Hydraulic geometry, described by depth, width, velocity, slope, and friction, is determined using three equations of continuity, resistance, and sediment transport and by satisfying the condition of minimum production of entropy in the channel system. This chapter discusses the methodology based on this condition.

Design of a stable alluvial channel is based on the hypothesis that the equilibrium state of a channel corresponds to maximum flow. The channel design can then be accomplished by employing the continuity equation, resistance law, sediment transport equation, and the channel cross-section shape. This chapter derives the channel hydraulic geometry for primarily three cross-sections, namely trapezoidal, rectangular, and triangular.

Rivers tend to follow the path of least action for transporting the sediment and water loads imposed on them. Because regime hydraulic geometry relations entail more unknowns than the equations of continuity, resistance, and sediment transport, optimization is utilized to determine the preferred cross-section from among many possible cross-sections and this cross-section satisfies the path of least action. This chapter discusses this principle and derives the hydraulic geometry based on this principle.