  As fractional diffusion equations can describe the early breakthrough and the
heavy-tail decay features observed in anomalous transport of contaminants in
groundwater and porous soil, they have been commonly employed in the related
mathematical descriptions. These models usually involve long-time range
computation, which is a critical obstacle for its application, improvement of
the computational efficiency is of great significance. In this paper, a
semi-analytical method is presented for solving a class of time-fractional
diffusion equations which overcomes the critical long-time range computation
problem of time fractional differential equations. In the procedure, the
spatial domain is discretized by the finite element method which reduces the
fractional diffusion equations into approximate fractional relaxation
equations. As analytical solutions exist for the latter equations, the burden
arising from long-time range computation can effectively be minimized. To
illustrate its efficiency and simplicity, four examples are presented. In
addition, the method is employed to solve the time-fractional
advection-diffusion equation characterizing the bromide transport process in a
fractured granite aquifer. The prediction closely agrees with the experimental
data and the heavy-tail decay of anomalous transport process is
well-represented.
