Pattern density function for reconstruction of three-dimensional porous media from a single two-dimensional image

Three-dimensional (3D) structures are useful for studying the spatial structures and physical properties of porous media. A 3D structure can be reconstructed from a single two-dimensional (2D) training image (TI) by using mathematical modeling methods. Among many reconstruction algorithms, an optimal-based algorithm was developed and has strong stability. However, this type of algorithm generally uses an autocorrelation function (which is unable to accurately describe the morphological features of porous media) as its objective function. This has negatively affected further research on porous media. To accurately reconstruct 3D porous media, a pattern density function is proposed in this paper, which is based on a random variable employed to characterize image patterns. In addition, the paper proposes an original optimal-based algorithm called the pattern density function simulation; this algorithm uses a pattern density function as its objective function, and adopts a multiple-grid system. Meanwhile, to address the key point of algorithm reconstruction speed, we propose the use of neighborhood statistics, the adjacent grid and reversed phase method, and a simplified temperature-controlled mechanism. The pattern density function is a high-order statistical function; thus, when all grids in the reconstruction results converge in the objective functions, the morphological features and statistical properties of the reconstruction results will be consistent with those of the TI. The experiments include 2D reconstruction using one artificial structure, and 3D reconstruction using battery materials and cores. Hierarchical simulated annealing and single normal equation simulation are employed as the comparison algorithms. The autocorrelation function, linear path function, and pore network model are used as the quantitative measures. Comprehensive tests show that 3D porous media can be reconstructed accurately from a single 2D training image by using the method proposed in this paper.

Figure 1

Morphological features of an image. (a) The core image; (b) a local morphological feature extracted using the 17×17 template; (c) a local morphological feature extracted using the 9×9 template.

Figure 2

Relationship between morphological feature and patterns.

Figure 3

Multiple-grid system.

Figure 4

Different images have different distributions of patterns. (a) Battery materials image with a size of 128×128; (b) core image with a size of 128×128; (c) distribution of patterns of the two images, using a 2×2 template.

Figure 5

PDFs of the multiple-grid system. (a) First grid, with a size of 128×128; (b) second grid, with a size of 64×64; (c) third grid, with a size of 32×32; (d) fourth grid, with a size of 16×16; (e) the pattern density functions of four grids, using a 2×2 template.

Figure 6

Eighteen changed patterns caused by two exchanged points, using a grid with a size of 16×16.

Figure 7

Schematic of adjacent grid and reversed phase method: (a) 4×4 grid; (b) 8×8 grid expanded by (a) and added points.

Figure 8

The results of core image reconstructed by temperature-controlled mechanism and simplified temperature-controlled mechanism, respectively. (a) Core sample; (b) the result reconstructed by temperature-controlled mechanism; (c) the result reconstructed by simplified temperature-controlled mechanism.

Figure 9

The results of a 2D artificial image reconstructed by different methods. (a) Original image; (b) the result reconstructed by HSA; (c) the result reconstructed by SNESIM; (d) the result reconstructed by PDFSIM; (e) an E-type image composed by 90 results reconstructed by PDFSIM.

Figure 10

Three-dimensional reconstructed results of the battery material. (a) Two-dimensional training image; (b) the result reconstructed by HSA; (c) the result reconstructed by SNESIM; (d) the result reconstructed by PDFSIM.

Figure 11

Morphological features of three orthogonal planes of 3D reconstructed results. (a–c) The slices of XY, XZ, and YZ planes of the 3D result reconstructed by HSA; (d–f) the slices of XY, XZ, and YZ planes of the 3D result reconstructed by SNESIM; (g–i) the slices of XY, XZ, and YZ planes of the 3D result reconstructed by PDFSIM.

Figure 12

Three-dimensional reconstructed results of the core. (a) Two-dimensional training image; (b) 3D structure of Micro-CT sample; (c) the perspective image of Micro-CT sample; (d) reconstructed result; (e) the perspective image of reconstructed result.

Figure 13

Comparison of autocorrelation functions, in (a) the $x$ direction, (b) the $y$ direction, and (c) the $z$ direction.

Figure 14

Comparison of linear path functions, in (a) the $x$ direction, (b) the $y$ direction, and (c) the $z$ direction.

Figure 15

Pore-throat network model. (a) The pore-throat network model of Micro-CT; (b) the pore-throat network model of reconstructed result.

Figure 16

Relative permeability curves of the 3D micro-CT image and the 3D reconstructed result. (a) Drainage and (b) imbibition.