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.