In order to model non-Fickian transport behaviour in groundwater aquifers, various forms of the time-space fractional advection-dispersion equation have been developed and used by several researchers in the last decade. The solute transport in groundwater aquifers in fractional time-space takes place by means of an underlying groundwater flow field. However, the governing equations for such groundwater flow in fractional time-space are yet to be developed in a comprehensive framework. In this study, a finite difference numerical scheme based on Caputo fractional derivative is proposed to investigate the properties of a newly developed time-space fractional governing equations of transient groundwater flow in confined aquifers in terms of the time-space fractional mass conservation equation and the time-space fractional water flux equation. Here, we apply these time-space fractional governing equations numerically to transient groundwater flow in a confined aquifer for different boundary conditions to explore their behaviour in modelling groundwater flow in fractional time-space. The numerical results demonstrate that the proposed time-space fractional governing equation for groundwater flow in confined aquifers may provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy-tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long-range dependence of the underlying groundwater flow field.