Instances are cited in which the pressure gradient in an oil pipe line has been reduced by the injection of water into the pipeline. A general mathematical analysis is presented for two immiscible liquids flowing (1) in two layers between wide parallel plates, and (2) concentrically in a circular pipe. This will form a basis for the further study of oil-water systems. Equations are derived relating the volumetric flow rates and the viscosities of the liquids with the pressure gradient. The conditions for which minimum pressure gradients and minimum power requirements occur were determined and these minimum values have been compared with known values for a pipeline flowing full with only a single liquid. The factors by which the pressure gradient and power requirement can be reduced are very large. For example, for an oil of viscosity 1,000 cp. flowing concentrically with water, the reduction factor is approximately 500. The pressure gradient reduction factors reported in the literature are compared with those predicted by theory, and conclusions are drawn regarding the position of the water phase.