Numerical upscaling for flows in heterogeneous porous media
Porous media are often highly heterogeneous. Solving porous media problems at the fine scale, which
resolves all the heterogeneities, is computationally expensive due to the
required memory and computational time.
Upscaling techniques, such as homogenization, averaging, etc., have to be used to make solving multiscale problems possible.
Essential success was achieved during the last decades in the studies of problems with clearly
separated fine and coarse scales (periodic microstructure, statistically
homogeneous porous media). When the fine and the coarse scales can be decoupled, solving a multiscale problem reduces
to one way two-stage procedure: i) solve fine scale “cell-problem” and use its
solution to upscale the effective properties of the multiscale media; ii) solve coarse scale equations with the calculated effective coefficients. The separation of scales, however, is not always possible, and developing numerical upscaling techniques for such problems is the subject of this presentation.
We consider pressure equation, obtained by combining the continuity equation
and Darcy's law, for steady state incompressible single-phase flow. Finite
volume discretization (Multi-Point Flux Approximation) for this problem is
developed in the case of jump discontinuities for the permeability. The effective properties of the media are calculated, two-grid method is
discussed and the results from numerical experiments are presented.