A Multi-Scale Fractal Approach for Coal Permeability Estimation via MIP and NMR Methods
Abstract
:1. Introduction
2. Experimental Implementations
2.1. Coal Samples
2.2. Experimental Facilities and Procedures
2.3. MIP and NMR Theory
2.4. Experimental Data Analysis
2.4.1. Experimental Data of MIP
2.4.2. Experimental Data of NMR
3. Permeability Formulas Derivation Based on MFU
3.1. Fractal Characteristics of Porous Media
3.2. Multi-Scale Fractal Dimension Characteristics Units Model
3.3. Multi-Scale Fractal Permeability Expression
4. Results and Discussion
4.1. Fractal Characteristics of Pore Size Distribution
4.1.1. Fractal Characteristics of Pore Size Distribution Base on MIP
4.1.2. Fractal Characteristics of Pore Size Distribution Base on NMR
4.2. The Permeability Contribution of the Different Fractal Dimension
4.3. Comparison of the Pore Structure from MFU and Experiments
4.4. The Permeability Predicted Results Base on the MFU
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
Acronyms and Symbols | Explanation |
MFU | Multi-scale fractal dimension characteristics units model |
MIP | Mercury injection porosimetry |
NMR | Nuclear magnetic resonance |
LTNA | Low temperature N2 adsorption |
Micro-CT | Micro X-ray computed tomography |
CPMG | Carr-Purcell-Meiboom-Gill |
T2 | Transverse relaxation time |
r | Pore radius |
rmax | Maximum pore radius |
rmin | Minimum pore radius |
γ | Interfacial tension |
θ | Three-phase contact angle |
P | Mercury intrusion pressure |
ρ | Surface relaxivity |
SA | Surface area |
V | Pore volume |
Vsample | Volume of the sample |
Vu | Volume of unit |
F | Shape factor of pore |
c | Shape factor of pore |
Pd | Displacement pressures |
ϕ | Porosity of coal sample |
α | Factor of fractal coefficient |
Df | Fractal dimension |
S | Cumulative pore volume fraction when the pore radius is less than or equal to the pore radius r |
μ | Fluid viscosity |
ΔPf | Pressure gradient along the tortuous capillary |
τ | Average tortuosity of the capillaries |
τi | Average tortuosity of the capillaries with radii between ri−1 and ri |
L | Straight length of the capillary |
Qi | Flow rate for the capillaries with radii between ri−1 and ri |
Q | Total flow rate for the whole cross-sectional area |
ϕi | Porosity of the capillaries with radii between ri−1 and ri |
k | Permeability |
DMIP | Fractal dimension of total pore structure measured by mercury injection porosimetry |
DNMR | Fractal dimension of total pore structure measured by nuclear magnetic resonance |
D1 | Fractal dimension of micropore |
D2 | Fractal dimension of mesopore |
D3 | Fractal dimension of macropore |
D | Theory fractal dimension |
dE | Euclidean space dimension |
Kj | Permeability contribution of capillaries of radius rj |
Ki | Cumulative permeability contribution of the capillaries with radii between ri−1 and ri |
Wj | Pore aperture distribution frequency of capillaries of radius rj |
WTj | T2 distribution frequency of capillaries of transverse relaxation time T2j |
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Sample | Pd (MPa) | rmax (μm) | Maximum Mercury Saturation (%) | Residual Mercury Saturation (%) | Mercury Withdrawal Efficiency (%) | Φ (%) | ||
---|---|---|---|---|---|---|---|---|
Micro-Pore | Meso-Pore | Macro-Pore | ||||||
M1 | 0.261 | 2.814 | 69.492 | 20.065 | 71.126 | 1.912 | 1.546 | 0.552 |
M2 | 0.261 | 2.813 | 82.271 | 28.761 | 65.041 | 1.819 | 2.233 | 0.538 |
M3 | 0.138 | 5.332 | 90.877 | 30.690 | 66.230 | 2.096 | 1.799 | 1.348 |
Sample | Φ (%) | ||
---|---|---|---|
Micropore | Mesopore | Macropore | |
N1 | 0.925 | 0.295 | 0.265 |
N2 | 1.814 | 0.614 | 0.940 |
N3 | 1.273 | 1.005 | 0.212 |
MIP | NMR | ||||||
---|---|---|---|---|---|---|---|
M1 | M2 | M3 | N1 | N2 | N3 | ||
Di | D1 | 1.160 | 1.128 | 1.212 | 1.984 | 2.002 | 2.089 |
D2 | 2.888 | 2.831 | 2.883 | 2.913 | 2.884 | 2.835 | |
D3 | 2.973 | 2.978 | 2.947 | 2.926 | 2.868 | 2.968 | |
k (mD) | Xu | 2.015 | 2.240 | 8.277 | 37.3351 | 37.409 | 46.958 |
multi-scale fractal permeability model | 0.018 | 0.024 | 0.275 | 0.0360 | 0.0302 | 0.0157 | |
Experiment | 0.026 | 0.073 | 0.336 | - | 0.0347 | 0.0099 |
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Ren, W.; Zhou, H.; Zhong, J.; Xue, D.; Wang, C.; Liu, Z. A Multi-Scale Fractal Approach for Coal Permeability Estimation via MIP and NMR Methods. Energies 2022, 15, 2807. https://doi.org/10.3390/en15082807
Ren W, Zhou H, Zhong J, Xue D, Wang C, Liu Z. A Multi-Scale Fractal Approach for Coal Permeability Estimation via MIP and NMR Methods. Energies. 2022; 15(8):2807. https://doi.org/10.3390/en15082807
Chicago/Turabian StyleRen, Weiguang, Hongwei Zhou, Jiangcheng Zhong, Dongjie Xue, Chaosheng Wang, and Zelin Liu. 2022. "A Multi-Scale Fractal Approach for Coal Permeability Estimation via MIP and NMR Methods" Energies 15, no. 8: 2807. https://doi.org/10.3390/en15082807