1-D steady state runoff production in light of queuing theory: Heterogeneity, connectivity, and scale

摘要

We used the frameworks of queuing theory and connectivity to study the runoff generated under constant rainfall on a one-dimensional slope with randomly distributed infiltrability. The equivalence between the stationary runoff-runon equation and the customers waiting time in a single server queue provides a theoretical link between the statistical description of infiltrability and that of runoff flow rate. Five distributions of infiltrability, representing soil heterogeneities at different scales, are considered: four uncorrelated (exponential, bimodal, lognormal, uniform) and one autocorrelated (lognormal, with or without a nugget). The existing theoretical results are adapted to the hydrological framework for the exponential case, and new theoretical developments are proposed for the bimodal law. Numerical simulations validate these results and improve our understanding of runoff-runon for all of the distributions. The quantities describing runoff generation (runoff one-point statistics) and its organization into patterns (patterns statistics and connectivity) are studied as functions of rainfall rate. The variables describing the wet areas are also compared to those describing the rainfall excess areas, i.e., the areas where rainfall exceeds infiltrability. Preliminary results concerning the structural and functional connectivity functions are provided, as well as a discussion about the origin of scale effects in such a system. We suggest that the upslope no-flow boundary condition may be responsible for the dependence of the runoff coefficient on the scale of observation. Queuing theory appears to be a promising framework for runoff-runon modeling and hydrological connectivity problems.