Triangular Fuzzy Finite Element Solution for Drought Flow of Horizontal Unconfined Aquifers
Abstract
1. Introduction
2. Materials and Methods
2.1. Crisp Model
Numerical Method
2.2. Fuzzy Model Definitions
2.2.1. Fuzzy Theory Definitions
- (a)
- is increasing, is decreasing as functions of α, and ;
- (b)
- is decreasing, is increasing as functions of α, and .
- is (i)-gH-differentiable at if
- (i)
- ;
- is (ii)-gH-differentiable at if
- (i)
- .
- is [(i)-p]-differentiable with respect to x at ( if
- is [(ii)-p]-differentiable with respect to x at ( if
2.2.2. Possibility Theory Definitions
2.3. Fuzzy Finite Elements and Corresponding Systems
2.3.1. Fuzzy Model
- Boundary conditions
- Initial condition
- System (1, 1):
- System (1, 2):
- System (1, 3):
- System (1, 4):
- System (2, 1):
- System (2, 2):
- System (2, 3):
- System (2, 4):
- Case a
- Case b
2.3.2. The Proposed Fuzzy Triangular Finite Element Solution
2.3.3. Outflow Volumes
2.4. Proposed Method for Solving the Crisp and Fuzzy Cases of Non-Dimensional Variables
Algorithem 1 A pseudocode solving process | |
Step 1: | τ = 0 The interval [s0, sN] is divided into N equal parts: |
Step 2: | τ = τ+Δτ, |
Step 3: | Initial values H(sr,0) sr, r = 1,2,…N + 1 |
Step 4: | Boundary values H(0,τ) = h0, |
Step 5: | Find coefficients |
Step 6: | Solve the tridiagonal system [36,37] |
Step 7: | Put HNEW into HINITIAL |
Step 8: | Compute outflow volume V(HNEW) |
Step 9: | Print τ, V(τ), HΝΕW values |
Step 10: | If go to 2 |
end |
3. Results
3.1. Boussinesq Analytical Solution
- (a)
- Neglecting the effect of capillary rise above the water table;
- (b)
- Accepting the Dupuit–Forcheimer approximation, i.e., the hydraulic head is independent of depth and therefore the streamlines are assumed to be approximately parallel to the bed;
- (c)
- His solution is valid when t is large, that is, when the water table at x = L is below the aquifer depth h0 (See Figure 1).
3.2. Initial Water Table
3.3. Storativity and Discharge
3.3.1. Dimensional Storativity
3.3.2. Non-Dimensional Storativity
3.3.3. Discharge
3.4. Fuzzy Triangular Finite Element Method Application and Comparison
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
FDM | Finite Difference Method |
FEM | Finite Element Method |
FVM | Finite Volume Method |
gH | Generalized Hukuhara |
XFEM | Extended Finite Element Method |
Appendix A
- Gelerkin’s Method
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ORTH Fuzzy FEM | TRIAN Fuzzy FEM | Boussinesq | |||||||
---|---|---|---|---|---|---|---|---|---|
a/a | 0.48959 | 0.20000 | 0.12002 | 0.48959 | 0.20000 | 0.12002 | 0.48959 | 0.20000 | 0.12002 |
0.00 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |
0.05 | 0.13991 | 0.20232 | 0.23077 | 0.13991 | 0.20240 | 0.23077 | 0.14041 | 0.20311 | 0.23169 |
0.10 | 0.19702 | 0.28490 | 0.32496 | 0.19702 | 0.28501 | 0.32496 | 0.19729 | 0.28541 | 0.32555 |
0.15 | 0.23987 | 0.34686 | 0.39564 | 0.23987 | 0.34700 | 0.39564 | 0.23990 | 0.34703 | 0.39586 |
0.20 | 0.27500 | 0.39766 | 0.45358 | 0.27500 | 0.39781 | 0.45357 | 0.27482 | 0.39758 | 0.45349 |
0.25 | 0.30493 | 0.44094 | 0.50294 | 0.30493 | 0.44111 | 0.50294 | 0.30460 | 0.44068 | 0.50263 |
0.30 | 0.33095 | 0.47856 | 0.54585 | 0.33095 | 0.47875 | 0.54585 | 0.33052 | 0.47815 | 0.54541 |
0.35 | 0.35382 | 0.51164 | 0.58358 | 0.35382 | 0.51184 | 0.58358 | 0.35335 | 0.51113 | 0.58308 |
0.40 | 0.37405 | 0.54089 | 0.61695 | 0.37405 | 0.54110 | 0.61695 | 0.37358 | 0.54039 | 0.61645 |
0.45 | 0.39197 | 0.56681 | 0.64651 | 0.39197 | 0.56703 | 0.64651 | 0.39152 | 0.56639 | 0.64606 |
0.50 | 0.40783 | 0.58974 | 0.67266 | 0.40784 | 0.58997 | 0.67266 | 0.40742 | 0.58935 | 0.67230 |
0.55 | 0.42182 | 0.60996 | 0.69572 | 0.42182 | 0.61020 | 0.69572 | 0.42146 | 0.60968 | 0.69547 |
0.60 | 0.43406 | 0.62766 | 0.71591 | 0.43406 | 0.62790 | 0.71591 | 0.43376 | 0.62746 | 0.71576 |
0.65 | 0.44467 | 0.64300 | 0.73341 | 0.44467 | 0.64325 | 0.73341 | 0.44442 | 0.64288 | 0.73336 |
0.70 | 0.45373 | 0.65610 | 0.74835 | 0.45373 | 0.65635 | 0.74835 | 0.45353 | 0.65609 | 0.74838 |
0.75 | 0.46130 | 0.66705 | 0.76083 | 0.46130 | 0.66731 | 0.76083 | 0.46113 | 0.66708 | 0.76093 |
0.80 | 0.46744 | 0.67593 | 0.77096 | 0.46744 | 0.67619 | 0.77096 | 0.46729 | 0.67600 | 0.77109 |
0.85 | 0.47219 | 0.68279 | 0.77878 | 0.47219 | 0.68305 | 0.77878 | 0.47203 | 0.68285 | 0.77892 |
0.90 | 0.47556 | 0.68767 | 0.78435 | 0.47556 | 0.68793 | 0.78435 | 0.47539 | 0.68769 | 0.78446 |
0.95 | 0.47759 | 0.69060 | 0.78769 | 0.47759 | 0.69087 | 0.78769 | 0.47740 | 0.69059 | 0.78777 |
1.00 | 0.47772 | 0.69087 | 0.78804 | 0.47772 | 0.69113 | 0.78804 | 0.47806 | 0.69156 | 0.78887 |
ORTH Fuzzy FEM | TRIAN Fuzzy FEM | Boussinesq | |||||||
---|---|---|---|---|---|---|---|---|---|
a/a | 0.97917 | 0.40000 | 0.24004 | 0.97917 | 0.40000 | 0.24004 | 0.97917 | 0.40000 | 0.24004 |
0.00 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |
0.05 | 0.09194 | 0.15466 | 0.19058 | 0.09194 | 0.15466 | 0.19058 | 0.09225 | 0.15523 | 0.19130 |
0.10 | 0.12947 | 0.21779 | 0.26836 | 0.12947 | 0.21779 | 0.26836 | 0.12963 | 0.21813 | 0.26880 |
0.15 | 0.15763 | 0.26515 | 0.32673 | 0.15763 | 0.26515 | 0.32673 | 0.15763 | 0.26522 | 0.32685 |
0.20 | 0.18071 | 0.30398 | 0.37457 | 0.18071 | 0.30399 | 0.37457 | 0.18057 | 0.30386 | 0.37444 |
0.25 | 0.20038 | 0.33707 | 0.41534 | 0.20038 | 0.33707 | 0.41534 | 0.20014 | 0.33680 | 0.41501 |
0.30 | 0.21748 | 0.36583 | 0.45078 | 0.21748 | 0.36583 | 0.45078 | 0.21717 | 0.36543 | 0.45033 |
0.35 | 0.23251 | 0.39112 | 0.48194 | 0.23251 | 0.39112 | 0.48194 | 0.23217 | 0.39064 | 0.48143 |
0.40 | 0.24581 | 0.41348 | 0.50949 | 0.24581 | 0.41348 | 0.50949 | 0.24546 | 0.41300 | 0.50898 |
0.45 | 0.25759 | 0.43329 | 0.53390 | 0.25759 | 0.43329 | 0.53390 | 0.25725 | 0.43288 | 0.53343 |
0.50 | 0.26801 | 0.45082 | 0.55551 | 0.26801 | 0.45082 | 0.55551 | 0.26770 | 0.45042 | 0.55510 |
0.55 | 0.27720 | 0.46628 | 0.57455 | 0.27720 | 0.46628 | 0.57455 | 0.27692 | 0.46596 | 0.57423 |
0.60 | 0.28524 | 0.47981 | 0.59122 | 0.28525 | 0.47981 | 0.59122 | 0.28501 | 0.47955 | 0.59099 |
0.65 | 0.29222 | 0.49154 | 0.60567 | 0.29222 | 0.49154 | 0.60567 | 0.29201 | 0.49133 | 0.60551 |
0.70 | 0.29817 | 0.50155 | 0.61801 | 0.29817 | 0.50155 | 0.61801 | 0.29799 | 0.50143 | 0.61792 |
0.75 | 0.30315 | 0.50992 | 0.62833 | 0.30315 | 0.50992 | 0.62833 | 0.30299 | 0.50983 | 0.62828 |
0.80 | 0.30719 | 0.51671 | 0.63669 | 0.30719 | 0.51671 | 0.63669 | 0.30704 | 0.51665 | 0.63667 |
0.85 | 0.31031 | 0.52196 | 0.64315 | 0.31031 | 0.52196 | 0.64315 | 0.31015 | 0.52188 | 0.64313 |
0.90 | 0.31253 | 0.52569 | 0.64775 | 0.31253 | 0.52569 | 0.64775 | 0.31236 | 0.52558 | 0.64771 |
0.95 | 0.31386 | 0.52793 | 0.65051 | 0.31386 | 0.52793 | 0.65051 | 0.31368 | 0.52780 | 0.65044 |
1.00 | 0.31392 | 0.52809 | 0.65075 | 0.31392 | 0.52809 | 0.65075 | 0.31411 | 0.52854 | 0.65134 |
TRIAN Fuzzy FEM vs. Boussinesq | |||
---|---|---|---|
0.0000011292 | 0.0000011775 | 0.0000011763 | |
0.0000012431 | 0.0000012289 | 0.0000011372 | |
0.0000013667 | 0.0000011672 | 0.0000011166 |
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Tzimopoulos, C.; Samarinas, N.; Papadopoulos, K.; Evangelides, C. Triangular Fuzzy Finite Element Solution for Drought Flow of Horizontal Unconfined Aquifers. Hydrology 2025, 12, 128. https://doi.org/10.3390/hydrology12060128
Tzimopoulos C, Samarinas N, Papadopoulos K, Evangelides C. Triangular Fuzzy Finite Element Solution for Drought Flow of Horizontal Unconfined Aquifers. Hydrology. 2025; 12(6):128. https://doi.org/10.3390/hydrology12060128
Chicago/Turabian StyleTzimopoulos, Christos, Nikiforos Samarinas, Kyriakos Papadopoulos, and Christos Evangelides. 2025. "Triangular Fuzzy Finite Element Solution for Drought Flow of Horizontal Unconfined Aquifers" Hydrology 12, no. 6: 128. https://doi.org/10.3390/hydrology12060128
APA StyleTzimopoulos, C., Samarinas, N., Papadopoulos, K., & Evangelides, C. (2025). Triangular Fuzzy Finite Element Solution for Drought Flow of Horizontal Unconfined Aquifers. Hydrology, 12(6), 128. https://doi.org/10.3390/hydrology12060128