Theory of modelling the solute transport through heterogeneous soil influenced by non-equilibrium is still limited to bi-continuum models, which often lacks to predict long tailings. The models based on dual domains do not embraces the effects of near grain boundary layer flow velocity. To fill the gap we presents one-dimensional semi-analytical solution for slow, fast transport (SFT) model for simulating sub-surface solute transport. This model accounts for both chemical and physical non-equilibrium conditions within the soil matrix. Advective phase closer to the soil grains will have lesser velocity due to boundary layer formation and is called slow region while region away from soil grains will have higher velocity called fast region. Both instantaneous and rate limited sorption are assumed at slow and immobile region to characterize chemical non-equilibrium. Semi-analytical solution of this model is derived into Laplace domain and finally inversion is carried out by robust and widely implemented De-Hoog numerical inversion algorithm. Generalized solutions are developed for both semi-infinite and finite domain with zero initial concentration of solute in the porous matrix. Present solution is validated using existing numerical solution for slow fast transport (SFT) model. Experimental BTC's available in literature are simulated using the current solution and comparison is made with existing models which lump advective regions. Analytical expressions of temporal moments are also derived for SFT model to compare effect of mass transfer and advective velocity of different regions. Sensitivity analysis is carried out to ascertain the effects of sorption parameters over solute evolution. It is observed that better simulation of solute phase concentration can be made using current solution developed. Also sensitivity analysis show that increasing value of sorption parameter, increases residence period of solute. This solution developed can be suitably applied to field conditions where much heterogeneity is present assuming triple porosity continua.