## 4 Verification
### Method Verification
#### 4.2.3 Nine Hundred and Thirty-Seven Grouting Holes  
During the treatment, a total of 937 grouting holes were drilled to a depth range of 0.2-0.3 m below the bottom of the karst caverns in a rectangular pattern at a spacing of 2 m. Figure 8 shows the layout of the grouting holes for a single karst cavern. The injected volume of grouts for each hole was recorded during the treatment. Figure 11 shows the relationship between the 937 grouting holes and the diameters of the karst caverns. Two straight line segments representing the data are expressed as follows:  
\[\left\{ \begin{array}{c}\ln N=-0.10\ln h+6.82\\ \ln N=-13.11\ln h+34.35\end{array} \right. \tag{13}\]  
By Equation (9), the volume of the karst caverns can be calculated and visualized, as shown in Figures 12 and 13. Table 1 summarizes the covering areas, fractal parameters, the calculated total volume of the karst caverns, and the karst grouting coefficients for the three datasets analyzed (the three datasets are called 'Case A', 'Case B', and 'Case C', respectively, in the following discussion). In fact, the three cases can represent three scenarios. Cases A and C represent inadequate drilling holes and adequate drilling holes, respectively, while Case B represents a scenario that often occurs during the ground investigation phase, in which the number of drilling holes is more than Case A but still less than Case C. In fact, the number of drilling holes for Case B is less than that of Case A. However, as Case B excluded drilling holes that did not detect karst caverns, its accuracy of detecting karst caverns is similar to cases having more drilling holes than Case A. Generally, the higher the number of geological drilling holes, the higher the accuracy of the calculated total volume (\(V\)) of karst caverns. This implies that the \(V\) value calculated from Case A could largely deviate from the actual value, and that derived from Case C should be close to the real value. However, due to a limited project budget and duration in most cases, it is not likely to provide adequate drilling holes for the detection of karst caverns. Thus, there is a higher chance to obtain the \(V\) value close to that from Case B, and the \(V_{R}\) value can then be calculated to be 0.26, taking into account \(V=8742.2\) m\({}^{3}\) and \(R_{C}=2292.1\) m\({}^{3}\). This higher \(R_{C}\) value than the actual \(V\) value (i.e., 2083.7 m\({}^{3}\)) implies that there was a loss of grouts injected, most likely due to leaking fractures or fissures of the surrounding geology. As the \(V\) value from Case B could lead to misleading inferences regarding the \(R_{C}\) value, an excessive injection of grouts may occur. As to the treatment also referred to as \"void-filling grouting\" [76; 77; 78], two types of slurry; namely, single slurry and binary slurry, were injected into the karst caverns. The single slurry consisted of water, cement, and sand at a mass ratio of 0.8:1:1, and was used to fill the central holes (Figure 8). The binary slurry consisted of water, cement, sand, and sodium silicate at a mass ratio of 0.8:1:1:0.1, and was used  
Figure 11: Fractal model describing the relationship between the 937 grouting holes and the diameters of the karst caverns. to fill the peripheral holes (Figure 8). The results indicated that the 28-day unconfined compressive strength of the recovered core samples, and the SPT values as well as the core recovery (CR) values are all above the engineering requirements, i.e., UCS \(>\) 0.2 MPa, SPT-N \(>\) 10, and CR \(>\) 90%, respectively. Additionally, there will be more construction projects located underground in buried karst in China in the near future. The distribution of karst caverns and the required volume of grouts would be the main concern for engineers and practitioners.