Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis
Abstract
1. Introduction
2. Experimental Approach
2.1. Sample Collection and Preparation
2.2. Testing Equipment
2.3. Experimental Procedure
- The samples were first dried at a constant temperature of 80 °C for 24 h to remove any moisture. After drying, the samples were weighed to establish their baseline dry mass. The samples were then vacuumed for 2 h and saturated with water at 15 MPa pressure for 24 h. Afterward, the samples were weighed again to determine the change in mass.
- The saturated samples were wrapped in heat-shrink tubing and placed into the loading chamber. Axial pressure and confining pressure of 3 MPa were applied, followed by a hydraulic pressure of 2 MPa. Once water was observed flowing from the outlet, T2 and NMRI tests were conducted under hydrostatic pressure conditions.
- The in situ NMR experiment under pseudo-triaxial loading was conducted, maintaining a constant confining pressure of 3 MPa and hydraulic pressure of 2 MPa, while the axial pressure was incrementally increased. Before reaching peak strength, stress-controlled loading was applied, with T2 and NMRI tests conducted at every 5 MPa increment. After peak strength was reached, strain-controlled loading was applied, with T2 and NMRI tests conducted at regular intervals.
- After replacing the sample, the in situ NMR experiment under pseudo-triaxial loading–unloading was conducted. The confining pressure was maintained at 3 MPa and the hydraulic pressure at 2 MPa, while the axial pressure was applied and unloaded. Before reaching peak strength, stress-controlled loading was applied, with the maximum axial pressure of each cycle being 5 MPa higher than the previous one. After reaching peak strength, strain-controlled loading was applied, and the pressure was unloaded to hydrostatic pressure at regular intervals. T2 and NMRI tests were conducted at the maximum axial pressure of each cycle and after unloading to hydrostatic pressure.
- After replacing the sample, steps (2) to (4) were repeated twice, completing three sets of repeat experiments.
2.4. Experimental Principles
3. Evolution Characteristics of the Pore and Fracture Structure in Coal
3.1. Evolution Characteristics of Spatial Distribution
- Stage I (Compaction of Initial PFS):
- 2.
- Stage II (Elastic Stage):
- 3.
- Stage III (Unstable Growth and Failure of PFS):
- 4.
- Stage IV (Residual Strength Stage):
3.2. Evolution Characteristics of Pore Size Distribution
3.2.1. Pore Classification
3.2.2. Triaxial Loading
3.2.3. Triaxial Loading and Unloading
3.3. Evolution Characteristics of Average Pore Size
3.4. Evolution Characteristics of Complexity
4. Evolution Characteristics of Energy in Coal
- In Stage I, some pores and fractures are compacted, most of which are irrecoverable. The closure of these pores and fractures results in plastic deformation, leading to a higher energy dissipation ratio.
- In Stage II, some pores and fractures are compacted but can recover after unloading, and most pores undergo elastic deformation. This results in a lower and stable energy dissipation ratio.
- In Stage III, a large number of pores and fractures gradually form and interconnect. The coal sample can no longer bear the entire load, leading to extensive plastic deformation. Consequently, its energy storage capacity gradually decreases, the stored elastic energy is reduced, and both the energy dissipation and the energy dissipation ratio increase sharply.
- In Stage IV, the coal sample’s energy storage capacity diminishes, and the generation of secondary pores and fractures consumes a large amount of energy, causing the energy dissipation ratio to remain at a high level.
5. Correlation Between Porosity of Seepage Pores and Axial Strain in Coal
6. Conclusions
- During the compaction phase of pseudo-triaxial loading and unloading, the primary mechanism is the compaction of the original pores and fractures, which are difficult to recover after unloading. In the elastic phase, the main mechanism involves the mutual transformation between SPs and APs, most of which can recover after unloading. After yielding, new pores and fissures are mainly generated, and unloading promotes the connectivity and opening of these pores and fractures.
- With the increase in loading cycles, the porosity of SPs in coal samples decreases with increasing axial strain before yielding, exhibiting a good negative exponential function relationship. After yielding, the porosity increases with increasing axial strain, showing a good logarithmic function relationship.
- Under different stress conditions, both APs and SPs exhibit distinct fractal characteristics. The fractal dimension of APs decreases during the compaction phase and gradually increases afterward, while the fractal dimension of SPs decreases before yielding and increases after yielding, reflecting changes in pore complexity and permeability.
- Before yielding, the energy dissipation ratio is high due to irreversible compaction and plastic deformation. After yielding, the energy dissipation ratio increases sharply as a result of extensive plastic deformation and pore-fracture generation, with the ratio remaining elevated during the residual strength stage. These findings indicate that energy dissipation and permeability are closely linked to pore structure evolution.
- However, this study primarily focused on mechanical loading effects. Future work should consider the combined influence of temperature, chemical factors, and more complex stress paths that reflect real-world mining scenarios. Expanding this research to include different coal types and deeper strata conditions would help improve the generalizability of the findings.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
PFS | Pore-Fracture Structure |
PSD | Pore Size Distribution |
CT | X-Ray Computed Tomography |
SEM | Scanning Electron Microscopy |
AE | Acoustic Emission |
NMR | Nuclear Magnetic Resonance |
NMRI | Nuclear Magnetic Resonance Imaging |
CPMG | Carr–Purcell–Meiboom–Gill |
AP | Adsorption Pores |
SP | Seepage Pores and Fractures |
References
- Flores, R.M. Coalbed Methane: From Hazard to Resource. Int. J. Coal Geol. 1998, 35, 3–26. [Google Scholar] [CrossRef]
- Cai, J.; Perfect, E.; Cheng, C.-L.; Hu, X. Generalized Modeling of Spontaneous Imbibition Based on Hagen–Poiseuille Flow in Tortuous Capillaries with Variably Shaped Apertures. Langmuir 2014, 30, 5142–5151. [Google Scholar] [CrossRef] [PubMed]
- Kozhevnikov, E.; Turbakov, M.; Riabokon, E.; Gladkikh, E.; Guzev, M.; Qi, C.; Li, X. The Mechanism of Porous Reservoir Permeability Deterioration Due to Pore Pressure Decrease. Adv. Geo-Energy Res. 2024, 13, 96–105. [Google Scholar] [CrossRef]
- Zou, M.J.; Wei, C.T.; Huang, Z.Q.; Wei, S. Porosity Type Analysis and Permeability Model for Micro-Trans-Pores, Meso-Macro-Pores and Cleats of Coal Samples. J. Nat. Gas Sci. Eng. 2015, 27, 776–784. [Google Scholar] [CrossRef]
- Jia, Q.; Liu, D.; Cai, Y.; Yao, Y.; Lu, Y.; Zhou, Y. Variation of Adsorption Effects in Coals with Different Particle Sizes Induced by Differences in Microscopic Adhesion. Chem. Eng. J. 2023, 452, 139511. [Google Scholar] [CrossRef]
- Liu, D.; Zhao, Z.; Cai, Y.; Sun, F. Characterizing Coal Gas Reservoirs: A Multiparametric Evaluation Based on Geological and Geophysical Methods. Gondwana Res. 2024, 133, 91–107. [Google Scholar] [CrossRef]
- Cai, Y.; Liu, D.; Pan, Z.; Yao, Y.; Li, J.; Qiu, Y. Pore Structure and Its Impact on CH4 Adsorption Capacity and Flow Capability of Bituminous and Subbituminous Coals from Northeast China. Fuel 2013, 103, 258–268. [Google Scholar] [CrossRef]
- Qin, X.; Wu, J.; Xia, Y.; Wang, H.; Cai, J. Multicomponent Image-Based Modeling of Water Flow in Heterogeneous Wet Shale Nanopores. Energy 2024, 298, 131367. [Google Scholar] [CrossRef]
- Wu, R.; Wei, B.; Li, S.; Zhang, Y.; Luo, Q. Enhanced Oil Recovery in Complex Reservoirs: Challenges and Methods. Adv. Geo-Energy Res. 2023, 10, 208–212. [Google Scholar] [CrossRef]
- Zhao, Y.; Wang, Y.; Wang, W.; Tang, L.; Liu, Q.; Cheng, G. Modeling of Rheological Fracture Behavior of Rock Cracks Subjected to Hydraulic Pressure and Far Field Stresses. Theor. Appl. Fract. Mech. 2019, 101, 59–66. [Google Scholar] [CrossRef]
- Zhang, M.; Duan, C.C.; Li, G.F.; Fu, X.; Zhong, Q.; Liu, H.; Dong, Z. Determinations of the Multifractal Characteristics of the Pore Structures of Low-, Middle-, and High-Rank Coal Using High-Pressure Mercury Injection. J. Pet. Sci. Eng. 2021, 203, 108656. [Google Scholar] [CrossRef]
- Zhang, B.; Chen, Y.L. Particle Size Effect on Pore Structure Characteristics of Lignite Determined via Low-Temperature Nitrogen Adsorption. J. Nat. Gas Sci. Eng. 2020, 84, 103633. [Google Scholar] [CrossRef]
- Meng, Y.; Li, Z.P.; Lai, F.P. Experimental Study on Porosity and Permeability of Anthracite Coal under Different Stresses. J. Pet. Sci. Eng. 2015, 133, 810–817. [Google Scholar] [CrossRef]
- Ju, Y.; Xi, C.D.; Zhang, Y.; Mao, L.; Gao, F.; Xie, H. Laboratory In Situ CT Observation of the Evolution of 3D Fracture Networks in Coal Subjected to Confining Pressures and Axial Compressive Loads: A Novel Approach. Rock Mech. Rock Eng. 2018, 51, 3361–3375. [Google Scholar] [CrossRef]
- Tan, L.; Ren, T.; Dou, L.; Sun, J.; Yang, X.; Qiao, M. Moisture Penetration and Distribution Characterization of Hard Coal: A µ-CT Study. Int. J. Coal Sci. Technol. 2024, 11, 55. [Google Scholar] [CrossRef]
- Xu, H.; Wang, G.; Fan, C.; Liu, X.; Wu, M. Grain-Scale Reconstruction and Simulation of Coal Mechanical Deformation and Failure Behaviors Using Combined SEM Digital Rock Data and DEM Simulator. Powder Technol. 2020, 360, 1305–1320. [Google Scholar] [CrossRef]
- Chalmers, G.R.; Bustin, R.M.; Power, I.M. Characterization of Gas Shale Pore Systems by Porosimetry, Pycnometry, Surface Area, and Field Emission Scanning Electron Microscopy/Transmission Electron Microscopy Image Analyses: Examples from the Barnett, Woodford, Haynesville, Marcellus, and Doig Units. Bulletin 2012, 96, 1099–1119. [Google Scholar] [CrossRef]
- Wang, T.; Liu, Z.; Liu, L. Investigating a Three-Dimensional Convolution Recognition Model for Acoustic Emission Signal Analysis during Uniaxial Compression Failure of Coal. Geomat. Nat. Hazards Risk 2024, 15, 2322483. [Google Scholar] [CrossRef]
- He, M.C.; Miao, J.L.; Feng, J.L. Rock Burst Process of Limestone and Its Acoustic Emission Characteristics under True-Triaxial Unloading Conditions. Int. J. Rock Mech. Min. Sci. 2010, 47, 286–298. [Google Scholar] [CrossRef]
- Huang, Z.; Gu, Q.X.; Wu, Y.F.; Wu, Y.; Li, S.; Zhao, K.; Zhang, R. Effects of Confining Pressure on Acoustic Emission and Failure Characteristics of Sandstone. Int. J. Coal Sci. Technol. 2021, 31, 963–974. [Google Scholar] [CrossRef]
- Zhao, Y.; Wang, C.; Ning, L.; Zhao, H.; Bi, J. Pore and Fracture Development in Coal under Stress Conditions Based on Nuclear Magnetic Resonance and Fractal Theory. Fuel 2022, 309, 122112. [Google Scholar] [CrossRef]
- Bi, J.; Ning, L.; Zhao, Y.; Wu, Z.; Wang, C. Analysis of the Microscopic Evolution of Rock Damage Based on Real-Time Nuclear Magnetic Resonance. Rock Mech. Rock Eng. 2023, 56, 3399–3411. [Google Scholar] [CrossRef]
- Lian, S.; Wan, W.; Zhao, Y.; Peng, W.; Du, C.; Hu, H. Study on the Damage Mechanism and Evolution Model of Preloaded Sandstone Subjected to Freezing–Thawing Action Based on the NMR Technology. Rev. Adv. Mater. Sci. 2024, 63, 20240034. [Google Scholar] [CrossRef]
- Li, X.; Fu, X.H.; Ranjith, P.G.; Xu, J. Stress Sensitivity of Medium- and High Volatile Bituminous Coal: An Experimental Study Based on Nuclear Magnetic Resonance and Permeability-Porosity Tests. J. Pet. Sci. Eng. 2019, 172, 889–910. [Google Scholar] [CrossRef]
- Cheng, M.; Fu, X.H.; Kang, J.Q. Compressibility of Different Pore and Fracture Structures and Its Relationship with Heterogeneity and Minerals in Low-Rank Coal Reservoirs: An Experimental Study Based on Nuclear Magnetic Resonance and Micro-CT. Energy Fuels 2020, 34, 10894–10903. [Google Scholar] [CrossRef]
- Zhou, H.W.; Liu, Z.L.; Zhong, J.C.; Chen, B.C.; Zhao, J.W.; Xue, D.J. NMRI Online Observation of Coal Fracture and Pore Structure Evolution under Confining Pressure and Axial Compressive Loads: A Novel Approach. Energy 2022, 261, 125297. [Google Scholar] [CrossRef]
- Zhao, Y.; Zhu, G.; Dong, Y.; Danesh, N.N.; Chen, Z.; Zhang, T. Comparison of Low-Field NMR and Microfocus X-Ray Computed Tomography in Fractal Characterization of Pores in Artificial Cores. Fuel 2017, 210, 217–226. [Google Scholar] [CrossRef]
- Yao, Y.B.; Liu, D.M.; Che, Y.; Tang, D.; Tang, S.; Huang, W. Petrophysical Characterization of Coals by Low-Field Nuclear Magnetic Resonance (NMR). Fuel 2010, 89, 1371–1380. [Google Scholar] [CrossRef]
- Cai, J.; Yu, B.; Zou, M.; Luo, L. Fractal Characterization of Spontaneous Co-Current Imbibition in Porous Media. Energy Fuels 2010, 24, 1860–1867. [Google Scholar] [CrossRef]
- Li, Z.T.; Liu, D.M.; Cai, Y.D.; Wang, Y.; Si, G. Evaluation of Coal Petrophysics Incorporating Fractal Characteristics by Mercury Intrusion Porosimetry and Low-Field NMR. Fuel 2020, 263, 116802. [Google Scholar] [CrossRef]
- Ouyang, Z.Q.; Liu, D.M.; Cai, Y.D.; Yao, Y. Fractal Analysis on Heterogeneity of Pore–Fractures in Middle–High Rank Coals with NMR. Energy Fuels 2016, 30, 5449–5458. [Google Scholar] [CrossRef]
- Zhou, S.D.; Liu, D.M.; Cai, Y.D.; Yao, Y. Fractal Characterization of Pore–Fracture in Low-Rank Coals Using a Low-Field NMR Relaxation Method. Fuel 2016, 181, 218–226. [Google Scholar] [CrossRef]
- Zheng, S.J.; Yao, Y.B.; Liu, D.M.; Cai, Y.; Liu, Y. Characterizations of Full-Scale Pore Size Distribution, Porosity and Permeability of Coals: A Novel Methodology by Nuclear Magnetic Resonance and Fractal Analysis Theory. Int. J. Coal Geol. 2018, 196, 148–158. [Google Scholar] [CrossRef]
- Cai, J.; Hu, X.; Standnes, D.C.; You, L. An Analytical Model for Spontaneous Imbibition in Fractal Porous Media Including Gravity. Colloids Surf. A Physicochem. Eng. Asp. 2012, 414, 228–233. [Google Scholar] [CrossRef]
- Zhou, H.; Xie, H. Fractal Description of Porosity and Specific Surface Area of Porous Media. J. Xi’an Inst. Min. Technol. 1997, 17, 97–102. [Google Scholar]
- Peng, L.; Zhang, C.; Ma, H.L.; Pan, H. Estimating Irreducible Water Saturation and Permeability of Sandstones from Nuclear Magnetic Resonance Measurements by Fractal Analysis. Mar. Pet. Geol. 2019, 110, 565–574. [Google Scholar] [CrossRef]
- Chen, H.; Tang, D.Z.; Li, S.; Xu, H.; Tao, S.; Wang, J.; Liu, Y. Dynamic Evaluation of Heterogeneity in Pore-Fracture System of Different Rank Coals under Different Confining Pressure Based on Low-Field NMR. Energy Sources Part A Recovery Util. Environ. Eff. 2021, 43, 1620–1634. [Google Scholar] [CrossRef]
- Zhou, H.W.; Liu, Z.L.; Zhao, J.W.; Chen, B.C.; Li, X.N.; Zhong, J.C. In-Situ Observation and Modeling Approach to Evolution of Pore-Fracture Structure in Coal. Int. J. Coal Sci. Technol. 2023, 33, 265–274. [Google Scholar] [CrossRef]
- Mitra, P.P.; Sen, P.N. Effects of Microgeometry and Surface Relaxation on NMR Pulsed-Field-Gradient Experiments: Simple Pore Geometries. Phys. Rev. B 1992, 45, 143–156. [Google Scholar] [CrossRef]
- Kenyon, W.E. Nuclear Magnetic Resonance as a Petrophysical Measurement. Int. J. Radiat. Appl. Instrumentation. Part E. Nucl. Geophys. 1992, 6, 153–171. [Google Scholar] [CrossRef]
- Xu, H.; Tang, D.Z.; Chen, Y.P.; Ming, Y.; Chen, X.; Qu, H.; Yuan, Y.; Li, S.; Tao, S. Effective Porosity in Lignite Using Kerosene with Low-Field Nuclear Magnetic Resonance. Fuel 2018, 213, 158–163. [Google Scholar] [CrossRef]
- Kleinberg, R.L.; Vinegar, H.J. NMR Properties of Reservoir Fluids. Log Anal. 1996, 37, 20–32. [Google Scholar]
- Cai, M.; Zhao, X.; Kaiser, P.K. On Field Strength of Massive Rocks. Chin. J. Rock Mech. Eng. 2014, 33, 1–13. [Google Scholar]
- Xu, L.; Li, Q.; Myers, M.; Tan, Y.; Chen, Q.; Li, X. Flow Behavior Characteristics and Residual Trapping of Supercritical Carbon Dioxide in Tight Glutenite by MRI Experiments. J. Nat. Gas Sci. Eng. 2020, 83, 103540. [Google Scholar] [CrossRef]
- Zhou, H.W.; Wang, X.Y.; Zhang, L.; Zhong, J.C.; Wang, Z.H.; Rong, T.L. Permeability Evolution of Deep Coal Samples Subjected to Energy-Based Damage Variable. J. Nat. Gas Sci. Eng. 2020, 73, 103070. [Google Scholar] [CrossRef]
- Dai, B.; Zhao, G.; Konietzky, H.; Wasantha, P.L.P. Experimental Investigation on Damage Evolution Behaviour of a Granitic Rock under Loading and Unloading. J. Cent. S. Univ. 2018, 25, 1213–1225. [Google Scholar] [CrossRef]
- Li, Y.; Xu, T.; Xin, X.; Xia, Y.; Zhu, H.; Yuan, Y. Multi-Scale Comprehensive Study of the Dynamic Evolution of Permeability during Hydrate Dissociation in Clayey Silt Hydrate-Bearing Sediments. Adv. Geo-Energy Res. 2024, 12, 127–140. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Liu, Z.; Xie, S.; Yin, Y.; Su, T. Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis. Fractal Fract. 2025, 9, 93. https://doi.org/10.3390/fractalfract9020093
Liu Z, Xie S, Yin Y, Su T. Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis. Fractal and Fractional. 2025; 9(2):93. https://doi.org/10.3390/fractalfract9020093
Chicago/Turabian StyleLiu, Zelin, Senlin Xie, Yajun Yin, and Teng Su. 2025. "Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis" Fractal and Fractional 9, no. 2: 93. https://doi.org/10.3390/fractalfract9020093
APA StyleLiu, Z., Xie, S., Yin, Y., & Su, T. (2025). Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis. Fractal and Fractional, 9(2), 93. https://doi.org/10.3390/fractalfract9020093