Geological formations are characterized by a **hierarchy of spatial and temporal scales**, which cannot be explicitly described due to prohibitive computational costs. Due to the lack of scale separations in critical flow regimes, traditional upscaling techniques are in general unable to describe nonlinear/nonequilibrium processes.

**Multiscale numerical methods** offer an alternative: they allow to remember fine-scale details where and if needed, and to correctly capture the large-scale response of the system. Our research focuses on the Multiscale Finite Volume Methods (MsFVM), which we have developed to improve the **computational efficiency** of large-reservoir simulators [3, 2,4]. Currently, we explore the possibility of using the MsFVM to capture flow **instability and non-equilibrium processes** (e.g., gravity fingers [1,6] due to the dissolution rate of supercritical CO2 in deep aquifers, reaction or phase transition kinetics). In this context, the method can also be used as a **downscaling technique**, which offer peculiar advantages [1,5] .

Contact: R. Künze, I. Lunati

Figure: Adaptive multiscale simulations of saltwater-freshwater instability with different front-detection criteria and grid refinements; shown is the normalized salt concentration (red corresponds to the highest concentration) at the same time step obtained with different adaptive criteria. From left to right the adaptive criterion becomes looser and fine-scales details of the flow are modeled in a smaller portion of the domain. A looser adaptive criterion leads to a partially coarsened description of the flow, but the overall behavior is still captured correctly–(b) and (c) [1].

**References**

[1] Künze, R., and I. Lunati, An adaptive multiscale method for density-driven instabilities, *Journal of Computational Physics*, 231, 5557–5570, 2012, doi:10.1016/j.jcp.2012.02.025

[2] Künze, R., I. Lunati, and S.H. Lee, A Multilevel Multiscale Finite Volume method, *Journal of Computational Physics*, 255, 502–520, 2013 doi:10.1016/j.jcp.2013.08.042

[3] Lunati, I., M. Tyagi, and S.H. Lee, An iterative Multiscale Finite-Volume algorithm con- verging to the exact solution, *Journal of Computational Physics*, 230(5), 1849-1864, 2011, doi:10.1016/j.jcp.2010.11.036

[4] Hajibeygi, H., S.H. Lee, and I. Lunati, Accurate and Efficient Simulation of Multiphase Flow in a Heterogeneous Reservoir by Using Error Estimate and Control in the Multiscale Finite-Volume Framework, *SPE Journal*, 17 (4), 1071–1083, SPE-141954-PA, 2012 doi:10.2118/141954-PA

[5] Tomin, P., and I. Lunati, A Hybrid Multiscale Method for Two-Phase Flow in Porous Media, *Journal of Computational Physics*, 250, 293–307, 2013 doi:10.1016/j.jcp.2013.05.019

[6] Künze, R., P. Tomin, and I. Lunati, Local modelling of instability onset for global finger evolution, *Advances in Water Research*, 70, 148–159, 2014 doi:10.1016/j.advwatres.2014.05.003

[7] Künze, R., *Multiscale Descriptions of Density Driven Instabilities*, PhD Thesis, University of Lausanne, 2014 [text]