Transport in Porous Media

, Volume 87, Issue 3, pp 717–737 | Cite as

Homogenization of Hydraulic Conductivity for Hierarchical Sedimentary Deposits at Multiple Scales

  • Ye Zhang
  • Baozhong Liu
  • Carl W. Gable


Based on a three-dimensional heterogeneous aquifer model exhibiting non-stationary, statistically anisotropic correlation, three hydrostratigraphic models (HSMs) are created within a sedimentary hierarchy. A geostatistical analysis of natural log conductivity (lnK) is conducted for the units of the HSMs. Hydraulic conductivity is then upscaled using numerical and analytical methods. Increasing lnK variances are evaluated. Results suggest that for the aquifer model tested: (1) the numerical method is capable of upscaling irregular domains with reasonable accuracy for a lnK variance up to 7.0. (2) Accuracy of the upscaled equivalent conductivities (K*) and associated performance of the HSMs are sensitive to homogenization level, heterogeneity variance, and boundary condition. Variance is found to be the most significant factor impacting the accuracy of the HSMs. (3) Diagonal tensor appears a good approximation for the full-tensor K*. (4) For the HSM units, when the variance is low (less than 1.0), all analytical methods are nearly equally accurate; however, when variance becomes higher, analytical methods generally are less accurate.


Hydraulic conductivity Heterogeneity Upscaling Equivalent conductivity Sedimentary hierarchy 


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  1. Ababou, R.: Identification of effective conductivity tensor in randomly heterogeneous and stratified aquifers. In: Proceedings of the 5th Canadian/American Conference on Hydrogeology: Parameter Idenfitication and Estimation for Aquifer and Reservoir Characterization, pp. 155–157 (1991)Google Scholar
  2. Ababou R.: Random porous media flow on large 3-D grids: numerics, performance and application to homogenization. In: Wheeler, M.F. (eds) Mathematics and its Applications: Environmental Studies—Math, Comput and Statistical Analysis, IMA vol 79, Chapter 1, pp. 1–25. Springer, New York (1996)Google Scholar
  3. Desbarats A.J.: Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media. Math. Geol. 24(3), 249–267 (1992)CrossRefGoogle Scholar
  4. Desbarats A.J., Srivastava R.M.: Geostatistical analysis of groundwater flow parameters in a simulated aquifer. Water Resour. Res. 27(5), 687–698 (1991)CrossRefGoogle Scholar
  5. Deutsch C.V.: Geostatistical Reservoir Modeling, pp. 376. Oxford University Press, NY, USA (2002)Google Scholar
  6. Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. In: Proceedings of the 8th International Forum on Reservoir Simulation. Stresa, Italy, 20–25 June 2005Google Scholar
  7. Gelhar L.W.: Stochastic Subsurface Hydrology. Prentice Hall, Englewood Cliffs, NJ, USA (1993)Google Scholar
  8. Gelhar L.W., Axness C.L.: Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 19(1), 161–180 (1983)CrossRefGoogle Scholar
  9. Journel, A.G., Deutsch, C., Desbarats, A.J.: Power averaging for block effective permeability. SPE Paper 15128 (1986)Google Scholar
  10. Knudby C., Carrera J.: On the relationship between indicators of geostatistical, flow and transport connectivity. Adv. Water Resour. 28(4), 405–421 (2005)CrossRefGoogle Scholar
  11. Knudby C., Carrera J.: On the use of apparent hydrauilc diffusivity as an indicator of connectivity. J. Hydrol. 329(3–4), 377–389 (2006)CrossRefGoogle Scholar
  12. Milliken, W., Levy, M., Strebelle, S., Zhang, Y.: The effect of geologic parameters and uncertainties on subsurface flow: deepwater depositional systems. SPE Paper 109950 (2007)Google Scholar
  13. Noetinger B., Haas A.: Permeability averaging for well tests in 3D stochastic reservoir models. SPE J. 36653, 919–925 (1996)Google Scholar
  14. Renard P., de Marsily G.: Calculating equivalent permeability: a review. Adv. Water Resour. 20(5-6), 253–278 (1997)CrossRefGoogle Scholar
  15. Ritzi, R.W., Allen-King, R.M.: Why did Sudicky (1986) find an exponential-like spatial correlation structure for hydraulic conductivity at the Borden research site? Water Resour. Res. (2007). doi: 10.1029/2006WR004935
  16. Sanchez-Vila X., Girardi J.P., Carrera J.: A synthesis of approaches to upscaling of hydraulic conductivities. Water Resour. Res 31(4), 867–882 (1995)CrossRefGoogle Scholar
  17. Sanchez-Vila X., Carrera J., Girardi J.P.: Scale effects in transimissivity. J. Hydrol. 183(1-2), 1–22 (1996)CrossRefGoogle Scholar
  18. Sanchez-Vila, X., Guadagnini, A., Carrera, J.: Representative hydraulic conductivities in saturated groundwater flow. Rev. Geophys. (2006). doi: 10.1029/2005RG000169
  19. Schlumberger: ECLIPSE, Technical Description 2009.2. Schlumberger (2009)Google Scholar
  20. Wen, X.-H., Gómez-Hernández, J.: Upscaling hydraulic conductivities in heterogeneous media: an overview. J. Hydrol. (1996). doi: 10.1016/S0022-1694(96)80030-8
  21. Wen X.H., Durlofsky L.J., Edwards M.G.: Use of border regions for improved permeability upscaling. Math. Geol. 35(5), 521–547 (2003)CrossRefGoogle Scholar
  22. Zhang D.: Stochastic methods for flow in porous media, coping with uncertainties. Academic Press, San Diego, CA (2002)Google Scholar
  23. Zhang, Y.: Hierarchical geostatistical analysis of an experimental stratigraphy. Math. Geosci. (2008). doi: 10.1007/s11004-008-9180-6
  24. Zhang, Y., Gable, C.W., Person, M.: Equivalent hydraulic conductivity of an experimental stratigraphy—implications for basin-scale flow simulations. Water Resour. Res. (2006). doi: 10.1029/2005WR004720
  25. Zhang Y., Gable C.W., Sheets B.: Equivalent hydraulic conductivity of three-dimensional heterogeneous porous media: an upscaling study based on an experimental stratigraphy. J. Hydrol. 388, 304–320 (2010). doi: 10.1016/j.jhydrol.2010.05.009 CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.University of WyomingLaramieUSA
  2. 2.University of WyomingLaramieUSA
  3. 3.Los Alamos National LabLos AlamosUSA

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