Boolean reconstructions of complex materials: Integral geometric approach

We show that for the Boolean model of random composite media one can, from a single image of a system at any particle fraction, define a set of parameters which allows one to accurately reconstruct the medium for all other phase fractions. The morphological characterization is based on a family of measures known in integral geometry which provides powerful formulas for the Boolean model. The percolation thresholds of either phase are obtained with good accuracy. From the reconstructions one can subsequently predict property curves for the material across all phase fractions from the single three-dimensional image. We illustrate this for transport and mechanical properties of complex Boolean systems and for experimental sandstone samples.

Figure 1

Minkowski measures over particle fraction for a mixture of cubes and sticks (${\lambda }_{1i}=2$, ${\lambda }_{21}=8$, ${\lambda }_{22}={\lambda }_{23}=1$, and $p_1=1/2$).

Figure 2

Scaling of the standard deviation $\sigma \left(L\right)$ of the local measures $V_{\nu }$ for Boolean models with cubes of $\lambda =8$ (left) and spheres of $r=4$ (right). Data points are from right to left for $L=150$, 200,300,400,600, and $L=800$. We show the scaling fit with the theoretical scaling $L^{-1.5}$. One obtains a very low standard deviation in the local measures from images measured at large $L$. Top: $f=0.2$, middle: $f=0.5$, and bottom: $f=0.8$. Each data points represents at least 100 realizations.

Figure 3

Original five-mix system of 10% sticks $\left(40\times 1\times 1\right)$, 10% plates $\left(20\times 20\times 1\right)$, 40% cubes $\left(8\times 8\times 8\right)$, 20% each of rectangular prisms of size $\left(10\times 5\times 2\right)$, and $\left(16\times 8\times 4\right)$ on a ${500}^3$ lattice at (a) 20% and (b) 80% grain fraction and (c) and (d) corresponding two-particle reconstructions.

Figure 4

(Color online) Comparison of the correlation functions of the original five-grain mixture (shown as spheres) of 10% sticks $\left(40\times 1\times 1\right)$, 10% plates $\left(20\times 20\times 1\right)$, 40% cubes $\left(8\times 8\times 8\right)$, 20% each of rectangular prisms of size $\left(10\times 5\times 2\right)$ and $\left(16\times 8\times 4\right)$ to the two-grain reconstruction ($B{G}^2$, shown as cubes) given in Table for particle fractions of (a) 20% and (b) 80%.

Figure 5

(Color online) Effective conductivity over fraction $f$ for the original five-particle mixture given in Table compared to its equivalent two-grain Boolean model defined in Table . The equivalent model defined from the single 3D image at $f=50\%$ matches the full $f$-conductance curve for both phases. Top: conductivity grain:pore contrasts 1:0 and 0:1. Bottom: conductivity contrasts 1:0.1 and 0.1:1. ${\sigma }_0$ is the larger conductivity of the constituent materials

Figure 6

Bulk and shear modulus over solid grain fraction $f$ for the heterogeneous five-particle mixture defined in Table compared to the $B{G}^2$ model with equivalent local measures $V_{\nu }$. The elasticity contrasts are 1:0 and 0:1 for the Young modulus. The Poisson’s ratio of the solid phase is set to ${\nu }_s=0.25$ in both cases.

Figure 7

Variation in the porosity distribution along the $x$, $y$, and $z$ axes for the Fontainebleau sandstone samples of size $2.7\times 2.7\times 2.7{\text{mm}}^3$. The Fontainebleau sample exhibits negligible directional heterogeneity. Mean of porosity and variance along the z-direction are for the Fontainebleau samples: $\varphi =8.29\pm .54\%$ for fb7.5, $\varphi =12.9\pm .95\%$ for fb13, and $\varphi =17.7\pm 1.0\%$ for fb15.

Figure 8

Global measures $v_{\nu }$ of the Fontainebleau sandstone data sets at different image sizes $\left({120}^3–{480}^3\right)$. The variability in the measures $v_{\nu }$ does increase with decreasing image size, but the values at ${120}^3$ are consistent with data at the large scale. On the left side we compare the prediction of the IOSC and OSC systems to the ${\text{ROS}}^2$ model on the right side.

Figure 9

Visual comparisons of Fontainebleau sandstone reconstructions ${240}^3$, $\varphi =18.2\%$ to the original data sets. (a) Original sandstone image. (b) reconstruction based on the ${\text{IOS}}^C$ model. (c) reconstruction based on the ${\text{ROS}}^2$ model. (d) reconstruction based on the OSC model.

Figure 10

Variation in the porosity distribution along the $z$ axis of the Boolean reconstructions to the Fontainebleau sandstone. Sample size is the same as in Fig. 7.

Figure 11

Comparison of the prediction for conductance of the three matching Boolean models to the Fontainebleau sandstone data. (a) fb7.5, (b) fb13, and (c) fb15. The overlapping sphere model reconstruction (OSC) based on covering radius gives a poor match to the data for the low porosity sample and the identical overlapping sphere model based on reconstruction through the correlation function (IOSC) is poorest for the high porosity sample.

Figure 12

Comparison of the prediction of the bulk modulus of the matching Boolean models to the water saturated Fontainebleau sandstone data. (a) fb7.5, (b) fb13, and (c) fb15.

Figure 13

Comparison of the prediction of the shear modulus of the matching Boolean models to the water saturated Fontainebleau sandstone data. (a) fb7.5, (b) fb13, and (c) fb15.

Figure 14

Illustration of the influence of rotation on the average of the 1D lattice projections of an ellipse with aspect ratio 10:1. (a) Projection length $V_1$ over rotation angle for an ellipse with aspect ratio of 10:1. $a$, $b$, and $\frac{1}{2}\left(a+b\right)$ are given as base cases of the continuum measures. (b) Projection length over rotation angle for ellipses with varying aspect ratios.

Figure 15

Probability density of the discretized spheres used for the reconstruction of the sandstone samples (the radius distribution is not evenly spaced). (a) fb7.5, (b) fb13, and (c) fb15.

Figure 16

Radius distribution functions of the discretized spheres used for the reconstruction of the sandstone samples.