Reservoir Slope Stability Analysis under Dynamic Fluctuating Water Level Using Improved Radial Movement Optimisation (IRMO) Algorithm
Abstract
:1. Introduction
2. Rigorous Limit Equilibrium Analysis Considering Seepage Force
2.1. Seepage Analysis
2.1.1. Relationship of Phreatic Surface and Fluctuating External Water Level
- The reservoir slope is considered a vertical slope. The reservoir slope within the declining amplitude is much smaller than the ground, so it is considered a vertical slope for simplification.
- The change in phreatic flow parallel to the slope surface is caused by the reservoir water-level fluctuation. (The change in phreatic flow perpendicular to the slope surface caused by the rainfall infiltration is not considered herein.)
- The external reservoir water level is decreasing or increasing at a constant speed of V0.
- The aquifer within the slope is homogeneous and isotropic with an infinite lateral extension.
2.1.2. Simplification of Seepage Field
2.1.3. Seepage Force Analysis
2.2. Rigorous Janbu Method Combined with Seepage Force
- Assuming ΔTi = 0 at the beginning, there is only FS unknown in Equation (18).
- Using the Newton iteration method, the initial FS1 can be obtained by Equation (18).
- Substitute the initial FS1 and assumed ΔTi into Equation (17) to obtain the value of ΔEi and Ei.
- According to Equation (14), the ΔTi and Ti can be updated iteratively with known ΔEi and Ei.
- Repeat steps (1)–(2) above, the FSk can also be updated with the Newton iteration method.
- If FSk − FSk−1 < δ, δ is the requirement of calculation accuracy set in advance, the FSk will be output as the final result. Otherwise, repeat steps (3)–(5) above until the calculation accuracy is satisfied.
2.3. Non-Circular Critical Failure Surface Model
3. Improved Radial Movement Optimisation (IRMO) Algorithm
3.1. Introduction of IRMO Algorithm
3.2. Computation Process of the IRMO Algorithm
3.2.1. Initialisation
3.2.2. Evolution: New Particles Updated
3.2.3. Evolution: Radial Movement of the Central Particle
3.3. Implementation of IRMO Algorithm for Reservoir Slope Stability Analysis
4. Case Studies
4.1. Comparative Analysis of the Impact of Seepage Force
4.2. Effect of Varying Fluctuation Direction and Rate
4.2.1. Effect of Reservoir Water Level Rising with Different Rates
4.2.2. Effect of Reservoir Water Level Dropping
4.3. Effect of Permeability Coefficient
5. Conclusions
- The studies herein demonstrate that both the increase and decrease in the reservoir water level will significantly impact the stability of the reservoir slope. With the rising reservoir water level, the minimum FS of silt slope and clay slope will increase, and the possible slide body will be also be larger. It can be argued that the rising reservoir water level significantly increases the hydrostatic pressure, which is beneficial for stability, compared to the unfavourable seepage force which increases less relatively due to the hysteresis of the seepage field. When the reservoir water level drops, the minimum Fs will decrease significantly. But, with the lower water level, the minimum Fs will decrease slowly and then increase slightly after reaching the most dangerous water level. Due to the relative hysteresis of the seepage changes, the phreatic surface will continue to decrease after reaching the lowest reservoir water level, which leads to a slight increase in the minimum FS of the reservoir slope.
- In the process of the reservoir water level rising and falling, the fluctuation rate of water level and soil permeability coefficient could influence the reservoir slope stability. With a higher change rate of water level rising or falling and a smaller permeability coefficient, the hysteresis effect of seepage will be more serious, and the corresponding minimum Fs will increase or decrease more rapidly.
- However, the proposed method in this study did not consider the influence of the matric suction of the saturated–unsaturated area. Compared to the results that consider the matric suction effect in the same slope model, the minimum FS results are higher during the water level rising and lower during the water level falling. This illustrates that the matric suction has a great influence on the stability of the slope. Therefore, it is necessary to figure out the change and effect of matric suction inside the slope in further studies for stability analysis.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Reference | LEMs | Seepage Analysis Methods | CFSs | Optimisation Techniques |
---|---|---|---|---|
Zolfaghari et al. [25] | Morgenstern–Price | Pore–water pressure from distribution steady seepage | Non-circular | GA |
Himanshu et al. [31] | Bishop | Pore–water pressure from distribution steady seepage | Circular | PSO |
Biniyaz et al. [35] | Simplified Bishop | Transient pore–water pressure distribution from saturated–unsaturated flow | Circular | FEM (FEniCS) |
Liu et al. [26] | Spencer | One-dimensional groundwater flow in the unconfined aquifer | Non-circular (spline curve) | GA |
Jin et al. [36] | Rigorous Janbu | Infiltration analysis from two-dimensional steady seepage | Non-circular | IRMO |
Cases | Reference | Optimisation Methods | LEMs | Minimum FS | Error |
---|---|---|---|---|---|
Homogeneous slope case | Pham and Fredlund [41] | SLOPE/W | Morgenstern–Price | 1.168 | −13.8% |
SLOPE/W | Simplified Bishop | 1.167 | −13.7% | ||
Qin [42] | Fortran | Fellenius | 1.070 | −12.9% | |
Fortran | Simplified Bishop | 1.185 | −33.5% | ||
Fortran | Rigorous Janbu | 1.178 | −32.3% | ||
Jin et al. [36] | IRMO | Rigorous Janbu | 1.007 | Average in −21.2% | |
Inhomogeneous slope case with two-layered soil | Pham and Fredlund [41] | SLOPE/W | Morgenstern–Price | 1.485 | −8.9% |
SLOPE/W | Simplified Bishop | 1.483 | −8.8% | ||
Qin [42] | Fortran | Fellenius | 1.376 1.489 | −1.7% | |
Fortran | Bishop | −9.1% | |||
Jin et al. [36] | IRMO | Rigorous Janbu | 1.353 | Average in −7.1% | |
Inhomogeneous slope case with a weak inter-layer | Pham and Fredlund [41] | SLOPE/W | Morgenstern–Price | 1.140 | −7.7% |
SLOPE/W | Simplified Bishop | 1.125 | −10.9% | ||
Chen et al. [43] | PSO & FEM | - | 1.053 | −0.1% | |
Jin et al. [36] | IRMO | Rigorous Janbu | 1.052 | Average in −6.2% | |
Inhomogeneous slope case in four soil layered | Zolfaghari et al. [25] | GA | Morgenstern–Price | 1.360 | −10.6% |
Cheng et al. [44] | SA | Spencer | 1.284 | −5.3% | |
GA | Spencer | 1.232 | −1.3% | ||
PSO | Spencer | 1.210 | +0.5% | ||
Tabu search | Spencer | 1.343 | −10.4% | ||
ACO | Spencer | 1.449 | −16.1% | ||
Kahatadeniya et al. [28] | ACO | Morgenstern–Price | 1.377 | −11.7% | |
Khajehzadeh et al. [29] | PSO | Morgenstern–Price | 1.203 | +1.1% | |
MPSO | Morgenstern–Price | 1.171 | +3.8% | ||
Singh et al. [45] | BBO | Bishop | 1.348 | −9.8% | |
BBO | Fellenius | 1.226 | −0.8% | ||
BBO | Janbu | 2.103 | −42.2% | ||
BBO | Janbu corrected | 2.104 | −42.2% | ||
Jin et al. [36] | IRMO | Rigorous Janbu | 1.216 | Average in −11.2% |
Optimisation Algorithms | Minimum Fs | Standard Deviation | Average CPU Time (ms) | |||
---|---|---|---|---|---|---|
Maximum | Minimum | Average | ||||
Homogeneous slope case [36,41] | IRMO | 1.0092 | 1.0042 | 1.0066 | 0.0013 | 755.85 |
RMO | 1.0302 | 1.0116 | 1.0173 | 0.0048 | 713.90 | |
DE | 1.0715 | 1.0342 | 1.0571 | 0.0092 | 431.30 | |
PSO | 1.0890 | 1.0293 | 1.0594 | 0.0196 | 2104.05 | |
Inhomogeneous slope case with a weak inter-layer [36,43] | IRMO | 1.0775 | 1.0412 | 1.0638 | 0.0093 | 665.50 |
RMO | 1.1497 | 1.0341 | 1.0932 | 0.0249 | 643.20 | |
DE | 1.1965 | 1.1048 | 1.1395 | 0.0216 | 395.90 | |
PSO | 1.2978 | 1.1138 | 1.1962 | 0.0505 | 2087.60 |
IRMO Algorithm: |
---|
Input:
|
Initialisation:
|
Evolution:
|
Output:
|
Population Size (M) | Number of Variables (N) | Generations (G) | Coefficients | ||
---|---|---|---|---|---|
C1 | C2 | ||||
IRMO | 250 | 50 | 250 | 0.4 | 0.5 |
Reference | Computation Techniques | Seepage Analysis | Slope Stability Analysis | Critical Failure Surface | Minimum FS |
---|---|---|---|---|---|
Pham and Fredlund [41] | FlexPDS | Distribution of pore–water pressures | DNYPROG (μ = 0.48) | Non-circular | 1.187 |
FlexPDS | DNYPROG (μ = 0.33) | Non-circular | 1.041 | ||
SIGMA/W and SEEP/W | Enhanced (μ = 0.48) | Circular | 1.171 | ||
SIGMA/W and SEEP/W | Enhanced (μ = 0.33) | Circular | 1.132 | ||
SLOPE/W | Morgenstern–Price | Circular | 1.168 | ||
SLOPE/W | Simplified Bishop | Circular | 1.167 | ||
Qin [42] | Fortran | Considering seepage force | Fellenius | Circular | 1.070 |
Simplified Bishop | Circular | 1.185 | |||
Rigorous Janbu | Circular | 1.178 | |||
This study | IRMO | Without considering seepage force | Rigorous Janbu | Non-circular | 1.181 |
Considering seepage force | Rigorous Janbu | Non-circular | 1.007 |
Soil | γ (kN/m3) | c (kPa) | φ (°) | Permeability Coefficient k (m/s) |
---|---|---|---|---|
clay | 20.2 | 10 | 20 | 5.4 × 10−7 |
silty | 20.2 | 5 | 30 | 5.4 × 10−6 |
Soil | γ (kN/m3) | c (kPa) | φ (°) |
---|---|---|---|
Upper silt layer | 20 | 10 | 25 |
Limestone bedrock | 25 | 67 | 42 |
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Jin, L.; Luo, C.; Wei, J.; Liu, P. Reservoir Slope Stability Analysis under Dynamic Fluctuating Water Level Using Improved Radial Movement Optimisation (IRMO) Algorithm. Mathematics 2024, 12, 2055. https://doi.org/10.3390/math12132055
Jin L, Luo C, Wei J, Liu P. Reservoir Slope Stability Analysis under Dynamic Fluctuating Water Level Using Improved Radial Movement Optimisation (IRMO) Algorithm. Mathematics. 2024; 12(13):2055. https://doi.org/10.3390/math12132055
Chicago/Turabian StyleJin, Liangxing, Chunwa Luo, Junjie Wei, and Pingting Liu. 2024. "Reservoir Slope Stability Analysis under Dynamic Fluctuating Water Level Using Improved Radial Movement Optimisation (IRMO) Algorithm" Mathematics 12, no. 13: 2055. https://doi.org/10.3390/math12132055
APA StyleJin, L., Luo, C., Wei, J., & Liu, P. (2024). Reservoir Slope Stability Analysis under Dynamic Fluctuating Water Level Using Improved Radial Movement Optimisation (IRMO) Algorithm. Mathematics, 12(13), 2055. https://doi.org/10.3390/math12132055