Permeability of Two-Component Granular Materials

Abstract

We expanded an existing model for permeability in mudrocks and shaly sands to include computation of effective grain radius and the Archie’s law parameter \(m\) in granular media composed of two different grain sizes. We found that the effective grain radius is the harmonic mean of the endmember grain radii and that \(m\) can be computed as the geometric mean of the endmember \(m\) values. We tested our model with three-dimensional lattice-Boltzmann simulations of flow through dilute and concentrated systems, and with comparison to measurements of laboratory-prepared and natural samples as well as field data. Modeled permeabilities matched the simulated and measured permeabilities over a wide range of porosities, grain sizes, and grain shapes. We additionally found that the model is independent of grain packing and aspect ratio, as these parameters only affect the endmember \(m\) values. Our predicted permeabilities generally fall within previously determined bounds, and we derive approximations for permeability as a function of endmember permeability for cases when endmember \(m\) values are equal and when endmember grain radii are very different. Our results advance our understanding of permeability in heterogeneous porous media.

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Acknowledgments

This work was supported by the University of Texas at Austin.

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Correspondence to Hugh Daigle.

Appendix: Generalized Mean

Appendix: Generalized Mean

The generalized mean \(M\) of a set of positive numbers \(x_{i}\) with associated weights \(w_{i}\) such that \(\sum w_{i}=1\) may be expressed as

$$\begin{aligned} M=\left( {\sum _i {w_i x_i^p } } \right) ^{\frac{1}{p}}, \end{aligned}$$
(13)

where \(p\) is a real number. The cases of \(p = 1\) and \(-\)1 correspond to the arithmetic and harmonic means, respectively. To prove that the case of \(p = 0\) corresponds to the geometric mean, we invoke Jensen’s inequality (Jensen 1906). Let \(f\) be a real function defined on the interval \(I = [a,b]\). For any numbers \(y_{i}~\in ~[a,~b]\) and \(\lambda _{i}\ge 0\) with \(\sum \lambda _{i}=1\),

$$\begin{aligned} f\left( {\sum _i \lambda _i y_i } \right) \ge \sum _i \lambda _i f\left( {y_i } \right) . \end{aligned}$$
(14)

In the case where \(f\) is the natural logarithm, by substituting \(y_{i}=x_{i}^{p}\) and \(\lambda _{i}=w_{i}\) Eq. 14 may be written as

$$\begin{aligned} \hbox {In}\left( {\sum _i {w_i x_i^p}}\right) \ge \sum _i {w_i \hbox {In } x_i^p}, \end{aligned}$$
(15)

or

$$\begin{aligned} \ln \left( {\sum _i {w_i x_i^p } } \right) \ge \ln \left( {\prod _i {x_i ^{w_i p}} } \right) . \end{aligned}$$
(16)

Applying the exponential function to both sides of Eq. 16,

$$\begin{aligned} \sum _i {w_i x_i^p } \ge \prod _i {x_i^{w_i p} } . \end{aligned}$$
(17)

Raising both sides of Eq. 17 to the power 1/\(p\) yields

$$\begin{aligned} \left( {\sum _i {w_i x_i^p } } \right) ^{\frac{1}{p}}\ge \prod _i {x_i^{w_i } } . \end{aligned}$$
(18)

By similar analysis, it can be shown additionally that

$$\begin{aligned} \left( {\sum _i {w_i x_i^p } } \right) ^{-\frac{1}{p}}\le \prod _i {x_i^{w_i } } . \end{aligned}$$
(19)

Combining Eqs. 18 and 19 yields

$$\begin{aligned} \left( {\sum _i {w_i x_i^{-p} } } \right) ^{-\frac{1}{p}}\le \prod _i {x_i^{w_i } } \le \left( {\sum _i {w_i x_i^p } } \right) ^{\frac{1}{p}}. \end{aligned}$$
(20)

In the limit \(p\rightarrow 0\), \(w_{i} x_{i}^{-p}=w_{i} x_{i}^{p}=w_{i}\), making the right- and leftmost terms in Eq. 20 equal. Therefore,

$$\begin{aligned} \mathop {\lim }\limits _{p\rightarrow 0} \left( {\sum _i {w_i x_i^p } } \right) ^{\frac{1}{p}}=\prod _i {x_i^{w_i } } , \end{aligned}$$
(21)

which is the case of the geometric mean corresponding to \(p = 0\).

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Daigle, H., Reece, J.S. Permeability of Two-Component Granular Materials. Transp Porous Med 106, 523–544 (2015). https://doi.org/10.1007/s11242-014-0412-6

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Keywords

  • Permeability
  • Shaly sand
  • Mudrocks