# Experimental Analysis of Pore and Permeability Characteristics of Coal by Low-Field NMR
## 3 Discussion
### NMR Pore Structure  
#### 3.3.1 PSD of NMR  
The preceding discussion shows that the \(T_{2}\) is linearly related to the equivalent pore size, and the conversion coefficient between them can be obtained by centrifugal experiments. The water in the pores of the coal sample can overcome the capillary pressure discharge coal sample under the centrifugal force. According to the National Standard of GB/T29171-2012 [24], the centrifugal force is related to the rotational speed, rotor radius, and sample length, and the centrifugal force was calculated by the following equation:  
\[P_{\rm c}=1.097\times 10^{-9}\Delta\rho L(R_{\rm c}-\frac{L}{2})n^{2} \tag{4}\]  
where \(P_{\rm c}\) is centrifugal force, MPa; \(\Delta\rho\) is two-phase fluid density difference, g/cm\({}^{3}\); \(L\) is the coal sample length, cm; \(Rc\) is the rotor radius, cm; and \(n\) is the rotational speed, r/min. The capillary force in the capillary is equal to the additional pressure (\(\Delta P\)) produced by the curved liquid surface, the \(\Delta P\) is related to the shape of the capillary, and the relationship between the \(\Delta P\) and the radius of the surface curvature can be deduced according to the Young-Laplace equation [25]:  
\[\Delta P=\sigma\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right) \tag{5}\]  
where \(\sigma\) is the liquid surface tension, N/m; and \(r_{1}\) and \(r_{2}\) are the radius of curvature of the curved liquid surface in two directions. When the capillary is a cylindrical tube or a plate fissure, there will appear \(r_{1}=r_{2}\) or \(r_{1}\gg r_{2}\), so the upper-equation is simplified to the following:  
\[\Delta P=2\sigma/r_{\rm p},\ \rm{and}\ \Delta P=2\sigma/d_{\rm f} \tag{6}\]  
where \(r_{\rm p}\), \(d_{\rm f}\) is the radius of cylindrical pore and the width of plate-like fissure, nm, respectively. The critical pore size of the centrifugal force can be expressed by the Washburn equation [26]. \[r_{\rm p}=\frac{2\sigma\cos\theta}{P_{\rm c}},\ \rm{and}\ d_{\rm f}=\frac{2 \sigma\cos\theta}{P_{\rm c}} \tag{7}\]  
where \(\sigma\) is the water surface tension, 0.072, N/m; and \(\theta\) is the wetting angle, 0\({}^{\circ}\). The centrifugal test instrument is the Eppendorf 5804R high-speed centrifuge (Eppendorf, Hamburg, Germany). The fixed-angle rotor, rotor radius, and maximum speed are F-34-6-38, 11.5 cm, and 12,000 rpm, respectively. The rotational speed of the centrifugal experiments is 8000 rpm. According to Equations (4)-(7), when the rotational speed of the centrifugal experiments is 8000 rpm, the corresponding pore radius is 45.64 nm. The shape and structure of the pores and fissures in coal are very complicated. The nanopores in coal are mainly spherical and cylindrical, whereas the micro-fissures and fissures are network-like, poorly connected, rough plate-like cracks. The above equations assumed that the pores in the coal are regular, smooth cylindrical pores or plate-like fissures. It differs greatly from the equivalent pore size of the different pores and fissures in the coal, so the PSD of 5-100 nm is corrected by the gas adsorption analysis results. The equivalent pore size corresponding to \(T_{2\rm C}\) is calculated by the centrifugal experiment. The pore size corresponding to other \(T_{2}\) values can be calculated by the proportional relationship between them. Considering the correction factor, the expression is as follows:  
\[d_{pi}=C\frac{d_{2\rm C}}{T_{2\rm C}}T_{2i} \tag{8}\]  
where \(d_{pi}\) is the equivalent pore size, nm; \(T_{2i}\) is the transverse relaxation time, ms; \(d_{2\rm C}\) is the pore size corresponding to the \(T_{2\rm C}\), nm; and \(C\) is the correction factor.