Multi-scale analysis of two-phase flow in porous media with complex heterogeneities
Final Report Abstract
This research project was very successful in different fields. With the common interest in two-phase flow equations and heterogeneities, three research teams could interact and collaborate, they could share useful methods and they succeeded to deepen todays knowledge on the two-phase flow system. Some specific questions turned out to be of particular interest and much of the research efforts have been directed to these fields: 1.) Numerical multi-scale methods, 2.) Homogenization of two-phase flow equations, 3.) Analysis of the degenerate system, 4.) Multilevel Monte Carlo methods. The first point embraces the analysis of multi-scale methods, the convergence of schemes and a posteriori estimates; reduced basis approaches have been discussed. In the second field, the first analytical homogenization result for the incompressible flow system was obtained. It extends previous results that were restricted to the one-dimensional case and gives a quantitative description of the effect of oil-trapping in heterogeneous media. The third field regards existence and uniqueness results for the degenerate system, and the fourth field is related to uncertainty quantification of multiscale problems at macroscopic level.
Publications
- The needle problem approach to non-periodic homogenization. Netw. Heterog. Media, 6(4):755-781, 2011
B. Schweizer and M. Veneroni
- Error control based model reduction for multi-scale problems. In Proceedings of Algoritmy 2012, Conference on Scientific Computing, Vysoke Tatry. Podbanske, September 9-14, 2012, pages 1-10. Slovak University of Technology in Bratislava, Publishing House of STU, 2012
M. Ohlberger
- On selected efficient numerical methods for multiscale problems with stochastic coefficients. PhD Thesis. Technical University of Kaiserslautern, 2012
C. Kronsbein
- The localized reduced basis multiscale method. In: Proceedings of Algoritmy 2012, Conference on Scientific Computing, Vysoke Tatry, Podbanske. September 9-14, 2012, 393-403. Slovak University of Technology in Bratislava, Publishing House of STU, 2012
F. Albrecht, B. Haasdonk, S. Kaulmann, and M. Ohlberger
- The Richards equation with hysteresis and degenerate capillary pressure. J. Differential Equations. 252(10):5594-5612, 2012
B. Schweizer
(See online at https://doi.org/10.1016/j.jde.2012.01.026) - Homogenization of the degenerate two-phase flow equations. Math. Models Methods Appl. Sci.
P. Henning, M. Ohlberger, and B. Schweizer
(See online at https://doi.org/10.1142/S0218202513500334) - Two-phase flow equations with a dynamic capillary pressure. European J. Appl. Math.. 24(1): 49-75. 2013
J. Koch, A. Rätz, and B. Schweizer
(See online at https://doi.org/10.1017/S0956792512000307)
