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6.1 Introduction
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high-saturation range increased non-linearly with decrease in saturation down
to a critical saturation, S cr , when dispersivity started to decrease with further
decrease in saturation. The maximum dispersivity, α max = 1.13 cm, occurred
at a saturation of S cr = 0.43.
Toride et al. [2003] have studied hydrodynamic dispersion in non-aggregated
dune sand to minimize the effect of immobile water. Experiments at unitgradient
flow (i.e., the water velocity is equal to the unsaturated conductivity)
were conducted to measure solute BTCs under steady-state flow conditions.
Transport parameters for the ADE and the mobile-immobile model (MIM) were
determined by fitting analytical solutions to the observed BTCs. They found
the maximum dispersivity, α max , of 0.97 cm occurring at S cr = 0.40, whereas
for saturated flow, dispersivity was equal to 0.1 cm, irrespective of pore-water
velocity, (which ranged from 2.08 to 58.78 m/d). The BTCs for unsaturated
flow showed considerable tailing compared with the BTCs for saturated flow.
This was also observed by Gupta et al. [1973] and Krupp and Elrick [1968].
The ADE described the observed data better for saturated than for unsaturated
conditions. Similar results were obtained by Padilla et al. [1999] for an
unsaturated sand with a mean particle size of 0.25 mm.
Since pore channels inside the porous medium are interconnected, solute particles
moving in different channels may meet after traveling different distances,
resulting in mixing of the solute. Hence, mixing length theory could be applied
to dispersion phenomena in a porous medium. Using mixing length theory,
Matsubayashi et al. [1997] found that, under unsaturated conditions, the dispersion
coefficient increases more rapidly with pore-water velocity compared to
saturated conditions, resulting in higher values of dispersivity. The dispersion
coefficient was expressed as [Matsubayashi et al., 1997]:
D (θ, v) = lσ vel = lc v (θ)v (6.3)
where l [L] is the mixing length, σ vel is the standard deviation of the porewater
velocity, v is the average velocity, and c v is the coefficient of variation of
pore-water velocity, defined as
c v = σ vel
v
=
(
v
′2 ) 1 2
v
(6.4)
where, v ′
≈ overlinev − v and v, is the micro-scale pore-water velocity. They
found an increase in dispersivity with a decrease in saturation, down to a critical
saturation (S cr ), beyond which the value of dispersivity was almost constant
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