Page 124 - Materials Chemistry, Second Edition

Plume Migration in Aquifer and Soil 107

The travel of the dissolved COC in the vadose zone can be described by an
advection–dispersion equation, and its one-dimensional form is
2
( ∂θ w C) = ∂θ w DC) − ( ∂θ w vC) ± RXNs (3.37)
(
t ∂ ∂ z 2 ∂ z

This equation is similar to the one for the saturated zone (i.e., Equation
3.22), except the volumetric water content, θ , is a variable, and the velocity
w
and dispersion coefficient depend on the moisture content. The dispersion
coefficient is analogous to the dispersion term in the saturated zone, except v
is a function of the moisture content, as
D = D + D = ξ × D + α × v(θ ) (3.38)
0
h
d
w
Example 3.23: Estimate the Hydraulic Conductivity in the Vadose Zone
A subsurface soil is relatively sandy and has a hydraulic conductivity of 500
gpd/ft when the soil is saturated. Estimate its hydraulic conductivity (a)
2
when the water saturation is 40% and (b) when the water saturation is 90%.
The relative permeability for sand at 40% saturation is 0.02, and that at 90%
saturation is 0.44.

Solution:
(a) Use Equation (3.36) to find the hydraulic conductivity at 40%
saturation:
K = (0.02)(500) = 10 gpd/ft 2
(b) Use Equation (3.36) to find the hydraulic conductivity at 90%
saturation:
K = (0.44)(500) = 220 gpd/ft 2

Discussion:
The water saturation is the percentage of the pore space that is occu-
pied by the water: 100% for saturated soil and 0% for dry soil. At 40%
water saturation, the hydraulic conductivity is close to zero, and at
90% water saturation, the hydraulic conductivity is 44% of the maxi-
mum value.

3.6.2 Gaseous Diffusion in the Vadose Zone

Under nonpumping conditions, the molecular diffusion is the prime mecha-
nism for gas-phase transport. The transport equation can be expressed by
Fick’s law, and its one-dimensional form is
∂ 2 G G)
ξφ D = ∂φ ( a (3.39)
aa
∂ x 2 t ∂