Cementing process is an important step in the construction of oil and gas wells. During cementing, it is necessary to displace the drilling mud by the cement slurry. Therefore, it is common to have three or more liquids owing through the eccentric annular space. Non-uniform liquid displacement may lead to severe problems to the well structure and safety. Gomes and Carvalho (2009) developed a lubrication based model coupled with a level-set type method to simulate this process. However, when the interface deformation is large, such as in situations where buoyancy driven instabilities occur, the method led to large errors. Here, we propose a method at which the interface is tracked by material points placed along the initial configuration of the meniscus. The method greatly improved the accuracy of the predictions and is able to describe ow situations with very large interface deformation.