A one-dimensional transport model for simulating water flow and solute transport in homogeneous-heterogeneous, saturated-unsaturated porous media is presented. The model is composed of a combination of accurate numerical algorithms for solving the nonlinear Richard's and advection-dispersion equations (ADE). The mixed form of Richard's equation is solved using a standard finite element method (FEM) with primary variable switching. The transport equation is solved using operator splitting, with the discontinuous finite element method (DFE) for discretization of the advective term. A slope limiting procedure for DFE avoids numerical instabilities but creates very limited numerical dispersion for high Peclet numbers. An implicit finite differences scheme (FD) is used for the dispersive term. The unsaturated flow and transport model (Wamos-T) is applied to a variety of rigorous problems including transient flow, heterogeneous medium and abrupt variations of velocity in magnitude and direction due to time-varying boundary conditions. It produces accurate and mass-conservative solutions for a very large range of grid Peclet numbers. The Wamos-T model is a good and robust alternative for the simulation of mass transport in unsaturated domain.