Recent developments and improvements in numerical techniques and computer capability enable more accurate numerical solutions of spatially varied flow problems such as various phases of urban storm runoffs. Consequently, it is desirable to re-examine the compatability of the flow equations used in solving unsteady spatially varied flow problems. The continuity, momentum, and energy equations for unsteady nonuniform flow of an incompressible viscous nonhomogeneous fluid with lateral flow into or leaving a channel of arbitrary geometry in cross section and alignment are formulated in integral form for a cross section by using the Leibnitz rule. The resulting equations are then transformed into one-dimensional form by introducing the necessary correction factors and these equations can be regarded as the unified open-channel flow equations for incompressible fluids. The flow represented by these equations can be turbulent or laminar, rotational or irrotational, steady or unsteady, uniform or nonuniform, gradually or rapidly varied, subcritical or supercritical, with or without spatially and temporally variable lateral discharge. (Author)