Page 129 - gas transport in porous media

Costanza-Robinson and Brusseau
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applied to the transport of gases in unsaturated systems, including the concepts of
dispersion. However, any rigorous or quantitative analysis of gas-phase systems
requires consideration of the unique complexities of unsaturated systems and the
properties of gases themselves. For example, unsaturated porous media have air-
filled porosities that may vary both spatially- and temporally, and depend on such
factors as soil-water content and particle-size distribution. Gas-phase diffusion coeffi-
cients are generally four to six orders of magnitude larger than aqueous-phase values,
and in contrast to water, gases are significantly affected by pressure-temperature rela-
tionships. Gases experience slip-flow along pore walls, often termed the Klinkenberg
effect, while water does not. In the following discussion, the authors use the terms
“gas” and “vapor” interchangeably, while the term “solute” refers very broadly to the
gas/vapor of interest.

7.2 THEORY

7.2.1 Diffusion
Dispersion includes diffusive and mechanical mixing components. Gas-phase diffu-
sionis often assumed to be dominated by molecular diffusion, the random spreading of
a solute along concentration gradients over time, described here by a one-dimensional
Fick’s second law:
2
∂C ∂ C
= D a (7.1)
∂t ∂x 2
−3
where C is the gas concentration (M·L ), t is time (T), D a is the binary molecular
2
diffusion coefficient in air (L ·T −1 ), and x is the distance along the axis of flow (L).
For molecular diffusion, molecule-molecule collisions are the only type of collisions
that occur, implying a system without walls. In some cases, more complex gas-phase
diffusion processes may also occur, including viscous, Knudsen, and nonequimolar
diffusion (e.g., Scanlon et al., 2000). The former two processes occur due to the
presence of pore walls and consequent molecule-wall collisions (Cunningham and
Williams, 1980), while the latter requires both the presence of walls and a multicom-
ponent gas (i.e., a mixture). Such conditions are present in porous media and may
lead to deviations from Fick’s law (e.g., Sleep, 1998). Baehr and Bruell (1990) report
that high vapor pressures, such as those achieved near organic liquid sources, also
cause deviation from Fick’s law. In accordance with the bulk of the literature, this
chapter will focus on molecular diffusion (e.g., assuming a system without walls),
while other diffusion processes are discussed in detail elsewhere.
Diffusion is a solute-dependent component of dispersion, due to the relationships
among average kinetic energy, velocity, and molecular mass. At a given temperature
the average kinetic energy of all gases will be equal and described as:
3 1 2
E k = kT = mv rms (7.2)
2 2