Abstract
This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures.
Highlights
This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach
To resolve some of the shortcomings mentioned above, we propose an enriched Galerkin (EG) discretization (Lee et al, 2016a; Zi et al, 2004; Khoei et al, 2018) to model fluid flow in fractured porous media using the mixed-dimensional approach
This study presents the EG discretization for solving a single-phase fluid flow in the fractured porous media using the mixed-dimensional approach
Summary
Modeling of fluid flow in fractured porous media is essential for a wide variety of applications including water resource management (Glaser et al, 2017; Peng et al, 2017), geothermal energy (Willems and Nick, 2019; Salimzadeh et al, 2019a; Salimzadeh and Nick, 2019), oil and gas (Wheeler et al, 2019; Kadeethum et al, 2019c; Andrianov and Nick, 2019; Kadeethum et al, 2020c), induced seismicity (Rinaldi and Rutqvist, 2019), CO2 sequestration (Salimzadeh et al, 2018), and biomedical engineering (Vinje et al, 2018; Ruiz Baier et al, 2019; Kadeethum et al, 2020a). Many approaches have been proposed to model the fractured porous media using the mixed-dimensional approach; (1) two-point flux approximation in unstructured control volume finitedifference technique (Karimi-Fard et al, 2004), (2) multi-point flux approximation using mixed finite element method on general quadrilateral and hexahedral grids (Wheeler et al, 2012), (3) eXtended finite element combined with mixed finite element formulation (D’Angelo and Scotti, 2012; Prevost and Sukumar, 2016; Sanborn and Prevost, 2011), (4) embedded discrete fracture-matrix (DFM) modeling with non-conforming mesh (Hajibeygi et al, 2011; Odsaeter et al, 2019), (5) mixed approximation such as mimetic finite difference (Flemisch and Helmig, 2008; Formaggia et al, 2018), (6) two-field formulation using mixed finite element (MFE) (Martin et al, 2005; Fumagalli et al, 2019), and (7) disconinuous Galerkin (DG) method (Rivie et al, 2000; Hoteit and Firoozabadi, 2008; Antonietti et al, 2019; Arnold et al, 2002). The numerical examples section presents five examples, and the conclusion is provided
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