Up scaling techniques allowing to use coarser meshes have rigorous fundations as well as the underlying heterogeneous porous medium presents some statistical homogeneity. In that situation, homogenization theory works well and provides closure problems, the solution of which gives rises to up scaled paramters. These assumptions break down in presence of a front corresponding for example to water flooding of an oil reservoir, or to situations leading to miscible viscous fingering. Breaking translational invariance leads to revisit existing up scaling techniques: it is possible to use standard up scaling far from the front, and to adapt the methods in coarse grid blocks close to the front? It is possible to use effective transport and pressure equations defined at coarse scale accounting for subgrid heterogeneities? It is possible to get a deeper understanding of the coupling between instabilities and heterogeneities?
In that presentation, we will present some approaches and some existing results or conjectures on that issue.
Artus, V., & Noetinger, B. (2004). Up-scaling two-phase flow in heterogeneous reservoirs: current trends. Oil & gas science and technology, 59(2), 185-195.
Nœtinger, B., Artus, V., & Ricard, L. (2004). Dynamics of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. 2–isotropic media. Transport in porous media, 56(3), 305-328.
Ouaki, F., Allaire, G., Desroziers, S., & Enchéry, G. (2015). A priori error estimate of a multiscale finite element method for transport modeling. SeMa Journal, 67(1), 1-37.
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