Environmental Engineering Reference
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main, then the use of effective parameters might
be a useful approximation (though the effective
value of hydraulic conductivity might not easily
be related to the distribution of point scale values).
However, where the subsurface interacted with
surface runoff then no consistent effective param-
eter values could be found.
There is a related issue as to whether the con-
tinuum equations described by Freeze and Harlan
are an adequate description of the actual flow
processes of surface and subsurface runoff in real
catchments. Irregularities in flow pathways lead-
ing to depth variability in surface runoff; the
effects of 3D channel geometry in streamflows;
and preferential flows in the soil might all mean
that the depth and velocity averaged St-Venant
equations andDarcy-Richards equationmight not
be adequate representations of the actual flow
processes (see, e.g., Beven and Germann, 1982;
Beven 1989, 2001b, 2006a, 2010). Some attempts
have been made to address these limitations, such
as using dual porosity soil characteristics or two
flow domains in Darcy-Richards (e.g. Brontstert
and Plate 1997) or simulations of hypothetical
hillslopes with preferential flow elements (e.g.
Weiler and McDonnell 2007; Neiber and
Sidle 2010; Klaus and Zehe 2010), but there is no
current agreement on what type of formulation
should be used for real hillslopes, or how the
(effective) parameters of a new formulation
should be identified in any application to a real
catchment.
The second issue is to know what the effective
parameter values might be for every element in
the model representation of a catchment. Even if
we assume that effective parameter values are
a useful approximation, and even if wemight have
some information about the variability in point-
scale soil and surface characteristics, then defin-
ing effective parameter values for every element in
the catchment discretization will again require
simplification and approximation. We cannot
make measurements for every element in the
discretization, and getting any information at all
becomes much more difficult as the hydrological-
ly active depth becomes deeper (especially if it
includes fractured bedrock layers).
with the advantage that the grid of solution
elements is easily made finer where more detail
is required in the representation of gradients or
simulation outputs.
The limitations of the Freeze and Harlan
blueprint
Freeze and Harlan blueprint provides a theoreti-
cally consistent continuum differential equation
approach to defining a distributed hydrological
model. As such it has inherent attractions in
providing a structure that goes from local contin-
uum to catchment scales. There are, however,
three essential limitations in applying these con-
cepts. The first is the solution of the differential
equations. Since the differential equations are
nonlinear and subject to arbitrary changes in
boundary conditions in both space and time, they
cannot be solved analytically but we necessarily
have to resort to approximate numerical solutions
on a spatial discretization of the catchment.
Except for very small catchments (e.g. Ebel and
Loague 2006; Ebel et al. 2008), the discretization
will bemuch larger than the 'point' scales atwhich
the flux equations, such as the Darcy-Richards
equation, hold. Thus, unless there is homogeneity
at the sub-discretization scale, gradient terms and
fluxes should not be expected to be well repre-
sented. The effects of heterogeneity in the subsur-
face gradients and hydraulic conductivity fields do
not average linearly so that even if the Darcy-
Richards equation applies at the point scale, the
physics suggests we should be using a different
equation at the element scale. Thus Beven (1989,
2002), for example, argued that this type of dis-
tributed model should be considered as lumped at
the element scale.
This is the case even if we assume that any
effects of sub-discretizationheterogeneities can be
allowed for by fitting an
effective
parameter value
at the grid scale (Beven 1989). The result will not
be a solution to the point scale equations where
the flow domain is heterogeneous (this applies to
both surface and subsurface fluxes). Binley
et al. (1989b) suggested that for the case of purely
Darcian subsurface flow in a heterogeneous do-
