Our purpose in this work is to analyse in some detail the influence of convection on the formation and the build-up of the solutal boundary layer during the initial composition transient. Our main assumption is that the alloy is sufficiently dilute for the variation of the interface temperature to be neglected, in other words that the solidification interval is small enough. In order to thoroughly characterise the problem, we use a coupled theoretical analytical and scaling analysis and numerical approach. Comparison with existing experimental data is also provided. The effect of convection is seen to be adequately accounted for owing to the convecto-diffusive parameter Delta. The overall agreement observed allows us to propose a simple rule of thumb expression for the extent of the initial composition transient.