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Effects of Multi-scale Heterogeneity

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Percolation Theory for Flow in Porous Media

Part of the book series: Lecture Notes in Physics ((LNP,volume 880))

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Abstract

At different measurement scales it is possible that different variability in local properties can dominate the process of upscaling. This chapter addresses a number of such complex media and shows how it may be possible to use percolation concepts at, e.g., both the pore scale and at the geologic scale in the same problem. At the same time it demonstrates that, for the case of nested heterogeneity, the scale dependence of the mean permeability can be understood only when it is clear that the effective permeability at a given scale has no relationship to the mean value of the permeability at smaller scales.

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Notes

  1. 1.

    The fact that these authors tended to replace equilibrium K values less than 5×10−8 cm/s with approximately this value (which represented a maximum experimental time) narrowed the widths of their K distributions drastically in the limit of small θ.

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Author information

Authors and Affiliations

  1. Department of Earth and Environmental Sciences, Wright State University, Dayton, OH, USA

    Allen Hunt

  2. Department of Agronomy, Iowa State University, Ames, IA, USA

    Robert Ewing

  3. Department of Earth and Environmental Sciences, Wright State University, Dayton, OH, USA

    Behzad Ghanbarian

Authors
  1. Allen Hunt
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  2. Robert Ewing
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  3. Behzad Ghanbarian
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Problems

Problems

  1. 12.1

    Repeat the calculations of K for nested heterogeneity that led to the entries of Table 12.1, but using a critical volume fraction of 0.10.

  2. 12.2

    Allow K to follow a log-uniform distribution with a prescribed width equal to that of the example in Sect. 12.3, and discretize the distribution as in the procedure there. Find the upscaled K as a function of an arbitrary critical volume fraction. Find the value of the critical volume fraction which yields the Matheron conjecture (Eq. (2.30)). How does this critical volume fraction compare with the typical value of about 0.16 quoted in the literature? Is it possible for the Matheron conjecture to be accurate for all values of the width parameter using the same critical volume fraction?

  3. 12.3

    Consider Problem 12.2 again, but allow nested heterogeneity analogously to the procedure of this chapter. Thus the critical volume fraction is, to a good approximation, independent of scale. Investigate the performance of the Matheron conjecture for the upscaling at both scales; does choice of the critical volume fraction of Problem 12.2, which guarantees equivalence to the Matheron conjecture at the lowest length scale, also guarantee equivalence to the Matheron conjecture at the next higher length scale?

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Hunt, A., Ewing, R., Ghanbarian, B. (2014). Effects of Multi-scale Heterogeneity. In: Percolation Theory for Flow in Porous Media. Lecture Notes in Physics, vol 880. Springer, Cham. https://doi.org/10.1007/978-3-319-03771-4_12

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