Improved recurrent generative model for reconstructing large-size porous media from two-dimensional images

Modeling the three-dimensional (3D) structure from a given 2D image is of great importance for analyzing and studying the physical properties of porous media. As an intractable inverse problem, many methods have been developed to address this fundamental problems over the past decades. Among many methods, the deep learning–(DL) based methods show great advantages in terms of accuracy, diversity, and efficiency. Usually, the 3D reconstruction from the 2D slice with a larger field-of-view is more conducive to simulate and analyze the physical properties of porous media accurately. However, due to the limitation of reconstruction ability, the reconstruction size of most widely used generative adversarial network–based model is constrained to ${64}^3$ or ${128}^3$. Recently, a 3D porous media recurrent neural network based method (namely, 3D-PMRNN) (namely 3D-PMRNN) has been proposed to improve the reconstruction ability, and thus the reconstruction size is expanded to ${256}^3$. Nevertheless, in order to train these models, the existed DL-based methods need to down-sample the original computed tomography (CT) image first so that the convolutional kernel can capture the morphological features of training images. Thus, the detailed information of the original CT image will be lost. Besides, the 3D reconstruction from a optical thin section is not available because of the large size of the cutting slice. In this paper, we proposed an improved recurrent generative model to further enhance the reconstruction ability (${512}^3$). Benefiting from the RNN-based architecture, the proposed model requires only one 3D training sample at least and generates the 3D structures layer by layer. There are three more improvements: First, a hybrid receptive field for the kernel of convolutional neural network is adopted. Second, an attention-based module is merged into the proposed model. Finally, a useful section loss is proposed to enhance the continuity along the $Z$ direction. Three experiments are carried out to verify the effectiveness of the proposed model. Experimental results indicate the good reconstruction ability of proposed model in terms of accuracy, diversity, and generalization. And the effectiveness of section loss is also proved from the perspective of visual inspection and statistical comparison.

Figure 1

Graphical illustration of training dataset construction, reconstruction from a thin section and the larger-size realization generated by original 3D-PMRNN. (a) The original procedure of creating the training dataset for DL-based methods. (b) The schematic diagram of 2D-to-3D reconstruction based on a thin section. (c) The visual result of synthetic realization (${512}^3$) generated by original 3D-PMRNN.

Figure 2

The architecture of the proposed model.

Figure 3

The architecture of generating module.

Figure 4

The architecture of CBAM.

Figure 5

The structure of section loss.

Figure 6

Definition of three morphological descriptors.

Figure 7

The 2D and 3D image of training dataset and the slice extracting from thin section.

Figure 8

Comparison of 3D structures between testing target and synthetic realizations. Yellow represents the pore phase. (a) The 3D structure and cross section of the target. (b) The 3D structures and cross sections of synthetic realizations.

Figure 9

Comparison of 2D slices between testing target and synthetic realizations at different layers.

Figure 10

Comparison of statistical functions among 10 reconstructions, their average, and testing target. The calculations of statistical functions are along the $X, Y$, and $Z$ directions and their average.

Figure 11

Relative permeability curves of the target and average. (a) Drainage and (b) imbibition.

Figure 12

Visual comparison of 3D structures and cross sections. (a) The 3D structure of testing target and its $XZ$ cross section. (b) The 3D structures of synthetic realizations and their $XZ$ cross sections without section loss. (c) The 3D structures of synthetic realizations and their $XZ$ cross sections with section loss.

Figure 13

Statistical comparison of $S_2, L$, and $C_2$ among testing target, average with section loss, reconstructions without section loss, and their average.

Figure 14

Visual inspection of the reference 2D slice, and 2D and 3D images of synthetic realizations. (a) Different extracting template on original thin section. (b) The 2D reference slice. (c) The 3D structures and different 2D layers of synthetic realizations.

Figure 15

Visual comparison between proposed method and original 3D-PMRNN

Figure 16

Statistical comparison of $S_2, L$, and $C_2$ among the target of the thin section, average of training samples, average of original 3D-PMRNN, 10 reconstructions, and their average along the $X$ and $Y$ directions.