Scaling Effects on Finite-Domain Fractional Brownian Motion

  • S. Cintoli
  • S. P. Neuman
  • V. Di Federico
Conference paper


Hydraulic Conductivity Power Model Fractional Brownian Motion Integral Scale Horizontal Well 


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  1. Abramowitz M, Stegun IA (1972) Handbook of Mathematical Functions. Dover, Mineola, New YorkGoogle Scholar
  2. Bellin A, Rubin Y (1996) HYDRO_GEN: A spatially distributed random field generator for correlated properties. Stochastic Hydrology and Hydraulics (10), 253–278CrossRefGoogle Scholar
  3. Boufadel MC, Lu S, Molz FJ, Lavalle D (2000) Multifractal scaling of the intrinsic permeability. Water Resources Research 36, 3211–3222Google Scholar
  4. Dagan G (1994) Significance of heterogeneity of evolving scales to transport in porous formations. Water Resources Research 30(12), 3327–3336CrossRefGoogle Scholar
  5. Desbarats AJ, Bachu S (1994) Geostatistical analysis of aquifer heterogeneity from the core scale to the basin scale: A case study. Water Resources Research 30(3), 673–684CrossRefGoogle Scholar
  6. Di Federico V, Neuman SP (1997) Scaling of random fields by means of truncated power variograms and associated spectra. Water Resources Research 33(5), 1075–1085Google Scholar
  7. Di Federico V, Neuman SP (1998a) Flow in multiscale log conductivity fields with truncated power variograms. Water Resources Research 34(5), 975–987CrossRefGoogle Scholar
  8. Di Federico V, Neuman SP (1998b) Transport in multiscale log conductivity fields with truncated power variograms. Water Resources Research 34(5), 963–973CrossRefGoogle Scholar
  9. Di Federico V, Neuman SP, Tartakovsky DM (1999) Anisotropy, lacunarity, upscaled conductivity and its covariance in multiscale fields with truncated power variograms. Water Resources Research 35(10), 2891–2908CrossRefGoogle Scholar
  10. Eggleston J, Rojstaczer S (1998) Inferring spatial correlation of hydraulic conductivity from sediment cores and outcrops. Geophys. Res. Lett. 25(13), 2321–2324CrossRefGoogle Scholar
  11. Gelhar LW (1993) Stochastic Subsurface Hydrology. Prentice-Hall, Englewood Cliffs, New JerseyGoogle Scholar
  12. Glimm J, Lindquist WB, Pereira F, Zhang Q (1993) A theory of macrodispersion for the scale-up problem. Transp. Porous Media 13(1), 97–122CrossRefGoogle Scholar
  13. Grindrod P, Impey MD (1992) Fractal field simulations of tracer migration within the WIPP Culebra Dolomite. Rep. IM2856-1, vers. 2, p. 62, Intera Inf. Technol., Denver, Colorado, March 1992Google Scholar
  14. Guzman AG, Geddis AM, Henrich MJ, Lohrstorfer CF, Neuman SP (1996) Summary of Air Permeability Data From Single-Hole Injection Tests in Unsaturated Fractured Tuffs at the Apache Leap Research Site: Results of Steady-State Test Interpretation. Rep. NUREG/CR-6360, prepared for U.S. Nuclear Regulatory Commission, Washington, D.C.Google Scholar
  15. Hewett TA (1986) Fractal distributions of reservoir heterogeneity and their influence on fluid transport. SPE Pap. 15386 presented at 61st Annual Technical Conference, Soc. Petrol. Engin., New Orleans, Los AngelesGoogle Scholar
  16. Liu HH, Molz FJ (1996) Discrimination of fractional Brownian movement and fractional Gaussian noise structures in permeability and related property distribution with range analysis. Water Resources Research 32(8), 2601–2605CrossRefGoogle Scholar
  17. Liu HH, Molz FJ (1997) Multifractal analysis of hydraulic conductivity distributions. Water Resources Research 33(11), 2483–2488CrossRefGoogle Scholar
  18. Molz FJ, Boman GK (1993) A stochastic interpolation scheme in subsurface hydrology. Water Resources Research 29(11), 3769–3774CrossRefGoogle Scholar
  19. Molz FJ, Boman GK (1995) Further evidence of fractal structure in hydraulic conductivity distribution. Geophys. Res. Lett. 22(18), 2545–2548CrossRefGoogle Scholar
  20. Molz FJ, Liu HH, Szulga J (1997) Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions. Water Resources Research 33(10), 2273–2286CrossRefGoogle Scholar
  21. Molz FJ, Hewett TA, Boman GK (1998) A pseudo-fractal model for hydraulic properties in porous medium. In: Baveye P, Parlange J-Y, Stewart BA (eds) Fractals in Soil Sciences. CRC Press, Boca Raton, Fla, 341–372Google Scholar
  22. Molz FJ, Rajaram H, Lu S (2003) Stochastic fractal-based models of heterogeneity in subsurface hydrology: Origins, applications, limitations, and future research questions. Reviews of Geophysics, in pressGoogle Scholar
  23. Neuman SP (1990) Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resources Research 26(8), 1749–1758CrossRefGoogle Scholar
  24. Neuman SP (1994) Generalized scaling of permeabilities: Validation and effect of support scale. Geophys. Res. Lett. 21(5), 349–352CrossRefGoogle Scholar
  25. Neuman SP (1995) On advective transport in fractal velocity and permeability fields. Water Resources Research 31(6), 1455–1460CrossRefGoogle Scholar
  26. Neuman SP, Di Federico V (2003) The multifaceted nature of hydrogeologic scaling and its interpretation. Rev. Geophys., in pressGoogle Scholar
  27. Painter S (1996a) Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations. Water Resources Research 32(5), 1183–1195Google Scholar
  28. Painter S (1996b) Stochastic interpolation of aquifer properties using fractional Levy motion. Water Resources Research 32(5), 1323–1332Google Scholar
  29. Painter S (1998) Numerical method for conditional simulation of Levy random fields. Math. Geol. 30(2), 163–179CrossRefGoogle Scholar
  30. Robin MJL, Sudicky EA, Gillham RW, Kachanoski RG (1991) Spatial variability on strontium distribution coefficients and their correlation with hydraulic conductivity in the Canadian Forces Base Borden aquifer. Water Resources Research 27(10), 2619–2632CrossRefGoogle Scholar
  31. Rubin Y, Bellin A (1998) Conditional Simulation of Geologic Media with Evolving Scales of Heterogeneity. In: Sposito G (ed), Scale Dependence and Scale Invariance in Hydrology, Cambridge University Press, 398–420Google Scholar
  32. Tubman KM, Crane SD (1995) Vertical versus horizontal well log variability and application to fractal reservoir modeling. In: Barton CC, La Pointe PL (eds) Fractals in Petroleum Geology and Earth Processes. Plenum, New York, 279–293Google Scholar
  33. Voss RF (1985) Random fractals: Characterization and measurement. In: Pynn R, Skjeltorp A (eds), Scaling Phenomena in Disordered Systems, NATO ASI Series, p. 133Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • S. Cintoli
    • 1
    • 2
  • S. P. Neuman
    • 1
  • V. Di Federico
    • 2
  1. 1.Department of Hydrology and Water ResourcesUniversity of ArizonaTucsonUSA
  2. 2.DISTARTUniversità di BolognaBolognaItaly

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