Reconstruction of three-dimensional porous media from a single two-dimensional image using three-step sampling

A random three-dimensional (3D) porous medium can be reconstructed from a two-dimensional (2D) image by reconstructing an image from the original 2D image, and then repeatedly using the result to reconstruct the next 2D image. The reconstructed images are then stacked together to generate the entire reconstructed 3D porous medium. To perform this successfully, a very important issue must be addressed, i.e., controlling the continuity and variability among adjacent layers. Continuity and variability, which are consistent with the statistics characteristic of the training image (TI), ensure that the reconstructed result matches the TI. By selecting the number and location of the sampling points in the sampling process, the continuity and variability can be controlled directly, and thus the characteristics of the reconstructed image can be controlled indirectly. In this paper, we propose and develop an original sampling method called three-step sampling. In our sampling method, sampling points are extracted successively from the center of $5\times 5$ and $3\times 3$ sampling templates and the edge area based on a two-point correlation function. The continuity and variability of adjacent layers were considered during the three steps of the sampling process. Our method was tested on a Berea sandstone sample, and the reconstructed result was compared with the original sample, using tests involving porosity distribution, the lineal path function, the autocorrelation function, the pore and throat size distributions, and two-phase flow relative permeabilities. The comparison indicates that many statistical characteristics of the reconstructed result match with the TI and the reference 3D medium perfectly.

Figure 1

Three-grid: (a) the coarsest grid; (b) the finer grid; (c) the finest grid.

Figure 2

The procedure of three-step sampling. (a) The TI with size of $128\times 128$ pixels. (b) The result after sampling using the $5\times 5$ template. (c) The result after sampling with the $3\times 3$ template. (d) The invalid sampling area and the edge area after the first two sampling steps. (e) The final selected sampling points.

Figure 3

Comparison between reconstructed results using different size sampling templates. The TI is shown in Fig. 2; (a) and (b) are, respectively, the No. 1 and No. 11 layers using only the $3\times 3$ sampling template; (c) and (d) are, respectively, the No. 1 and No. 11 layers using only the $5\times 5$ sampling template; (e) and (f) are, respectively, the No. 1 and No. 11 layers using the $5\times 5$ and $3\times 3$ sampling templates combined; and (g) and (h) are, respectively, the No. 1 and No. 11 layers of the CT.

Figure 4

Five initial steps for constructing the external view of the Berea sandstone.

Figure 5

Visual comparison of spatial pore connectivity of the micro-CT image of Berea sandstone with that of a corresponding 3D reconstruction. (a) 3D exterior view of the reconstructed result, (b) a cross section of the reconstructed result, and (c) the perspective image of the reconstructed result. (d) Micro-CT sample, (e) a cross section of micro-CT sample, and (f) the perspective image of the micro-CT sample.

Figure 6

Comparison of the porosity distribution in the reconstructed Berea sandstone with that of its micro-CT sample.

Figure 7

Comparison of ACFs for three images, in (a) the $x$ direction, (b) the $y$ direction, and (c) the $z$ direction (perpendicular to the page and reference with the $y$ direction of TI).

Figure 8

Comparison of the LPFs for TI, micro-CT image, and reconstructed result of the Berea sandstone in the (a) $x$, (b) $y$, and (c) $z$ directions.

Figure 9

Pore network model: (a) micro-CT sample and (b) reconstructed result.

Figure 10

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