The hybrid dimensional representation of permeability tensor: A reinterpretation of the discrete fracture model and its extension on nonconforming meshes

Abstract The discrete fracture model (DFM) has been widely used in the simulation of fluid flow in fractured porous media. Traditional DFM uses the so-called hybrid-dimensional approach to treat fractures explicitly as low-dimensional entries (e.g. line entries in 2D media and face entries in 3D media) on the interfaces of matrix cells and then couple the matrix and fracture flow systems together based on the principle of superposition with the fracture thickness used as the dimensional homogeneity factor. Because of this methodology, DFM is considered to be limited on conforming meshes and thus may raise difficulties in generating high quality unstructured meshes due to the complexity of fracture's geometrical morphology. In this paper, we clarify that the DFM actually can be extended to non-conforming meshes without any essential changes. To show it clearly, we provide another perspective for DFM based on hybrid-dimensional representation of permeability tensor to describe fractures as one-dimensional line Dirac delta functions contained in permeability tensor. A finite element DFM scheme for single-phase flow on non-conforming meshes is then derived by applying Galerkin finite element method to it. Analytical analysis and numerical experiments show that our DFM automatically degenerates to the classical finite element DFM when the mesh is conforming with fractures. Moreover, the accuracy and efficiency of the model on non-conforming meshes are demonstrated by testing several benchmark problems. This model is also applicable to curved fracture with variable thickness.

[1]  Hussein Hoteit,et al.  Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures , 2008 .

[2]  Jun Yao,et al.  A Two-Phase Flow Simulation of Discrete-Fractured Media using Mimetic Finite Difference Method , 2014 .

[3]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

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

[5]  Abbas Firoozabadi,et al.  Control-Volume Model for Simulation of Water Injection in Fractured Media: Incorporating Matrix Heterogeneity and Reservoir Wettability Effects , 2007 .

[6]  Abbas Firoozabadi,et al.  Comparison of fully implicit and IMPES formulations for simulation of water injection in fractured and unfractured media , 2007 .

[7]  Alessio Fumagalli,et al.  A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media , 2016 .

[8]  J. Noorishad,et al.  An upstream finite element method for solution of transient transport equation in fractured porous media , 1982 .

[9]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[10]  Yueying Wang,et al.  Accurate multiscale finite element method for numerical simulation of two-phase flow in fractured media using discrete-fracture model , 2013, J. Comput. Phys..

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

[12]  Yushu Wu,et al.  Coupled numerical approach combining X-FEM and the embedded discrete fracture method for the fluid-driven fracture propagation process in porous media , 2020 .

[13]  G. I. Barenblatt,et al.  Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata] , 1960 .

[14]  L. K. Thomas,et al.  Fractured Reservoir Simulation , 1980 .

[15]  R. Glowinski,et al.  A fictitious domain method for Dirichlet problem and applications , 1994 .

[16]  Sebastian Geiger,et al.  A Novel Multi-Rate Dual-Porosity Model for Improved Simulation of Fractured and Multiporosity Reservoirs , 2013 .

[17]  R. Helmig,et al.  A mixed-dimensional finite volume method for two-phase flow in fractured porous media , 2006 .

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

[19]  Rainer Helmig,et al.  A discrete fracture model for two-phase flow in fractured porous media , 2017 .

[20]  Hussein Hoteit,et al.  Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media , 2005 .

[21]  R. Helmig,et al.  Black-Oil Simulations for Three-Component -- Three-Phase Flow in Fractured Porous Media , 2007 .

[22]  Cornelis Vuik,et al.  Algebraic dynamic multilevel method for embedded discrete fracture model (F-ADM) , 2018, J. Comput. Phys..

[23]  Hossein Kazemi,et al.  Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution , 1969 .

[24]  E. M. Baskin,et al.  Anomalous diffusion and drift in a comb model of percolation clusters , 1991 .

[25]  Hadi Hajibeygi,et al.  Projection-based Embedded Discrete Fracture Model (pEDFM) , 2017 .

[26]  Milind Deo,et al.  Finite element, discrete‐fracture model for multiphase flow in porous media , 2000 .

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

[28]  Hussein Hoteit,et al.  An efficient numerical model for incompressible two-phase flow in fractured media , 2008 .

[29]  Liehui Zhang,et al.  Numerical simulation of water flooding in natural fractured reservoirs based on control volume finite element method , 2016 .

[30]  Alessio Fumagalli,et al.  Benchmarks for single-phase flow in fractured porous media , 2017, ArXiv.

[31]  Mats G. Larson,et al.  A simple embedded discrete fracture–matrix model for a coupled flow and transport problem in porous media , 2018, Computer Methods in Applied Mechanics and Engineering.

[32]  S. Geiger,et al.  Black-Oil Simulations for Three-Component, Three-Phase Flow in Fractured Porous Media , 2009 .

[33]  Peter Hansbo,et al.  Cut finite elements for convection in fractured domains , 2018, Computers & Fluids.

[34]  Mayur Pal,et al.  Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model , 2015, J. Comput. Phys..

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

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

[37]  Mixed‐primal Discontinuous Galerkin approximation of flows in fractured porous media on polygonal and polyhedral grids , 2019, PAMM.

[38]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[39]  Hossein Kazemi,et al.  Analytical and numerical solution of oil recovery from fractured reservoirs with empirical transfer functions , 1992 .

[40]  Yang Li,et al.  An efficient embedded discrete fracture model based on mimetic finite difference method , 2016 .

[41]  Seong H. Lee,et al.  Efficient Field-Scale Simulation of Black Oil in a Naturally Fractured Reservoir Through Discrete Fracture Networks and Homogenized Media , 2008 .

[42]  Kamy Sepehrnoori,et al.  Development of an Efficient Embedded Discrete Fracture Model for 3D Compositional Reservoir Simulation in Fractured Reservoirs , 2014 .

[43]  O. C. Zienkiewicz,et al.  The Finite Element Method: Its Basis and Fundamentals , 2005 .

[44]  Jiamin Jiang,et al.  An improved projection-based embedded discrete fracture model (pEDFM) for multiphase flow in fractured reservoirs , 2017 .

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

[46]  Jean E. Roberts,et al.  A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures , 2019, GEM - International Journal on Geomathematics.

[47]  Nasser Khalili,et al.  Fully Coupled XFEM Model for Flow and Deformation in Fractured Porous Media with Explicit Fracture Flow , 2016 .

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

[49]  Abbas Firoozabadi,et al.  Three-Phase Compositional Modeling with Capillarity in Heterogeneous and Fractured Media , 2013 .

[50]  Rolf Krause,et al.  3D non-conforming mesh model for flow in fractured porous media using Lagrange multipliers , 2019, Comput. Geosci..

[51]  Paola F. Antonietti,et al.  Discontinuous Galerkin Approximation of Flows in Fractured Porous Media on Polytopic Grids , 2019, SIAM J. Sci. Comput..

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

[53]  J. E. Warren,et al.  The Behavior of Naturally Fractured Reservoirs , 1963 .

[54]  Bailian Chen,et al.  A hybrid embedded discrete fracture model for simulating tight porous media with complex fracture systems , 2019, Journal of Petroleum Science and Engineering.

[55]  R. Arnett,et al.  Modelling fluid flow in fractured‐porous rock masses by finite‐element techniques , 1984 .

[56]  Jérôme Jaffré,et al.  A Lagrange multiplier method for a discrete fracture model for flow in porous media , 2018, Computational Geosciences.

[57]  Hussein Hoteit,et al.  Compositional Modeling of Discrete-Fractured Media Without Transfer Functions by the Discontinuous Galerkin and Mixed Methods , 2006 .

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

[59]  A. Firoozabadi,et al.  Control‐volume method for numerical simulation of two‐phase immiscible flow in two‐ and three‐dimensional discrete‐fractured media , 2004 .

[60]  Jan M. Nordbotten,et al.  An efficient multi-point flux approximation method for Discrete Fracture-Matrix simulations , 2012, J. Comput. Phys..

[61]  Abbas Firoozabadi,et al.  Higher-order compositional modeling of three-phase flow in 3D fractured porous media based on cross-flow equilibrium , 2013, J. Comput. Phys..

[62]  A. Firoozabadi,et al.  An efficient numerical model for multicomponent compressible flow in fractured porous media , 2014 .

[63]  Na Zhang,et al.  Locally conservative Galerkin and finite volume methods for two-phase flow in porous media , 2013, J. Comput. Phys..

[64]  Abbas Firoozabadi,et al.  Numerical Study of Natural Convection and Diffusion in Fractured Porous Media , 2000 .

[65]  Hanqiao Jiang,et al.  A discrete modeling framework for reservoirs with complex fractured media: Theory, validation and case studies , 2017, Journal of Petroleum Science and Engineering.

[66]  Joachim Moortgat,et al.  Implicit finite volume and discontinuous Galerkin methods for multicomponent flow in unstructured 3D fractured porous media , 2016, 1608.06894.

[67]  Alexandru-Bogdan Tatomir From discrete to continuum concepts of flow in fractured porous media , 2012 .