For each soil to be modeled, point values of θu
2. The porous media are nondeformable as far
and hp are determined in a laboratory moisture
as moisture flux is concerned; i.e., consoli-
retention test (Ingersoll 1981). Gardner's eq 2 is
dation is negligible.
then fit to the data using a least squares approach
3. All processes are single valued; i.e., hyster-
to determine the best fit parameters Aw and α. Test
esis is not present in relationships such as
results for the Mn/DOT materials are given in the
the soil water characteristic curve.
first report of this series (Bigl and Berg 1996a).
4. Water flux is primarily as liquid; i.e., vapor
Unsaturated hydraulic conductivity is also ap-
flux is negligible.
proximated in FROST using a Gardner's equa-
The governing equation used in FROST to de-
tion:
scribe soil moisture flow is derived by substitut-
ing the extended Darcy moisture-flow law into
ks
KH =
the one-dimensional continuity equation for an
β
(3)
AK hp + 1
incompressible fluid flowing through porous
media, i.e.,
h θ
ρ
θi
(cm/hr)
KH = u + i
x
(1)
x
t ρw
ks = saturated hydraulic conductivity
t
(cm/hr)
where KH = unsaturated hydraulic conductivity
hp = pore pressure head (cm of water)
(permeability) (cm/hr)
AK = Gardner's multiplier for hydraulic
h = total hydraulic head (cm water)
conductivity
β = Gardner's exponent for hydraulic
x = depth (cm)
θu = volumetric unfrozen water content (%)
conductivity.
ρi = density of ice (g/cm3)
ρw = density of water (g/cm3)
Point values of KH and hp for each soil are deter-
θi = volumetric ice content (%)
mined in the laboratory by an unsaturated hydrau-
lic conductivity test (Ingersoll 1981), and, again,
t = time (hr).
Gardner's eq 3 is fit to the data using a least
squares approach to determine the best fit param-
The total hydraulic head h equals the sum of the
eters AK and β. See Bigl and Berg (1996a) for the
pore pressure head (hp = u/ρw), where u is the pore
test results on the Mn/ROAD materials.
water pressure, plus the elevation head (he = x),
Within the partially frozen zone, FROST re-
where x is measured vertically downward. The ice
sink term, ρi θi /ρw t, exists only in freezing or
duces the unsaturated hydraulic conductivity us-
ing an empirical constant, termed the E-factor,
thawing zones, and in these zones, eq 1 is coupled
combined with the ice content according to the
to the heat transport equation. The ice sink term
assumes that θi is a continuous function of time.
following equation:
In FROST, the soil water characteristics are
()
KF = KH hp 10- Eθi , Eθi ≥ 0
(4)
represented using a relationship in the form of
Gardner's (1958) equation:
θo
in a partially frozen element (cm/hr)
θu =
α
E = an empirical constant, dimensionless
(2)
+1
Aw hp
θi = volumetric ice content (%).
where θu =
volumetric unfrozen water content (%)
For this study, the E-factor was determined within
θo =
soil porosity (%)
the FROST program using an empirically derived
hp =
pore pressure head (cm of water)
equation based on the saturated hydraulic conduc-
Aw =
Gardner's multiplier for the mois-
tivity, ks, in centimeters/hour:
ture characteristics
α = Gardner's exponent for the moisture
(ks - 3)2 + 6 .
5
E=
(5)
characteristics.
4
3
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