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. 2026 Mar 16;11(12):18731–18743. doi: 10.1021/acsomega.5c09552

Study on Water Seepage Characteristics of Coal under Triaxial Stress Based on Computed Tomography Reconstruction and Three-Dimensional Printing

Shilong Cui †, Xiangyu Chu †,‡,*, Juntao Liu †
PMCID: PMC13044834  PMID: 41939360

Abstract

Coal seam water injection is a crucial technique for mining disaster prevention, and its effectiveness depends on an understanding of precise seepage mechanisms. Current three-dimensional (3D) printing methods cannot accurately replicate coal’s complex pore structure, leading to unreliable seepage results. Through computed tomography (CT) 3D reconstruction and 3D printing technology, we prepared gypsum samples that precisely mimic natural coal’s internal structure and then conducted uniaxial compression and triaxial seepage tests to analyze fluid–solid coupling characteristics. The results show that (i) the stress–strain curves of 3D printed samples were similar to those of natural coal samples, and the mechanical parameters were close to each other; (ii) the permeability of 3D printed coal samples gradually decreases and stabilizes with the increase of axial and circumferential pressures, while the seepage volume increases linearly; (iii) the initial permeability of the 3D printed samples is similar, but the slope of permeability decrease increases with loading rate; and (iv) the permeability changes conform to the power law function and monoexponential decay function, with R 2 higher than 0.94 and 0.99, respectively. This work has provided a new repeatable experimental method for the study of coal seam water injection and a reliable means for improving the coal seam water injection seepage theory.

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1. Introduction

Due to the resource endowment characteristics of “rich in coal, poor in oil, and short in gas” of China, the “ballast stone” status of coal in the energy system will remain for quite a long term. , As shallow coal resources are gradually exhausted, the coal mining depth continuously increases. The deep coal seams are generally characterized by poor fracture/pore development and low water content, which not only exacerbates the problem of mine pressure manifestation during mining but also leads to increasingly prominent safety hazards, such as dust pollution. Coal seam water injection can effectively suppress dust generation, prevent and control gas disasters and dynamic disasters, and greatly improve the safety level of coal mines and protect the workers. However, in actual studies, due to the complexity of the coal structure and the variability of stress–strain, the study of the water injection seepage process is relatively challenging.

The seepage behavior of coal seams under triaxial stress conditions has been widely investigated, with substantial advancements achieved in this field. Building upon Biot’s pioneering work, which established a three-dimensional consolidation theory for porous media by modeling coal and rock as continuous permeable materials, subsequent researchers have made considerable contributions. Thallak and colleagues employed a discrete element approach incorporating seepage pressure to examine hydraulic fracturing mechanisms. Meanwhile, Witherspoon created a finite element analysis tool to investigate the interactions between seepage-induced stress, geostatic stress, and external loading conditions. Junqing’s research team systematically investigated the seepage patterns of natural coal samples during various loading stages while designing an innovative experimental apparatus capable of maintaining constant gravitational loading during permeability measurements. Further developments were made by Xue, who enhanced fluid–solid coupling theory through deriving an effective stress principle and corresponding numerical algorithms. Guoxin’s work yielded a finite element formulation for seepage loading that enabled the simulation of stress-deformation behavior in fluid-saturated formations. These investigations have primarily focused on elucidating permeability evolution throughout the complete stress–strain history, with particular attention to parameters including mechanical stress, strain development, pore fluid pressure, void ratio characteristics, fluid properties, and particulate dimensions. Nevertheless, it should be noted that rock failure mechanisms and fracture patterns exhibit significant dependence on loading rate conditions. Empirical evidence − demonstrates that both compressive strength and failure intensity exhibit positive correlations with increased loading rates. Specific experimental investigations by Liang and Liu on granite specimens under quasi-static loading conditions revealed a linear relationship between uniaxial compressive strength and applied loading rate. Complementary numerical simulations conducted by Li et al. using RFPA2D software further established proportional relationships between loading rate and both acoustic emission event frequency and energy release magnitude. While existing literature provides a comprehensive understanding of loading rate effects on mechanical response and fracture characteristics, research addressing the influence of loading rate variations on petrophysical propertiesparticularly permeability behaviorremains notably limited. This knowledge gap underscores the need for dedicated investigation into the coupled hydromechanical response of coal-bearing formations under diverse loading rate conditions.

Raw coals and shaped coals prepared by a conventional compaction method are often used for the study of seepage characteristics of coal. However, the anisotropies, such as differences in primary fractures and pore structures, can affect the experimental results. With the development of 3D printing technology and materials, it has become possible to quickly produce complex 3D models, and the technology has gradually been applied to geotechnical mechanics and engineering. − In terms of material and method innovation, Suzuki et al. prepared a smooth fracture network model with randomly distributed fracture parameters. Jiang reported that inorganic materials could be used to achieve accurate mimics of rock microvoids and reconstruct the complex macroscopic fracture networks and joint configurations of rock. Wei et al. solved the strength and stiffness problems of rock-like materials by vacuum infiltration and low-temperature treatment. These studies provide great support for the application of 3D printing technology in geotechnical engineering.

Many applications of 3D printing in the study of mechanics and seepage characteristics of coal rock have also been reported. Wang et al. made rough cross-fracture models with different angles and studied their fracture mechanisms. Jiang et al. printed uniaxial compression test pieces containing artificial pores and fractures using powder materials and made molds to cast shear test pieces with different surface morphologies. Na et al. prepared a 3D rough-surface fracture network model and verified the influence of surface roughness on permeability. Boutt et al. pointed out that rough fracture walls were prone to non-Darcy flow and emphasized the importance of the geometric description of rough joint surfaces of fractures. Song et al. undulated and roughened the natural structural surfaces using 3D laser scanning. Cui et al. analyzed the relationships between flow rate and pressure head, fracture aperture, and fractal dimension using a transparent fracture model. Most of these studies on the seepage characteristics of a coal seam based on 3D printing randomly or artificially created fractures using modeling software to characterize the pore/fracture structure, and the degree of restoration of the real complex pore/fracture network in a coal seam was low.

In summary, all prior 3D printing-based studies on coal seepage have relied on artificial or simplified fracture networks created via modeling software, which fail to replicate the true spatial distribution and complexity of pores/fractures in natural coal. In this study, we reconstructed the 3D pore/fracture structure of real coal samples by CT scanning and prepared samples using gypsum-based composites by high-fidelity 3D printing. The prepared coal samples exhibited mechanical properties similar to those of raw coal, and the strength met the requirements of seepage experiments. The influence of different triaxial stresses on the seepage characteristics of printed coal samples was systematically studied using a triaxial seepage test system. This method realized the accurate reproduction of coal structure, provided a new repeatable and quantifiable strategy for coal seepage experiments, and was of great significance to the improvement of the coal seam water injection seepage theory.

2. 3D Printing of Gypsum Powder-Based Coal Samples

2.1. Construction of 3D Printing Model

The lignite from the Xinjiang Wudong Coal Mine was selected as the research object. The bulk coal sample was cut into ø25 mm × 50 mm coal pillar test pieces using a corer and grinder for subsequent CT scanning using a Xradia 510 Versa high-resolution X-ray scanner. Table lists the specific instrument settings for the scanning. A total of 1500 grayscale images with a size of 800 pixels × 800 pixels and a resolution of 20 μm were recorded. CT scanning is a nondestructive imaging method. Due to the differences in their densities, the X-ray absorption capacities of different components in coal vary, which can reflect the spatial distribution of the internal structure of the coal sample. −

1. CT Scanning Parameters.

Scanning time/h Voltage/kV Current/μA Scanning Frames Resolution/μm
4.3 60 80 998 20
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Figure shows the flowchart for the 3D printing of the coal model based on 3D CT reconstruction. Noises are inevitably generated during the CT scanning due to the interference of the instrument itself and external factors. Therefore, the original CT images were denoised using a median filter based on the noise characteristics to improve the accuracy of imaging.

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3D printing of coal models based on 3D CT reconstruction.

The processed CT images clearly reveal that the coal sample is mainly composed of primary fractures, coal matrix, and high-density white impurities. Their grayscale values are significantly different due to their different densities. The grayscale value of high-density white impurities is the highest, followed by that of the coal matrix, and the grayscale value of the primary fracture is the lowest.

The original CT images are 16-bit grayscale images with grayscale values in the range of 0 to 65535. Because the grayscale values of each component in the CT images of different slices are different, the component was accurately identified in different slices by the “half-watershed” threshold segmentation method using AVIZO software. For the CT image of the same slice, the areas with grayscale values between 0 and 15035 are defined as primary fractures, those with grayscale values between 15036 and 33207 are considered coal matrix, and the areas with grayscale values between 33208 and 65535 are high-density impurities. The blue area in Figure represents the part selected for threshold segmentation.

The spatial distribution of the primary fractures in the test piece can greatly affect its strength and the water injection seepage path. Since the two-dimensional (2D) CT image can only show the local distribution of the fractures, the original 2D CT images are reconstructed by 3D visualization using the AVIZO three-dimensional visualization software.

To more clearly show the content, size distribution, and complexity of the spatial distribution of the mesoscopic pore/fracture structure in coal, volume proportion (v), fractal dimension (D), and porosity (ø) are used to jointly characterize the pore structure. As shown in eqs and , the volume proportion (v) is defined as the ratio of the sum of the pixel volumes of each component in each 2D CT image (V i ) to the total pixel volume of the test piece (V). Fractal dimension (D) establishes a cubic box with a side length of a for target coverage by the box counting method. The relationship between the side length and the minimum number N (a) in the box is obtained by changing the size of the box. The slope of the lg­(a) vs lgN­(a) curve is fitted by the least-squares method, which is the fractal dimension.

V=∑i=11000ViV 1
D=−lgN(a)lg(a) 2

The fractal dimension of a 3D space is generally in the range of 2–3. The volume portion and fractal dimension gradually increase with the increase in the fracture number and complexity. The fractal dimension of the coal sample with a porosity of 13.21% is calculated to be 2.474.

The shapes and sizes of the pores/fractures in the coal seam are significantly different. To quantitatively characterize the pore size distribution, the parameter equivalent diameter is introduced, which refers to the diameter of a sphere with the same pore volume. It can be calculated as

ϕeq=6Vπ3 3

where φ eq is the equivalent diameter and V is the pore volume.

The volume of each pore can be directly measured using AVIZO software, and the equivalent diameter of the pore is then calculated by substituting the measured volume into eq . Figure shows the volume proportions of different equivalent diameters.

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Distribution of equivalent diameters of pores.

The pore/fracture equivalent diameters of the test piece are in the range of 0–3080 μm (Figure ), and 97.6% of the pores/fractures are smaller than 100 μm, with their volume proportion being more than 24%. The total volume of the pores/fractures with equivalent diameters greater than 1000 μm is the largest, accounting for 42.26% of the total pore volume. The volume proportions conform to normal distribution with the determination of correlation (R²) greater than 90%.

The preliminary experiment on the lignite of Wudong Coal Mine using a high-temperature and high-pressure nuclear magnetic resonance seepage system measured its porosity to be 13.37%, 0.16% different from the porosity calculated by 3D reconstruction. The measured fractal dimension of the natural coal sample is 2.483, 0.009 different from that of the reconstructed model. The volumes of the pores with diameters greater than 1000 μm account for 43.81% of the total pore volume, 1.55% different from the model. The number of pores with diameters smaller than 100 μm is also the largest, accounting for 98.2% of the total pore number, 0.8% different from that of the model. These results indicate that the reconstructed 3D model is accurate and similar to the pore structure of the natural coal sample.

2.2. 3D Printing of Coal Samples

Coal samples were printed using a ProJet 860 Pro printer by injection molding, as shown in Figure . The printing accuracy was 0.1 mm, and the resolution was 600 × 540 dpi. The maximum print size could reach 381 × 508 × 229 mm, which met the printing requirements of this experiment. Gypsum powder with a particle size of 0.1 mm was used as the particle raw material, and VisiJet Binder glue was used as the binder. Printing was conducted following the steps below: (i) The 3D model was imported into 3DSPRINT software in STL format for slicing and generating a printing path (G-code). (ii) The printer automatically laid gypsum powder layer by layer (0.089–0.102 mm thick) according to the received printing instructions, and the printing nozzle sprayed the bonding material on the surface of the gypsum powder layer according to the layered shape. The process was repeated until the test piece was completed. (iii) The printed specimens were dried in a drying oven at 50 °C and relative humidity ≤45% for 2 h to ensure that the gypsum binder was completely cured without cracks.

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Principle of 3D printing and a photo of a printed test piece.

3. Mechanical Properties of 3D Printed Coal Samples

To evaluate the mechanical similarity between 3D printed gypsum powder specimens and natural coal samples, comparative testing was performed using an electronic universal material testing system with an 800 kN load capacity. This experimental setup enabled synchronous measurement of key mechanical parameters, including displacement and stress response during uniaxial compression testing. The displacement-controlled loading protocol was implemented at a constant rate of 0.002 mm/s, which offers distinct advantages over force-controlled methods by providing more precise deformation control and enabling the acquisition of comprehensive stress–strain relationships throughout the complete loading process.

3.1. Mechanical Properties of 3D Printed Gypsum Powder-Based Coal Samples

Uniaxial compression testing was conducted on both 3D printed specimens and natural coal samples, with their respective stress–strain behaviors illustrated in Figure . Mechanical analysis revealed that the printed specimens and natural coal shared remarkably similar elastoplastic characteristics, with both materials displaying four distinct deformation phases: The initial compaction phase (OA) featured nonlinear concave-upward stress–strain behavior as inherent microstructural voids and fractures progressively closed. Subsequently, the linear elastic phase (AB) demonstrated near-perfect Hookean behavior, where compressed fractures generated sufficient frictional resistance to maintain predominantly elastic deformation. Transitioning into the plastic deformation phase (BC), the stress–strain relationship became nonlinear as the material yielded, with primary fractures coalescing into macroscopic cracks until peak strength was attained. The final postpeak phase (CD) showed rapid stress reduction accompanied by extensive fracture propagation, leading to complete brittle failure and load-bearing capacity loss.

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Comparison of stress–strain curves of 3D printed and natural coal samples. (a) Stress–strain curve of the natural coal sample. (b) Stress–strain curve of the 3D printed specimen.

A comparative analysis of key mechanical parameters between gypsum-based 3D printed coal analogues and natural coal specimens is presented in Table . The printed samples demonstrate close agreement with natural coal, showing marginal variations of just 0.78 MPa in peak strength, 0.006 in peak strain, 0.062 GPa in elastic modulus, and 1.02 MPa in yield strength. These minimal discrepancies confirm the high mechanical fidelity of the 3D printed replicas to that of their natural counterparts.

2. Mechanical Properties of 3D Printed and Natural Coal Samples.

Specimen Peak strength/MPa Ultimate strain/% Elastic modulus/GPa Yield strength/MPa
3D printed 17.52 3.6 0.658 12.53
Coal 16.74 3 0.72 11.51
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3.2. Failure Mode of Gypsum Powder-Based 3D Printed Coal Sample

Based on the movement direction of the coal rock and the fracture direction, the displacement parallel to the fracture surface is regarded as shear failure, and that perpendicular to the fracture surface is considered as tensile failure. , Figure shows the failure mode of the 3D printed coal sample under a load exceeding its ultimate strength in the uniaxial compression test. Under uniaxial compression, the fractures in the test piece are compacted. Many fine cracks are generated on the specimen, mainly on the loading end and the middle, with the increase in loading. The cracks propagate and connect, resulting in small broken particles. The energy is gradually transformed into kinetic energy and shock waves. After the load reaches its peak strength, the test piece is damaged, and a fracture sloping surface parallel to the bedding plane appears. The degree of damage is similar to the shear slip along the weak surface at an angle of 45°, and macroscopic damages are observed. Yet, the main body of the test piece is relatively intact, and a small broken block is found along the circumference. The height of the block is roughly equivalent to the height of the 3D printed specimen. At least two main failure fractures run through the entire specimen, with obvious sounds recorded during the breaking. The impact tendency is weak, and thus, it can be considered as shear failure, similar to that of the natural coal sample.

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Comparison of failure modes of 3D printed specimens and natural coal samples.

3.3. Energy Evolution of Gypsum Powder-Based 3D Printed Coal Samples under Uniaxial Compression

Coal failure is a state of instability phenomenon driven by energy. Herein, we analyzed the instability failure of coal from the perspective of energy, explored the relationship between energy evolution, strength, and failure in the uniaxial compression deformation failure process, and assessed the similarity between 3D printed and natural coal samples from the perspective of energy evolution.

The coal sample is conceptualized as consisting of inherent defects (such as pores and fractures) and the surrounding solid matrix that forms the load-bearing structure. Based on the first law of thermodynamics and considering an adiabatic condition where no heat is exchanged with the external environment during mechanical loading, the external work is converted to energy within the coal sample. This energy manifests as both dissipated energy, which drives plastic deformation and the evolution of defects, and elastic energy stored in the solid matrix. The dissipated energy density (U d ) can be determined by subtracting the elastic energy density (U e ) from the total input energy density (U o ).

Ud=U0−Ue=∫0εcσidε−12Eσc2 4

where σ i is the stress at any point on the axial stress-axial strain curve, MPa; σ c is the peak stress, MPa; ε c is the peak strain; and E is the unloading modulus of the coal sample, MPa, and it is replaced with the elastic modulus for calculation.

The postpeak release energy density (U f ) corresponds to the area under the axial stress–strain curve between the strain limits ε e and ε f , representing the energy dissipated during structural failure. This value can be computed as follows:

Uf=∫εcεfσidε 5

where ε f is the maximum strain on the axial stress-axial strain curve. In the postpeak stage, a part of U e is converted to U f and the other part is converted to residual energy density (U y ) that can be calculated using eq .

Uy=Ue−Uf=12Eσc2−∫εcεfdε 6

The energy densities during loading are then calculated using eqs –, as shown in Table .

3. Energy Densities during Loading.

Specimen U 0 (J·mm–3) U e (J·mm–3) U y (J·mm–3) U f (J·mm–3)
3D printed coal 0.05881 0.04913 0.04472 0.00614
Natural coal 0.05359 0.04375 0.03849 0.00773
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During the prepeak stage, most of the input energy is accumulated as U e within the coal’s load-bearing framework, with minimal energy dissipation. Structural failure occurs once U e exceeds the storage capacity of this framework. Notably, 3D printed specimens exhibit a 10.95% greater U e value compared to natural coal samples, demonstrating their enhanced structural integrity and superior energy storage capability, as shown in Figure .

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Energy densities of 3D printed and natural coal samples.

In the postpeak stage, U f is the main energy driving the extension and penetration of macrofractures. The ratio of U f to U e of the 3D printed test piece is 0.1217 lower, and its U y is 13.93% higher than those of the natural coal sample. It can be explained that under the same conditions, more U e in the natural coal sample is converted into U f , and U y decreases, resulting in a small dynamic manifestation intensity when the coal sample fails. A large amount of U e is used for the aggregation and penetration of macrofractures in the coal sample, and the coal sample mainly undergoes delayed destruction and continuously releases energy, showing a step-like decline in the postpeak stage of the stress–strain curve.

4. Seepage Experiment of 3D Printed Coal Samples under Triaxial Stress

4.1. Experiment Setup

The seepage experiments were carried out using an independently constructed multiscale seepage real-time online loading experimental system (Figure ). The system includes a confining pressure pump, an axial pressure pump, a back pressure pump, and two fluid pressure pumps. The pressure of each pump can be freely adjusted in the range of 0–15 MPa. The test system is equipped with clamps, and the one with a diameter of 5 mm was selected in this work. The seepage quantity was recorded in real time.

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Schematic diagram for the multiscale seepage real-time online loading experimental system.

Currently, coal permeability is predominantly determined through two approaches: the steady-state method and the transient method. The steady-state approach derives permeability by measuring fluid flow rates at both ends of the coal sample and applying a theoretical calculation. In contrast, the transient method determines permeability by analyzing real-time pressure variations recorded by sensors at the sample’s inlet and outlet. For this study, the steady-state measurement technique was employed to evaluate coal permeability, which can be quantified through Darcy’s law as presented in eq .

k=QμLAΔP 7

where k is permeability, 10–3μm2; Q is the flow rate of the fluid, 10–10/ml·s–1; μ is the fluid viscosity, Pa·s; L is the length of the medium, mm; A is the cross-sectional area of the medium, μm2; and ΔP is the pressure difference of the fluid when it passes through the medium, MPa.

The initial permeability of the coal sample under hydrostatic pressure can be calculated when the sample is in a hydrostatic state and the water flow through the sample becomes stable. During the seepage-stress coupling test, the water flow through the coal specimen during loading was recorded to explore the change in the law of permeability.

4.2. Experimental Design

Deep mining operations necessitate a comprehensive investigation into the mechanical behavior and seepage characteristics of coal seams. In this study, the mechanical properties of 3D printed coal samples were evaluated through displacement-controlled loading at a constant rate of 0.002 mm/s. For the triaxial stress seepage experiments, axial pressures ranging from 3 to 6 MPa and confining pressures from 4 to 7 MPa were systematically applied across four distinct gradient levels, while seepage pressures between 2 and 5 MPa were tested at five gradient levels with 1 MPa increments. Based on preliminary experimental results, four loading rate gradients were implemented (0.001, 0.01, 0.1, and 1 MPa/s), with complete test parameters detailed in Table . A series of eight seepage experiments were conducted under strictly controlled temperature conditions (25 ± 1 °C) to ensure data accuracy by eliminating thermal interference. Each test maintained a 2 h pressurization period during which injection flow rates and cumulative flow volumes were recorded at 1 min intervals. The steady-state seepage flow measurements were specifically utilized for subsequent permeability calculations of the test specimens.

4. Parameters for Triaxial Seepage Test.

Test piece Water injection pressure/MPa Confining Pressure/MPa Axial pressures/MPa Loading rate/(MPa/s)
1# 3 4 3, 4, 5, 6 1
2# 3 5 3, 4, 5, 6 1
3# 3 6 3, 4, 5, 6 1
4# 3 7 3, 4, 5, 6 1
5# 2, 4, 5, 6 6 5 1
6# 3 6 3, 4, 5, 6 0.001
7# 3 6 3, 4, 5, 6 0.01
8# 3 6 3, 4, 5, 6 0.1
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4.3. Effects of Triaxial Stress on Seepage Characteristics of 3D Printed Samples

Figure illustrates the permeability evolution of 3D printed specimens under varying triaxial stress conditions as the axial stress increases. The permeability initially exhibits a progressive decline due to stress-induced compression of internal voids and fractures. During this compaction phase, the printed coal undergoes volumetric contraction, where the diminished connectivity of flow pathways impedes fluid movement. With continued axial loading, the permeability reduction rate slows progressively until stabilizing at a constant value.

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Changes of permeability with axial stress at the seepage pressure of 3 MPa.

Under a constant water injection pressure of 3 MPa, the 3D printed specimens exhibit initial permeability values of 4.416, 2.612, 1.82, and 1.21 mD at confining pressures of 4, 5, 6, and 7 MPa, respectively. These measurements reveal significant permeability reductions of 40.85%, 58.78%, and 72.59% at 5, 6, and 7 MPa confining pressures compared to the 4 MPa baseline. Notably, the rate of permeability decline diminishes with increasing confining pressure.

During axial loading, the specimens demonstrate a progressive permeability reduction characteristic of the initial fracture compaction and linear elastic deformation phases. This behavior stems from the gradual closure of primary microfractures, which obstructs flow pathways and consequently decreases the permeability. The magnitude of this reduction is inversely proportional to the confining pressure. At an axial stress of 6 MPa, measured permeability values are 0.995, 0.967, 0.958, and 0.95 mD for confining pressures of 4–7 MPa, representing modest decreases of 2.81%, 3.71%, and 4.52% relative to the 4 MPa reference condition.

The seepage flow demonstrates an initial linear increase during the 0–4000 s period, followed by a progressive decline as the confining pressure rises, as illustrated in Figure . This reduction behavior closely parallels the observed permeability response to confining pressure, with both exhibiting gradually diminishing attenuation rates. Under a constant 6 MPa confining pressure condition, the seepage flow shows a systematic decrease from 0.11404 × 10–10 ml·s–1 to 0.06825 × 10–10 ml·s–1, then to 0.04108 × 10–10 ml·s–1, and ultimately stabilizes at 0.03361 × 10–10 ml·s–1 as axial stress increases incrementally from 3 to 6 MPa in 1 MPa steps. These measurements reveal an inverse relationship between seepage flow and axial stress magnitude, with the rate of flow reduction continuously slowing as stress intensifies until reaching a stable state upon complete fracture compaction within the test specimens.

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Real-time change of seepage quantity with axial stress under different confining pressures. (a) Confining pressure 4 MPa. (b) Confining pressure 5MPa. (c) Confining pressure 6MPa. (d) Confining pressure 7MPa.

Under constant axial stress (5 MPa) and confining pressure (6 MPa) conditions, the evolution of seepage characteristics in 3D printed coal samples with increasing water injection pressure was systematically investigated (Figure ). Experimental results demonstrate that as the water injection pressure increased incrementally from 2 to 6 MPa in 1 MPa steps, the cumulative seepage volume showed substantial growth, with respective increases of 29.81%, 80.09%, 356.51%, and 887.15% (Table ). Correspondingly, the permeability exhibited a progressive enhancement from 0.972 to 1.09, 1.414, 2.688, and ultimately 4.65 mD (Figure ). This pressure-dependent behavior reveals two key characteristics: a consistent positive correlation between permeability and injection pressure and an accelerating growth rate at higher pressure levels.

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Variation of seepage characteristics of the 3D printed samples with water injection pressure. (a) Variation of permeability with water injection pressure. (b) Variation of seepage quantity with water injection pressure.

5. Cumulative Seepage and Permeability.

Water injection pressure/MPa 2 3 4 5 6
Measured cumulative seepage Q p  × 10–10/ml·s–1 0.02281 0.02961 0.04108 0.10413 0.22517
Permeability/(10–3μm2) 0.972 1.09 1.414 2.688 4.65
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The observed phenomenon can be attributed to water injection pressure-induced fracture propagation within the 3D printed coal matrix, which transitions the system into a fracture development phase that significantly improves flow path connectivity. Quantitative analysis indicates that compared to the baseline at 2 MPa injection pressure, the permeability under 3–6 MPa injection pressures increased by 12.13%, 45.47%, 176.54%, and 378.39%, respectively, demonstrating the remarkable enhancement effect of water injection pressure on permeability.

Figure b shows the seepage quantity variations of the 3D printed coal samples with the water injection pressure in the range of 2–5 MPa, and the record starts 4000 s from the beginning of the stable permeability. The seepage quantity increases linearly under all water injection pressures initially. At 4000 s, the seepage quantities under the water injection pressures of 2, 3, 4, 5, and 6 MPa are 0.02281 × 10–10/ml·s–1, 0.02961 × 10–10/ml·s–1, 0.04108 × 10–10/ml·s–1, 0.10413 × 10–10/ml·s–1, and 0.22517 × 10–10/ml·s–1, respectively. The increasing trend is similar to that of the permeability.

4.4. Effect of Loading Rate on Seepage Characteristics of 3D Printed Coal Samples

Figure shows the changes in permeability of the 3D printed coal samples with axial stress at different loading rates. The permeability change undergoes two stages at all loading rates, a sharply declining stage and a slowly declining and fluctuating stage. The initial permeabilities at different axial stress loading rates are similar, with values of 1.32 mD, 1.58 mD, 1.72 mD, and 1.8 mD at the loading rates of 0.001 MPa/s, 0.01 MPa/s, 0.1 MPa/s, and 1 MPa/s, respectively. The slope of the sharp decline curve of permeability gradually increases with the increase of the axial stress loading rate. The reason is that the relative motion rate of the pore wall in the 3D printed coal sample increases with the increase of the loading rate. Due to the rough pore wall of 3D printed coal, under the action of axial pressure, the higher relative motion speed may produce more fine debris through the friction and wear between the pore walls, thus blocking the seepage channel and reducing the fluid flow. Previous studies on rock mechanics have mentioned that the increase in loading rate promotes the generation of fine particles in the process of fracture sliding. , The smaller the debris generated, the higher the degree of blockage and the more the permeability is reduced.

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Permeability curves at different loading rates.

The stabilization threshold of permeability exhibits a loading-rate-dependent hysteresis effect relative to the maximum principal stress. Specifically, 3D printed coal specimens reach stable permeability at 4.8 MPa axial stress under 0.001 MPa/s loading, whereas at 1 MPa/s loading, permeability approaches but does not fully stabilize, even at 6 MPa. Minimum permeability values of 0.901, 0.913, 0.938, and 0.958 mD were recorded at 6 MPa axial stress for loading rates of 0.001–1 MPa/s, respectively, demonstrating an inverse relationship between loading rate and permeability reduction.

At identical axial stress levels, the seepage flow demonstrates a loading-rate dependence that parallels the permeability behavior, with both parameters exhibiting increasing trends (Figure ). For instance, at 3 MPa axial stress, measured seepage flows are 0.03834 × 10–10, 0.0459 × 10–10, 0.04997 × 10–10, and 0.05229 × 10–10 ml·s–1 corresponding to loading rates of 0.001–1 MPa/s. This represents progressive enhancements of 19.71%, 30.33%, and 36.38% relative to the 0.001 MPa/s baseline, although the magnitude of improvement diminishes at higher loading rates.

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Real-time seepage quantity versus axial stress at different loading rates. (a) Loading rate 0.001 MPa/s. (b) Loading rate 0.01 MPa/s. (c) Loading rate 0.1 MPa/s. (d) Loading rate 1 MPa/s.

5. Discussion

5.1. Permeability Variation of 3D Printed Coal Samples with Confining Pressure

To establish a comprehensive characterization of 3D printed coal’s pressure-dependent permeability under triaxial conditions, three constitutive relationships were derived through the integration of prior research findings with fluid–solid coupling experimental data fitting analyses.

Power law function relationship: y = axb,

Exponential function relationship: y = aebx,

Polynomial function relationship: y = ax 2 + bx + c,

where y is the permeability of rock, 10–3μm2; x is the confining pressure (MPa); and a, b, and c are undetermined coefficients.

Existing studies predominantly employ exponential or polynomial functions to characterize the permeability-confining pressure relationship in rocks. , However, these investigations primarily utilized raw coal specimens, while neglecting the structural damage-induced permeability alterations under increasing confining pressures. Our stress-seepage coupling experiments with 3D printed coal analogues demonstrate that the power law function (y = ax b ) achieves optimal goodness-of-fit (R 2), accurately captures experimental trends, and satisfies all testing requirements. Consequently, this function was selected for seepage data fitting, with the detailed results presented in Table . These findings provide novel insights for investigating coal permeability-pressure relationships.

6. Power Law Function Fitting Results of the Relationship between the Permeability of the 3D Printed Coal Samples and the Confining Pressure under Different Triaxial Conditions.

    Undetermined coefficient
  
Confining pressure (MPa) Axial stress (MPa) a b GoF (R 2)
4–7 3 104.022 –2.28 0.99922
4 21.596 –1.533 0.9964
5 4.391 –0.771 0.95988
6 1.109 –0.081 0.94412
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Table demonstrates excellent fitting performance of the permeability-confining pressure relationship for 3D printed coal samples, with all R 2 values exceeding 0.94 across test conditions. While the fitting curves vary under different axial stresses, their undetermined coefficients (a and b) exhibit systematic variations: coefficient a shows a decreasing trend, while b increases proportionally with axial stress elevation. This pattern reflects progressive fracture compression within the coal matrix under increasing axial loading, with coefficient b displaying particularly consistent sensitivityincreasing approximately 0.7 units per 1 MPa axial stress increment.

Figure compares the power law function fitting curve of permeability against the confining pressure of the 3D printed coal samples with the variation trends of the experimental data. As can be seen, the fitting curve is highly consistent with the experimental data, suggesting the fitting effect and accuracy are good and the influence of the structural anisotropy of raw coal on the test is avoided. Therefore, the power law function relationship can be used to more accurately reflect the change law of coal permeability with confining pressure.

13.

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Comparison of the fitting curves of the permeability of 3D printed coal samples with confining pressure and the variations of experimental data.

5.2. Permeability Variation of 3D Printed Coal Samples with Loading Rate

The plot shape of permeability against loading rate in Figure was analyzed, and the experimental data were fitted to qualitatively and quantitatively characterize the relationship between the permeability of the 3D printed coal samples and the loading rate. The single exponential decay function shows the highest GoF (R 2), suggesting that it can best reflect the variation law of the experimental data and meet the requirements of the samples. Therefore, the single exponential decay function, y = y 0 + A 1 e –x/t1, was used to fit the experimental data of the permeability changes with loading rate during the seepage process of the 3D printed test piece.

The triaxial seepage test results in Table demonstrate exceptional fitting accuracy (R 2 > 0.99) for the permeability-loading rate relationship of 3D printed coal samples across all test conditions. Analysis reveals that while the undetermined coefficient y 0 shows minimal variation with loading rate reduction, exhibiting only a slight upward trend, coefficient A 1 experiences substantial increases accompanied by significant decreases in t 1 as loading rates diminish.

7. Single Exponential Decay Function Fitting Results of the Relationship between the Permeability of the 3D Printed Coal Samples and the Loading Rate.

  Undetermined coefficients
  
Loading rate y 0 A 1 t 1 GoF (R 2)
1 0.868 5.003 1.969 0.99874
0.1 0.883 13.101 1.09 0.99998
0.01 0.902 17.31 0.924 0.99547
0.001 0.916 48.301 0.671 0.99193
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The single exponential decay function fitting curves of the permeability against the loading rate are compared with the trend curves of the experimental data in Figure . The fitting curves roughly coincide with the trend curves of the experimental data, with a certain deviation in the position of the inflection point, yet the overall fitting effect is good, and the fitting accuracy is high. The GoF (R 2) is higher than 0.99, indicating that the permeability change law of the 3D printed coal samples with the loading rate can be roughly reflected by the single exponential decay function.

14.

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Comparison of the fitting curves of the 3D coal sample permeability with the loading rate and the trend curves of the experimental data.

In addition to the fine debris plugging mechanism proposed above, the loading-rate-dependent permeability evolution of 3D printed coal samples may also be synergistically regulated by the viscoelastic response of the matrix and the time scale of fracture compaction. The natural coal samples and 3D printed samples used in this study showed inherent weak viscoelasticity. At higher loading rates, the deformation of the specimen lags behind the applied axial stress, resulting in the incomplete closure of the microcracks inside the specimen. The heterogeneity of fracture closure further aggravates the discontinuity of the seepage channel, thus accelerating the decline rate of permeability. At the same time, the time scale required for fracture compaction also plays a crucial role. At low loading rates, the fracture has enough time to be uniformly compacted so that the fluid is evenly redistributed in the remaining pore space, resulting in a gentle decline in permeability. On the contrary, high loading rates lead to rapid closure of fractures, limited time for fluid redistribution and pore pressure adjustment, and a rapid decline in permeability. These factors are not mutually exclusive but dynamic interactions, which together affect the fluid–solid coupling behavior of 3D printed coal samples under triaxial stress conditions.

The 3D printed coal samples for the experiment contain the exact same internal structures, which avoid the influence of the anisotropy of raw coal on the experimental results. In addition, the internal structure of the 3D printed test piece is obtained by CT scanning and 3D reconstruction, which is more realistic than those of the coal samples with prefabricated fractures in the previous reports. Therefore, the results of this work are more accurate. This work provides a new idea for subsequent seepage experiments in complex fractured coal and rock masses and shows certain guiding significance for the improvement of coal seam water injection seepage theory.

In future studies, the particle size and concentration of debris will be analyzed to verify the mechanism of fine debris blockage. In addition, CT scanning will be carried out during the triaxial seepage test to directly observe the dynamic evolution process of pore/fracture closure and debris accumulation and further improve the understanding of the fluid–solid coupling mechanism.

6. Conclusion

In this study, we obtained a 3D digital model of a natural coal sample through CT scanning combined with AVIZO 3D reconstruction technology. Based on this model, coal-like specimens were fabricated using 3D printing technology. Uniaxial compression tests were conducted to characterize and compare the mechanical properties and failure modes between the natural coal samples and 3D printed coal-like specimens. Additionally, triaxial seepage experiments were performed to investigate the deformation and seepage characteristics of the 3D printed specimens under different confining pressures and axial loading rates. The main conclusions drawn from these experimental results are as follows:

  • Both 3D printed and natural coal specimens exhibit identical four-stage mechanical behavior under uniaxial compression. Comparative analysis reveals that while the 3D printed specimens demonstrate marginally higher peak strength, ultimate tensile strength, and yield strength, along with a slightly lower elastic modulus, their overall mechanical characteristics closely resemble those of natural coal. During loading, the 3D printed coal undergoes fracture compaction followed by shear failure at peak stress, a failure mode indistinguishable from that of natural coal. Energy evolution analysis further shows that the 3D printed specimens accumulate 8.87% more elastic strain energy (U e ) and 13.93% greater yield strain energy (U y ), while displaying a 0.1217 reduction in the U f /U e ratio compared to natural counterparts.

  • The 3D printed coal specimens exhibit a progressive permeability reduction with increasing axial stress and confining pressure, with the most pronounced decrease occurring during initial compaction. This attenuation effect demonstrates confining pressure-dependence, showing reduced decline rates between 4 and 7 MPa. Conversely, permeability displays positive correlation with injection pressure, increasing from 0.972 mD to 4.65 mD (at 1 MPa intervals from 2 to 5 MPa) with accelerating growth rates.

  • The 3D printed coal samples exhibit similar initial permeability at different axial stress loading rates, yet the slope of the permeability decline curve under triaxial stress increases with the increase of loading rate. The axial stress corresponding to the permeability stable stage shows a lagging trend with the increase of the maximum principal stress loading rate.

  • The power law function fits the permeability change of the 3D printed coal samples with confining pressure best, with the GoF (R 2) higher than 0.94, and the single exponential decay function fits the permeability change with loading rate best, with the GoF (R 2) higher than 0.99, and both exhibit high fitting accuracies.

Acknowledgments

This work was financially supported by the National Science and Technology Major Project (No. 2024ZD1700103), the National Natural Science Foundation of China (Nos. 52504241 and 52174194), and the Shandong Provincial Natural Science Foundation (No. ZR2025QC459).

The data used to support the findings of this study are available from the corresponding author upon request.

Ethics approval: not applicable. Consent to participate: not applicable. Consent for publication: not applicable.

The authors declare no competing financial interest.

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Data Availability Statement

The data used to support the findings of this study are available from the corresponding author upon request.

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