Geoscience Reference
In-Depth Information
Time
h
i
-1
j
-1
h
i
-1
j
h
i
-1
j
+1
∆
z
u
∆
z
i
Depth
j
-1
h
i
j
h
i
j
+1
h
i
∆
z
j
-1
h
i
+1
j
+1
h
i
+1
j
h
i
+1
∆
t
j
-1
∆
t
j
Known
Unknown
Figure 9.2
Spatial and temporal discretization used to solve Richards' equation.
that the vertical column is divided in compartments (don't confuse with natural soil
layers!) with calculation nodes in the centre. Both the time and space steps may vary
in length. The subscript
i
is used for the node number (increasing with depth) and
superscript
j
for the time level (increasing with time). At a certain time level
j
all the
state variables (
h, k
and
θ
) are known in the nodes depicted as a star in
Figure 9.2
.
The task of the numerical scheme is to calculate the new state variables at time level
j
+ 1 (depicted as open circles).
We may calculate the new state variables by solving for each compartment the
water balance. A irst approximation is given in
Figure 9.3
. All the values of the state
variables at time level
j
are known. We want to calculate
h
i
j
+ 1
. We may use the Darcy
luxes at the top and bottom of the compartment at time level
j
. The new pressure head
h
i
j
+1
is provided by the mass balance, as depicted in
Figure 9.3
. This scheme is a so-
called explicit inite difference scheme, which will work if the time steps are small. At
larger time steps this explicit scheme becomes unstable. The reason is that the water
luxes at time level
j
are used, although in fact the average water luxes during time
step
Δt
j
should be used. At larger time steps and longer simulation periods this may
result in substantial errors.
The inite difference scheme can be made stable if we use the luxes at the new
time level. Our starting point is again Richards' equation (
Chapter 4
):
∂
∂
+
h
z
∂
kh
()
1
∂
∂
=
h
t
Ch
()
−
Sz
()
(9.1)
∂
z
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