This is a short version of the introduction to my Cand. Scient thesis, On Solving a Three Phase Flow Model with Capillary Forces [ PS ] [ PDF ]
Mathematical models that involve a combination of advection and diffusion processes are among the most widespread in all of science, engineering and other fields where mathematical modelling is important. Very often the models are advection dominated with sharp fronts building up. Because of these fronts difficulties will be experienced with standard numerical approximations. Thus many different methods have been proposed to overcome the difficulties. Advection-diffusion equations occur for example in the study of three phase flow, atmospheric pollution and ground water transport.
In this thesis we will investigate the three phase flow model. The model describes the flow of oil, gas and water in a porous media. The solution of this 2 x 2 system of (coupled) equations are well understood for some cases. However, we will assume that the flow of one of the phases is independent of the other leading to a system of equations where one equation is decoupled from the other, i.e., a triangular system. This assumption is not trivial, but it seems that at least for some systems it is a good approximation. We are going to solve the system with a corrected operator splitting method. In the first step we solve for the hyperbolic part of the problem, and in the second step for the diffusion part. In this process we make a splitting error. We therefore try to find a correction term that can reduce the splitting error. This leads to the corrected operator splitting method . The advection part will be solved by the means of front tracking , and the diffusive part will be solved by finite difference approximations. In this paper only one spatial dimension is considered. Generalisations to more space dimensions can be done using dimensional splitting.
The main objectives of this thesis are to investigate the corrected operator splitting method and to compare the triangular model with the fully coupled model. The triangular model without diffusion is well understood. We will build on this work when we introduce diffusion terms and the corrected operator splitting. We want to quantify when a fully coupled system can be approximated by a triangular system. Investigating this approximation is interesting since solving the triangular system is much faster than solving the fully coupled one.