We present the results of developing a multi-fluid model for a transient multiphase flow in a pipe with application to oil and gas well cleanup after drilling or hydraulic fracturing. The model is an extension of the earlier work on multi-fluid models for near-horizontal pipes to cover any inclination angle and include additional phases. The 1D multi-fluid model governs a transient isothermal flow of three compressible fluids in a horizontal, deviated, or a vertical pipe. The multi-fluid model research code is based on well-known SIMPLE schemes, and features implementation of either a geometry or a mass conservation based algorithms. The multi-fluid model is able to predict the flow regimes with a single interface between gas and liquid mixture: stratified flow, flow with rolling waves, and flow with slugs. The code is validated against an analytical solutions and flow pattern maps. Agreement with theory is found in predicting the transition between the regimes. The final application of the model will be in transient wellbore dynamics with flow regime transitions.