The development of a macroscopic model for solute transport coupled with unsaturated water flow in double-porosity media is presented in this work, by using the asymptotic homogenization method. The model was derived for the case in which the medium exhibits a strong contrast of transport properties by upscaling rigorously the transport mechanisms from micro-scale to macro-scale. It consists of two coupled equations for dispersion-convection processes at macroscopic level and diffusion intervention from local scale that can be described by a non-Fickian behaviour of solute concentration breakthrough. The proposed model was numerically implemented in the environment of a finite element code (commercial software) and applied to 2D examples with different boundary conditions. To validate, a comparative analysis between the results obtained from the homogenized model and the fine scale model (reference solution obtained from explicit heterogeneous representation of the medium structure) was carried out. The obtained numerical tool for the two-scale implementation enables treating various types of two-equation models to study the macroscopic non-Fickian transport and also non-equilibrium evolution of concentration fields inside the micro-porous medium.