## 2 Theory of Slug Test on Determining Permeability Coefficient of Anisotropic Fractured Rock Mass
### The Calculation of the Fractured Rock Mass Permeability Coefficient Tensor
 \tag{33}\]  
The equivalent permeability coefficient of the second fracture group in the whole borehole test section is as follows:  
\[k_{e2}=\frac{b_{2}}{L\cdot\cos\varphi_{2}}\cdot k_{e2}+\frac{b_{3}}{L\cdot \cos\varphi_{3}}\cdot k_{e3}+\frac{b_{4}}{L\cdot\cos\varphi_{4}}\cdot k_{e4} \tag{34}\]  
If Equations (24) and (25) are placed into Equation (34), then  
\[ \begin{array}{c}k_{e2}=\frac{b_{2}}{L\cdot\cos\varphi_{2}}\cdot\frac{k_{3} \cdot L_{2}\cdot\cos\varphi_{2}}{b_{3}}+\frac{b_{3}}{L\cdot\cos\varphi_{3}} \cdot\frac{k_{3}\cdot L_{3}\cdot\cos\varphi_{2}}{b_{3}}+\frac{b_{4}}{L\cdot \cos\varphi_{4}}\cdot\frac{k_{4}\cdot L_{4}\cdot\cos\varphi_{4}}{b_{4}}\\ =\frac{L_{2}}{L}\cdot k_{3}+\frac{L_{2}}{L}\cdot k_{3}+\frac{L_{4}}{L}\cdot k _{4}\end{array}  \tag{35}\]  
\[=\frac{0.41}{9.62}\times 1.0\times 10^{-4}+\frac{0.4}{9.62}\times 1.0\times 10^{- 4}+\frac{0.5}{9.62}\times 1.45\times 10^{-4}\]  
The equivalent permeability coefficient of the third fracture group in the whole borehole test section is as follows:  
\[k_{e3}=\frac{b_{5}}{L\cdot\cos\varphi_{5}}\cdot k_{e5}+\frac{b_{6}}{L\cdot \cos\varphi_{6}}\cdot k_{e6}+\frac{b_{9}}{L\cdot\cos\varphi_{9}}\cdot k_{e9} \tag{36}\]  
If Equations (26), (27) and (30) are placed into Equation (36), then  
\[ \begin{array}{c}k_{e3}=\frac{b_{5}}{L\cdot\cos\varphi_{5}}\cdot\frac{k_{5} \cdot L_{5}\cdot\cos\varphi_{5}}{b_{5}}+\frac{b_{6}}{L\cdot\cos\varphi_{6}} \cdot\frac{k_{e}\cdot L_{4}\cdot\cos\varphi_{6}}{b_{6}}+\frac{b_{9}}{L\cdot \cos\varphi_{9}}\cdot\frac{k_{9}\cdot L_{4}\cdot\cos\varphi_{9}}{b_{9}}\\ =\frac{L_{4}}{L}\cdot k_{5}+\frac{L_{4}}{L}\cdot k_{6}+\frac{L_{4}}{L}\cdot k _{9}\end{array}  \tag{37}\]  
\[=\frac{0.63}{9.62}\times 6.69\times 10^{-5}+\frac{0.73}{9.62}\times 1.27\times 1 0^{-5}+\frac{1.07}{9.62}\times 2.28\times 10^{-5}\]  
\[=7.88\times 10^{-6}\text{cm/s}\]The equivalent permeability coefficient of the fourth fracture group in the whole borehole section is as follows:  
\[k_{e\alpha 4}=\frac{b_{7}}{L\cdot\cos\varphi_{7}}\cdot k_{e7}+\frac{b_{8}}{L \cdot\cos\varphi_{8}}\cdot k_{e8}+\frac{b_{10}}{L\cdot\cos\varphi_{10}}\cdot k_{e10} \tag{38}\]  
If Equations (29) and (31) are placed into Equation (38), then  
\[k_{e\alpha 4}=\frac{b_{7}}{L\cdot\cos\varphi_{7}}\cdot\frac{k_{7} \cdot L\cdot\cos\varphi_{7}}{2\cdot b_{7}}+\frac{b_{8}}{L\cdot\cos\varphi_{8}} \cdot\frac{k_{7}\cdot L\cdot\cos\varphi_{7}}{2\cdot b_{7}}+\frac{b_{10}}{L\cdot \cos\varphi_{10}}\cdot\frac{k_{10}\cdot L_{0}\cdot\cos\varphi_{10}}{b_{10}}\] \[=\frac{L_{7}}{L\cdot k_{78}}+\frac{L_{7}}{2\cdot L}\cdot k_{78}+ \frac{L_{6}}{L}\cdot k_{10}=\frac{1.6}{2\times 9.62}\times 2.08\times 10^{-5}+ \frac{1.6}{2\times 9.62}\times 6.0\times 10^{-6} \tag{39}\] \[=6.14\times 10^{-6}\text{cm/s}\]  
According to the average dip direction \(\beta_{i}\) and dip angle \(\theta_{i}\) of the four fracture groups in Table 6 and the Equation (16), the fractured rock mass permeability coefficient tensor in the whole borehole test section is calculated, and the following is obtained:  
\[K=\sum_{i=1}^{4}K_{ei} \begin{bmatrix}1-\cos^{2}\beta_{i}\sin^{2} \theta_{i}&-\sin\beta_{i}\sin^{2}\theta_{i}\cos\beta_{i}&-\cos\beta_{i}\sin \theta_{i}\cos\theta_{i}\\ -\sin\beta_{i}\cos\beta_{i}\sin^{2}\theta_{i}&1-\sin^{2}\beta_{i}\sin^{2} \theta_{i}&-\sin\beta_{i}\sin\theta_{i}\cos\theta_{i}\\ -\cos\beta_{i}\sin\theta_{i}\cos\theta_{i}&-\sin\beta_{i}\sin\theta_{i}\cos \theta_{i}&1-\cos^{2}\theta_{i}\end{bmatrix} \] \[=3.53\times 10^{-8}\times \begin{bmatrix}0.82372&0.101775&0.367216\\ 0.101775&0.94124&-0.21201\\ 0.367216&-0.21201&0.23504\end{bmatrix} +1.61\times 10^{-5}\times \begin{bmatrix}0.786667 &-0.3285&-0.24476\\ -0.3285&0.494148&-0.3769\\ -0.24476&-0.3769&0.719186\end{bmatrix} \] \[+7.88\times 10^{-6}\times \begin{bmatrix}0.798042&0.349801&0.197003\\ 0.349801&0.394127&-0.34122\\ 0.197003&-0.34122&0.807831\end{bmatrix} +6.14\times 10^{-6}\times \begin{bmatrix}0.385403 &0.119466&0.4718\\ 0.119466&0.97678&-0.09171\\ 0.4718&-0.09171&0.637819\end{bmatrix} \] \[=