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.
  1. 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.
  2. 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.
  3. 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.
  4. 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.