Retrofitting Upstream Fish Passage at Culverts through a Modular Bypass: A Controlled Flow-Division Approach

Longitudinal river connectivity can be disrupted by culverts due to excessive flow velocities, insufficient flow depths, abrupt hydraulic transitions, or inaccessible outlets for upstream-migrating fish. In this regard, a modular bypass constructed adjacent to an existing culvert may provide a practical retrofit solution without requiring complete replacement of the crossing structure. This study develops a conceptual redesign and mathematical framework for a hydraulically controlled fish-passage bypass, with particular emphasis on predicting the discharge division between the existing culvert and the bypass branch. The overall system is represented as two parallel hydraulic pathways connecting common upstream and downstream control sections, where mass conservation is coupled with the equality of branch-wise head losses. The hydraulic resistances of the two branches are formulated by considering distributed friction, entrance and exit losses, relevant geometric parameters, and the effective roughness of the bypass. Based on these relationships, a general implicit formulation is established for the discharge division. An exact closed-form solution is further derived for the case in which the branch head losses vary quadratically with discharge. The resulting expression demonstrates that the bypass discharge fraction is primarily governed by the ratio of bypass to culvert hydraulic resistance. Accordingly, changes in bypass geometry, roughness, and local-loss characteristics directly influence the amount of flow diverted into the ecological branch. The proposed formulation provides a transparent preliminary design tool for estimating and controlling ecological discharge allocation within a culvert-bypass system prior to detailed numerical simulations, laboratory experiments, field validation, or species-specific biological assessment.