Linking structural and functional connectivity in a simple runoff-runon model over soils with heterogeneous infiltrability

Runoff production on a hillslope during a rainfall event may be simplified as follows. Given a soil of constant infiltrability I, which is the maximum amount of water that the soil can infiltrate, and a constant rainfall intensity R, runoff is observed wherever R is greater than I. The infiltration rate equals the infiltrability where runoff is produced, R otherwise. When ponding time, topography, and overall spatial and temporal variations of physical parameters, such as R and I, are neglected, the runoff equation remains simple. In this study, we consider soils of spatially variable infiltrability. As runoff can re-infiltrate on down-slope areas of higher infiltrabilities (runon process), the resulting process is highly non-linear. The stationary runoff equation is: Qn+1 = max (Qn + (R - In)*Δx , 0) where Qn is the runoff arriving on pixel n of size Δx [L2/T], R and In the rainfall intensity and infiltrability on that same pixel [L/T]. The non-linearity is due to the dependence of infiltration on R and Qn, that is runon. This re-infiltration process generates patterns of runoff along the slope, patterns that organise and connect differently to each other depending on the rainfall intensity and the nature of the soil heterogeneity. In order to characterize the runoff patterns and their connectivity, we use the connectivity function defined by Allard (1993) in Geostatistics. Our aim is to assess, in a stochastic framework, the runoff organization on 1D and 2D slopes with random infiltrabilities (log-normal, exponential and bimodal distributions) by means of numerical simulations. Firstly, we show how runoff is produced and organized in patterns along a 2D slope according to the infiltrability distribution. We specifically illustrate and discuss the link between the statistical nature of the infiltrability and that of the flow-rate, with a special focus on the relations between the connectivities of both fields: the structural connectivity (infiltrability patterns) and the functional connectivity (runoff patterns). In a second step, we demonstrate how, on a 1D geometry defined by different uncorrelated infiltrability distributions, these interactions between infiltrability and resulting runoff field can be quantified by the Queueing Theory.