## 2 Methodology
\[N=Ch^{-D} \tag{5}\]  
Figure 3: (**a**) Projection of karst cavern onto longitudinal section; (**b**) Projection of karst cavern onto ground surface plane (after [PERSON] et al., 2010) [35]. Figure 2: (**a**) Plane distribution of geological drilling holes; (**b**) Plane distribution of karst caverns. where \(h\) is the diameter of a karst cavern, which is the top level of the karst cavern minus its bottom level; \(N\) is the accumulated number of karst caverns, and the maximum of \(N\) is the total number of geological drilling holes; \(C\) is constant, which equals the number of geological drilling holes when the diameter of the karst cavern is minimal; and \(D\) is fractal dimension. Provided that \(a\) and \(b\) are small enough, it is expected that the variation of the diameter of a karst cavern is continuous and that the volume of a karst cavern can be calculated by integration. For a simple fractal model, volumes of karst caverns can be expressed by:  
\[V=\int_{h_{\min}}^{h_{\max}}ab\frac{\mathrm{d}N(\geq h)}{\mathrm{d}h}h \mathrm{d}h=\frac{AD}{(1-D_{f})h_{\min}^{D}}(h_{\max}^{1-D}-h_{\min}^{1-D})(D> 0,D\
eq 1) \tag{6}\]  
where \(V\) is the volume of the karst cavern; \(h_{\max}\) is the maximum diameter of the karst cavern; \(h_{\min}\) is the minimum diameter of the karst cavern; \(A\) is the total area. For a bifractal model, the volumes of karst caverns become:  
\[ \begin{split}& V=\int_{h_{\min}}^{h_{\max}}ab\frac{\mathrm{d}N( \geq h)}{\mathrm{d}h}h\mathrm{d}h=\sum\limits_{i=1}^{n-1}\frac{ab\mathrm{C}_{ i}D_{i}}{1-D_{i}}(H_{i+1}^{1-D_{i}}-H_{i}^{1-D_{i}})\\ &=\frac{ab\mathrm{C}_{1}D_{f(1)}}{1-D_{f(1)}}(H_{2}^{1-D_{f(1)} }-H_{1}^{1-D_{f(1)}})+\sum\limits_{i=2}^{n-1}\frac{ab\mathrm{C}_{1}D_{f(1)} \frac{h_{i=2}^{n-1}}{H_{i}^{D_{f(1)}-D_{f(i-1)}}}}{1-D_{f(i)}}(H_{i+1}^{1-D_{ f(i)}}-H_{i}^{1-D_{f(i)}})\end{split}  \tag{7}\]  
\[C_{n}=C_{i}\prod\limits_{i=2}^{n-1}H_{i}^{D_{i}-D_{i(i-1)}} \tag{8}\]  
\[V=AH_{1}^{D_{1}}\left[\frac{D_{1}(H_{2}^{1-D_{1}}-H_{1}^{1-D_{1}})}{1-D_{1}}+ \sum\limits_{i=2}^{n-1}\frac{D_{i}\prod\limits_{i=2}^{n-1}H_{i}^{D_{i}-D_{(i- 1)}}}{1-D_{i}}(H_{i+1}^{1-D_{i}}-H_{i}^{1-D_{i}})\right] \tag{9}\]  
where \(H_{1}=h_{\min}\an area of about 488.2 km\({}^{2}\) is widely distributed within the Guanghua Basin area. The quantity of karst water will vary, largely depending on the stage of development of the karst caverns. Because of the alluvial silt soil in the overlying Quaternary sediments, the underlying clasolite strata located at a depth range of 0.82-4.55 m can thus be deemed a confined aquifer. The piezometric level of this confined aquifer is close to the ground surface. Additionally, in the hilly region, the bedrock fissure water widely distributed across an area of 743.3 km\({}^{2}\) usually has a very high piezometric level [52; 53; 54; 55; 56; 57]. It is worth noting that the well-developed river network combined with the high precipitation provides abundant water resources that may worsen the geological conditions during shield tunneling [58; 59; 60; 61; 62; 63; 64; 65; 66]. The strata in Guangzhou can be divided into nine categories, from top to bottom: (1) back fill layer; (2) silt layer; (3) alluvial-diluvial sand layer; (4) alluvial-diluvial clay layer; (5) residual soil layer; (6) completely weathered bedrock layer; (7) highly weathered bedrock layer; (8) moderately weathered bedrock layer; and (9) slightly weathered bedrock layer. Based upon past experience [68], during shield tunneling the excavation face may encounter mixed ground containing a soft layer (e.g., clay) at the top and a hard layer (e.g., granite) at the bottom, namely the so-called upper-soft lower-hard strata. It is anticipated that while shield tunneling within the upper-soft lower-hard strata, an excessive wearing of the cutter bits and an upward alignment deviation would result in a reduction in the advance rate and additional frictional resistance, respectively, thus affecting construction progress [69; 70; 71; 72].