A pair of equations, one for continuity and one for momentum, that describe unsteady one-dimensional flow in open channels such as rivers and sewers.
The Saint-Venant equations describe how depth and discharge change over time and along the length of an open channel. They are the basis of most one-dimensional unsteady river and sewer models, including the unsteady flow solver in HEC-RAS and the dynamic wave option in SWMM. They are named after the French engineer Adhémar Barré de Saint-Venant, who published them in 1871. There are two equations. The continuity equation says water is conserved: if more water enters a stretch of channel than leaves it, the flow area must grow. The momentum equation balances the forces on the water: local acceleration, convective acceleration, the pressure force from changes in depth, gravity from the bed slope, and friction. In the formula, A is flow area, Q discharge, y depth, q side inflow per unit length, S₀ bed slope and Sf friction slope, which is usually computed with the Manning equation. A simple continuity example: if 120 m³/s enters a 1 km reach and 110 m³/s leaves it, with no side inflow, then ∂A/∂t = -(110 - 120) / 1,000 = 0.01 m² per second. Over one hour the flow area grows by 36 m², which in a 40 m wide channel is a rise of about 0.9 m. The equations have no general exact solution, so models solve them numerically, often with implicit finite difference schemes such as the Preissmann scheme. Simpler forms are common. The kinematic wave form keeps only gravity and friction, the diffusion wave form adds the pressure term, and the full dynamic wave form keeps every term. Backwater, tides and flow reversal need the full form. The equations assume one-dimensional flow, hydrostatic pressure, a small bed slope and the same velocity across each section. They fit poorly where water spreads across wide floodplains, where 2D shallow water models are used instead. Results depend strongly on cross-section surveys, roughness values and the boundary conditions at each end.