Improved multipoint statistics method for reconstructing three-dimensional porous media from a two-dimensional image via porosity matching

Reconstructing a three-dimensional (3D) structure from a single two-dimensional training image (TI) is a challenging issue. Multiple-point statistics (MPS) is an effective method to solve this problem. However, in the traditional MPS method, errors occur while statistical features of reconstruction, such as porosity, connectivity, and structural properties, deviate from those of TI. Due to the MPS reconstruction mechanism that the voxel being reconstructed is dependent on the reconstructed voxel, it may cause error accumulation during simulations, which can easily lead to a significant difference between the real 3D structure and the reconstructed result. To reduce error accumulation and improve morphological similarity, an improved MPS method based on porosity matching is proposed. In the reconstruction, we search the matching pattern in the TI directly. Meanwhile, a multigrid approach is also applied to capture the large-scale structures of the TI. To demonstrate its superiority over the traditional MPS method, our method is tested on different sandstone samples from many aspects, including accuracy, stability, generalization, and flow characteristics. Experimental results show that the reconstruction results by the improved MPS method effectively match the CT sandstone samples in correlation functions, local porosity distribution, morphological parameters, and permeability.

Figure 1

Template and pattern: (a) template, (b) $15\times 15$ TI, and (c) pattern.

Figure 2

Pattern matching: (a) TI and (b) simulated 3D structure.

Figure 3

Three-grid system: (a) the third grid, (b) the second grid, and (c) the first grid.

Figure 4

Microstructures obtained from the CT sample of sandstone, the traditional MPS method, and the improved MPS method: (a) TI; (b) CT sample; (c) reconstructed result by the improved MPS method; (d) reconstructed result by the traditional MPS method; (e) 3D slice of the CT sample; (f) 3D slice of the reconstructed result by the improved MPS method; and (g) 3D slice of the reconstructed result by the traditional MPS method.

Figure 5

Comparison of autocorrelation functions for the CT sample, reconstructed results of the traditional MPS method, and the improved MPS method in $x, y$, and $z$ directions corresponding to (a)–(c), and the lineal path functions corresponding to (d)–(f). The numbers in parentheses indicate the $L_2$ norm errors.

Figure 6

Comparison of the local porosity distributions for the CT sample, reconstructed results of the traditional MPS method, and the improved MPS method.

Figure 7

Comparison of the average volume of pore in (a) the average shape factor in (b) and the average coordination number in (c) among ten sets of microstructures from the same TI, ten sets of microstructures from ten different slices, and the CT sample.

Figure 8

Pore-throat network model: (a) the pore-throat network model of the CT sample; (b) the pore-throat network model of a reconstructed result by the traditional MPS method; and (c) the pore-throat network model of the improved MPS method.

Figure 9

Comparison of relative permeability among the CT sample, the traditional MPS method, and the improved MPS method: (a) drainage and (b) imbibition.