A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media

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.

[1]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

[2]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[3]  I. V. Radhakrishna Murthy,et al.  The density difference and generalized programs for two- and three-dimensional gravity modeling , 1990 .

[4]  Gudmundur S. Bodvarsson,et al.  Lubrication theory analysis of the permeability of rough-walled fractures , 1991 .

[5]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[6]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[7]  W. Hackbusch,et al.  Composite finite elements for the approximation of PDEs on domains with complicated micro-structures , 1997 .

[8]  Jan Lunze.  Qualitative modelling of dynamical systems motivation, methods, and prospective applications , 1998 .

[9]  Jérôme Jaffré,et al.  Domain Decomposition for Some Transmission Problems in Flow in Porous Media , 2000 .

[10]  Ted Belytschko,et al.  Discontinuous enrichment in finite elements with a partition of unity method , 2000 .

[11]  Jérôme Jaffré,et al.  Modeling fractures as interfaces for flow and transport in porous media , 2001 .

[12]  P. Hansbo,et al.  An unfitted finite element method, based on Nitsche's method, for elliptic interface problems , 2002 .

[13]  Philippe Angot.  A model of fracture for elliptic problems with flux and solution jumps , 2003 .

[14]  Décomposition de domaine pour un milieu poreux fracturé , 2005 .

[15]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[16]  P. Hansbo,et al.  A finite element method for the simulation of strong and weak discontinuities in solid mechanics , 2004 .

[17]  Peter Hansbo,et al.  Nitsche's method for interface problems in computa‐tional mechanics , 2005 .

[18]  Vincent Martin,et al.  Modeling Fractures and Barriers as Interfaces for Flow in Porous Media , 2005, SIAM J. Sci. Comput..

[19]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[20]  Yazid Abdelaziz,et al.  Review: A survey of the extended finite element , 2008 .

[21]  S. Mohammadi.  Extended Finite Element Method , 2008 .

[22]  T. Fries.  A corrected XFEM approximation without problems in blending elements , 2008 .

[23]  Maxim A. Olshanskii,et al.  A Finite Element Method for Elliptic Equations on Surfaces , 2009, SIAM J. Numer. Anal..

[24]  Philippe Angot,et al.  ASYMPTOTIC AND NUMERICAL MODELLING OF FLOWS IN FRACTURED POROUS MEDIA , 2009 .

[25]  Alessio Fumagalli,et al.  Numerical modelling of multiphase subsurface flow in the presence of fractures , 2011 .

[26]  Thierry Gallouët,et al.  A model for conductive faults with non-matching grids , 2011, Computational Geosciences.

[27]  Ted A. Long,et al.  On the use of enriched finite element method to model subsurface features in porous media flow problems , 2011 .

[28]  Mats G. Larson,et al.  Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D , 2012, 1210.7076.

[29]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[30]  Vincent Martin,et al.  Modeling fractures as interfaces with nonmatching grids , 2012, Computational Geosciences.

[31]  C. D'Angelo,et al.  A mixed finite element method for Darcy flow in fractured porous media with non-matching grids , 2012 .

[32]  Stefano Berrone,et al.  On Simulations of Discrete Fracture Network Flows with an Optimization-Based Extended Finite Element Method , 2013, SIAM J. Sci. Comput..

[33]  Alessio Fumagalli,et al.  A numerical method for two-phase flow in fractured porous media with non-matching grids , 2013 .

[34]  Alessio Fumagalli,et al.  A Reduced Model for Flow and Transport in Fractured Porous Media with Non-matching Grids , 2013 .

[35]  Stefano Berrone,et al.  The virtual element method for discrete fracture network simulations , 2014 .

[36]  Simona Perotto,et al.  Efficient geometric reconstruction of complex geological structures , 2014, Math. Comput. Simul..

[37]  Alessio Fumagalli,et al.  An Efficient XFEM Approximation of Darcy Flows in Arbitrarily Fractured Porous Media , 2014 .

[38]  Alessio Fumagalli,et al.  A reduced model for Darcy’s problem in networks of fractures , 2014 .

[39]  Stefano Berrone,et al.  An optimization approach for large scale simulations of discrete fracture network flows , 2014, J. Comput. Phys..

[40]  Rainer Helmig,et al.  Dimensionally reduced flow models in fractured porous media: crossings and boundaries , 2015, Computational Geosciences.

[41]  Peter Hansbo,et al.  CutFEM: Discretizing geometry and partial differential equations , 2015 .

[42]  Nicolas Schwenck,et al.  An XFEM-based model for fluid flow in fractured porous media , 2015 .

[43]  Jmrj Jacques Huyghe,et al.  The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials , 2015 .

[44]  Anna Scotti,et al.  MIMETIC FINITE DIFFERENCE APPROXIMATION OF FLOWS IN FRACTURED POROUS MEDIA , 2016 .

[45]  Oliver Sander,et al.  Simulation of Deformation and Flow in Fractured, Poroelastic Materials , 2016, 1606.05765.

[46]  Sabine Fenstermacher,et al.  Numerical Approximation Of Partial Differential Equations , 2016 .

[47]  Alessio Fumagalli,et al.  Well Posedness of Fully Coupled Fracture/Bulk Darcy Flow with XFEM , 2017, SIAM J. Numer. Anal..

[48]  Katja Bachmeier,et al.  Finite Elements Theory Fast Solvers And Applications In Solid Mechanics , 2017 .

[49]  Alessio Fumagalli,et al.  A Double-Layer Reduced Model for Fault Flow on Slipping Domains with an Hybrid Finite Volume Scheme , 2017, J. Sci. Comput..

[50]  Jan M. Nordbotten,et al.  Robust Discretization of Flow in Fractured Porous Media , 2016, SIAM J. Numer. Anal..