## A review of the XFEM-based approximation of flow in fractured porous media.(English)Zbl 1371.76093

Ventura, Giulio (ed.) et al., Advances in discretization methods. Discontinuities, virtual elements, fictitious domain methods, Ferrara, Italy, September 2015. Selected contributions based on the presentations at the international conference “eXtended Discretization MethodS”, X-DMS. Cham: Springer (ISBN 978-3-319-41245-0/hbk; 978-3-319-41246-7/ebook). SEMA SIMAI Springer Series 12, 47-76 (2016).
Summary: This paper presents a review of the available mathematical models and corresponding non-conforming numerical approximations which describe single-phase fluid flow in a fractured porous medium. One focus is on the geometrical difficulties that may arise in realistic simulations such as intersecting and immersed fractures. Another important aspect is the choice of the approximation spaces for the discrete problem: in mixed formulations, both the Darcy velocity and the pressure are considered as unknowns, while in classical primal formulations, a richer space for the pressure is considered and the Darcy velocity is computed a posteriori. In both cases, the extended finite element method is used, which allows for a complete geometrical decoupling among the fractures and rock matrix grids. The fracture geometries can thus be independent of the underlying grid thanks to suitable enrichments of the spaces that are able to represent possible jumps of the solution across the fractures. Finally, due to the dimensional reduction, a better approximation of the resulting boundary conditions for the fractures is addressed.
For the entire collection see [Zbl 1357.76007].

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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### References:

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