Fast Upscaling of the Hydraulic Conductivity of Three-Dimensional Fractured Porous Rock for Reservoir Modeling

  • Tawfik RajehEmail author
  • Rachid Ababou
  • Manuel Marcoux
  • Israel Cañamon
Special Issue


A fast upscaling procedure for determining the equivalent hydraulic conductivity of a three-dimensional fractured rock is presented in this paper. A modified semi-analytical superposition method is developed to take into account, at the same time, the hydraulic conductivity of the porous matrix (KM) and the fractures (KF). The connectivity of the conductive fracture network is also taken into account. The upscaling approach has been validated by comparison with the hydraulic conductivity of synthetic samples calculated with full numerical procedures (flow simulations and averaging). The extended superposition approach is in good agreement with numerical results for infinite size fractures. For finite size fractures, an improved model that takes into account the connectivity of the fracture network through multiplicative connectivity indexes determined empirically is proposed. This improved model is also in good agreement with the numerical results obtained for different configurations of fracture networks.


Fractured porous medium Three-dimensional upscaling Equivalent hydraulic conductivity Numerical simulations Darcy Random fracture sets Network connectivity 



The first three authors wish to acknowledge the financial support of ADEME (France) in the framework of the GeotRef project “Géothermie haute énergie en Reservoirs fracturés” ( We thank the anonymous reviewers for their careful reading of the manuscript and for their insightful comments and suggestions.


  1. Ababou R (1990) Identification of effective conductivity tensor in randomly heterogeneous and stratified aquifers. In: Bachu S (ed) Proceedings Fifth Canadian–American conference on hydrogeology: parameter identification and estimation for aquifers and reservoirs, Calgary, Alberta, Canada, Sep. 18–20, 1990. Nat. Water Well Assoc., Water Well Journal Publish. Co., Dublin, Ohio, 1990, pp 155-157Google Scholar
  2. Ababou R, Bagtzoglou AC(1993) BIGFLOW: a numerical code for simulating flow in variably saturated, heterogeneous geologic media (theory & user’s manual, Version 1.1). Report NUREG/CR-6028, U.S. Nuclear Regul. Commission, Gov. Printing Office, Washington, DCGoogle Scholar
  3. Ababou R., Renard P (2011) Equivalent permeability tensor in fractured media: an algebraic approach. In: Amaziane B, Barrera D, Mraoui H, Rodriguez ML, Sbibih D (eds) Proceedings MAMERN11: 4th internat. conf. on approx. methods and numer. model. in envir. and natur. resour. (Saïdia, Morocco, May 23–26, 2011). Univ. Granada (2011), ISBN: 078-84-338-5230-4, 2011Google Scholar
  4. Ababou R, McLaughlin D, Gelhar LW, Tompson AF (1989) Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media. Transp Porous Media 4(6):549–565CrossRefGoogle Scholar
  5. Ababou R, Millard A, Treille E, Durin M, Plas F (1994) Continuum modeling of coupled thermo-hydro-mechanical processes in fractured rock. In: Peters XA et al (eds) Computational methods in water resources. Kluwer Academic Publishers, Netherlands, pp 651–658CrossRefGoogle Scholar
  6. Ababou R, Cañamón Valera I, Poutrel A (2011) Macro-permeability distribution and anisotropy in a 3D Fissured and fractured clay rock: ‘Excavation Damaged Zone’ around a cylindrical drift in Callovo-Oxfordian Argilite (Bure). J Phys Chem Earth (Spec. Issue “Clays in Natural & Engineer. Barriers for Radioact. Waste Confin.”: CLAYS 2010, Nantes, 29 March–1st April 2010). 36(17–18):1932–1948. ISSN 1474-7065.
  7. Adler PM, Thovert J-F (1999) Fractures and fracture networks. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  8. Adler PM, Thovert J-F, Mourzenko VV (2012) Fractured porous media. Oxford University Press, OxfordCrossRefGoogle Scholar
  9. Balberg I, Anderson CH, Alexander S, Wagner N (1984) Excluded volume and its relation to the onset of percolation. Phys Rev B 30:3933CrossRefGoogle Scholar
  10. Bamberger A (1977) Approximation des coefficients d’opérateurs elliptiques, stables pour la G-convergence. Rapport du Centre de mathématiques appliquées, École polytechnique, n° MAP/15Google Scholar
  11. Berkowitz B, Adler PM (1998) Stereological analysis of fracture network structure in geological formations. J Geophys Res [Solid Earth] 103:15339–15360CrossRefGoogle Scholar
  12. Bouwer H (1969) Planning and interpreting soil permeability measurements. J Irrig Drain Div, ASCE 95:391–402Google Scholar
  13. Brown SR (1987) Fluid flow through rock joints: the effect of surface roughness. J Geophys Res 92(B2):1337–1347. CrossRefGoogle Scholar
  14. Budiansky B (1965) On the elastic moduli of some heterogeneous materials. J Mech Phys Solids 13:223CrossRefGoogle Scholar
  15. Cañamón I (2006) Analysis and modeling of coupled thermo-hydro-mechanical phenomena in three-dimensional fractured media. PhD thesis, Institut National Polytech. de Toulouse & Univ. Politécnica de MadridGoogle Scholar
  16. Cardwell WT, Parsons RL (1945) Average permeabilities of heterogeneous oil sands. Trans Am Inst Mining Metall Pet Eng 160:34–42Google Scholar
  17. Charlaix E, Guyon E, Rivier N (1984) A criterion for percolation threshold in a random array of plates. Solid State Commun 50(11):999–1002CrossRefGoogle Scholar
  18. Dagan G (1979) Models of groundwater flow in statistically heterogeneous porous formations. Water Resour Res 15(1):47–63CrossRefGoogle Scholar
  19. Desbarats AJ (1992) Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media. Math Geol 24(3):249–267CrossRefGoogle Scholar
  20. Deutsch C (1989) Calculating effective absolute permeability in sandstone/shale sequences. SPE Form Eval 4:343–348CrossRefGoogle Scholar
  21. Dimitrakopoulos R, Desbarats AJ (1997) Geostatistical modelling of grid block permeabilities for 3D reservoir simulators. SPE Reservoir Eng 8:13–18CrossRefGoogle Scholar
  22. Farmer CL (2002) Upscaling: a review. Int J Numer Meth Fluids 40:63–78CrossRefGoogle Scholar
  23. Hashin Z, Shtrikman S (1963) A variational approach to the theory of elastic behaviour of multiphase materials. J Mech Phys Solid 11:127–140. CrossRefGoogle Scholar
  24. Journel AG, Deutsch, C, Debarats AJ (1986) Power averaging for block effective permeability: SPE 15128, presented at the 56th California Regeonal Meeting of the SPE, Oakland, California, April 2–4, 1986Google Scholar
  25. Kfoury M, Ababou R, Noetinger B, Quintard M (2006) Upscaling fractured heterogeneous media: permeability and mass exchange coefficient. J Appl Mech (JAM), Trans ASME 73(1):41–46CrossRefGoogle Scholar
  26. Kiraly L (1969) Anisotropie et hétérogénéité de la perméabilité dans les calcaires fissurés. Eclogae Geol Helv 62(2):613–619Google Scholar
  27. Lang PS, Paluszny A, Zimmerman RW (2014) Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions. J Geophys Res Solid Earth 119:6288–6307. CrossRefGoogle Scholar
  28. Le Loc’h, G (1988) An efficient strategy for combining the permeabilities: practical application on a simulated reservoir. In: Proc. of the 3rd internat. Geostatistics congress, Avignon, Sept 5–9Google Scholar
  29. Li L, Zhou H, Gómez-Hernández JJ (2011) A comparative study of three dimensional hydraulic conductivity upscaling at the macro-dispersion experiment (MADE) site, Columbus Air Force Base, Mississippi (USA). J Hydrol 404:278–293. CrossRefGoogle Scholar
  30. Long JCS, Remer JS, Wilson CR, Witherspoon PA (1982) Porous media equivalents for networks of discontinuous fractures. Water Resour Res 18(3):645–658CrossRefGoogle Scholar
  31. Marchant.J (1977) Sur la résistance équivalente d’un réseau aléatoire de structure irrégulière. CR Acad Sci Paris, t.284, Série B-85:88Google Scholar
  32. Matheron G (1967) Eléments pour une Théorie des Milieux Poreux. Masson et Cie, Paris, p 166Google Scholar
  33. Mourzenko VV, Thovert JF, Adler PM (2005) Percolation of three-dimensional fracture networks with power-law size distribution. Phys Rev E 72:036103CrossRefGoogle Scholar
  34. Mourzenko V, Thovert J-F, Adler PM (2009) Proceedings of the international conference on rock joints and jointed rock masses, Tucson, ArizonaGoogle Scholar
  35. Oda M (1985) Permeability tensor for discontinuous rock masses. Géotechnique 35(4):483–495. CrossRefGoogle Scholar
  36. Oda M (1986) An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock masses. Water Resour Res 22(13):1845–1856CrossRefGoogle Scholar
  37. Pouya A, Fouché O (2009) Permeability of 3D discontinuity networks: new tensors from boundary-conditioned homogenization. Adv Water Resour 32:303–314CrossRefGoogle Scholar
  38. Pozdniakov S, Tsang C-F (2004) A self-consistent approach for calculating the effective hydraulic conductivity of a binary, heterogeneous medium. Water Resour Res 40:W05105. CrossRefGoogle Scholar
  39. Renard P, de Marsily G (1997) Calculating equivalent permeability: A review. Adv Water Resour 20:253–278CrossRefGoogle Scholar
  40. Renard P, Ababou R (2009) Relation between the definition and properties of the equivalent permeability tensor in heterogeneous and fractured porous media. In: Amaziane B et al (eds) Proceedings MAMERN 09: 3rd international conference on approximation methods and numerical modeling in environment and natural resources (Pau, France, 8–11 June 2009), Editorial Univ. de Granada, ISBN: 978-84-338-5006-5Google Scholar
  41. Renard Ph, Le Loc’h G, Ledoux E, de Marsily G, Mackay R (2000) A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous porous media. Water Resour Res 36(12):3567–3580CrossRefGoogle Scholar
  42. Sævik PN, Berre I, Jakobsen M, Lien M (2013) A 3D computational study of effective medium methods applied to fractured media. Transp Porous Media 100(1):115–142CrossRefGoogle Scholar
  43. Sahimi M (1995) Flow and transport in porous media and fractured rock. VCH, New YorkGoogle Scholar
  44. Snow DT (1969) Anisotropic permeability of fractured media. Water Resour Res 5(6):1273–1289CrossRefGoogle Scholar
  45. Tsang YW (1984) The effect of tortuosity on fluid flow through a single fracture. Water Resour Res 20:1209–1215CrossRefGoogle Scholar
  46. Vanmarcke E (1983) Random fields (analysis and synthesis). The MIT Press, CambridgeGoogle Scholar
  47. Warren JE, Price HS (1961) Flow in heterogeneous porous media. Soc Pet Eng J 1:153–169CrossRefGoogle Scholar
  48. Wen XH, Gomez-Hernandez JJ (1996) Upscaling hydraulic conductivities in heterogeneous media: an overview. J Hydrol 183:ix–xxxiiCrossRefGoogle Scholar
  49. Wiener O (1912) Abh. Math.-Phys. Klasse Königlich Sächsischen Des Wiss. Leipzig 32:509–604Google Scholar
  50. Zinn B, Harvey CF (2003) When good statistical models of aquifer heterogeneity go bad: a comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields. Water Resour Res 39(3):1051. CrossRefGoogle Scholar

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© International Association for Mathematical Geosciences 2019

Authors and Affiliations

  1. 1.Institut de Mécanique des Fluides de Toulouse (IMFT)CNRS and Université de ToulouseToulouseFrance
  2. 2.ETSI de Minas y Energía, Departamento de Ingeniería Geológica y MineraUniversidad Politécnica de MadridMadridSpain

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