## 3 Theoretical Volume Considerations in Assessing P30 and P32 and Rock Strength and Deformability
### Unit Volume  
In the quantified V-GSI chart (Figure 6 and discussed in more detail Section 4), a unit volume is assumed to be a sphere with a radius of 0.62 m. VFC (equivalent to P30) and P32 included on the vertical axis of the V-GSI chart are estimated via a new method that only uses 'true' spacing data to estimate VFC. It is assumed in the V-GSI approach that joints sizes or persistence are comparable to a REV dimension and therefore are larger than the diameter of unit volume sphere (1 m\({}^{3}\)) and fully intersect the unit sphere.  
P30\({}_{\rm{UNIT\ VOLUME}}\) and P32\({}_{\rm{UNIT\ VOLUME}}\) is the nomenclature adopted here (Table 1) to describe the results obtained using V-GSI method of quantification. The equations and method of quantification of P30\({}_{\rm{UNIT\ VOLUME}}\) and P32\({}_{\rm{UNIT\ VOLUME}}\) are fully described in [3] and are summarized in Section 4 below.  
In the V-GSI system, the ratio of P32\({}_{\rm{unit\ volume}}\)/P30\({}_{\rm{unit\ volume}}\) (which results in m\({}^{2}\)/number fractures) gives the surface area of a single fracture intersecting a unit sphere volume of rock on the assumption that the single fracture has a persistence or size greater than the diameter of a unit volume sphere or cube of rock mass. It is also implicit in this that as the value of P32\({}_{\rm{unit\ volume}}\) and P30\({}_{\rm{unit\ volume}}\) increases, so does the summed fracture area with an increasing degree of blockiness. In the V-GSI chart (Figure 6), the ratio simply implies that one fracture with greater persistence than the diameter of a unit volume that intersects a unit volume of rock mass will have an area close to \(\pi r^{2}\) (approximately 1.3 m\({}^{2}\)) for an assumed sphere with a radius of 0.62 m. This does not mean that the fracture area is only 1.3 m\({}^{2}\), it is only the area of intersection of a fracture in the unit sphere and, as described above, the surface area can increase up to the REV or up to the size of the structural domain.  
As an additional point of clarification, the parameter VFC (Figure 6) in the V-GSI chart is used interchangeably with P30\({}_{\rm{unit\ volume}}\) in this paper and both have the same definition and method of quantification (summarized in Table 1).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Nomenclature** & **Definition** \\ \hline P30\({}_{\rm{REV}}\) & Total fracture count (No.) per total volume (Number/m\({}^{3}\)) of a field \\  & established rock mass REV. Structural Domain \textgreater{} REV \textgreater{} Unit Volume \\ P30\({}_{\rm{DFN}}\) & Fracture count (No.) per total model block volume (m\({}^{3}\)) in a computer \\  & derived DFN model \\ P30\({}_{\rm{UNIT\ VOLUME}}\) or VFC\({}_{\rm{UNIT\ VOLUME}}\) & Fracture count (No.) per unit volume (1 m\({}^{3}\) sphere) using the new method of estimation included on the vertical axis of the V-GSI chart [3] \\ P32\({}_{\rm{REV}}\) & Total area of fractures (m\({}^{2}\)) per total volume (m\({}^{3}\)) of a field established REV. \\ P32\({}_{\rm{DFN}}\) & Structural Domain \textgreater{} REV \textgreater{} Unit Volume \\ P32\({}_{\rm{DFN}}\) & Area of fractures (m\({}^{2}\)) per total model block volume (m\({}^{3}\)) in a computer-derived DFN model \\ P32\({}_{\rm{UNIT\ VOLUME}}\) & Area of fractures (m\({}^{2}\)) per unit volume (1 m\({}^{3}\) sphere) using the new method of estimation included on the vertical axis of the V-GSI chart [3] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nomenclature and units used to define P30 (VFC) and P32 using field REV approach, DFN approach, or unit volume approach.  
The research presented below shows that the same or similar results for P30 and P32 are obtained using an REV-defined volume, DFN block volume, or a V-GSI unit volume sphere approach, with some limitations. This provides validation that the vertical axis of the V-GSI chart can be quantified using real field measurements.