Page 132 - gas transport in porous media

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Chapter 7: Gas-Phase Dispersion in Porous Media
and Webber, 1953). Effective diffusion coefficients can be estimated using tortuosi-
ties (see above) or can measured directly in the laboratory (Taylor, 1949; Bruce and
Webber, 1953; Currie, 1961; Shearer et al., 1973; Sallam et al., 1984; Johnson and
Perrott, 1991; Barone et al., 1992; Jin and Jury, 1996; Batterman et al., 1996; Schaefer
et al., 1997; Moldrup et al., 1998, 2000) and field (Raney, 1949; Lai et al., 1976;
Weeks et al., 1982; Kreamer et al., 1988).
The longitudinal dispersivity term scales with the degree of heterogeneity of the
physical system and is often measured using column-scale nonreactive tracer tests.
Gas-phase longitudinal dispersivities have been measured in laboratory porous media
systems and are found to range approximately between 0.2 and 2.9 cm (Popovi˘cová
and Brusseau, 1997; Ruiz et al., 1999; Garcia-Herruzo et al., 2000; Costanza-
Robinson and Brusseau, 2002). Dispersivities measured in the field tend to be
larger due to increased system heterogeneity. Ideally, dispersivities should be solute-
independent and insensitive to changes in carrier gas velocity or nonequilibrium
effects. However, if the data analysis fails to consider all relevant transport processes,
dispersivity can become a “lumped” solute-dependent parameter that no longer solely
represents physical heterogeneity of the porous medium (Costanza-Robinson and
Brusseau, 2002).
The Peclet number, P e , is a dimensionless measure of the degree of dispersion
experienced by a solute, defined alternatively as the ratio between the advective and
dispersive processes or the ratio of advective to diffusive processes (Rose, 1973). The
former definition is more encompassing and will be used here. The Peclet number is
usually obtained by fitting a solute breakthrough curve with an advective-dispersive
solute transport model. The magnitude of the Peclet number is inversely proportional
to the degree of dispersion. Thus, low Peclet numbers correspond to a large degree
of solute spreading. The Peclet number, also termed the Brenner number (Rose and
Passioura, 1971), can be related to the dispersion coefficient as follows:
vL
P e = (7.4)
D
where L is a characteristic length of the system (L). The characteristic length can
be defined at small- or macro-scales (e.g., grain diameter or column length) (Rose,
1973). The specific formulation of the Peclet number varies by application and field
of study. The macroscale length is used commonly in the fields of soil physics and
hydrology, while grain-scale lengths are often used in engineering disciplines. Thus,
caution should be exercised when interpreting absolute values of Peclet numbers or
when comparing values from different studies.

7.2.4 Functional Dependence of Dispersion and the Dispersion Coefficient
The dispersion coefficient is a function of the solute velocity, v. While the diffusion
term is independent of v, the mechanical mixing component of dispersion is propor-
tional to velocity (see Equation 7.3). Thus, D decreases with decreases in velocity.
However, slower velocities lead to larger residence times, thereby allowing more time