X-Ray Computed Tomography-Based Three-Dimensional Fractal Characterization of Bedding-Fracture-Controlled Porosity and Permeability Anisotropy in LGS Shale Oil Cores

Abstract

Bedding fractures strongly influence pore structure and anisotropic flow capacity in laminated shale oil reservoirs, but conventional porosity–permeability relationships cannot adequately explain permeability differences caused by bedding orientation and fracture connectivity. This problem represents an important gap in shale oil reservoir evaluation because cores with similar porosity may exhibit markedly different permeability when bedding-fracture connectivity and flow direction differ. The main question addressed in this study is how bedding-fracture structures in paired horizontal and vertical LGS shale oil cores selected from the same depth intervals influence porosity, permeability, and permeability anisotropy. To answer this question, this study establishes a quantitative framework linking X-ray computed tomography-derived bedding-fracture structure, three-dimensional fractal dimension, and stress-sensitive permeability anisotropy in LGS shale oil cores. Paired horizontal and vertical cores from the same depth intervals were tested under confining pressures of 10–50 MPa. X-ray computed tomography reconstruction was used to extract bedding-fracture volume fraction $V_f,$ fracture number $N_b$, fracture density ${\rho }_b$, connectivity index $C_b$, and three-dimensional box-counting fractal dimension $D_3$. The H-series cores exhibit much higher bedding-parallel permeability than the V-series cores, although their porosity ranges partly overlap. At 10 MPa, the average permeability of the H-series is 0.24402 mD, approximately 21.7 times that of the V-series 0.01127 mD. As confining pressure increases from 10 to 50 MPa, the average permeability decreases by approximately 97.1% for the H-series and 96.5% for the V-series, indicating strong stress sensitivity of bedding-fracture-controlled flow channels. The $D_3$ values range from 2.16 to 2.63 for the H-series and from 2.12 to 2.56 for the V-series. Higher $D_3$, $V_f$, and $C_b$ enhance permeability when bedding fractures are aligned with the flow direction, whereas complex but discontinuous bedding structures may still result in low bedding-normal permeability. A fractal-corrected porosity–permeability model incorporating $\phi$, $V_f$, $C_b$, and $D_3$ is proposed to improve permeability interpretation beyond porosity alone. This study demonstrates that permeability anisotropy in LGS shale oil cores is controlled by the combined effects of pore–fracture volume, directional connectivity, fractal complexity, and stress-induced fracture closure.

1. Introduction

Shale oil reservoirs are commonly characterized by low porosity, low permeability, strong heterogeneity, and pronounced bedding-controlled anisotropy. In such reservoirs, storage and flow capacity are not governed only by matrix pores, but are also strongly affected by bedding-parallel fractures, lamination-related microfractures, and connected pore–fracture networks. These structures may provide preferential seepage pathways and local storage space, especially when their orientation is favorable to the principal flow direction. Backeberg et al. [1] used X-ray tomography-based models to quantify anisotropy and tortuosity in clay-rich mudstones and showed that permeable pathways are strongly affected by pores, laminae, and microfractures. Tan et al. [2] experimentally investigated shale-fracture permeability supported with proppant and demonstrated the strong directional dependence of fracture-controlled flow. Gu [3] further emphasized that lamination- and natural-fracture-induced anisotropy can influence reservoir development and operational design. Therefore, quantitative characterization of bedding-fracture structures is essential for understanding porosity–permeability relationships and anisotropic flow behavior in shale oil reservoirs.

Bedding fractures are important structural discontinuities in laminated shale formations. They may originate from sedimentary lamination, diagenetic shrinkage, tectonic reactivation, or stress-induced opening along weak bedding interfaces. Compared with randomly distributed matrix pores, bedding fractures usually show stronger directionality and greater sensitivity to mechanical closure. Their contribution to reservoir quality is therefore controlled not only by their volume, but also by their density, spatial continuity, connectivity, and orientation relative to the flow direction. Jiang et al. [4] reviewed the formation mechanisms and geological significance of bedding-parallel fractures in shale systems and highlighted their implications for hydrocarbon accumulation and flow capacity. Li et al. [5] showed that bedding orientation significantly affects fracture toughness and failure patterns in anisotropic shale. Zhang et al. [6] experimentally investigated the combined effects of bedding anisotropy and matrix heterogeneity on hydraulic fracturing behavior in shale. Liu et al. [7] further demonstrated that bedding anisotropy influences hydraulic-fracture initiation and propagation. Lin et al. [8] used real-time computed tomography to examine failure and fractal characteristics of anisotropic oil shale, whereas Huang et al. [9] analyzed anisotropic crack evolution and fractal failure mechanisms in shale under compression. These studies indicate that bedding-related pore–fracture structures are critical components of shale oil reservoirs. However, conventional porosity-based evaluation methods often cannot fully explain the large permeability differences observed among cores with similar porosity, especially when bedding-fracture connectivity differs significantly.

The comparison between horizontal and vertical coring directions provides a direct way to evaluate bedding-controlled permeability anisotropy. In horizontally cored samples, the core axis is generally parallel to the bedding plane, and the measured permeability mainly reflects flow along bedding-parallel pore–fracture pathways. In vertically cored samples, the core axis is perpendicular to bedding, and the measured permeability is more strongly affected by cross-bedding connectivity, bedding-interface sealing, and discontinuous fracture intersections. As a result, permeability parallel to bedding may differ markedly from permeability perpendicular to bedding, even when the difference in bulk porosity is limited. Rose [10] recognized the role of sedimentary bedding in causing permeability anisotropy in petroleum reservoirs. Takada [11] further demonstrated that bedding planes can affect permeability and diffusivity anisotropies in porous rocks. More recently, Gu [3] showed that shale lamination and natural fractures can affect reservoir development strategies and operational design, while Liu et al. [12] reported that fracture propagation in structurally complex shale zones is closely related to bedding-plane characteristics. In addition, Teklu et al. [13] experimentally investigated permeability and porosity hysteresis in tight formations, and Hashemi and Zoback [14] showed that fracture permeability in shale evolves strongly under stress and fluid–rock interaction. These observations suggest that permeability anisotropy in shale oil cores should be interpreted from the coupled perspective of bedding-fracture geometry, connectivity, coring direction, and stress sensitivity.

With the development of digital rock physics and high-resolution imaging techniques, X-ray computed tomography has become an effective tool for reconstructing pore–fracture structures in three dimensions. Compared with two-dimensional thin-section or surface observations, CT reconstruction can reveal the internal distribution, geometry, and connectivity of fractures without destroying the core. Qi et al. [15] used three-dimensional imaging to characterize microfractures in shale reservoir rocks. Zhang et al. [16] combined in situ micro-CT and digital volume correlation to investigate the effect of bedding orientation on shale damage evolution. Jiang et al. [17] reconstructed shale hydraulic-fracture geometry using CT scanning and quantified fracture propagation characteristics. Hou et al. [18] developed a multiscale reconstruction method for fractured shale and analyzed the influence of fracture morphology on shale gas flow. Wu et al. [19] integrated CT scanning and box-counting analysis to reconstruct three-dimensional fracture networks and quantify their fractal characteristics. Wang et al. [20] applied μ-CT imaging and multifractal analysis to characterize explosion-induced fractures in bedding shale. Beyond shale, You et al. [21] used three-dimensional CT reconstruction to study the distribution and evolution of pore–fracture systems in deep sandstone. Zhao et al. [22] examined pore–fracture structure and permeability evolution in oil shale. Sun et al. [23] investigated pore and fracture evolution in coal using CT-based three-dimensional reconstruction. Wang et al. [24] combined X-ray CT imaging and fractal theory to model macro-pore structures in coal. These studies demonstrate that CT imaging provides a reliable basis for identifying connected pore–fracture structures. Nevertheless, for laminated shale oil cores, the direct linkage among CT-identified bedding fractures, horizontal/vertical coring direction, porosity, stress-sensitive permeability, and permeability anisotropy remains insufficiently quantified.

Fractal theory provides a useful mathematical framework for describing irregular pore–fracture structures that cannot be fully represented by Euclidean geometric parameters alone. The fractal dimension reflects the space-filling ability, morphological complexity, and multiscale heterogeneity of a pore or fracture network. Yu and Cheng [25] developed a fractal permeability model for bi-dispersed porous media and demonstrated that permeability is affected by pore-size distribution and pore-space geometry. Yu and Liu [26] further established a fractal analysis framework for permeability in porous media, showing that tortuosity and pore structure are essential controls on flow. In shale and tight reservoirs, Zhou and Zhao [27] evaluated nanoscale shale pore fractal dimensions and discussed their implications for permeability. Zhang et al. [28] analyzed pore-type-dependent fractal features of shale and showed that different pore types contribute differently to permeability. Jiang et al. [29] investigated the fractal dimension of marine shale pore structures in the Niutitang Formation and linked fractal characteristics with pore heterogeneity. Li et al. [30] integrated nitrogen adsorption, mercury intrusion, and deep learning-assisted FIB–SEM reconstruction to characterize lacustrine shale pore structures and fractal features. Zhang et al. [31] examined pore-throat structure and fractal characteristics of tight sandstone and applied them to permeability prediction. Wang et al. [32] studied shale pore structure fractal characteristics and their influence on seepage flow. Zhao and Zhang [33] compared NMR- and SEM-derived fractal dimensions to explore shale pore structure in the Ordos Basin. Xu et al. [34] investigated fractal characteristics of different shale lithofacies in the Dalong Formation. Hu et al. [35] analyzed marine shale pore structures in the Sichuan Basin and showed that fractal descriptors can reflect pore complexity and heterogeneity. These studies confirm that fractal analysis is useful for characterizing pore-scale complexity, but most of them focus on matrix pores, nanopores, or pore-throat systems rather than CT-resolvable bedding-fracture networks.

Fractal theory has also been extended to fracture-dominated and dual-porosity media. Zong et al. [36] proposed a fractal fracture-permeability model considering fracture morphology and spatial distribution. Hu et al. [37] developed a fractal permeability model for dual-porosity media containing rough tree-like fracture networks. Yang et al. [38] established a fractal gas–water relative permeability model for inorganic shale considering water occurrence state. Yang et al. [39] proposed a fractal permeability model for Newtonian fluids in rough fractured dual-porous media. Xia et al. [40] developed a fractal model for complex tortuous fracture networks and emphasized the effects of tortuosity and fracture-network complexity. Hu et al. [41] proposed a porous-media permeability model based on fractal theory and idealized pore-space geometry. Ge et al. [42] incorporated effective stress into a fractal pore-permeability model, highlighting the importance of stress-dependent pore deformation. Xia et al. [43] developed a fractal-theory-based permeability model for coal fracture networks. Zhang et al. [44] investigated three-dimensional fracture evolution induced by CO₂ phase-transition fracturing and quantified its fractal characteristics. Wang et al. [45] investigated hydration, shrinkage, pore structure, and fractal dimension of silica-fume-modified low-heat Portland cement-based materials and showed that fractal dimension can effectively describe pore-structure complexity and material heterogeneity. These studies demonstrate that incorporating fractal descriptors into permeability models can improve the interpretation of flow behavior in heterogeneous porous and fractured media. However, most existing fractal permeability models are not specifically established for paired horizontal and vertical shale oil cores in which bedding-fracture orientation, CT-resolvable connectivity, and stress-sensitive anisotropic permeability are considered simultaneously.

Despite these advances, several limitations remain. First, many existing studies focus on nanopore structure, two-dimensional pore images, or isolated fracture morphology, whereas fewer studies quantitatively characterize bedding-fracture networks in three-dimensional shale oil cores. Second, the influence of horizontal and vertical coring directions is often discussed qualitatively, but the relationships among bedding-fracture density, pore–fracture volume fraction, connectivity, and permeability anisotropy are not fully established. Third, conventional porosity–permeability relationships cannot adequately explain permeability differences caused by bedding orientation and fracture connectivity, especially when cores from comparable depth intervals have similar porosity but different directional flow capacities. Fourth, although fractal dimension has been widely used to describe pore complexity, its role in linking CT-derived bedding-fracture structure with porosity, stress-sensitive permeability, and anisotropic flow remains unclear. Consequently, a systematic framework is needed to integrate CT reconstruction, bedding-fracture identification, three-dimensional fractal characterization, and porosity–permeability modeling for paired horizontal and vertical shale oil cores.

Unlike studies based on unpaired cores, single coring directions, or isolated pore/fracture descriptors, this study used paired horizontal and vertical LGS shale oil cores selected from the same depth intervals to address the key question of how bedding-fracture structures influence porosity, permeability, and permeability anisotropy. This paired-core design reduces the influence of lithologic and burial-depth differences and allows the directional effect of bedding-fracture connectivity to be evaluated more directly. Porosity and stress-sensitive permeability were measured under multiple confining pressures, and CT scanning was used to reconstruct the three-dimensional pore–fracture structure of each core. Bedding fractures were identified from the reconstructed volumes, and quantitative parameters, including fracture number, fracture density, pore–fracture volume fraction, connectivity, and three-dimensional box-counting fractal dimension D₃, were extracted. Based on these parameters, the relationships among bedding-fracture structure, porosity, permeability, and stress sensitivity were analyzed. Finally, a fractal-corrected porosity–permeability model was proposed by incorporating CT-derived structural parameters and D₃.

Compared with previous studies that mainly focused on nanopore-scale fractal characteristics, isolated fracture morphology, unpaired samples, or qualitative bedding-controlled anisotropy, the originality of the present study lies in the paired-core design and the integrated evaluation of CT-derived bedding-fracture structure, three-dimensional fractal dimension, and stress-sensitive permeability anisotropy. The present study has three specific differences. First, paired horizontal and vertical LGS shale oil cores from the same depth intervals were used to isolate the influence of bedding-fracture orientation and directional connectivity on porosity, permeability, and permeability anisotropy. This design minimizes the effects of lithologic variation and burial-depth differences, making the comparison between bedding-parallel and bedding-normal flow more direct. Second, CT-derived bedding-fracture parameters, including $V_f$, $N_b$, ${\rho }_b$, $C_b$, and $D_3$, were jointly analyzed with stress-sensitive permeability measured under 10–50 MPa confining pressures. Third, a fractal-corrected porosity–permeability model was developed by integrating porosity, CT-resolvable fracture volume, connectivity, and three-dimensional fractal dimension. Therefore, this study does not use fractal dimension only as a geometric descriptor, but further evaluates its hydraulic significance under different bedding-related flow directions.

2. Samples and Experimental Methods

2.1. Shale Oil Core Samples

The paired sampling strategy was designed to answer the main research question of whether, and to what extent, bedding-fracture structures in cores from the same depth intervals produce different porosity–permeability responses in bedding-parallel and bedding-normal directions. Representative shale oil cores from the LGS Formation were selected to investigate the influence of bedding-fracture structure on porosity, permeability, and permeability anisotropy. The LGS shale is characterized by well-developed lamination, bedding-related weak interfaces, and heterogeneous pore–fracture systems. These features make it suitable for evaluating the role of bedding-controlled pore–fracture structures in shale oil storage and flow. The overall workflow of this study is shown in Figure 1. The workflow includes shale oil core sampling, definition of horizontal and vertical coring directions, porosity and permeability measurements, CT scanning, three-dimensional reconstruction, bedding-fracture identification, three-dimensional fractal-dimension calculation, and fractal-corrected porosity–permeability modeling.

To examine the effect of bedding orientation, the samples were divided into two groups according to the coring direction relative to the bedding plane. The first group consisted of horizontally cored samples, denoted as the H-series, in which the core axis was approximately parallel to the bedding plane. The second group consisted of vertically cored samples, denoted as the V-series, in which the core axis was approximately perpendicular to the bedding plane. The relationship between coring direction, bedding orientation, and flow direction is illustrated in Figure 2. In this study, the permeability measured from the H-series cores is regarded as the bedding-parallel permeability, k_h, whereas the permeability measured from the V-series cores is regarded as the bedding-normal permeability, k_v. This sampling design allows the anisotropic effect of bedding-fracture structures on flow capacity to be evaluated directly.

A total of 24 cylindrical shale oil cores were used, including 12 H-series samples and 12 V-series samples. Each sample was subjected to porosity measurement, stress-sensitive permeability testing, and X-ray CT scanning. The same sample-numbering system was used throughout the experimental and image-analysis workflow to ensure consistency among physical properties, CT-derived structural parameters, and fractal dimensions.

Before testing, the core surfaces were cleaned, and loose particles were removed. The macroscopic bedding orientation and visible fractures were recorded. The samples were then dried under controlled conditions to minimize the influence of movable fluids on porosity and CT image segmentation. Basic sample information, including sample ID, coring direction, diameter, length, and bedding orientation, is summarized in

Table 1. Basic information of H- and V-series LGS shale oil cores.

Sample ID	Coring Direction	Bedding Orientation Relative to Core Axis	Diameter (cm)	Length (cm)
H-1#	Horizontal	Parallel to bedding	2.53	4.96
H-2#	Horizontal	Parallel to bedding	2.53	4.97
H-3#	Horizontal	Parallel to bedding	2.52	4.95
H-4#	Horizontal	Parallel to bedding	2.53	4.97
H-5#	Horizontal	Parallel to bedding	2.53	4.96
H-6#	Horizontal	Parallel to bedding	2.52	4.97
H-7#	Horizontal	Parallel to bedding	2.54	4.97
H-8#	Horizontal	Parallel to bedding	2.54	4.98
H-9#	Horizontal	Parallel to bedding	2.53	4.96
H-10#	Horizontal	Parallel to bedding	2.53	4.96
H-11#	Horizontal	Parallel to bedding	2.52	4.99
H-12#	Horizontal	Parallel to bedding	2.52	5.01
V-1#	Vertical	Perpendicular to bedding	2.52	4.96
V-2#	Vertical	Perpendicular to bedding	2.52	4.95
V-3#	Vertical	Perpendicular to bedding	2.52	4.94
V-4#	Vertical	Perpendicular to bedding	2.54	4.96
V-5#	Vertical	Perpendicular to bedding	2.53	4.96
V-6#	Vertical	Perpendicular to bedding	2.53	4.97
V-7#	Vertical	Perpendicular to bedding	2.54	4.97
V-8#	Vertical	Perpendicular to bedding	2.54	4.97
V-9#	Vertical	Perpendicular to bedding	2.54	4.97
V-10#	Vertical	Perpendicular to bedding	2.53	4.95
V-11#	Vertical	Perpendicular to bedding	2.52	4.99
V-12#	Vertical	Perpendicular to bedding	2.52	5.01

.

2.2. Porosity and Permeability Measurements

Porosity and permeability measurements were performed to quantify the storage and flow capacity of the LGS shale oil cores. Porosity was measured before permeability testing and was used as the basic volumetric parameter for evaluating the pore–fracture storage capacity. Because shale oil cores commonly contain both matrix pores and bedding-related microfractures, the measured porosity represents the combined contribution of the matrix pore system and CT-resolvable bedding-fracture structures. To ensure the comparability between bedding-parallel and bedding-normal flow measurements, the H- and V-series samples were prepared as paired cores from the same depth intervals. The sample size in this study was determined by the availability of paired horizontal and vertical cores with complete porosity, stress-sensitive permeability, X-ray computed tomography reconstruction, and fractal-dimension data. The 24 samples include 12 H-series cores and 12 V-series cores, corresponding to 12 paired depth intervals. Although a larger sample size would further improve statistical robustness, the paired-core design was selected to minimize lithologic and burial-depth differences and to isolate the influence of bedding orientation and bedding-fracture connectivity on permeability anisotropy. Therefore, the present dataset provides a controlled basis for evaluating bedding-fracture-controlled anisotropic flow, while future studies should include more paired cores from additional depth intervals to further validate the proposed model. For example, H-1 and V-1 were drilled from the same depth interval of the same downhole core, and the same pairing principle was applied to the remaining numbered samples. In each pair, the H-series core was drilled approximately parallel to the bedding plane, whereas the corresponding V-series core was drilled approximately perpendicular to the bedding plane. This paired sampling design minimizes the influence of lithologic variation, mineral composition, burial depth, and diagenetic heterogeneity, allowing the observed differences in porosity, permeability, and CT-derived bedding-fracture structure to be attributed primarily to bedding orientation and anisotropic pore–fracture connectivity. Permeability was measured under multiple confining pressures to evaluate stress-sensitive flow behavior. The confining pressure was increased stepwise, and permeability was measured after the flow response reached a stable state at each pressure level. The tested pressure levels included 10, 20, 30, 40, and 50 MPa. These pressure stages were selected to capture the progressive closure of bedding fractures and microfractures under increasing effective stress. For each sample, the permeability measured at low confining pressure reflects the initial connected flow capacity, whereas the permeability measured at higher pressure reflects the residual flow capacity after stress-induced fracture closure. For the H-series samples, the flow direction was approximately parallel to the bedding plane. Therefore, the measured permeability mainly reflects bedding-parallel flow through lamination-related pore–fracture pathways. For the V-series samples, the flow direction was approximately perpendicular to bedding. In this case, the measured permeability is controlled by cross-bedding pore connectivity, fracture intersections, and the sealing effect of bedding interfaces. This distinction provides the basis for evaluating permeability anisotropy between k_h and k_v. The porosity and stress-sensitive permeability data of the paired H- and V-series LGS shale oil cores are listed in

Table 2. Porosity and stress-sensitive permeability of H- and V-series LGS shale oil cores.

Sample ID	Porosity (%)	k at 10 MPa (mD)	k at 20 MPa (mD)	k at 30 MPa (mD)	k at 40 MPa (mD)	k at 50 MPa (mD)
H-1#	1.2456	0.13963	0.05682	0.02374	0.01037	0.00491
H-2#	1.6842	0.25837	0.10159	0.04128	0.01664	0.00679
H-3#	0.7358	0.07318	0.03192	0.01417	0.00642	0.00308
H-4#	1.0387	0.17852	0.07116	0.02963	0.01227	0.00523
H-5#	0.6845	0.06173	0.02746	0.01259	0.00583	0.00286
H-6#	1.3569	0.15184	0.06147	0.02582	0.01096	0.00514
H-7#	1.8745	0.28391	0.11268	0.04613	0.01876	0.00752
H-8#	2.4368	0.41684	0.16153	0.06517	0.02608	0.01024
H-9#	2.7124	0.54269	0.21037	0.08394	0.03341	0.01283
H-10#	1.9362	0.29647	0.11734	0.04762	0.01938	0.00786
H-11#	1.7526	0.33761	0.13024	0.05137	0.02023	0.00774
H-12#	1.0835	0.18743	0.07386	0.03019	0.01208	0.00527
V-1#	0.0311	0.03982	0.00907	0.00324	0.00083	0.00021
V-2#	0.1457	0.00858	0.00323	0.00131	0.00059	0.00028
V-3#	0.3552	0.01073	0.00408	0.00176	0.00082	0.00039
V-4#	0.2513	0.00741	0.00286	0.00134	0.00061	0.00031
V-5#	2.0555	0.00492	0.00234	0.00118	0.00072	0.00043
V-6#	0.1465	0.01276	0.00472	0.00193	0.00079	0.00038
V-7#	2.9349	0.01863	0.00684	0.00268	0.00112	0.00052
V-8#	1.1264	0.00614	0.00279	0.00147	0.00076	0.00048
V-9#	2.1214	0.01517	0.00563	0.00224	0.00094	0.00042
V-10#	1.9866	0.00428	0.00207	0.00113	0.00063	0.00039
V-11#	1.3248	0.00364	0.00182	0.00097	0.00058	0.00041
V-12#	2.1189	0.00313	0.00157	0.00091	0.00056	0.00038

.

The stress sensitivity of permeability was described using an exponential attenuation model:

$$k\left(\sigma \right)=k_0\mathrm{e}\mathrm{x}\mathrm{p}(-\alpha \sigma )$$

(1)

where $k\left(\sigma \right)$ is the permeability under confining pressure or effective stress, $k_0$ is the extrapolated initial permeability, and $\alpha$ is the stress-sensitivity coefficient. Taking the natural logarithm gives

$$\mathrm{l}\mathrm{n} k=\mathrm{l}\mathrm{n} k_0-\alpha \sigma$$

(2)

A larger $\alpha$ indicates stronger stress sensitivity and more significant closure of bedding-fracture-controlled flow channels. Similar stress-dependent permeability behavior has been observed in tight formations and shale fractures, where permeability evolution is strongly influenced by microfracture closure and pore–fracture deformation under increasing stress [13,14].

2.3. CT Scanning and Three-Dimensional Reconstruction

X-ray computed tomography scanning was performed to characterize the internal bedding-related structures of the LGS shale oil cores. CT imaging provides a non-destructive method for observing the three-dimensional distribution of bedding planes, bedding fractures, lamination-related microfractures, and connected voids inside the core. Because this study focuses on the influence of bedding on porosity and permeability anisotropy, the CT interpretation mainly emphasized the identification and reconstruction of bedding-related fracture structures rather than isolated matrix pores. Compared with two-dimensional surface observations, CT reconstruction allows the spatial continuity, orientation, and connectivity of bedding-controlled structures to be evaluated in three dimensions, which is essential for interpreting anisotropic flow behavior in laminated shale. Each cylindrical core was scanned along its axial direction using an X-ray computed tomography system. For shale plugs with a diameter of approximately 25 mm and a length of approximately 50 mm, representative scanning parameters were selected to balance image resolution, penetration capability, and signal-to-noise ratio. The scanning voltage, current, exposure time, voxel size, and image resolution were approximately 160 kV, 100 μA, 500 ms, 25 μm, and 2000 × 2000 pixels, respectively. The scan was conducted over 360°, with approximately 1440 projections collected for each core. The slice interval was set equal to the voxel size, and a metal filter was used to reduce beam-hardening artifacts. The CT slices were reconstructed using a filtered back-projection algorithm. The obtained grayscale CT slices were first inspected to identify bedding planes, bedding-related low-density bands, lamination-parallel fractures, high-density mineral components, and local microfracture zones. Image preprocessing was then performed to reduce noise and improve the contrast between the shale matrix and the CT-resolvable bedding-fracture space. The reconstructed grayscale slices were corrected for grayscale drift and filtered to reduce random noise. A threshold-based segmentation method was used to extract low-density bedding-related pore–fracture features from the shale matrix. Manual correction was applied where necessary to remove high-density mineral artifacts, edge artifacts, and isolated noise voxels. Connected-component analysis was then performed to identify bedding-fracture clusters and calculate the volume of the largest connected component. The photographs and CT-reconstructed pore–fracture structures of all H- and V-series LGS shale oil cores are shown in Figure 3.

2.4. Bedding-Fracture Identification and Structural Parameters

Bedding fractures were identified from the CT-reconstructed pore–fracture volumes based on their morphology, orientation, continuity, and spatial relationship with lamination. In laminated shale cores, bedding fractures generally appear as elongated, planar, or band-like low-density features distributed along or near bedding interfaces. Compared with isolated pores, bedding fractures show stronger directionality and larger aspect ratios. Compared with randomly distributed microcracks, bedding fractures are more closely associated with lamination-parallel or lamination-controlled structures.

The identification process consisted of three steps. First, the CT-resolvable pore–fracture phase was segmented from the grayscale volume. Second, connected-component analysis was used to distinguish isolated pores, small microcracks, and larger connected bedding-fracture clusters. Third, bedding-fracture components were extracted based on geometric criteria, such as orientation, length, thickness, continuity, and volume contribution. The extracted bedding-fracture network was then used for quantitative structural analysis.

The procedure for CT image segmentation, bedding-fracture extraction, and three-dimensional box-counting fractal analysis is shown in Figure 4. This procedure provides a consistent workflow for transforming grayscale CT images into quantitative structural and fractal parameters.

The following CT-derived parameters were defined.

The pore–fracture volume fraction, $V_f$ was calculated as

$$V_f={\displaystyle \frac{V_{pf}}{V_{core}}}$$

(3)

where $V_{pf}$ is the volume of the segmented pore–fracture phase and $V_{core}$ is the total analyzed core volume. This parameter reflects the volumetric contribution of CT-resolvable pore–fracture space.

The bedding-fracture number, $N_b$, was defined as the number of identifiable bedding-fracture bands or connected bedding-fracture components within the analyzed core volume. The bedding-fracture density, ${\rho }_b$, was calculated as

$${\rho }_b={\displaystyle \frac{N_b}{L}}$$

(4)

where L is the core length or the length of the analyzed volume along the core axis. This parameter normalizes fracture abundance by sample length and allows comparison among different samples.

The connectivity index, $C_b$, was defined as

$$C_b={\displaystyle \frac{V_{max}}{V_{pf}}}$$

(5)

where $V_{max}$ is the volume of the largest connected pore–fracture cluster. A larger $C_b$ indicates that a greater proportion of the pore–fracture space belongs to a connected network, which is expected to contribute more strongly to permeability.

The permeability anisotropy coefficient, $A_k$, was defined as

$$A_k={\displaystyle \frac{k_h}{k_v}}$$

(6)

where $k_h$ is the bedding-parallel permeability measured from H-series cores and $k_v$ is the bedding-normal permeability measured from V-series cores. When direct one-to-one pairing between H- and V-series samples is not available, the anisotropy coefficient can be evaluated using grouped statistical averages or samples from comparable lithologic intervals.

These structural parameters provide quantitative descriptors of bedding-fracture abundance, storage contribution, and flow connectivity. They are used in later sections to examine the relationships among CT-derived pore–fracture structure, porosity, permeability, and fractal dimension. The CT-derived bedding-fracture parameters of the H- and V-series LGS shale oil cores are listed in

Table 3. CT-derived bedding-fracture parameters of H- and V-series LGS shale oil cores.

Sample ID	Vf (%)	Nb	${\mathit{\rho }}_{\mathit{b}}$ (cm−1)	Cb	D3
H-1#	1.18	3	6.05	0.43	2.27
H-2#	1.61	5	10.1	0.59	2.42
H-3#	0.74	2	4.04	0.36	2.18
H-4#	1.07	4	8.07	0.51	2.34
H-5#	0.67	2	4.03	0.33	2.16
H-6#	1.32	3	6.04	0.45	2.29
H-7#	1.83	5	10.06	0.57	2.40
H-8#	2.39	7	14.08	0.69	2.54
H-9#	2.71	8	16.11	0.77	2.63
H-10#	1.91	5	10.09	0.56	2.43
H-11#	1.74	6	12.04	0.63	2.48
H-12#	1.11	4	7.99	0.5	2.33
V-1#	0.46	4	8.07	0.54	2.30
V-2#	0.31	2	4.04	0.34	2.12
V-3#	0.63	3	6.07	0.41	2.21
V-4#	0.97	5	10.09	0.57	2.36
V-5#	1.84	7	14.12	0.64	2.48
V-6#	0.56	3	6.03	0.39	2.19
V-7#	1.36	4	8.06	0.46	2.31
V-8#	1.69	6	12.07	0.6	2.44
V-9#	1.12	4	8.05	0.44	2.27
V-10#	1.59	7	14.13	0.59	2.45
V-11#	1.91	8	16.05	0.68	2.52
V-12#	2.08	9	17.95	0.72	2.56

.

2.5. Three-Dimensional Box-Counting Fractal Dimension

The geometry of bedding-fracture networks in shale is highly irregular and cannot be fully described by simple Euclidean parameters, such as fracture number or volume fraction. Therefore, three-dimensional box-counting fractal analysis was used to quantify the spatial complexity and space-filling ability of CT-resolvable pore–fracture networks.

The box-counting method was applied to the binarized three-dimensional pore–fracture volume obtained from the CT image-processing workflow shown in Figure 4. The analyzed domain was covered by cubic boxes with side length $\epsilon$. For each box size, the number of boxes containing at least one pore–fracture voxel was counted and denoted as $N\left(\epsilon \right)$. If the pore–fracture network exhibits fractal scaling within the selected scale range, $N\left(\epsilon \right)$ follows:

$$N\left(\epsilon \right)\sim {\epsilon }^{-D_3}$$

(7)

where D₃ is the three-dimensional box-counting fractal dimension. Taking the logarithm of both sides gives

$$logN\left(\epsilon \right)=D_{3}log\left({\displaystyle \frac{1}{\epsilon }}\right)+C$$

(8)

where C is a constant. Therefore, D₃ can be obtained from the slope of the linear fitting relationship between $logN\left(\epsilon \right)$ and $log\left({\displaystyle \frac{1}{\epsilon }}\right)$.

For each core, a series of box sizes was selected according to the voxel resolution and the size of the analyzed volume. Very small box sizes that were strongly affected by image noise and very large box sizes that contained insufficient statistical information were excluded from the fitting range. The coefficient of determination, R², was used to evaluate the reliability of the fractal fitting. A higher R² indicates that the CT-derived pore–fracture structure follows a clearer fractal scaling relationship within the selected scale range. In a three-dimensional system, the theoretical range of D₃ is between 0 and 3. For bedding-fracture networks in shale, a higher D₃ generally indicates a more complex, more spatially distributed, and more space-filling pore–fracture structure. However, D₃ should not be interpreted as a direct substitute for porosity or connectivity. Porosity mainly reflects the volume of pore–fracture space, whereas connectivity controls whether the pore–fracture space contributes to effective flow. The fractal dimension provides complementary information by describing the spatial complexity and multiscale distribution of the network. Three-dimensional fractal analysis based on CT reconstruction has been used to quantify fracture networks, pore structures, and damage evolution in rocks [19,20,24]. In this study, D₃ was combined with CT-derived structural parameters, porosity, and permeability to establish a fractal-corrected porosity–permeability model for LGS shale oil cores. The three-dimensional fractal dimension of the CT-derived pore–fracture networks is also summarized in Table 3.

3. Results Analysis

3.1. Porosity and Permeability Characteristics

The porosity and stress-sensitive permeability of the H- and V-series LGS shale oil cores are listed in Table 2. The H-series samples show porosity values ranging from 0.6845% to 2.7124%, with an average value of 1.5451%. The V-series samples show a wider porosity range, from 0.0311% to 2.9349%, with an average value of 1.2165%. Although the average porosity of the H-series is slightly higher than that of the V-series, the two groups exhibit overlapping porosity ranges, indicating that porosity alone cannot fully distinguish the storage characteristics of the two coring directions. In contrast, permeability shows a much stronger directional difference. At 10 MPa confining pressure, the permeability of the H-series samples ranges from 0.06173 to 0.54269 mD, with an average value of 0.24402 mD. The V-series samples show significantly lower permeability, ranging from 0.00313 to 0.03982 mD, with an average value of 0.01127 mD. This contrast indicates that bedding-parallel flow is much more favorable than bedding-normal flow in the LGS shale oil cores. The paired comparison between corresponding H- and V-series samples further confirms this anisotropic behavior. The permeability anisotropy coefficient A_k at 10 MPa varies among sample pairs but is consistently greater than unity, demonstrating the dominant contribution of bedding-parallel pore–fracture pathways to flow capacity.

As confining pressure increases from 10 to 50 MPa, permeability decreases continuously for both H- and V-series samples. For the H-series, the average permeability decreases from 0.24402 mD at 10 MPa to 0.00696 mD at 50 MPa. For the V-series, the average permeability decreases from 0.01127 mD at 10 MPa to 0.00040 mD at 50 MPa. The continuous permeability reduction reflects progressive closure of bedding fractures, lamination-related microfractures, and connected pore–fracture pathways under increasing confining pressure. However, the H-series samples maintain higher absolute permeability than the V-series samples at all pressure levels, suggesting that bedding-parallel flow paths retain higher residual flow capacity even after stress-induced closure. Figure 5 shows the porosity and permeability characteristics of the H- and V-series samples. Figure 6 further illustrates the stress-sensitive permeability evolution under increasing confining pressure.

3.2. CT Reconstruction of Bedding-Fracture Structures

CT reconstruction was used to visualize the internal bedding-related structures of all H- and V-series LGS shale oil cores. As shown in Figure 3, the red phase represents the segmented bedding-related structural features, including bedding planes, bedding fractures, lamination-related microfractures, and connected voids along bedding interfaces, whereas the uncolored or transparent region represents the shale matrix. Clear differences can be observed between the two coring directions. In the H-series cores, the bedding-related red phase is mainly distributed along or subparallel to the core axis, indicating relatively continuous bedding-parallel structural features. In contrast, the V-series cores show more segmented lamination bands, bedding-normal intersections, and discontinuous layered features. These CT observations provide a direct structural basis for distinguishing bedding-parallel permeability k_h from bedding-normal permeability k_v. The hydraulic implication of these structural differences is further discussed in Section 5.

3.3. Quantitative Bedding-Fracture Parameters

The CT-derived bedding-fracture parameters are listed in Table 3. These parameters include bedding-fracture volume fraction V_f, bedding-fracture number N_b, bedding-fracture density ${\rho }_b$, connectivity index C_b, and three-dimensional fractal dimension D₃. For the H-series samples, V_f ranges from 0.67% to 2.71%, with an average value of 1.5233%. The bedding-fracture number varies from 2 to 8, and the bedding-fracture density ranges from 4.03 to 16.11 cm⁻¹. The connectivity index ranges from 0.33 to 0.77, indicating variable degrees of connected bedding-fracture development among the H-series samples. H-8 and H-9 exhibit relatively high V_f, ${\rho }_b$, and C_b, consistent with their high permeability values. For the V-series samples, V_f ranges from 0.31% to 2.08%, with an average value of 1.2100%. The bedding-fracture number varies from 2 to 9, and the bedding-fracture density ranges from 4.04 to 17.95 cm⁻¹. The connectivity index ranges from 0.34 to 0.72. Some V-series samples, such as V-11 and V-12, show relatively high bedding-fracture density and connectivity index, but their measured permeability remains low. This indicates that the CT-derived abundance and connectivity of bedding-related structures must be interpreted together with the flow direction. For vertically cored samples, bedding-related structures may be well developed but may not form effective continuous pathways along the bedding-normal flow direction. Figure 7 summarizes the CT-derived bedding-fracture parameters of the H- and V-series samples. These results indicate that both groups contain CT-resolvable bedding-related structures, but the effect of these structures on permeability depends strongly on their orientation and flow-direction connectivity.

3.4. Three-Dimensional Fractal Dimension

The three-dimensional box-counting fractal dimension D₃ was calculated from the CT-derived bedding-fracture structures. As shown in Table 3, the D₃ values of the H-series samples range from 2.16 to 2.63, with an average value of 2.3725. The D₃ values of the V-series samples range from 2.12 to 2.56, with an average value of 2.3508. The similar average values indicate that both coring directions contain spatially complex bedding-related structures. For the H-series samples, relatively high D₃ values are generally associated with higher V_f, ρ_b, and C_b. For example, H-8 and H-9 have high D₃ values of 2.54 and 2.63, respectively, corresponding to well-developed bedding-fracture volume fraction, density, and connectivity. For the V-series samples, some cores such as V-11 and V-12 also show relatively high D₃ values, but their permeability remains low. This contrast indicates that D₃ should be interpreted together with flow direction and connectivity. Figure 8 shows the distribution of D₃ and its relationships with CT-derived bedding-fracture parameters.

3.5. Relationship Between Fractal Dimension and Porosity

The relationship between porosity and CT-derived bedding-fracture parameters was analyzed to evaluate the storage contribution of bedding-related structures. As shown in Figure 9, porosity generally increases with increasing V_f, ρ_b, Cb, and D₃ in the H-series samples. Samples with higher D₃, such as H-8 and H-9, also show higher porosity values, indicating that CT-resolvable bedding-fracture development contributes to measurable pore volume. For the V-series samples, the relationship between porosity and CT-derived parameters is more scattered. Some V-series samples show relatively high porosity but low permeability, suggesting that pore–fracture volume does not necessarily correspond to effective bedding-normal flow capacity. These results indicate that porosity primarily reflects volumetric storage contribution, whereas permeability requires further evaluation of directional connectivity and bedding-fracture orientation.

3.6. Relationship Between Fractal Dimension and Permeability

The relationship between CT-derived parameters and permeability was evaluated using the permeability measured at 10 MPa as the representative low-stress flow capacity. Because permeability varies over a wide range, logarithmic permeability was used for correlation analysis and visualization. As shown in Figure 10 and

Table 4. Statistical evaluation of relationships between permeability and CT-derived parameters.

Model or Parameter	H-Series	V-Series	All Samples
Correlation between log k10 and ρb	0.9331	−0.2905	0.2380
Correlation between log k10 and Vf	0.9386	−0.7220	0.2558
Correlation between log k10 and Cb	0.9760	−0.6214	0.0618
Correlation between log k10 and D3	0.9794	−0.6739	0.1120
Conventional model R2	0.9231	0.3216	0.0623
Fractal-corrected model R2	0.9945	0.7016	0.8804
RMSE of conventional model	0.1746	0.5904	1.6721
RMSE of fractal-corrected model	0.0465	0.3916	0.5971
MAE of conventional model	0.1441	0.4793	1.5415
MAE of fractal-corrected model	0.0414	0.2754	0.5155

, the H-series samples show strong positive correlations between logk₁₀ and CT-derived structural parameters. The correlation coefficients between logk₁₀ and ρ_b, V_f, C_b, and D₃ are 0.9331, 0.9386, 0.9760, and 0.9794, respectively. These results indicate that bedding-fracture abundance, connectivity, and fractal complexity are closely related to bedding-parallel permeability. In contrast, the V-series samples show negative or weak correlations between logk₁₀ and these parameters. The correlation coefficients between logk₁₀ and V_f, C_b, and D₃ are −0.7220, −0.6214, and −0.6739, respectively. This indicates that high bedding-fracture abundance or complexity does not necessarily enhance bedding-normal permeability. Therefore, the hydraulic significance of V_f, C_b, and D₃ depends strongly on flow direction. The model comparison in Table 4 further shows that the fractal-corrected model improves permeability prediction relative to the conventional porosity-only model. For all samples, the model R² increases from 0.0623 to 0.8804, while RMSE decreases from 1.6721 to 0.5971 in logarithmic space.

To further support the observed trends, Pearson correlation coefficients and model-evaluation metrics were calculated in logarithmic permeability space. The results are summarized in Table 4. The statistical results confirm that CT-derived bedding-fracture parameters have different hydraulic significance in the two coring directions. They are strongly positively correlated with bedding-parallel permeability in the H-series, but they do not show the same positive relationship with bedding-normal permeability in the V-series.

3.7. Stress Sensitivity of Bedding-Fracture-Controlled Permeability

Permeability decreases continuously with increasing confining pressure for both H- and V-series samples. For the H-series, the average permeability decreases from 0.24402 mD at 10 MPa to 0.00696 mD at 50 MPa, corresponding to a reduction of approximately 97.1%. For the V-series, the average permeability decreases from 0.01127 mD to 0.00040 mD, corresponding to a reduction of approximately 96.5%. These results indicate strong stress sensitivity of bedding-fracture-controlled flow channels. Although the H-series cores have higher initial permeability, they also experience substantial permeability attenuation under increasing confining pressure. The V-series cores maintain much lower absolute permeability throughout the tested pressure range. The fitted stress-sensitivity coefficients and their relationships with CT-derived parameters are shown in Figure 11. The physical mechanism of stress-sensitive anisotropic flow is discussed in Section 5.3.

4. Fractal-Corrected Porosity–Permeability Model

4.1. Conventional Porosity–Permeability Relationship

Porosity is commonly used as the primary parameter for evaluating reservoir storage capacity and predicting permeability. For conventional porous media, permeability is often expressed as a power-law function of porosity:

$$k=A{\varphi }^m$$

(9)

where k is permeability, $\varphi$ is porosity, and A and m are empirical fitting parameters. Taking the logarithm gives

$$\mathrm{l}\mathrm{n} k=\mathrm{l}\mathrm{n} A+m\mathrm{l}\mathrm{n} \varphi$$

(10)

This relationship assumes that an increase in pore volume generally enhances flow capacity. However, for laminated shale oil cores, permeability is not controlled only by the total pore volume. Bedding fractures, lamination-related microfractures, pore–fracture connectivity, and flow direction relative to bedding can strongly affect the effective seepage pathway. Therefore, samples with similar porosity may exhibit significantly different permeability. In this study, the conventional porosity–permeability relationship was first fitted using the permeability measured at 10 MPa, k₁₀, as the representative low-stress permeability. The results show that porosity alone can partially describe permeability variation but cannot adequately capture the large difference between the H- and V-series samples. The H-series samples generally show much higher permeability than the V-series samples, even when their porosity ranges overlap. This indicates that porosity mainly represents storage capacity, whereas permeability is more sensitive to bedding-fracture orientation and connectivity. The limitation of the conventional porosity–permeability relationship is especially evident for V-series samples. Some V-series cores have relatively high porosity but low permeability, suggesting that their pore–fracture space does not form effective continuous pathways in the bedding-normal direction. Therefore, additional structural descriptors derived from CT reconstruction and fractal analysis are required to improve permeability prediction.

4.2. Fractal Correction Factor

To incorporate the influence of bedding-fracture complexity into permeability prediction, a fractal correction factor was introduced. The three-dimensional box-counting fractal dimension D₃ quantifies the spatial complexity and space-filling ability of the CT-resolvable bedding-fracture network. However, D₃ alone does not directly represent permeability because flow capacity also depends on pore–fracture volume and connectivity. Therefore, the fractal correction should be combined with CT-derived structural parameters. The fractal correction factor is defined as

$$F_D=\mathrm{e}\mathrm{x}\mathrm{p}\left[\lambda \right(D_3-D_0\left)\right]$$

(11)

where D₀ is a reference fractal dimension and $\lambda$ is the fractal sensitivity coefficient. A larger F_D indicates stronger contribution of spatially complex bedding-fracture structures to the effective flow system. In this study, D₀ can be taken as the mean D₃ value of all samples or treated as a regression reference value.

Because bedding-fracture-controlled permeability is also affected by volume fraction and connectivity, a combined structural correction factor can be written as

$$F_b=V_f^n{C}_b^{\eta }\mathrm{e}\mathrm{x}\mathrm{p}\left[\lambda \right(D_3-D_0\left)\right]$$

(12)

where V_f is the CT-derived bedding-fracture volume fraction, C_b is the connectivity index, and n, $\eta$, and $\lambda$ are fitting parameters. This formulation links the volumetric contribution, connected-network contribution, and fractal complexity of bedding-related structures.

Physically, V_f describes the amount of CT-resolvable bedding-fracture space, C_b describes the proportion of the connected pore–fracture network, and D₃ describes the spatial complexity of that network. Therefore, the correction factor provides a more comprehensive structural descriptor than porosity alone. Figure 12 shows the distribution of F_D and its relationships with D₃ and permeability.

4.3. Fractal-Corrected Permeability Model

Based on the conventional porosity–permeability relationship and the fractal correction factor, the permeability of LGS shale oil cores can be expressed as

$$k=A{\varphi }^m{V}_f^n{C}_b^{\eta }\mathrm{e}\mathrm{x}\mathrm{p}\left[\lambda \right(D_3-D_0\left)\right]$$

(13)

where A, m, n, $\eta$, and $\lambda$ are regression coefficients. Taking the logarithm gives the linearized form:

$$\mathrm{l}\mathrm{n} k=\mathrm{l}\mathrm{n} A+m\mathrm{l}\mathrm{n} \varphi +n\mathrm{l}\mathrm{n} V_f+\eta \mathrm{l}\mathrm{n} C_b+\lambda (D_3-D_0)$$

(14)

This model explicitly incorporates porosity, CT-derived bedding-fracture volume fraction, connectivity, and fractal dimension. Compared with the conventional model, the fractal-corrected model can better reflect the fact that permeability is controlled by both pore volume and effective flow-channel geometry.

For the H-series samples, the bedding-fracture network is generally aligned with the flow direction. Therefore, higher V_f, C_b, and D₃ tend to increase permeability. In this case, the fractal-corrected model can be interpreted as a bedding-parallel flow-enhancement model. For the V-series samples, bedding-fracture structures may be abundant and geometrically complex, but they are not necessarily connected in the bedding-normal direction. Therefore, high D₃ or high V_f does not always correspond to high vertical permeability. This suggests that the model coefficients may differ between bedding-parallel and bedding-normal flow directions. Accordingly, the model can be fitted separately for the H- and V-series samples, or a group indicator can be introduced to represent the effect of flow direction. A practical direction-dependent form can be written as

$$\mathrm{l}\mathrm{n} k_i=\mathrm{l}\mathrm{n} A_i+m_i\mathrm{l}\mathrm{n} \varphi +n_i\mathrm{l}\mathrm{n} V_f+{\eta }_i\mathrm{l}\mathrm{n} C_b+{\lambda }_i(D_3-D_0)$$

(15)

where the subscript i represents either the H-series or V-series group. This formulation allows the same structural parameters to have different hydraulic significance depending on bedding orientation.

4.4. Permeability Anisotropy Model

The paired sampling design allows permeability anisotropy to be evaluated using corresponding H- and V-series samples drilled from the same depth intervals. The permeability anisotropy coefficient is defined as Equation (6). Because paired H–V samples were taken from the same depth interval, the difference in A_k mainly reflects the influence of bedding orientation and directional connectivity rather than lithologic or burial-depth variation. The anisotropy coefficient can be linked to paired differences in CT-derived parameters:

$$\Delta D_3=D_{3,h}-D_{3,v}$$

(16)

$$\Delta C_b=C_{b,h}-C_{b,v}$$

(17)

$$\Delta V_f=V_{f,h}-V_{f,v}$$

(18)

A permeability anisotropy model can then be expressed as

$$\mathrm{l}\mathrm{n} A_k=b_0+b_1\Delta D_3+b_2\Delta V_f+b_3\Delta C_b$$

(19)

where b₀, b₁, b₂, and b₃ are fitted coefficients. This model reflects the concept that permeability anisotropy is controlled by the contrast in bedding-fracture complexity, volume contribution, and connectivity between bedding-parallel and bedding-normal flow systems.

When the H-series core has a more connected and spatially complex bedding-fracture structure than its paired V-series core, A_k is expected to increase. However, if the V-series core contains abundant bedding-related structures that are not connected in the vertical flow direction, its high D₃ may not reduce anisotropy effectively. Therefore, D₃ should be interpreted together with C_b, V_f, and the flow direction. The paired permeability anisotropy model and its relationship with CT-derived structural differences are shown in Figure 13.

4.5. Model Evaluation

The performance of the conventional and fractal-corrected permeability models was evaluated using measured versus predicted permeability, coefficient of determination R², root-mean-square error (RMSE), and mean absolute error (MAE). Because permeability spans several orders of magnitude, model fitting and error evaluation were performed in logarithmic space. This approach avoids excessive weighting of high-permeability samples and provides a more balanced assessment of prediction accuracy for both H- and V-series cores. The conventional model used only porosity as the predictor and followed the commonly used porosity–permeability power-law relationship. This type of model assumes that permeability is mainly controlled by pore volume and has been widely used as a first-order empirical description of porous-media flow. However, in laminated shale oil cores, pore volume alone cannot fully represent effective flow capacity because permeability is also controlled by bedding-fracture orientation, connected pore–fracture pathways, and stress-sensitive fracture aperture. Therefore, the conventional model was used as the baseline model to evaluate whether additional CT-derived structural parameters can improve permeability prediction. The fractal-corrected model incorporates porosity $\varphi$, bedding-fracture volume fraction V_f, connectivity index C_b, and three-dimensional fractal dimension D₃. This model is conceptually consistent with previous fractal permeability models, which have shown that permeability is affected not only by porosity but also by pore-size distribution, tortuosity, fracture-network geometry, and pore–fracture complexity. Compared with these existing models, the present model is specifically designed for CT-resolvable bedding-fracture systems in paired horizontal and vertical shale oil cores. Thus, it extends the conventional porosity–permeability framework by explicitly considering bedding-fracture abundance, connected-network contribution, and three-dimensional spatial complexity. The comparison between the two models is important because the present data show that samples with similar porosity can have markedly different permeability depending on bedding orientation. For example, some V-series cores have relatively high porosity or high CT-derived structural abundance, but their bedding-normal permeability remains low because the pore–fracture network is not effectively connected along the flow direction. In contrast, H-series cores generally show higher permeability when bedding fractures are aligned with the flow direction. Therefore, model reliability should be evaluated not only by statistical fitting metrics, such as R², RMSE, and MAE, but also by whether the model can explain the physical mechanism of bedding-controlled anisotropic flow.

A lower prediction error and higher R² for the fractal-corrected model indicate that CT-derived bedding-fracture structure and fractal complexity provide essential information beyond porosity. In particular, V_f reflects the volumetric contribution of CT-resolvable bedding-fracture space, C_b represents the connected-network contribution, and D₃ characterizes the spatial complexity and space-filling ability of the pore–fracture network. When these parameters are combined with porosity, the model provides a more physically meaningful interpretation of permeability variation in laminated shale oil cores than the porosity-only model. The model-evaluation results are summarized in Table 4. The conventional porosity-only model shows limited predictive ability when all samples are considered together, with an R² of only 0.0623 in logarithmic permeability space. This low value reflects the inability of porosity alone to explain the large permeability contrast between bedding-parallel and bedding-normal flow. After incorporating V_f, C_b, D₃, and the bedding-related flow-direction effect, the fractal-corrected model improves the all-sample R² to 0.8804 and reduces RMSE from 1.6721 to 0.5971.

It should also be noted that the proposed fractal-corrected model is a core-scale semi-empirical model. Its applicability is strongest for laminated shale oil cores in which CT-resolvable bedding fractures and lamination-related microfractures exert significant control on flow capacity. For samples dominated by sub-resolution nanopores, clay-bound pores, or matrix diffusion, additional nanoscale characterization methods, such as NMR, mercury intrusion, gas adsorption, or FIB–SEM, may be required. Therefore, the model should be recalibrated before being applied to other shale formations, different CT resolutions, different stress conditions, or multiphase-flow systems.

This interpretation is also consistent with studies on other porous materials, where permeability and transport properties are shown to depend not only on total porosity but also on pore-structure evolution, connectivity, and fractal characteristics. For example, Wang et al. [45] showed that silica fume modifies the pore structure and fractal dimension of low-heat Portland cement-based materials, supporting the use of fractal dimension to describe pore-structure complexity. Yu et al. [46] modeled the permeability and thermal conductivity of low-heat Portland cement using thermodynamics and fractal theory, showing that hydration-induced pore-structure evolution strongly affects transport properties. Li et al. [47] further demonstrated that water-to-cement ratio, curing temperature, and MgO admixture can modify the pore structure and porosity of low-heat Portland cement pastes, thereby influencing their transport-related properties. Although these studies focus on cement-based materials rather than shale, they support the broader porous-media concept that porosity alone is insufficient to determine transport behavior.

5. Discussion

5.1. Physical Meaning of D₃

The three-dimensional fractal dimension D₃ provides a quantitative measure of the spatial complexity and space-filling ability of CT-resolvable bedding-fracture structures. A higher D₃ indicates that the bedding-related pore–fracture network occupies the three-dimensional space in a more complex and distributed manner. However, D₃ should not be interpreted as a direct equivalent of porosity or permeability. Porosity is a volumetric parameter, whereas permeability is a directional transport property controlled by connectivity, aperture, tortuosity, and stress state. This distinction is evident in the V-series samples, where several cores show relatively high D₃ but low permeability. Therefore, D₃ should be regarded as a complementary structural descriptor. Its hydraulic significance becomes meaningful only when it is interpreted together with V_f, C_b, bedding orientation, and flow direction.

5.2. Bedding-Fracture Control on Pore–Fracture Structure

The hydraulic contribution of bedding fractures differs between storage and flow. Bedding-related structures can increase pore–fracture volume and thus contribute to porosity, but their contribution to permeability depends on whether they form continuous pathways along the flow direction. Mechanistically, this difference is controlled by the directional arrangement of bedding fractures relative to the imposed flow direction. In the H-series cores, the core axis is approximately parallel to the bedding plane. Therefore, bedding fractures, lamination-related microfractures, and connected voids are more likely to form continuous or semi-continuous flow channels along the axial direction. These connected bedding-parallel pathways reduce flow resistance and explain the relatively high permeability of the H-series cores. In contrast, in the V-series cores, the flow direction is approximately perpendicular to bedding. Fluid flow must pass across bedding interfaces and depends on limited cross-bedding connections among discontinuous fractures, matrix pores, and local fracture intersections. As a result, bedding-related structures may contribute to pore–fracture volume and structural complexity but may not form effective bedding-normal flow pathways. In H-series cores, bedding fractures are favorably aligned with the flow direction, so bedding-parallel connectivity enhances permeability. In V-series cores, bedding interfaces are intersected by the flow direction, and effective flow requires cross-bedding connection among discontinuous fractures and matrix pores. This explains why some V-series cores have relatively high porosity, V_f, or D₃, but still exhibit low bedding-normal permeability. Therefore, bedding-fracture evaluation should distinguish between storage-related parameters, such as porosity and V_f, and flow-related parameters, such as directional connectivity and permeability anisotropy.

5.3. Stress Sensitivity and Anisotropic Flow Mechanism

The stress-sensitive permeability behavior can be attributed to progressive closure of bedding fractures, reduction in effective fracture aperture, and weakening of connected pore–fracture pathways. The mechanism of stress sensitivity is related to the mechanical compliance of bedding-related fractures. At low confining pressure, bedding-parallel fractures in the H-series cores retain relatively large effective apertures and provide preferential flow channels. With increasing confining pressure, fracture surfaces are progressively pressed into contact, the effective hydraulic aperture decreases, and narrow flow channels are compressed or partially closed. This process causes strong permeability attenuation even in samples with high initial permeability. For the V-series cores, bedding-normal flow is already restricted by poor cross-bedding connectivity at low confining pressure. Increasing confining pressure further compresses the limited cross-bedding connections, leading to very low residual permeability. Therefore, the observed permeability reduction is caused not only by pore-volume change, but also by directional loss of connected flow pathways. Bedding-parallel flow paths in H-series cores provide high initial permeability but remain sensitive to closure under increasing confining pressure. Bedding-normal flow paths in V-series cores are limited even at low confining pressure because vertical flow requires cross-bedding connectivity. The permeability reductions of approximately 97.1% for the H-series and 96.5% for the V-series from 10 to 50 MPa confirm that both bedding-parallel and bedding-normal flow systems are strongly stress-sensitive. However, the absolute permeability of the H-series remains higher throughout the tested pressure range. Therefore, the anisotropic flow mechanism can be summarized as directional enhancement by bedding-parallel connectivity and stress-sensitive attenuation by fracture closure.

5.4. Implications for Shale Oil Reservoir Evaluation

The practical significance of the proposed approach lies in its ability to distinguish storage-related pore–fracture volume from flow-effective pore–fracture connectivity. In the present dataset, the H- and V-series cores have partly overlapping porosity ranges, but the average permeability of the H-series at 10 MPa is approximately 21.7 times that of the V-series. This large difference cannot be explained by porosity alone. By incorporating V_f, C_b, and D₃, the proposed approach provides additional structural information for identifying whether CT-resolvable bedding-fracture space contributes to effective flow. This is useful for interpreting core permeability data, evaluating bedding-controlled permeability anisotropy, selecting representative permeability inputs for reservoir simulation, and improving the assessment of shale oil sweet spots where bedding-parallel flow pathways are well developed.

The results have important implications for shale oil reservoir evaluation. First, porosity alone is insufficient for predicting permeability in laminated shale oil cores. The average permeability of the H-series at 10 MPa is approximately 21.7 times that of the V-series, although the porosity ranges of the two groups partly overlap. This indicates that reservoir quality evaluation should incorporate bedding orientation and CT-derived structural parameters. Second, three-dimensional fractal analysis provides a useful method for quantifying bedding-fracture complexity. When D₃ is combined with V_f and C_b, the resulting structural description can help distinguish whether CT-resolvable bedding-fracture space contributes to storage only or to effective flow. Third, permeability anisotropy should be considered in shale oil development. Bedding-parallel flow pathways may enhance lateral seepage capacity, whereas bedding-normal flow may remain restricted by discontinuous cross-bedding connectivity. These findings are relevant for core-scale permeability interpretation, sweet-spot evaluation, hydraulic-fracturing design, and permeability input selection for reservoir simulation.

Overall, the findings can be explained by a coupled mechanism of bedding-fracture orientation, directional connectivity, and stress-induced aperture closure. Bedding-parallel permeability is enhanced when lamination-related fractures and connected voids are aligned with the flow direction. Bedding-normal permeability is restricted when bedding-related structures are laterally developed but vertically discontinuous. Increasing confining pressure reduces effective fracture aperture in both directions, resulting in stress-sensitive permeability attenuation. This mechanism is schematically summarized in Figure 14.

5.5. Limitations

Several limitations should be noted. First, CT resolution controls the minimum detectable pore–fracture size. Nanopores and sub-resolution microfractures may not be fully captured by the reconstructed bedding-fracture phase. Therefore, the CT-derived parameters mainly represent resolvable bedding-related pore–fracture structures rather than the complete multiscale pore system. Second, threshold segmentation may influence the calculated V_f, C_b, and D₃. Although manual correction was applied to reduce misclassification, segmentation uncertainty remains an inherent limitation of CT-based analysis. Third, the permeability model was established at the core scale. Upscaling from core-scale bedding-fracture networks to reservoir-scale flow systems requires additional constraints, such as in situ stress state, natural fracture distribution, mineral composition, and hydraulic-fracture connectivity. Fourth, the current model mainly focuses on dry or laboratory-measured permeability. In actual shale oil reservoirs, fluid saturation, wettability, capillary pressure, clay swelling, and multiphase flow may further affect permeability. Future work should combine CT reconstruction with NMR, mercury intrusion, FIB–SEM, and multiphase-flow experiments to establish a more complete multiscale pore–fracture flow model.

The proposed CT-fractal permeability interpretation is not intended to be universally applicable to all shale or tight rocks. It is most suitable for laminated shale oil cores in which bedding-related fractures, lamination-parallel microfractures, and connected voids can be resolved by CT imaging and have a measurable influence on permeability. The method is less applicable to matrix-dominated samples, where permeability is controlled mainly by nanopores below the CT resolution, to samples with severe beam-hardening or mineral-density artifacts, and to rocks in which fracture connectivity is controlled by large-scale natural fractures beyond the core scale. In addition, the model coefficients are core-scale empirical parameters and should be recalibrated before being applied to other formations, different CT resolutions, different stress states, or multiphase-flow conditions.

The sample size is another limitation of this study. The present dataset includes 24 samples, consisting of 12 paired H–V core sets from the same depth intervals. This paired design improves the comparability between bedding-parallel and bedding-normal flow measurements, but the number of samples is still limited for broad statistical generalization. Future work should expand the dataset to include more paired cores from additional depth intervals and lithofacies, which would further improve the robustness, uncertainty evaluation, and transferability of the CT-fractal permeability model.

6. Conclusions

This study addressed the question of how bedding-fracture structures in paired horizontal and vertical LGS shale oil cores selected from the same depth intervals influence porosity, permeability, permeability anisotropy, and stress sensitivity. By integrating X-ray computed tomography reconstruction, three-dimensional box-counting fractal analysis, stress-sensitive permeability testing, and porosity–permeability modeling, this work establishes a quantitative framework for evaluating bedding-fracture-controlled anisotropic flow in laminated shale oil cores. The originality of this study lies in using paired H- and V-series cores from comparable depth intervals to reduce lithologic and burial-depth effects and to isolate the directional influence of bedding-fracture connectivity. The engineering relevance is that the proposed CT-fractal evaluation approach helps explain permeability differences that cannot be captured by conventional porosity–permeability relationships alone. The main conclusions are as follows:

(1). The H- and V-series LGS shale oil cores show partly overlapping porosity ranges but significantly different permeability. At 10 MPa, the average permeability of the H-series is 0.24402 mD, approximately 21.7 times that of the V-series 0.01127 mD. This indicates that bedding-parallel pore–fracture pathways provide much more favorable flow channels than bedding-normal pathways, and porosity alone cannot explain permeability anisotropy in laminated shale oil cores.

(2). CT reconstruction reveals clear bedding-related structural differences between the two coring directions. In H-series cores, bedding fractures and lamination-related microfractures are commonly distributed along or subparallel to the flow direction, which favors bedding-parallel flow. In V-series cores, bedding-related structures are more segmented and discontinuous in the bedding-normal direction. This confirms that bedding-fracture orientation and directional continuity are key controls on permeability anisotropy.

(3). The CT-derived parameters V_f, N_b, ${\rho }_b$, and C_b provide quantitative descriptors of bedding-fracture abundance, density, and connectivity. For the H-series samples, V_f ranges from 0.67% to 2.71%, and C_b ranges from 0.33 to 0.77. For the V-series samples, V_f ranges from 0.31% to 2.08%, and C_b ranges from 0.34 to 0.72. These parameters help distinguish storage-related pore–fracture volume from flow-effective directional connectivity.

(4). The three-dimensional fractal dimension D₃ quantifies the spatial complexity and space-filling ability of CT-resolvable bedding-fracture structures. The D₃ values range from 2.16 to 2.63 for the H-series, and from 2.12 to 2.56 for the V-series. However, D₃ should be regarded as a complementary structural descriptor rather than a direct permeability predictor. Its hydraulic significance depends on its coupling with V_f, C_b, bedding orientation, and flow direction.

(5). Permeability decreases continuously with increasing confining pressure for both H- and V-series samples, indicating strong stress sensitivity of bedding-fracture-controlled flow channels. From 10 to 50 MPa, the average permeability decreases by approximately 97.1% for the H-series and 96.5% for the V-series. Although bedding-parallel flow paths provide higher initial permeability, they remain sensitive to stress-induced fracture closure.

(6). The fractal-corrected porosity–permeability model incorporating φ, V_f, C_b, and D₃ provides a more physically meaningful framework than the conventional porosity-only model. For all samples, the model R² increases from 0.0623 for the conventional model to 0.8804 for the fractal-corrected model, while RMSE decreases from 1.6721 to 0.5971 in logarithmic permeability space. This demonstrates that CT-derived bedding-fracture structure and fractal complexity provide essential information beyond porosity.

(7). From an engineering perspective, the proposed CT-fractal evaluation approach can support core-scale permeability anisotropy interpretation, reservoir quality classification, shale oil sweet-spot evaluation, hydraulic-fracturing design, and permeability-parameter selection for reservoir simulation. The method is most suitable for laminated shale oil cores where CT-resolvable bedding fractures and lamination-related microfractures exert significant control on flow capacity.