Viable skin drug transport is an important concept to consider as it can have a significant impact on the local concentration of a drug. The concentration becomes even more critical for toxicological issues when implementing different permeability enhancement techniques. For this reason, it is important to develop models that can predict drug transport in the viable skin. This paper expands upon previous capillary modelling by representing the convective transport of a solute that has permeated into the capillary loops. As a result, convective transport caused the concentration profile to plateau within the deeper dermal layers, effectively matching the trend of previous experimental data. Furthermore, the new model also has a significantly quicker transient profile as the time required to reach steady-state is five-fold faster than predicted in previous homogenous models.