# Digital Core Modeling Based on Pretrained Generative Adversarial Neural Networks

Authors: Qing Zhang; Benqiang Wang; Xusheng Liang; Yizhen Li; Feng He; Yuexiang Hao
Journal: Geofluids
 (2022)

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## Abstract

Accurately establishing a 3D digital core model is of great significance in oil and gas production. The physical experiment method and numerical modeling method are common modeling methods. With the development of deep learning technology, a variety of deep learning algorithms have been applied to digital core modeling. The digital core modeling method based on generative adversarial neural networks (GANs) has attracted wide attention due to its good quality and simple generation process. The disadvantage of this method is that the network needs thousands of trainings to achieve acceptable results. For this reason, this paper proposes to use the pretrained GANs for digital core modeling training, which can greatly reduce the number of network training while ensuring the core modeling effect. We can use the presented method to quickly complete the training and use the trained generator model to obtain multiple digital cores. By analyzing the quality of the generated cores from multiple aspects, it is revealed that the properties of the generated cores are in good agreement with the ones of the real core samples. The results indicate the reliability of the pretrained GAN method.


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## Body

## 1. Introduction

In order to find and exploit oil and gas reservoirs better, the demand for fine reservoir characterization is constantly increasing [1, 2]. It is of great significance to find a method that can accurately describe the internal structure of complex rocks [3]. With the development of computers and various scanning imaging technologies, the digital core technology that reflects the internal microstructure of rocks with 3D digital images has developed rapidly [4]. The digital core can accurately reflect the spatial structure inside the rock [5] and carry out the simulation of rock electrical characteristics [6–10], elastic properties [11–15], seepage characteristics, and NMR simulation response [16–20] and obtain a variety of physical properties of the rock for better reservoir evaluation. The physical experiment method and numerical reconstruction method are currently commonly used digital core modeling methods [21]. The physical experiment method uses a scanning imaging instrument to obtain multiple 2D images of cores and then obtains 3D core images through mathematical algorithms. The physical experiment method mainly includes the sequence slice superposition method [22], X-ray CT scanning method [23, 24], and focused ion-scanning electron double beam microscope method [25]. The digital core model obtained by the physical experiment methods can accurately describe the internal structure and mineral composition of the rock. The disadvantage is that it requires professional equipment and software, and the high cost limits the wide use of this method. Considering the cost, using core analysis data and 2D thin slices for digital core modeling is a good choice. Numerical reconstruction methods include stochastic simulation method and process-based method. The stochastic simulation method obtains information from the 2D rock images as constraint conditions to establish a digital core model, including the random Gaussian field method [26], simulated annealing method [27], Markov Chain-Monte Carlo Method [28], and multiple point statistics method [29]. The process-based method determines the radius of sedimentary particles through rock particle size analysis data, simulates the formation process of rocks, and establishes a 3D digital core model [30]. When using the above-mentioned numerical methods for digital core modeling, it is necessary to manually select various characteristic functions and constraint conditions, and it is necessary to repeat the complete process many times when digital core modeling. Generally, as the size of the generated core increases, the amount of calculation required for modeling also increases exponentially.With the rapid development of deep learning research, generative learning using the powerful feature extraction and expression capabilities of neural networks has become a research hotspot. The convolutional neural network has a wide range of applications in image processing under the characteristics of local connection and weight sharing [31, 32]. As a typical unsupervised learning model, autoencoders are widely used in image editing and generation [33]. Goodfellow et al. proposed the generative adversarial neural networks (GANs) in 2014. The proposal of this network has greatly promoted the research in the field of unsupervised learning and image generation [34]. The digital core is used as a 3D digital image, and a variety of deep learning algorithms have achieved good results in digital core modeling. Mosser et al. used DCGANs to achieve rapid modeling of a variety of digital cores, and the obtained core samples are in good agreement with the real core samples [35, 36]. Valsecchi et al. used DCGANs to generate 3D digital cores from 2D images of rocks, which is fast and accurate [37]. Zha et al. used WG-GANs to generate high-quality shale pictures, which are consistent and diverse with real shale samples [38]. Feng et al. used CGANs to obtain a complete core image from a limited image [39]. Many researchers combined AE and GANs to realize the generation of 3D digital core from 2D core thin slices [40, 41]. Feng et al. used Bicycle GANs to realize the process from 2D core slicing to 3D digital core reconstruction [42]. Wang et al. used the CNN network to achieve high-resolution 3D digital core modeling [43]. Ong et al. proposed generative encoding in combination with GANs and AE, which realized the use of 2D slices to generate 3D digital cores [44]. Compared with traditional numerical modeling methods, the use of deep learning algorithms can save researchers from paying too much attention to the extraction of feature information, and the training process is relatively simple. The parameters obtained during the training process can be saved, and the subsequent core reconstruction process can be directly completed quickly using the parameters obtained during the training.Previous studies have shown that high-quality digital core models can be obtained by using network models that generative adversarial neural networks and its variants. However, the training of GANs is a complicated process, and it is easy to encounter problems such as gradient disappearance or mode collapse during the training process [38]; it usually takes thousands or even tens of thousands of training times to achieve a better generation effect. Therefore, it is of great significance to find a way to effectively reduce the number of trainings of GANs. This paper is based on the DCGAN model proposed by Radford et al. [45], using the pretrained GANs for digital core generation training can effectively reduce the number of network training while ensuring the quality of the generated core samples. The follow-up arrangement of the paper is as follows: in the second part, the structure of the generative adversarial neural networks and the parameter setting of the training process of this paper are introduced; the training data and training process are shown in the third part, and the generation effect of the model is analyzed. The fourth part elaborates the conclusions obtained in this paper.


## 2. Methods

### 2.1. GANs

GANs are mainly composed of two parts, the generator network (G) and the discriminator network (D), which are in a state of confrontation during training [34]. The goal of the generator is to learn the data distribution characteristics of real samples and generate fake samples with a high degree of similarity to real samples, so that the discriminator cannot distinguish the authenticity of the input samples. The goal of the discriminator is to accurately distinguish the input samples from the real training samples or fake samples from the generator. As the number of training increases, the capabilities of the generator and the discriminator continue to improve. Eventually reaching a balance between the generator and discriminator, that is, when the discriminator has a strong discriminatory ability but cannot correctly determine the source of the data, a generator network model that can generate samples that conform to the characteristic distribution of real data is obtained. The generator takes a random vector as input and outputs the generated pseudo samples. The discriminator takes real samples and pseudo samples from generator as input and outputs 0 and 1. The output result is 0 for pseudo samples and 1 for real samples. According to the loss function of the discriminator and the generator, the gradient descent method is used to update the parameters of the network in the training of the generated confrontation neural network, so as to achieve the purpose of optimizing the network parameters of the model. The following will briefly introduce the optimization function of GANs:
(1)minGmaxDVG,D=Ex~prlogDx+Ex~pglog1−Dx,where x~pr represents the real sample and x~pg represents the pseudo sample generated by the generator. When optimizing the parameters of the discriminator, keep the generator parameters fixed. The objective function of the discriminator consists of two parts: (1) for all real samples, the discriminator should judge them as true samples, so that Dx is 1, which maximizes Ex~prlogDx; (2) for the pseudo sample from the generator, the discriminator judges it as a pseudo sample, makes Dx 0, and maximizes Ex~pglog1−Dx, that is to maximize VG,D. Generate parameter optimization, fix the parameters of the trained discriminator, and optimize the parameters of the generatorGz;θ. The goal of generator optimization parameters is to make the discriminator judge the output of the generator as a true sample; let Dx be 1, which minimizes Ex~pglog1−Dx. It can be seen from the training of GANs that the parameter update information of the generator comes from the result of the discriminator; that is, the distance between the data distribution and the model distribution is obtained by neural network fitting, which avoids the solution of the sample probability density function.


### 2.2. Network Structure and Parameter Setting

The pretrained generative adversarial neural network model proposed in this paper is used for digital core modeling. The network structure is shown in Figure1.Figure 1
Model structure diagram.In training, the generatorGz;θ through 5 transposed convolutional layers, a random noise z in a latent space is mapped to the image domain to generate a three-dimensional digital core. d is the dimension of random noise, which is set to 512 in this paper, and N0,1 represents a standard normal distribution with a mean value of 0 and a standard deviation of 1.
(2)z~N0,1d×1×1×1,(3)Gθ:z⟶R64×64×64.The discriminatorDx;φ maps the real core samples and the generated core samples to the interval [0,1] through 5 convolutional layers to complete the sample determination:
(4)Dφ:R64×64×64⟶0,1.The network structure parameters of the generator and the discriminator are shown in Table1.Table 1
Network structure parameter table.
LayerTypeConvolution kernelStridePaddingNormalizedActivation functionGenerator1ConvTransp3D(256,4,4,4)10YesReLU2ConvTransp3D(128,4,4,4)21YesReLU3ConvTransp3D(64,4,4,4)21YesReLU4ConvTransp3D(32,4,4,4)21YesReLU5ConvTransp3D(1,4,4,4)21NoTanhDiscriminator1Conv3D(16,4,4,4)21NoLeakyReLU2Conv3D(32,4,4,4)11YesLeakyReLU3Conv3D(64,4,4,4)11YesLeakyReLU4Conv3D(128,4,4,4)11YesLeakyReLU5Conv3D(1,4,4,4)10YesSigmoid


## 2.1. GANs

GANs are mainly composed of two parts, the generator network (G) and the discriminator network (D), which are in a state of confrontation during training [34]. The goal of the generator is to learn the data distribution characteristics of real samples and generate fake samples with a high degree of similarity to real samples, so that the discriminator cannot distinguish the authenticity of the input samples. The goal of the discriminator is to accurately distinguish the input samples from the real training samples or fake samples from the generator. As the number of training increases, the capabilities of the generator and the discriminator continue to improve. Eventually reaching a balance between the generator and discriminator, that is, when the discriminator has a strong discriminatory ability but cannot correctly determine the source of the data, a generator network model that can generate samples that conform to the characteristic distribution of real data is obtained. The generator takes a random vector as input and outputs the generated pseudo samples. The discriminator takes real samples and pseudo samples from generator as input and outputs 0 and 1. The output result is 0 for pseudo samples and 1 for real samples. According to the loss function of the discriminator and the generator, the gradient descent method is used to update the parameters of the network in the training of the generated confrontation neural network, so as to achieve the purpose of optimizing the network parameters of the model. The following will briefly introduce the optimization function of GANs:
(1)minGmaxDVG,D=Ex~prlogDx+Ex~pglog1−Dx,where x~pr represents the real sample and x~pg represents the pseudo sample generated by the generator. When optimizing the parameters of the discriminator, keep the generator parameters fixed. The objective function of the discriminator consists of two parts: (1) for all real samples, the discriminator should judge them as true samples, so that Dx is 1, which maximizes Ex~prlogDx; (2) for the pseudo sample from the generator, the discriminator judges it as a pseudo sample, makes Dx 0, and maximizes Ex~pglog1−Dx, that is to maximize VG,D. Generate parameter optimization, fix the parameters of the trained discriminator, and optimize the parameters of the generatorGz;θ. The goal of generator optimization parameters is to make the discriminator judge the output of the generator as a true sample; let Dx be 1, which minimizes Ex~pglog1−Dx. It can be seen from the training of GANs that the parameter update information of the generator comes from the result of the discriminator; that is, the distance between the data distribution and the model distribution is obtained by neural network fitting, which avoids the solution of the sample probability density function.


## 2.2. Network Structure and Parameter Setting

The pretrained generative adversarial neural network model proposed in this paper is used for digital core modeling. The network structure is shown in Figure1.Figure 1
Model structure diagram.In training, the generatorGz;θ through 5 transposed convolutional layers, a random noise z in a latent space is mapped to the image domain to generate a three-dimensional digital core. d is the dimension of random noise, which is set to 512 in this paper, and N0,1 represents a standard normal distribution with a mean value of 0 and a standard deviation of 1.
(2)z~N0,1d×1×1×1,(3)Gθ:z⟶R64×64×64.The discriminatorDx;φ maps the real core samples and the generated core samples to the interval [0,1] through 5 convolutional layers to complete the sample determination:
(4)Dφ:R64×64×64⟶0,1.The network structure parameters of the generator and the discriminator are shown in Table1.Table 1
Network structure parameter table.
LayerTypeConvolution kernelStridePaddingNormalizedActivation functionGenerator1ConvTransp3D(256,4,4,4)10YesReLU2ConvTransp3D(128,4,4,4)21YesReLU3ConvTransp3D(64,4,4,4)21YesReLU4ConvTransp3D(32,4,4,4)21YesReLU5ConvTransp3D(1,4,4,4)21NoTanhDiscriminator1Conv3D(16,4,4,4)21NoLeakyReLU2Conv3D(32,4,4,4)11YesLeakyReLU3Conv3D(64,4,4,4)11YesLeakyReLU4Conv3D(128,4,4,4)11YesLeakyReLU5Conv3D(1,4,4,4)10YesSigmoid


## 3. Digital Core Modeling

### 3.1. Sample Data and Training Process

The selected training data is the Berea sandstone X-CT scan sample of Imperial College London, the sample size is4003 voxels, and the resolution is 3 μm/voxel [46]. The sample image has been thresholded and only contains pores and skeleton parts. In order to ensure the number of training samples and reduce the amount of calculation in the model training process, the porosity is used as a constraint to find the representative volume element that can reflect the nature of the original sample. Figure 2(a) shows the variation of porosity with the size of the subsample. It can be seen that when the side length of the training sample is greater than 60 voxels, the porosity gradually stabilizes. Therefore, the training sample is set as a cube with a side length of 64 voxels, the sample interval is 32 voxels, and a total of 1331 subsamples are obtained. Figure 2(b) is a three-dimensional section view of an arbitrarily selected representative volume element.Figure 2
Representative volume element analysis and schematic diagram of Berea sandstone.
(a)
REV analysis results(b)
REV section viewPrevious scholars used generative adversarial neural networks to model digital cores and directly carried out network training on the basis of all training data. In the beginning, both the generator network and the discriminator network are networks initialized with parameters, and the latter two always keep the same training times until the generator network can obtain satisfactory results, and the training of the network is ended. The changes in this paper are mainly in the network training process. The specific process of using the pretrained generative adversarial neural networks for digital core modeling proposed in this paper is as follows:(1)
Randomly select 300 subsamples from the training sample as sample data for network pretraining, set the learning rate to 0.0003 and batch size to 16, use the Adam optimizer, and set the momentum to 0.5. There is no need to pay attention to the quality of the network during pretraining, so you can train hundreds of times. The number of pretrainings in this paper is set to 300(2)
After the pretraining is completed, use the discriminator pretrained 300 times to train the network. At this time, the generator in the network is initialized with parameters, and the discriminator is trained a certain number of times. Use the divided 1331 subsamples as training data and set the learning rate to 0.0005, batch size to 32, optimizer to Adam, and momentum to 0.5 to start network training. Through many experiments, it is found that when training 800 times, the model can achieve a better generation effect. At the same time, it is found that when the network model achieves a good generation effect using this method, continued training may cause the generation effect to deteriorate


### 3.2. Comparison of Modeling Effects

We implemented the training of digital core modeling with both the method presented in this paper and the previous methods, keeping the same training parameters and training times, and used the generator obtained by training 800 times to generate some digital core models of the same size as the original training core sample. A comparative analysis of the modeling effects of digital cores is carried out in various aspects.


#### 3.2.1. Image Appearance Comparison

The cross-section, three-dimensional section, outer surface, and pore structure of the generated sample and the real sample are compared and analyzed. Figure3 is a list of images of the real sample, Figure 4 is a list of core images obtained by normal training 800 times, and Figure 5 is a list of core images obtained by using a pretrained discriminator. It can be seen that using the pretrained discriminator, after the network model is trained 800 times, the pore shape and pore distribution have good consistency with the original training sample, and there is good connectivity between the pores, indicating the core generation effect better. Without pretraining, the core generated after 800 training is very different from the original training sample in terms of pore shape, distribution, and connectivity, indicating that the quality of the generated core is poor. It can be seen intuitively from the appearance that the generation effect of the neural network model without pretraining is very poor, so in the subsequent analysis, the digital core model obtained by the non-pretraining neural network is not analyzed.Figure 3
Real core images.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structureFigure 4
Digital core modeling based on no pretrained GANs.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structureFigure 5
Digital core modeling based on pretrained discriminator.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structure


#### 3.2.2. Morphological Parameters

Porosity and specific surface area are common morphological parameters in core comparison, which can quantitatively evaluate the morphological characteristics of cores. Porosity is used to reflect the degree of pore development of the rock and is the ratio of the sum of the volume of pore spaces in the rock to the total volume of the rock sample. The specific surface area refers to the ratio of the total area of the pores in the core to the total volume, which is a parameter related to the pore size. Randomly select 5 digital cores obtained by using the pretrained generative adversarial neural network, and calculate the porosity and specific surface area of these cores and the original cores. The calculation results are shown in Table2. It can be seen from Table 2 that the porosity and specific surface area of the generated core are very close to the real sample, indicating that the generated core samples are close to the training samples in nature, and the model generation effect is better.Table 2
Comparison of parameters between real core sample and generated core samples.
SamplesPorosity (%)Specific area (μm-1)Real core sample19.650.438Generated sample 119.510.387Generated sample 219.680.384Generated sample 319.500.389Generated sample 419.520.386Generated sample 519.550.385


#### 3.2.3. Two-Point Covariance Function

The two-point covariance functionS2r can be used to characterize the spatial structure of the core. The two-point covariance function can be expressed as follows:
(5)S2r=Px∈P,x+r∈Px,r∈Rd.That is, the probability is that the two pointsx and x+r separated by the separation distance r are located in the pore phase P. At the origin, S20 is equal to the porosity ∅; when r⟶∞, S2r will stabilize at ∅2. Since the core is anisotropic, we calculate the S2r of the training sample and the generated digital core along the x, y, and z coordinate axis directions and the radial average to evaluate the effect of the model.Figures6 and 7 are the comparison of radial covariance functions and covariance functions in x, y, and z directions, respectively. It can be seen from the figures that the covariance function of the generated core samples and the training sample in the radial direction and the x, y, and z directions has a good consistency, indicating that the spatial structure of the generated lithology is close to the training sample.Figure 6
Radial covariance functions of generated core samples and training sample.Figure 7
Comparison of covariance functions inx, y, and z directions.
(a)
Covariance function inx direction(b)
Covariance function iny direction(c)
Covariance function inz direction


## 3.1. Sample Data and Training Process

The selected training data is the Berea sandstone X-CT scan sample of Imperial College London, the sample size is4003 voxels, and the resolution is 3 μm/voxel [46]. The sample image has been thresholded and only contains pores and skeleton parts. In order to ensure the number of training samples and reduce the amount of calculation in the model training process, the porosity is used as a constraint to find the representative volume element that can reflect the nature of the original sample. Figure 2(a) shows the variation of porosity with the size of the subsample. It can be seen that when the side length of the training sample is greater than 60 voxels, the porosity gradually stabilizes. Therefore, the training sample is set as a cube with a side length of 64 voxels, the sample interval is 32 voxels, and a total of 1331 subsamples are obtained. Figure 2(b) is a three-dimensional section view of an arbitrarily selected representative volume element.Figure 2
Representative volume element analysis and schematic diagram of Berea sandstone.
(a)
REV analysis results(b)
REV section viewPrevious scholars used generative adversarial neural networks to model digital cores and directly carried out network training on the basis of all training data. In the beginning, both the generator network and the discriminator network are networks initialized with parameters, and the latter two always keep the same training times until the generator network can obtain satisfactory results, and the training of the network is ended. The changes in this paper are mainly in the network training process. The specific process of using the pretrained generative adversarial neural networks for digital core modeling proposed in this paper is as follows:(1)
Randomly select 300 subsamples from the training sample as sample data for network pretraining, set the learning rate to 0.0003 and batch size to 16, use the Adam optimizer, and set the momentum to 0.5. There is no need to pay attention to the quality of the network during pretraining, so you can train hundreds of times. The number of pretrainings in this paper is set to 300(2)
After the pretraining is completed, use the discriminator pretrained 300 times to train the network. At this time, the generator in the network is initialized with parameters, and the discriminator is trained a certain number of times. Use the divided 1331 subsamples as training data and set the learning rate to 0.0005, batch size to 32, optimizer to Adam, and momentum to 0.5 to start network training. Through many experiments, it is found that when training 800 times, the model can achieve a better generation effect. At the same time, it is found that when the network model achieves a good generation effect using this method, continued training may cause the generation effect to deteriorate


## 3.2. Comparison of Modeling Effects

We implemented the training of digital core modeling with both the method presented in this paper and the previous methods, keeping the same training parameters and training times, and used the generator obtained by training 800 times to generate some digital core models of the same size as the original training core sample. A comparative analysis of the modeling effects of digital cores is carried out in various aspects.


### 3.2.1. Image Appearance Comparison

The cross-section, three-dimensional section, outer surface, and pore structure of the generated sample and the real sample are compared and analyzed. Figure3 is a list of images of the real sample, Figure 4 is a list of core images obtained by normal training 800 times, and Figure 5 is a list of core images obtained by using a pretrained discriminator. It can be seen that using the pretrained discriminator, after the network model is trained 800 times, the pore shape and pore distribution have good consistency with the original training sample, and there is good connectivity between the pores, indicating the core generation effect better. Without pretraining, the core generated after 800 training is very different from the original training sample in terms of pore shape, distribution, and connectivity, indicating that the quality of the generated core is poor. It can be seen intuitively from the appearance that the generation effect of the neural network model without pretraining is very poor, so in the subsequent analysis, the digital core model obtained by the non-pretraining neural network is not analyzed.Figure 3
Real core images.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structureFigure 4
Digital core modeling based on no pretrained GANs.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structureFigure 5
Digital core modeling based on pretrained discriminator.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structure


### 3.2.2. Morphological Parameters

Porosity and specific surface area are common morphological parameters in core comparison, which can quantitatively evaluate the morphological characteristics of cores. Porosity is used to reflect the degree of pore development of the rock and is the ratio of the sum of the volume of pore spaces in the rock to the total volume of the rock sample. The specific surface area refers to the ratio of the total area of the pores in the core to the total volume, which is a parameter related to the pore size. Randomly select 5 digital cores obtained by using the pretrained generative adversarial neural network, and calculate the porosity and specific surface area of these cores and the original cores. The calculation results are shown in Table2. It can be seen from Table 2 that the porosity and specific surface area of the generated core are very close to the real sample, indicating that the generated core samples are close to the training samples in nature, and the model generation effect is better.Table 2
Comparison of parameters between real core sample and generated core samples.
SamplesPorosity (%)Specific area (μm-1)Real core sample19.650.438Generated sample 119.510.387Generated sample 219.680.384Generated sample 319.500.389Generated sample 419.520.386Generated sample 519.550.385


### 3.2.3. Two-Point Covariance Function

The two-point covariance functionS2r can be used to characterize the spatial structure of the core. The two-point covariance function can be expressed as follows:
(5)S2r=Px∈P,x+r∈Px,r∈Rd.That is, the probability is that the two pointsx and x+r separated by the separation distance r are located in the pore phase P. At the origin, S20 is equal to the porosity ∅; when r⟶∞, S2r will stabilize at ∅2. Since the core is anisotropic, we calculate the S2r of the training sample and the generated digital core along the x, y, and z coordinate axis directions and the radial average to evaluate the effect of the model.Figures6 and 7 are the comparison of radial covariance functions and covariance functions in x, y, and z directions, respectively. It can be seen from the figures that the covariance function of the generated core samples and the training sample in the radial direction and the x, y, and z directions has a good consistency, indicating that the spatial structure of the generated lithology is close to the training sample.Figure 6
Radial covariance functions of generated core samples and training sample.Figure 7
Comparison of covariance functions inx, y, and z directions.
(a)
Covariance function inx direction(b)
Covariance function iny direction(c)
Covariance function inz direction


## 3.2.1. Image Appearance Comparison

The cross-section, three-dimensional section, outer surface, and pore structure of the generated sample and the real sample are compared and analyzed. Figure3 is a list of images of the real sample, Figure 4 is a list of core images obtained by normal training 800 times, and Figure 5 is a list of core images obtained by using a pretrained discriminator. It can be seen that using the pretrained discriminator, after the network model is trained 800 times, the pore shape and pore distribution have good consistency with the original training sample, and there is good connectivity between the pores, indicating the core generation effect better. Without pretraining, the core generated after 800 training is very different from the original training sample in terms of pore shape, distribution, and connectivity, indicating that the quality of the generated core is poor. It can be seen intuitively from the appearance that the generation effect of the neural network model without pretraining is very poor, so in the subsequent analysis, the digital core model obtained by the non-pretraining neural network is not analyzed.Figure 3
Real core images.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structureFigure 4
Digital core modeling based on no pretrained GANs.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structureFigure 5
Digital core modeling based on pretrained discriminator.
(a)
Cross-section(b)
Three-dimensional section(c)
Outer surface(d)
Pore structure


## 3.2.2. Morphological Parameters

Porosity and specific surface area are common morphological parameters in core comparison, which can quantitatively evaluate the morphological characteristics of cores. Porosity is used to reflect the degree of pore development of the rock and is the ratio of the sum of the volume of pore spaces in the rock to the total volume of the rock sample. The specific surface area refers to the ratio of the total area of the pores in the core to the total volume, which is a parameter related to the pore size. Randomly select 5 digital cores obtained by using the pretrained generative adversarial neural network, and calculate the porosity and specific surface area of these cores and the original cores. The calculation results are shown in Table2. It can be seen from Table 2 that the porosity and specific surface area of the generated core are very close to the real sample, indicating that the generated core samples are close to the training samples in nature, and the model generation effect is better.Table 2
Comparison of parameters between real core sample and generated core samples.
SamplesPorosity (%)Specific area (μm-1)Real core sample19.650.438Generated sample 119.510.387Generated sample 219.680.384Generated sample 319.500.389Generated sample 419.520.386Generated sample 519.550.385


## 3.2.3. Two-Point Covariance Function

The two-point covariance functionS2r can be used to characterize the spatial structure of the core. The two-point covariance function can be expressed as follows:
(5)S2r=Px∈P,x+r∈Px,r∈Rd.That is, the probability is that the two pointsx and x+r separated by the separation distance r are located in the pore phase P. At the origin, S20 is equal to the porosity ∅; when r⟶∞, S2r will stabilize at ∅2. Since the core is anisotropic, we calculate the S2r of the training sample and the generated digital core along the x, y, and z coordinate axis directions and the radial average to evaluate the effect of the model.Figures6 and 7 are the comparison of radial covariance functions and covariance functions in x, y, and z directions, respectively. It can be seen from the figures that the covariance function of the generated core samples and the training sample in the radial direction and the x, y, and z directions has a good consistency, indicating that the spatial structure of the generated lithology is close to the training sample.Figure 6
Radial covariance functions of generated core samples and training sample.Figure 7
Comparison of covariance functions inx, y, and z directions.
(a)
Covariance function inx direction(b)
Covariance function iny direction(c)
Covariance function inz direction


## 4. Conclusions

In order to solve the problem of tens of thousands of times required for digital core generation training of the generative adversarial neural networks, this paper proposes a pretrained discriminator to train the digital core modeling of the generative adversarial neural network. The method proposed in the paper was used to train the network about 800 times, and a number of digital core models were generated. Through various analyses of the generated core samples, the following conclusions can be drawn:(1)
Comparing generated cores and real core from visual and quantitative analysis, it can be seen that the generated digital cores have good consistency with the real core, which proves that the use of generative adversarial neural networks can generate high-quality digital core models(2)
The method in this paper can achieve a good modeling effect in about 800 network trainings, which proves the reliability of the method. Under the premise of ensuring the effect of model generation, the training times of the network can be greatly reduced, thereby greatly shortening the training time of the network



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*Source: 9159242-2022-09-05.xml*