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Transport in Porous Media

, Volume 12, Issue 2, pp 161–183 | Cite as

Upscaling of conductivity of heterogeneous formations: General approach and application to isotropic media

  • Peter Indelman
  • Gedeon Dagan
Article

Abstract

The numerical simulation of flow through heterogeneous formations requires the assignment of the conductivity value to each numerical block. The conductivity is subjected to uncertainty and is modeled as a stationary random space function. In this study a methodology is proposed to relate the statistical moments of the block conductivity to the given moments of the continuously distributed conductivity and to the size of the numerical blocks. After formulating the necessary conditions to be satisfied by the flow in the upscaled medium, it is found that they are obeyed if the mean and the two-point covariance of the space averaged energy disspation function over numerical elements in the two media, of point value and of upscaled conductivity, are identical. This general approach leads to a systematic upscaling procedure for uniform average flow in an unbounded domain. It yields the statistical moments of upscaled logconductivity that depend only on those of the original one and on the size and shape of the numerical elements.

The approach is applied to formations of isotropic heterogeneity and to isotropic partition elements. After a general discussion based on dimensional analysis, the procedure is illustrated by using a first-order approximation in the logconductivity variance. The upscaled logconductivity moments (mean, two-point covariance) are computed for two and three dimensional flows, isotropic heterogeneous media and elements of circular or spherical shape. The asymptotic cases of elements of small size, which preserve the point value conductivity structure on one hand, and of large blocks for which the medium can be replaced by one of deterministic effective properties, on the other hand, are analyzed in detail. The results can be used in order to generate the conductivity of numerical elements in Monte Carlo simulations.

Key words

Heterogeneous porous media homogenization upscaling numerical simulation stochastic modeling effective conductivity 

Nomenclature

C

covariance

e

rate of dissipation of mechanical energy per unit weight of fluid

E

total rate of energy dissipation in the flow domain

H

overlap function

K

hydraulic conductivity

KG

geometrical mean of conductivity

I

integral scale

J=∇〈P

mean head gradient

L

characteristic size of Ω

l

characteristic size ofΩ also diameter of circle and sphere

n

number of dimensions

P

pressure head

Q

total fluid discharge

SA,SB

inlet and outlet boundaries of flow domain

v

velocity

Y

logconductivity

λ

characteristic scale of flow nonuniformity

ρ

autocorrelation function

σ2

variance

Ω

flow domain

Ω

partition element

Overlining

ā

space averaged overΩ

Ã

upscaled quantity

â

Fourier transform ofa

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References

  1. Ababou, R., McLaughlin, D., Gelhar, L. W., and Tompson, A. F. B., 1989, Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media,Transport in Porous Media 4, 549–565.Google Scholar
  2. Ababou, R. and Wood, E. F., 1990, Comment on ‘Effective groundwater model parameter values: influence of spatial variability of hydraulic conductivity, leakance, and recharge’ by J. J. Gomez-Hernandez, and S. M. Gorelick,Water Resour. Res. 26, 1843–1846.Google Scholar
  3. Bakr, A. A., Gelhar, L. W., Gutjahr, A. L., and MacMillan, J. R., 1978, Stochastic analysis of spatial variability in subsurface flow, 1, Comparison of one- and three-dimensional flows,Water Resour. Res. 14, 263–271.Google Scholar
  4. Beran, M. J., 1968,Statistical Continuum Theories, Interscience, New York.Google Scholar
  5. Dagan, G., 1981, Analysis of flow through heterogeneous random aquifers by the method of embedding matrix 1. Steady flow,Water Resour. Res. 17, 107–122.Google Scholar
  6. Dagan, G., 1982, Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 1. Conditional simulation and direct problem,Water Resour. Res. 18, 813–833.Google Scholar
  7. Dagan, G., 1986, Statistical theory of groundwater flow and transport; pore to laboratory, laboratory to formation and formation to regional scale,Water Resour. Res. 22, 120S-135S.Google Scholar
  8. Dagan, G., 1989,Flow and Transport in Porous Formations, Springer-Verlag, Berlin.Google Scholar
  9. Dagan, G. and Indelman, P., 1990, On numerical simulation of flow through heterogeneous formations, in Gambolati (ed),Computational Methods in Subsurface Hydrology, Proc. of the VIII Inter. Conf. on Comput. Methods in Water Resour., Computational Mechanics Publications and Springer-Verlag, pp. 445–455.Google Scholar
  10. Delhomme, J. P., 1979, Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach,Water Resour. Res. 15, 269–280.Google Scholar
  11. Desbarats, A. J. 1987, Numerical estimation of effective conductivity in sand-shale formations,Water Resour. Res. 23, 273–286.Google Scholar
  12. Desbarats, A. J. and Srivastava, R. M., 1991, Geostatistical characterization of groundwater flow parameters in a simulated aquifer,Water Resour. Res. 27, 687–698.Google Scholar
  13. Desbarats, A. J. 1992, Spatial averaging of transmissivity in heterogeneous fields with flow toward a well,Water Resour. Res. 28, 757–767.Google Scholar
  14. Dykhne, A. M., 1970, Conductivity of two-dimensional two-phase system,J. Experimental and Theoretical Physics (in Russian)59, 111–115.Google Scholar
  15. Freeze, R. A., 1975, A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media,Water Resour. Res. 11, 725–741.Google Scholar
  16. Gelhar, L. J., 1986, Stochastic subsurface hydrology from theory to applications,Water Resour. Res. 22, 135S-145S.Google Scholar
  17. Gelhar, L. W. and Axness, C. L., 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers,Water Resour. Res. 19, 161–180.Google Scholar
  18. Gomez-Hernandez, J. J. and Gorelick, S. M., 1989, Effective groundwater model parameter values: influence of spatial variability of hydraulic conductivity, leakance, and recharge,Water Resour. Res. 25, 405–419.Google Scholar
  19. Gutjahr, A. L., Gelhar, L. W., Bakr, A. A., and MacMillan, J. R., 1978, Stochastic analysis of spatial variability in subsurface flow, 2, Evaluation and application,Water Resour. Res. 14, 953–959.Google Scholar
  20. Hoeksema, R. J. and Kitanidis, P. K., 1985, Analysis of patial structure of properties of selected aquifers,Water Resour. Res. 21, 563–572.Google Scholar
  21. Indelman, P., Kats, R. M., and Shvidler, M. I., 1979, Numerical simulation of unstable displacement in porous media,Fluid Dynamics 14(2), 185–190.Google Scholar
  22. Indelman, P. and Dagan, G., 1993, Upscaling of permeability of anisotropic formations. Part 1: The general framework,Water Resour. Res.,29, 917–923.Google Scholar
  23. Indelman, P. and Dagan, G., 1993, Upscaling of permeability of anisotropic formations. Part 2: General structure and small perturbation analysis,Water Resour. Res.,29, 924–934.Google Scholar
  24. Indelman, P., 1993, Upscaling of permeability of anisotropic formations. Part 3: Applications,Water Resour. Res.,29, 935–943.Google Scholar
  25. King, P. R., 1989, The use of renormalization for calculating effective conductivity,Transport in Porous Media 4, 37–58.Google Scholar
  26. Koplik, J., 1982, Creeping flow in two-dimensional networks,J. Fluid Mech. 119, 219–247.Google Scholar
  27. Landauer, R., 1978, Electrical conductivity in inhomogeneous media, in Garland, J. C., and Tanner, D. B. (eds.)Electrical Transport and Optical Properties of Inhomogeneous Media, AIP Conference Proceedings No. 40, American Institute of Physics.Google Scholar
  28. Matheron, G., 1967,Elements pour une theorie des milieux poreux, Masson, Paris.Google Scholar
  29. Noetinger, B., The effective conductivity of a heterogeneous porous medium,Transport in Porous Media, submitted.Google Scholar
  30. Rubin, Y., and Gomez-Hernandez, J. J., 1990, A stochastic approach to the problem of upscaling of conductivity in disordered media: theory and unconditional numerical simulations,Water Resour. Res. 26, 691–701.Google Scholar
  31. Shvidler, M. I., 1985,Stochastic Hydrodynamics of Porous Media (in Russian), Nedra, Moscow.Google Scholar
  32. Smith, L., and Freeze, R. A., 1979, Stochastic analysis of steady state groundwater flow in bounded domain, Two-dimensional simulations,Water Resour. Res. 15, 1543–1559.Google Scholar
  33. Vanmarcke, E., 1983,Random Fields: Analysis and Synthesis, MIT Press, Cambridge, Mass., U.S.A.Google Scholar
  34. Warren, J. E. and Price, H. S., 1961, Flow in heterogeneous porous media,Soc. Pet. Eng. J., 153–169.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Peter Indelman
    • 1
  • Gedeon Dagan
    • 1
  1. 1.Department of Fluid Mechanics and Heat Transfer, Faculty of EngineeringTel Aviv UniversityTel AvivIsrael

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