Page 359 - gas transport in porous media

Chapter 22: Environmental Remediation of Volatile Organic Compounds
361
static, and at hydrostatic pressure. Therefore, the vertical water pressure gradient is
∂p l /∂z = ρ l g. Becausethegaspressureisrelatedtothewaterpressurebythecapillary
pressure, the vertical gas pressure gradient is
∂p g ∂p l ∂p c lg
= + (22.6)
∂z ∂z ∂z
If the vertical capillary pressure gradient is small compared to hydrostatic pressure
gradient, then the vertical gas pressure gradient can be approximated as the hydrostatic
water pressure gradient. Substituting into the multiphase Darcy’s law and neglecting
the gas density gives the approximate vertical sparge gas velocity

kk rg
v g ≈ ρ l g (22.7)
µ g
rearranging, the gas phase relative permeability is found to be linearly proportional
to the gas Darcy velocity, and inversely proportional to the intrinsic permeability:
v g µ g
k rg (S g ) = (22.8)
kρ l g
Thus, for a given darcy velocity, the relative permeability, and hence gas saturation
must be large if the intrinsic permeability is small, and the relative permeability and
gas saturation must be small if the intrinsic permeability is large.

22.3.4 Mass Transfer Issues at Different Scales
A number of laboratory air sparging experiments using dissolved VOCs have shown
strong nonequilibrium effects (Braida and Ong, 1998; Hein et al., 1998; Semer and
Reddy, 1998; Adams and Reddy, 1999). Attempts to simulate these laboratory experi-
ments assuming local chemical equilibrium between the gas and aqueous phases tend
to greatly overpredict the rate of removal of the dissolved VOC. Figure 22.6 shows the
effluent gas concentration of trichloroethylene (TCE) measured during a column air
sparging experiment by Hein et al. (1998). The experimental data are represented by
the open circles, and they show an initial concentration peak, followed by extensive
tailing. An attempt to simulate this experiment using a conventional multiphase flow
approach assuming local chemical equilibrium produces the solid line in the figure.
The local equilibrium model predicts rapid and complete removal of the TCE from
the column with no tailing.
The tailing of VOC concentrations during column sparging tests is believed to be
due to aqueous phase diffusion limitations [Clayton, 1998; Semer and Reddy, 1998;
Elder and Benson, 1999; Falta, 2000a]. A common approach for modeling this type
of kinetic interphase mass transfer is through a first order mass transfer reaction in
each gridblock (see, e.g., Sleep and Sykes, 1989; Gierke et al., 1992; Braida and Ong,
1998; Elder et al., 1999). Considering a two-phase gas/aqueous (no NAPL) system,
the rate of interphase mass transfer can be calculated by:
Q imt = k imt a(C l H − C g ) (22.9)