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.

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 $p = 0$.