# Further Discussion on the Influence Radius of a Pumping Well: A Parameter with Little Scientific and Practical Significance That Can Easily Be Misleading

^{*}

## Abstract

**:**

## 1. Origin of the Issue

## 2. Birth and Application History of Influence Radius

#### 2.1. Birth: A Misunderstanding

^{3}T

^{−1}]; K is the hydraulic conductivity, [LT

^{−1}]; H

_{0}is the thickness of the aquifer, [L]; h

_{w}is the water level of the pumping well, [L]; R is the influence radius, [L]; and r

_{w}is the radius of the pumping well, [L].

_{w}and Q-R in Equation (1) is further analyzed in Figure 2. In the Q-h

_{w}relationship curve (Figure 2(1)), the lower confining bed of the aquifer is considered as the base level. When h

_{w}is zero, the drawdown in the well reaches the maximum, and the hydraulic gradient also reaches the maximum. Because the hydraulic conductivity of the aquifer remains unchanged, the flow velocity in the aquifer is also maximized according to Darcy’s law. If the water sectional area is not considered, the water yield of the pumping well will be maximized. When h

_{w}is equal to H

_{0}, there is no drawdown in the well; therefore, the hydraulic gradient cannot be formed, the flow velocity is correspondingly zero, and the water yield of the pumping well is also zero. In the Q-R relationship curve (Figure 2(2)), when the radius of the round island (R) is the same as the radius of the pumping well (r

_{w}), the pumping well is equivalent to pumping directly from the sea. In this case, the pumping well can extract an infinite amount of water without considering the limitation of pumping power. When the radius of the round island is infinite, the distance between the sea (source) and the pumping well is also infinite. In this case, even if the pumping time is sufficiently long, the seawater cannot reach the pumping well, and the amount of water from the sea in the pumping well will be zero.

_{w}is the water level of the pumping well, [L]; Q is the water yield of the pumping well, [L

^{3}T

^{−1}]; K is the hydraulic conductivity, [LT

^{−1}]; r is the distance between the pumping well and observation well, [L]; r

_{w}is the radius of the pumping well, [L]; h

_{1}and h

_{2}are the groundwater levels of the two observation wells, respectively, [L]; and r

_{1}and r

_{2}are the distances between the two observation wells and the pumping well, respectively, [L].

#### 2.2. Application: Crude Simplification

_{0}, and μ, but also time factors. This indicates that some studies have realized that the influence radius is not a fixed hydrogeological parameter, but instead changes with time. The others are empirical formulas [15], which not only contain hydrogeological parameters but also include pumping variables such as s

_{w}and Q in the calculation of the influence radius. These formulas only consider one of the time variables and pumping condition variables, and some even consider the influence radius as a given hydrogeological parameter [15].

## 3. Gap between Theory and Practice

#### 3.1. Substance of Influence Radius: Distance

_{w}and water yield Q [16]. Studies have given the empirical values of R for aquifers with different particle structures (Table 3) and have considered that greater aquifer permeability—that is, larger aquifer particles—results in a larger R value. When pumping water in an aquifer without external recharge, the cone of depression will expand with the increase of the water yield and the advancement of time. If the empirical influence radius value is used, such as in the case of the coarse gravel aquifers mentioned in some papers, the empirical value of the influence radius will be 1500–3000 m (Table 3). In other words, regardless of how large the amount of exploitation is and how long the exploitation period is, the cone of depression of the aquifer will not continue to expand outward after extending to 1500–3000 m. Accordingly, it is assumed that, in a certain pumping well group in an aquifer, the water yield of n single pumping wells is Q

_{m1}, Q

_{m2},…, Q

_{mn}, respectively, their influence radius is R (Figure 4(1)), and a regional cone of depression will be formed. If the sum of the water yield of n single wells in the well group is provided by a single well, then a cone of depression with an influence radius R will be formed (Figure 4(2)). In this case, owing to the decrease in the number of wells, the area affected by pumping will be much smaller compared with when the well group is pumped. However, without external recharge, this phenomenon is completely impossible in an aquifer; otherwise, the aquifer will become an inexhaustible resource, which is contrary to common sense.

_{w}. For a particular aquifer with thickness H

_{0}, each drawdown of a pumping well in the aquifer has a corresponding hydraulic gradient I. Notably, there are two extremes: one is that the drawdown of the well is zero. At this point, the hydraulic gradient is zero and the groundwater does not flow. In another case, the drawdown of the well reaches H

_{0}; that is, the water level in the well drops to the lower confining bed. In this case, the hydraulic gradient reaches the maximum and the groundwater flow velocity also reaches the maximum. However, the drawdown s

_{w}is closely related to the water yield Q; that is, it increases with the water yield. Therefore, the influence radius R is not only affected by the hydrogeological conditions reflecting the natural properties of the aquifer, such as the porosity, specific yield, and hydraulic conductivity, but is also controlled by the pumping conditions. Therefore, it is inappropriate to consider the influence radius R in the Thiem model as a fixed parameter of the aquifer, which means that this parameter is constant for a specific aquifer and does not change with the changes of the water yield and drawdown [7].

#### 3.2. The Continuity Principle of Fluids Reverses the Rationality of the Influence Radius

#### 3.3. Essential Difference between Theory and Practice

_{w}is considered as a function of x, namely z = s

_{w}(x). When pumping water, the drawdown of the aquifer is larger when the distance to the pumping well is shorter. As x approaches ∞, the drawdown tends towards zero (Figure 6). Assuming that the radius of the pumping well is r

_{w}= 0, then the water yield of the pumping well is the volume of the rotating body obtained by rotating the curvilinear trapezoid bounded by a continuous curve z = s

_{w}(x), z-axis, line x = +∞, and line z = 0 once around the z-axis. For convenience of description, the object of investigation was considered to be a unit width aquifer passing through the axis; then, the value of its volume is $V={\displaystyle \int {s}_{w}}(x)dx,\left(0,\text{}\infty \right)$. When water is pumped continuously, x tends towards ∞; therefore, V also tends towards ∞. This means that the cone of depression will expand infinitely; therefore, a stable state cannot be formed. What is commonly referred to as the seemingly stable state does not mean that the groundwater level is stable, but that the change of the water level cannot be observed, which thus gives the illusion of stability. The phenomenon whereby the change of the water level cannot be observed is caused by the means of observation and other factors (external recharge). This is similar to the detection limit in analytical chemistry, which is a relative concept. With the improvement of the observation means, the observed drawdown range of the water level will increase. Therefore, the influence radius R cannot be considered as an intrinsic parameter of the aquifer beyond which there is no drawdown of the water level.

## 4. The Dilemma of Practice

#### 4.1. Misleading the Management of Sustainable Groundwater Development

_{w}is the total water yield in a certain period, [L

^{3}]; ΔV

_{r}is the increment of recharge in the same period, [L

^{3}]; ΔV

_{d}is the increment of discharge in the same period, [L

^{3}]; and ΔV

_{s}is the increment of storage in the same period, [L

^{3}]. A riverside well is a typical example. Under natural conditions, groundwater may recharge rivers. After the addition of pumping wells close to the river, the recharge relationship between them is the fact that the river recharges groundwater and the water being pumped comes from the river and the aquifer [32]. If the recharge and discharge of the aquifer remain unchanged before and after pumping water, then V

_{w}= −ΔV

_{s}. This indicates that all the water being pumped comes from the consumption of the aquifer storage, and it is impossible to produce an influence radius. To achieve a steady state, the sum of the increment of recharge and the reduction of discharge (sustainable yield) should always be equal to the pumping yield in the pumping state. Notably, when evaluating the amount of groundwater resources, the safe yield is usually adopted, which means that the water yield does not exceed the natural recharge amount [33]. This ignores the exploitation potential of the aquifer, which includes the increment of recharge and the reduction of discharge caused by pumping.

#### 4.2. Misleading the Safeguarding of Groundwater Quality

## 5. Summary and Prospects

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Symbol | Description | Dimension |

R | Influence radius | L |

K | Hydraulic conductivity | LT^{−1} |

H_{0} | Thickness of aquifer | L |

s_{w} | Drawdown of pumping well | L |

Q | Water yield of pumping well | L^{3}T^{−1} |

r_{w} | Radius of pumping well | L |

T | Transmissibility coefficient of aquifer | L^{2}T^{−1} |

t | Time from beginning of pumping to formation of stable cone of depression of groundwater level | T |

μ | Specific yield | / |

I | Hydraulic gradient of groundwater level | / |

r | Distance between pumping well and observation well | L |

h | Groundwater level at distance r from pumping well | L |

h_{w} | Water level of pumping well | L |

e | Empirical coefficient | / |

Q_{0} | Water yield of single pumping well | L^{3}T^{−1} |

B | Width of aquifer | L |

Q_{m} | Maximum water yield of pumping well | L^{3}T^{−1} |

S | Distance in physics | L |

v | Seepage velocity | LT^{−1} |

n | Porosity of porous media (aquifer) | / |

## References

- Darcy, H. Les Fontaines Publiques de la Ville de Dijon [The Public Fountains of the City of Dijon]; Victor Dalmont: Paris, France, 1856. [Google Scholar]
- Dupuit, J. Etudes Théoriques et Pratiques sur le Mouvement des Eaux dans les Canaux Découverts et à Travers les Terrains Perméables [Theoretical and Practical Studies on the Movement of Water in Open Channels and Permeable Ground]; Dunod: Paris, France, 1863. [Google Scholar]
- Thiem, G.A. Hydrologische Methoden; Gebhardt: Leipzig, Germany, 1906. [Google Scholar]
- Chen, Y.S.; Yao, D.S.; Hua, R.R. Try to discuss the influence radius. Geotech. Invest. Survey
**1976**, 6, 5–33. (In Chinese) [Google Scholar] - Todd, D.K. Ground Water Hydrology; Wiley: New York, NY, USA, 1959. [Google Scholar]
- Wang, J.H.; Wang, F. Discussion on the range of groundwater depression cone, radius of influence and scope of environmental impacts during pumping. J. Hydraul. Eng.
**2020**, 51, 827–834, (In Chinese with English abstract). [Google Scholar] - Wang, X.M.; Wang, X.H.; Wen, W.; Li, G.Y. Model analysis of Dupuit’s steady well flow formula. Coal. Geol. Explor.
**2014**, 6, 73–75, (In Chinese with English abstract). [Google Scholar] - Chen, C.X.; Lin, M.; Cheng, J.M. Groundwater Dynamics, 5th ed.; Geological Publishing House: Beijing, China, 2011. (In Chinese) [Google Scholar]
- Chen, C.X. Stable well flow model of influence radius and sustainable yield: Differences of a basic theoretical problem in Groundwater Dynamics—Discuss with academician Xue Yuqun. J. Hydraul. Eng.
**2010**, 41, 1003–1008. (In Chinese) [Google Scholar] - Xue, Y.Q. Discussion on the stable well flow model and Dupuit formula—Reply to professor Chen Chongxi’s article “Discussion”. J. Hydraul. Eng.
**2011**, 39, 1252–1256. (In Chinese) [Google Scholar] - Chen, C.X. Rediscuss on stable well flow model of influence radius and sustainable yield: Differences of a basic theoretical problem in Groundwater Dynamics—Reply to academician Xue Yuqun’s discussion. J. Hydraul. Eng.
**2014**, 45, 117–121. (In Chinese) [Google Scholar] - Xue, Y.Q.; Wu, J.C. Groundwater Dynamics, 3rd ed.; Geological Publishing House: Beijing, China, 2010. (In Chinese) [Google Scholar]
- Wu, C.M.; Yeh, T.C.J.; Zhu, J.; Lee, T.H.; Hsu, N.S.; Chen, C.H.; Sancho, A.F. Traditional analysis of aquifer tests: Comparing apples to oranges? Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef] [Green Version] - Theis, C.V. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Eos Trans. AGU
**1935**, 16, 519–524. [Google Scholar] [CrossRef] - Bear, J. Hydraulics of Groundwater; McGraw-Hill, Inc.: New York, NY, USA, 1979. [Google Scholar]
- China Geological Survey. Handbook of Hydrogeology, 2nd ed.; Geological Publishing House: Beijing, China, 2012. (In Chinese)
- Zhang, H.R. The steady state of groundwater movement. Hydrogeol. Eng. Geol.
**1986**, 6, 18–21. (In Chinese) [Google Scholar] - Zhu, D.T. Discussion on Dupuit formula and Forchheimer formula. J. Hydraul. Eng.
**2012**, 43, 502–504. (In Chinese) [Google Scholar] - Zhang, H.R. A couple of interesting inference of Theis’s equation. Hydrogeol. Eng. Geol.
**1985**, 5, 30–32. (In Chinese) [Google Scholar] - Chang, P.Y.; Hsu, S.Y.; Chen, Y.W.; Liang, C.; Wen, F.; Lu, H.Y. Using the resistivity imaging method to monitor the dynamic effects on the vadose zone during pumping tests at the Pengtsuo site in Pingtung, Taiwan. Terr. Atmos. Ocean. Sci.
**2016**, 27, 059. [Google Scholar] [CrossRef] [Green Version] - Chang, P.Y.; Chang, L.C.; Hsu, S.Y.; Tsai, J.P.; Chen, W.F. Estimating the hydrogeological parameters of an unconfined aquifer with the time-lapse resistivity-imaging method during pumping tests: Case studies at the Pengtsuo and Dajou sites. Taiwan. J. Appl. Geophys.
**2017**, 144, 134–143. [Google Scholar] [CrossRef] - Craik, A.D.D. “Continuity and change”: Representing mass conservation in fluid mechanics. Arch. Hist. Exact Sci.
**2013**, 67, 43–80. [Google Scholar] [CrossRef] - Tóth, J. A theoretical analysis of groundwater flow in small drainage basin. J. Geophys. Res.
**1963**, 68, 4795–4812. [Google Scholar] [CrossRef] - Tóth, J. Cross-formation gravity flow of groundwater: A mechanism of the transport and accumulation of petroleum (The generalized hydraulic theory of petroleum migration). AAPG Stud. Geol.
**1980**, 10, 121–167. [Google Scholar] - Mádl-Szónyi. From the Artesian Paradigm to Basin Hydraulics: The Contribution of József Tóth to Hungarian Hydrogeology; Publishing Company of Budapest University of Technology and Economics: Budapest, Hungary, 2008. [Google Scholar]
- Zhang, R.Q.; Liang, X.; Jin, M.G.; Wan, L.; Yu, Q.C. Fundamentals of Hydrogeology, 7th ed.; Geological Publishing House: Beijing, China, 2018. (In Chinese) [Google Scholar]
- Jiang, X.W.; Wan, L.; Wang, J.Z.; Yin, B.X.; Fu, W.X.; Lin, C.H. Field identification of groundwater flow systems and hydraulic traps in drainage basins using a geophysical method. Geophys. Res. Lett.
**2014**, 41, 2812–2819. [Google Scholar] [CrossRef] - Zha, Y.Y.; Shi, L.S.; Liang, Y.; Tso, C.H.M.; Zeng, W.Z.; Zhang, Y.G. Analytical sensitivity map of head observations on heterogeneous hydraulic parameters via the sensitivity equation method. J. Hydrol.
**2020**, 591, 125282. [Google Scholar] [CrossRef] - Liu, Y.P.; Yamanaka, T.; Zhou, X.; Tian, F.Q.; Ma, W.C. Combined use of tracer approach and numerical simulation to estimate groundwater recharge in an alluvial aquifer system: A case study of Nasunogahara area, central Japan. J. Hydrol.
**2014**, 519, 833–847. [Google Scholar] [CrossRef] - Yuan, R.Q.; Song, X.F.; Han, D.M.; Zhang, L.; Wang, S.Q. Upward recharge through groundwater depression cone in piedmont plain of North China Plain. J. Hydrol.
**2013**, 500, 1–11. [Google Scholar] [CrossRef] - Chen, C.X. Improvement of Dupuit model: With infiltration recharge. Hydrogeol. Eng. Geol.
**2020**, 47, 1–4, (In Chinese with English abstract). [Google Scholar] - Zhu, Y.G.; Zhai, Y.Z.; Du, Q.Q.; Teng, Y.G.; Wang, J.S.; Yang, G. The impact of well drawdowns on the mixing process of river water and groundwater and water quality in a riverside well field, Northeast China. Hydrol. Process.
**2019**, 33, 945–961. [Google Scholar] [CrossRef] - Wood, W.W. Groundwater “durability” not “sustainability”? Groundwater
**2020**. [Google Scholar] [CrossRef] - Ahmed, K.; Shahid, S.; Demirel, M.C.; Nawaz, N.; Khan, N. The changing characteristics of groundwater sustainability in Pakistan from 2002 to 2016. Hydrogeol. J.
**2019**, 27, 2485–2496. [Google Scholar] [CrossRef] - Jia, X.Y.; Hou, D.Y.; Wang, L.W.; O’Connor, D.; Luo, J. The development of groundwater research in the past 40 years: A burgeoning trend in groundwater depletion and sustainable management. J. Hydrol.
**2020**, 587, 125006. [Google Scholar] [CrossRef] - White, E.K.; Costelloe, J.; Peterson, T.J.; Western, A.W.; Carrara, E. Do groundwater management plans work? Modelling the effectiveness of groundwater management scenarios. Hydrogeol. J.
**2019**, 27, 1–24. [Google Scholar] [CrossRef] - Famiglietti, J.S. The global groundwater crisis. Nat. Clim. Chang.
**2014**, 4, 945–948. [Google Scholar] [CrossRef] - Giordano, M. Global groundwater? Issues and solutions. Ann. Rev. Environ. Resour.
**2009**, 34, 153–178. [Google Scholar] [CrossRef] - Konikow, L.; Kendy, E. Groundwater depletion: A global problem. Hydrogeol. J.
**2005**, 13, 317–320. [Google Scholar] [CrossRef] - Wada, Y.; van Beek, L.P.H.; van Kempen, C.M.; Reckman, J.W.T.M.; Vasak, S.; Bierkens, M.F.P. Global depletion of groundwater resources. Geophys. Res. Lett.
**2010**, 37, L20402. [Google Scholar] [CrossRef] [Green Version] - Erban, L.E.; Gorelick, S.M.; Zebker, H.A.; Fendorf, S. Release of arsenic to deep groundwater in the Mekong Delta, Vietnam, linked to pumping-induced land subsidence. Proc. Natl. Acad. Sci. USA
**2013**, 110, 13751–13756. [Google Scholar] [CrossRef] [Green Version] - Galloway, D.L.; Burbey, T.J. Review: Regional land subsidence accompanying groundwater extraction. Hydrogeol. J.
**2011**, 19, 1459–1486. [Google Scholar] [CrossRef] - Gleeson, T.; Alley, W.M.; Allen, D.M.; Sophocleous, M.A.; Zhou, Y.X.; Taniguchi, M.; VanderSteen, J. Towards sustainable groundwater use: Setting long-term goals, backcasting, and managing adaptively. Groundwater
**2012**, 50, 19–26. [Google Scholar] [CrossRef] - Qian, H.; Chen, J.; Howard, K.W.F. Assessing groundwater pollution and potential remediation processes in a multi-layer aquifer system. Environ. Pollut.
**2020**, 263, 114669. [Google Scholar] [CrossRef] [PubMed] - Scanlon, B.R.; Reedy, R.C.; Gates, J.B.; Gowda, P.H. Impact of agroecosystems on groundwater resources in the Central High Plains, USA. Agric. Ecosyst. Environ.
**2010**, 139, 700–713. [Google Scholar] [CrossRef] - Feng, W.; Zhong, M.; Lemoine, J.M.; Biancale, R.; Hsu, H.T.; Xia, J. Evaluation of groundwater depletion in North China using the Gravity Recovery and Climate Experiment (GRACE) data and ground-based measurements. Water Resour. Res.
**2013**, 49, 2110–2118. [Google Scholar] [CrossRef] - Hakan, A. Application of multivariate statistical techniques in the assessment of groundwater quality in seawater intrusion area in Bafra Plain. Turkey
**2013**, 185, 2439. [Google Scholar] - Jia, Y.F.; Xi, B.D.; Jiang, Y.H.; Guo, H.M.; Yang, Y.; Lian, X.Y.; Han, S.B. Distribution, formation and human-induced evolution of geogenic contaminated groundwater in China: A review. Sci. Total Environ.
**2018**, 643, 967–993. [Google Scholar] [CrossRef] [PubMed] - Lei, S.; Jiao, J.J. Seawater intrusion and coastal aquifer management in China: A review. Environ. Earth Sci.
**2014**, 72, 2811–2819. [Google Scholar] - Narasimhan, T.N. Hydrogeology in North America: Past and future. Hydrogeol. J.
**2005**, 13, 7–24. [Google Scholar] [CrossRef] - Chinese Academy of Science. Development Strategy of Chinese Subjects: Groundwater Science; Science Press: Beijing, China, 2019. (In Chinese) [Google Scholar]
- Zhang, F.G.; Huang, G.X.; Hou, Q.X.; Liu, C.Y.; Zhang, Y.; Zhang, Q. Groundwater quality in the Pearl River Delta after the rapid expansion of industrialization and urbanization: Distributions, main impact indicators, and driving forces. J. Hydrol.
**2019**, 577, 124004. [Google Scholar] [CrossRef] - Kurwadkar, S. Emerging trends in groundwater pollution and quality. Water Environ. Res.
**2014**, 86, 1677–1691. [Google Scholar] [CrossRef] - Zhang, B.; Song, X.; Zhang, Y.; Han, D.; Tang, C.; Yu, Y.; Ma, Y. Hydrochemical characteristics and water quality assessment of surface water and groundwater in Songnen plain, Northeast China. Water Res.
**2012**, 46, 2737–2748. [Google Scholar] [CrossRef] [PubMed] - Zhou, Y.; Khu, S.T.; Xi, B.; Su, J.; Hao, F.; Wu, J.; Huo, S. Status and challenges of water pollution problems in China: Learning from the European experience. Environ. Earth Sci.
**2014**, 72, 1243–1254. [Google Scholar] [CrossRef] - Datta, B.; Prakash, O.; Cassou, P.; Valetaud, M. Optimal unknown pollution source characterization in a contaminated groundwater aquifer—evaluation of a developed dedicated software tool. J. Geosci. Environ. Prot.
**2014**, 2, 41–51. [Google Scholar] [CrossRef] - Ayvaz, M.T. A linked simulation—optimization model for solving the unknown groundwater pollution source identification problems. J. Contam. Hydrol.
**2010**, 117, 46–59. [Google Scholar] [CrossRef] [PubMed] - Sun, A.Y.; Painter, S.L.; Wittmeyer, G.W. A constrained robust least squares approach for contaminant release history identification. Water Resour. Res.
**2006**, 42, 263–269. [Google Scholar] [CrossRef] [Green Version] - Mirghani, B.Y.; Mahinthakumar, K.G.; Tryby, M.E.; Ranjithan, R.S.; Zechman, E.M. A parallel evolutionary strategy based simulation—optimization approach for solving groundwater source identification problems. Adv. Water Resour.
**2009**, 32, 1373–1385. [Google Scholar] [CrossRef] - Singh, R.M.; Datta, B.; Jain, A. Identification of unknown groundwater pollution sources using artificial neural networks. J. Water Res. Plan. Manag.
**2004**, 130, 506–514. [Google Scholar] [CrossRef] - Singh, R.M.; Datta, B. Identification of groundwater pollution sources using GA-based linked simulation optimization model. J. Hydrol. Eng.
**2006**, 11, 101–109. [Google Scholar] [CrossRef] - Hintze, S.; Gaétan, G.; Hunkeler, D. Influence of surface water-groundwater interactions on the spatial distribution of pesticide metabolites in groundwater. Sci. Total Environ.
**2020**, 733, 139109. [Google Scholar] [CrossRef] - Zhang, H.; Jiang, X.W.; Wan, L.; Ke, S.; Liu, S.A.; Han, G.L.; Guo, H.M.; Dong, A.G. Fractionation of Mg isotopes by clay formation and calcite precipitation in groundwater with long residence times in a sandstone aquifer, Ordos Basin, China. Geochim. Cosmochim. Acta
**2018**, 237, 261–274. [Google Scholar] [CrossRef] - Kim, J.W. Optimal pumping time for a pump-and-treat determined from radial convergent tracer tests. Geosci. J.
**2014**, 18, 69–80. [Google Scholar] [CrossRef] - Bortone, I.; Erto, A.; Nardo, A.D.; Santonastaso, G.F.; Chianese, s.; Musmarra, D. Pump-and-treat configurations with vertical and horizontal wells to remediate an aquifer contaminated by hexavalent chromium. J. Contam. Hydrol.
**2020**, 235, 103725. [Google Scholar] [CrossRef] - Cecconet, D.; Sabba, F.; Devecseri, M.; Callegari, A.; Capodaglio, A.G. In situ groundwater remediation with bioelectrochemical systems: A critical review and future perspectives. Environ. Int.
**2020**, 137C, 105550. [Google Scholar] [CrossRef] - Margalef-Marti, R.; Carrey, R.; Vilades, M.; Carrey, R.; Viladés, M.; Jubany, I.; Vilanova, E.; Grau, R.; Soler, A.; Otero, N. Use of nitrogen and oxygen isotopes of dissolved nitrate to trace field-scale induced denitrification efficiency throughout an in-situ groundwater remediation strategy. Sci. Total Environ.
**2019**, 686, 709–718. [Google Scholar] [CrossRef] [PubMed] - Gierczak, R.F.D.; Devlin, J.F.; Rudolph, D.L. Field test of a cross-injection scheme for stimulating in situ denitrification near a municipal water supply well. J. Contam. Hydrol.
**2007**, 89, 48–70. [Google Scholar] [CrossRef] [PubMed] - Ministry of Environmental Protection, the People’s Republic of China. Technical Guideline for Delineating Source Water Protection Areas (HJ 338-2018); China Environment Press: Beijing, China, 2018. (In Chinese)
- Zhou, Y. Sources of water, travel times and protection areas for wells in semi-confined aquifers. Hydrogeol. J.
**2011**, 19, 1285–1291. [Google Scholar] [CrossRef] - Ameli, A.A.; Craig, J.R. Semi-analytical 3D solution for assessing radial collector well pumping impacts on groundwater–surface water interaction. Hydrol. Res.
**2017**, 49, 17–26. [Google Scholar] [CrossRef] [Green Version] - Valois, R.; Cousquer, Y.; Schmutz, M.; Pryet, A.; Delbart, C.; Dupuy, A. Characterizing stream-aquifer exchanges with self-potential measurements. Groundwater
**2018**, 56, 437–450. [Google Scholar] [CrossRef] - Zhu, Y.G.; Zhai, Y.Z.; Teng, Y.G.; Wang, G.Q.; Du, Q.Q.; Wang, J.S.; Yang, G. Water supply safety of riverbank filtration wells under the impact of surface water-groundwater interaction: Evidence from long-term field pumping tests. Sci. Total Environ.
**2020**, 711, 135141. [Google Scholar] [CrossRef] [PubMed] - Qian, H.; Zheng, X.L.; Fan, X.F. Numerical modeling of steady state 3-D groundwater flow beneath an incomplete river caused by riverside pumping. J. Hydraul. Eng
**1999**, 30, 32–37, (In Chinese with English abstract). [Google Scholar] - Liu, S.; Zhou, Y.; Tang, C.; McClain, M.; Wang, X.S. Assessment of alternative groundwater flow models for Beijing Plain, China. J. Hydrol.
**2021**, 596, 126065. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of Dupuit model and Thiem model. Q is the water yield of the pumping well, [L

^{3}T

^{−1}]; s

_{w}is the drawdown of the pumping well, [L]; h is the groundwater level at distance r from the pumping well, [L]; H

_{0}is the thickness of the aquifer, [L]; h

_{w}is the water level of the pumping well, [L]; r

_{w}is the radius of the pumping well, [L]; r is the distance between the pumping well and observation well, [L]; and R is the influence radius, [L].

**Figure 2.**Relationship between the water yield of the pumping well and water level of the pumping well, and between the water yield of the pumping well and the influence radius in the Dupuit model. Q is the water yield of the pumping well, [L

^{3}T

^{−1}]; Q

_{m}is the maximum water yield of the pumping well, [L

^{3}T

^{−1}]; H

_{0}is the thickness of the aquifer, [L]; h

_{w}is the water level of the pumping well, [L]; r

_{w}is the radius of the pumping well, [L]; and R is the influence radius, [L].

**Figure 3.**Influence radius of ripples in water and dominoes (used as an analogy to describe the influence radius discussed in this paper). R is the influence radius, [L]; S is the distance in physics, [L]; and t is time, [T].

**Figure 4.**Influence radius in well group pumping and single well pumping in influence radius theory. R is the influence radius, [L]; Q

_{mi}is the maximum water yield of pumping well i, [L

^{3}T

^{−1}]; and n is the number of pumping wells.

**Figure 5.**Schematic flow lines of a practical site according to the distribution of apparent resistivity, with a hypothetical pumping well added (modified from [27]).

**Figure 6.**Formation of cone of depression of groundwater level during pumping. Q is the water yield of the pumping well, [L

^{3}T

^{−1}].

**Figure 7.**Drawing water from cup (used as analogy for pumping in bounded aquifer). Q is the water yield of the pumping well, [L

^{3}T

^{−1}]; and t is the time, [T].

Equation Name | Equation | Application Condition | Author, Year | Parameter |
---|---|---|---|---|

Weber equation | $R=74\sqrt{\frac{6K{H}_{0}t}{\mu}}$ | Unconfined aquifer | Schultze, 1924 | R: influence radius, [L]; K: hydraulic conductivity, [LT^{−1}]; H_{0}: thickness of aquifer, [L]; t: time from beginning of pumping to formation of stable cone of depression of groundwater level, [T]; μ: specific yield; s_{w}: drawdown of pumping well, [L]; Q: water yield of pumping well, [L^{3}T^{−1}]; I: hydraulic gradient of groundwater level |

Kusakin equation | $R=2{s}_{w}\sqrt{{H}_{0}K}$ | Unconfined or confined aquifer | Chertousov, 1949 | |

$R=47\sqrt{\frac{6K{H}_{0}t}{\mu}}$ | Unconfined aquifer | Aravin and Numerov, 1953 | ||

Siechardt equation | $R=10{s}_{w}\sqrt{K}$ | Preliminary stage of pumping in unconfined or confined aquifer | Chertousov, 1962 | |

Wilbur equation | $R=3\sqrt{\frac{K{H}_{0}t}{\mu}}$ | Unconfined aquifer | Chen, 1976 | |

Kelgay equation | $R=\frac{Q}{2K{H}_{0}I}$ | Completely penetrating well in unconfined aquifer | Chen, 1976 |

Model/Equation Name | Equations Group | Application Condition | Author, Year | Parameter |
---|---|---|---|---|

Forward model/equation | R: influence radius, [L]; K: hydraulic conductivity, [LT^{−1}]; H_{0}: thickness of aquifer, [L]; t: time from beginning of pumping to formation of stable cone of depression of groundwater level, [T]; μ: specific yield; s_{w}: drawdown of pumping well, [L]; Q: water yield of pumping well, [L^{3}T^{−1}]; I: hydraulic gradient of groundwater level. | |||

Plotnikov equation | $Q=e\frac{{Q}_{0}}{2R}B$ | Well group pumping | Chen et al., 1976 | |

Dupuit–Forchheimer equation | ${h}^{2}={H}_{0}^{2}-\frac{Q}{\pi K}\mathrm{ln}\frac{R}{{r}_{w}}$ | Unconfined aquifer | Poehls and Smith, 2009 | |

s_{w}-calculate equation | $Q=\frac{2\pi T{s}_{w}}{\mathrm{ln}\frac{R}{{r}_{w}}}$ | Confined aquifer | China Geological Survey, 2012 | |

Inversion model/equation | ||||

Siechardt equation | $T=\frac{Q}{2\pi {s}_{w}}\mathrm{ln}\frac{R}{{r}_{w}}$ | Confined aquifer | China Geological Survey, 2012 | |

Wilbur equation | $K=\frac{Q}{\pi \left[{H}_{0}^{2}-{({H}_{0}-{s}_{w})}^{2}\right]}\mathrm{ln}\frac{R}{{r}_{w}}$ | Unconfined aquifer |

K (m/d) | R (m) |
---|---|

0.5–1 | 25–50 |

1–5 | 50–100 |

5–20 | 100–300 |

20–50 | 300–400 |

50–100 | 400–500 |

75–150 | 500–600 |

100–200 | 600–1500 |

200–500 | 1500–3000 |

^{−1}]; and R is the influence radius, [L].

Viewpoint | Reference |
---|---|

(1) Dupuit’s R is an abstract parameter that reflects the well supply conditions and is recommended as a reference recharge radius. (2) There is still a considerable amount of drawdown beyond the range that we used to think of as R. | [4] |

R should be interpreted as a parameter indicating the distance beyond which the drawdown is negligible or unobservable. | [15] |

R does not exist in an infinite aquifer. | [17] |

(1) In theory, R does not exist in a confined aquifer that extends indefinitely without overcurrent recharge. (2) In practice, R should be considered as the horizontal distance from the pumping well to the point where the water level cannot actually be observed to drop and can be used as the basis for designing reasonable distances between wells. | [12] |

(1) Dupuit’s R is different from Thiem’s R. (2) Confusion between them has led to theoretical errors and incorrect methods of groundwater resource evaluation. | [8] |

(1) The magnitude of R has assumed properties making it essentially the same as unsteady flow. (2) The Kusakin equation with a time factor should be applied to the calculation of R. | [18] |

(1) Dupuit’s R is different from Thiem’s R. (2) Dupuit’s R is simply the radius of the round island, while Thiem’s R is a variable related to the cone of depression of the groundwater level. | [7] |

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## Share and Cite

**MDPI and ACS Style**

Zhai, Y.; Cao, X.; Jiang, Y.; Sun, K.; Hu, L.; Teng, Y.; Wang, J.; Li, J.
Further Discussion on the Influence Radius of a Pumping Well: A Parameter with Little Scientific and Practical Significance That Can Easily Be Misleading. *Water* **2021**, *13*, 2050.
https://doi.org/10.3390/w13152050

**AMA Style**

Zhai Y, Cao X, Jiang Y, Sun K, Hu L, Teng Y, Wang J, Li J.
Further Discussion on the Influence Radius of a Pumping Well: A Parameter with Little Scientific and Practical Significance That Can Easily Be Misleading. *Water*. 2021; 13(15):2050.
https://doi.org/10.3390/w13152050

**Chicago/Turabian Style**

Zhai, Yuanzheng, Xinyi Cao, Ya Jiang, Kangning Sun, Litang Hu, Yanguo Teng, Jinsheng Wang, and Jie Li.
2021. "Further Discussion on the Influence Radius of a Pumping Well: A Parameter with Little Scientific and Practical Significance That Can Easily Be Misleading" *Water* 13, no. 15: 2050.
https://doi.org/10.3390/w13152050