Multi-scale analysis of two-phase flow in porous media with complex heterogeneities

• Applicants: Professor Dr. Oleg Iliev; Professor Dr. Mario Ohlberger; Professor Dr. Ben Schweizer
• Subject Area: Mathematics
• Term: from 2009 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 142009460

The goal of this project is the multi-scale analysis of porous media equations. We will start from a mesoscopic description of two phase flow in porous media and consider a wide range of heterogeneities. We plan to study: (1) problems with heterogeneities at a large scale such that subdomains with homogeneous properties can be introduced at the cost of interface conditions between the subdomains; (2) problems where the scales can not be separated but a proper multi-scale discretization technique can be exploited in order to obtain a macro-scale solution enriched by meso-scale information; (3) problems where the meso-scale solution is the target and multi-scale preconditioning techniques can serve as a basis for robust and efficient solution methods. In case (1) with a clear scale separation, we want to adapt and apply homogenization techniques in order to justify and analyze effective models and efficient approximation schemes. We want to develop numerical methods that exploit the multi-scale character of the problem and we wish to accompany the practical implementation with a rigorous analysis. As an extension of the basic model, we will consider interface conditions between different porous materials; this can lead to either coupling conditions between one- and two-phase flow equations, to effective Dirichlet conditions or to outflow conditions. Furthermore, in case (2) of more complex heterogeneities, when the scales can not be separated but it is still too expensive or impossible to solve a full meso-scale problem, we want to further develop and analyze multi-scale discretization techniques, such as the heterogeneous multiscale method, HMM, and the multiscale finite volume method, MSFV. When, as in (3), the full meso-scale solution is the target, we plan to develop proper multi-scale preconditioners. Having in mind that there will be a significant synergy effect and reuse of the discretization/ preconditioning components and of the analytical tools for problems with different levels of heterogeneity, we suggest to consider them in one common research project. We plan also to benefit from a tight connection between the studies of the continuous multiscale problems, the approaches for their discretization, and robust methods for solving the discretized problems. In particular, we also aim at evaluating numerically the range of applicability of certain multi-scale schemes and to come up with a generalized framework in order to be able to treat a wide range of heterogeneities within a common setting.

• DFG Programme: Research Grants