Abstract: This study shows in detail how the classical time fractional advection-dispersion equation (TFADE) can be generalized using the concept of tempering. The generalized TFADE model is then approximated by a new spatiotemporal splitting method, which is computationally more efficient than the classical Eulerian solver due to the logic tempering of the time nonlocal dependence in solute transport. Numerical experiments show that the generalized TFADE model captures a broad range of non-Fickian diffusion, where the tempering parameter λ (which is the inverse of the characteristic time), fractional index γ, and mobile/immobile capacity coefficient β can control the nuance of transport behavior. The model also efficiently distinguishes the mobile phase from the total phase for solute transport through heterogeneous media, which is critical for practical applications.