Conventional assumptions of gas kicks concern one “dry gas bubble“ over a wellbore length equivalent to the total influx volume. The aim of this thesis is to show that this assumption is not right. Because several investigations in history as well as observations on the rig site have shown that there is no single bubble heading to surface. Therefore, the central part of this work is to demonstrate that a certain amount of gas, percolating through the wellbore, is at least dispersed into several smaller bubbles. The focus of this thesis lies on the rheology model and the resulting dispersion of the void fraction. Therefore, the wellbore model used to simulate and prove the behavior is a rough simplification of the real wellbore. As basis for further research this work presents several off-the shelf solvers and their ability to solve this problem. It will describe the path from early research to real data simulation, including rheology as well as geometry and solver changes. A final simulation will prove that results produced by this off-the shelf solver match data from former investigations (e.g. traveling speed of the bubbles – 300 [m/hr]). The data produced during this thesis will be visualized in a movie, showing the behavior and shape of gas in these particular cases. Finally this thesis will show that in case of this simplified model the assumption of one “dry gas bubble” is not correct As part of a multi-year research project this thesis should create a basis for further investigation of the total wellbore/gas system. These results will be used to start research on the influence of sour environments onto high strength steel alloys. Therefore this work will be continued as a PhD thesis providing these further steps.