# Experimental Study on the Influence of Low Temperature on the Gas Permeability of Granite
## 2 Materials and Methods
### Two-Dimensional Gas Permeability Test  
Three hollow specimens, each of coarse-grained and fine-grained granite, were first tested to determine the 2D gas permeability under ambient conditions. Following this, 2D gas permeability tests were conducted under real-time low-temperature conditions at \(-20\)\({}^{\circ}\)C, \(-40\)\({}^{\circ}\)C, \(-60\)\({}^{\circ}\)C, \(-90\)\({}^{\circ}\)C, and \(-120\)\({}^{\circ}\)C in sequence. Prior to each test, the specimens were oven-dried to ensure uniform starting conditions, allowing for the determination of the real-time low-temperature 2D gas permeability at each of the specified temperatures.  
In this study, the 2D gas permeability of granite was investigated using the radial seepage method without applying confining pressure. The test apparatus consists of two primary components: a gas transmission system and a fixture that holds the specimens without confining pressure. Each granite specimen was drilled through the center and placed in the fixture, which was secured with four fastening bolts. The fixture was sealed with upper and lower covers to prevent any gas leakage. The upper cover was designed with a gas seepage channel that was connected to the gas transmission system, allowing controlled gas flow during testing, and the simplified diagram of the test principle is shown in Figure 4.  
During the 2D radial permeation tests, higher inlet gas pressures were applied within the cavities of the hollow granite specimens, and the pressure decrease over time was monitored. The gas passed through the annular specimen in a radial direction, and its permeability was considered as the 2D radial gas permeability [35].  
The 2D radial gas permeability was calculated using the following equation:  
\[K_{r}=\frac{\mu V}{\pi h}\frac{\ln(R)}{P^{2}-1}\frac{\Delta p}{\Delta t},\bigg{(} R=\frac{r_{2}}{r_{1}},P=\frac{p_{1}}{p_{a}}\bigg{)} \tag{3}\]  
where \(K_{r}\) is the 2D radial gas permeability coefficient; \(V\) is the volume of the gas buffer bottle; \(h\) is the height of the specimen; \(r_{1}\) and \(r_{2}\) are the inner and outer radii of the annular cavity, respectively, and \(p_{1}\) and \(p_{a}\) are the corresponding pressures at the inner and outer boundary conditions, respectively.  
The gaseous fluid flows radially out of the annular cavity, with fluid flow from the top and bottom of the cavity ideally neglected.  
The Knudsen number \(K_{n}\) is commonly used to categorize flow regimes in small pores [36]. It is expressed using the following form:  
\[K_{n}=\frac{\lambda}{d} \tag{4}\]  
\[\lambda=\frac{\mu}{p}\sqrt{\frac{\pi R_{g}T}{2M}} \tag{5}\]  
where \(K_{n}\) is the Knudsen number; \(\lambda\) is the average free range of gas molecules; \(d\) is the average pore diameter of the specimen; \(\mu\) is the gas viscosity coefficient of the gas at \(P\) (pressure) and \(T\) (temperature); \(R_{g}\) is the universal gas constant; \(M\) is the gas molecular mass. According to the Knudsen number \(K_{n}\), the gas flow regime can be categorized into continuous/Darcy flow (\(K_{n}<0.001\)); slip flow (\(0.001<K_{n}<0.1\)); transition flow (\(0.1<K_{n}<10\)); and free molecular (Knudsen) flow (\(K_{n}>10\)) [37].  
The temperature range for the 2D gas permeability test was from 20 \({}^{\circ}\)C to \(-120\) \({}^{\circ}\)C, with an air pressure range of 5 to 15 bar. Based on Equations (4) and (5), the calculated Knudsen number ranged from 0.00223 to 0.0128 for coarse-grained granite and from 0.0685 to 0.395 for fine-grained granite. Both ranges can be corrected for permeability using the Klinkenberg equation.