Rationale
Lateral near-surface flow has been envisaged as generated, either where
rainfall intensity exceeds the infiltration capacity of the soil surface
(infiltration excess overland flow: Horton, 1931), or where the soil
becomes saturated so that additional rainfall is diverted laterally
(saturation excess overland flow: Dunne and Black, 1970). It is now
recognised (Cammeraat, 2002; McDonell, 2003) that rainfall intensity may
exceed the capacity of the soil to percolate downwards at either the
surface or within the soil (Figure 1), and that significant spatial
variations in infiltration rate can lead to a ‘fill and spill’ pattern
of saturated patches which overflow and connect during storms to drive
surface and/or subsurface lateral flow. In this paper these spatial
patterns are simulated for infiltration excess overland flow with the
saturated patches at the surface, but it is recognised that there may be
strong similarities between surface and subsurface patterns (McDonnell,
2013).
The SCS Curve number has been the most widely used model to forecast
total storm runoff from total storm rainfall and is embedded within a
number of other models that are used to forecast erosion and solute
behaviour at range of scales. Despite its widespread application, the
curve number method has little theoretical underpinning, and is not
explicitly related to spatial scale or topography. Various attempts to
partially remedy these deficiencies (Porporato et al, 2016, Williams et
al, 2012) have been proposed.
Rainstorms are rarely simple in profile or in antecedent conditions, so
that any simple rainfall-runoff relationship is likely to show wide
variations. The bar for acceptability is therefore low, and the approach
is only justifiable if kept simple, facilitating inclusion in other
models.
The approach taken here has been applied for conditions of infiltration
excess overland flow, although comparisons may be drawn with sub-surface
fill-and-spill configurations. Flow patterns and slope-base output from
a square sloping plot have been simulated for isolated storms of
constant intensity. The plot has a roughened surface and randomly
distributed infiltration parameters. During a storm, areas of low
infiltration create patches of saturation that generate overland flow
which may connect with other saturated patches or re-infiltrate
downslope in patches of higher infiltration. Saturated areas close to
the slope base provide some outflow, even in brief and/or low intensity
storms. In progressively larger storms, overland flow connections with
the slope base are established farther and farther up the slope. After
rainfall stops, there is no more flow contribution from the top of the
slope, but connected flow persists downslope in areas of flow
concentration and shapes the recession limb of the outlet hydrograph.
The detailed model that has been used (Kirkby, 2014) generates
infiltration excess overland flow on a grid representing either a
fractally roughened uniform slope or a more topographically structured
surface with distinct valleys. Green Ampt (1911) infiltration parameters
are randomly assigned for each cell (nominally of 2.5m in the 128x128
cell grid), with values drawn from a specified distribution based on
field measurements in S.E. Spain. Overland flow is routed downslope
across the surface, according to a probability distribution of overland
flow ‘droplets’ based on D8 flow directions, a method that has also been
applied for saturation excess overland flow (Gao et al 2016,2017).
Overland flow is generated in a source cell wherever inflow and rainfall
exceed infiltration. 50 instances of this overland flow are treated as
droplets, each of which travels towards a neighbouring cell randomly
chosen from the probabilities assigned to each downslope direction. The
droplet mean velocity is calculated from the local gradient and overland
flow depth to determine the probability of stopping, at the end of the
time step, in the receiving cell. If not stopping this process is
repeated until the droplet comes to rest. This process is repeated for
the 50 droplets and their mean is used to define the redistribution of
the overland flow generated in the source cell at each time step.
The previous work (Kirkby, 2014) simulated total storm runoff, rat the slope base from simple storms of constant intensity and total
storm rainfall R , on initially dry surfaces.
The expressions derived are well behaved at extreme values in the
following ways.
First r = 0 when R = 0. This is a self-evident
requirement. Second, there is very low runoff for small storms, for
which equation (1) behaves like r ~
Rn +1 . This seems to be a more
appropriate response than the sharp lower threshold for runoff in the
SCS curve number method, since both model and field data (Cammeraat,
2002) show that, in even small storms, patches of low infiltration near
the outlet boundary are able to deliver small amounts of runoff before
their flow contribution is absorbed in higher infiltration areas.
Thirdly, at high storm amounts, runoff asymptotically approaches
rainfall, a behaviour in common with the SCS approach. However, the
expression previously proposed seemed to suggest that the volume of
infiltrated water stored in the soil, together with the volume of water
in detention upon the surface decreased as total storm rainfall was
increased, appearing to violate the requirements of mass balance, and
leading to the alternative formulations proposed here, that differ
appreciably in the forecast volume of runoff for the largest storms.