# An Application of Inverse Problem and Universal Solutions for Pumping Wells in Unconfined Aquifers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{r}and K

_{z}can be quantified.

## 2. Materials and Methods

#### 2.1. Mathematical Model

_{r}the radial velocity, v

_{z}the vertical velocity, K

_{r}the radial hydraulic conductivity and K

_{z}the vertical hydraulic conductivity.

_{r}= K

_{z}= K, where K is the isotropic hydraulic conductivity), reduces to

_{w}the well radius, H the aquifer thickness, h

_{w}water height in the well and h

_{s}the seepage surface. The well wall is the vertical contact between the aquifer and the well, and its length is the aquifer thickness. This length includes the water height in the well and the seepage surface, and it has to be permeable.

#### 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

_{w}and R, can be related in a discriminated monomial

_{w}, can also be obtained:

_{1}(or an equivalent one) would be obtained again.

_{1}means that radial flow is more important than vertical one, and the opposite happens if the value is low. Referring to π

_{2}, the lower the value of this monomial is, the more relevant the horizontal flow. Finally, π

_{3}shows the importance of vertical flow, as if considering that the well only works pumping water out of the aquifer, low values of this group are equivalent to a higher importance of vertical flow.

_{α}is a length along the circumferential perimeter while L

_{r}and L

_{z}are lengths in the radial and vertical directions, respectively. In short, we can define the dimensional flow as

_{s}− h

_{w}in Figure 1), which is the wet length over the water height in the well through which water also flows. This is the length of the well wall in which the hydraulic potential is just higher or equal than its vertical position. In this study, well entry loss is not included, and the seepage surface appears because both vertical and radial flow are considered. The variable h

_{s}is the whole contact surface between the well and the aquifer, and in this surface the potential is higher than the position, as authors such as [17] have considered in their work. The deduction of the dimensionless group which includes the seepage surface is simpler than that of the water flow, as it must include somehow a ratio of this variable and one of the boundary hydraulic potentials. After trying different options to set the dimensionless form of h

_{s}, the one which leads to more understandable results is

#### 2.2.3. Verification of the Dimensionless Groups

_{1}, π

_{2}and π

_{3}) are verified, i.e., it is demonstrated that they work as independent groups to rule the solution of the problem. In addition, it is also confirmed that they are the least number of independent groups that influence the solution. To do this, we choose a significate number of typical different scenarios for which the groups take the same value, for different values of the physical and geometrical parameters involved. After simulation, for all these scenarios, the dimensionless value of the unknown must remain the same.

_{1}= 1, π

_{2}= 0.1 and π

_{3}= 0.5. For Cases 3 and 4, π

_{1}is changed from 1 to 0.2, keeping π

_{2}and π

_{3}the same values as in Cases 1 and 2. In this way, the interaction between the global vertical and radial hydraulic conductivities is modified. Cases 5 and 6 have the same values of π

_{1}and π

_{3}as cases 1 and 2, while π

_{2}takes a value of 0.25. With this change, the well radius has been decreased. Finally, cases 7 and 8 have the same values of π

_{1}and π

_{2}as in cases 1 and 2 while π

_{3}is 0.7, so scenarios with a higher value of water height in the well are represented.

_{hs}in Cases 7 and 8, but this small difference (below 1%) is attributable to the numerical simulation employing the Network Method, since dividing the scenarios in cell or volume elements may lead to small deviations in the results. Furthermore, it is also interesting to remark the little difference between the values of π

_{Q}for π

_{1}= 1, π

_{2}= 0.1 and π

_{3}= 0.5 and for π

_{1}= 0.2, π

_{2}= 0.1 and π

_{3}= 0.5, which are 0.295 and 0.297, respectively. The effect of monomial π

_{2}is the following: as the well radius is increased, then higher values of dimensional and dimensionless pumping flow are obtained. In the case of π

_{3}, its effect is the opposite. As h

_{w}increases (which means that π

_{3}also does), the pumping rate decreases because the water potential difference value, H − h

_{w}, is lower. This fact is easily observed studying Dupuit’s expression (Equation (5), giving h the value of h

_{w}). In the study of the dimensionless pumping rate, Equation (15) is considered. If the dimensional flow value is kept constant, increasing h

_{w}leads to a decrease in the reference flow (lower part of the ratio in Equation (15)) and, therefore, an increase in the dimensionless flow.

## 3. Results and Discussion

#### 3.1. Universal Curves

_{3}, 0.001, is employed to model the problem of a well where the hydraulic potential is zero.

_{Q}in relation to π

_{1}, for the same values of π

_{2}and π

_{3}, were low enough to consider that the group involving hydraulic conductivities does not affect the dimensionless water flow (the highest difference is around 2.3%). This fact leads to a simplification of the type-curve (Figure 2) as it only includes one curve for each π

_{2}.

_{1}monomial, it is also relevant to know how the geometry (reflected in π

_{2}and π

_{3}) influences the solution. Referring to the effect on water flow, it can differ whether the real or the dimensionless water flow is being considered. Studying the impact of π

_{2}, it is observed that, as its value increases (which means that the well radius is wider), the pumping capacity is also increased. This influences both real and dimensionless water flow.

_{3}affects differently the behavior of real and dimensionless water flow. If studying the real variable, the lower the value of π

_{3}, the higher the hydraulic potential difference is, which generates more water flow. Nevertheless, when comparing this with the reference flow, Q

_{ref}, as it also grows with the decrease of π

_{3}, the difference between the two water flows also increases. This is translated into a reduction of π

_{Q}.

_{hs}, and, unlike the dimensionless groundwater flow, this dimensionless variable is affected by the permeability monomial, π

_{1}.

_{1}is the one expected: the higher the horizontal flow is, the higher the value of the seepage surface, because it is easier for the flow to go on in the horizontal direction. Monomial π

_{2}has the opposite effect: higher values of π

_{hs}are obtained as π

_{2}decreases. This means that the seepage surface increases as the well radius presents a lower value because the flow has a longer horizontal length to keep flowing in that direction. Finally, the effect of π

_{3}is the same as that of π

_{2}:π

_{hs}increases its value as π

_{3}decreases, which means that the seepage surface is higher as the water height in the well has a lower value. It occurs because the potential difference increases and there is a longer length to develop the seepage surface.

#### 3.2. Illustrative Example—Influence of Measure Deviations in the Calculation of Conductivity Values

- H = 10 m
- h
_{w}= 5 m - R = 10 m
- r
_{w}= 1 m - Q = 0.002308 m
^{3}/s - h
_{s}= 8.471 m

- K
_{r}= 0.0000225 m/s (named K_{r,real}, as it is the real value of the parameter in the example). - K
_{z}= 0.00001 m/s (named K_{z,real}, as it is the real value of the parameter in the example).

_{2}, π

_{3}and π

_{hs}.

_{2}and π

_{3}in Figure 2, we obtain the value of π

_{Q}, which is 0.295. As the pumping rate, Q, is part of the problem data, the horizontal conductivity, K

_{r}, is calculated from Equation (15).

_{r,cal}. Once it is known, the deviation must be calculated.

_{r}allows calculating K

_{z}from the type-curve in Figure 3, from which, for the values of π

_{hs}, π

_{2}and π

_{3}, π

_{1}is 1.5. Therefore, according to Equation (8), K

_{z}is calculated as

_{z,cal,}and its deviation is calculated according to Equation (23).

## 4. Conclusions

_{1}(conductivity ratio affected by an aspect ratio).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c_{1} | constant in Dupuit deduction (m) |

H | aquifer thickness (m) |

h | hydraulic potential (m) |

h′ | dimensionless hydraulic potential |

h_{s,real} | real value of seepage surface (m) |

h_{s} | seepage surface (m) |

h_{w} | water height in the well (m) |

K | hydraulic conductivity (m/s) |

K_{r,real} | real value of radial hydraulic conductivity (m/s) |

K_{r} | radial hydraulic conductivity (m/s) |

K_{z,real} | real value of vertical hydraulic conductivity (m/s) |

K_{z} | vertical hydraulic conductivity (m/s) |

L_{r} | radial length quantity (m) |

L_{wc} | water column quantity (m) |

L_{z} | vertical length quantity (m) |

L_{α} | angular length quantity (m) |

Q | pumping flow (m^{3}/s) |

Q_{real} | real value of pumping flow (m^{3}/s) |

Q_{ref} | reference pumping flow (m^{3}/s) |

R | aquifer radius (m) |

r | radial coordinate (m) |

r′ | dimensionless radial coordinate |

r_{w} | well radius (m) |

S_{ref} | reference surface (m^{2}) |

T | time quantity (s) |

v_{r} | radial velocity (m/s) |

v_{ref} | reference velocity (m/s) |

v_{z} | vertical velocity (m/s) |

z | vertical coordinate (m) |

z′ | dimensionless vertical coordinate |

Δh | potential variation (m) |

ξ | statistic error (%) |

π_{1}, π_{2}, π_{3}, π_{Q}, π_{hs} | discriminated dimensionless monomials |

∂ | partial derivative |

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Case | K_{r} (m/s) | K_{z} (m/s) | R (m) | r_{w} (m) | H (m) | h_{w} (m) | Q (m^{3}/s) | h_{s} (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 | π_{1} | π_{2} | π_{3} | π_{Q} | π_{hs} |
---|---|---|---|---|---|

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 (%) | K_{r} × 10^{−5} (m/s) | Max Deviations K_{r} (%) | K_{z} × 10^{−5} (m/s) | Max Deviations K_{z} (%) |
---|---|---|---|---|

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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Martínez-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