# 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:

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**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