Modelled and observed stage-discharge relationships for cobble leaky barriers with and without pipes

Flooding is costly and disruptive in the UK and worldwide. Leaky barriers (LBs), small-scale blockages to streamflow, provide multiple environmental benefits. Depending on design, and if installed in sufficient numbers, they could also play an important role in reducing downstream flooding. Leaky barrier installation is proceeding at pace, thousands of cobble dams have been installed in peat gullies across the South Pennines (UK). However, the hydraulics of LBs in general and these cobble barriers in particular is poorly understood. Here we develop a simple model coupling two classical engineering flux estimates: Darcy/Casagrande equations for matrix flow and Colebrook equation for pipe flow (where drains are installed). We test this model against observed stage and discharge measurements for four study features with and without drains to: identify stage-discharge relationships; evaluate model performance for individual features; and apply it to model chains of features of varying design (i.e., LB density, matrix permeability, and pipe diameter). We find that: 1) stage-discharge relationships for cobble dams are concave up and are generally well captured by our simple model; 2) current designs offer relatively little attenuation because they are too permeable; 3) instead, optimal designs have low matrix permeability with pass-forward pipes at their base of a diameter tuned to design flow. Based on these results we hypothesise that LBs will perform best where they are designed to have negative permeability-depth relationships (and thus convex up stage-discharge relationships) and where the form and magnitude of the relationship is optimised to accommodate peak flood discharges.