Upscaling from Darcy Scale to Field Scale

  • Adam Szymkiewicz
Part of the GeoPlanet: Earth and Planetary Sciences book series (GEPS)


Porous media often are often characterized by spatially varying properties such as porosity, permeability or capillary function. The scale of variability is often much smaller than the size of the considered domain, so it is necessary to develop an upscaled description of flow processes, which captures the field-scale behaviour of the heterogeneous medium in an average sense. In this chapter, an overview of upscaling approaches is presented, with particular focus on the methods applicable to binary media, consisting of porous inclusions dispersed in the background material. Various approaches for permeability upscaling in steady state flow are discussed and numerical examples are presented to show the differences between methods. In the last part of the chapter, the problem of transient flow in heterogeneous media is examined. The approaches based on the assumptions of local equilibrium and local non-equilibrium are compared and contrasted.


Representative Elementary Volume Matrix Block Effective Permeability Cell Problem Effective Medium Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ahmadi A, Quintard M (1996) Large-scale properties for two-phase flow in random porous media. J Hydrol 183(1–2):69–99. doi: 10.1016/S0022-1694(96)80035-7 Google Scholar
  2. 2.
    Amaziane B, Koebbe J (2006) JHomogenizer: a computational tool for upscaling permeability for flow in heterogeneous porous media. Comput Geosci 10(4):343–359. doi: 10.1007/s10596-006-9028-4 Google Scholar
  3. 3.
    Amaziane B, Bourgeat A, Koebbe J (1991) Numerical simulation and homogenization of two-phase flow in heterogeneous porous media. Transp Porous Media 6(5–6):519–547. doi: 10.1007/BF00137848
  4. 4.
    Arbogast T (1993) Gravitational forces in dual-porosity systems: I. Model derivation by homogenization. Transp Porous Media 13(2):197–203. doi: 10.1007/BF00654409 Google Scholar
  5. 5.
    Arbogast T, Douglas J, Hornung U (1990) Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J Math Anal 21:823–836Google Scholar
  6. 6.
    Artus V, Nœtinger B (2004) Up-scaling two-phase flow in heterogeneous reservoirs: current trends. Oil Gas Sci Technol 59(2):185–195. doi: 10.2516/ogst:2004014 Google Scholar
  7. 7.
    Auriault JL (2011) Heterogeneous periodic and random media. Are the equivalent macroscopic descriptions similar? Int J Eng Sci 49(8):806–808. doi: 10.1016/j.ijengsci.2011.01.005
  8. 8.
    Auriault JL, Boutin C, Geindreau C (2009) Homogenization of coupled phenomena in heterogeneous media. Wiley, HobokenGoogle Scholar
  9. 9.
    Bakker M, Nieber J (2004) Analytic element modeling of cylindrical drains and cylindrical inhomogeneities in steady two-dimensional unsaturated flow. Vadose Zone J 3(3):1038–1049. doi: 10.2136/vzj2004.1038 Google Scholar
  10. 10.
    Barenblatt G (1963) On some boundary-value problems for the equation of filtration of fluid in fissurized rocks. J Appl Math Mech (PMM) 27(2):513–518Google Scholar
  11. 11.
    Barenblatt G, Zheltov I, Kochina I (1960) Basic concepts in the theory of seepage of homogeneous liquids in fissurized rocks. J Appl Math Mech (PMM) 24(5):1286–1303Google Scholar
  12. 12.
    Barker J, Thibeau S (1997) A critical review of the use of pseudo relative permeabilities for upscaling. SPE Reservoir Eng 12(2):138–143. doi: 10.2118/35491-PA
  13. 13.
    Barnes R, Janković I (1999) Two-dimensional flow through large numbers of circular inhomogeneities. J Hydrol 226(3–4):204–210. doi: 10.1016/S0022-1694(99)00142-0 Google Scholar
  14. 14.
    Bear J, Tsang CF, de Marsily G (eds) (1993) Flow and contaminant transport in fractured rock. Academic Press, San DiegoGoogle Scholar
  15. 15.
    Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. North-Holland, AmsterdamGoogle Scholar
  16. 16.
    Berkowitz B (2002) Characterizing flow and transport in fractured geological media: A review. Adv Water Resour 25(8–12):861–884. doi: 10.1016/S0309-1708(02)00042-8
  17. 17.
    Bielski W, Telega J (1997) Effective properties of geomaterials: rocks and porous media. Publ Inst Geophys Pol Acad Sci A-26(285):1–120Google Scholar
  18. 18.
    Bierkens M, Finke P, de Willigen P (eds) (2000) Upscaling and downscaling methods for environmental research. Kluwer, DordrechtGoogle Scholar
  19. 19.
    Bøe Ø (1994) Analysis of an upscaling method based on conservation of dissipation. Transp Porous Media 17(1):77–86. doi: 10.1007/BF00624051
  20. 20.
    Braun C, Helmig R, Manthey S (2005) Macro-scale effective constitutive relationships for two phase flow processes in heterogeneous porous media with emphasis on the relative permeability-saturation relationship. J Contam Hydrol 76(1–2):47–85. doi: 10.1016/j.jconhyd.2004.07.009 Google Scholar
  21. 21.
    Cardwell W, Parsons R (1945) Average permeabilities of heterogeneous oil sands. Trans Am Inst Mining Eng 160:34–42Google Scholar
  22. 22.
    Chen Y, Durlofsky L, Gerritsen M, Wen XH (2003) A coupled local global upscaling approach for simulating flow in highly heterogeneous formations. Ad Water Resour 26(10):1041–1060. doi: 10.1016/S0309-1708(03)00101-5
  23. 23.
    Choquet C (2009) Homogenized model for flow in partially fractured media. Electron J Differ Equ 2009:1–27Google Scholar
  24. 24.
    Christie M (2001) Flow in porous media - scale up of multiphase flow. Curr Opin Colloid Interface Sci 6(3):236–241. doi: 10.1016/S1359-0294(01)00087-5 Google Scholar
  25. 25.
    Cushman J, Bennethum L, Hu B (2002) A primer on upscaling tools for porous media. Adv Water Resour 25(8–12):1043–1067. doi: 10.1016/S0309-1708(02)00047-7 Google Scholar
  26. 26.
    Dagan G (1989) Flow and transport in porous formations. Springer, New YorkGoogle Scholar
  27. 27.
    Darman N, Pickup G, Sorbie K (2002) A comparison of two-phase dynamic upscaling methods based on fluid potentials. Comput Geosci 6(1):5–27. doi: 10.1023/A:1016572911992 Google Scholar
  28. 28.
    Dietrich P, Helmig R, Sauter M, Hötzl H, Köngeter J, Teutsch G (eds) (2005) Flow and transport in fractured porous media. Springer, BerlinGoogle Scholar
  29. 29.
    Douglas J, Peszyńska M, Showalter R (1997) Single phase flow in partially fissured media. Transp Porous Media 28(3):285–306. doi: 10.1023/A:1006562120265
  30. 30.
    Durlofsky L (1998) Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques. Comput Geosci 2(2):73–92. doi: 10.1023/A:1011593901771 Google Scholar
  31. 31.
    Durner W (1994) Hydraulic conductivity estimation for soils with heterogeneous pore system. Water Resour Res 30(2):211–233. doi: 10.1029/93WR02676
  32. 32.
    Dykhuizen R (1990) A new coupling term for dual-porosity models. Water Resour Res 26(2):351–356. doi: 10.1029/WR026i002p00351
  33. 33.
    Ekrann S, Aasen J (2000) Steady-state upscaling. Transp Porous Media 41(3):245–262. doi: 10.1023/A:1006765424927
  34. 34.
    Famy C, Bourbiaux B (2005) Accurate modeling of matrix-fracture transfers in dual-porosity models: optimal subgridding of matrix blocks. In: SPE Reservoir Simulation Symposium, 31 January-2 Feburary 2005. The Woodlands, Texas. doi: 10.2118/93115-MS
  35. 35.
    Flint A, Flint L, Bodvarsson G, Kwicklis E, Fabryka-Martin J (2001) Evolution of the conceptual model of unsaturated zone hydrology at Yucca Mountain. Nevada. J Hydrol 247(1–2):1–30. doi: 10.1016/S0022-1694(01)00358-4
  36. 36.
    Fokker P (2001) General anisotropic effective medium theory for the effective permeability of heterogeneous reservoirs. Transp Porous Media 44(2):205–218. doi: 10.1023/A:1010770623874
  37. 37.
    Furman A, Warrick A (2005) Unsaturated flow through spherical inclusions with contrasting sorptive numbers. Vadose Zone J 4(2):255–263. doi: 10.2136/vzj2004.0076 Google Scholar
  38. 38.
    Gasda S, Nordbotten J, Celia M (2009) Vertical equilibrium with sub-scale analytical methods for geological co-2 sequestration. Comput Geosci 13(4):469–481. doi: 10.1007/s10596-009-9138-x Google Scholar
  39. 39.
    Gelhar L (1993) Stochastic subsurface hydrology. Prentice Hall, Engelwood Cliffs, New JerseyGoogle Scholar
  40. 40.
    Gerke H (2006) Preferential flow descriptions for structured soils. J Plant Nutr Soil Sci 169:382–400Google Scholar
  41. 41.
    Gerke H, van Genuchten M (1993) A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour Res 29(2):305–319. doi: 10.1029/92WR02339 Google Scholar
  42. 42.
    Gerke H, van Genuchten M (1993) Evaluation of a first-order water transfer term for variably saturated dual-porosity flow models. Water Resour Res 29(4):1225–1238. doi: 10.1029/92WR02467 Google Scholar
  43. 43.
    Gerke H, van Genuchten M (1996) Macroscopic representation of structural geometry for simulating water and solute movement in dual-porosity media. Adv Water Resour 19(6):343–357. doi: 10.1016/0309-1708(96)00012-7
  44. 44.
    Guéguen Y, Le Ravalec M, Ricard L (2006) Upscaling: effective medium theory, numerical methods and the fractal dream. Pure Appl Geophys 163:1175–1192. doi: 10.1007/s00024-006-0053-y Google Scholar
  45. 45.
    Harter T, Hopmans J (2004) Role of vadose-zone flow processes in regional-scale hydrology: review, opportunities and challenges. In: Unsaturated zone modelling: progress, challenges and applications, Kluwer, Dordrecht.Google Scholar
  46. 46.
    Harter T, Knudby C (2004) Effective conductivity of periodic media with cuboid inclusions. Adv Water Resour 27(10):1017–1032. doi: 10.1016/j.advwatres.2004.07.004 Google Scholar
  47. 47.
    Hashin Z, Shtrikman S (1962) A variational approach to the theory of effective magnetic permeability of multiphase materials. J Appl Phys 33:3125–3131Google Scholar
  48. 48.
    Hornung U (1991) Homogenization of miscible displacement in unsaturated aggregated soils. In: Dal Masse G and Dell'Antonio GF (ed) Composite media and homogenization theory, Birkhäuser, Boston, pp 143-153Google Scholar
  49. 49.
    Indelman P, Dagan G (1993) Upscaling of heterogeneous formations: General approach and application to isotropic media. Transp Porous Media 12(2):161–184. doi: 10.1007/BF00616978
  50. 50.
    Janković I, Barnes R (1999) Three-dimensional flow through large numbers of spheroidal inhomogeneities. J Hydrol 226(3–4):224–233. doi: 10.1016/S0022-1694(99)00141-9 Google Scholar
  51. 51.
    Jarvis N (1994) The MACRO model (version 3.1). Technical description and sample simulation. Reports and Dissertations 19. Department of Soil Science, Swedish University of Agricultural Science, UppsalaGoogle Scholar
  52. 52.
    Jones S, Friedman S (2000) Particle shape effects on the effective permittivity of anisotropic or isotropic media consisting of aligned or randomly oriented ellipsoidal particles. Water Resour Res 36(10):2821–2833. doi: 10.1029/2000WR900198 Google Scholar
  53. 53.
    Journel A, Deutsch C, Desbarats A (1986) Power averaging for block effective permeability. In: SPE California Regional Meeting, 2–4, April 1986, Oakland, California. doi: 10.2118/15128-MS
  54. 54.
    Karimi-Fard M, Gong B, Durlofsky L (2006) Generation of coarse-scale continuum flow models from detailed fracture characterizations. Water Resour Res 42:W10423. doi: 10.1029/2006WR005015
  55. 55.
    Knudby C, Carrera J, Bumgardner J, Fogg G (2006) Binary upscaling - the role of connectivity and a new formula. Adv Water Resour 29(4):590–604. doi: 10.1016/j.advwatres.2005.07.002
  56. 56.
    Köhne J, Mohanty B, Simunek J, Gerke H (2004) Numerical evaluation of a second-order water transfer term for variably saturated dual-permeability models. Water Resour Res 40:W07409. doi: 10.1029/2004WR003285
  57. 57.
    Köhne J, Köhne S, Šimunek J (2009) A review of model applications for structured soils: (a) Water flow and tracer transport. J Contam Hydrol 104(1–4):4–35. doi: 10.1016/j.jconhyd.2008.10.002
  58. 58.
    Lemonnier P, Bourbiaux B (2010) Simulation of naturally fractured reservoirs. State of the art. Part 1 Physical mechanisms and simulator formulation. Oil Gas Sci Technol 65(2):239–262. doi: 10.2516/ogst/2009066
  59. 59.
    Lewandowska J, Laurent JP (2001) Homogenization modeling and parametric study of moisture transfer in an unsaturated heterogeneous porous medium. Transp Porous Media 45(3):321–345. doi: 10.1023/A:1012450327408 Google Scholar
  60. 60.
    Lewandowska J, Szymkiewicz A, Burzyński K, Vauclin M (2004) Modeling of unsaturated water flow in double-porosity soils by the homogenization approach. Adv Water Resour 27(3):283–296. doi: 10.1016/j.advwatres.2003.12.004
  61. 61.
    Lewandowska J, Szymkiewicz A, Boutin C (2005) Modelling of unsaturated conductivity of double-porosity soils. In: Actes de 17ème Congrès Francais de Mécanique, TroyesGoogle Scholar
  62. 62.
    Lewandowska J, Szymkiewicz A, Vauclin M (2005) Study of the exchange term in the non-equilibrium water flow model for double-porosity soils. In: Abousleiman Y et al. (ed) Poromechanics III, Taylor and Francis, pp 365–370Google Scholar
  63. 63.
    Li D, Beckner B, Kumar A (2001) A new efficient averaging technique for scaleup of multimillion-cell geologic models. SPE Reservoir Eval Eng 4(4):297–307. doi: 10.2118/72599-PA Google Scholar
  64. 64.
    Lichtner P (2000) Critique of dual continuum formulations of multicomponent reactive transport in fractured porous media. Technical Report, National Laboratory, Los AlamosGoogle Scholar
  65. 65.
    Lux J (2010) A non-periodic closure scheme for the determination of effective diffusivity in real porous media. Trans Porous Media 82(2):299–315. doi: 10.1007/s11242-009-9423-0
  66. 66.
    Manthey S, Hassanizadeh M, Helmig R (2005) Macro-scale dynamic effects in homogeneous and heterogeneous porous media. Transp Porous Media 58(1–2):121–145. doi: 10.1007/s11242-004-5472-6 Google Scholar
  67. 67.
    Milton G (2002) The theory of composites. Cambridge University Press, CambridgeGoogle Scholar
  68. 68.
    Neuweiler I (2006) Scale dependence of flow and transport parameters in porous media. University of Stuttgart, StuttgartGoogle Scholar
  69. 69.
    Neuweiler I, Cirpka O (2005) Homogenization of Richards equation in permeability fields with different connectivities. Water Resour Res 41:W02009. doi: 10.1029/2004WR003329
  70. 70.
    Neuweiler I, Vogel HJ (2007) Upscaling for unsaturated flow for non-Gaussian heterogeneous porous media. Water Resour Res 43:W03443. doi: 10.1029/2005WR004771
  71. 71.
    Nœtinger B, Zargar G (2004) Multiscale description and upscaling of fluid flow in subsurface reservoirs. Oil Gas Sci Technol 59(2):119–139. doi: 10.2516/ogst:2004010 Google Scholar
  72. 72.
    Nœtinger B, Artus V, Zargar G (2005) The future of stochastic and upscaling methods in hydrogeology. Hydrogeol J 13:184–201Google Scholar
  73. 73.
    Novák V, Knava K, Šimunek J (2011) Determining the influence of stones on hydraulic conductivity of saturated soils using numerical method. Geoderma 161:177–181. doi: 10.1016/j.geoderma.2010.12.016 Google Scholar
  74. 74.
    Penuela G, Hughes R, Civan F, Wiggins M (2002) Time-dependent shape factors for secondary recovery in naturally fractured reservoirs. In: SPE/DOE Improved Oil Recovery Symposium, 13–17, April 2002. Tulsa, Oklahoma. doi: 10.2118/75234-MS
  75. 75.
    Peters R, Klavetter E (1988) A continuum model for water movement in an unsaturated fractured rock mass. Water Resour Res 24(3):416–430. doi: 10.1029/WR024i003p00416
  76. 76.
    Pinder G, Gray W (2008) Essentials of multiphase flow and transport in porous media. Wiley, HobokenGoogle Scholar
  77. 77.
    Poley A (1988) Effective permeability and dispersion in locally heterogeneous aquifers. Water Resour Res 24(11):1921–1926. doi: 10.1029/WR024i011p01921
  78. 78.
    Pozdniakov S, Tsang CF (2004) A self-consistent approach for calculating the effective hydraulic conductivity of a binary, heterogeneous medium. Water Resour Res 40:W05105. doi: 10.1029/2003WR002617
  79. 79.
    Pruess K (2004) A composite medium approximation for unsaturated flow in layered sediments. J Contam Hydrol 70(3–4):225–247. doi: 10.1016/j.jconhyd.2003.09.007 Google Scholar
  80. 80.
    Pruess K, Narasimhan T (1985) A practical method for modelling fluid and heat flow in fractured porous media. Soc Pet Eng J 25(1):14–26. doi: 10.2118/10509-PA
  81. 81.
    Quintard M, Whitaker S (1988) Two-phase flow in heterogeneous porous media: The method of large scale averaging. Transp Porous Media 3(4):357–413. doi: 10.1007/BF00233177
  82. 82.
    Radcliffe D, Šimunek J (2010) Soil physics with Hydrus. Modeling and applications. CRC Press, Boca Raton, FloridaGoogle Scholar
  83. 83.
    Reichenberger V, Helmig R, Jakobs H, Bastian P, Niessner J (2004) Complex gas-water processes in discrete fracture-matrix system: up-scaling, mass conservative solution and efficient multilevel solution. Institut für Wasserbau, Universität Stuttgart, StuttgartGoogle Scholar
  84. 84.
    Renard P, de Marsily G (1997) Calculating equivalent permeability: a review. Adv Water Resour 20(5–6):253–278. doi: 10.1016/S0309-1708(96)00050-4 Google Scholar
  85. 85.
    Ross P, Smettem K (2000) A simple treatment of physical nonequilibrium water flow in soils. Soil Sci Soc Am J 64(6):1926–1930. doi: 10.2136/sssaj2000.6461926x Google Scholar
  86. 86.
    Rucker D, Warrick A, Ferré T (2005) Parameter equivalence for the Gardner and van Genuchten soil hydraulic conductivity functions for steady vertical flow with inclusions. Adv Water Resour 28(7):689–699. doi: 10.1016/j.advwatres.2005.01.004
  87. 87.
    Saez A, Otero C, Rusinek I (1989) The effective homogeneous behavior of heterogeneous porous media. Transp Porous Media 4(3):213–238. doi: 10.1007/BF00138037
  88. 88.
    Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory, Lecture Notes in Physics, vol 127, Springer, BerlinGoogle Scholar
  89. 89.
    Sanchez-Villa X, Guadagnini A, Carrera J (2006) Representative hydraulic conductivities in saturated groundwater flow. Rev Geophys 44(3):RG3002. DOI: 10.1029/2005RG000169
  90. 90.
    Schaetzl R, Anderson S (2005) Soils - genesis and geomorphology. Cambridge University Press, CambridgeGoogle Scholar
  91. 91.
    Scheidegger A (1957) The physics of flow through porous media. University of Toronto Press, TorontoGoogle Scholar
  92. 92.
    Šimunek J, Jarvis N, van Genuchten M, Gärdenäs A (2003) Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. J Hydrol 272(1–4):14–35. doi: 10.1016/S0022-1694(02)00252-4 Google Scholar
  93. 93.
    Stephen K, Pickup G, Sorbie K (2001) The local analysis of changing force balances in immiscible incompressible two-phase flow. Transp Porous Media 45(1):63–88. doi: 10.1023/A:1011850618324 Google Scholar
  94. 94.
    Sviercoski R (2010) An analytical effective tensor and its approximation properties for upscaling flows through generalized composites. Adv Water Resour 33(7):728–739. doi: 10.1016/j.advwatres.2010.03.011
  95. 95.
    Sviercoski R, Warrick A, Winter C (2009) Two-scale analytical homogenization of Richards equation for flows through block inclusions. Water Resour Res 45:W05403. doi: 10.1029/2006WR005598
  96. 96.
    Szymkiewicz A (2005) Calculating effective conductivity of heterogeneous soils by homogenization. Arch Hydro Eng Environ Mech 52(2):111–130Google Scholar
  97. 97.
    Szymkiewicz A (2008) Modelowanie przeplywu wody w utworach o podwójnej porowatosci (Modeling water flow in double porosity geological formations). Biuletyn Państwowego Instytutu Geologicznego 431:251–258Google Scholar
  98. 98.
    Tatomir A, Szymkiewicz A, Class H, Helmig R (2011) Modeling two phase flow in large scale fractured porous media with an extended multiple interacting continua method. Comput Model Eng Sci 77(2):81–112Google Scholar
  99. 99.
    Torquato S (2002) Random heterogeneous materials: microstructure and macroscopic properties. Springer, New YorkGoogle Scholar
  100. 100.
    Trykozko A, Zijl W, Bossavit A (2001) Nodal and mixed finite elements for the numerical homogenization of 3D permeability. Comput Geosci 5(1):61–84. doi: 10.1023/A:1011621529611 Google Scholar
  101. 101.
    Vereecken H, Kasteel R, Vanderborght J, Harter T (2007) Upscaling hydraulic properties and soil water flow processes in heterogeneous soils: a review. Vadose Zone J 6(1):1–28. doi: 10.2136/vzj2006.0055 Google Scholar
  102. 102.
    Virnovsky G, Friis H, Lohne A (2004) A steady-state upscaling approach for immiscible two-phase flow. Transp Porous Media 54(2):167–192. doi: 10.1023/A:1026363132351 Google Scholar
  103. 103.
    Warren J, Root P (1963) The behavior of naturally fractured reservoirs. Soc Pet Eng J 3(3):245–255. doi: 10.2118/426-PA
  104. 104.
    Warrick A, Knight J (2002) Two-dimensional unsaturated flow through a circular inclusion. Water Resour Res 38(7):1113. doi: 10.1029/2001WR001041 Google Scholar
  105. 105.
    Warrick A, Knight J (2004) Unsaturated flow through a spherical inclusion. Water Resour Res 40:W05101. doi: 10.1029/2003WR002890
  106. 106.
    Wen XH, Gómez-Hernández J (1996) Upscaling hydraulic conductivities in heterogeneous media: An overview. J Hydrol 183(1–2):ix–xxxii, doi: 10.1016/S0022-1694(96)80030-8
  107. 107.
    Wen XH, Durlofsky L, Edwards M (2003) Use of border regions for improved permeability upscaling. Math Geol 35(5):521–547Google Scholar
  108. 108.
    Whitaker S (1999) The method of volume averaging. Kluwer, DordrechtGoogle Scholar
  109. 109.
    White C, Horne R (1987) Computing absolute transmissibility in the presence of fine-scale heterogeneity. In: SPE Symposium on Reservoir Simulation, 1–4, February 1987. San Antonio, Texas. doi: 10.2118/16011-MS
  110. 110.
    Wiener O (1912) Die Theorie des Mischkörpers für das Feld der Stationären Strömung: I. Die Mittelwertsatze für Kraft, Polarisation und Energie. Die Abhandlungen der Mathematisch-Physischen Klasse der Königlichen Sachsischen Gesellschaft der Wissenschaften 32:509–604Google Scholar
  111. 111.
    Wood B (2000) Review of upscaling methods for describing unsaturated flow. Technical Report, Pacific Northwest, National LaboratoryGoogle Scholar
  112. 112.
    Wu X, Efendiev Y, Hou T (2002) Analysis of upscaling absolute permeability. Discrete Continuous Dyn Syst Ser B 2(2):185–204Google Scholar
  113. 113.
    Wu YS (2002) Numerical simulation of single-phase and multiphase non-Darcy flow in porous and fractured reservoirs. Transp Porous Media 49(2):209–240. doi: 10.1023/A:1016018020180
  114. 114.
    Zhang D (2002) Stochastic methods for flow in porous media: coping with uncertainties. Academic Press, San DiegoGoogle Scholar
  115. 115.
    Zhu J (2012) Effect of layered structure on anisotropy of unsaturated soils. Soil Sci 177(2):139146, DOI: 10.1097/SS.0b013e31824114f6 Google Scholar
  116. 116.
    Zijl W, Nawalany M (1993) Natural groundwater flow. Lewis, Boca Raton, FloridaGoogle Scholar
  117. 117.
    Zijl W, Trykozko A (2001) Numerical homogenization of the absolute permeability using the conformal-nodal and mixed-hybrid finite element method. Transp Porous Media 44(1):33–62. doi: 10.1023/A:1010776124186
  118. 118.
    Zimmerman R (1996) Effective conductivity of a two-dimensional medium containing elliptical inhomogeneities. Proc Royal Soc A 452:1713–1727Google Scholar
  119. 119.
    Zimmerman R, Bodvarsson G, Kwicklis E (1990) Absorption of water into porous blocks of various shapes and sizes. Water Resour Res 26(11):2797–2806. doi: 10.1029/WR026i011p02797 Google Scholar
  120. 120.
    Zimmerman R, Chen G, Hadgu T, Bodvarsson G (1993) A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow. Water Resour Res 29(7):2127–2137. doi: 10.1029/93WR00749 Google Scholar
  121. 121.
    Zimmerman R, Hadgu T, Bodvarsson G (1996) A new lumped-parameter model for flow in unsaturated dual-porosity media. Adv Water Resour 19(5):317–327. doi: 10.1016/0309-1708(96)00007-3
  122. 122.
    Zyvoloski G, Robinson B, Viswanathan H (2008) Generalized dual porosity: A numerical method for representing spatially variable sub-grid scale processes. Adv Water Resour 31(3):535–544. doi: 10.1016/j.advwatres.2007.11.006

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringGdansk University of TechnologyGdanskPoland

Personalised recommendations