2
(
)
K = K sat w ⎡1 - 1 - w1/ mvG
⎤
mvG
0.5
(5.13)
⎢
⎥
⎣
⎦
where
θw - θr
w=
.
(5.14)
θmax - θr
A plot of Equation (5.13) as a function of θ is presented in Figure 7.2 along with the
hydraulic gradient, ∂θw/∂ψ used in Equation (5.11).
In a layered system, the average hydraulic conductivities Ki1/2 and Ki+1/2 found in
Equations (5.7), (5.9) and (5.10) equal
Ki Ki-1∆zi ∆zi-1
Ki-1/ 2 =
Ki ∆zi-1 + Ki-1∆zi
(5.15)
Ki Ki+1∆zi ∆zi+1
Ki+1/ 2 =
.
Ki ∆zi+1 + Ki+1∆zi
Their derivatives with respect to the pressure head are of the general form
∂Ki-1/ 2 ∂Ki-1/ 2 ∂Ki-1 ∂wi-1 ∂θwi-1
=
∂Ki-1 ∂wi-1 ∂θwi-1 ∂Ψi-1
∂Ψi-1
∂Ki-1/ 2 ∂Ki-1/ 2 ∂Ki ∂wi ∂θwi
=
∂Ki ∂wi ∂θwi ∂Ψ i
∂Ψi
(5.16)
∂Ki+1/ 2 ∂Ki+1/ 2 ∂Ki+1 ∂wi+1 ∂θwi+1
=
∂Ki+1 ∂wi+1 ∂θwi+1 ∂Ψ i+1
∂Ψ i+1
∂Ki+1/ 2 ∂Ki+1/ 2 ∂Ki ∂wi ∂θwi
=
∂Ki ∂wi ∂θwi ∂Ψ i
∂Ψi
The problem is now fully developed and comparisons may be made between the model
predictions and actual field measurements.
5.1.4 Model Validation
As for the soil temperature model verification, we used data from the SWOE program to
validate the model. The specific locations are Grayling, MI, and Yuma, AZ. In both
cases, soil moisture by percent weight measurements were taken essentially once a day.
Researchers at CRREL (Cold Regions Research and Engineering Laboratory) and WES
(Waterways Experiment Station) made independent measurements. The results for
Grayling, MI, are shown in Figure 5.3. The percent weight measurements were converted
to volumetric soil moisture values to be consistent with the model output. In doing so, we
assumed a soil density of 1.49 g/cm3. Also for the Grayling comparisons, an initial
surface soil moisture of 0.05 m3/m3 was assumed based on the measured data. The model
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