Development of Novel Simplified Empirical Equations for the Estimation of Peak Discharge due to Dam Failures

Abstract This study presents a novel methodology for estimating the peak discharge resulting from the failure of earth-fill dams through the development of simplified empirical equations. The proposed method integrates parametric sensitivity analysis, numerical simulations, and meta-heuristic optimization techniques. The historical failure of the Teton Dam in Idaho, USA, was selected as a reference study area due to the availability of reliable spatial and hydraulic data, and the developed models are therefore mainly applicable to piping-type failures in earth-fill dams. Five critical parameters were initially considered in the dam breach analysis: bottom breach width ( BBW ), breach side slope ( SS ), breach development time ( BDT ), volume of water above breach invert ( V w ), and water height at piping initiation ( H w ). A total of 2000 dam failure scenarios were simulated using the Hydrologic Engineering Center's – River Analysis System (HEC-RAS), and the results were analyzed to derive empirical models. Correlation analysis revealed that while BBW and BDT have strong correlation with Peak Breach Flow ( Q p ) and clear directional influence, SS had a minor direct effect. In contrast, H w and V w exhibited complex and nonlinear behavior due to their interdependence with BDT . Although they physically affect Q p , their influence is non-monotonic and scenario-dependent. For empirical equation development approach, their contribution was found to be minimal compared to BBW and BDT . As a result, simplified empirical equations were developed using the most influential parameters. Multiple empirical equation structures were investigated: (i) single-parameter models, (ii) two- and three-parameter models, and (iii) a novel table-driven model where the coefficients adapt dynamically based on discrete values of the third parameter. Among these, the table-driven empirical equations performed the best, with a maximum absolute percent error ranging from 20.3% to 25.1%, significantly improving the accuracy compared to fixed-coefficient models. Coefficients of the empirical models were optimized using the Teaching–Learning-Based Optimization (TLBO) algorithm, which proved effective in calibrating the nonlinear relationships between breach parameters and peak flow. The proposed empirical equation family, named Simplified Empirical Equation for Dam failure (SEED), provides a rapid and flexible estimation framework that can support early warning systems, dam breach risk assessment, and emergency action planning. The study demonstrates that well-designed empirical equations, when supported by robust simulation and optimization techniques, can offer reliable approximations for complex dam breach hydraulics, especially in data-scarce or time-critical scenarios.