# Estimating RQD for Rock Masses Based on a Comprehensive Approach
## 2 The Introduced Approach for Estimating the Representative RQD Based on Class Ratio Analysis
### Step 1  
In this paper, the RQD is derived by laying virtual boreholes in a three-dimensional fracture network model, the parameters of which can be estimated from data obtained via the scanline method, three-dimensional laser scanning, photogrammetry and other approaches. Figure 2 depicts the specific principle of computing the RQD for a single virtual borehole.  
As shown in Figure 2, to facilitate laying virtual boreholes, the mesh is divided on the upper surface of the model according to the number of boreholes. By taking one of the virtual boreholes, which intersects and splits into segments with numerous discs within the model size \(D\), the RQD calculation formula is as follows [9]:  
\[\text{RQD}=\frac{\sum_{i=1}^{n}d_{i}}{D} \tag{1}\]  
where \(d_{i}\) is the length of the \(i\)th line segment longer than 100 mm, and \(D\) is the length of the virtual borehole.  
Figure 1: Methodology workflow.  
The RQD values obtained from virtual boreholes (indexed as \(i=1,2,\ldots,n\)) within a model of size \(D\), sharing an identical orientation and length, collectively form a multi-valued set \(R_{n}=(r_{1},\)\(r_{2},\)\(r_{3}\)\(\ldots,\)\(r_{n})\). When each \(r_{i}\) in \(R_{n}\) follows a normal distribution, the RQD sample set \(R_{n}\) can be transformed into the Confidence Neutrosophic Number Cubic Value (CNNCV) [39], which includes the confidence interval and the mean. This method can ensure that \((1-\alpha)\%\) of the data in \(R_{n}\) falls within the confidence intervals, where \(\alpha\) is the significance level, and the mean can reveal the central tendency of the data. Thus, from the perspective of probability estimation, this conversion method reflects the advantage of rationality and credibility [39]. The calculation equations of the mean \(\xi\) and standard deviation \(\beta\) of the RQD sample set \(R_{n}\) are as follows:  
\[\xi=\frac{1}{n}{\sum\limits_{i=1}^{n}{r_{i}}} \tag{2}\]  
\[\beta=\sqrt{\frac{1}{n-1}{\sum\limits_{i=1}^{n}{(r_{i}-\xi)}^{2}}} \tag{3}\]  
Then, the RQD sample dataset \(R_{n}\) can be converted into the CNNCV \(E_{D}\) (\(n\), \(\alpha\)) using the following equation:  
\[E_{D}(n,\alpha)=\left\langle\left[E_{D}(n)^{-},E_{D}(n)^{+}\right]\text{, }M_{D}(n)\right\rangle=\left\langle\left[\xi-\frac{\beta}{\sqrt{n}}t_{\alpha/2 },\xi+\frac{\beta}{\sqrt{n}}t_{\alpha/2}\right]\text{, }\xi\right\rangle \tag{4}\]  
where \(E_{D}(n)^{-}\), \(E_{D}(n)^{+}\) and \(M_{D}(n)\) are the lower confidence limit (LCL), upper confidence limit (UCL) and mean of the RQD sample dataset \(R_{n}\), respectively. \(t_{\alpha/2}\) is the critical value adopted from [40] in view of the confidence levels of \((1-\alpha)\%\), as detailed in Table 1 below.  
It should be noted that, when calculating confidence intervals for sample data sets obtained from different numbers of virtual boreholes, the same confidence level must be selected for the case study.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Confidence Level (\(1-\alpha\))\%** & \(t_{\alpha/2}\) \\ \hline
80\% & 1.282 \\
85\% & 1.440 \\
90\% & 1.645 \\
95\% & 1.960 \\
99\% & 2.576 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relationship between \(t_{\alpha/2}\) and confidence level \((1-\alpha)\%\).  
Figure 2: Virtual borehole layout and RQD calculation principle.