Mathematical Analysis of Nonlinear Theories in Flow and Transport in Porous media

he research group at Utrecht University (formerly at Delft University of Technology) has played a prominent role in the development of new nonlinear theories for flow and transport in prous media. In general these theories were developed because carefully conducted laboratory experiments could not be modelled using the classical theories. We mention two typical examples: 1) A new nonlinear theory for density dependent flow in heterogeneous porous media. 2) A new nonlinear theory for two-phase flow, including the dynamic capillary pressure effect. Both theories include new nonlinear terms and corresponding new parameters as compared to the classical equations. On one hand, these theories are validated using the results of laboratory experiments, while on the other hand analytical and numerical methods are applied to analyze and interpret the consequences of the presence of the new terms in the partial differential equations. The nonlinear nature of the aformentioned new theories raised challenging mathematical questions. The mathematical analysis of the governing equations turned out to be of great value for the physical interpretation of the new theories. Moreover, mathematical techniques like the homogenization method are utilized to upscale local-scale phenomena towards a higher averaged scale. This resulted in intensive and inspiring collaboration between hydrogeologists (Utrecht University) and mathematicians (Eindhoven Univ., Vrije Univ., Kazan State University). Many new and interesting results were obtained. Amongst others, we mention the application of the new dynamic capillary pressure theory to the problem of the prediction of instabilities in wetting fronts in unsaturated porous media, and the application of homogenization to upscale the stable movement of brine in weakely heterogeneous media. However, it must be emphasized that the recently obtained results are of preliminary nature. They are the first steps into the direction establishing physically based theories for two-phase flow and for density dependent flow and transport in heterogeneous media. Many open (fundamental) questions still need to answered. To find these answers we propose to intensify and expand the collaboration between the mathematicians (Kazan State Univ. & Vrije Univ.) and the hydrogeologists (Utrecht Univ. & Moscow Stae Univ.). This collabroration involves (as in te past) both young researchers (PhD students, postdocs) and senior researchers. The role of the young researchers is quite prominent. They are stimulated to participate actively in the research program and in the bi-lateral collaboration. The goals of this proposed project are the development of physically based theories for both two-phase flow and density dependent flow in heterogeneous porous media using mathematical techniques and tools.