https://orcid.org/0000-0002-2515-815X
SUMMARY
Digital rock physics (DRP) is important for characterizing the pore characteristics and petrophysical parameters of rocks from a microscopic perspective. Among the digital rock reconstruction methods, the most widely used is the computer tomography (CT) scan method. However, the pore structure of rocks has multiscale features, and CT scan images with a single resolution cannot completely describe the pore structure characteristics of rocks. In this paper, we analysed reconstructed multisource and multiscale 3-D digital rocks based on the cycle-consistent generative adversarial network (CycleGAN) method. This method fully integrates the advantages of the large field of view of low-resolution images and the high-precision features of high-resolution images. To investigate the performance of the method, two sequences of CT scan images of a sandstone (sample A) and a carbonate rock (sample B) collected from oilfields were studied. Moreover, due to the high resolution of scanning electron microscope (SEM) images, we also constructed 3-D digital rocks with different resolutions of the Berea sandstone based on the combination of CT scan images and SEM images. Finally, the statistical properties and absolute permeabilities were calculated to evaluate the accuracies of reconstructed multisource and multiscale 3-D digital rocks. The results show that the reconstructed multiscale digital rocks based on the CycleGAN method have good accuracy in terms of statistical properties and petrophysical properties. Considering the computational cost and computational accuracy, for high- and low-resolution CT scan images and CT scan images with SEM images as training image data sources, we suggest that the resolution of the reconstructed multiscale digital cores is 4–8 times and 4–16 times higher than that of the low-resolution CT scan images, respectively. The findings of our research will be helpful in gaining insight into the petrophysical properties of heterogeneous rocks.
1 INTRODUCTION
Digital rock physics (DRP) has become an important tool for characterizing the pore spaces of reservoirs and studying petrophysical properties (Andrä et al. 2013a,b; Zhu et al. 2019; Wu et al. 2020; Yang et al. 2021; Cai et al. 2022). Among the 3-D digital rock construction methods, CT scan technology (Blunt et al. 2013; Nie et al. 2019; Fang et al. 2020; Sun et al. 2021) is currently one of the most commonly used methods. However, there is an inherent conflict between the field of view and resolution for CT scan technology. At a low resolution, a large physical size rock can be obtained, but it is difficult to characterize the microstructural information of the rock. At a high resolution, microscopic pore features can be obtained, but the representation of the rock is poor because of its small physical size. Moreover, with the exploration and development of unconventional oil and gas resources (Ali et al. 2023; Baouche et al. 2023), it is difficult to fully characterize the nanopores and micropores only by using the CT scan technique. Other advanced techniques, such as scanning electron microscopy (SEM; Wu et al. 2019; Xu et al. 2020; Golsanami et al. 2022) and focused ion beam scanning electron microscopy (FIB-SEM; Jiang et al. 2016; Zhao et al. 2020, 2022), have also been applied to characterize pore structures. Due to the broad range of pore size distributions and multiscale characteristics of a rock, pore structures cannot be fully characterized by one source of data.
Multisource data are needed for constructing a multiscale 3-D digital rock. Gerke et al. (2015) proposed a general technique capable of utilizing multiscale spatial information and reconstructing a multiscale shale rock model with randomly generated 2-D images. Yao et al. (2013) reconstructed low-resolution 3-D digital rocks by an annealing simulation algorithm based on 2-D SEM images at different resolutions, and then they reconstructed high-resolution 3-D digital rocks by the Markov Chain Monte Carlo method. Finally, they reconstructed multiscale carbonate digital rocks by combining low-resolution and high-resolution digital rocks. Tahmasebi et al. (2015) generated multiscale 3-D shale digital rocks using 2-D SEM images with different resolutions. Liu et al. (2017) investigated pore structures from multiscale and multimineral digital rocks constructed by combining CT, SEM and energy dispersive spectroscopy techniques. Cui et al. (2020) constructed a multiscale multicomponent digital rock of sandstone by aligning and analysing CT scan images with different resolutions. However, the multiscale digital rocks constructed by these methods only considered the binarized pore space and did not investigate the greyscale features of the images.
In recent years, several deep learning methods have been applied to multiscale digital rock reconstructions (Wang et al. 2021). The super resolution convolutional neural network (SRCNN; Dong et al. 2014; Wang et al. 2019), enhanced deep super resolution (EDSR; Lim et al. 2017; Jackson et al. 2022), and super resolution generative adversarial network (SRGAN; Ledig et al. 2017; Wang et al. 2020) were used for CT image enhancement. The aforementioned studies were based on supervised learning methods, aiming to use a large number of aligned high-resolution images and low-resolution images as training data to improve the quality of low-resolution images. In practical applications, it is difficult to obtain a large number of aligned high- and low-resolution images of rocks. As an unsupervised learning method, the cycle-consistent generative adversarial network method (CycleGAN; Chen et al. 2020) can be used for training unpaired CT images, providing great operational convenience for improving the quality of low-resolution images. Moreover, the cycle-in-cycle generative adversarial network method (CinCGAN; Yuan et al. 2018; Niu et al. 2020) is capable of denoising and deblurring images and generating high-quality super-resolution images. The above methods can enhance the information of low-resolution CT images and basically achieve image fusion of different resolutions. Liu & Mukerji (2022) used style-based GAN and CycleGAN for the integration of pore structures of images from CT and SEM images to reconstruct multiscale digital rocks.
Past studies have only used subsampling high-resolution CT images to acquire low-resolution CT images and used them for validation of super-resolution studies. However, there is a large difference between the real low-resolution CT images and the low-resolution CT images obtained by the subsampling method (Fig. 1). Affected by the CT scanning equipment, real low-resolution CT images often have a large loss of information. In other words, relying on super-resolution techniques for fusion of CT images at different resolutions is not enough; what needs to be considered is how to convert existing low-resolution (LR) CT images into high-resolution (HR) CT images, or even SEM images, that are indistinguishable from real images. In addition, even if the images have the same resolution and size, the information displayed by the images can vary widely. For example, although Figs 1(a) and (c) have the same number of pixels, it is clearly evident that Fig. 1(c) contains more information. Therefore, how to present more information with the least number of pixels in an image is the key issue to be considered in multiscale digital rock reconstructions.
2-D image of a rock. (a) Real LR image, 162 × 162 pixels, 16 μm pixel−1. (b) Real HR image, 648 × 648 pixels, 4 μm pixel−1. (c) LR image obtained by HR image, 162 × 162 pixels, 16 μm pixel−1.
This study attempts to discuss the effectiveness of reconstructing multisource and multiscale 3-D digital rocks based on the cycle-consistent generative adversarial network method. Two sequences of CT scan images of a homogeneous sandstone and a heterogeneous carbonate rock were first studied. Then, CT scan images and SEM images were used to reconstruct the multiscale 3-D Berea sandstone. Finally, the statistical properties and absolute permeabilities were calculated to evaluate the accuracies of reconstructed multisource and multiscale 3-D digital rocks.
2 METHODOLOGIES
2.1 Basic theory of CycleGAN
In practice, it is difficult to obtain a large number of aligned real high- and low-resolution images, which can greatly increase the workload of the study. Therefore, we choose CycleGAN for conversion between high- and low-resolution images so that a large number of unaligned high- and low-resolution images can be used for training.
CycleGAN is a well-known unpaired image conversion algorithm. Fig. 2 shows the schematic diagram of CycleGAN (Zhu et al. 2017), in which, given a data domain X in an image set and another image set in domain Y, the images in a domain X are transformed to the target domain Y. CycleGAN consists of two generators (G1 and G2) and two discriminators (DX and DY). Specifically, the model consists of two mappings G1: X→Y and G2: Y→X. DX is used to distinguish image x from the transformed image G2(y), and DY is used to distinguish y from G1(x). Cycle consistency loss is used to prevent mappings G1 and G2 learned by the model from contradicting each other. This ensures consistency between G1 and G2, as shown in Figs 2(b) and (c). Due to the good application of CycleGAN in unpaired image conversion, we try to apply it to convert rock images from low to high resolution.
![(a) Two mapping functions of CycleGAN (G1: X → Y and G2: Y → X) and two associated adversarial discriminators DY and DX. (b) Forward cycle-consistency loss: G2[G1(x)] ≈ x. (c) Backward cycle-consistency loss: G1[G2(x)] ≈ y.](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/gji/235/1/10.1093_gji_ggad271/1/m_ggad271fig2.jpeg?Expires=1749128075&Signature=zcK2Mc4a~59w9AEoR2bwAUjA1pAdQOHKJFQE8NFuYrIWY4ARkSqWIlk2k5EvVJmvk8NjEWEgECBYSfXuGPUdcxn~NXinclti7ZxmHKKPOCzHnj5Pztqi1Q3hwrjFOr9ZyLwbp88ZM5CpW8leCh5Q5~LIMCGeGhRFBDJcHi4YwkwGDAVI9F4lNM4kCRV94yU90hqlJ5mruCLO5GI285l1nxiiUjNvwTYPK9NDO0hGcROeKYBaPbz7jCuGd8hngKlqiVCc~q-LtjWhFRxQ7nD5s0cjE1NVK-IPDKsGBJmUFV8QWsK9p8hB9BGf1Nqziu~l8aR5SB8y3kZwrZ9OS1RegQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a) Two mapping functions of CycleGAN (G1: X → Y and G2: Y → X) and two associated adversarial discriminators DY and DX. (b) Forward cycle-consistency loss: G2[G1(x)] ≈ x. (c) Backward cycle-consistency loss: G1[G2(x)] ≈ y.
Similarly, the CycleGAN method used for digital rock reconstruction consists of four networks (Fig. 3), including a generator G1 that converts LR images to HR images, a generator G2 that converts HR images to LR images, a discriminator DY that aims to make G1 generate more realistic HR images, and a discriminator DX that encourages G2 to generate more realistic LR images. Unaligned high- and low-resolution images are used to train four networks, and adversarial loss is applied to two mapping functions (Goodfellow et al. 2014), which is expressed as follows:
The framework of CycleGAN: G1 and G2 are two generators that predict the HR image from the LR image and recover the LR image from the HR image, respectively; DY and DX are two discriminators corresponding to G1 and G2, respectively.
The two mapping functions should be cyclically consistent. For each LR in domain X, it should be recoverable to the initial image, that is x→G1(x)→G2(G1(x)) ≈ x, which is called forward circular consistency. Similarly, for each HR in domain Y, the reverse cyclic consistency should be satisfied, i.e. y→G2(x)→G1(G2(x)) ≈ y. The above behaviour is implemented using cycle-consistency loss (Lcyc).
In addition, the training process uses identity mapping loss, which constrains the two generators to behave as identity mapping. The process uses real samples from the target domain as input to the generator, which is given by:
The sizes of the HR image and LR image are not the same, which leads to different sizes of input and output images for the two generators. Therefore, the real images entered in this process need to be preprocessed. The HR or LR images are resampled to the size of the generator input image, and then the images are input to the generator for processing. The identity mapping loss is expressed as:
where yres and xres are the resampled HR and LR images, respectively. The total loss is given by:
where λ1, λ2, λ3 control the relative importance of the three terms on the right-hand side of the equal sign. The final goal is to solve for two generators.
Fig. 4 shows the network architectures of the two generators G1 and G2. Since the two sets of images input to the model are of different sizes, the architectures of G1 and G2 are also different. Nine residual blocks are used in both G1 and G2, and each residual block consists of two convolutional layers and an intermediate activation layer. G1 is used to convert LR images to HR images. In addition, upsampling and convolutional layers are added so that the generated HR image has the same size as the real HR image. For G2, the upsampling part is removed so that the generated LR image has the same size as the real LR image. The two discriminators DY and DX adopt essentially the same architectures as described by Zhu et al. (2017).
Network architectures of generators. (a) Generator G1. (b) Generator G2.
2.2 Multisource rock data collection
To reconstruct multisource and multiscale 3-D digital rocks, we used nanoVoxel-2000 CT scanning equipment to scan the rocks at different resolutions to obtain HR and LR images. One sandstone and one carbonate CT scan images with different resolutions were studied. The sandstone (sample A) was collected from the Member 2, Upper Permian Shihezi Formation in Huagu area, Bohai Bay Basin, China (Figs 5a and b), with a measured porosity of 8.11 per cent and a permeability of 0.1754 mD. It is a grey blocky sandstone without layering. Analysis of the cast thin sections revealed that the pore space is predominantly composed of intergranular micropores, with occasional secondary dissolution pores and interparticle pores observed locally. The mineral analysis was conducted on the advanced mineral identification and characterization equipment (AmicSCAN) at the Institute of Geology and Geophysics, Chinese Academy of Sciences. This equipment determines the distribution of various minerals based on X-ray energy spectra. The analysis reveals that the main components of sample A are quartz (78.84 per cent) and illite (17.78 per cent), with small amounts of other minerals such as rutile, zircon and others. For the CT scan experiments, the resolution of the HR image was 4.0 μm, while the resolution of the LR image was 16.0 μm. In our research, 3024 images were taken from each of the HR and LR images for training, where the size of the HR images was 256 × 256 and the size of the LR images was 64 × 64. The carbonate (sample B) was collected from the Member 4, Upper Sinian Dengying Formation in Gaoshiti–Moxi area, Sichuan Basin, China (Figs 5a and c), with a measured porosity of 4.67 per cent and a permeability of 0.0106 mD. It is a brown crystalline clotted limestone, with 95.20 per cent of its mineral composition being dolomite (matrix dolomite and crystalline dolomite), along with a small amount of other minerals. In addition, the pores are primarily composed of intercrystalline pores and dissolved vugs. The resolutions of its HR and LR images were 3.0 and 24.0 μm, respectively. A total of 3528 images were taken from each of the HR and LR images for training, where the size of the HR images was 512 × 512 and the size of the LR images was 64 × 64. In the above data set, although aligned HR and LR images were available, they were used for testing rather than training (Fig. 6).
Geographic locations and stratigraphic columns of two study areas. (a) Geographic locations of samples A and B, with the sampling locations indicated by red dots. (b) and (c) Positions of the two samples on the stratigraphic columns, indicated by yellow star symbols.
Aligned HR and LR images of two samples. (a)–(b) LR sample A images. (c)–(d) HR sample A images. (e)–(f) LR sample B images. (g)–(h) HR sample B images.
To further test the adaptability of the CycleGAN-based 3-D digital rock reconstruction method to data from different sources. A Berea sandstone with 17.3 per cent porosity and 109.85 mD permeability was collected for CT scanning and ion polishing SEM scanning. The solid matrix of this Berea sandstone is composed of 74.68 per cent quartz, 4.14 per cent feldspar, 3.10 per cent kaolinite, 1.69 per cent illite, 1.43 per cent albite, as well as small amounts of muscovite, rutile, zircon and other minerals. The equipment used for SEM scanning was the FEI QEMSCAN 650F, and CT scan images with 8.0 μm resolution and SEM images with 0.1 μm resolution were obtained. Obviously, the size difference between these two sets of images is too large for us to support such a large computational volume (80 GB of memory on the workstation used in this study, and the GPU model is NVIDIA GeForce RTX 3090 with 24.0 GB memory). Moreover, the high resolution provides information that is not always needed; for example, in the present numerical simulation, some of the microstructures can be ignored. Therefore, we compressed the SEM images by decimating pixels to narrow the gap with the CT images, as shown in Fig. 7. From left to right are images of the rock at different resolutions. The leftmost column shows the CT scan images with a resolution of 8.0 μm. On the right are SEM images with resolutions compressed to 2.0, 1.0 and 0.5 μm. When the image resolution is reduced from 0.5 to 1.0 μm, only minor changes occur, such as blurring of the internal boundaries of the clay minerals (kaolinite) and the micropores of the quartz particles. The clay minerals and microfracture boundaries also become blurred when the image resolution is reduced to 2.0 μm. In this study, 4220 different CT images and SEM images were selected separately to evaluate the effect of reconstructed digital rocks at different resolutions. In this study, the SEM images were cropped from a large SEM image with a size of 90 542 × 84 996 pixels. Each individual SEM image had dimensions of 1024 × 1024 pixels. The cropping size was determined based on the magnification of the images and the capabilities of our computer system.
Berea images. From left- to right-hand side, CT images with 8.0 μm resolution (size of 64 × 64 pixels), SEM images with 2.0 μm resolution (size of 256 × 256 pixels), SEM images with 1.0 μm resolution (size of 512 × 512 pixels) and SEM images with 0.5 μm resolution (size of 1024 × 1024 pixels).
3 RESULTS AND DISCUSSION
3.1 Digital rocks reconstructed by two sequences of CT scan images
Sandstone from the Bohai Bay Basin (sample A) was first studied. To minimize the introduction of artefacts, a fourfold interpolation of the LR image in the vertical direction was performed, and the reconstructed 3-D digital rock had the same resolution in all three directions. For example, for a cubic low-resolution reconstructed sandstone with a side length of 162 voxels, we increase the quantity in the Z-axis fourfold to match the size of the high-resolution reconstruction. We then reconstruct the CycleGAN-based digital rocks, which ensures the highest possible quality of the reconstruction results. Figs (a) and (b) show the 2-D LR and HR real slices, respectively. Figs 8 (e) and (f) show their related reconstructed 3-D digital rocks. To compare the effectiveness of the CycleGAN method of reconstructing 3-D digital rocks, we used the SRGAN method to perform super-resolution reconstruction of LR images. The SRGAN-based digital rock was obtained by training with aligned LR and HR images. It is obvious that the results of SRGAN (Figs 8c and g) do not improve the overall image quality much, although they increase the sharpness of the boundaries of the rock components. For clay-bearing sandstones with complex minerals, the error introduced by the CT scanning instrument makes the difference between the HR image and the LR image much larger than four times (e.g. Fig. 1). Figs 8(d) and (h) show the results of reconstructing multiscale digital rock using the CycleGAN method, and it can be visualized that the results obtained by this method are closer to the real HR images. The black part, dark part and light part represent pores, illite and quartz, respectively. Therefore, the CycleGAN-based digital rock construction method not only makes the boundary between minerals and pores clearer but also restores the structure of pores and minerals more realistically, which indicates that the method is reliable for the fusion of HR and LR images.
Sample A images. (a) 2-D LR real slice (size of 162 × 162 pixels). (b) 2-D HR real slice (size of 648 × 648 pixels). (c) SRGAN generated slice (size of 648 × 648 pixels). (d) CycleGAN generated slice (size of 648 × 648 pixels). (e)–(h) 3-D digital rocks constructed from slices (a)–(d), respectively.
Then, we studied the heterogeneous carbonate (sample B) from the Sichuan Basin. The difference in resolution between real LR images (Figs 9a and d) and real HR images (Figs 9b and e) was eight times. Figs 9(c) and (f) are the generated 2-D carbonate image and the corresponding 3-D digital rock based on the CycleGAN method. Among these figures, the black part, dark part and light part are the pore, crystalline dolomite and matrix dolomite, respectively. It is obvious that the HR image generated from the LR image is very similar to the real HR image. Therefore, even if the high- and low- resolutions of CT scan images differ by a factor of eight, the CycleGAN method can be well applied to achieve multiscale 3-D digital rock reconstruction.
Sample B images. (a) 2-D LR real slice (size of 73 × 73 pixels). (b) 2-D HR real slice (size of 584 × 584 pixels). (c) CycleGAN generated slice (size of 584 × 584 pixels). (d)–(f) 3-D digital rocks constructed from slices (a)–(c), respectively.
Based on the results in Figs 8 and 9, it can be concluded that the CycleGAN-based multiscale 3-D digital rock reconstruction method is valid for two sequences of CT scan images with a fourfold or eightfold difference in resolution. The HR images generated using LR images are very close to the real HR images. In addition, this study also demonstrates that the multiscale 3-D digital rock reconstruction method is feasible for homogeneous sandstones and heterogeneous carbonates.
3.2 Digital rocks reconstructed by CT scan images and SEM images
In general, SEM images have a higher resolution than CT scan images, but the disadvantage of SEM images is that they cannot characterize the 3-D spatial features of rocks. Therefore, we combined CT scan images (LR images) and SEM images (HR images) to reconstruct multisource and multiscale 3-D digital rocks.
The CycleGAN method was applied to the collected Berea sandstone, as shown in Fig. 10. From top to bottom are the 3-D visualization of different digital rock models, the 2-D greyscale cross section and the binarized segmented image. The first column shows the original CT scan results, and the next three columns are the 3-D digital rocks reconstructed by applying the CycleGAN method based on the difference in resolution between the CT scan and SEM image of 4.0 times (×4), 8.0 times (×8) and 16.0 times (×16), respectively. The original CT-based 3-D digital rock contains 200 × 200 × 200 voxels, and the three CycleGAN-based 3-D digital rocks contain 400 × 400 × 400 voxels, 800 × 800 × 800 voxels and 1600 × 1600 × 1600 voxels.
3-D Berea digital rocks. From left- to right-hand side, the CT-based model and CycleGAN-based models for multisource image data differ by factors of 4.0, 8.0 and 16.0, respectively.
The reconstructed 3-D digital rocks add more detail as the SEM image resolution differs more from the CT scan image resolution. For example, compared with other models, when the SEM image resolution differs from the CT scan image resolution by a factor of 16.0, the reconstructed 3-D digital rock contains more details, including dissolved pores and more feldspars (white part). With the increased resolution of the applied SEM images, the reconstructed results compensate for the missing information in the CT images and enhance the connectivity of the pores. Since the aligned HR images were not available, we combined the results of petrophysical experiments and digital rock analysis techniques to validate the reconstructed 3-D digital rock.
3.3 Evaluations of the CycleGAN-based digital rocks
The goal of CycleGAN is to generate high-resolution images with statistics similar to the original measured high-resolution images. As sample A and sample B were obtained from high- and low-resolution CT scan images used as training image data, the statistical characteristics of these two rock were analysed. To characterize the structures of reconstructed CycleGAN-based digital rocks, a representative elementary volume (REV) analysis of each digital rock was performed. A voxel was arbitrarily selected within the digital rock, and a cube with a side length of L was constructed with this voxel as the centre. A specific rock component was chosen as the marker, and its content within the cube was calculated. By gradually increasing the side length L, the relationship between the marker content and the cube size was determined. For sample A, clay mineral (mainly illite) was selected as the marker due to its high content, while for sample B, crystalline dolomite was chosen as the marker. The REV analysis of sample A and sample B are shown in Figs 11(a) and (b). As the normalized length increases, both the marker content of sample A and sample B show instability, indicating a dispersed distribution of the markers in 3-D rock space. The marker content obtained from low-resolution CT scanning of the digital cores differs significantly from that of high-resolution CT scanning. However, for both sample A and sample B, the marker content of the 3-D digital rocks generated using CycleGAN closely matches the results obtained from high-resolution CT scanning. The autocorrelation function (ACF) represents the probability that two points in the 3-D digital rock that are separated by H are both in the marker space. Figs 11 (c) and (d) show the ACF of sample A and sample B. As the normalized length increases, the autocorrelation function gradually approaches stability. Moreover, ACF values of the CycleGAN-based digital rocks show similarities with the results obtained from high-resolution CT-based digital rocks. Therefore, both REV analysis and ACF demonstrate the excellent performance of CycleGAN in reconstructing 3-D digital rock. It should be noted that due to the low porosity of sample A and sample B, even the 3-D digital rocks reconstructed from high-resolution CT scanning images do not possess connected pores, making it impossible to simulate permeability. The effectiveness of CycleGAN in reconstructing these two digital rocks cannot be analysed based on permeability.
Evaluation of digital rocks with different samples. (a)–(b) REV analysis of sample A and sample B, respectively. (c)–(d) ACF of sample A and sample B, respectively.
It is also necessary to characterize the statistics of the constructed 3-D digital rocks with different resolutions of the Berea sandstone based on the combination of CT scan images and SEM images. Pore was selected as the marker in the REV analysis. The normalized length versus porosity curves for different digital rock models are shown in Fig. 12(a). With the increase in normalized length, the porosity of digital rocks of different resolutions gradually tends to be stable. The ACF of digital rocks with different resolutions were calculated separately, as shown in Fig. 12(b), and the digital rocks with different resolutions have similar ACF values. The pore distribution of the generated multiscale digital rock is similar to that of the original model. Due to the low accuracy of CT scans, micropores smaller than resolution were difficult to identify. Therefore, the porosity of the generated multiscale digital rock is larger than that of the digital rock constructed by the CT scan.
Evaluation of digital rocks with different resolutions of Berea. (a) REV analysis, (b) ACF, (c) LPD and (d) APP.
As the Berea sandstone has a relatively high porosity (17.3 per cent), two additional methods were analysed, including the local porosity theory and the local seepage probability function. The main idea of local porosity theory is to reflect the porosity characteristic parameters of the rock by measuring physical quantities such as porosity in a small selected area inside the digital rock and then evaluate the validity and accuracy of the constructed rock. Let K(r, L) be the cube inside the digital rock centred at the end of vector r with side length L. The porosity ϕ(r, L) is defined as (Liu et al. 2009):
where V(G) is the volume of a certain set G ⸦ Rd and P denotes the pore space. The local porosity distribution (LPD) function is defined as:
where m is the number of measured units K(r, L) in the 3-D digital rock, and δ(x) is the Dirac function. Local porosity theory was also used to evaluate the reconstructed digital rocks with different resolutions. Fig. 12(c) shows the local porosity distribution function when L is equal to one-fourth of the digital rock side length. The comparison shows that the local porosity distribution functions of the three newly reconstructed digital rocks are basically the same. However, the local porosity distribution function of the digital rock reconstructed from the original CT images is shifted to the left, indicating that the reconstructed digital rocks have higher porosities.
The local seepage probability function λα(ϕ, L) characterizes the pore connectivity of a measurement unit with side length L and porosity ϕ. The seepage characteristic function is defined as (Liu et al. 2009).
The measured unit K(r, L) is permeable in the x, y and z directions, which means that the fluid can permeate from one side of the unit to the other side. Ʌ3=1 means that K(r, L) is permeable in the x, y and z directions. The local percolation probability function is expressed as:
λα(ϕ, L) defines the proportion of K(r, L) with permeability in the α direction to all measured units with porosity ϕ and side length L. Based on the local seepage probability function, the average permeability probability (APP) function of the rock, pα(L), is obtained by using the weighted integration of the LPD.
The APP function characterizes the pore connectivity by measuring the permeability of a measured unit within the rock in a specific direction. The degree of pore connectivity of a digital rock is proportional to the slope of the APP curve, which means that the faster the APP function curve rises, the better the pore connectivity of the rock is. Fig. 12(d) shows the APP functions of digital rocks with different resolutions, and the results are the average of the APP function in the x, y and z directions. Compared with the digital rocks reconstructed from the original CT images, the CycleGAN-based digital rocks have better pore connectivity. In addition, the higher the resolution is, the better the connectivity.
To quantitatively evaluate the accuracy of CycleGAN-based reconstructed digital rocks, the geometric mean values of ACF, LPD and APP were calculated and are shown in Table 1. A1–A4 are the CT-based digital rocks and the CycleGAN-based digital rocks with 4 times, 8 times and 16 times higher resolutions, respectively. As the resolution increases, the geometric mean values of the ACF and LPD gradually increase, and the geometric mean values of the APP gradually decrease. When the resolution of the 3-D digital rock is increased to 8 times, the above parameters tend to be stable, and the difference is smaller than that at 16 times. In other words, when the resolution of the Berea digital rock is increased to 8 times, it basically meets the requirements of pore structure analysis and numerical simulations. By combining the above methods, the accuracy of the multisource and multiscale digital rock reconstruction method is verified from the perspective of the statistical properties of 3-D digital rocks.
Statistical characteristics and petrophysical properties of Berea with different resolutions.
| ACF | LPD | APP | Por. (per cent) | Por. RE (per cent) | Perm. (mD) | Perm. RE (per cent) | |
|---|---|---|---|---|---|---|---|
| A1 | 0.2041 | 0.1136 | 0.3508 | 13.2 | 23.70 | 33.31 | 69.68 |
| A2 | 0.2193 | 0.1411 | 0.3087 | 15.7 | 9.25 | 74.67 | 32.03 |
| A3 | 0.2340 | 0.1579 | 0.2981 | 17.0 | 1.73 | 98.53 | 10.30 |
| A4 | 0.2331 | 0.1544 | 0.2912 | 17.6 | 1.73 | 107.13 | 2.48 |
| ACF | LPD | APP | Por. (per cent) | Por. RE (per cent) | Perm. (mD) | Perm. RE (per cent) | |
|---|---|---|---|---|---|---|---|
| A1 | 0.2041 | 0.1136 | 0.3508 | 13.2 | 23.70 | 33.31 | 69.68 |
| A2 | 0.2193 | 0.1411 | 0.3087 | 15.7 | 9.25 | 74.67 | 32.03 |
| A3 | 0.2340 | 0.1579 | 0.2981 | 17.0 | 1.73 | 98.53 | 10.30 |
| A4 | 0.2331 | 0.1544 | 0.2912 | 17.6 | 1.73 | 107.13 | 2.48 |
Statistical characteristics and petrophysical properties of Berea with different resolutions.
| ACF | LPD | APP | Por. (per cent) | Por. RE (per cent) | Perm. (mD) | Perm. RE (per cent) | |
|---|---|---|---|---|---|---|---|
| A1 | 0.2041 | 0.1136 | 0.3508 | 13.2 | 23.70 | 33.31 | 69.68 |
| A2 | 0.2193 | 0.1411 | 0.3087 | 15.7 | 9.25 | 74.67 | 32.03 |
| A3 | 0.2340 | 0.1579 | 0.2981 | 17.0 | 1.73 | 98.53 | 10.30 |
| A4 | 0.2331 | 0.1544 | 0.2912 | 17.6 | 1.73 | 107.13 | 2.48 |
| ACF | LPD | APP | Por. (per cent) | Por. RE (per cent) | Perm. (mD) | Perm. RE (per cent) | |
|---|---|---|---|---|---|---|---|
| A1 | 0.2041 | 0.1136 | 0.3508 | 13.2 | 23.70 | 33.31 | 69.68 |
| A2 | 0.2193 | 0.1411 | 0.3087 | 15.7 | 9.25 | 74.67 | 32.03 |
| A3 | 0.2340 | 0.1579 | 0.2981 | 17.0 | 1.73 | 98.53 | 10.30 |
| A4 | 0.2331 | 0.1544 | 0.2912 | 17.6 | 1.73 | 107.13 | 2.48 |
In addition, the accuracy of the digital rock was evaluated by comparing the petrophysical property numerical simulation results with the experimental results. We constructed four 3-D digital rocks with the same physical dimensions and different resolutions. The CT-based digital rock contains 80 × 80 × 80 voxels, and the 3-D digital rocks with 4 times, 8 times and 16 times higher resolution contain 320 × 320 × 320 voxels, 640 × 640 × 640 voxels and 1280 × 1280 × 1280 voxels, respectively. In this study, the lattice Boltzmann method (LBM; Saxena et al. 2018; Yan et al. 2021) was used to simulate the permeabilities of the four digital rocks. The method is based on a statistical approach to describe macroscopic physical phenomena and depicts fluid motion through particle collisions. The biggest advantage of this method is that it can accurately simulate fluid flow in complex pore media without simplifying and modifying the porous media. The D3Q15 LBM program was used to simulate the flow of a single-phase fluid in a digital rock to calculate the absolute permeability, and the simulation results are shown in Fig. 13. LBM simulations provide flow velocity values for each voxel in the pore space. Due to the wide range of flow velocities, we performed a linear transformation on the velocity values to enhance the 3-D visualization of the flow velocity distribution. The minimum and maximum velocity values were mapped to the range of 0–255, representing an 8-bit greyscale image format. In the image, the converted flow velocity greyscale values are ranging from 0 to 50 and do not have specific units. Only the main channels are connected in the CT-based digital rock, while the CycleGAN-based digital rocks have more seepage channels. The higher the resolution is, the richer the seepage channels. This is because the higher the resolution of the model is, the better the pore connectivity and the richer the seepage channels. The relative errors between the digital rocks with different resolutions and the experimental measurements are calculated and shown in Table 1. The high-resolution digital rocks are in good agreement with the experimental measurements, which verifies the accuracy of the multisource and multiscale 3-D digital rock reconstruction method from the perspective of the petrophysical properties. Therefore, the reconstructed digital rocks have good accuracy in terms of both statistical properties and petrophysical properties, and the CycleGAN method is reliable for generating multisource and multiscale digital rocks.
3-D seepage fields of Berea with different resolutions.
4 CONCLUSIONS
This paper discusses the effectiveness of a CycleGAN-based multisource and multiscale 3-D digital rock reconstruction method for different types of rocks and different image sources. The method uses cyclic consistency loss to solve the problem of lacking paired training images by using unaligned LR images and HR images as training images. Through model training, the features of LR images and HR images are fused, and the generated digital rock has the advantages of a large field of view of LR images and the high accuracy features of HR images. This method was validated using aligned sandstone and carbonate samples.
In addition to CT images of different resolutions for multiscale digital rock reconstruction, the method can also be applied to digital rock reconstruction of CT images and SEM images. The unaligned CT images and SEM images are used as training images, and the features of CT images and SEM images are fused to construct 3-D digital rocks with pore structure features of SEM images. In our research, the resolution of SEM images was compressed to a size 4–16 times that of CT images. Three tests were conducted, and the CycleGAN-based digital rocks have good accuracies in terms of statistical properties and petrophysical attributes.
For multiscale digital rock reconstruction, it is undesirable to pursue high-resolution images. The larger the difference in resolution, the higher the requirements on computer performance and computation time in training. For example, in the work of this paper, we recommend the use of images with an eight times difference in resolution for multiscale digital rock reconstruction, which maximally preserves the field of view of CT images and the microstructure of SEM images and has high accuracy and computational efficiency in the subsequent digital rock analysis and numerical simulations. In summary, this method is effective for multisource and multiscale digital rock reconstruction, which is important for accurate numerical simulations of rock properties.
ACKNOWLEDGEMENTS
This research work was funded by the National Natural Science Foundation of China (No. 42004098, No. 52074251, No. 42121005, No. 42174143 and No. 92058211), Shandong Provincial Natural Science Foundation of China (No. ZR2020QD054), Fundamental Research Funds for the Central Universities (No. 862201013140) and 111 project (No. B20048), the National Natural Science Foundation of Shaanxi Province of China (grant no. 2022JQ-293), the High-level Innovation and Entrepreneurship Talent Program of Qinchuangyuan (grant no. QCYRCXM-2022–24).
DATA AVAILABILITY
The data or code that support the findings of this study are available from the corresponding author upon reasonable request.
CONFLICT OF INTEREST
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
