The objective of this study is to investigate the dispersion of solute matter introduced into a fully developed laminar flow in a circular capillary tube under preasymptotic regimes. For this purpose, we have used the method of volume averaging to upscale microscale balance equations for the case of diffusion-convection transport system while the initial condition was kept as a non-zero term. More precisely, any arbitrary space function can be considered as an initial condition for the system. Upon scaling, a non-conventional additional source term appears in one of the closure variables that accounts for the initial configuration. An important feature of this rigorous theory is that it is able to predict the effective transport properties from initial to asymptotic times. In this study, we compare the asymptotic solution for the effective dispersion tensor with that of calculated by Taylor-Aris diffusion-convection transport equation as a validation for the developed theory. The results for the effective dispersion tensor and the second moment of initial concentration is presented for a wide Peclet range.