Internal runoff structure
This paper explores the internal structure and implicit connectivity
within the modelled hillslope, in order to further generalise and
improve the rainfall-runoff model previously presented and examine its
internal structure. The model output that underlies this relationship
can be illustrated through examples. The storms have been generated on a
roughened surface draped across a uniform slope. Figure 2(a) shows the
contours on an example surface used here. Figure 2(b) shows total storm
discharge at every point on this surface, after applying a 120mm storm
at an intensity of 30 mm per hour. Discharge is in units of mm x cell
length
. The random convergences are sufficient to create local concentrations
of catchment area and discharge towards the base of the slope, with
discharge generally increasing with area, both downslope and laterally,
in convergent areas.
The relationship between discharge and area is illustrated explicitly in
figure 3, for two contrasting storm sizes, and with higher infiltration
rates in (b). Values for every cell across the grid are plotted for
three cross-slope transects near the top, middle and bottom of the
slope. The horizontal axis shows the areas drained to each individual
cell across the transect, with their average equal to the transect
value, x . It can be seen that the roughening of the surface
produces a wide range of areas within the transect. In each case there
are strong overall relationships, and the trend within individual
transects differs slightly from the overall trend. It is also clear from
the regression lines that discharge increases less than linearly with
area, and more strongly so for the smaller storm ad higher infiltration
(in figure 3b), so that runoff (discharge per unit area) is decreasing
with area drained. In a simulated storm, four stages of response can be
distinguished. Figure 4 shows two example hydrographs that illustrate
these stages.
- At the very start of a storm, infiltration capacity is theoretically
very large, following the Green-|Ampt expression\(f=A+B/S\) (1)
where f is the instantaneous infiltration rate (mm.
hr-1),S is the conceptual near-surface storage (mm: initially zero)
and A, B are the parameter values that are randomly and
independently distributed across grid cells. A is the steady
infiltration rate that conceptually leaks from the near-surface store
until it is exhausted, and represents the steady final long-term
infiltration rate. B controls the initial rapid infiltration
onto the near-surface store, S .
In this first stage, almost all rain infiltrates into the near surface
store, and there is only very limited runoff from saturated patches
close to the outlet.
- Quite soon near-surface storage increases, and, in the second stage,
infiltration rate is controlled by equation (1) over an increasing
proportion of the area. Average detention depths increase but
slope-base runoff increases only slowly, since much of the ponded
water is not connected to the slope base. In large storms, runoff may
reach an almost steady state, in which rainfall intensity is
partitioned between infiltration and runoff (figure 4b), whereas in
smaller storms (figure 4a) runoff continues to increase.
- A third stage begins when storm rainfall ends. Existing detention
continues to support infiltration, though over a shrinking area. This
allows further addition to the near-surface store for a while from the
shrinking ponded area. Average detention and runoff both decrease
sharply, with losses due to the runoff itself and the continuing
infiltration.
- In the final stage, all remaining water in the slope has infiltrated,
and the near-surface stores gradually drain into the soil beneath.
-
Figure 5 helps to further illustrate these stages of runoff and storage
for storms of different total size (8 – 480 mm) and intensity. In (a)
storms all have a duration of two hours, and so widely varying
intensities. In (b) the storms are at constant intensity of 60
mm.hour-1 and differing duration. In each case,
infiltration initially absorbs almost all rainfall, and the small volume
of runoff behaves as a power function of rainfall. For the fixed
duration storms in (a) the final storage increases only very slightly
with storm size and almost all additional rainfall is converted into
runoff. With the storms of fixed intensity in (b), final storage rises
significantly with increasing storm size., and only 82% of additional
rainfall contributes to runoff. These differences are primarily due to
the different durations of infiltration during rainfall and
post-rainfall saturation. In Figure 6, slope length has been added as an
additional variable. The effect of increasing slope length is seen in an
increase in the total available storage depth for large storms, and in
the exponent of runoff for small storms.
The relationships seen here may be described as showing two asymptotic
behaviours. For small storms infiltration approaches 100% of rainfall.
For large storms, total infiltrated storage approaches an upper limit
that increases primarily with storm duration, but also with slope
length, which is linked to the duration of runoff after the storm ends.
These two extreme behaviours are described by the relationships:
S = R for R <<Θ (2)
S = Θ for R >>Θ (3)
Where R = storm rainfall (mm),
S = Storm cumulative infiltration (mm)
and Θ = Storage threshold for cumulative infiltration (mm)
Empirically, the storage threshold may be expressed as
Θ=b +aT + c log2(L/L0) (4)
Where T is storm duration (hours),
and L is slope length (m)
For the simulation the constants a,b,c, L0 take
the values
a = 10 mm.hr -1; b = 10 mm; c= 2
mm; L0 = 2.5 m.
Repeated runs suggest that the constant [b – clog2(L0)] reflects the initially
declining infiltration rate [B in equation (1)]; the constanta reflects the long-term final infiltration rate [A in
equation (1)]. The constant c reflects the duration of runoff
after the end of rainfall, perhaps also reflecting long term
infiltration rate.
Combining the asymptotic expression of equation (2) and (3), It is
proposed to use the Michaelis-Mentem (Michaelis and Menten, 1913)/
Budyko (Budyko and Gerasimov, 1961) family of expressions, which take
the form
\(\frac{1}{S^{m}}=\ \frac{1}{R^{m}}+\frac{1}{\Theta^{m}}\) (5)
r=R-S (6)
where r = storm total runoff (mm), for some exponent m>1.
For runoff. this expression behaves asymptotically like
\(r=\frac{R^{m+1}}{\text{m\ }\Theta^{m}}\) for
R<<Θ (7)
\(r=R-\Theta+\frac{\Theta^{m+1}}{\text{m\ }R^{m}}\) for
R>> Θ (8)
and, at the cross-over point (R= Θ),
r = R.2-1/m for R= Θ (9)
These expression [equations (4)- (6)] provide an adequate
description of the runoff response across the range of storms. There is
a power law response for small storms, with the exponent m = 3
-5, and the runoff coefficient (r/ R) approaches 100% for the
largest storms. Figure 7 compares values of (a) total storm storage and
(b) total storm runoff obtained from the full simulation and from the
regression equations (4) to (6) above. It can be seen that there is a
satisfactory level of agreement in runoff over almost 6 orders of
magnitude. Figure 7(c) compares the full model storage with the SCS
method for Curve Numbers of 80 and 90, showing much greater divergences
from the simulated storage. Substantial improvements in forecast runoff
are also evident, particularly for smaller storms, although, to provide
a useful forecast of storm runoff, the effect for large storms is seen
as the more important. These expressions in equations (4) to (6) are
considered to provide an enhanced replacement for the SCS curve number
method.
The expression is relatively insensitive to the topography of the
sloping surface. If similar storms are applied to the roughened surface
of figure 1 and to a more strongly valleyed surface, estimated runoff
values lie within the confidence bands, perhaps because runoff
generation is a near-linear process. However, the different surfaces
have a profound influence on sediment transport. If, as a first
approximation, sediment transport is estimated as proportional to
discharge squared multiplied by gradient the pattern of sediment
transport strongly reflects the structure of ridges and valleys and is
then generally dominated by the differences in area drained.