## 2 Theory of Slug Test on Determining Permeability Coefficient of Anisotropic Fractured Rock Mass  
On the basis of the slug test theory of the horizontal layered aquifer with different relative damping coefficients, which is proposed by [PERSON] [16], a new mathematical model of the vibration equation of a well different dip angle fractured aquifer system is established by combining the groundwater continuous equation, the law of mass conservation and momentum conservation, as shown in Figure 1. In Figure 1a, the center of the top hole of the fractured aquifer is the origin \(o\) of the cylindrical coordinate, the radial direction is \(r\), and the vertical upward direction is the positive direction of the \(z\) axis. In the Cartesian coordinate system, A and B are two points on the elliptic fracture plane, the projection of the maximum oblique line on the fracture surface passing through the point \(o\) on the horizontal plane is the positive \(x\) axis of the horizontal plane xoy, \(Ox\) is the positive axial of \(x\) axis, and \(Oy\) is the positive axial of y axis. Where \(\varphi\) is the dip angle of the fracture surface (\(\varphi\
eq 0\)), \(\theta\) is the center angle, which takes \(ox\) as the initial angle, \(A^{\prime}\) is the projection of A on the horizontal plane xoy and \(C^{\prime}\) is the projection of C on the horizontal plane xoy. The well structure schematic is shown in Figure 1b. If the cartesian coordinates of A are \((x,y,z)\) and the cylindrical coordinates of A are \((r,\theta,z)\), then \(x=rcos\theta\), \(y=rsin\theta\), \(z=xtan\varphi=rcos\theta tan\varphi\). A new theoretical model of the slug test under different dip angle fractures is proposed. A new calculation method of determining the different angle fracture permeability coefficient, equivalent permeability coefficient and rock mass permeability coefficient tensor is proposed. Figure 1: Slug test schematic on the dip fracture surface. (**a**) Coordinate system schematic on the dip fracture surface, (**b**) well structure schematic. The groundwater continuous equation for the cylindrical coordinate is as follows:  
\[\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial h}{\partial r} \bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}h}{\partial\theta^{2}}+\frac{ \partial^{2}h}{\partial z^{2}}=\frac{\mu_{s}}{k}\frac{\partial h}{\partial t} \tag{1}\]  
Among them, \(r\) is the radial distance of the cylindrical coordinate (\(L\)); \(z\) is the vertical axis of the cylindrical coordinate (\(L\)); \(\theta\) is the center angle; \(h\) is the water head of the aquifer (\(L\)); \(\mu_{s}\) is storage rate of the fracture (\(L^{-1}\)); \(k\) is the permeability coefficient of the fracture (\(LT^{-1}\)); \(t\) is time (\(T\)). The corresponding initial conditions are as follows: when \(t=0\), \(h=h_{0}\); for the boundary condition, when \(r\rightarrow\infty\), \(h\to h_{0}\). By converting the Cartesian coordinates for dip fracture surfaces, \(z=r\cos\theta\tan\varphi\), is related to \(r\), \(\theta\) and then Equation (1) is converted to  
\[\Big{(}1+\tan^{2}\varphi\Big{)}\frac{\partial^{2}h}{\partial z^{2}}=\frac{\mu _{s}}{k}\frac{\partial h}{\partial t} \tag{2}\]  
According to the law of mass conservation, the well flow caused by the slug test is equal to the radial flow from the surrounding fractured aquifer. The fractured aquifer surrounding boundary of the wall of the well approximates the circle from the ellipse. The mass conservation equation is as follows:  
\[\pi r_{c}^{2}\frac{dw}{dt}=k\cdot\frac{b}{\cos\varphi}\cdot 2\pi r_{s}\cdot \frac{\overline{\partial h}}{\partial r}|_{r=r_{s},\text{is relative with }\theta} \tag{3}\]  
Among them, \(\frac{\overline{\partial h}}{\partial r}|_{r=r_{s},\text{is relative with }\theta}\) is the average hydraulic gradient of the fractured aquifer around the wall of the well; \(w\) is the variation in water level in the well (\(L\)); \(r_{s}\) is the screen radius or well radius of the test section (\(L\)); \(r_{c}\) is the casing radius (\(L\)); b is the fracture width (\(L\)).