## 4 Fractal Study on Microscopic Pore Structure of Zhanjiang Formation Structural Clay  
### Menger Sponge Model  
The Menger sponge model is derived from the two-dimensional Sierpinski carpet fractal model (Figure 6). The construction starts with a solid cube with an edge length of \(b\), which is divided into \(n^{3}\) equal smaller cubes with an edge length of \(b/n\). Then, according to a set pattern, a number of small cubes are removed, leaving behind \(n^{3}-a\) small cubes to form the first iteration (\(n=1\)). For each of the \(n^{3}-a\) small cubes in the first iteration, the same process is applied: they are further divided into \(n^{3}\) smaller cubes with an edge length of \(b/n^{2}\). At this stage, there are \((n^{3}-a)^{2}\) small cubes remaining, which form the second iteration (\(n=2\)). Similarly, after the \(k\)-th iteration, the remaining small cubes have an edge length of \(b/n^{k}\) and there are \((n^{3}-a)^{k}\) of them. This procedure is iterated to generate the Menger sponge model, as depicted in Figure 7.  
In the Menger sponge model, let the edge length of the initial cube be \(b_{0}=1\), and its volume \(V_{0}=1\). At the first iteration, \(n=1\), \(b_{1}=1/3\), the cube is divided into nine squares on each face, which splits the cube into 27 smaller cubes, similar to a Rubik's cube. The small cube at the center of the larger cube and the six adjacent small cubes connected to it are removed. The remaining volume of the 20 small cubes is \(V_{1}=20\times(1/3)^{3}=20/27\). At the second iteration, \(n=2\), \(b_{2}=(1/3)^{2}\), and the remaining volume of the 400 small cubes is \(V_{2}=20^{2}\times(1/3)^{6}=400/729\). At the \(n\)-th iteration, \(n=i\), \(b=(1/3)^{i}\), the volume of the remaining \(20^{i}\) small cubes is \(V_{i}=20^{i}\times[(1/3)^{i}]^{3}\). The volume of the Menger sponge model is thus determined by this iterative process, which is given by Equation (4). Figure 5: Relationship between the fitted value and the test value of the unconfined compressive strength. Figure 6: Sierpinski carpet fractal model [35,36]: (**a**) \(n=0\); (**b**) \(n=1\); (**c**) \(n=2\); (**d**) \(n=3\). \[V(b_{i})=N_{i}\times{b_{i}}^{3} \tag{4}\]  
In conjunction with [PERSON]'s concept of \"fractional dimension [37; 38]\", when \(b\) approaches 0, the measure corresponding to the Hausdorff dimension tends towards a finite value, which is Equation (5). \[N(b)b^{D_{H}}=C^{*}, \tag{5}\]  
where \(D_{H}\) is the Hausdorff dimension. By combining Equations (4) and (5), Equation (6) can be derived. \[V(b_{i})=C^{*}{b_{i}}^{3-D} \tag{6}\]  
In the equation, \(C\) represents a constant, while \(D\) denotes the fractal dimension. Therefore, the porosity \(\phi\) corresponding to \(N(b)\) pores can be expressed as (7). \[\phi=\frac{V(b_{i})}{{b_{0}}^{3}}=\frac{C^{*}{b_{i}}^{3-D}}{{b_{0}}^{3}}=C^{*}{ b_{0}}^{-D}\bigg{(}\frac{b_{i}}{b_{0}}\bigg{)}^{3-D} \tag{7}\]  
Let \(C=C^{*}{b_{0}}^{-D}\), then Equation (7) can be derived to yield Equation (8). \[\phi=C\bigg{(}\frac{b_{i}}{b_{0}}\bigg{)}^{3-D} \tag{8}\]  
When \(b_{i}=b_{0}\), the porosity \(\phi=1\); hence, \(C=1\). Therefore, Equation (9) is obtained. \[\phi=\bigg{(}\frac{b_{i}}{b_{0}}\bigg{)}^{3-D} \tag{9}\]  
When observed at the scale of \(b_{i}\), only particles with diameters greater than or equal to \(b_{i}\) can be detected, and the volume should be the total volume minus the corresponding pore volume:  
\[V_{i}(\geq b)=V_{a}(1-\phi) \tag{10}\]  
Substituting (9) into (10) yields Equation (11). Figure 5:0  
Equation (11) can be rephrased as Equation (12). Figure 5:1  
Taking the logarithm on both sides yields Equation (13). Figure 5:2  
In the formula, \(V_{i}(\geq b)\) represents the total pore volume with pores greater than or equal to \(b\), \(V_{a}\) represents the total pore volume within the measurement range, \(b\) represents the observation scale (pore diameter), \(b_{0}\) represents the scale of the observation range, and \(D\) is the fractal dimension of the pore volume.