A numerical scheme is designed and implemented to solve a simplified set of equations modeling 1-D multi-phase flow based on drift flux model in an isothermal setup with phase dissolution. The difficulty in obtaining the analytical Jacobian of the fluxes leads to the difficulty in obtaining an efficient linearized Riemann solver which in turn affects the accuracy in capturing the contact wavefront/gas migration. To address this issue a fully explicit second order finite volume solver based on flux corrected transport (FCT) is implemented. The choice of variables used for limiting the fluxes affects the amount of numerical diffusion and an appropriate choice of the gradient in volume fraction is used. Practical test cases while drilling in the oil and gas industry, of gas injection inside a well annulus and shut-in of the vertical well are presented. The results conclude that the FCT solver is better and efficient for accurately capturing gas migration for multi-phase models with phase behaviour involving slip velocities given by algebraic relations. (C) 2015 Elsevier Ltd. All rights reserved.