Geoscience Reference
In-Depth Information
+
( )
−
()
hz
∆
zt
,
hzt
,
∂
∂
≡
∂
h
z
h
z
(4.17)
=
li
z0
∂
∆
z
∆
→
t
where ( )
t
means that the derivative is evaluated at constant
t
. Partial derivatives
are required for the mathematical description of transient (time-dependent) low. If
the system is at steady state, the partial derivatives reduce to an ordinary derivative
because in steady state
h
depends only on
z
.
The unsaturated hydraulic conductivity is a strongly nonlinear function of water
content or soil water pressure head.
Figure 4.19
shows typical curves for a coarse
textured (sandy) and a ine-textured (clay) soil. At saturation, the coarse textured soil
has a higher conductivity than the ine-textured soil, because it contains large pore
spaces, which are illed with water. However, these pores drain at modest suctions,
producing a dramatic decrease in hydraulic conductivity in the sandy soil. Eventu-
ally, the curves will cross and the sandy soil will actually have a lower hydraulic
conductivity than the clayey soil at the same matrix potential, because the latter will
retain considerable more water and will contain a larger number of illed pores (Jury
et al.,
1991
).
4.6 Richards' Equation for Water Flow in Variably
Saturated Soils
Let's consider the water balance of a small, cubic volume of soil with one-dimen-
sional, vertical water low (
Figure 4.20
). The amount of water lowing into the ele-
mentary cubic at the bottom,
Q
bottom
(kg d
-1
), equals:
Qq
=
ρ
dd
xy
(4.18)
bottom
where
ρ
is the water density (kg m
-3
) and d
x
and d
y
are the horizontal cube
sides (m).
When the vertical cube side d
z
(m) approaches zero, the amount of water lowing
out of the cubic at the top,
Q
top
, can be calculated with the irst derivative only:
=
∂
( )
∂
q
z
ρ
Qq
ρ
ddd
zxy
(4.19)
top
The water balance of the cube can than thus be written as:
∂
( )
∂
− +
∂
( )
∂
θρ
q
z
ρ
ddd
xyzq xy
=
ρ
d d
q
ρ
d
zxyS xyz
dd
−
ρ
ddd
(4.20)
z
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