Applicability of 2D algorithms for 3D characterization in digital rocks physics: an example of a machine learning-based super resolution image generation

Abstract

Digital rock physics is based on imaging, segmentation and numerical computations of rock samples. Due to challenges regarding the handling of a large 3-dimensional (3D) sample, 2D algorithms have always been attractive. However, in 2D algorithms, the efficiency of the pore structures in the third direction of the generated 3D sample is always questionable. We used four individually captured µCT-images of a given Berea sandstone with different resolutions (12.922, 9.499, 5.775, and 3.436 µm) to evaluate the super-resolution 3D images generated by multistep Super Resolution Double-U-Net (SRDUN), a 2D algorithm. Results show that unrealistic features form in the third direction due to section-wise reconstruction of 2D images. To overcome this issue, we suggest to generate three 3D samples using SRDUN in different directions and then to use one of two strategies: compute the average sample (reconstruction by averaging) or segment one-directional samples and combine them together (binary combination). We numerically compute rock physical properties (porosity, connected porosity, P- and S-wave velocity, permeability and formation factor) to evaluate these models. Results reveal that compared to one-directional samples, harmonic averaging leads to a sample with more similar properties to the original sample. On the other hand, rock physics trends can be calculated using a binary combination strategy by generating low, medium and high porosity samples. These trends are compatible with the properties obtained from one-directional and averaged samples as long as the scale difference between the input and output images of SRDUN is small enough (less than about 3 in our case). By increasing the scale difference, more dispersed results are obtained.

Introduction

In rock physics, physical equations and empirical relations, known as rock physics models, are employed to translate elastic parameters into geological characteristics of rocks. With the introduction of high-resolution micro-computed tomography (CT) scanners, a new window was opened into the pore-scale world of rocks, resulting in the development of digital rock physics (DRP). In DRP, many image processing algorithms and numerical computations have been developed for segmentation and computation of effective parameters of the rock, respectively (Andrä et al. 2013b, a; Kadyrov et al. 2022). Recent advances in artificial intelligence and machine learning (ML) have also been increasingly applied in DRP, with many examples ranging from classification (Karimpouli and Tahmasebi 2019a; Phan et al. 2021; Li et al. 2022) to estimation purposes (Kamrava et al. 2020; Tembely et al. 2021). However, some well-known challenges in most of the standard and ML-based DRP studies are the high memory needed for loading and storing images and data and the need for advanced computing systems for solving the equations, especially in large three-dimensional (3D) samples. For example, suppose that solving Hock’s linear elastic equations in a 2D sample with a size of 128² pixels takes about one minute on a standard computer. It takes more than eight hours for a 3D sample with 128³ voxels on the same computing system (personal experiences). According to Liu et al., (2016), the handling of large samples, for example, with 1000³ voxels, is challenging even for commercial software such as Avizo, no matter how powerful the computational system is. The other challenge, which is mostly found in ML methods, is the high number of training data sets (hundreds to thousands). There are two problems here: first, such a huge number of real samples is not easily available, especially in 3D. Second, imaging such a dataset is both time-consuming and expensive, even if we have access to a massive memory to store it all.

In some cases where the only problem is a small amount of data, data augmentation methods can be applied. They could be either simple geometrical transformations such as image cropping, rotation and flipping or more complex approaches such as generative adversarial networks, neural style transfer, and meta-learning (Khalifa et al. 2022). For example, Karimpouli and Tahmasebi (2019b) increased the number of training images (grayscale input and segmented output images) using a cross-correlation-based simulation algorithm to train a multiphase segmentation autoencoder network.

Nevertheless, a bypass to jump over these problems is to implement the algorithms in 2D, and then, find a reconstruction method to combine them into a 3D sample. This 3D reconstruction is not always required, and 3D parameters can be estimated from 2D computations using semi-empirical or rock physics models. For example, to estimate 3D elastic moduli of rock from 2D thin section images, Saxena and Mavko (2016) represented simple power-law 2D-to-3D transforms and Karimpouli et al. (2018) proposed a method based on Differential Effective Medium (DEM) theory. Similarly, to estimate 3D permeability from 2D thin section images, Saxena et al. (2017) and Hussaini and Dvorkin (2022) proposed some estimation methods based on the Kozeny-Carman equation. In the other cases where the output of an algorithm is a 2D image, reconstruction of the 3D sample from 2D generated images is a common option (Ahuja et al. 2022; Niu et al. 2022). This seems rational, but in some cases, the results of 3D output suffer from 2D reconstruction. For example, Karimpouli and Kadyrov (2022) proposed a Super Resolution Double-U-Net (SRDUN) to generate high-resolution (HR) images from their low-resolution (LR) counterparts. Although the results in 2D images are promising, the 3D reconstructed model is still controversial. Figure 1 schematically shows what happens in other planes during a 2D SRDUN. The generated 2D section (xy in Fig. 1) contains the high-resolution structure of the sample. However, in the other two planes, some discontinuous traces and unrealistic features are seen (xz and yz in Fig. 1). The main purposes of this paper are to understand the influence of the 2D-3D reconstruction on the physical properties of the model and to find an optimal reconstruction strategy to reduce such effects for 3D characterization.

Fig. 1

The normal process for implementing a section-wise 2D algorithm such as SRDUN to generate HR images from their LR counterparts. Although results are promising in the processing section (here, xy), other sections (xz and yz) contain unrealistic features

To answer these questions, we use multiscale µCT images of a Berea sandstone as the data set and multiscale SRDUN as a 2D algorithm (Karimpouli and Kadyrov 2022). We try to understand the effects of such unrealistic features in other planes by computing P- and S-wave velocities, permeability, and formation factor. Also, we propose some combination strategies, such as averaging one-directional reconstructions and binary combinations, to produce more realistic 3D characterizations of the Berea sample.

Methodology

Data

When scanning with a laboratory microtomograph, the resolution (and hence the detail) of the 3D image will mainly depend on the size of the sample being studied; the smaller the sample, the higher the resolution of the resulting image. This is called the upscaling problem: it is possible to obtain either a 3D image for a large volume with a low resolution or a high resolution (for example, sufficient to work on a pore scale) with a meager volume. To investigate this problem, a dataset of 3D images with increasing resolutions was created for a sample of Berea sandstone.

The Berea Sandstone has received widespread recognition in the petroleum industry and has been employed as an etalon reservoir rock in many lab investigations and computer modeling. Numerous publications have discussed its characteristics (Churcher et al. 1991; Gong et al. 2020; Schön 2015). A core sample of Berea sandstone from the "Split rock" type of the Berea formation in Ohio, USA, was utilized for this study. According to XRD analysis, it consists of quartz (83%), microcline (8%), albite (4%), mica (3%), and kaolinite (2%). The sample has a porosity of 19.35% and a water permeability of 81.22 millidarcy. Pores with equivalent diameters of 50–100 µm are dominant in the porous structure of the studied sample. We successively cut the sample into cubes with sides of 8, 6, 4, 2 mm and proceeded µCT scans for each size with resolutions of 12.922, 9.499, 5.775, and 3.509 µm, respectively (Fig. 2). The stair-step spatial registration was applied to the 3D images, starting from the smallest cube to the next larger one. Finally, we obtained a cubic region of interest (ROI) from the smallest 3D image (2 mm) and cropped the same ROI from all lower-resolution 3D images. Thus, we achieved 3D images with different voxel sizes of the same ROI of Berea sandstone. The full procedure of data preparation is described in (Karimpouli and Kadyrov 2022).

Fig. 2

Illustration of all 3D images captured individually from a Berea sandstone with increasing geometrical magnifications (top) and resolutions (bottom). A digital 3D model was obtained for an 8-mm cube sample using microCT, after which the sample was physically reduced to 6, 4, and 2 mm and rescanned each time. The region of interest was excised digitally in the 2 mm sample and was found in all lower resolution 3D images (8, 6, and 4 mm)

SRDUN

Recently, several Super Resolution (SR) networks have been developed, most of them are based on convolutional neural networks (CNN) such as super resolution convolutional neural networks (SRCNN) (da Wang et al. 2019) and Super Resolution Generative Adversarial Networks (SRGAN) such as Super Resolution Cycle-Consistent Generative Adversarial Networks (SRCycleGAN) (Zhu et al. 2017; Chen et al. 2020). Among them, U-net-based networks have been rarely used, mostly due to their poor results. Karimpouli and Kadyrov (2022) explain that the encoder part of U-nets, fed by the LR input image, is not informative enough to reconstruct the detail gaps in the decoder part even if skip connections are used. To overcome this problem, they proposed a SRDUN with two individual U-nets (Fig. 3). The first offline U-net is trained by HR images for both input and output. The idea is to generate informative skip connections. In fact, skip connections in this network contain the detailed information required to reconstruct the rock microstructure in the decoder part. These pretrained connections are then imported into a second U-net with the LR image as input and the HR counterpart as output. Unlike traditional U-nets, offline skip connections, enhance the quality of LR images by adding more details about the microstructure of rock.

Fig. 3

Architecture of SRDUN consisting of two U-nets: First U-net is trained with HR images. The purpose is to train and save informative skip connections for reconstructing small details in the second U-net which transforms LR images into their HR counterparts. Each U-net consists of three downsampling and upsampling steps in the encoder and decoder parts. In the encoder part, one convolutional layer is used in each step which is followed by a residual block and dropout layers. In each step of the decoder part, the pre-trained skip connections are added with a convolutional layer followed by one more convolutional and a dropout layer. For more details see the text (sec "SRDUN") and Karimpouli and Kadyrov (2022)

Combination strategies

According to Fig. 3, the image should be resized or, let’s say, interpolated and, then, SRDUN enhances the interpolation for the sake of more details. However, SRDUN can be implemented along one direction (or a 2D section), thus three cubes are generated in three directions (Fig. 4). It should be noted that these two steps could also be done interchangeably. It means the SRDUN could enhance the resolution of the 2D sections and then an interpolation method would resize the third direction. Our pre-testing results showed that these two arrangements generate very similar results, thus we report the results by interpolation and SRDUN (Fig. 4). After generating three cubes in different directions, averaging is a normal way of combining these cubes. Supposing that A, B and C are three individual cubes, different methods such as arithmetic, harmonic and geometrical averaging are applied as follows:

$$Avg_{A} = \frac{A + B + C}{3}$$

(1)

$$Avg_{H} = ~\left( {\frac{{A^{{ - 1}} + B^{{ - 1}} + C^{{ - 1}} }}{3}} \right)^{{ - 1}}$$

(2)

$$Avg_{G} = \sqrt[3]{A \times B \times C}.$$

(3)

Fig. 4

A normal flowchart for a 2D algorithm to generate a 3D cube. The 2D algorithm is applied in different directions, leading to three cubes, which can be combined, for example, by different averaging methods

The Peak Signal to Noise Ratio (PSNR) is a popular criterion for assessing the similarity of average images to the main original image, as follows:

$${\text{PSNR}} = 10\log \frac{{L^{2} }}{{\frac{1}{N}\mathop \sum \nolimits_{i = 1}^{N} \left( {Org\_im_{i} - Out\_im_{i} } \right)^{2} }},$$

(4)

where $Org\_im$ and $Out\_im$ are original and average images with a length of $L$. The mean square error term is a dominator, which means the higher the PNSR, the more accurate the averaging method.

An essential step for numerically computing physical properties is segmentation. Figure 5 illustrates that this step could be directly done on either the average cube or the directional cubes. In the latter case, averaging methods could not be used anymore. In fact, some rules should be defined to combine these binary values. Suppose that porosity is 0 and grain is 1 in the segmented samples (A, B and C). A summation of these cubes (D = A + B + C) contains four different values (i.e., 0, 1, 2, 3). In the D cube, each voxel is a pore:

Fig. 5

Segmentation is done on either the average cube or the directional cubes. In the latter case, a further binary combination method is needed

- If the voxel is a pore in at least one cube: high porosity model.

$$D_{ijk} = \left\{ {\begin{array}{*{20}c} {0, A_{ijk} + B_{ijk} + C_{ijk} \le 2} \\ {1, A_{ijk} + B_{ijk} + C_{ijk} = 3} \\ \end{array} } \right. \;{\text{and}}\; i, j, k = 1, 2, 3$$

(5)

- If the voxel is a pore in at least two cubes: medium porosity model.

$$D_{ijk} = \left\{ {\begin{array}{*{20}c} {0, A_{ijk} + B_{ijk} + C_{ijk} \le 1} \\ {1, A_{ijk} + B_{ijk} + C_{ijk} \ge 2} \\ \end{array} } \right. \;{\text{and}}\; i, j, k = 1, 2, 3$$

(6)

- If the voxel is a pore in all three cubes: low porosity model.

$$D_{ijk} = \left\{ {\begin{array}{*{20}c} {0, A_{ijk} + B_{ijk} + C_{ijk} = 0} \\ {1, A_{ijk} + B_{ijk} + C_{ijk} \ge 1} \\ \end{array} } \right. \;{\text{and}}\; i, j, k = 1, 2, 3$$

(7)

Numerical computations

Velocity

A dynamic pulse propagation method based on the rotated-staggered-grid finite difference technique for modeling elastic wave propagation (Saenger et al. 2000) is used. We investigate the speeds of elastic waves through heterogeneous materials in the long wavelength limit (pore size ⪡ wavelength) in this approach (Saenger et al. 2004). The technique has been used since 2004 in several papers and was benchmarked in (Andrä et al. 2013b) and (Saxena et al. 2019).

Permeability

In order to determine the permeability of the rock samples, the stationary Stokes equations

$$\nabla \cdot u = 0$$

(8)

$$\nabla \cdot \left( {\mu ~\nabla u} \right) = \nabla p$$

(9)

are solved numerically with a finite volume solver (Siegert et al. 2022). In our simulations, air is considered the pore fluid and the dynamic viscosity $\mu$ is set accordingly. The target variables of the simulation are the pressure field $p$ and the velocity field ${u}$ of the pore space. The computational mesh is built on the information of the µCT scans, whereby only the connected pore space is taken into account. As boundary conditions, the following setup is used: A pressure difference $\Delta p$ is specified at two opposite faces of the cubic rock model; these faces are defined as the inlet and outlet. The remaining surfaces are modelled as walls. During the computation, the resulting linear system of equations is solved with a conjugate gradient method and a stopping criterion of at least 10^–8 for the normalized L1-norm of the residual vector is set. Once the flow fields are converged, the volumetric flow rate $Q$ in the direction of the inlet and outlet is determined. Taking into account the geometric distance $L$ between the inlet and outlet, the cross-section $A$, as well as the pressure difference $\Delta P$, the permeability of the sample in the prescribed flow direction can be determined utilizing the one-dimensional Darcy-Law (Darcy 1856):

$$k = \frac{{Q \cdot \mu \cdot L}}{{A \cdot \nabla p}}.$$

(10)

Formation factor (conductivity)

The simulations for the determination of the electrical conductivity are performed with water-saturated rock samples. We assume that the conductivity of the pore fluid dominates that of the minerals. More precisely, the conductivity of the pore fluid is set to $\sigma _{{pore}} = 1$ and that of the minerals to $\sigma _{{\min eral}} = 0$. The formation factor is thus obtained as:

$$f = \frac{{\sigma _{{pore}} }}{{\sigma _{{sample}} }} = \frac{1}{{\sigma _{{sample}} }}.$$

(11)

In order to determine the effective conductivity $\sigma_{{{\text{sample}}}}$, we compute the electric potential field $\varphi$ of each rock sample. A stationary electric potential is assumed and the calculations are performed by solving the current continuity equation:

$$\nabla \cdot \left( {\sigma ~\nabla \varphi } \right) = 0.$$

(12)

Since the conductivity of the minerals is set to zero, only the connected pore space is considered in the numerical meshes. As boundary conditions, two opposing faces of the cubic rock geometry are prescribed with a potential difference $\Delta \varphi$. The remaining faces are modelled as insulated walls. With this setup, the resulting boundary value problem is solved numerically with a newly developed finite volume solver (for validation, see Appendix A). As a stopping criterion, a L1-residual of at least 10^–8 has to be reached by each simulation. Once the potential field is known, the electric current $I$ is determined. Via Ohm's law (Saslow 2002), the resulting conductivity is computed as:

$$\sigma _{{sample}} = \frac{{I \cdot L}}{{A \cdot \Delta \varphi }}.$$

(13)

Results and discussions

Based on the samples in this study (8, 6, 4 and 2 mm), three multiscale SRDUNs are used, as they are mentioned in Table 1. For each SRDUN, three individual cubes are generated in three directions and then combination strategies are evaluated. Since our aim here is to investigate numerical characteristics, we avoid repetition and show all the results of the SRDUNs. Extensive visualizations are found in Karimpouli and Kadyrov (2022). According to Table 1, our investigations focus on SRDUN-62 with a scale difference of 2.71 and after coming to conclusions about combination strategies, we will seek the effects of scale difference for two other SRDUNs with lower (1.65) and higher (3.96) scale differences (i.e., SRDUN-42 and SRDUN-82).

Table 1 Scale difference of three SRDUNs

SRDUN-62

The multistep SRDUN-62 contains two steps of high-resolution sample generations, one from 6-mm to 4-mm and the other from 4-mm to 2-mm. In every step, the image is first resized and interpolated in all three directions and then the SRDUN is applied to enhance the details. According to Sect. "Combination strategies", the combined cube is generated by arithmetic, harmonic and geometric averaging, which is also assumed as the input image for the SRDUN in the next step. Figure 6 illustrates the results for three averaging methods. Visually speaking, they are very similar, with no obvious differences. However, in some parts (for example, zoomed-in parts in Fig. 6), harmonic averaging results in more sharp borders between pore and grain phases. In the other two averaging methods, those borders are too ambiguous to be easily differentiated during segmentation. As a quantitative evaluation, the PNSR of all three cubes is also computed as 42.74, 42.82, and 42.71. According to these values, it can be concluded that the harmonic averaging is slightly better than the others, although this difference is not very large. Therefore, we assume harmonic averaging as the best method of combining three cubes in different directions and use it for numerical computations in the remainder of the paper.

Fig. 6

Results for (a) arithmetic, (b) harmonic and (c) geometry averaging, with a zoomed area to highlight the sharpness of the harmonic averaging. The PNSR is 42.74, 42.82 and 42.71, respectively

As it is mentioned in Sect. "Combination strategies", the binary combination (Eqs. 5–7) leads to three low, medium, and high porosity models. Figure 7 depicts the sum of these three models (D = A + B + C) for SRDUN-62 in a single section. Since all voxels in each segmented cube have a value of 0 or 1, the summation cube has four values: 0, 1, 2 and 3. Among them, 0 values are pore spaces in the low porosity model, 0 and 1 are pore spaces in the medium porosity model and 0, 1 and 2 are pore spaces in the high porosity model. Those voxels with a value of 3 are grain in all models. Figure 7 implies, that from a high to a low porosity model, small pores and narrow pore throats are removed, leading to a model with higher isolated pore spaces and smaller connected porosities.

Fig. 7

Summation of three cubes segmented from a directional 2D SRDUN (D = A + B + C) (see Fig. 5). The binary combination of 0, 1, and 2 means that the voxel is pore in three, two, and one cubes, respectively

Table 2 contains all parameters computed for the direct and combined models of SRDUN-62 as well as the original 2-mm sample. Among them, SRDUN-62 is used for generating models in one direction (x, y and z; see Fig. 4); then, harmonic averaging is used for generating the average model. Based on the binary combination, one-directional models are first segmented and then combined via a low, medium and high porosity strategy. Porosity computations (Table 2) show that all three models based on the reconstruction of 2D algorithms (SRDUN-62 in one direction) contain roughly 14% porosity, which is more than 5% lower than the original sample with 19.58% porosity. When these models are averaged, the porosity increases to 16.24%, which is closer to the original value. As it was expected, the porosity of a binary combination ranges from low to high (7.02, 13.05 and 21.87%).

Table 2 Numerical computations of SRDUN-62

Because these models have different porosities, we use rock physics trends and models to make a more accurate assessment. Figure 8 illustrates the P- and S-wave velocities of all models, where the dashed line is the modified upper Hashin–Shtrikman or stiff sand model for the sandstones (Dvorkin and Nur 1996). Also, Fig. 9 shows permeability and formation factor, while the main trends (dashed lines) are Kozeny-Carman (KC) and Archie relationships (Mavko et al. 2009) (for rock physics models, see Appendix B). Since fluid flow and electrical conductivity are mainly based on the connected porosity, this parameter has also been reported for each model in Table 2. The difference between this value and the porosity reveals that in almost all models there are some isolated pore spaces ranging from 0.6 to 4.34%; however, this difference for the original sample is only 0.2%, which is much smaller. This shows that the SRDUN algorithm generates models with larger isolated pore spaces.

Fig. 8

Velocity-porosity plots for (a) P-wave and (b) S-wave velocities of different models of the SRDUN-62 as well as the original 2-mm sample

Fig. 9

a Permeability–porosity and b formation factor—porosity plots of different models of SRDUN-62 as well as the original 2-mm sample. The permeability trend is based from the well-known KC relations with a gain size of 95 µm. The formation factor trend is obtained from the Archie relationship, where a = 0.3 and m = 2.9. The formation factor of the binary combination model with low porosity is 7688.97, which is not shown in its plot

According to Figs. 8 and 9 (circles), 3D models obtained from the reconstruction of 2D section-wise SRDUN show similar properties that are relatively far from the original values. Due to their similarities, they are close to the main trend, but the trend is not well understood only based on their arrangements. However, when these models are combined via harmonic averaging, both porosity and velocity enhance and a much closer result is obtained to the original values. Thus, it can be concluded that an average model from a 2D algorithm in different directions could produce more realistic results than the 3D model reconstructed in one direction.

The other models are obtained from the binary combination strategy as low, medium and high porosity models (diamonds in Figs. 8 and 9). Due to the wide coverage of porosity by these models, one of them is usually closer to the original sample; for example, here the high porosity sample is much closer to the 2-mm sample. Nevertheless, the main advantage of these models is the possibility of determining a trend. Unlike one directional SRDUN models, the arrangement of these models is best suited for fitting the rock physics trend. This means they are beneficial for extracting trends and predicting physical parameters of a rock if 3D characterization of the sample is intended.

SRDUN-42 and SRDUN-82

To evaluate the results obtained for SRDUN-62, we use two other SRDUNs, but with a lower (SRDUN-42) and a higher (SRDUN-82) scale difference between input and output (original) images. Figures 10 and 11 illustrate the results for computing wave velocities (P and S), permeability and the formation factor for SRDUN-42. All trends (dashed lines) are similar to what was extracted from SRDUN-62. Compared to the last results (Figs. 8 and 9), more porous models are obtained in both strategies. The average model of SRDUNs shows a higher porosity than the original 2-mm sample and this time the binary combination model with medium porosity is the closest model to the original one. Although these values are different from each other, an interesting result is that the main trends are still the same as in SRDUN-62. This means, by decreasing the scale difference during a SR generation algorithm, the same results may not be obtained, but all of them will obey the main trends of the sample. It shows that the main structure of pore spaces is preserved, but more details may be added.

Fig. 10

Same as Fig. 8, but for SRDUN-42

Fig. 11

Same as Fig. 9, but for SRDUN-42

All computations are repeated once more, but this time for SRDUN-82, which has the highest scale difference compared to other SRDUNs. Results are shown in Figs. 12 and 13. According to Fig. 12, the porosities of the reconstructed models are much closer to the SRDUN-62 than the SRDUN-42, but the velocity values are located a little above the main trends. This means that the pore shape and structure may change a little bit, which is expected due to the low resolution, but the arrangement is such that they are very close to the main trends.

Fig. 12

Same as Fig. 8, but for SRDUN-82

Fig. 13

Same as Fig. 9, but for SRDUN-82

Computing the connected porosity, however, shows that the binary combination model with low porosity has a connected porosity value of zero. This means that although the porosity value does not change with respect to the SRDUN-62, an increase in the scale difference leads to isolated pores. In addition, the arrangement of these results does not follow the main trends of the original sample. In this case, we defined two upper and lower trends to show the variability of these samples. According to these trends, permeability values vary in a Kozeny-Carman trend with grain sizes ranging from 38 to 95 mm. These values for the Archie trend are equal to an “a” value from 0.5 to 3 (Appendix B).

In addition to the parameters considered in this study, anisotropy is another factor that can significantly affect these computations. In our case, the Berea sandstone is assumed to be an isotropic medium (Andreä et al. 2013a), leading to relatively similar results in different directions. However, this assumption cannot be extended to anisotropic media. We expect that the reconstruction of a 3D sample would not yield a representative sample by 2D slices using any one direction or even in a combination of three directions (e.g., binary combination). The extent of this mismatch is unclear and should be evaluated as a future work using an anisotropic sample.

Overall, DRP has its limitations, primarily stemming from the trade-off between field of view and resolution of images, as well as computational challenges (Karimpouli et al. 2020). To address some of these limitations, researchers have explored the application of ML algorithms, specifically for tasks like image segmentation and SR image generation. Karimpouli and Kadyrov (2022) demonstrated that additional enhanced networks, such as SRDUN, may be required to overcome the applicability of ML to real rock images across different scales.

In this study, we aimed to address the challenges associated with reconstructing a 3D image of rock by combining 2D images using different methods, such as averaging and binary combination. Although our results are promising, it is crucial to conduct future studies that consider all rock physics parameters, including anisotropy, multiphase flow, and saturation, to develop a universally applicable method.

One significant limitation of this approach is its dependence on the specific samples and images for which the network has been trained. For example, if we have a carbonate sample, the SRDUN network trained solely on sandstone samples may not yield satisfactory results. It is crucial to recognize that different rock types may require dedicated ML networks, and from an applied perspective, training a ML network with a large dataset encompassing various rock types could lead to a more generalized network. In summary, while our study addresses some challenges in DRP, further investigations are necessary to encompass a broader range of rock physics parameters and ensure the development of robust and universally applicable methods.

Conclusions

In this work, 3D reconstruction and characterization based on section-wise 2D algorithms have been investigated as an important challenge in DRP. To achieve this aim, we used four individually captured µCT scan images (8, 6, 4 and 2 mm) with different resolutions (12.922, 9.499, 5.775, and 3.436 µm) from a given Berea sandstone. A multistep SRDUN was applied as a 2D algorithm to generate the HR image of a 2 mm sample from the other samples. Results show that if SRDUN is applied to 2D sections along one direction, the generated 3D image contains unrealistic features. To tackle this problem, we proposed two strategies: (I) reconstruction by averaging over three cubes, where each of them is generated along one direction; and (II) binary combination, where the one-directional cubes are first segmented and, then, combined together. Numerical computations of P- and S-wave velocities, permeability and formation factor show that:

• Samples generated by 2D SRDUN in different directions indicate close properties to each other but are far from the original sample. For example, their porosities could be up to 5% less than the original one and consequently different properties are obtained.

• The averaging strategy leads to a sample with closer properties to the original sample. Results reveal that among arithmetic, harmonic and geometrical averaging, the harmonic method generates more reasonable samples.

• A binary combination generates low, medium and high porosity samples, determining the rock physics trend. As long as the scale difference between the input and output images of SRDUN is small, this trend is compatible with other results (one-directional and averaged samples). However, by increasing the scale difference (in our case, greater than roughly 3), we can get results that are dispersed from the main trend and define an area of variation between two trends.

Appendices

Appendix A: Validation of the conductivity solver

The validation of the conductivity solver is performed with the benchmark cases published in (Andrä et al. 2013b, a). Two sandstone samples (Berea and Fontainebleau) as well as one carbonate sample (Grosmont) are investigated. For each sample, anisotropic effects are considered and the formation factor is determined in the x, y and z directions of the corresponding models. The results of our solver and three already published investigations are given in Table 3. Overall, the simulations show comparable results. A general trend regarding overestimation or underestimation of our solver is not observable. Whereas our solver computes the highest formation factors in the cases of the Berea and Grosmont samples, the Fraunhofer solver gives the highest results for the Fontainebleau sample. Regarding anisotropic effects, one can observe, that all solvers predict similar trends. The only exception can be found in the results of the Fontainebleau sample. Here, the Stanford solver predicts the highest formation factor in the z-direction, while the other three solvers predict the highest formation factor in the y-direction.

See Table 3.

Table 3 Validation of the conductivity solver, based on Andrä et al. (2013a, b)

Appendix B: Rock physics models

Stiff sand model

Stiff sand model is a modified Hashin–Shtrikman upper bound model. Similar to the soft sand model, the endmember is assumed to be an uncemented sand (dense random pack) with a critical porosity of ${\phi }_{0}=$ and a coordination number of $C = 9$. The effective bulk ($K_{HM}$) and shear ($\mu_{HM}$) moduli are obtained using the contact Hertz-Mindlin theory a hydrostatic pressure $P$ (Mavko et al. 2009):

$$K_{HM} = \left[ {\frac{{C^{2} \left( {1 - \phi_{0} } \right)^{2} \mu^{2} }}{{18\pi^{2} \left( {1 - \upsilon } \right)^{2} }}P} \right]^{1/3}$$

(B1)

$$\mu_{HM} = \frac{5 - 4\nu }{{5\left( {2 - \nu } \right)}}\left[ {\frac{{3C^{2} \left( {1 - \phi_{0} } \right)^{2} \mu^{2} }}{{2\pi^{2} \left( {1 - \upsilon } \right)^{2} }}P} \right]^{1/3},$$

(B2)

where $\nu$ is the Poisson ratio of grain. The effective moduli ($K_{eff}$ and $\mu_{eff}$) of the spherical grains with bulk and shear moduli of $K$ and $\mu$ at different porosity are obtained by:

$$K_{eff} = \left[ {\frac{{\phi /\phi_{0} }}{{K_{HM} + \frac{4}{3}\mu }} + \frac{{1 - \phi /\phi_{0} }}{{K + \frac{4}{3}\mu }}} \right]^{ - 1} - \frac{4}{3}\mu$$

(B3)

$$\mu_{eff} = \left[ {\frac{{\phi /\phi_{0} }}{{\mu_{HM} + \frac{\mu }{6}\left( {\frac{9K + 8\mu }{{K + 2\mu }}} \right)}} + \frac{{1 - \phi /\phi_{0} }}{{\mu + \frac{\mu }{6}\left( {\frac{9K + 8\mu }{{K + 2\mu }}} \right)}}} \right]^{ - 1} - \frac{\mu }{6}\left( {\frac{9K + 8\mu }{{K + 2\mu }}} \right).$$

(B4)

Kozeny-Carman model

The Kozeny-Carman relation for permeability ($k$) calculation is based on flow through a pipe having a circular cross section, which can be extended as a packing of identical spheres of diameter $d$ and porosity of $\phi$ (Mavko et al. 2009).

$$k = B\frac{{\phi^{3} }}{{\left( {1 - \phi } \right)^{2} }}d^{2},$$

(B5)

where $B$ is a constant value depending on, for example, tortuosity.

Archie model

Archie derived an empirical relation for the formation factor based on the porosity $\phi$ of the reservoir rock (without shale), which forms the basis for resistivity log interpretation (Archie 1942; Mavko et al. 2009).

$$F=a{\phi }^{-m},$$

(B6)

where $m$ is the cementation factor and $a$ is an empirical constant.