Ionic transport through ionic exchange membrane cannot be interpreted by the Nernst-Planck equation if the ion density is high, particularly in the membrane. In order to extend the density range, excess terms must be added to the chemical potential. These terms are computed by considering charged hard spheres embedded in a dielectric continuum. In this aim and owing to heterogeneity of the electrolytic solution the density function theory (DFT) is used. A previous work has been carried out with a homogeneous and amorphous solvent. Here an extension including the finite size of the solvent molecule and the local dielectric properties via the dielectric coefficient is presented. The electro-osmosis is also examined. The selectivity at equilibrium, the density profiles and the voltammograms are analysed. The numerical results obtained with NaCl and CaCl2 show that the physical properties of the solvent decrease the selectivity and increases the conductivity of the membrane systems. In the same time, the dielectric properties increase the electro-osmotic effects on the ionic transport. The approach described in this work can be used to study the solvent confinement effect inside the membrane on the ionic transport.