Seminari di Analisi Numerica: “Flexible discretizations for mixed-dimensional single-phase flow in fractured porous media”

Venerdì 2 Marzo 2018, ore 12:30 - Aula 2AB40 - Alessio Fumagalli


Venerdì 2 Marzo 2018 alle ore 12:30 in Aula 2AB40, Alessio Fumagalli (Department of Mathematics, University of Bergen, Norway) terrà un seminario dal titolo “Flexible discretizations for mixed-dimensional single-phase flow in fractured porous media”.

Fractures are planar or sub-planar discontinuities along which a rock has been broken, and represent important conduits or barriers for fluid flow. Fracture intersections can have a significant impact as well and, because of infilling process, may behave differently from the surrounding fractures. Since fracture aperture is order of magnitude smaller than other characteristic sizes, a mixed-dimensional representation is preferable to the full three-dimensional [1]. We focus on incompressible single-phase flow described by a Darcy-type model. Fractures may impose severe constraints to the gridding, possibly resulting in high number of cells and low quality grids. The former is more evident for complex fracture networks [2]. In the first part of the presentations, we consider the equations discretized by the mixed virtual element method (MVEM) [3-4]. MVEM is locally mass conservative, robust with respect to strong permeability jumps, and able to handle cells with (almost) arbitrary geometries thus perfectly suited for our problem.
In the second part of the presentation, we introduce a new framework able to handle in a natural way heterogeneity in term of geometrical conformity among the different dimensions [5]. The key ingredient is a conservative mortar-like approach for the intra-dimensional coupling. The flexibility of the non-conforming geometrical coupling given by the mortar is crucial to solve realistic problems in presence of complex fracture networks decreasing the computational cost possibly associated to over-refined conforming grids. This approach is of particular importance for geothermal energy extraction, CO2
sequestration, enhance oil recovery, and nuclear waste disposal. Illustrative bi-dimensional examples and realistic three-dimensional geometries are considered to emphasize the potentialities and a high freedom of the proposed approaches.

[1] Boon, W. M.; Nordbotten, J. M. & Yotov, I. Robust Discretization of Flow in Fractured Porous Media. arXiv:1601.06977v2 [math.NA], 2017
[2] Flemisch, B.; Berre, I.; Boon, W.; Fumagalli, A.; Schwenck, N.; Scotti, A.; Stefansson, I. & Tatomir, A. Benchmark of single-phase flow in fractured porous media with non-conforming discretization methods Advances in Water Resources, 2018, 111, 239-258
[3] Beirao da Veiga, L.; Brezzi, F.; Marini, L. D. & Russo, A. Mixed virtual element methods for general second order elliptic problems on polygonal meshes ESAIM: M2AN, 2016, 50, 727-747
[4] Fumagalli, A. & Keilegavlen, E. Dual Virtual Element Methods for Discrete Fracture Matrix Models. arXiv:1711.01818 [math.NA], 2017
[5] Nordbotten, J. M.; Boon, W.; Fumagalli, A. & Keilegavlen, E. Unified approach to discretization of flow in fractured porous media 2018. In preparation