# Pore Structure of Coals by Mercury Intrusion, N\({}_{2}\) Adsorption and NMR: A Comparative Study
## 2 Experiments and Methods
### Basic Methodology for NMR  
The NMR analysis is based on the NMR relaxation behavior of the spin hydrogen nuclei in rocks in uniform static magnetic field and radio frequency field. According to previous studies, the expression of NMR transverse relaxation time, T\({}_{2}\), can be simplified as follows [5]:  
\[\frac{1}{\text{T}_{2}}=\rho_{2}\bigg{(}\frac{\text{S}}{\text{V}}\bigg{)} \tag{1}\]  
where, T\({}_{2}\) represents the relaxation time of the pore fluid obtained in the experiment; \(\rho_{2}\) is the transverse surface relaxation, constant; S is the surface relaxation; and V stands for pore volume. Therefore, according to the Equation (1), the T\({}_{2}\) value of coal sample is in positive ratio to pore radius, and the longer relaxation time represent the larger the pore radius. Therefore, the NMR signal corresponding to T\({}_{2}\) value can be used to directly reflect the pore distribution [6; 7].  
In order to determine the corresponding relation between T\({}_{2}\) value and pore diameter, the movable fluid in pores is removed by means of centrifugation technology. According to Washburn equation, when the pore fluid in coal is water, surface tension is 0.076 N/m, contact angle of 60\({}^{\circ}\), after iterative:  
\[r_{\text{c}}=\frac{0.14}{P_{\text{c}}} \tag{2}\]  
The Equation (2) is satisfied between the centrifugal force and the pore radius. NMR signal from coal samples was re-measured after centrifugation. The amount of signal displayed corresponds to that of the fluid constrained in the pores with diameter smaller than r\({}_{\text{c}}\). The overall distribution of pore size distribution in coals can be reflected by the corresponding relationship between value r\({}_{\text{c}}\) and T\({}_{2\text{C}}\).  
Through the application of the Equation (2), the water in the pores with diameter greater than 100 nm under the centrifugal force of 200 psi can be centrifuged. The corresponding T\({}_{2}\) value is equivalent to T\({}_{2\text{C}}\). Therefore, the relationship between T\({}_{2}\) value and pore size can be established:  
\[\frac{\text{T}_{2\text{C}}}{\text{100 nm}}=\frac{\text{T}_{2}}{r} \tag{3}\]  
where \(r\) represents the coal pore radius, in units of nm.  
In addition, porosity could be calculated by the calibration of NMR signal of water content in NMR. The results of the nuclear magnetic T\({}_{2}\) spectra of saturated and centrifugal coal samples can be used to reflect the pore distribution characteristics of coal reservoirs. There is a good linear relationship between the mass of the water sample and NMR signal amplitude:  
\[y=10^{4}\;x\;\left(R^{2}=0.997\right) \tag{4}\]  
where y represents NMR signal amplitude, dimensionless; x represents the water content, in units of g; the slope of the standard equation is in units g\({}^{-1}\). The Equation of nuclear magnetic porosity can be established by combining the T\({}_{2}\) spectrum of the coal samples saturated with water and the Equation (4):  
\[\varnothing=\frac{T(\mathrm{S_{w}})/10^{4}}{\rho\mathrm{V}} \tag{5}\]  
where \(\emptyset\) represents the NMR porosity, dimensionless; T(S\({}_{\mathrm{w}}\)) is the nuclear magnetic resonance signal amplitude of saturated water coal sample, dimensionless; \(\varrho\) represents water density, and the experiment distilled water is set to 1 g/m\({}^{3}\); V is the volume of coal pillar. The cylindrical coal samples with a diameter of 2.5 cm and a length of 5 cm were used uniformly, defining V as a constant of 24.54, unit cm\({}^{3}\).