Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells
Abstract
:1. Introduction
2. Simplified Calculation of Multi-Well System for Elliptical Pit
2.1. Basic Assumptions of Simplified Calculations
- Owing to the function of the cut-off wall, the water inflow of drainage relief depends on the average water level difference between the inside and outside of the pit post-relief and is independent of the specific arrangement of the relief well. Hence, the seepage field is divided into two sections—inside and outside the pit by the cut-off wall. This paper simplifies the calculation method by treating these two parts separately, first calculating the internal seepage field. Subsequently, the pit is equated into a circle based on the area [18], and the equivalent internal seepage field is solved in series with the external seepage field.
- The seepage field on the inner side of the cut-off wall is pressurized.
- The primary aquifer layers outside the cut-off wall are categorized into two scenarios: the pressure-bearing mode and the submerged mode. In cases where there are multiple layers of highly permeable strata with negligible variations in permeability coefficients, equivalent permeability coefficients can be employed.
- The material of each soil layer and cut-off wall is homogeneous and isotropic [32].
- The inner edge line of the cut-off wall is an equal head line.
2.2. Seepage Field Inside a Circular Pit
2.3. Ellipse to Circle Mapping
2.4. Inside-Outside Seepage Field Solved in Series
2.5. Processes to Simplify Calculations
- 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 , the height of the edge of the area of influence , the thickness of the gravelly sand layer , the thickness of the powdery clay layer , the depth of the relief wells and the thickness of the water-reducing layer , the thickness of the cut-off wall in the pit , the permeability coefficients of the highly permeable layer and the water-reducing layer , the permeability coefficient of the weakly permeable layer , the permeability coefficient of the highly permeable layer , the permeability coefficient of the cut-off wall , the thickness of the weakly permeable layer at the bottom of the cut-off wall , and the shape of the pit, etc. Finally, the location of the relief wells, the diameter of the wells , and the height of the wellhead are adjusted.
- 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.
- 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 .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 , where A is an matrix representing wellbore information, and B is a matrix representing the difference in water head. Solve this system of equations to obtain the 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.
- 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 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.
3. Algorithm Validation
3.1. Arithmetic Parameters for Simplified Calculations
3.2. Flow Verification
3.3. Hydraulic Pressure Distribution Verification
3.4. Error Analysis of a Pit Simplified to an Ellipse
4. IDPR Anti-Floating Design and Case Application
4.1. Control Parameters for Anti-Floating Design
4.2. Verification of Actual Engineering Case
5. Conclusions
- For the elliptical pit, it is transformed into a unit circle by conformal mapping, and after obtaining the equivalent coordinates after the transformation, the seepage field of the pit is solved by combining with the resistance coefficient method, so as to obtain the simplified calculation method for the multi-well system of the elliptical pit.
- By comparing the results of the simplified calculation method with the seepage field distribution obtained from finite element calculations, as well as the total flow rate and head, it is observed that the seepage field distribution inside the pit is nearly identical. This demonstrates the high accuracy of the simplified calculation method for the multi-well system of elliptical pits under IDPR conditions. Furthermore, the simplified calculation method was successfully applied to real-world engineering cases. By comparing the results of the simplified calculation with both the finite element calculations and the actual measured data, the reasonableness and practicality of the simplified calculation method were further validated.
- This simplified calculation method is suitable for sites with relatively uniform soil thickness and permeability, such as sites with artificial hydrophobic layers, but its accuracy needs to be further improved for sites with large variations in soil distribution.
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
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R0 (m) | n | hw (m) | H0 (m) | D (m) | K (cm/s) | K0 (cm/s) | K1 (cm/s) | Kw (cm/s) |
---|---|---|---|---|---|---|---|---|
400.00 | 20 | 0.00 | 6.00 | 2.50 |
a (m) | b (m) | T0 (m) | T1 (m) | Distribution | rw (m) | T (m) | d (m) |
---|---|---|---|---|---|---|---|
90 | 70 | 2.00 | 5.00 | A | 0.50 | 0.60 | 0.80 |
100 | 60 | 3.00 | 4.00 | B | 1.00 | 1.00 | 1.60 |
120 | 50 | 4.00 | 3.00 | C | - | - | - |
a (m) | b (m) | T0 (m) | T1 (m) | z0 (m) | Distribution | rw (m) | T (m) | T2 (m) | d (m) |
---|---|---|---|---|---|---|---|---|---|
90 | 70 | 2.50 | 7.50 | 7.50 | A | 0.50 | 0.50 | 3.50 | 0.80 |
100 | 60 | 3.00 | 7.00 | 7.00 | B | 1.00 | 0.80 | 3.20 | 1.60 |
120 | 50 | 3.50 | 6.50 | 6.50 | C | - | - | - | - |
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Luo, G.; Yang, F.; Li, H.; Pan, H.; Cao, H. Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells. Appl. Sci. 2023, 13, 12647. https://doi.org/10.3390/app132312647
Luo G, Yang F, Li H, Pan H, Cao H. Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells. Applied Sciences. 2023; 13(23):12647. https://doi.org/10.3390/app132312647
Chicago/Turabian StyleLuo, Guanyong, Fei Yang, Haoxi Li, Hong Pan, and Hong Cao. 2023. "Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells" Applied Sciences 13, no. 23: 12647. https://doi.org/10.3390/app132312647
APA StyleLuo, G., Yang, F., Li, H., Pan, H., & Cao, H. (2023). Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells. Applied Sciences, 13(23), 12647. https://doi.org/10.3390/app132312647