Abstract
Because of the complex internal porous structures, the reconstruction of porous media has encountered many difficulties in practical research and development. At present, numerical simulation methods are widely used in the field of reconstructing porous media, which can reconstruct results similar to the true pore structure, but generally are CPUintensive and timeconsuming. Recently, with the rapid development of deep learning, the powerful ability of feature extraction and structure prediction owned by deep learning can be transferred to reconstruct porous media. The convolutional neural network (CNN) is one of the classical methods in deep learning. A traditional CNN is unable to specially focus on the effective features in learning and possibly has degradation problem with the increase of layers. To address the degradation problem and make CNNs extract important features, residual networks and attention mechanisms are combined in CNNs to respectively alleviate network degradation and focus on the important features in the reconstruction of porous media. Besides, the bilinear interpolation is used to enlarge the size of compressed data. Compared with some traditional numerical reconstruction methods, CNN, GAN and some variants of GAN, our method has shown its practicability and effectiveness.
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Funding
This work is supported by the National Natural Science Foundation of China (Nos. 41672114, 41702148).
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Appendices
Appendix 1: The main functions and parameters of our method
As shown in Fig. 5, our method is based on CNNs, combining a CBAM with ResNets as well as bilinear interpolation to realize porous media reconstruction. The specific parameters and functions of our method are shown in Table
9. Some parameters and terms are described below.

1.
Conv2d: a 2D convolution operation that is used to extract features of 2D data.

2.
Hidden layer of MLP: the hidden layer in the fullyconnected network MLP.

3.
Output layer of MLP: the output layer in the fullyconnected network MLP.

4.
Units: the number of neural units in a layer of the network.

5.
batch_normalization: a regularization function that normalizes the input data, alleviates the problem of scattered distribution of data features, and makes the network model more stable.

6.
kernel_size: the size of a convolution kernel in a layer.

7.
Stride: the step size of each convolution kernel.

8.
leaky_relu: an activation function that alleviates the problem of gradient vanishing.

9.
Filters: the number of filters in one layer of networks.

10.
Scale: a magnification (scale > 1) or contraction (0 < scale < 1) factor for a bilinear interpolation, e.g. ‘scale = 2’ means that the original feature map is enlarged to twice using bilinear interpolation.

11.
leaky_relu: an activation function.

12.
Padding = '‘same’: “padding” means the filling of image edges, and “same” is a filling attribute, which means the output size and input size of the layer are same after they are divided by the stride size and rounded up.
Note that in Conv_1 and Conv_2 there are altogether six layers, and in Conv_3 and Conv_4 there are also exactly the same six layers. The reason is that in Conv_1 and Conv_2 the convolution operation gradually reduces the size of feature maps for six times; on the contrary, bilinear interpolation used as the reverse operation of reducing the size of feature maps enlarges the feature maps gradually for six times, too. Although convolution operations are used in Conv_3 and Conv_4, the size of feature maps is not influenced since their kernel size is 1 × 1 and the stride is 1.
Appendix 2: Discussion on the influence of different network layers
As shown in Table 9 and discussed in Sect. 2.3, Conv_1 and Conv_2 altogether have six layers; Conv_3 and Conv_4 also have the same number of layers because the same number of bilinear interpolation operations are used to offset the effect of convolution operations used for the reduction of feature maps in Conv_1 and Conv_2. Normally the layer numbers of Conv_1 (layer number = 1) and Conv_4 (layer number = 2) in our method are fixed. Let the total layer number in the Conv_1 and Conv_2 be i, so the total layer number in Conv_3 and Conv_4 is i, too. We will discuss the influence of i on the reconstruction time and quality.
Table
10 shows the reconstruction time for only one result when i = 3, 4, 5, 6, respectively. Note that when i = 3, since the layer numbers of Conv_1 (layer number = 1) and Conv_4 (layer number = 2) are fixed, it can be inferred that the layer number of Conv_2 is 2 and that of Conv_3 is 1. Similarly, when i = 4, the layer number of Conv_2 is 3 and that of Conv_3 is 2; when i = 5, the layer number of Conv_2 is 4 and that of Conv_3 is 3; when i = 6, Conv_2’s layer number is 5 and Conv_3’s is 4.
It can be seen from Table 10 that with the increase of i, the reconstruction time also increases. Figures
15a–c show MPC curves of using different i layers in three directions. As shown in Fig. 15, it can be seen that the reconstructed result is closest to TI when i = 6. In real reconstruction, the reconstruction time and quality should be balanced. Generally, increasing the number of layers can improve the reconstruction quality, but the reconstruction time also increases dramatically. In our experiment, i = 6 seems to be an appropriate balance between reconstruction quality and time.
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Zhang, T., Xia, P. & Lu, F. 3D stochastic reconstruction of porous media based on attention mechanisms and residual networks. Stoch Environ Res Risk Assess (2021). https://doi.org/10.1007/s0047702102092z
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Keywords
 Porous media
 Reconstruction
 Convolutional neural network
 Convolution block attention module
 Residual network