to
as molecular diffusivity .
Length prolongation and tortuosity are connected to the formation factor , which
is determined by electrical resistivity measurements. Using this technique, Archie
( 1942 ) found a power law relationship between porosity and formation factor,
which in terms introduced above can be noted as

m D mol or t ¼ y m 1

D eff ¼ y (3.10)

and can be found as Archie's Law in several publications (Sahimi 1993 ; Boudreau
1996 ).
Archie ( 1942 ) reports values for m (Eq. 3.10 ) of 1.8-2 for consolidated
sandstones, 1.3 for unconsolidated sand in a laboratory experiment, and 1.3-2 for
partly consolidated sand. For theoretical or conceptual work the value m ¼ 2is

considered, which may be justified if there is no further information. From ( 3.10 )
then follows:

2 D mol with formation factor

2 .
Boudreau ( 1996 ) provides an extensive overview of papers about tortuosity and
porosity and their relationship. Several fixed relationships between

t ¼ y , and from ( 3.9 ): D eff ¼ y y

have
been proposed. The relations given by Archie ( 1942 ), Weissberg ( 1963 ) and Iversen
and Jørgensen ( 1993 ) contain parameters that can be estimated based on measured
data. The latter propose the relation

t and '

2

# ¼ 1 þ n 1 ð y Þ with a typical value of

n ¼ 2 for sandy sediments. As Boudreau

( 1996 ) already noted, the resulting parameter curves are identical to those given
by the Burger-Frieke equation

3 for clay-silt sediments and of n ¼

2

with parameter a . Figure 3.2
depicts the curves for Archie's Law and the Iversen-Jørgensen equation for the
main parameter range.

# ¼ y þ a 1 ð y Þ

3

ϑ , which is here called the length prolongation factor, is often introduced as tortuosity (Boudreau
1996 ; Drewer 1997 ).