Spatially Averaged Conservation Equations for Interacting Rill‐Interrill Area Overland Flows

Abstract

The modeling of flows at hillslope scale over land surfaces that contain rills and interrill areas is presented. The flow in rills is treated as one‐dimensional “channel flow.” The flow at interrill areas is treated as two‐dimensional “sheet flow.” Sheet‐flow equations are averaged along the interrill area width (local averaging) to account for the interaction between interrill and rill sections. The flow depth at the interrill area outlet section is derived by assuming that the flow profile of the interrill area section has the profile of a sine function. To minimize computational efforts and economize on the number of model parameters, locally averaged rill and sheet‐flow equations are then averaged over the transects of a hillsope (large‐scale averaging). The expectations of the terms containing more than one variable are obtained through the Taylor series expansion around their mean values. Only the first two moments of the series expansion are considered. The results of the model are in good agreement with the field observations. The effects of average local slopes and rill occurrence probability on flow rates are shown to be quite pronounced. On a rilled surface, most of the flow occurs in rill sections. Comparisons of model results with observed results over experimental hillslopes are satisfactory.