Although fractional integration and differentiation have found many applications in various fields of science, such as physics, finance, bioengineering, continuum mechanics, and hydrology, their engineering applications, especially in the field of fluid flow processes, are rather limited. In this study, a finite difference numerical approach is proposed to solve the time-space fractional governing equations of 1-dimensional unsteady/non-uniform open channel flow process. By numerical simulations, results of the proposed fractional governing equations of the open channel flow process were compared with those of the standard Saint-Venant equations. Numerical simulations showed that flow discharge and water depth can exhibit heavier tails in downstream locations as space and time fractional derivative powers decrease from 1. The fractional governing equations under consideration are generalizations of the well-known Saint-Venant equations, which are written in the integer differentiation framework. The new governing equations in the fractional-order differentiation framework have the capability of modelling nonlocal flow processes both in time and in space by taking the global correlations into consideration. Furthermore, the generalized flow process may possibly shed light on understanding the theory of the anomalous transport processes and observed heavy-tailed distributions of particle displacements in transport processes.