24 November – Grégory Etangsale: A primal hybridizable discontinuous Galerkin method for modelling flows in fractured porous media

Grégory Etangsale: Wednesday 24th November at 10:30

 

ABSTRACT: Modeling fluid flow in fractured porous media has received tremendous attention from engineering, geophysical, and other research fields over the past decades. We focus here on large fractures described individually in the porous medium, which act as preferential paths or barriers to the flow. Two different approaches are available from a computational aspect:

  • The first one, and definitively the oldest, consists of meshing inside the fracture. In this case, the flow is governed by a single Darcy equation characterized by a large scale of variation of the permeability coefficient within the matrix region and the fracture, respectively. However, this description becomes quite challenging since it requires a considerable amount of memory storage, severely increasing the CPU time.
  • A more recent approach differs by considering the fracture as an encapsulated object of lower dimension, i.e., (d − 1)-dimension. As a result, the flow process is now governed by distinctive equations in the matrix region and fractures, respectively. Thus, coupling conditions are added to close the problem. This mathematical description of the fractured porous media has been initially introduced by Martin et al. in [4] and is referred to as the Discrete Fracture-Matrix (DFM) model.

The DFM description is particularly attractive since it significantly simplifies the meshing of fractures and allows the coupling of distinctive discretizations such as Discontinuous and Continuous Galerkin methods inside the bulk region and the fracture network, respectively. For instance, we refer the reader to the recent works of Antonietti et al. [1] (and references therein), where the authors coupled the Interior Penalty DG method with the (standard) H1-Conforming finite element method to solve the DFM problem (see e.g., [3]). However, it is well-known that DG methods are generally more expensive than most other numerical methods due to their high number of coupled degrees of freedom (DOFs) and their large stencils. Therefore, in the present work, we favour families of Hybridizable Discontinuous Galerkin (HDG) methods which are more performant and competitive (thanks to the static condensation) than standard DG counterparts. Furthermore, HDG methods are particularly relevant in the DFM model due to the localization of their DOFs on the mesh skeleton. To our knowledge, only Chave et al. [2] recently designed a discontinuous skeletal approach based on the Hybrid High-Order (HHO) method to address this issue. Numerical experiments are then presented to corroborate our assertions. First, we measure the estimated convergence rates in simple situations characterized by a single fracture proving that the proposed discretization method converges optimally. We, therefore, investigate the ability of the HDG scheme to handle more complex geometries characterized by intersecting and immersed fractures by comparing our discrete solutions to those of commercial software such as COMSOL.

[1] Antonietti P. F., Facciola C. and Verani M., Unified analysis of discontinuous Galerkin approximations of flows in fractured porous media on polygonal and polyhedral grids, Mathematics in Engineering, 2, 340–385 (2020)

[2] Chave F., Di Pietro D. A. and Formaggia L., A Hybrid High-Order method for Darcy flows in fractured porous media, SIAM Journal on Scientific Computing, 40, A1063–A1094 (2018)

[3] Kadeethum T., Nick H. M., Lee S. and Ballarin F., Flow in porous media with low dimensional fractures by employing enriched Galerkin method, Advances in Water Resources, 142, 103620 (2020)

[4] Martin V., Jaffre J. and Roberts J. E., Modeling fractures and barriers as interfaces for flow in porous media, SIAM Journal on Scientific Computing, 26, 1667–1691 (2005)

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