The Hybrid-dimensional Darcy's Law: A Reinterpreted Discrete Fracture Model for Fracture and Barrier Networks on Non-conforming Meshes

In this paper, we extend the reinterpreted discrete fracture model for flow simulation of fractured porous media containing flow blocking barriers on non-conforming meshes. The methodology of the approach is to modify the traditional Darcy’s law into the hybrid-dimensional Darcy’s law where fractures and barriers are represented as Dirac-δ functions contained in the permeability tensor and resistance tensor, respectively. As a natural extension of the reinterpreted discrete fracture model [21] for highly conductive fractures, this model is able to account for the influence of both highly conductive fractures and blocking barriers accurately on non-conforming meshes. The local discontinuous Galerkin (LDG) method is employed to accommodate the form of the hybrid-dimensional Darcy’s law and the nature of the pressure/flux discontinuity. The performance of the model is demonstrated by several numerical tests.

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

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

[3]  B. Berkowitz.  Characterizing flow and transport in fractured geological media: A review , 2002 .

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

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

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

[7]  James P. Evans,et al.  Fault zone architecture and permeability structure , 1996 .

[8]  H. Rashid,et al.  Three-Dimensional Projection-Based Embedded Discrete-Fracture Model for Compositional Simulation of Fractured Reservoirs , 2020 .

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

[10]  M. F. Lough,et al.  Hierarchical modeling of flow in naturally fractured formations with multiple length scales , 2001 .

[11]  Abbas Firoozabadi,et al.  Numerical Simulation of Water Injection in 2D Fractured Media Using Discrete-Fracture Model , 2001 .

[12]  K. Sepehrnoori,et al.  Development of an Embedded Discrete Fracture Model for Field-Scale Reservoir Simulation With Complex Corner-Point Grids , 2019, SPE Journal.

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

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

[15]  F. Brezzi,et al.  Mathematical models and finite elements for reservoir simulation : single phase, multiphase and multicomponent flows through porous media , 1988 .

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

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

[18]  J. Latham,et al.  Modelling stress-dependent single and multi-phase flows in fractured porous media based on an immersed-body method with mesh adaptivity , 2018, Computers and Geotechnics.

[19]  Louis J. Durlofsky,et al.  An Efficient Discrete Fracture Model Applicable for General Purpose Reservoir Simulators , 2003 .

[20]  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 .

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

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

[23]  Michel Quintard,et al.  A two-phase flow simulation of a fractured reservoir using a new fissure element method , 2001 .

[24]  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..

[25]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

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

[27]  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.

[28]  A. Firoozabadi,et al.  Reservoir simulation of fractured media in compressible single-phase flow in 2D, 2.5D and 3D unstructured gridding , 2018, Advances in Water Resources.

[29]  Sebastian Geiger,et al.  A Novel Multi-rate Dual-porosity Model for Improved Simulation of Fractured and Multi-porosity Reservoirs , 2011 .

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

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

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

[33]  Alberto Cominelli,et al.  Simulation of Miscible Gas Injection in a Fractured Carbonate Reservoir using an Embedded Discrete Fracture Model , 2014 .

[34]  Hussein Hoteit,et al.  Nuclear Waste Disposal Simulations: Couplex Test Cases , 2004 .

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

[36]  Alessio Fumagalli,et al.  An upscaling procedure for fractured reservoirs with embedded grids , 2016 .

[37]  Yang Yang,et al.  The hybrid dimensional representation of permeability tensor: A reinterpretation of the discrete fracture model and its extension on nonconforming meshes , 2020, J. Comput. Phys..