Geoscience Reference
In-Depth Information
∆
t
j
h
i-1
j
j
j
h
i
-1
-
h
i
j
q
i
=
-
K
i-
1/2
-
K
i
-1/2
q
i
j
∆
z
u
∆
z
u
soil compartment
i
h
i
j
h
i
j
+
1
S
i
j
∆
z
i
∆
z
ℓ
j
j
h
i
-
h
i
+1
j
q
i
+1
=
-
K
i
+1/2
-
K
i
+1/2
q
i
+
1
j
∆
z
ℓ
Mass change = In - Out
h
i
+
1
j
j
(
j
+1
j
j
)
)
(
j
j
∆
t
j
C
i
h
i
-
h
i
∆
z
i
=
q
i
+1
q
i
S
i
--
Figure 9.3
Straightforward numerical discretization of Richards' equation.
where
C
is the differential water capacity (m
-1
),
h
the soil water pressure head (m),
k
the hydraulic conductivity (m d
-1
) and
S
the root water extraction rate (d
-1
). A straight-
forward, inite difference scheme with luxes at the new time level
j
+ 1 and the differ-
ential water capacity halfway the new and old time level (
j
+ 1/2), is:
j
h
j
+
1
,
p
−
h
j
+
1
,
p
h
j
+
1
,
p
−
h
j
+
1
,
p
∆
∆
t
z
( )
=
j
+
½
j
+
1
j
j
j
j
j
Ch h
k
i
−
1
i
+
k
−
k
i
i
+
1
−
k
i
i
i
i
−
½
i
−
½
i
+
½
i
+
½
∆
z
∆
z
i
u
j
j
−
∆
tS
(9.2)
i
We call this scheme implicit, as the new pressure head
h
i
j
+1
is a function of itself and
can be solved only by iteration (the subscript
p
indicates the iteration step). Although
this scheme may work well for ordinary ield conditions, it is not accurate to simulate
rapid hydrological events such as intensive rain showers on dry soils or fast luctuations
of the groundwater table near the soil surface. In such cases, numerical errors origi-
nate from two main sources: the averaging of the hydraulic conductivity
k
between the
nodes, and the averaging of the water capacity
C
during the time step. Let's see how
we can address these error sources.
To determine the average
k
between the nodes, different methods can be used,
for example, arithmetic, geometric and harmonic. If we view the system as a series
of layers with different
k
, the harmonic average would seem the most appropriate
(
Chapter 4
). Especially at sharp wetting fronts, the averaging method may have a
large impact on the calculated soil water lux, as illustrated in
Question 9.1
.
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