# Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells
## 2 Simplified Calculation of Multi-Well System for Elliptical Pit
### Processes to Simplify Calculations  
Based on the previous discussion, the simplified calculation can be summarized into the following steps.  
1. Determine the parameters: first, determine whether the seepage calculation mode of the outside of the cut-off wall is pressure-bearing or submerged; second, determine the seepage parameters of the site, such as the area of influence of the seepage field \(R_{0}\), the height of the edge of the area of influence \(H_{0}\), the thickness of the gravelly sand layer \(T_{0}\), the thickness of the powdery clay layer \(T_{1}\), the depth of the relief wells and the thickness of the water-reducing layer \(T\), the thickness of the cut-off wall in the pit \(d\), the permeability coefficients of the highly permeable layer and the water-reducing layer \(K\), the permeability coefficient of the weakly permeable layer \(K_{1}\), the permeability coefficient of the highly permeable layer \(K_{2}\), the permeability coefficient of the cut-off wall \(K_{w}\), the thickness of the weakly permeable layer at the bottom of the cut-off wall \(D\), and the shape of the pit, etc. Finally, the location of the relief wells, the diameter of the wells \(r_{w}\), and the height of the wellhead \(h_{w}\) are adjusted.
2. Simplify an arbitrarily shaped pit to an ellipse according to its area and map it to a unit circle. First, according to the shape of the target pit, which is equivalent to the ellipse that it most closely resembles, the target pit is transformed to a unit circle by conformal mapping.
3. Internal seepage field calculation. After obtaining the equivalent coordinates of the pit through conformal transformation, assuming the water head height on the inner side of the cut-off wall and the wellhead elevation, let \(\varphi=H_{R}-h_{w}\).Together with the transformed equivalent coordinates, substitute them into Equation (6) for calculating the potential function of the internal seepage field in the circular pit with a curtain. This will yield a system of equations \(Ax=B\), where \(A\) is an \(n\times n\) matrix representing wellbore information, and \(B\) is a \(1\times n\) matrix representing the difference in water head. Solve this system of equations to obtain the \(1\times n\) matrix \(x\), which represents the discharge flow rate of each well under the assumed water head conditions, i.e., the ratio of discharge flow rates between wells. Finally, the distribution of the seepage field inside the target pit's curtain under the assumed water head can be obtained by the superposition principle.
4. Serial solution of seepage fields on the inner and outer sides. The elliptical pit is transformed into an equivalent circular pit based on the equal area, obtaining the equivalent radius \(R_{d}\) of the pit. By using the resistance coefficient method, the assumed water head in step 2 is related to the actual water head at the site. The resistance coefficient formulas for internal seepage Equation (19), external seepage Equation (14), and seepage and bypass resistance at the cut-off wall Equation (18) are connected in series using the resistance coefficient method formula Equation (20). Finally, the distribution of actual water head heights at the site, as well as the total flow rate and the actual discharge of each well, are solved.  
In summary, the application process of this simplified calculation in the design of IDPR anti-floating is shown in Figure 5. See Appendix A for details of calculation codes.  
Figure 5: Design flowchart for IDPR anti-floating system.