A new upscaling method for fractured porous media

We present a method to determine equivalent permeability of fractured porous media. Inspired by the previous flow-based upscaling methods, we use a multi-boundary integration approach to compute flow rates within fractures. We apply a recently developed multi-point flux approximation Finite Volume method for discrete fracture model simulation. The method is verified by upscaling an arbitrarily oriented fracture which is crossing a Cartesian grid. We demonstrate the method by applying it to a long fracture, a fracture network and the fracture network with different matrix permeabilities. The equivalent permeability tensors of a long fracture crossing Cartesian grids are symmetric, and have identical values. The application to the fracture network case with increasing matrix permeabilities shows that the matrix permeability influences more the diagonal terms of the equivalent permeability tensor than the off-diagonal terms, but the off-diagonal terms remain important to correctly assess the flow field.

[1]  Pierre M. Adler,et al.  PERMEABILITY OF THREE-DIMENSIONAL FRACTURE NETWORKS , 1998 .

[2]  Michael G. Edwards,et al.  Finite volume discretization with imposed flux continuity for the general tensor pressure equation , 1998 .

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

[4]  Jerome Droniou,et al.  FINITE VOLUME SCHEMES FOR DIFFUSION EQUATIONS: INTRODUCTION TO AND REVIEW OF MODERN METHODS , 2014, 1407.1567.

[5]  L. Durlofsky,et al.  An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators , 2004 .

[6]  P. Renard,et al.  Calculating equivalent permeability: a review , 1997 .

[7]  Robert W. Zimmerman,et al.  Permeability tensor of three‐dimensional fractured porous rock and a comparison to trace map predictions , 2014 .

[8]  L. Durlofsky.  Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media , 1991 .

[9]  J. Jaime Gómez-Hernández,et al.  Three-dimensional hydraulic conductivity upscaling in groundwater modeling , 2010, Comput. Geosci..

[10]  J. Gómez-Hernández,et al.  Stochastic characterization of gridblock permeabilities , 1994 .

[11]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[12]  M. F. Lough,et al.  An Efficient Boundary Integral Formulation for Flow Through Fractured Porous Media , 1998 .

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

[14]  C. Clauser.  Permeability of crystalline rocks , 1992 .

[15]  Sebastian Geiger,et al.  Static and Dynamic Assessment of DFN Permeability Upscaling , 2012 .

[16]  Alireza Baghbanan,et al.  Stress effects on permeability in a fractured rock mass with correlated fracture length and aperture , 2008 .

[17]  Lingli Wei,et al.  Integration of Discrete Feature Network Methods With Conventional Simulator Approaches , 2000 .

[18]  Christian Vogt,et al.  Estimating the permeability distribution and its uncertainty at the EGS demonstration reservoir Soultz‐sous‐Forêts using the ensemble Kalman filter , 2012 .

[19]  Ronald A. Nelson,et al.  Geologic Analysis of Naturally Fractured Reservoirs , 1986 .

[20]  C. L. Farmer,et al.  Upscaling: a review , 2002 .

[21]  M. Oda,et al.  Permeability tensor for discontinuous rock masses , 1985 .

[22]  P. Witherspoon,et al.  Porous media equivalents for networks of discontinuous fractures , 1982 .

[23]  L. Durlofsky,et al.  Generation of coarse‐scale continuum flow models from detailed fracture characterizations , 2006 .

[24]  Yu-Shu Wu,et al.  On the Effective Continuum Method for Modeling MultiphaseFlow, Multicomponent Transport and Heat Transfer in FracturedRock , 2013 .

[25]  Stein Krogstad,et al.  Open-source MATLAB implementation of consistent discretisations on complex grids , 2012, Computational Geosciences.

[26]  Jonny Rutqvist,et al.  Stress-dependent permeability of fractured rock masses: A numerical study , 2004 .

[27]  Jean-Claude Berthet,et al.  Numerical scheme to simulate flow through anisotropic rocks in TOUGH2 , 2014, Comput. Geosci..

[28]  Mikhail Shashkov,et al.  Solving Diffusion Equations with Rough Coefficients in Rough Grids , 1996 .

[29]  O. Gosselin,et al.  Computation of Effective Dynamic Properties of Naturally Fractured Reservoirs: Comparison and Validation of Methods , 2013 .

[30]  Pierre M. Adler,et al.  Effective permeability of fractured porous media in steady state flow , 2003 .

[31]  Haiyan Zhou,et al.  Steady-state saturated groundwater flow modeling with full tensor conductivities using finite differences , 2010, Comput. Geosci..

[32]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[33]  D. T. Snow,et al.  Anisotropie Permeability of Fractured Media , 1969 .

[34]  I. Aavatsmark,et al.  An Introduction to Multipoint Flux Approximations for Quadrilateral Grids , 2002 .

[35]  X. Sanchez‐Vila,et al.  Representative hydraulic conductivities in saturated groundwater flow , 2006 .

[36]  S. P. Neuman,et al.  Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal , 1993 .

[37]  J. Gómez-Hernández,et al.  Upscaling hydraulic conductivities in heterogeneous media: An overview , 1996 .