An Analytical Treatment of the Hysteretic Storage–Discharge Relation

The storage–discharge ($S\text{-}Q$) relation, widely employed in hydrology and hydraulics, is often derived from empirical data and is found to be hysteretic or looped, although approximated by a straight line or a curve in hydrologic modeling. The nature of hysteresis depends on geometric and hydraulic conditions. Despite its ubiquitous use, an analytical treatment discussing what causes the storage–discharge relationship to exhibit a looped behavior does not seem to have been reported. This study analyzes the $S\text{-}Q$ relation analytically using kinematic wave approximation for a watershed represented by a plane, considering simultaneously rainfall, infiltration, and surface runoff or overland flow. For purposes of simplicity and tractability of analytical solutions, both rainfall intensity and infiltration rate are assumed to be constant. Depending on the duration of rainfall, two cases—equilibrium and partial equilibrium—are distinguished. The hysteretic $S\text{-}Q$ relationship is different for these two cases and requires close scrutiny, which is pursued in this study. It is emphasized that the assumptions of constant rainfall intensity and infiltration rate, rectangular geometry, and kinematic wave approximation undermine the dynamics of hysteresis.