## 6 A Discussion of the Qualitative Aspects (Joint Conditions) of Existing GSI Charts
### Consideration of a Persistence Correction Factor (Pi) in P32 Estimates for V-GSI  
Table 4 below shows the results using a suggested new method that considers the estimated means' size (disc diameters) and spacing means per joint set as a correction factor for P32 estimates using field-measured data. [PERSON]'s data (Table 2) are used again to validate Equations (8) and (9) below. In essence the Pi factor is used to adjust the true spacings of joint sets that have less persistence than the diameter of a unit volume to correct the overestimate of the VFC parameter. The Pi ratio is the mean field-assessed area per joint set over the maximum area possible in a unit volume sphere. The adjusted spacings are then used to recalculate VFC and P32 for use in the V-GSI chart. The geometry used in this factor is already explicitly considered in Equations (1)-(3) and (5). Pi comes from \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Set** & \(n\) & **Mean** & **SD** & **1.96 \(\ast\) SE** & **VFC\({}_{\text{icont}}\)** & **VFC\({}_{\text{icont}}\)** & **VFC\({}_{\text{icont}}\)** & **P32\({}_{\text{UNIT VOLUME}}\)** & **P32\({}_{\text{UNIT VOLUME}}\)** & **P32\({}_{\text{UNIT VOLUME}}\)** \\  & & & & **(LD)** & **(Mean)** & **(UL)** & **(LL)** & **(Mean)** & **(UL)** \\ \hline S1 & 199 & 0.93 & 3.46 & 0.48 & 0.71 & 1.08 & 2.23 & 0.86 & 1.299 & 2.688 \\ S2 & 80 & 2.71 & 8.72 & 1.91 & 0.22 & 0.37 & 1.25 & 0.26 & 0.446 & 1.511 \\ S3 & 153 & 1.03 & 1.83 & 0.29 & 0.76 & 0.97 & 1.35 & 0.92 & 1.173 & 1.632 \\ S4 & 44 & 4.98 & 3 & 0.89 & 0.171 & 0.20 & 0.24 & 0.21 & 0.245 & 0.295 \\  & & & & **(L)** & **1.85** & **2.62** & **5.07** & **2.24** & **3.16** & **6.13** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of VFC (P32\({}_{\text{UNIT VOLUME}}\)) and P32\({}_{\text{UNIT VOLUME}}\) estimates using true spacing data provided by [29]—see Table 2 above (some data in Table 3 were provided independently by [PERSON]). concepts proposed by [PERSON] et al. (2004 [17]) and [PERSON] and is adapted here for the ith set of non-persistent joints, such that \"equivalent\" [true] spacings (_Sti_) [can be] \"adjusted\" based on the mean true spacing of the ith joint set (Sti) and the mean persistence of fractures, as per Equation (8)  
\[St_{i}^{\prime}=\frac{St_{i}}{\sqrt[3]{P_{i}}} \tag{8}\]  
where _Pi_ is the persistence factor that can be considered the ratio of the mean fracture area (assumed to be a disc) for the ith discontinuity set (from Table 2) divided by the maximum disc area for a 1 m\({}^{3}\) unit volume sphere, as used in the unit volume approach in the V-GSI chart as follows:  
\[P_{i}=\frac{Field\ Assessment\ Mean\ Disc\ area\ \left(\pi r^{2}\right)_{i}}{Maximum\ Disc\ Area\ \left(\pi r^{2}\right)\ for\ unit\ sphere\ volume\ where\ r=0.62m} \tag{9}\]  
As a point of clarification, the denominator in Equation (9) does not imply that every field-measured disc has a constant area--rather on average most joints of consequence will intersect the unit sphere and each disc per set that fully intersects the theoretical unit volume will have an area close to 1.2 to 1.3 m\({}^{2}\). This is not to be confused with the actual size of the joints. As can be seen in Table 4 below, the estimated ratio Pi varies between 0.5 and 0.13. When the ratio Pi from Equation (9) is used in Equation (8), the Pi-adjusted true spacings per set give an adjusted Sti\({}^{\prime}\) for the _ith_ discontinuity set for \"equivalent persistence planes\". For example, the mean true spacing for Set 1 increases from 0.93 m to 1.2 m. Using these adjusted spacings and estimating the VFC contribution per set for an adjusted VFC\({}^{\prime}\) of 1.8 fractures/m\({}^{3}\) and then using Equation (3) provides an adjusted (reduced) P32 estimate of 2.2\({}^{-1}\) compared to 3.16\({}^{-1}\) (Table 3).