Geoscience Reference
In-Depth Information
as we have seen, they make significant simplifying assumptions to allow a model that is computationally
feasible. They also have the important advantage that they make predictions that are distributed in space
so that the effects of partial changes to the catchment and the predicted spatial dynamics of the processes
can be assessed. They have important disadvantages in the computational resources required and in the
problems of specifying the huge numbers of parameters required over all the spatial elements of the model.
These disadvantages have led to the investigation of simplified distributed models of two main types: those
based on kinematic wave theory (considered in the remainder of this chapter) and probability distributed
and semi-distributed models (considered in Chapter 6), in which elements with similar characteristics
are grouped together to reduce the calculations required.
5.5.1 Kinematic Wave Models
Kinematic wave models are simplified versions of the surface or subsurface flow equations, resulting
from making additional approximations. The first such model reported in the literature was, in fact, a
grid-based model of surface runoff developed by Merrill Bernard in 1937 (see Hjelmfelt and Amerman,
1980). In another early study, Keulegan (1945) analysed the magnitude of the various terms of the
St. Venant equations for shallow surface runoff over a sloping plane and concluded that a simplified
equation, essentially the kinematic wave equation, was an adequate approximation. The flow routing
of the Huggins and Monke (1968) grid-based model was also effectively a kinematic wave solution.
There are some problems in applying kinematic wave principles in two-dimensional cases (see Section
5.5.5) and most models based on the assumptions have used a catchment discretisation based on one-
dimensional hillslope planes (as in Figure 5.4). Early models used fixed width planes or planes with
radial symmetry to allow analytical solutions to be made, but it is now easy to implement variable width
planes in numerical solutions (see Box 5.7).
All kinematic wave models are combinations of the continuity equation with a storage-flow relationship
(Box 5.7). Generally some simple mathematical function is used for the storage-flow relationship but this
is not strictly necessary. Numerical solutions can use any function represented as a look-up table (even
hysteretic functions, although to my knowledge hysteretic functions have never been tried in rainfall-
runoff modelling based on the kinematic wave approximation). However, the resulting models are flexible
and relatively easy to implement, with many analytical solutions available for simple boundary conditions.
A comprehensive coverage of kinematic wave theory and its application in surface hydrology has been
published by Singh (1996). The general kinematic wave equation for a variable width flow domain has
the form:
W
x
∂h
∂t
=−
c
∂W
x
h
+
W
x
r
(5.6)
∂x
where
h
is a depth of flow,
W
x
is the width of the slope or channel,
r
is an inflow rate per unit area of slope
or channel and
c
is the kinematic wave velocity or
celerity
which is, in general, a function of flow depth
(but may be a constant in some special cases). The form of that function will vary with the relationship
between downslope flow rate and depth of flow (see Box 5.7 for surface and subsurface flow examples).
There is one important limitation of using kinematic wave models, even in applications to one-
dimensional systems. Unlike both the Richards equation for subsurface flow and the St. Venant equations
and diffusion wave analogy for surface flow, the kinematic wave equation cannot reproduce the effects
of a downstream boundary condition on the flow. Essentially, the effects of any disturbance to the flow
generate a kinematic wave but the equation can only predict the downslope or downstream movement
of these waves. Thus, a kinematic wave description cannot predict the effects of the drawdown of a wa-
ter table due to an incised channel at the base of a hillslope or the backwater effects of an obstruction to
the flow for a surface flow. This has led to a number of theoretical studies of the conditions under which
the kinematic approximation is a valid approximation to a more complete description (see Box 5.7),
