Geoscience Reference
In-Depth Information
(Akay
et al.
, 2008); and CATFLOW (Zehe
et al.
, 2005; Klaus and Zehe, 2010). The complexities of
obtaining accurate solutions are such that developing solution techniques is best left to the specialist
numerical analyst, but the following points are worth remembering in assessing any package:
•
All the solution techniques for this type of nonlinear problem are approximate (see Box 5.3) and,
because of the nonlinearities involved, it is difficult to generalise about whether one method will be
more accurate than another for a particular problem.
•
For any solution algorithm that is consistent with the differential equation, accuracy will depend on
the time and space discretisation used. The finer the space increments (or the smaller the elements
used), then the shorter the time steps will have to be to ensure stability of the solution.
•
It may be necessary to use a very large number of nodes to represent the flow domain, especially in
three dimensions. This will then require the solution of a very large sparse matrix equation at least once
at each time step, together with the calculation of the nonlinear functions at each node. The computer
time required mounts rapidly with the number of nodes.
•
Problems where steep hydraulic gradients are expected, such as a wetting front during infiltration or
around a pumped well, will require small elements (and consequently small time steps) to represent
the gradient and flow velocities adequately in that part of the flow domain. This may seem obvious
but has not always been evident in published applications of distributed models.
•
Solutions with soil moisture
θ
as the dependent variable tend to be better for dry soil conditions;
solutions with capillary potential
ψ
as the dependent variable tend to be better for wet soil conditions.
•
Some solutions, particularly those known as
explicit
solutions in which the solution at time step
t
depends only on values of the nonlinear functions calculated at time step
t
−
1 (see Box 5.3), produce
unstable solutions if the time step is too big. Instability in the solution will normally be seen as
increasingly large oscillations in the solutions at some nodes. A good implementation of an explicit
solution scheme should check for stability, albeit at the expense of additional calculations, and adjust
the time step accordingly.
•
Implicit solutions
, which use variable and function values at both
t
and
t
−
1 (see Box 5.3), are generally
more stable and can use much longer time steps, but may involve a number of iterations at each time
step to converge on the solution at time
t
.
•
There are some stability problems inherent in the solution of the Richards equation, even using an
implicit time stepping scheme. This is because the nonlinear soil moisture capacity function (
C
(
ψ
)in
Equation (5.1)) peaks at a certain value of
ψ
. Thus it is possible for the solution at a node to oscillate
either side of that value of
ψ
, while still calculating an appropriate value of
C
(
ψ
).
•
In heterogeneous flow domains, the values of the parameters required at the grid or element scale may
be dependent on the scale of the elements. A Darcian description of the flow using effective parameter
values at the model element scale is not necessarily an adequate representation of the flow processes.
The effects of spatial heterogeneity of soil characteristics, time variability due to crusting and other
processes and preferential flow in structured soils remain topics of research with no generally accepted
descriptive models.
•
Always test the model, using different time and space steps, against some simple test cases. There is
no absolute guarantee that an approximate solution of a nonlinear partial differential equation will be
stable and accurate under all circumstances. Validation against some test cases will at best (but also at
least) give a guide.
5.1.2 Surface Runoff and Channel Routing
The physical basis of models of overland and channel flow is essentially the same. In both cases, in
catchment scale rainfall-runoff modelling, one-dimensional flows downslope or downstream are usually
