Patrick Jenny (Institute of Fluid Dynamics, ETH Zürich, Suisse). Adaptive Multiscale Methods to Solve Elliptic PDEs with Applications to Subsurface Flow.

In order to model flow and transport in heterogeneous porous media, various upscaling and multiscale methods have been devised. One of them is the multiscale finite-volume (MSFV) method, for which was shown that it can rigorously handle complex wells, discrete fractures, gravity and compressibility. The iterative MSFV method can be regarded as a domain decomposition solver with the particular advantage that conservative fluxes are obtained at any iteration level. Therefore, usually full converge of the flow solution is not required, which is an attractive property for the construction of adaptive, efficient simulation tools. The main purpose of adaptivity is to minimize local fine-scale calculations, which are responsible for most of the computational cost. For example, once computed, the basis functions can be reused as long as the local mobility changes remain small. Moreover, it was shown that MSFV solutions can dramatically be improved by enrichment strategies. Besides efficient and adaptive flow computations, also transport has to be addressed. On one hand, a multiscale solver can take great advantage form recently developed unconditionally stable transport schemes, which allow for very large time steps, and from an adaptive multi-resolution approach for saturation transport. It has to be emphasised that such multiscale methods can be employed for many difficult practical problems in other application areas.
 

Figure: Water invading a heterogeneous oil reservoir from the bottom left corner. Red and blue indicate 100% and 0% water, respectively. The solution on the left was obtained by a standard finite volume method using 12,100 dof and the solution on the right by the MSFV method using only 100 coupled dof.