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


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 





rate of dissipation of mechanical energy per unit weight of fluid


total rate of energy dissipation in the flow domain


overlap function


hydraulic conductivity


geometrical mean of conductivity


integral scale


mean head gradient


characteristic size of Ω


characteristic size ofΩ also diameter of circle and sphere


number of dimensions


pressure head


total fluid discharge


inlet and outlet boundaries of flow domain






characteristic scale of flow nonuniformity


autocorrelation function




flow domain


partition element



space averaged overΩ


upscaled quantity


Fourier transform ofa


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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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