The Localized Reduced Basis Multiscale method for two-phase flows in   porous media
  In this work, we propose a novel model order reduction approach for two-phase flow in porous media by introducing a formulation in which the mobility, which realizes the coupling between phase saturations and phase pressures, is regarded as a parameter to the pressure equation. Using this formulation, we introduce the Localized Reduced Basis Multiscale method to obtain a low-dimensional surrogate of the high-dimensional pressure equation. By applying ideas from model order reduction for parametrized partial differential equations, we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two-phase flow simulation, which is thereby largely accelerated. Usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online step to reach an ideal runtime at acceptable error increase for the two-phase flow simulation. 
Sven Kaulmann, Bernd Flemisch, Bernard Haasdonk, Knut-Andreas Lie, and   Mario Ohlberger
math.NA