Development of a mathematical solution for injection/falloff problem of an oil-water system for transient data analysis is complicated due to the presence of two mobile phases that create a moving discontinuity. The solutions that have been developed in the past are based on a series of assumptions that make the models limited. The previous models have neglected one or more of the key reservoir and well parameters such as the variation of saturation in the invaded zone, temperature effects, wellbore storage, skin and outer boundary conditions. In this work, a generalized two-step semi-analytical solution method is presented to precisely simulate the pressure transient behaviour for injection and falloff tests following cold water injection for a fully-penetrating vertical well in a warm and bounded oil reservoir in the presence of skin and wellbore storage. The formulated model uses two simplistic steps to solve the complex mathematical problem and improve computational efficiency. The usage of the Laplace-transform finite-difference method to solve the system of equations eliminates the need for temporal discretization, thereby resolving issues related to convergence and time iterations by allowing an unlimited timestep size without the loss of stability and accuracy. The accuracy and consistency of the proposed solution method are assessed by comparing the results with that of ECLIPSE 100 and an excellent agreement is observed. The results suggest that the injection period pressure behaviour is vastly influenced by the injected water propagation under cold conditions, whereas, the falloff period pressure behaviour provides information of the water bank under cold conditions and the uninvaded zone under hot conditions. The comparison of the non-isothermal and isothermal pressure behaviour shows that the isothermal model can be improved using modified fluid viscosities to match the non-isothermal pressure behaviour that not only could minimize the mathematical complications, but also the computation time. © 2020 Elsevier B.V.