# Topological Representative Element Volume of Fractured Rock Mass
## 4 Discussion
### The Permeability Representative Element Volume  
Methods for estimating the hydraulic conductivity of a two-dimensional fracture network have been widely discussed [27; 28; 29]. [PERSON] [27] provides the calculation formula for two-dimensional steady seepage based on the water balance principle in a unit area; it is assumed that the fracture is straight and smooth and the flow in it is laminar: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Average Fracture**} & \multicolumn{5}{c}{**Fracture Length/Persistence (m)**} \\ \cline{2-6}  & **Very Low** & **Low** & **Moderate** & **High** & **Very High** \\  & **L \textless{} 1** & **1 \textless{} L \textless{} 3** & **3 \textless{} L \textless{} 10** & **10 \textless{} L \textless{} 20** & **L \textgreater{} 20** \\ \hline Very closely & \multirow{2}{*}{1.0 \(\times\) 1.0} & \multirow{2}{*}{3.0 \(\times\) 3.0} & \multirow{2}{*}{4.0 \(\times\) 4.0} & \multirow{2}{*}{\textgreater{}8.0 \(\times\) 8.0} & \multirow{2}{*}{3.0 \(\times\) 3.0} \\ (0.02 \(<\) C \(<\) 0.06) & & & & & \\ \hline Closely & \multirow{2}{*}{2.0 \(\times\) 2.0} & \multirow{2}{*}{4.0 \(\times\) 4.0} & \multirow{2}{*}{9.0 \(\times\) 9.0} & \multirow{2}{*}{8.0 \(\times\) 8.0} & \multirow{2}{*}{13.0 \(\times\) 13.0} \\ (0.06 \(<\) C \(<\) 0.2) & & & & & \\ \hline Moderately & \multirow{2}{*}{6.0 \(\times\) 6.0} & \multirow{2}{*}{10.0 \(\times\) 10.0} & \multirow{2}{*}{8.0 \(\times\) 8.0} & \multirow{2}{*}{12.0 \(\times\) 12.0} & \multirow{2}{*}{12.0 \(\times\) 12.0} \\ (0.2 \(<\) C \(<\) 0.6) & & & & & \\ \hline Moderate to blocky & \multirow{2}{*}{-} & \multirow{2}{*}{18.0 \(\times\) 18.0} & \multirow{2}{*}{26.0 \(\times\) 26.0} & \multirow{2}{*}{32.0 \(\times\) 32.0} & \multirow{2}{*}{50.0 \(\times\) 50.0} \\ (0.6 \(<\) C \(<\) 2) & & & & & \\ \hline Blocky to massive & \multirow{2}{*}{-} & \multirow{2}{*}{58.0 \(\times\) 58.0} & \multirow{2}{*}{34.0 \(\times\) 34.0} & \multirow{2}{*}{64.0 \(\times\) 64.0} \\ (2 \(<\) C \(<\) 6) & & & & & \\ \hline Massive (C \(>\) 6) & - & - & - & - & 54.0 \(\times\) 54.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: TREV of the fractured rock mass. \[\left\{ \begin{array}{l}\mathbf{A_{1}TA_{1}^{T}H_{1}+A_{1}TA_{2}^{T}H_{2}+A_{1}TA_{3 }^{T}H_{3}+Q_{1}}=0\\ \mathbf{A_{2}TA_{1}^{T}H_{1}+A_{2}TA_{2}^{T}H_{2}+A_{2}TA_{3}^{T}H_{3}+Q_{2}}=0 \\ \mathbf{A_{3}TA_{1}^{T}H_{1}+A_{3}TA_{2}^{T}H_{2}+A_{3}TA_{3}^{T}H_{3}+Q_{3}}=0 \end{array} \right., \tag{15}\]  
where, \(\mathbf{A_{1}}\), \(\mathbf{A_{2}}\), \(\mathbf{A_{3}}\) is the cohesion matrix which describes the connected relationship between nodes, branches, and boundaries and the numbers in the matrices are 0, 1, or \(-1\). \(\mathbf{T}\) is a diagonal matrix, and the numbers in the matrix are \(\left(\rho gb^{3}h_{\mathrm{j}}\right)/\left(12\mu l_{\mathrm{j}}\right)\); \(\rho\) is the density of water, \(g\) is the gravitational acceleration, \(b\) is fracture width, \(h_{\mathrm{j}}\) is the water head difference between the two ends of branch j, \(\mu\) is the dynamic viscosity, \(l_{\mathrm{j}}\) is the length of a branch j, \(\mathbf{H_{1}}\) is the hydraulic head of the internal node, \(\mathbf{H_{2}}\) is the hydraulic head of the Upper and Lower boundary node, \(\mathbf{H_{3}}\) is the hydraulic head of the left and right boundary node, \(\mathbf{Q_{1}}\) is the source and sink item of the internal node, \(\mathbf{Q_{2}}\) is the upper/lower boundary flux, \(\mathbf{Q_{3}}\) is the left/right boundary flux. Equation (15) contains six unknown parameters (\(\mathbf{H_{1-3}}\), \(\mathbf{Q_{1-3}}\)), and if three of them are known, the others can be calculated. Once \(\mathbf{H_{3}}\), \(\mathbf{Q_{1}}\), and \(\mathbf{Q_{2}}\) are provided, \(\mathbf{H_{1}}\), \(\mathbf{H_{2}}\), and \(\mathbf{Q_{3}}\) can be obtained using Equation (14). Then, the equivalent hydraulic conductivity of the fractured rock mass can be obtained according to [PERSON]'s law. The permeability representative element volume (PREV) plays a vital role in studying groundwater seepage.