# Experimental Analysis of Pore and Permeability Characteristics of Coal by Low-Field NMR
## 3 Discussion
### NMR Pore Structure
#### 3.3.2 NMR Permeability  
Although the NMR test cannot directly determine reservoir permeability, it can be used to obtain distribution and structural parameters of the pores in the reservoir and then to calculate the permeability with them. There are two main kinds of calculation models for NMR permeability of conventional oil and gas reservoir: the Timur-Coates (TC) model [27; 28] and the mean \(T_{2}\) model (SDR) model [29]. In the two models, the NMR porosity was all used, and the TC model also took into account the saturation of the irreducible and movable water in the reservoir. The equation is as follows:  
\[k_{\text{TC}}=a\cdot\Big{(}\frac{\varphi_{\text{NMR}}}{100}\Big{)}^{4}\cdot \Big{(}\frac{\text{FFI}}{\text{BVI}}\Big{)}^{2} \tag{9}\]  
where \(k_{\text{TC}}\) is NMR permeability calculated by the TC model, mD; \(a\) is the empirical statistic constant, which is 42,000 in this paper; and \(\varphi_{\text{NMR}}\) is the NMR porosity, %. The TC model has been modified by many scholars to adapt to various conventional reservoirs under different geologic backgrounds [30; 31]. [PERSON] has also modified this model and applied it to the permeability calculation of coal samples [11]. Figure 8: Comparison of pore characteristics of the YJL coal samples by mercury intrusion porosimetry (MIP) and NMR. Besides considering the NMR porosity, the SDR model also considers the geometric mean of transverse relaxation time, and the equation is as follows [29]:  
\[k_{\text{SDR}}=b\cdot\Big{(}\frac{\varphi_{\text{NMR}}}{100}\Big{)}^{4}\Big{(}T _{\text{2, log mean}}\Big{)}^{2} \tag{10}\]  
where \(k_{\text{SDR}}\) is the NMR permeability calculated by SDR model, mD; and \(T_{\text{2,log mean}}\) is the geometric mean of transverse relaxation time, ms.  
[PERSON] et al. [32] used the SDR model to predict the permeability of the limestone reservoir, but the calculation effect in the coal reservoir was poor. Based on the NMR porosity of movable water, [PERSON] et al. [11] proposed a PP model to calculate the permeability of coal reservoir. \[k_{\text{PP}}=0.49\exp(\frac{\varphi_{\text{NMR}}}{1.35})-0.54 \tag{11}\]  
[PERSON] et al. [15] compared and analyzed the reliability of the above three models according to the permeability and NMR parameters of more than 30 coal samples and proposed a coal rank (CR) model based on the coal rank. The above equations only discuss the relationship between permeability and porosity. Because the pore structure of coal is complex and changeable, the structure parameters, such as pore size, tortuosity, and fractal dimension, will also affect the permeability. Based on the Kozeny-Carman equation, the permeability is related to the porosity, tortuosity, and effective fluid radius [33; 34]. \[k=\Big{(}\frac{\varphi_{\text{NMR}}}{\tau}\Big{)}\frac{(r_{\text{eff}})^{2}}{8} \tag{12}\]  
where \(\tau\) is the tortuosity; and \(r_{\text{eff}}\) is the effective fluid radius, nm. The above equation is based on the assumption that the wall of the pore is smooth. Considering the fractal characteristics of the pore structure, the relation between the average particle size, and surface area, the following equation (NMRF model) can be deduced [16]:  
\[k_{\text{NMRF}}=\frac{r_{\text{g}}}{8}^{2}\frac{\varphi_{\text{FFI}}}{\tau}[ \frac{2\varphi_{\text{FFI}}}{3\tau(1-\varphi_{\text{FFI}})}]^{\frac{2}{D_{ \text{NMR}}^{2}-1}} \tag{13}\]  
where \(\varphi_{\text{FFI}}\) is the NMR movable fluid porosity, %; \(D_{\text{NMR}}\) is the NMR fractal dimension of seepage fissures; and \(r_{\text{g}}\) the average grain radius, nm. The average grain radius can be expressed as follows [35]:  
\[r_{g}=\frac{3}{S_{\text{p}}}\frac{1-\varphi_{\text{FFI}}}{\varphi_{\text{FFI}}} \tag{14}\]  
where \(S_{\text{p}}\) is the ratio of pore specific surface area to movable water pore volume, m\({}^{-1}\).