An Application of Inverse Problem and Universal Solutions for Pumping Wells in Unconfined Aquifers
Abstract
:1. Introduction
2. Materials and Methods
2.1. Mathematical Model
2.2. Discriminated Nondimensionalization Technique for Flow towards Pumping Wells
2.2.1. Discriminated Nondimensionalization Procedure
- New groups that remove classical ones appear. This occurs as a consequence of including spatial and general discrimination.
- Correct references for the problem variables are chosen, since ratios of lengths must include parameters in the same spatial direction, and aspect ratios can only appear modified by the ratio of other parameters.
- Universal solutions can be presented as universal type-curves in which the relationship between unknowns and data is depicted.
- Inverse problem methodologies can be developed using the universal solutions.
2.2.2. Discriminated Dimensionless Groups Achievement
2.2.3. Verification of the Dimensionless Groups
3. Results and Discussion
3.1. Universal Curves
3.2. Illustrative Example—Influence of Measure Deviations in the Calculation of Conductivity Values
- H = 10 m
- hw = 5 m
- R = 10 m
- rw = 1 m
- Q = 0.002308 m3/s
- hs = 8.471 m
- Kr = 0.0000225 m/s (named Kr,real, as it is the real value of the parameter in the example).
- Kz = 0.00001 m/s (named Kz,real, as it is the real value of the parameter in the example).
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
c1 | constant in Dupuit deduction (m) |
H | aquifer thickness (m) |
h | hydraulic potential (m) |
h′ | dimensionless hydraulic potential |
hs,real | real value of seepage surface (m) |
hs | seepage surface (m) |
hw | water height in the well (m) |
K | hydraulic conductivity (m/s) |
Kr,real | real value of radial hydraulic conductivity (m/s) |
Kr | radial hydraulic conductivity (m/s) |
Kz,real | real value of vertical hydraulic conductivity (m/s) |
Kz | vertical hydraulic conductivity (m/s) |
Lr | radial length quantity (m) |
Lwc | water column quantity (m) |
Lz | vertical length quantity (m) |
Lα | angular length quantity (m) |
Q | pumping flow (m3/s) |
Qreal | real value of pumping flow (m3/s) |
Qref | reference pumping flow (m3/s) |
R | aquifer radius (m) |
r | radial coordinate (m) |
r′ | dimensionless radial coordinate |
rw | well radius (m) |
Sref | reference surface (m2) |
T | time quantity (s) |
vr | radial velocity (m/s) |
vref | reference velocity (m/s) |
vz | vertical velocity (m/s) |
z | vertical coordinate (m) |
z′ | dimensionless vertical coordinate |
Δh | potential variation (m) |
ξ | statistic error (%) |
π1, π2, π3, πQ, πhs | discriminated dimensionless monomials |
∂ | partial derivative |
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Case | Kr (m/s) | Kz (m/s) | R (m) | rw (m) | H (m) | hw (m) | Q (m3/s) | hs (m) |
---|---|---|---|---|---|---|---|---|
1 | 0.0001 | 0.0001 | 10 | 1 | 10 | 5 | 0.0103 | 7.736 |
2 | 0.000225 | 0.0001 | 15 | 1.5 | 10 | 5 | 0.0231 | 7.736 |
3 | 0.0001 | 0.0001 | 50 | 5 | 10 | 5 | 0.0104 | 5.343 |
4 | 0.0001 | 0.000025 | 50 | 5 | 5 | 2.5 | 0.0260 | 2.672 |
5 | 0.0001 | 0.0001 | 10 | 2.5 | 10 | 5 | 0.0171 | 7.349 |
6 | 0.0005 | 0.0001 | 22.36 | 5.59 | 10 | 5 | 0.0854 | 7.349 |
7 | 0.0001 | 0.0001 | 10 | 1 | 10 | 7 | 0.0070 | 8.240 |
8 | 0.0003 | 0.0001 | 10 | 1 | 5.77 | 4.07 | 0.0070 | 4.753 |
Case | π1 | π2 | π3 | πQ | πhs |
---|---|---|---|---|---|
1 | 1 | 0.1 | 0.5 | 0.295 | 0.274 |
2 | 1 | 0.1 | 0.5 | 0.295 | 0.274 |
3 | 0.2 | 0.1 | 0.5 | 0.297 | 0.034 |
4 | 0.2 | 0.1 | 0.5 | 0.297 | 0.034 |
5 | 1 | 0.25 | 0.5 | 0.408 | 0.235 |
6 | 1 | 0.25 | 0.5 | 0.408 | 0.235 |
7 | 1 | 0.1 | 0.7 | 0.334 | 0.124 |
8 | 1 | 0.1 | 0.7 | 0.334 | 0.123 |
Deviations ξ (%) | Kr × 10−5 (m/s) | Max Deviations Kr (%) | Kz × 10−5 (m/s) | Max Deviations Kz (%) |
---|---|---|---|---|
0 | 2.24 | 0.44 | 0.996 | 0.40 |
0.5 | 2.23/2.25 | 0.89 | 1.07/0.99 | 7.02 |
1 | 2.22/2.26 | 1.33 | 1.12/0.89 | 11.92 |
1.5 | 2.21/2.28 | 1.78 | 1.19/0.84 | 18.69 |
2 | 2.20/2.29 | 2.22 | 1.24/0.78 | 24.36 |
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Martínez-Moreno, E.; Alhama, F.; Alhama, I.; García-Ros, G. An Application of Inverse Problem and Universal Solutions for Pumping Wells in Unconfined Aquifers. Water 2023, 15, 2524. https://doi.org/10.3390/w15142524
Martínez-Moreno E, Alhama F, Alhama I, García-Ros G. An Application of Inverse Problem and Universal Solutions for Pumping Wells in Unconfined Aquifers. Water. 2023; 15(14):2524. https://doi.org/10.3390/w15142524
Chicago/Turabian StyleMartínez-Moreno, Encarnación, Francisco Alhama, Iván Alhama, and Gonzalo García-Ros. 2023. "An Application of Inverse Problem and Universal Solutions for Pumping Wells in Unconfined Aquifers" Water 15, no. 14: 2524. https://doi.org/10.3390/w15142524