The study of solute transport processes in open channel flows is important for water quality management and environment protection. Solute transported in open channel flows is usually uncertain because of the underlying stochastic flow and uncertain solute source/sink conditions. Here a stochastic one-dimensional nonreactive transport model based on the Fokker-Planck equation (FPE) approach is developed. The governing equation for the developed stochastic model is a two-dimensional FPE that can explain the effect of both uncertain flow fields and solute source/sink conditions on the stochastic solute transport behavior. The proposed model can provide the spatiotemporal probability density function (PDF) of the solute concentration. Ensemble mean and standard deviation behavior in space and time can then be easily obtained through the corresponding PDF. A numerical experiment is conducted to demonstrate the capabilities of the two-dimensional stochastic transport model. The two-dimensional ULTIMATE QUICKEST finite difference scheme is applied to solve the FPE. Monte Carlo simulation is performed to validate the results obtained from the proposed approach. The validation results indicate that the proposed FPE approach can capture the complete probabilistic dynamics of a one-dimensional solute transport system under uncertain flow fields and solute source/sink conditions by providing the evolutionary PDF of solute concentration in both time and space in a computationally efficient way.