# The Radius of Influence Myth

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Sichardt Formula

_{w}(m) the well radius.

^{3}/d), Equation (1) is introduced into the Dupuit equation [76] for steady well-flow in a homogeneous unconfined aquifer [4,5,6]:

_{0}(m) the constant initial head, which defines the saturated thickness of the aquifer before the extraction starts. Since drawdown is positive in Sichardt’s formula [36] (1), pumping rate Q in (2) is positive.

_{0}at the well is taken if the aquifer is phreatic. Note that distance R according to Sichardt’s formula [36] (1) sometimes is defined as the distance to the well face, in which case r

_{w}must be subtracted from R. At distance R, a constant-head boundary is defined in (2) and (3). Since drawdown is zero at R, this distance is the radius of influence according to the Dupuit [76] and Thiem [70] Equations (2) and (3), respectively.

_{w}, to find the two unknown variables: drawdown at the well face and the radius of influence. For the sake of simplicity, we replace the Dupuit Equation (2) [76] by the Thiem Equation (3) [70]. Multiplying each side of (3) by $\left(10.206\sqrt{K}/{r}_{w}\right)$, and dividing both sides of the Sichardt Formula (1) [36] by r

_{w}, gives:

_{w}; hence, dimensionless drawdown (dotted line) cannot be smaller than 1, as it equals the dimensionless radius of influence. The minimum of the curve is found by solving $\frac{\mathrm{d}{Q}^{*}}{\mathrm{d}{s}^{*}}=0$, which gives ${Q}^{*}=\frac{1}{e}$. Hence, there is no solution if dimensionless discharge is smaller than 1/e, exactly one solution if ${Q}^{*}=\frac{1}{e}$ and two solutions otherwise. In the case of (5), dimensionless discharge virtually equals 1 when dimensionless drawdown (solid line) is smaller than 0.1, as $\mathrm{ln}\left(x+1\right)\to x$ if $x\to 0$. There is no solution if dimensionless discharge is smaller than 1.

## 3. The de Glee Equation

_{0}is the zero-order modified Bessel function of the second kind.

## 4. The Theis Equation

^{s}D, with S

^{s}the specific elastic storage (m

^{−1}). Several approaches may be followed to derive (9) [92,93,94]. Appendix A shows how (9) is derived using the Laplace transform. In hydrogeological literature, function W is called the Theis’ well function; in mathematics, it is called the exponential integral [88]:

_{y}, which is also dimensionless. Like the de Glee Equation (6) [71,86], the Theis solution [72] (9) allows superposition if initial heads are steady.

## 5. The Hantush-Jacob Model

_{0}holds [103]:

^{3}/d) gives a better idea when leakage becomes relevant and when it is maximal:

^{−x}approximates 1 if x < 0.01 and 0 if x > 10, the Theis equation [72] (9) may be used if t < 0.01Sc and the de Glee equation [71,86] (6) if t > 10Sc. These are also useful rules of thumb to verify if estimating the radius of influence using (12) or (7), respectively, is justified. Figure 2 compares these approximations of the radius of influence with the radius of influence according to the Sichardt Formula (1) [36]. It is seen that the latter tends to underestimate the extent of the cone of depression after a period of pumping, and therefore, its use in assessing the environmental impact of permanent extractions must be avoided at all costs.

## 6. The Ernst Model

_{1}is the first order modified Bessel function of the second kind. Distance r

_{d}is the boundary between the proximal zone without drainage and the distal zone with drainage. This means s

_{1}in (18) is the drawdown in the proximal zone, whereas s

_{2}in (18) is the drawdown in the distal zone. Boundary r

_{d}is found by solving equation s

_{2}(r

_{d}) = Nc, which is straightforward applying a non-linear solver. The left plot in Figure 3 is a graphical representation of the solution of this equation, and the right plot shows drawdown according to (18) expressed in dimensionless form.

_{2}in (18) is zero, and s

_{1}is reduced to the well-known solution for a well in a circular infiltration pond with radius r

_{d}[67]:

_{2}in the distal zone is zero, the radius of influence R equals r

_{d}. The dotted straight line on the left plot in Figure 3 shows that in this case:

_{d}is negligibly small. In this case, Solution (18) simplifies to the de Glee solution [71,86] (6), which is clearly illustrated in the right plot of Figure 3. From this plot, we may derive the rule of thumb that using the de Glee equation [71,86] (6) is justified if Q/(πNKDc) < 1.

## 7. Transient State Solution of the Ernst Model

_{i}/c, and the recharge flux is multiplied by A

_{i}, where A

_{i}(m

^{2}) is the horizontal surface area of the ring represented by cell i.

## 8. Discussion

**Table 1.**Summary of the analytical models discussed in the paper applied to simulate axisymmetric flow towards a fully penetrating well with infinitesimal radius and constant pumping rate in a homogeneous aquifer with impervious base. From the solutions of these models, equations and rules of thumb are derived to estimate the radius of influence R, with KD the transmissivity, c the resistance, S the storage coefficient, N the infiltration flux, Q the pumping rate, and t the time. See text for explanation and definitions.

Model | Flow Regime | Outer Boundary | Upper Boundary | Initial Flow | Super-Position | Radius of Influence R |
---|---|---|---|---|---|---|

Dupuit [76] | Steady | Finite | Water table | None | No^{4} | Outer boundary (=input parameter) |

Thiem [70] | Steady | Finite | Impervious ^{1} | Steady | Yes | Outer boundary (=input parameter) |

de Glee [71,86] | Steady | Infinite | Leaky ^{2} | Steady | Yes | $R=4\sqrt{cKD}$ |

Theis [72] | Transient | Infinite | Impervious ^{1} | Steady | Yes | $R=1.5\sqrt{\frac{tKD}{S}}$ |

Hantush-Jacob [73] | Transient | Infinite | Leaky ^{2} | Steady | Yes | $\begin{array}{c}R=1.5\sqrt{\frac{tKD}{S}}\mathrm{if}t0.01Sc\\ R=4\sqrt{cKD}\mathrm{if}t10Sc\end{array}$ |

Ernst [74] | Steady | Infinite | Drainage + Recharge | None ^{3} | No ^{4} | $\begin{array}{c}R=\sqrt{\frac{Q}{\pi N}}\mathrm{if}\frac{Q}{\pi NKDc}100\\ R=4\sqrt{cKD}\mathrm{if}\frac{Q}{\pi NKDc}1\end{array}$ |

Transient Ernst (Appendix A) | Transient | Infinite | Drainage + Recharge | None ^{3} | No ^{4} | See Figure 5 |

^{1}Or water table if drawdown is less than 10% of initial saturated thickness.

^{2}Leakage through incompressible aquitard or linear surface water interaction (cfr. MODFLOW river).

^{3}Initial heads equal to Nc are relative to the steady drainage levels, which are set to zero for convenience.

^{4}Unless the solution may be approximated by its corresponding linear equation.

_{max}, which is independent of the aquifer transmissivity KD:

_{max}at distance R

_{max}. Additionally, Formula (21) derived from the de Glee equation [71,86] (6), requires resistance c, whereas Formula (22) derived from the Theis equation [72] (9), requires storage coefficient S. The latter is also time dependent. Appendix B explains how these formulas are derived.

_{max}is necessary in (21) and (22), because the model of de Glee [71,86] and the model of Theis [72] have a boundary condition at infinity. Only at this boundary condition, drawdown is exactly zero, by definition, whereas drawdown is nonzero at all other distances from the well. Mathematically, the radius of influence is thus infinitely large. Recall that Formulas (7) and (12) estimating the radius of influence are approximations that ensure drawdown is zero at distance R. The error induced by these approximations is proportional to Q/(KD) [63].

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Deriving the Transient State Solution of the Ernst Model

_{d}). It simulates flow in a bounded homogeneous aquifer with constant thickness and infiltration at the top. The second model calculates drawdown in the distal zone where drainage is still active (r > r

_{d}). It simulates flow in a homogeneous leaky aquifer with constant thickness and infiltration at the top. The inner boundary condition in the first model defines a constant discharge equal to the pumping rate at the well, whereas the outer boundary is a constant-head equal to the drainage level. The latter is set to zero; hence, the drawdown is equal to the initial head h

_{0}= Nc. This outer boundary coincides with the inner boundary of the second model. Therefore, drawdown in both models must be the same at this boundary, and continuity of flow requires that the inflow to the first model must be equal to the outflow from the second model. The latter determines the constant flux at the inner boundary of the second model. The outer boundary condition in the second model is a constant head at an infinite large distance.

^{2}/T] and constant storativity S [–]. The aquifer is recharged at the top by a constant flux N [L/T]. This flux is positive in case of infiltration. The leaky top and bottom boundary have constant heads [L] h

_{top}and h

_{bot}and hydraulic resistances [L] c

_{top}and c

_{bot}, respectively. The inner boundary is a well with radius r

_{w}[L] from which water is extracted at constant pumping rate Q [L

^{3}/T]:

_{out}[L] has a constant head h

_{out}[L]:

^{−1}]. The Laplace transform of the constants is:

_{out}, then the Thiem equation [70] (3) is obtained. If Q = 0 and r

_{w}= 0, then (A20) gives the solution for a circular infiltration area. If r

_{w}= 0, h

_{out}= 0, and r

_{out}= r

_{d}, then Equation (18a) is obtained by subtracting (A20) from h

_{0}= Nc, and setting c to zero gives (19).

_{out}given by (A14). If ${r}_{out}\to \infty $ and the lower boundary is confined, i.e., c

_{bot}= ∞, the steady state solution is:

_{top}= 0 and Q = 0, then (A28) simplifies to expression (17) with c

_{top}= c. If ${r}_{out}\to \infty $, ${r}_{w}\to 0$, and N = 0, then (A28) simplifies to the de Glee solution [71,86]:

_{top}, then Equation (6) is obtained, with c = c

_{top}. The generalized equation for an aquifer with leaky top and bottom layer is:

_{out}given by (A13). This solution may be inverted numerically applying the Stehfest [101] algorithm. If ${r}_{w}\to 0$, ${r}_{out}\to \infty $, and N = 0, and if the system is confined, i.e., c

_{top}= c

_{bot}= ∞, the following transient state solution in Laplace space is obtained:

_{0}, then Equation (9) is obtained. If ${r}_{w}\to 0$, ${r}_{out}\to \infty $, N = 0, and if the system is leaky with h

_{0}= h

_{top}and c

_{bot}= ∞, the following transient state solution in Laplace space is obtained:

_{0}, then Equation (13) is obtained, with c = c

_{top}.

^{3}/T] is the amount of water per unit of time that flows horizontally through the cylindrical surface with radius r, at time t:

_{out}given by (A14). If the aquifer is confined, then a = 0, and (A37) simplifies to:

_{0}= Nc, with h

_{top}= 0 and c = c

_{top}, after which r

_{w}is substituted by r

_{d}, and Q by Q

^{r}(r

_{d}) according to (A38), with r

_{w}= 0.

_{out}given by (A13). Expression (A39) may be inverted numerically. In case of transient flow, the storage change ${Q}^{s}\left({r}_{1},{r}_{2},t\right)$ is defined as the amount of water per unit of time released by or stored in the ring determined by radii r

_{1}and r

_{2}at time t:

_{out}given by (A13). Expression (A42) may be inverted numerically. Total storage change dV/dt is equal to ${Q}^{s}\left({r}_{w},{r}_{out},t\right)$. In case of the Theis model [72], this simplifies to Q/p, which is inverted to Q. To find expression (16), i.e., the total storage change in case of the Hantush and Jacob model [73], expression (A42) is evaluated for r

_{1}= 0 and r

_{2}= ∞. This simplifies to Q/(p+1/S/c), which is inverted to [112]:

_{bot}= c

_{top}= ∞, r

_{out}= r

_{d}, h

_{out}= 0, and h

_{0}= Nc, whereas in the distal zone, c

_{bot}= ∞, c

_{top}= c, r

_{w}= r

_{d}, h

_{top}= h

_{out}= 0, and h

_{0}= Nc. Using (A13), the exact solution in Laplace space is:

_{d}between the proximal and the distal zone is time dependent. However, it is not transformed, as radial distance r is not transformed either. It is assumed ${r}_{w}\to 0$ and ${r}_{out}\to \infty $, hence, constants ${\alpha}_{1}$ and ${\beta}_{1}$ are given by (A23) and (A24), respectively, and constant ${\beta}_{2}$ by (A26):

_{d}is found by solving h

_{2}(r

_{d},t) = 0 using a non-linear solver. To find the head h

_{2}at distan r

_{d}, ${\overline{h}}_{2}\left({r}_{d},p\right)=0$ according to (A44) is evaluated using the Stehfest algorithm [101]. The non-linear solver finds the value of r

_{d}that corresponds to the root of this numerically inverted equation. Note that the logarithm of r

_{d}is evaluated to avoid negative values. Once r

_{d}is found, drawdown s

_{1}and s

_{2}in the proximal and distal zone are found by numerically inverting (A44), and subtracting the calculated head h from the initial head h

_{0}= Nc. As r

_{d}is time dependent, this routine must be applied for each time t.

## Appendix B. Finding the Maximum Radius of Influence

^{3}/T], T = KD the transmissivity [L

^{2}/T], r the radial distance [L], and P

_{i}an independent variable or hydraulic parameter. In case of the de Glee equation [71,86] (6), function f is defined as $f\left(x\right)={\mathrm{K}}_{0}\left(\sqrt{x}\right)$ with K

_{0}the zero-order modified Bessel function of the second kind, whereas f is the Theis Well function W in case of the Theis equation [72] (9).

_{max}is found for a given maximum drawdown s

_{max}using the definition of dimensionless drawdown s*:

_{max}is found using the definition of dimensionless transmissivity T* and substituting r

_{max}by (A52):

_{max}given a maximum allowable drawdown s

_{max}is:

_{max}given a maximum allowable drawdown s

_{max}at time t [L] is:

## References

- 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. [Google Scholar] [CrossRef] - Janssen, G.J.M. Bemaling van Bouwputten; Van Marken Delft Drukkers: Delft, The Netherlands, 2003. (In Dutch) [Google Scholar]
- Bot, B. Grondwaterzakboekje, 2nd ed.; Bot Raadgevend Ingenieur: Rotterdam, The Netherlands, 2016. (In Dutch) [Google Scholar]
- Willems, E.; Monseré, T.; Dierckx, J. Geactualiseerd Richtlijnenboek Milieueffectrapportage–Basisrichtlijnen per Activiteitengroep–Landbouwdieren; Dienst MER, Afdeling Milieu-, Natuur- en Energiebeleid, Departement Leefmilieu, Natuur en Energie: Brussels, Belgium, 2009. (In Dutch) [Google Scholar]
- VMM. Richtlijnen Bemalingen Ter Bescherming van Het Milieu; Vlaamse Milieumaatschappij: Aalst, Belgium, 2019. (In Dutch) [Google Scholar]
- VMM. Handleiding Berekeningsinstrument Bemalingen; Vlaamse Milieumaatschappij: Aalst, Belgium, 2020. (In Dutch) [Google Scholar]
- OVAM. Standaardprocedure Bodemsaneringsproject; Openbare Vlaamse Afvalstoffenmaatschappij: Mechelen, Belgium, 2018. (In Dutch) [Google Scholar]
- Mansur, C.I.; Kaufman, R.I. Chapter 3–Dewatering. In Foundation Engineering; Leonards, G.A., Ed.; McGraw-Hill: New York, NY, USA, 1962; pp. 241–350. [Google Scholar]
- Fruco and Associates Inc. Dewatering and Groundwater Control for Deep Excavations. NAVFAC Manual; Department of the Navy: Washington, DC, USA, 1971.
- Powers, J.P. Construction Dewatering: New Methods and Applications, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1992. [Google Scholar]
- Powers, J.P. Construction Dewatering–A Guide to Theory and Practice; John Wiley & Sons: New York, NY, USA, 1981. [Google Scholar]
- Powers, J.P.; Corwin, A.B.; Schmall, P.C.; Kaeck, W.E. Construction Dewatering and Groundwater Control: New Methods and Applications; John Wiley & Sons: New York, NY, USA, 2007. [Google Scholar]
- Sanglerat, G.; Olivari, G.; Cambou, B. Practical Problems in Soil Mechanics and Foundation Engineering, 1–Developments in Geotechnical Engineering; Elsevier Science Publishers B.V.: Amsterdam, The Netherlands, 1984; Volume 34A. [Google Scholar]
- Punmia, B.C.; Ashok, K.J.; Arun, K.J. Water Supply Engineering, 2nd ed.; Laxmi Publications (P) LTD.: New Delhi, India, 1995. [Google Scholar]
- Dachroth, W. Handbuch Der Baugeologie Und Geotechnik; Springer: Berlin/Heidelberg, Germany, 2017. (In German) [Google Scholar]
- Puller, M. Deep Excavations: A Practical Manual; Thomas Telford: London, UK, 2003. [Google Scholar]
- Merkl, G. Technik Der Wasserversorgung: Praxisgrundlagen Für Führungskräfte; Deutscher Industrieverlag: Oldenbourg, Germany, 2008. (In German) [Google Scholar]
- Smoltczyk, U. Ground Dewatering. In Geotechnical Engineering Handbook; Smoltczyk, U., Ed.; John Wiley & Sons: Hoboken, NJ, USA, 2003; Volume 2, pp. 365–398. [Google Scholar]
- Powrie, W. Soil Mechanics: Concepts and Applications, 2nd ed.; Taylor & Francis: New York, NY, USA, 2004. [Google Scholar]
- Coduto, D.P.; Yeung, M.R.; Kitch, W.A. Geotechnical Engineering: Principles & Practices, 2nd ed.; Pearson Prentice Hall: New York, NY, USA, 2011. [Google Scholar]
- Cashman, P.M.; Preene, M. Groundwater Lowering in Construction–A Practical Guide to Dewatering. Volume 6 of Applied Geotechnics, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Desodt, C.; Reiffsteck, P. Géotechnique-Excercices et Problèmes Corrigés de Mécanique Des Sols, Avec Rappels de Cours, 2nd ed.; Dunod: Malakoff, France, 2020. (In French) [Google Scholar]
- Ergun, M.U.; Nalçakan, M.S. Dewatering of a Large Excavation Pit by Wellpoints. In Proceedings of the Third International Conference on Case Histories in Geotechnical Engineering, St. Louis, MO, USA, 1–5 June 1993; pp. 707–711. [Google Scholar]
- Kumar, B.A.; Sudip, B.; Prabir, M. An Innovative Methodology for Groundwater Management with Reference to Saline Water Intrusion. IOSR J. Eng.
**2012**, 2, 1473–1486. [Google Scholar] [CrossRef] - Rabie, M.H. Comparative Study between Predicted and Observed Records of Implementation Dewatering Systems at Abu Qir Intake Power Plant, Alexandria. J. Am. Sci.
**2013**, 9, 106–114. [Google Scholar] - Yihdego, Y. Engineering and Enviro-Management Value of Radius of Influence Estimate from Mining Excavation. J. Appl. Water Eng. Res.
**2018**, 6, 329–337. [Google Scholar] [CrossRef] - Yihdego, Y.; Paffard, A. Predicting Open Pit Mine Inflow and Recovery Depth in the Durvuljin Soum, Zavkhan Province, Mongolia. Mine Water Environ.
**2017**, 36, 114–123. [Google Scholar] [CrossRef] - Yihdego, Y.; Drury, L. Mine Dewatering and Impact Assessment in an Arid Area: Case of Gulf Region. Environ. Monit. Assess.
**2016**, 188, 634. [Google Scholar] [CrossRef] - De Filippi, F.M.; Iacurto, S.; Ferranti, F.; Sappa, G. Hydraulic Conductivity Estimation Using Low-Flow Purging Data Elaboration in Contaminated Sites. Water
**2020**, 12, 898. [Google Scholar] [CrossRef] [Green Version] - Khadka, D.B. Experimental Design of Physical Unconfined Aquifer for Evaluation of Well Abstraction Effects: Laboratory Approach. J. Eng. Res. Rep.
**2021**, 20, 90–102. [Google Scholar] [CrossRef] - Niu, W.; Wang, Z.; Chen, F.; Li, H. Settlement Analysis of a Confined Sand Aquifer Overlain by a Clay Layer Due to Single Well Pumping. Math. Probl. Eng.
**2013**, 2013, 789853. [Google Scholar] [CrossRef] - Masoud, M. Groundwater Resources Management of the Shallow Groundwater Aquifer in the Desert Fringes of El Beheira Governorate, Egypt. Earth Syst. Environ.
**2020**, 4, 147–165. [Google Scholar] [CrossRef] - Bair, E.S.; O’Donnell, T.P. Uses of Numerical Modeling in the Design and Licensing of Dewatering and Depressurizing Systems. Ground Water
**1983**, 21, 411–420. [Google Scholar] [CrossRef] - Moh, Z.-C.; Chuay, H.-Y.; Hwang, R.N. Large Scale Pumping Test and Hydraulic Characteristics of Chingmei Gravels. In Proceedings of the Twelfth Southeast Asian Geotechnical Conference and the Fourt International Conference on Tropcial Soils, Kuala Lumpur, Malaysia, 6–10 May 1996; Volume 1, pp. 119–124. [Google Scholar]
- Fileccia, A. Some Simple Procedures for the Calculation of the Influence Radius and Well Head Protection Areas (Theoretical Approach and a Field Case for a Water Table Aquifer in an Alluvial Plain). Acque Sotter. Ital. J. Groundw.
**2015**, 4, 3. [Google Scholar] [CrossRef] - Kyrieleis, W.; Sichardt, W. Grundwasserabsenkung Bei Fundierungsarbeiten; Springer: Berlin/Heidelberg, Germany, 1930. (In German) [Google Scholar]
- Sichardt, W. Das Fassungsvermögen von Rohrbrunnen Und Seine Bedeutung Für Die Grundwasserabsenkung, Insbesondere Für Größere Absenkungstiefen; Springer: Berlin/Heidelberg, Germany, 1928. (In German) [Google Scholar]
- Narasimhan, T.N. Hydraulic Characterization of Aquifers, Reservoir Rocks, and Soils: A History of Ideas. Water Resour. Res.
**1998**, 34, 33–46. [Google Scholar] [CrossRef] - Weber, H. Die Reichweite von Grundwasserabsenkungen Mittels Rohrbrunnen; Springer: Berlin/Heidelberg, Germany, 1928. (In German) [Google Scholar]
- Bredehoeft, J.D.; Papadopulos, S.S.; Cooper, H.H., Jr. The Water Budget Myth. In Scientific Basis of Water Resources Management Studies in Geophysics; National Academy Press: Washington, DC, USA, 1982; pp. 51–57. [Google Scholar]
- Bredehoeft, J.D. The Water Budget Myth Revisited: Why Hydrogeologists Model. Ground Water
**2002**, 40, 340–345. [Google Scholar] [CrossRef] [PubMed] - Theis, C.V. The Source of Water Derived from Wells: Essential Factors Controlling the Response of an Aquifer to Development. Civ. Eng.
**1940**, 10, 277–280. [Google Scholar] - Konikow, L.F.; Leake, S.A. Depletion and Capture: Revisiting “The Source of Water Derived from Wells”. Groundwater
**2014**, 52, 100–111. [Google Scholar] [CrossRef] - Lohman, S.W. Definitions of Selected Ground Water Terms–Revisions and Conceptual Refinements; USGS Water Supply Paper 1988; U.S. Government Printing Office: Washington, DC, USA, 1972.
- Bredehoeft, J.; Durbin, T. Ground Water Development-The Time to Full Capture Problem. Ground Water
**2009**, 47, 506–514. [Google Scholar] [CrossRef] - Seward, P.; Xu, Y.; Turton, A. Investigating a Spatial Approach to Groundwater Quantity Management Using Radius of Influence with a Case Study of South Africa. Water SA
**2015**, 41, 71–78. [Google Scholar] [CrossRef] [Green Version] - Barlow, P.M.; Leake, S.A.; Fienen, M.N. Capture Versus Capture Zones: Clarifying Terminology Related to Sources of Water to Wells. Groundwater
**2018**, 56, 694–704. [Google Scholar] [CrossRef] [Green Version] - Brown, R.H. The Cone of Depression and the Area of Diversion around a Discharging Well in an Infinite Strip Aquifer Subject to Uniform Recharge. In Shortcuts and Special Problems in Aquifer Tests; USGS Water-Supply Paper 1545C; U.S. Government Printing Office: Washington, DC, USA, 1963; pp. C69–C85. [Google Scholar]
- Sophocleous, M. Retracted: On Understanding and Predicting Groundwater Response Time. Groundwater
**2012**, 50, 528–540. [Google Scholar] [CrossRef] - Devlin, J.F.; Sophocleous, M. The Persistence of the Water Budget Myth and Its Relationship to Sustainability. Hydrogeol. J.
**2005**, 13, 549–554. [Google Scholar] [CrossRef] - Kalf, F.R.P.; Woolley, D.R. Applicability and Methodology of Determining Sustainable Yield in Groundwater Systems. Hydrogeol. J.
**2005**, 13, 295–312. [Google Scholar] [CrossRef] - Zhou, Y. A Critical Review of Groundwater Budget Myth, Safe Yield and Sustainability. J. Hydrol.
**2009**, 370, 207–213. [Google Scholar] [CrossRef] - Hansen, C.V. Description and Evaluation of Selected Methods Used to Delineate Wellhead-Protection Areas around Public-Supply Wells near Mt. Hope, Kansas; USGS Water-Resources Investigations Report 90-4102; U.S. Geological Survey: Denver, CO, USA, 1991. [CrossRef] [Green Version]
- Bear, J. Hydraulics of Groundwater; McGraw-Hill Series in Water Resources and Environmental Engineering; McGraw-Hill: New York, NY, USA, 1979. [Google Scholar]
- Kruseman, G.P.; de Ridder, N.A. Analysis and Evaluation of Pumping Test Data, 2nd ed.; ILRI Publication 47: Wageningen, The Netherlands, 1990. [Google Scholar]
- Neuman, S.P.; Witherspoon, P.A. Applicability of Current Theories of Flow in Leaky Aquifers. Water Resour. Res.
**1969**, 5, 817–829. [Google Scholar] [CrossRef] - MacDonald, T.R.; Kitanidis, P.K. Modeling the Free Surface of an Unconfined Aquifer Near a Recirculation Well. Ground Water
**1993**, 31, 774–780. [Google Scholar] [CrossRef] - Gefell, M.J.; Thomas, G.M.; Rossello, S.J. Maximum Water-Table Drawdown at a Fully Penetrating Pumping Well. Ground Water
**1994**, 32, 411–419. [Google Scholar] [CrossRef] - Chu, S.T. Transient Radius of Influence Model. J. Irrig. Drain. Eng.
**1994**, 120, 964–969. [Google Scholar] [CrossRef] - Soni, A.K.; Sahoo, L.K.; Ghosh, U.K.; Khond, M.V. Importance of Radius of Influence and Its Estimation in a Limestone Quarry. J. Inst. Eng. India Ser. D
**2015**, 96, 77–83. [Google Scholar] [CrossRef] - Castellazzi, P.; Martel, R.; Galloway, D.L.; Longuevergne, L.; Rivera, A. Assessing Groundwater Depletion and Dynamics Using GRACE and InSAR: Potential and Limitations. Groundwater
**2016**, 54, 768–780. [Google Scholar] [CrossRef] [Green Version] - Bresciani, E.; Shandilya, R.N.; Kang, P.K.; Lee, S. Well Radius of Influence and Radius of Investigation: What Exactly Are They and How to Estimate Them? J. Hydrol.
**2020**, 583, 124646. [Google Scholar] [CrossRef] - Dragoni, W. Some Considerations Regarding the Radius or Influence or a Pumping Well. Hydrogéologie
**1998**, 3, 21–25. [Google Scholar] - 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] - Pfannkuch, H.-O.; Mooers, H.D.; Siegel, D.I.; Quinn, J.J.; Rosenberry, D.O.; Alexander, S.C. Review: “Jacob’s Zoo”—How Using Jacob’s Method for Aquifer Testing Leads to More Intuitive Understanding of Aquifer Characteristics. Hydrogeol. J.
**2021**, 29, 2001–2015. [Google Scholar] [CrossRef] - Haitjema, H. The Role of Hand Calculations in Ground Water Flow Modeling. Ground Water
**2006**, 44, 786–791. [Google Scholar] [CrossRef] - Haitjema, H. Analytic Element Modeling of Groundwater Flow; Academic Press: San Diego, CA, USA, 1995. [Google Scholar]
- Bakker, M.; Strack, O.D.L. Analytic Elements for Multiaquifer Flow. J. Hydrol.
**2003**, 271, 119–129. [Google Scholar] [CrossRef] - Bruggeman, G.A. Analytical Solutions of Geohydrological Problems. Developments in Water Science 46; Elsevier: Amsterdam, The Netherlands, 1999. [Google Scholar]
- Thiem, G. Hydrologische Methoden; Gebhardt: Leipzig, Germany, 1906. (In German) [Google Scholar]
- De Glee, G.J. Over Grondwaterstroomingen Bij Wateronttrekking Door Middel van Putten. Ph.D. Thesis, Technische Universiteit Delft, Delft, The Netherlands, 1930. (In Dutch). [Google Scholar]
- 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. Trans. Am. Geophys. Union
**1935**, 16, 519–524. [Google Scholar] [CrossRef] - Hantush, M.S.; Jacob, C.E. Non-Steady Radial Flow in an Infinite Leaky Aquifer. Trans. Am. Geophys. Union
**1955**, 36, 95–100. [Google Scholar] [CrossRef] - Ernst, L.F. Analysis of Groundwater Flow to Deep Wells in Areas with a Non-Linear Function for the Subsurface Drainage. J. Hydrol.
**1971**, 14, 158–180. [Google Scholar] [CrossRef] - Louwyck, A.; Vandenbohede, A.; Bakker, M.; Lebbe, L. Simulation of Axi-Symmetric Flow towards Wells: A Finite-Difference Approach. Comput. Geosci.
**2012**, 44, 136–145. [Google Scholar] [CrossRef] - Dupuit, J. Etude Théoriques et Pratiques Sur Le Mouvement Des Eaux Dans Les Canaux Découverts et à Travers Les Terrains Perméables; Dunot: Paris, France, 1863. (In French) [Google Scholar]
- Wang, X.-W.; Yang, T.-L.; Xu, Y.-S.; Shen, S.-L. Evaluation of Optimized Depth of Waterproof Curtain to Mitigate Negative Impacts during Dewatering. J. Hydrol.
**2019**, 577, 123969. [Google Scholar] [CrossRef] - Wu, Y.-X.; Shen, S.-L.; Lyu, H.-M.; Zhou, A. Analyses of Leakage Effect of Waterproof Curtain during Excavation Dewatering. J. Hydrol.
**2020**, 583, 124582. [Google Scholar] [CrossRef] - Zheng, Y.; Lei, J.; Wang, F.; Xiang, L.; Yang, J.; Xue, Q. Investigation on Dewatering of a Deep Shaft in Strong Permeable Sandy Pebble Strata on the Bank of the Yellow River. Geofluids
**2021**, 2021, 9994477. [Google Scholar] [CrossRef] - Zhang, X.; Wang, X.; Xu, Y. Influence of Filter Tube of Pumping Well on Groundwater Drawdown during Deep Foundation Pit Dewatering. Water
**2021**, 13, 3297. [Google Scholar] [CrossRef] - Zeng, C.-F.; Wang, S.; Xue, X.-L.; Zheng, G.; Mei, G.-X. Evolution of Deep Ground Settlement Subject to Groundwater Drawdown during Dewatering in a Multi-Layered Aquifer-Aquitard System: Insights from Numerical Modelling. J. Hydrol.
**2021**, 603, 127078. [Google Scholar] [CrossRef] - Lyu, H.-M.; Shen, S.-L.; Wu, Y.-X.; Zhou, A.-N. Calculation of Groundwater Head Distribution with a Close Barrier during Excavation Dewatering in Confined Aquifer. Geosci. Front.
**2021**, 12, 791–803. [Google Scholar] [CrossRef] - Zeng, C.-F.; Zheng, G.; Xue, X.-L. Responses of Deep Soil Layers to Combined Recharge in a Leaky Aquifer. Eng. Geol.
**2019**, 260, 105263. [Google Scholar] [CrossRef] - Kooper, J. Beweging van Het Water in Den Bodem Bij Onttrekking Door Bronnen. De Ingenieur
**1914**, 29, 697–716. (In Dutch) [Google Scholar] - Hemker, C.J. Groundwater Flow in Layered Aquifer Systems. Ph.D. Thesis, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands, 2000. [Google Scholar]
- Jacob, C.E. Radial Flow in a Leaky Artesian Aquifer. Trans. Am. Geophys. Union
**1946**, 27, 198–208. [Google Scholar] [CrossRef] - Bakker, M. An Analytic, Approximate Method for Modeling Steady, Three-Dimensional Flow to Partially Penetrating Wells. Water Resour. Res.
**2001**, 37, 1301–1308. [Google Scholar] [CrossRef] [Green Version] - Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. In Dover Books on Advanced Mathematics; Abramowitz, M., Stegun, I.A., Eds.; Dover Publications: New York, NY, USA, 1965. [Google Scholar]
- De Smedt, F. Groundwater Hydrology; Department of Hydrology and Hydraulic Engineering, Faculty of Applied Sciences, Free University Brussels: Brussels, Germany, 2006. [Google Scholar]
- Harbaugh, A.W. MODFLOW-2005, the U.S. Geological Survey Modular Ground-Water Model-the Ground-Water Flow Process; USGS Techniques and Methods; U.S. Geological Survey: Reston, VA, USA, 2005. [CrossRef] [Green Version]
- Bredehoeft, J.D. An Interview with C.V. Theis. Hydrogeol. J.
**2008**, 16, 5. [Google Scholar] [CrossRef] - Loàiciga, H.A. Derivation Approaches for the Theis (1935) Equation. Ground Water
**2010**, 48, 2–5. [Google Scholar] [CrossRef] - Perina, T. Derivation of the Theis (1935) Equation by Substitution. Ground Water
**2010**, 48, 6–7. [Google Scholar] [CrossRef] - Masoodi, R.; Ghanbari, R.N. Derivation of the Theis (1935) Equation by Substitution. Ground Water
**2012**, 50, 8–9. [Google Scholar] [CrossRef] - Cooper, H.H.; Jacob, C.E. A Generalized Graphical Method for Evaluating Formation Constants and Summarizing Well-Field History. Trans. Am. Geophys. Union
**1946**, 27, 526–534. [Google Scholar] [CrossRef] - Jacob, C.E. Flow of Groundwater. In Engineering Hydraulics; Rouse, H., Ed.; Wiley: New York, NY, USA, 1950; pp. 321–386. [Google Scholar]
- Bear, J. Dynamics of Fluids in Porous Media; American Elsevier Publishing Company: New York, NY, USA, 1972. [Google Scholar]
- Hantush, M.S. Modification of the Theory of Leaky Aquifers. J. Geophys. Res.
**1960**, 65, 3713–3725. [Google Scholar] [CrossRef] - De Smedt, F. Constant-Rate Pumping Test in a Leaky Aquifer with Water Release from Storage in the Aquitard. Groundwater
**2020**, 58, 487–491. [Google Scholar] [CrossRef] - Neuman, S.P.; Witherspoon, P.A. Theory of Flow in a Confined Two Aquifer System. Water Resour. Res.
**1969**, 5, 803–816. [Google Scholar] [CrossRef] - Stehfest, H. Algorithm 368: Numerical Inversion of Laplace Transforms [D5]. Commun. ACM
**1970**, 13, 47–49. [Google Scholar] [CrossRef] - Shampine, L.F. Vectorized Adaptive Quadrature in MATLAB. J. Comput. Appl. Math.
**2008**, 211, 131–140. [Google Scholar] [CrossRef] [Green Version] - Veling, E.J.M.; Maas, C. Hantush Well Function Revisited. J. Hydrol.
**2010**, 393, 381–388. [Google Scholar] [CrossRef] [Green Version] - Louwyck, A.; Vandenbohede, A.; Bakker, M.; Lebbe, L. MODFLOW Procedure to Simulate Axisymmetric Flow in Radially Heterogeneous and Layered Aquifer Systems. Hydrogeol. J.
**2014**, 22, 1217–1226. [Google Scholar] [CrossRef] - Hemker, C.J. Steady Groundwater Flow in Leaky Multiple-Aquifer Systems. J. Hydrol.
**1984**, 72, 355–374. [Google Scholar] [CrossRef] - Hemker, C.J. Transient Well Flow in Leaky Multiple-Aquifer Systems. J. Hydrol.
**1985**, 81, 111–126. [Google Scholar] [CrossRef] - Hunt, B. Flow to a Well in a Multiaquifer System. Water Resour. Res.
**1985**, 21, 1637–1641. [Google Scholar] [CrossRef] - Hunt, B. Solutions for Steady Groundwater Flow in Multi-Layer Aquifer Systems. Transp. Porous Media
**1986**, 1, 419–429. [Google Scholar] [CrossRef] - Maas, C. The Use of Matrix Differential Calculus in Problems of Multiple-Aquifer Flow. J. Hydrol.
**1986**, 88, 43–67. [Google Scholar] [CrossRef] - Hemker, C.J.; Maas, C. Unsteady Flow to Wells in Layered and Fissured Aquifer Systems. J. Hydrol.
**1987**, 90, 231–249. [Google Scholar] [CrossRef] - Bakker, M. Semi-Analytic Modeling of Transient Multi-Layer Flow with TTim. Hydrogeol. J.
**2013**, 21, 935–943. [Google Scholar] [CrossRef] - Hantush, M.S. Hydraulics of Wells. In Advances in Hydroscience; Chow, V.T., Ed.; Academic Press: New York, NY, USA; London, UK, 1964; Volume 1, pp. 281–432. [Google Scholar]

**Figure 1.**Dimensionless discharge Q* as a function of dimensionless drawdown s* resulting from combining the Sichardt formula and the Thiem equation. The dimensionless parameters are defined in the text. Solid and dotted lines give the solution in which the radius of influence according to the Sichardt formula equals the distance to the well face and to the center of the well, respectively.

**Figure 2.**Plot showing the ratio of the radius of influence R approximated using the Theis or the de Glee equations and the radius of influence R

_{Sichardt}according to the empirical Sichardt formula as a function of parameter t*, which is defined in the figure’s legend. Parameter t is the time, D is the saturated thickness, S is the storage coefficient, c is the resistance, and s(r

_{w}) is the drawdown at the face of the well with radius r

_{w}. See text for definitions. The Sichardt radius of influence underestimates the hydraulic impact of the extraction if the ratio is larger than 1, i.e., above the horizontal dotted line. See text for a more detailed explanation.

**Figure 3.**Solution of the Ernst model. (

**a**) Dimensionless extent of the no-drainage zone r

_{d}/(KDc)

^{1/2}versus dimensionless pumping rate Q/(πNKDc). The dotted line is the asymptotic solution for zero resistance. (

**b**) Dimensionless drawdown (2πKDs)/Q versus dimensionless distance r/(KDc)

^{1/2}for different values of dimensionless pumping rate Q/(πNKDc). The solid line is the de Glee solution. K is the aquifer conductivity, D the saturated thickness, c the drainage resistance, N the infiltration flux, Q the pumping rate, s the drawdown, r the radial distance, and r

_{d}the boundary between the zones without and with drainage. See text for definitions and a more detailed explanation.

**Figure 4.**Transient state solution of the Ernst model developed in this study. (

**a**) Dimensionless drawdown (2πKDs)/Q as a function of dimensionless distance r/(KDc)

^{1/2}for dimensionless time t/(Sc) equal to 100 and for different dimensionless pumping rates Q/(πNKDc). (

**b**) Dimensionless drawdown (2πKDs)/Q as a function of dimensionless time t/(Sc) for dimensionless distance r/(KDc)

^{1/2}equal to 0.1 and for different dimensionless pumping rates Q/(πNKDc). K is the aquifer conductivity, S the storativity, D the saturated thickness, c the drainage resistance, N the infiltration flux, Q the pumping rate, s the drawdown, r the radial distance, and t the time. See text for definitions. The solution is verified against the finite-difference approach (circles), and against the asymptotic solutions developed by de Glee, Theis, Hantush and Jacob, and Ernst.

**Figure 5.**Contour plot of dimensionless storage change dV/dt/Q as a function of dimensionless time t/(Sc) and dimensionless pumping rate Q/(πNKDc) for the transient state solution of the Ernst model developed in this study, with dV/dt the storage change, Q the pumping rate, t the time, S the storage coefficient, c the drainage resistance, N the infiltration flux, K the aquifer conductivity, and D the saturated thickness. See text for definitions. The dotted lines indicate the rules of thumb derived in this paper.

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**MDPI and ACS Style**

Louwyck, A.; Vandenbohede, A.; Libbrecht, D.; Van Camp, M.; Walraevens, K.
The Radius of Influence Myth. *Water* **2022**, *14*, 149.
https://doi.org/10.3390/w14020149

**AMA Style**

Louwyck A, Vandenbohede A, Libbrecht D, Van Camp M, Walraevens K.
The Radius of Influence Myth. *Water*. 2022; 14(2):149.
https://doi.org/10.3390/w14020149

**Chicago/Turabian Style**

Louwyck, Andy, Alexander Vandenbohede, Dirk Libbrecht, Marc Van Camp, and Kristine Walraevens.
2022. "The Radius of Influence Myth" *Water* 14, no. 2: 149.
https://doi.org/10.3390/w14020149