Solution to the Unsteady Seepage Model of Phreatic Water with Linear Variation in the Channel Water Level and Its Application
Abstract
:1. Introduction
2. Basic Model and Its Linearization
- (1)
- A homogeneous isotropic submerged aquifer with a horizontal lower confining bed and infinite spatial extension;
- (2)
- A channel with a completely cut aquifer whose water level rapidly rises to a certain height and then remains constant for a long time with a water level rise of ΔH;
- (3)
- An initial water level of the phreatic water h(x,0) is horizontal;
- (4)
- Phreatic flow can be regarded as a one-dimensional flow;
- (5)
- The intensity of vertical water exchange ε is equally distributed throughout the region.
3. Theoretical General Solution
4. Solutions for Several Types of Linear Functions
4.1. Constant Value Function
4.2. Step Function
4.3. Lagrange Linear Interpolation Function
5. Verification and Application of the Solution
5.1. Numerical Verification of Analytical Solutions
5.2. Calculation Method for the Model Parameters
5.2.1. Curve-Fitting Method of φ(x,t) − t with ε ≠ 0 and ΔH = 0
5.2.2. Curve-Fitting Method for h(x,t) − t with ε = 0 and ΔH ≠ 0
5.2.3. Inflection Point Method of φ(x,t) − t with ε = 0 and ΔH ≠ 0
5.2.4. Ratio Method of φ(x,t) with ε ≠ 0 and ΔH ≠ 0
5.3. Case Study
6. Conclusions
- (1)
- For the unsteady flow model of phreatic water near rivers and canals under the influence of vertical water exchange, the analytic equations of the boundary water level f(t) and the vertical water exchange intensity ε were separately solved based on the properties of the Laplace transform, and the two analytic equations were then superimposed to obtain the solution.
- (2)
- When the boundary function f(t) is a simple function, such as a constant value function or a linear function, the image function after the positive transformation of the constant is 1/s, and the solution after the inverse transformation is the product of the constant term and the error function. For the linear function term with time t, the transformed image function is the 1/s2 term. Regarding the “integral property” [39] of the transformation, when solving the superposition function, attention should be paid to the application of the Heaviside function.
- (3)
- By using the process of variation in the phreatic water level h(x,t) over time and the process of variation in the speed of change in the phreatic water level 𝜕h(x,t)/𝜕t over time, the inflection point method and the curve-fitting method can be used to calculate the model parameter a under different conditions.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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t/h | 3 | 6 | 9 | 12 | 18 | 24 | 36 | 48 | 60 | 72 |
---|---|---|---|---|---|---|---|---|---|---|
h(x,t) (m) | 27.61 | 27.65 | 27.69 | 27.73 | 27.80 | 27.87 | 28.01 | 28.12 | 28.23 | 28.33 |
φ(x,t) (m/h) | 0.360 | 0.360 | 0.336 | 0.304 | 0.280 | 0.291 | 0.275 | 0.220 | 0.220 | 0.200 |
t/h | 3 | 5 | 7 | 9 | 12 | 15 | 16 | 17 | 18 | 21 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|
h(x,t) (m) | 25.80 | 25.81 | 25.83 | 25.88 | 25.97 | 26.08 | 26.12 | 26.16 | 26.20 | 26.31 | 26.41 |
φ(x,t) (mm/h) | 0.34 | 1.27 | 2.29 | 3.15 | 3.64 | 3.76 | 3.77 | 3.76 | 3.68 | 3.52 |
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Wu, D.; Tao, Y.; Yang, J.; Kang, B. Solution to the Unsteady Seepage Model of Phreatic Water with Linear Variation in the Channel Water Level and Its Application. Water 2023, 15, 2834. https://doi.org/10.3390/w15152834
Wu D, Tao Y, Yang J, Kang B. Solution to the Unsteady Seepage Model of Phreatic Water with Linear Variation in the Channel Water Level and Its Application. Water. 2023; 15(15):2834. https://doi.org/10.3390/w15152834
Chicago/Turabian StyleWu, Dan, Yuezan Tao, Jie Yang, and Bo Kang. 2023. "Solution to the Unsteady Seepage Model of Phreatic Water with Linear Variation in the Channel Water Level and Its Application" Water 15, no. 15: 2834. https://doi.org/10.3390/w15152834