capture zone of a partially penetrating well with skin effects in

Transp Porous Med (2012) 91:437–457DOI 10.1007/s11242-011-9853-3

Capture Zone of a Partially Penetrating Well with SkinEffects in Confined Aquifers

B. Ataie-Ashtiani · B. Shafei · H. Rashidian-Dezfouli ·M. Mohamadzadeh

Received: 29 August 2009 / Accepted: 25 August 2011 / Published online: 16 September 2011© Springer Science+Business Media B.V. 2011

Abstract The analysis of the capture zone of wells is useful to design pumping systemsand wellhead protection programs. In this study, an analytical solution for the head distribu-tion of a partially penetrating well, and semi-analytical methods to determine the geometryof the capture surface are presented. The analytical solution is derived based on an infin-itesimal radius under a constant pumping rate in a two-zone confined aquifer. Using thedeveloped solution, a sensitivity analysis is performed to study the influence of skin onthe drawdown, location of the stagnation point, maximum horizontal and vertical extent ofthe capture surface. The results show the efficiency of a partially penetrating well in theplume removal process is different for positive and negative skins. Also it can be concluded,the skin effect will decrease as the degree of penetration and pumping discharge of the wellincrease. Generally, the thickness of the skin zone is an influential factor in well hydraulicsand hydraulic head distribution. However, it can be seen where the skin zone is less than somespecific values, the skin effect is not a significant factor, thus, the geometry of the capturesurface can be obtained by assuming a single-zone aquifer. Moreover, the effects of differentparameters (pumping rates, degree of penetration of the well, thickness of the skin zone, etc.)on the capture zone are studied.

Keywords Semi-analytical solution · Capture zone · Skin effect · Stagnation point ·Pump and treat · Maximum horizontal extent · Maximum vertical extent

B. Ataie-Ashtiani (B) · B. Shafei · H. Rashidian-DezfouliDepartment of Civil Engineering, Sharif University of Technology, Tehran, Irane-mail: [email protected]

B. Ataie-AshtianiNational Centre for Groundwater Research and Training, Flinders University,Adelaide, SA, Australia

M. MohamadzadehDepartment of Physics, University of Tehran, Tehran, Iran

123

438 B. Ataie-Ashtiani et al.

List of SymbolsQ Constant pumping rate [L3T−1]Kr Horizontal hydraulic conductivity of the aquifer [LT−1]Kz Vertical hydraulic conductivity of the aquifer [LT−1]K Hydraulic conductivity of isotropic aquifer [LT−1]H(r, z) Drawdown [L]H̃(r, n) Drawdown in Fourier domainR Radial distance from the well [L]Z Vertical distance [L]U Darcy velocity [LT−1]rs Radius of the skin zone [L]rw Radius of the capture well [L]R Radius of the influence of the well [L]L Thickness of the aquifer [L]B1 Upper limit of the screen of the well [L]B2 Lower limit of the screen of the well [L]n Fourier transform of variable zφu Velocity potential due to regional flow [L2T−1]φw Velocity potential due to pumping well [L2T−1]U Discharge component in x direction [LT−1]V Discharge component in y direction [LT−1]W Discharge component in z direction [LT−1]I0(u) Modified Bessel function of the first kind of order zeroK0(u) Modified Bessel function of the second kind of order zero

1 Introduction

Pump and treat is one of the most widely used ground water remediation technologies.Conventional methods of pump and treat involve pumping the contaminated water to theground surface for the treatment. This system is a hydraulic containment, which controlsthe movement of contamination. The capture zone of a well is referred to the portion of theaquifer where the directions of the flow vectors are toward the well.

The size and shape of the capture zone depend on the interaction of numerous factors,such as: (1) the hydraulic gradient and hydraulic conductivity of the aquifer, (2) the extentto which the aquifer is heterogeneous, (3) the aquifer anisotropy, (4) whether the aquifer isconfined or unconfined, (5) the pumping rate, and (6) whether the screened interval of thewell fully or partially penetrates the aquifer.

When the screened portion of the well fully penetrates the aquifer, a two-dimensionalanalysis is adequate to delineate the horizontal capture zone. Although, when a pumpingwell only partially penetrates an aquifer, a vertical capture zone analysis is required becauseof the induced three-dimensional flow. In order to delineate the capture surface we need toobtain the locations of the stagnation points and the maximum horizontal and vertical extentof the capture surface.

Stagnation points refer to the points, where the discharge components are zero. They canbe considered as critical points since they separate the regions of the flow. Several numer-ical, analytical, and semi-analytical methods have been used to delineate the capture zone

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Capture Zone of a Partially Penetrating Well 439

of the wells. In addition, analytical and semi-analytical solutions are more accurate and lesscomplicated than numerical.

A wide range of analytical and semi-analytical solutions has been developed to define thecapture zone (e.g., Bakker and Strack 1996; Shan 1999; Christ and Goltz 1999, 2002, 2004;Eradmann 2000; Grubb 1993; Javandel and Tsang 1986; Yang et al. 1995; Zhan 1999), inmost of which, the theory of complex discharge potential (Strack 1989) has been employed.Complex potential is a convenient method to solve the Laplace equation in two dimen-sions for horizontal flow in a homogeneous and isotropic medium with no vertical gradients(Fienen et al. 2005). Therefore, in most of the previous developed analytical solutions, afully penetrating well in a confined or unconfined aquifer under the Dupuit–Forchheimerassumption has been considered. Javandel and Tsang (1986) used the concept of complexdischarge potential and the method of type curves to determine the shape of the capture zone.Christ and Goltz (2002) determined the capture zone for a number of arbitrary positionedextraction wells with different pumping rates. Using the particle tracking scheme, Fienen etal. (2005) plotted the capture zone curves by introducing the concept of terminal points andthe Hessian matrix.

Mostly, we encounter to the cases in which contaminated flow is not scattered throughoutthe whole thickness of the aquifer. Therefore, instead of using a fully penetrating well, itwill be much more effective to use a partially penetrating one. There are some analyticalsolutions with different assumptions for determining the drawdown distribution of a partiallypenetrating well (e.g., Cassiani and Kabala 1998; Cassiani et al. 1999; Chang and Chen 1999,2003). Faybishenko et al. (1995) developed a semi-analytical method that can be used fordetermining the geometry of the capture zone for a steady state flow to a partially penetratingwell that is screened from the top of a confined aquifer.

A well bore skin of finite thickness is the region that is developed due to the process ofdrilling, installing, and developing a well that causes the properties of the material in theimmediate vicinity of the well to be altered from the original formation. This can occur,because of the invasion of the mud into this formation or displacement of the particles of theadjacent formation. The skin effect reflects the connection between the aquifer and the well.Pressure drop in the vicinity of the wellbore can evaluate different values and this differencecan be interpreted by using: (1) infinitesimal skin method; (2) finite thickness skin method; or(3) the effective radius method (Bourdarot 1998). In this article, the second approach will beapplied to model the skin effect. In this case two types of skin effect are considered. If the skinzone has less hydraulic conductivity than the undisturbed part of the aquifer, there will be apositive skin effect; else it will be a negative one. This zone can be treated as a homogeneousformation, which may have a large impact on estimating the different characteristics of theaquifer during the different well tests. Different analytical solutions have been developedto express the effect of the skin on the distribution of the drawdown (e.g., Chang and Chen1999, 2002; Chen and Chang 2006; Novakowski 1993; Perina and Lee 2006; Yang and Yeh2002, 2005). Using the Laplace and Finite Fourier Cosine Transform (FFCT), Yang and Yeh(2005) developed a Laplace-domain solution for the dimensionless flow rate at the well boreand drawdown in the skin and formation zones.

The primary objective of this article is to develop a new analytical solution for a steadystate drawdown distribution in a confined aquifer due to the constant pumping rate froma partially penetrating well with an infinitesimal radius while considering the skin effect.Moreover, in this article the effects of the skin on the geometry of the capture zone are stud-ied. A sensitivity analysis was done, which considered the variation in discharge, the degreeof penetration of the well, and the thickness of the skin zone. Furthermore, the efficiency of

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the partially penetrating well in the removal of the contaminated groundwater for differenttypes of the skin (negative or positive skin) is emphasized.

2 Analytical Solution of the Drawdown

The schematic diagram of the problem is shown in Fig. 1a. This figure shows a partiallypenetrating well with an infinitesimal radius, screened in arbitrary depths, pumping in aconfined, anisotropic, and two-zone aquifer, with a finite thickness and large radial extent.Pumpage is performed under constant discharge Q, where steady state has been reached. Itis further assumed that a uniform regional groundwater flow is in the negative x-axis direc-tion with a Darcy velocity of U . It should be noted, since there is no flow towards the well,through the unscreened portion of the well, and the column of original soil under the wellarea (Javandel 1982; Perina and Lee 2006), the hydraulic conductivity of these portions haveno effect on the head distribution around the well, and thus, can be considered as arbitraryvalues. In this case to simplify our solution we assumed that the hydraulic conductivitiesof the two portions are equal to hydraulic conductivity of the screen portion. By using thisassumption for a given size of screen portion, there would be no deference between the caseof partially penetrating well (Fig. 1a) and the fully penetrating one (Fig. 1b).

Because of the partially penetrating well in the aquifer, a three-dimensional flow is induced.Therefore, the governing equation will be a single well equation that may be written as (Toddand Mays 2005):

Kr1

(∂2 H1

∂r2 + 1

r

∂ H1

∂r

)+ Kz1

∂2 H1

∂z2 = 0; 0 < r ≤ rs (1)

Kr2

(∂2 H2

∂r2 + 1

r

∂ H2

∂r

)+ Kz2

∂2 H2

∂z2 = 0; rs ≤ r ≤ R (2)

where Kr and Kz are the hydraulic conductivity in the horizontal and vertical plane, respec-tively, H is drawdown, r and z are radial and vertical distances, R is the radius of the influenceof the well and rs is the skin radius. The subscripts 1 and 2 are denoted for the screened portionof the skin zone and the aquifer, respectively. Related boundary conditions are expressed as:

H2(R, z) = 0 (3)

H1(rs, z) = H2(rs, z) (4)∂ H1(rs, z)

∂r= Kr2

Kr1

∂ H2(rs, z)

∂r(5)

∂ H1(r, 0)

∂z= ∂ H2(r, 0)

∂z= 0 (6)

∂ H1(r, L)

∂z= ∂ H2(r, L)

∂z= 0 (7)

In which L is the thickness of the confined aquifer. The boundary conditions (4) and (5)impose the continuity of drawdown and flux between the skin zone and the aquifer. Theno-flow boundary conditions, at the bottom and upper impervious boundaries of a confinedaquifer, are expressed by Eqs. 6 and 7. The condition of the constant discharge in a partially

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Fig. 1 a Schematic diagram, showing a partially penetrating well in a confined aquifer with a skin that isscreened from arbitrary depths. b Schematic diagram, showing a fully penetrating well in a confined aquiferwith a skin that is screened from arbitrary depths

penetrating well can be written as:

lim(

r ∂ H1∂r

)= − Q

2π Kr1 (B2−B1); B1 < z < B2

r → 0(8)

lim(

r ∂ H1∂r

)= 0; 0 < z < B1, B2 < z < L

r → 0(9)

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where B1 and B2 are the upper and lower limits of the screened portion. It is obvious, that theassumption of a uniform radial gradient along the well screen has been applied in derivingEqs. 8 and 9. Applying FFCT to Eqs. 1 and 2 and using the boundary conditions (6) and (7),related solutions in Fourier domain are obtained as (details of applying FFCT and solvingthe equations are presented in Appendix):

∂2 H̃1

∂r2 + 1

r

∂ H̃1

∂r− α1w

2n H̃1 = 0; 0 < r ≤ rs (10)

H̃1(r, n) = An I0(X1nr) + Bn K0(X1nr) (11)

∂2 H̃2

∂r2 + 1

r

∂ H̃2

∂r− α2w

2n H̃2 = 0; rs ≤ r ≤ R (12)

H̃2(r, n) = Dn I0(X2nr) + Cn K0(X2nr) (13)

where α1 = Kz1Kr1

, α2 = Kz2Kr2

, wn = nπL , X2

1n = α1w2n, X2

2n = α2w2n , and H̃(r, n) is the FFCT

of the drawdown, I0(u) and K0(u) are modified Bessel functions of the first and second kindof order zero, respectively. Using the boundary conditions (3), (4), (5), (8), and (9) in theFourier domain and following equations

lim [r X1n I1(X1nr)] = 0r → 0

(14)

lim [r X1n K1(X1nr)] = 1r → 0

(15)

We can obtain the coefficients An, Bn, Cn , and Dn as below:

An = rs (β1 Dn + β2Cn) (16)

Bn = rs (β3 Dn + β4Cn) (17)

Cn = − Q A(wn)

π Kr1 L (B2 − B1) rs× I0(X2n R)

β3 K0(X2n R) − β4 I0(X2n R)(18)

and

Dn = Q A(wn)

π Kr1 L (B2 − B1) rs× K0(X2n R)

β3 K0(X2n R) − β4 I0(X2n R)(19)

In which, A(wn) = sin(wn B2)−sin(wn B1)wn

(other dimensionless parameters have been definedin Table 1).

By applying the relations below

H̃1(r, 0) = lim

{Q A(wn)

π Kr1 L (B2 − B1)

×[

β1 K0(X2n R) − β2 I0(X2n R)

β3 K0(X2n R) − β4 I0(X2n R)I0(X1nr) + K0(X1nr)

]}

n → 0

= Q

π Kr1 L

[1

γln

(R

rs

)+ ln

(rs

r

)](20)

H̃2(r, 0) = lim

{Q A(wn)

π Kr1 Lrs (B2 − B1)× K0(X2n R)I0(X2nr) − I0(X2n R)K0(X2nr)

β3 K0(X2n R) − β4 I0(X2n R)

}

n → 0

= Q

π Kr1 L

1

γln

(R

r

)(21)

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Table 1 Dimensionlessexpressions

Symbol Illustration

Qd Q/(L2U )

xd x/L

yd y/L

zd z/L

ud u/U

vd v/U

wd w/U

α Kz/Kr

γ Kr2/Kr1

b1 B1/L

b2 B2/L

β1 X1n I0(X2nrs)K1(X1nrs) + γ X2n I1(X2nrs)K0(X1nrs)

β2 X1n K0(X2nrs)K1(X1nrs) − γ X2n K1(X2nrs)K0(X1nrs)

β3 X1n I0(X2nrs)I1(X1nrs) − γ X2n I1(X2nrs)I0(X1nrs)

β4 X1n K0(X2nrs)I1(X1nrs) + γ X2n K1(X2nrs)I0(X1nrs)

η2 wn K0(wnrs)K1(wnrs) − γwn K1(wnrs)K0(wnrs)

η4 wn K0(wnrs)I1(wnrs) + γwn K1(wnrs)I0(wnrsrs)

A(wn)sin(wn B2)−sin(wn B1)

wn

wn nπ/L

Xn√

αwn

lim [β3 K0(X2n R) − β4 I0(X2n R)] = − γrs

; lim β1 = 1rs

n → 0 n → 0(22)

And approximation of modified Bessel function of the second kind of order zero (seeAbramowitz and Stegun 1964)

K0(u) ∼ − ln(u

2

)(23)

Solutions for the drawdown in each of the zones will be obtained as:

H1(r, z) = Q

2π Kr1 L

{1

γln

(R

rs

)+ ln

(rs

r

)+ 2

B2 − B1

×∞∑

n=1

[β1 K0(X2n R) − β2 I0(X2n R)

β3 K0(X2n R) − β4 I0(X2n R)I0(X1nr)

+K0(X1nr)A(wn) cos(wnz)]} (24)

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and

H2(r, z) = Q

2π Kr1 L

{1

γln

(R

r

)+ 2

rs(B2 − B1)

×∞∑

n=1

[K0(X2n R)I0(X2nr) − I0(X2n R)K0(X2nr)

β3 K0(X2n R) − β4 I0(X2n R)

×A(wn) cos(wnz)]} (25)

It is observed that, if R/L � 1.5 then

K0(X2n R)

I0(X2n R)≈ 0 (26)

The solutions (24) and (25) are reduced to

H1(r, z) = Q

2π Kr1 L

{1

γln

(R

rs

)+ ln

(rs

r

)+ 2

B2 − B1

∞∑n=1

[β2

β4I0(X1nr) + K0(X1nr)

]

×A(wn) cos(wnz)} (27)

H2(r, z) = Q

2π Kr1 L

{1

γln

(R

r

)+ 2

rs(B2 − B1)

∞∑n=1

K0(X2nr)

β4A(wn) cos(wnz)

}(28)

3 Verification Versus Other Solutions

3.1 Faybishenko et al. Solution

In order to compare the solution of Faybishenko et al. (1995), we assumed a partially pene-trating well in an isotropic, homogeneous, and single-zone aquifer. These assumptions canbe written as:

α1 = α2 = 1; X1n = X2n = wn; γ = 1; B1 = 0; B2 = l; Kr1 = Kr2 = k (29)

β2 = β3 = 0; β1 = β4 = 1

rs(30)

Applying above equations, we have:

H1(r, z) = H2(r, z) = Q

2πkL

{ln

(R

r

)+

∞∑n=1

2 sin(wnl)

wnlK0(wnr) cos(wnz)

}(31)

3.2 Theim Solution

In another case, we consider a fully penetrating well. Theim solution can be reached bysubstituting l by L in (31) as follows:

∞∑n=1

2 sin(wn L)

wn LK0(wnr) cos(wnz) = 0 (31-a)

And thus:

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H1(r, z) = H2(r, z) = Q

2πkLln

(R

r

)(32)

Which is the Theim solution (Todd and Mays 2005).

4 Capture Surface

In order to delineate capture surface, one needs to obtain the location of the stagnation point.The analytical solution to find the location of the stagnation point and the semi analyticalmethod to delineate the capture surface are presented in Sects. 4.1 and 4.2, respectively.

4.1 Finding the Location of the Stagnation Point

Applying the concept of the velocity potential and superposition principle, an expression forthe combined flow field of regional flow and the flow due to pumping is developed for anisotropic aquifer. The solution can be generalized for an anisotropic aquifer (Fienen et al.2005). In this article, our concern is to develop a semi-analytical solution to delineate thecapture surface for a partially penetrating well installed in an isotropic aquifer, thus:

α1 = α2 = 1; X1n = X2n = wn; Kr1 = Kz1 = k1; Kr2 = Kz2 = k2, and B1 = 0.

For a single-zone aquifer one can write:

φ = φu + φw (33)

φu = −U x + Cu (34)

φw = k H + Cw (35)

where φu and φw are, respectively, the velocity potentials due to the regional flow and pump-ing well, H is drawdown, U is the Darcy velocity of uniform regional flow, and Cu and Cw

are constants. Substituting the solutions (27) and (28) into the combined velocity potentialequations gives:

φ1 = k1 H1 − U x + C1 (36)

φ2 = k2 H2 − U x + C2 (37)

where φ1 and φ2 are the combined velocity potentials of the skin and formation zone. Dis-charge components are defined as the partial derivatives of the velocity potential, which arerepresented by u, v, and w in x, y, and z directions, respectively. For convenience, follow-ing dimensionless parameters are defined: ud = u

U , vd = vU , wd = w

U , b1 = B1L , b2 =

B2L , xd = x

L , yd = yL , and zd = z

L . Therefore, the components of the skin zone are obtainedas:

u1 = ∂φ1

∂x= − Q

2π L

{x

x2 + y2 + 2x

(B2 − B1)√

x2 + y2

∞∑n=1

[−η2

η4I1(wnr) + K1(wnr)

]

A(wn)wn cos(wnz)} − U (38)

u1d = − Qd

2π

xd

x2d + y2

d

{1 + 2

(x2

d + y2d

)0.5

(b2 − b1)

∞∑n=1

[sin(nπb2) − sin(nπb1)] cos(nπ zd)

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[−η2

η4I1

(nπ

(x2

d + y2d

)0.5)

+ K1

(nπ

(x2

d + y2d

)0.5)]

− 1 (39)

v1 = ∂φ1

∂y= − Q

2π L

{y

x2 + y2 + 2y

(B2 − B1)√

x2 + y2

∞∑n=1

[−η2

η4I1(wnr) + K1(wnr)

]

× cos(wnz)A(wn)wn} (40)

v1d = − Qd

2π

yd

x2d + y2

d

{1 + 2

(x2

d + y2d

)0.5

(b2 − b1)

∞∑n=1

[sin(nπb2) − sin(nπb1)] cos(nπ zd)

[−η2

η4I1

(nπ

(x2

d + y2d

)0.5)

+ K1

(nπ

(x2

d + y2d

)0.5)]}

(41)

w1d = − Qd

π(b2 − b1)

{ ∞∑n=1

[sin(nπb2) − sin(nπb1)] sin(nπ zd)η2

η4I0

(nπ

(x2

d + y2d

)0.5)

+K0

(nπ

(x2

d + y2d

)0.5)

(42)

(η2 and η4 are defined in Table 1).The components of formation zone are:

u2 = ∂φ2

∂x= − Q

2π L

[x

x2+y2 + 2γ x

(B2−B1)rs√

x2+y2

∞∑n=1

A(wn)wn

η4K1(wnr) cos(wnz)

]−U

(43)

u2d = − Qd

2π

xd

x2d+y2

d

⎧⎪⎨⎪⎩1+

2γ(

x2d+y2

d

)0.5

rs(b2−b1)

∞∑n=1

1

η4[sin(nπb2)− sin(nπb1)] cos(nπ zd )

×K1

(nπ

(x2

d+y2d

)0.5)

−1

}(44)

v2 = ∂φ2

∂y= − Q

2π L

[y

x2+y2 + 2γ y

(B2−B1)rs√

x2+y2

∞∑n=1

A(wn)wn

η4K1(wnr) cos(wnz)

]

(45)

v2d = − Qd

2π

yd

x2d+y2

d

{1+2γ (x2

d+y2d )0.5

rs(b2−b1)

∞∑n=1

1

η4[sin(nπb2)− sin(nπb1)] cos(nπ zd ) (46)

×K1

(nπ

(x2

d+y2d

)0.5)}

w2d = − Qdγ

πrs(b2−b1)

⎧⎪⎪⎨⎪⎪⎩

∞∑n=1

[sin(nπb2)− sin(nπb1)] sin(nπ zd )

K0

(nπ

(x2

d+y2d

)0.5)

η4

⎫⎪⎪⎬⎪⎪⎭

(47)

The stagnation point is defined as a point where discharge components are zero. Consequently,by substituting Eqs. 38–47 one can obtain the equations for calculating the coordinates of

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the stagnation points. As a result, the two sets of equations are:

xd1 = − Qd

2π

{1 + 2

∣∣xd1

∣∣b2 − b1

∞∑n=1

[sin(nπb2) − sin(nπb1)]

×[−η2

η4I1

(nπ

∣∣xd1

∣∣) + K1(nπ

∣∣xd1

∣∣)]}(48)

yd1 = 0 (49)

zd1 = 0 (50)

and

xd2 = − Qd

2π

{1 + 2γ

∣∣xd2

∣∣rs(b2 − b1)

∞∑n=1

1

η4K1

(nπ

∣∣xd2

∣∣) [sin(nπb2) − sin(nπb1)]

}(51)

yd2 = 0 (52)

zd2 = 0 (53)

As can be seen from these equations, the coordinates of the stagnation points depend onγ, Qd , and the screen interval of the well. For different values of these parameters the stag-nation point might be in each point of the mentioned zones, therefore, to determine the exactlocation of the stagnation points, the method of trial and error is applied.

4.2 Delineating the Capture Surface

After determining the location of the stagnation point(s), the next step is to define the capturesurface. By definition, a streamline is a line in which the tangent at any point on that line isin the direction of the velocity vector at that point. According to represented definition, thedifferential equations that express the streamline can be written as:

dx

u= dy

v= dz

w(54)

Equation 54 is decomposed into two equations; that each of them can be applied to delineatethe horizontal and vertical extent of the capture surface. Since we have a two-zone aquifer,we will have two sets of equations: for the skin zone and for the formation zone.

Skin zone equations:

dyd

dxd= vd1

ud1

= f1(xd , yd , zd) (55)

dzd

dxd= wd1

ud1

= g1(xd , yd , zd) (56)

f1(xd , yd , zd) = yd

xd

⎡⎣1 − 1

1 + Qd xd2π(x2

d+y2d )

(1 + P1)

⎤⎦ (57)

g1(xd , yd , zd) =Qd P2

π(b2−b1)

1 + Qd xd2π(x2

d+y2d )

(1 + P1)(58)

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P1 = 2(x2d+y2

d )0.5

b2−b1

∞∑n=1

[−η2

η4I1

(nπ(x2

d + y2d )0.5

)+ K1

(nπ(x2

d + y2d )0.5

)]cos(nπ zd)

× [sin(nπb2) − sin(nπb1)]

(59)

P2 =∞∑

n=1

[η2

η4I0

(nπ(x2

d + y2d )0.5

)+ K0

(nπ(x2

d + y2d )0.5

)][sin(nπb2) − sin(nπb1)]

× sin(nπ zd)

(60)

Formation zone equations:

dyd

dxd= vd2

ud2

= f2(xd , yd , zd) (61)

dzd

dxd= wd2

ud2

= g2(xd , yd , zd) (62)

f2(xd , yd , zd) = yd

xd

⎡⎣1 − 1

1 + Qd xd2π(x2

d+y2d )

(1 + P1)

⎤⎦ (63)

g2(xd , yd , zd) =Qdγ P2

πrs(b2−b1)

1 + Qd xd2π(x2

d+y2d )

(1 + P1)(64)

P1 = 2γ (x2d + y2

d )0.5

rs(b2 − b1)

∞∑n=1

1

η4[sin(nπb2) − sin(nπb1)] K1

(nπ(x2

d + y2d )0.5

)cos(nπ zd)

(65)

P2 =∞∑

n=1

1

η4[sin(nπb2) − sin(nπb1)] sin(nπ zd)K0(nπ(x2

d + y2d )0.5) (66)

Since Eqs. 55, 56, 61, and 62 are first order differential equations, a starting point is requiredto solve them simultaneously, therefore, we have used the points located on the capturesurface. If the selected starting points are closed enough to the stagnation point, the resultswill be streamlines that lie on the capture surface. The numerical Runge–Kutta procedure isapplied to solve the equations (Faybishenko et al. 1995).

The procedure applied to plot the capture zone can be summarized in three steps:

(1) Calculating stagnation points by using Eqs. 48, 51 and the trial and error method.(2) Identifying the maximum horizontal extent of the capture zone, by applying Eqs. 55

or 61.(3) Identifying the maximum vertical extent of the capture zone using either of Eqs. 56

or 62.

To plot the capture zone curves, the plotting area is divided into two parts:

(1) The area between the boundaries of the skin zone,(2) The area outside the boundaries of the skin zone.

It should be noted that, in plotting the capture zone curves, Eqs. 55 and 56 are used in area (1)and as the same Eqs. 61 and 62 in area (2).

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Table 2 Coordinates of the stagnation points along the xd axis for negative and positive skin. (α1 = 1, α2 =1, rs = 1 m, L = 30 m)

ld Qd = 1 Qd = 2 Qd = 6 Qd = 10

γ = 0.1 γ = 10 γ = 0.1 γ = 10 γ = 0.1 γ = 10 γ = 0.1 γ = 10

0.01 2567 0.4656 −0.4158 −0.6497 −1.0004 −1.1675 −1.6049 −1.6736

0.1 −0.2506 −0.4601 −0.4125 −0.6451 −0.9998 −1.1651 −1.6048 −1.6725

0.3 −0.2168 −0.4147 −0.3906 −0.6104 −0.9938 −1.1461 −1.603 −1.6637

0.5 −0.189 −0.3363 −0.3629 −0.5438 −0.9837 −1.1085 −1.6002 −1.6475

0.7 −0.1727 −0.2525 −0.3405 −0.4552 −0.9715 −1.055 −1.5966 −1.6259

1 −0.1592 −0.1592 −0.3183 −0.3183 −0.9549 −0.9549 −1.5915 −1.5915

5 Discussion of the Results

In this section the effects of the skin on drawdown, location of the stagnation point(s), max-imum horizontal and vertical extent of the capture surface and the efficiency of the pumpingwell are discussed. Moreover the sensitivity analysis is done considering the variation in thedischarge, the degree of penetration of the well and the thickness of the skin zone.

5.1 Drawdown

The drawdown that is induced because of the partially penetrating well under the assump-tions, which have mentioned before, is calculated from Eqs. 27 and 28. Figure 2a, b, and cshows the plot of the drawdown versus the radial distance by considering the first condi-tions: rs = 3 m, R = 500 m, B1 = 20 m, B2 = 25 m, z = 22 m, Q = 5 × 10−4 m3/sand Kr2 = 6 × 10−5 m/s for negative (γ = 0.1), without (γ = 1), and positive (γ = 10)

skin, respectively. The graphs show that the drawdown decreases with the increase of radialdistance. In the case of no skin, the drawdown gradually decreases as the radial distanceincreases. Under negative and positive skin conditions, according to the contrast of hydraulicconductivity, the slopes of the curves are different at the interface of the skin and aquifer. Inthe case of the negative skin, the slope of the curve is smaller than the one for the undisturbedzone and this is because of the larger hydraulic conductivity of the skin. Furthermore, thesmaller hydraulic conductivity in the positive skin case causes larger slopes in the skin zone.It is obvious from the results that anisotropy induces a larger hydraulic gradient in each ofthe cases. These results are compatible with the results of Yang and Yeh (2005). It should bementioned that although the solution by Yang and Yeh (2005) is for an infinite aquifer, thesolution presented in this article is for a finite one.

5.2 Locating the Stagnation Points

Coordinates of the stagnation points are determined by solving either of the two sets ofEqs. 48 or 51. This is the preliminary step for determining the horizontal and vertical extentof the capture zone. As well as the efficiency of the pump and treat system. For differentvalues of Qd , well screen interval, and negative or positive skin conditions (Table 2). It isobserved from the results that the stagnation points, move downstream as the positive skinis considered, and moves towards the well in the case of the negative skin.

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Fig. 2 a Plot of drawdown versus radial distance in a two-zone aquifer (negative skin, γ = 0.1, rs = 3,

L = 30m) for isotropic (α1 = α2 = 1) case. b Plot of drawdown versus radial distance (without skin,γ = 1, rs = 3, L = 30 m) for isotropic case. c Plot of drawdown versus radial distance in a two-zone aquifer(positive skin, γ = 10, rs = 3, L = 30 m) for isotropic case

123


Table 3 Comparing the results of our solution to results of Faybishenko et al. (1995) for rs = 1 m andL = 30 m

ld Qd = 1 Qd = 2

γ = 0.95 γ = 1 γ = 1.05 Faybishenko γ = 0.95 γ = 1 γ = 1.05 Faybishenko

0.01 −0.3997 −0.4028 −0.4053 −0.403 −0.5754 −0.5785 −0.5821 −0.579

0.1 −0.3933 −0.3964 −0.3992 −0.396 −0.5711 −0.5743 −0.5778 −0.574

0.3 −0.3477 −0.3508 −0.3536 −0.351 −0.5374 −0.5411 −0.5442 −0.541

0.5 −0.279 −0.2815 −0.2838 −0.281 −0.4784 −0.4813 −0.4841 −0.481

0.7 −0.2187 −0.2201 −0.2213 −0.220 −0.4091 −0.4110 −0.4128 −0.411

1 −0.1592 −0.1592 −0.1592 −0.159 −0.3183 −0.3183 −0.3183 −0.318

Qd = 6 Qd = 10

γ = 0.95 γ = 1 γ = 1.05 Faybishenko γ = 0.95 γ = 1 γ = 1.05 Faybishenko

0.01 −1.1087 −1.111 −1.1136 −1.111 −1.6458 −1.6470 −1.648 −1.647

0.1 −1.1065 −1.1092 −1.1117 −0.109 −1.645 −1.6462 −1.6472 −1.646

0.3 −1.0911 −1.0935 −1.0958 −1.093 −1.6389 −1.6400 −1.6409 −1.640

0.5 −1.0616 −1.0637 −1.0656 −1.064 −1.6277 −1.6285 −1.6292 −1.628

0.7 −1.0217 −1.023 −1.0244 −1.023 −1.6134 −1.6139 −1.6144 −1.614

1 −0.9549 −0.9549 −0.9549 −0.955 −1.5915 −1.5915 −1.5915 −1.592

To compare the results with Faybishenko et al. (1995)’s results we assumed that the wellis screened from the top to depth B2. Also, the concept of the dimensionless degree of pen-etration of the well is introduced as ld = B2/L . In Table 3, the results of this solution arecompared to the results of the Faybishenko et al. (1995) for the case of γ = 0.95, γ = 1and γ = 1.05. It should be noted since Faybishenko et al. did not consider the skin effect(which equal to assume γ = 1 in new solution). This range of γ has been chosen to showthe validity of the solution.

For a constant Qd , as the degree of penetration increases, the effect of skin on the resultsbecomes less important. Also, the effects of the skin on the results is reduced by increasingthe amount of Qd . It can be concluded that the skin effect is more considerable when thecondition of the three-dimensional flow is dominant and the well partially penetrates theaquifer. As a result, in the case of the two-dimensional condition, the skin effect might benegligible.

5.3 Maximum Horizontal Extent of the Capture Zone

Figure 3a–c shows the capture zone at the top of the aquifer (zd = 0) for different cases.Figure 3a and b shows the capture zone of the well with negative skin and positive skin(γ = 0.1 and γ = 10) for different values of the degree of penetration. The curves have beenplotted for Qd = 1 and rs = 3 m. As demonstrated in these figures, the maximum horizontalextent of the capture zone increases as the degree of the penetration of the well decreases.This result is compatible with the result of Faybishenko et al. (1995). Also, the comparisonbetween the capture zones of a single and two-zone aquifer has been done in Fig. 3c forld = 0.01, ld = 0.6 and γ = 0.1, γ = 1, and γ = 10. It is apparent that the differencebetween the curves of the single and two-zone aquifer becomes larger when the degree of

123


penetration of the well decreases. This difference is the result of the contrast of the hydraulicconductivity and gradient at the interface of the zones, consequently, increases the velocityof the flow field. It can be seen that the zone of containment of the partially penetrating wellbecomes larger in the case of the positive skin than the case of the negative skin zone. It canbe concluded that a larger hydraulic gradient at the interface of the two zones in the case ofthe positive skin causes more extension of the horizontal extent of the capture zone than thenegative skin.

The skin effect decreases as the dimensionless pumping rate exceeds 10 and as the degreeof the penetration of the well becomes larger. Therefore, if the skin effect is not considered,the calculated horizontal extent of the capture zone is not accurate in the case of the positiveskin and/or negative skin. In fact the contamination can be extracted with less pumpingdischarge in the case of the positive skin. The effects of the thickness of the skin zone arepresented in Fig. 4, where rs = 1 m, rs = 3 m, and rs = 5 m for Qd = 1, ld = 0.1). It isobserved that for a constant discharge and well penetration the effect of the skin is increasedby increasing the thickness of the skin zone. In practice where the skin thickness is less thanmentioned values, its effect cannot be a significant factor, therefore, the capture zone can beobtained by using the assumption of a single-zone aquifer. The variations of the geometry ofthe horizontal capture zone for different values of Qd are shown in Fig. 5. It is obvious thatthe horizontal capture zone is extended while the pumping rate is increased and the resultstend to the results of the capture zone of a single-zone aquifer.

5.4 Maximum Vertical Extent of the Capture Zone

Considering the constant discharge of the well, it is concluded that as the horizontal extent ofthe capture surface increases, the vertical extent of the containment zone decreases. There-fore, either of the discussions in Section 5.3 can be inversely applied for the maximum verticalextent. For example, it was concluded that the positive skin effect increases the maximumhorizontal extent of the capture zone; therefore, inversely the maximum vertical extent willbe decreased. Figure 6 shows that as the degree of the penetration increases, the verticalextent of the capture surface increases too. When the positive skin is neglected, its negativeimpact on the vertical extent of the capture zone is not taken into account; therefore, a portionof the contamination may not be captured by the pumping well. However, in the case of thenegative skin, the efficiency of the partially penetrating well is increased.

To have a better understanding of the capture zone, a 3-D graph of the capture zone isshown in Fig. 7 which is plotted using dimensionless parameters.

A half-circle with a radius of 0.05 m has been assumed around the actual stagnation point(−0.473, 0, 0). Each point gives one line on the capture surface of the zone and several pointson this half-circle are used to draw extra lines of the capture surface.

6 Conclusion

In this article, a new analytical solution for drawdown of a partially penetrating well con-sidering the skin effect has been developed. The three-dimensional discharge potential isobtained by superposing the discharge potential due to well pumping and regional flow. It isconcluded from the results that the maximum horizontal extent of the capture zone increasesas the degree of the penetration of the well decreases, while the maximum vertical extentdecreases. The positive skin effect increases the maximum horizontal extent and decreasesthe maximum vertical extent of the capture surface.

123


Fig. 3 a Maximum horizontal extent of capture zone for different degrees of well penetration (Qd = 1, γ =0.1, rs = 3, L = 30 m). b Maximum horizontal extent of capture zone for different degrees of well pen-etration (Qd = 1, γ = 10, rs = 3, L = 30 m). c Comparison between the capture zones of a single andtwo-zone aquifer, for ld = 0.01, ld = 0.6 and γ = 0.1, γ = 1 and γ = 10

Neglecting the positive skin effect has a negative influence on the efficiency of the pumpand treat systems, Because a portion of the plume may not be captured by the system. Byselecting a suitable combination of pumping rate(s) and degree of penetration for a specialthickness of skin zone, the required efficiency of the pumping well(s) can be obtained. In thecase where the skin thickness is small, its influence might be of no importance. The well’s

123


Fig. 4 Maximum horizontal extent of capture zone for different values of skin thickness for (Qd = 1 andγ = 10, L = 30 m)

Fig. 5 Maximum horizontal extent of capture zone for different values of well discharge (γ = 10, rs = 1and ld = 0.1, L = 30 m)

skin effect can be omitted while the pumping rate and degree of penetration of the well isincreased.

The presented analytical solution can be used for the purpose of designing optimumpumping systems and wellhead protection programs.

Appendix

The FFCT of H(r, z) and ∂2 H∂z2 with respect to the variable z is defined as

c {H(r, z) : z → n} = H̃(r, n) (A-1)

c

{∂2 H

∂z2

}= 2

L

[(−1)n ∂ H(r, 0)

∂z− ∂ H(r, L)

∂z

]− n2π2

L2 H̃(r, n) (A-2)

123


Fig. 6 Maximum vertical extent of capture zone for different degrees of well penetration (Qd = 1, γ =10, rs = 1, L = 30 m)

Fig. 7 3-D extent of the capture zone (Qd = 1, γ = 10, rs = 3, Ld = 0.1, L = 30 m)

According to boundary conditions (6) and (7), Eq. A-2 is reduced to

c

{∂2 H

∂z2

}= −w2

n H̃(r, n) (A-3)

Imposing boundary conditions (3), (4), and (5) to Eqs. 11 and 13 gives:

Cn = −DnI0(X2n R)

K0(X2n R)(A-4)

An I0(X1nrs) + Bn K0(X1nrs) = Cn K0(X2nrs) + Dn I0(X2nrs) (A-5)

X1n [An I1(X1nrs) − Bn K1(X1nrs)] = γ X2n [−Cn K1(X2nrs) + Dn I1(X2nrs)] (A-6)

Solving for An and Bn results in Eqs. 16 and 17. Substituting them in Eq. 11 gives:

H̃1 (r, n) = rs [(β1 Dn + β2Cn) I0(X1nr) + (β3 Dn + β4Cn) K0(X1nr)] (A-7)

123


Substituting Eqs. 8 and 9 in A-7 gives:

lim(

r ∂ H̃1∂r

)= − Q

2π Kr1 (B2−B1)× 2

L

B2∫B1

cos (wnz)dz

r → 0

(A-8)

lim {rs (β1 Dn + β2Cn) X1nr I1(X1nr) − rs (β3 Dn + β4Cn) X1nr K1(X1nr)}= − Q A(wn)

π Kr1 L (B2 − B1)r → 0

(A-9)

Applying Eqs. 14 and 15 simplifies A-9 to

β3 Dn + β4Cn = Q A(wn)

π Kr1 L (B2 − B1) rs(A-10)

Solving for Cn and Dn results in Eqs. 18 and 19. According to the definition of the inverseof the Fourier transform, the solutions of the H1 and H2 will be

H1(r, z) = H̃1(r, 0)

2+

∞∑n=1

H̃1(r, n) cos(wnz) (A-11)

H2(r, z) = H̃2(r, 0)

2+

∞∑n=1

H̃2(r, n) cos(wnz) (A-12)

Using Eqs. 20 and 21 the solutions for H1(r, z) and H2(r, z) are obtained.

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capture zone of a partially penetrating well with skin effects in

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