Geoscience Reference
In-Depth Information
For example, if the Philip approximation to the Richards
equation is employed (with the numerous caveats noted
above), the time to ponding (
t
p
[T]) can be estimated as
et al.
, 2006). The reality will inevitably lie somewhere
between these two extreme possibilities, depending on
surface conditions. The Darcy-Weisbach, Chezy and
Manning equations are the most widely used empiri-
cal resistance equations employed for velocity calcula-
tion in hydrological and erosion models (Jetten, de Roo
and Guerif, 1998; Takken
et al.
, 2005), but these equa-
tions were developed from either experiments in pipes or
observations from open-channel flows. Smith, Cox and
Bracken (2007) chart the historical development of these
equations and underline the difficulty of developing re-
sistance equations for steep channels and overland flows
where the assumptions underlying the more conventional
approaches limit their range of applicability. Equations to
predict resistance coefficients from measures of surface
roughness typically relate the velocity-profile 'roughness
height' of the Keulegan (1938) equation to grain-size mea-
surements (Robert, 1990; Clifford, Robert and Richards,
1992; Lawless and Robert, 2001), and so should not be
applied uncritically to overland flows where the velocity
profile is likely to be distinct from that predicted by the
'law of the wall' (Prandtl, 1935). The Keulegan equation
relates flow resistance to the ratio of flow depth
d
and a
roughness height
2
S
t
p
=
(11.4)
2[
r
−
i
f
]
where
S
is a parameter relating to the unsaturated infil-
tration rate [L/T
0.5
],
r
is the rainfall intensity [L/T] and
i
f
is the final infiltration rate at saturation [L/T]. A further
constraint of this equation is that it is only valid for con-
stant rainfall intensities, which, as noted above, are rare in
dryland storms. In reality, such calculations may be use-
ful for evaluating relative spatial differences in the onset
of runoff, but the complexities of the infiltration process
mean that absolute estimates derived in this way need to
be treated with caution.
At the plot scale, microtopography influences surface
depression storage (Onstad, 1984), infiltration and its vari-
ability (Fox, Le Bissonnais and Bruand, 1998), evapo-
ration (Allmaras
et al.
, 1977), solar radiation reflection
(Allmaras, Nelson and Hallauer, 1972), overland flow hy-
draulics (Gilley and Finkner, 1991; Takken and Govers,
2000), soil erosion and sediment transport (Helming,
Romkens and Prasad, 1998; Gomez and Nearing, 2005)
and water routing (Dunne, Moore and Taylor, 1975; Dar-
boux
et al.
, 2002). The detention of water at the soil sur-
face is particularly important where the infiltration rate is
slightly lower than the rainfall intensity and plays a reg-
ulatory role in the generation of surface runoff (Abedini,
Dickinson and Rudra, 2006). This situation of precipita-
tion excess is often found in semi-arid areas where high-
intensity storms fall on soils that may exhibit a relatively
low infiltration capacity. At larger scales surface rough-
ness also affects flowpaths of subsequent runoff, the or-
ganisation of drainage patterns and the connectivity of the
landscape (Govers, Takken and Helming, 2000; Bracken
and Croke, 2007). Somewhere within these complex rela-
tions, feedbacks and interrelationships between vegetation
including type, structure, age and pattern (Thornes, 1976;
Domingo
et al.
, 1998), surface gradient (Govers, 1991)
and soil (including type, distribution and crusting) (Poe-
sen, Ingelmo Sanchez and Mucher, 1990; Le Bissonnais,
1996) also influence runoff and flowpath generation.
) (see
also Ferguson, 2007). Similarly, the Manning-Strickler
approach (Manning, 1891; Strickler, 1923) calculates re-
sistance as a function of
ε
(defined as the inundation ratio
with the added approximation
that conveyance (the inverse of resistance) increases with
the sixth-root of the hydraulic radius (see Smith, Cox and
Bracken, 2007).
Where resistance formulae are applied to open chan-
nel flows, the focus is often simply on grain resistance.
However, studies of overland flows have shown that the
other components (e.g. form and wave resistance) make
a substantial contribution to total resistance (Abrahams,
Parsons and Hirsch, 1992; Parsons, Abrahams and Wain-
wright, 1994). The hydraulic equations for the friction
factor and laminar/turbulent flow are used to determine
the influence of roughness on surface flow. When rough-
ness elements are fully submerged, hydraulic rough-
ness theoretically declines with flow depth (an inverse
relationship between the Darcy-Weisbach friction fac-
tor and the Reynolds number; see Savat, 1980; Govers,
1992b; Bryan, 2000). However, when roughness elements
equal or exceed flow depth, the Darcy-Weisbach fric-
tion factor-Reynolds number relationship becomes pos-
itive (Abrahams, Parsons and Luk, 1986; Abrahams and
Parsons, 1991b; Nearing
et al.
, 1998). This results in the
decreasing importance of grain resistance (Govers and
Rauws, 1986; Abrahams, Parsons and Luk, 1986), and
sediment movement is controlled by form resistance. The
11.4.2
Flow hydraulics
Overland flow may be dominated by flow dynamics
(where infiltration can be ignored) or, alternatively, flow
