Transport in Porous Media

, Volume 67, Issue 3, pp 395–412 | Cite as

Permeability up-scaling using Haar Wavelets

  • Vera PancaldiEmail author
  • Kim Christensen
  • Peter R. King
Original Paper


In the context of flow in porous media, up-scaling is the coarsening of a geological model and it is at the core of water resources research and reservoir simulation. An ideal up-scaling procedure preserves heterogeneities at different length-scales but reduces the computational costs required by dynamic simulations. A number of up-scaling procedures have been proposed. We present a block renormalization algorithm using Haar wavelets which provide a representation of data based on averages and fluctuations.

In this work, absolute permeability will be discussed for single-phase incompressible creeping flow in the Darcy regime, leading to a finite difference diffusion type equation for pressure. By transforming the terms in the flow equation, given by Darcy’s law, and assuming that the change in scale does not imply a change in governing physical principles, a new equation is obtained, identical in form to the original. Haar wavelets allow us to relate the pressures to their averages and apply the transformation to the entire equation, exploiting their orthonormal property, thus providing values for the coarse permeabilities.

Focusing on the mean-field approximation leads to an up-scaling where the solution to the coarse scale problem well approximates the averaged fine scale pressure profile.


Up-scaling Real-space Renormalization Haar Wavelets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aldroubi A., Unser M.: Wavelets in Medicine and Biology. CRC Press (1996)Google Scholar
  2. Aziz K., Settari A. (1979) Petroleum Reservoir Simulation. Kluwer Academic Publishers, NovisibirskGoogle Scholar
  3. Begg S.H., King P.R.: Modelling the effect of shales in reservoir performance: calculation of vertical effective permeability. 1985. Presented at the SPE Reservoir Simulation Symposium, Dallas, TX, Feb. 10–13, SPE Paper 13529.Google Scholar
  4. Best C. (2000) Wavelet induced renormalization group for the landau-ginzburg model. Nuclear Physics B, (Proc. Suppl.), 83–84: 848–850Google Scholar
  5. Daubechies I. (1992) Ten Lectures on Wavelets. CBMS-NSF Regional conference series in applied mathematics. SIAM, PhiladelphiaGoogle Scholar
  6. Durlofsky L.J.: Upscaling and gridding of fine scale geological models for flow simulation. Paper presented at the 8th International Forum on Reservoir Simulation, Iles Borromees, Stresa, Italy, June 20–24 (2005)Google Scholar
  7. Ebrahimi F., Sahimi M. (2004) Multiresolution wavelet scale up of unstable miscible displacements in flow through heterogeneous porous media. Trans. Porous Media 57, 75–102CrossRefGoogle Scholar
  8. Farmer C.L. (2002) Upscaling: a review. Int. J. for Num. Meth. in Fluids 40(1–2): 63–78CrossRefGoogle Scholar
  9. Haar A. Zur Thaeorie der Orthogonalen Funktionensysteme. PhD thesis, Goettingen, (1909)Google Scholar
  10. Hastings J.J., Muggeridge A.H. (2001) Upscaling uncertain permeability using small cell renormalization. Math. geol. 33: 491CrossRefGoogle Scholar
  11. Hristopulos D.T. (2003) Renormalization group methods in subsurface hydrology: overview and applications in hydraulic conductivity upscaling. Adv. in Water Resour. 26(12): 1279–1308CrossRefGoogle Scholar
  12. Hristopulos D.T., Christakos G. (1999) Renormalization group analysis of permeability upscaling. Stochastic Environmental Research and Risk Assessment 13(12): 131–160Google Scholar
  13. Ismail A.E., Rutledge G.C., Stephanopoulos G. (2003) Multiresolution analysis in statistical mechanics, part i. using wavelets to calculate thermodynamic properties. J. Chem. Phys. 118(10): 4414–4423CrossRefGoogle Scholar
  14. Ismail A.E., Rutledge G.C., Stephanopoulos G. (2005) Using wavelet transforms for multiresolution materials modeling. Comp. Chem. Eng. 29: 689CrossRefGoogle Scholar
  15. Kadanoff L.P. (1966) Scaling laws for ising models near tc. Physica 2, 263–272Google Scholar
  16. King P.R. (1989) The use of renormalization for calculating effective permeability. Trans. Porous Media 4, 37–58CrossRefGoogle Scholar
  17. King P.R. (1996) Upscaling permeability: Error analysis for renormalization. Trans. Porous Media 23, 337–354CrossRefGoogle Scholar
  18. King P.R., Muggeridge A.H., Price W.G. (1993) Renormalization calculations of immiscible flow. Trans. Porous Media 12, 237–260CrossRefGoogle Scholar
  19. Larsonneur J.L., Morlet J.: Wavelets and Seismic Interpretation. In: Wavelets. Time-Frequency Methods and Phase Space (1989)Google Scholar
  20. Noetinger B., Artus V., Zargar G. (2005) The future of stochastic and upscaling methods in hydrogeology. Hydrogeology Journal 13(1): 184–201CrossRefGoogle Scholar
  21. O’Sullivan, A.E., Christie M.A.: Solution error models: a new approach for coarse grid history matching. 2005. Presented at the SPE Reservoir Simulation Symposium, Houston, TX, Jan. 31–Feb. 2, SPE Paper 93268.Google Scholar
  22. Renard P., deMarsily G. (1997) Calculating equivalent permeability: a review. Adv. Water Resour. 20(5–6): 253–278CrossRefGoogle Scholar
  23. Wallstrom T.C., Hou S.L., Christie M.A., Durlofsky L.J., Sharp D.H. (1999) Accurate scale up of two phase flow using renormalization and nonuniform coarsening. Comp. Geosci. 3, 69–87CrossRefGoogle Scholar
  24. Wen X.H., Gomez-Hernandez J.J.: Upscaling hydraulic conductivities in heterogeneous media: An overview. J. Hydrol. 183, ix–xxxii (1996)Google Scholar
  25. Williams J.K.: Simple renormalization schemes for calculating effective properties of heterogeneous reservoirs. In P.R. King, editor, Mathematics of Oil Recovery, p. 281. Clarendon Press, Oxford (1992)Google Scholar
  26. Yeo I., Zimmerman R.W. (2001) Accuracy of the renormalization method for computing effective conductivities of heterogeneous media. Transp. Porous Media 45, 129–138CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Vera Pancaldi
    • 1
    • 2
    Email author
  • Kim Christensen
    • 2
  • Peter R. King
    • 1
  1. 1.Department of Earth Science and EngineeringSouth Kensington Campus, Imperial CollegeLondonUK
  2. 2.Department of PhysicsSouth Kensington Campus, Imperial CollegeLondonUK

Personalised recommendations