Using dimensional principles, three dimensionless variables can be defined for designing a regime channel. These variables contain six characteristic parameters that reflect fluid, sediment, and geometric characteristics of a channel. This chapter discusses the hydraulic geometry of regime channels using these dimensional principles and illustrates the application of these principles to channel design.

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.

A regime channel geometry can be computed using the second law of thermodynamics and the Gibbs equation which constitute the foundation of the thermodynamic method. With the use of a regime width relation, the need for a sediment transport rate relation can be obviated. This chapter discusses the thermodynamic methdology for deriving the hydraulic geometry of regime channels.