Geoscience Reference
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1000
R : arithmetic and geometric K , distance 0.1 cm
S 1: arithmetic K , distance 1 cm
S 3: arithmetic K , distance 5 cm
800
600
400
200
S2: geometric K ,
distance 1 cm
S 4: geometric K , distance 5 cm
0
0
0.02
0.04
Time (d)
0.06
0.08
0.1
Figure 9.4 Iniltration rate of sand in case of intensive rain at a dry soil as simulated
with geometric and arithmetic averages of hydraulic conductivity k at nodal distances
of 1 and 5 cm.
Question 9.1: Consider two adjacent nodes near the iniltration front in a sandy soil. The
upper node is in the wetted part and its state variables have the values h i -1 = 0 cm, θ i -1 =
0.431 cm 3 cm -3 and k i -1 = 9.65 cm d -1 . The lower node is still ahead of the wetting front with
state variables h i = -100 cm, θ i = 0.260 cm 3 cm -3 and k i = 0.12 cm d -1 . The vertical distance
between the nodes is 10 cm. Which soil water lux would you calculate between both nodes
if the hydraulic conductivity is arithmetically averaged? Which soil water lux in the case of
a geometric average of the hydraulic conductivity? And in the case of a harmonic average?
Above exercise shows that the arithmetic average inclines to the largest k , resulting
in high luxes, while the geometric and harmonic average tend to the lowest k , result-
ing in low luxes. If we would use different methods of averaging for runoff calcula-
tions, the results may deviate by a factor of 20 or more! The most suitable method
for averaging has been evaluated for extreme hydrological events with SWAP (Van
Dam and Feddes, 2000 ). One of the extreme events was an intensive rain shower of
100 mm in 0.1 d on a dry sand soil with θ = 0.1.
Figure 9.4 shows the calculated iniltration rates at the soil surface during the 0.1-
day period for various nodal distances and methods of averaging. The bold continu-
ous line is the theoretical iniltration curve. Initially the rain may iniltrate at a rate of
1000 mm d -1 into the dry sand soil. At t = 0.008 d, h at the soil surface becomes zero.
Gradually the iniltration rate declines, ultimately reaching the value of the saturated
hydraulic conductivity of the top soil. The total amount of iniltration is 39 mm out
of 100 mm of rainfall, the remaining amount is runoff. As Figure 9.4 shows, use of
arithmetic averages results in larger hydraulic conductivities and thus larger iniltra-
tion luxes than use of geometric averages.
In case of ∆ z i = 5 cm, arithmetic averages of k seriously overestimate the iniltration
rate (total = 47 mm) whereas geometric averages seriously underestimate the iniltra-
tion rate (total = 27 mm). The very steep wetting front due to low geometric k -averages
causes iniltration rate oscillations when the geometric average is used. When smaller
nodal distances are used (1 cm instead of 5 cm) the iniltration luxes calculated by
arithmetic and geometric averaging approach the theoretical curve. At ∆ z i = 0.1 cm,
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