## 4 Discussion  
### NMR-Based Pore Fractal Modelling  
The damage to basalt caused by microwave radiation can be assessed by examining the pore size distribution through the concept of fractal dimension. The fractal dimension quantifies the complexity of the pore structure. A lower fractal dimension indicates that the pores are more evenly distributed, suggesting less damage to the rock. Conversely, a higher fractal dimension points to a more irregular pore structure, often resulting from significant microwave-induced damage. This irregularity may arise from intense heating, which leads to larger or more interconnected pores. [PERSON] et al. [40] describe that fractal structures exhibit self-similarity, meaning they follow a power-law relationship between pore size (\(r\)) and the number of pores (\(N\)). This relationship is mathematically expressed as follows:  
\[N=C_{1}r^{-D_{f}} \tag{9}\]  
where \(C_{1}\) is the constant of proportionality, \(D_{f}\) is the fractal dimension of the fractal structure, and r is the scale parameter of the rock, i.e., the pore size. According to the Kozeny-Carman model, the pores in rock can be regarded as a series of capillaries, and the number of capillaries satisfies the following function:  
\[N=\frac{V_{porc}(r)}{\pi r^{2}L} \tag{10}\]  
where \(V_{porc}\) is the total volume of pores in the rock, and r and L are the radius and length of the capillaries, respectively. For simplicity, \(L=C_{2}r\). Combined with Equation (10), the total volume of pores can also be expressed as follows:  
\[V_{porc}=C\pi r^{3-D_{f}} \tag{11}\]  
In addition, the following relationships can be easily deduced:  
\[\frac{dV_{porc}}{dr}=C(3-D_{f})\pi r^{2-D_{f}} \tag{12}\]  
Therefore, the cumulative volume of pores in a rock sample with a radius less than \(r\) can be described as follows:  
\[V_{porc}(<r)=\int_{r_{min}}^{r}C\Big{(}3-D_{f}\Big{)}\pi r^{2-D_{f}}dr=C\pi(r ^{3-D_{f}}-r_{min}^{3-D_{f}}) \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Microwave Radiant** & **Microwave** & **Percentage of Shear** & **Percentage of Tensile** \\
**Power** & **Radiation Duration** & **Cracks** & **Cracks** \\ \hline
0 kW & 0 s & 32.5\% & 67.5\% \\ \hline  & 30 s & 32.8\% & 68.2\% \\
3 kW & 60 s & 22.8\% & 77.2\% \\  & 90 s & 21.6\% & 78.4\% \\  & 120 s & 18.0\% & 82.0\% \\ \hline  & 30 s & 23.3\% & 76.7\% \\
6 kW & 60 s & 18.3\% & 81.1\% \\  & 90 s & 17.1\% & 82.9\% \\  & 120 s & 15.4\% & 84.6\% \\ \hline \end{tabular}
\end{table} 
Table 6: Proportion of basalt tensile and shear cracks exposed to different microwave radiation in uniaxial compression experiments. where \(r_{min}\) is the radius of the smallest pore. Similarly, it is easy to obtain the following relationship:  
\[V_{pore}(<r_{max})=C\pi(r_{max}^{3-D_{f}}-r_{min}^{3-D_{f}}) \tag{14}\]  
where \(r_{max}\) is the radius of the largest pore. Combining Equations (13) and (14) gives Equation (15):  
\[\frac{V_{pore}(<r)}{V_{pore}(<r_{max})}=\frac{r^{3-D_{f}}-r_{min}^{3-D_{f}}}{r_{ max}^{3-D_{f}}-r_{min}^{3-D_{f}}}=\frac{\left(\frac{r}{r_{max}}\right)^{3-D_{f}}- \left(\frac{r_{min}}{r_{max}}\right)^{3-D_{f}}}{1-\left(\frac{r_{min}}{r_{max} }\right)^{3-D_{f}}} \tag{15}\]  
When \(r_{min}\) is much smaller than \(r_{max}\), combining Equations (14) and (15), the further calculation can be simplified as follows:  
\[\frac{V_{pore}(<r)}{V_{pore}(<r_{max})}=\left(\frac{r}{r_{max}}\right)^{3-D_{f }}=\left(\frac{T_{2}}{T_{2 max}}\right)^{3-D_{f}} \tag{16}\]  
\[\frac{V_{pore}(<r)}{V_{pore}(<r_{max})}=\frac{\varphi_{acc}(T_{2})}{\varphi_{N}} \tag{17}\]  
Thus, it is easy to obtain Equation (18):  
\[\ln(\varphi_{acc}(T_{2}))-\ln(\varphi_{N})=\left(3-D_{f}\right)\ln(\frac{T_{ 2}}{T_{2 max}}) \tag{18}\]  
The fractal dimension of pores within basalt specimens subjected to microwave radiation offers a detailed understanding of the structural changes induced by the treatment. Equation (18), which likely outlines a method for calculating fractal dimensions, was applied to analyze the pore structures using NMR-derived \(T_{2}\) relaxation times.