## 3 Methods
### Karst Pipes Model and Determination of Its Hydraulic Conductivity  
Groundwater flow in karst pipes is generally non-Darcy flow owing to the fast water velocity. The hydraulic head loss, \(h_{\rm I}\), can be given by Darcy-Weisbach equation:  
\[h_{1}=\lambda\frac{L}{d}\frac{u^{2}}{2g} \tag{2}\]  
where \(\lambda\) is head loss coefficient along the conduit, \(L\) is the length of the tube [L], \(d\) is the conduit diameter [L], \(u\) is the average velocity [L/T], and \(g\) is the gravitational acceleration [L/T\({}^{2}\)]. If the porosity (\(n\)) of the conduit is assumed to be 1, the seepage velocity, \(V\), is equal to the average velocity. In addition, the hydraulic gradient (\(J\)) can be expressed by head loss: (according to Abbreviations). \[J\;=\!\frac{h_{1}}{L} \tag{3}\]  
Substituting Equation (3) and the seepage velocity (\(V\)) into Equation (2), it can be rewritten as:  
\[JL\;=\;\lambda\frac{L}{d}\frac{VV}{2g} \tag{4}\]Therefore, Equation (4) can be written in the form of Darcy's Law:  
\[V= \frac{2 gd}{\lambda V}J \tag{5}\]  
A converted hydraulic conductivity, \(K_{c}\), is defined as:  
\[K_{c}=\frac{2 gd}{\lambda V} \tag{6}\]  
In order to calculate \(K_{c}\), the key is to calculate the corresponding \(\lambda\) according to the groundwater flow pattern. The \(\lambda\) values depend on Reynolds number (Re) and relative roughness (\(\Delta\)) of the tube walls. \(\lambda\) values can be given as [41]:  
For laminar flow (Re \(<\) 2320),  
\[\lambda= \frac{64}{\text{Re}} \tag{7}\]  
For turbulent flow (Re \(\geq\) 2320),  
\[\lambda= \left[1.74-21\text{g}\left(\frac{\Delta}{d}+\frac{21.25}{\text{Re}^{0.9 }}\right)\right]^{-2} \tag{8}\]  
The seepage velocity can be calculated according to the flux and diameter of the karst conduit. Then Re and \(\lambda\) values are obtained by the seepage velocity and Equations (7) and (8). Finally, the converted hydraulic conductivity from Equation (6) can be determined. The \(K_{c}\) is the main hydraulic conductivity along the direction of the karst pipe. In fact, there is a certain angle between the direction of the pipe and the coordinate axis. Therefore, the \(K_{c}\) value must be converted to tensor form in the global coordinate system, that is, geodetic coordinate system, of the computational domain. A local coordinate system, ox\({}^{\prime}\)y\({}^{\prime}\)z\({}^{\prime}\), is established to parallel the positive direction of the ox\({}^{\prime}\)-axis with the axial direction of karst pipe. The permeabilities of karst conduit are mainly along the ox\({}^{\prime}\)-axis owing to its strong anisotropy. So, under the local coordinate system, ox\({}^{\prime}\)y\({}^{\prime}\)z\({}^{\prime}\), the hydraulic conductivity tensors of karst pipe, \([K]_{L}^{e}\), can be expressed as [41]:  
\[[K]_{L}^{e}=\left[ \begin{array}{ccc}K_{c}&0&0\\ 0&0&0\\ 0&0&0\end{array} \right]=diag(K_{c},0,0) \tag{9}\]  
Rotating the direction of the local coordinate system with the same as that of the global coordinate system, based on the matrix correlation theory and conception, the hydraulic conductivity tensors of the global coordinate system, \([K]_{G}^{e}\), can be calculated with that of the local coordinate system:  
\[[K]_{G}^{e}=[P][K]_{L}^{e}[P]^{T} \tag{10}\]  
where \([P]\) is the orthogonal matrix, and it can be calculated using the local coordinates of karst pipe. Assuming  
\[\left\{\begin{array}{c}\alpha_{1}= \left\{ \begin{array}{ccc}\cos\alpha,&\cos\beta,&\cos\gamma\\ \alpha_{2}=\left\{\begin{array}{ccc}\cos\alpha^{\prime},&\cos\beta^{\prime },&\cos\gamma^{\prime}\\ \alpha_{3}=\left\{\begin{array}{ccc}\cos\alpha^{\prime\prime},&\cos\beta^ {\prime\prime},&\cos\gamma\end{array}\right\}\end{array}\right\}\end{array} \right. \tag{11}\]And assuming that the starting and ending coordinates of the axis of karst pipe are \(M_{1}\) (\(x_{1},y_{1}\), \(z_{1}\)), \(M_{2}\) (\(x_{2},y_{2}\), \(z_{2}\)), respectively. The directional cosine of vector (\(\alpha_{1}\)) or ox'-axis can be given as:  
\[\left\{