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Effect of wellbore storage and finite thickness skin on flow to apartially penetrating well in a phreatic aquifer

M. Pasandi a, N. Samani a,*, D.A. Barry b,1

a Department of Earth Sciences, College of Sciences, Shiraz University, Shiraz 71454, Iranb

Laboratoire de technologie ecologique, Institut des sciences et technologies de lenvironnement, Station No. 2, Ecole Polytechnique Federale

de Lausanne, CH-1015 Lausanne, Switzerland

Received 1 June 2007; received in revised form 17 September 2007; accepted 19 September 2007Available online 25 September 2007

Abstract

An analytical model is presented for the analysis of constant flux tests conducted in a phreatic aquifer having a partially penetratingwell with a finite thickness skin. The solution is derived in the Laplace transform domain for the drawdown in the pumping well, skin andformation regions. The time-domain solution in terms of the aquifer drawdown is then obtained from the numerical inversion of theLaplace transform and presented as dimensionless drawdowntime curves. The derived solution is used to investigate the effects ofthe hydraulic conductivity contrast between the skin and formation, in addition to wellbore storage, skin thickness, delayed yield, partialpenetration and distance to the observation well. The results of the developed solution were compared with those from an existing solu-tion for the case of an infinitesimally thin skin. The latter solution can never approximate that for the developed finite skin. Dimension-less drawdowntime curves were compared with the other published results for a confined aquifer. Positive skin effects are reflected in theearly time and disappear in the intermediate and late time aquifer responses. But in the case of negative skin this is reversed and thenegative skin also tends to disguise the wellbore storage effect. A thick negative skin lowers the overall drawdown in the aquifer and

leads to more persistent delayed drainage. Partial penetration increases the drawdown in the case of a positive skin; however its effectis masked by the negative skin. The influence of a negative skin is pronounced over a broad range of radial distances. At distant obser-vation points the influence of a positive skin is too small to be reflected in early and intermediate time pumping test data and conse-quently the type curve takes its asymptotic form. 2007 Elsevier Ltd. All rights reserved.

Keywords: Ground water; Analytical solution; Pumping test; Unsteady flow; Laplace transform

1. Introduction

Pumping tests remain the standard method to evaluatethe hydrodynamic properties of subsurface material forwater-supply purposes. Aquifer response data from apumping test can be employed to estimate subsurface prop-erties using schemes based on analytical or numerical solu-tions to the appropriate flow conditions. This methodologyis founded on a series of assumptions required to reduce

the complex natural system to a mathematically tractableone. The estimated hydraulic parameters can be erroneousif the configuration of the aquifer and the well is differentfrom the one assumed.

Traditionally, pumping test analysis has been limited toapplications involving aquifers whose properties areassumed uniform in space. For example, wellbore storageacts to delay the transmission of fluid from the formationto the well. By ignoring wellbore storage and using theline-source approximation to analyze the results of an aqui-fer test, values of Twill be underestimated and values of Swill be overestimated (symbol definitions are given inNomenclature).

0309-1708/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2007.09.001

* Corresponding author. Tel.: +98 711 228 2380; fax: +98 711 228 0926.E-mail addresses: [email protected] (M. Pasandi), [email protected]

(N. Samani), [email protected] (D.A. Barry).1 Tel.: +41 21 693 5576; fax: +41 21 693 5670.

www.elsevier.com/locate/advwatres

Available online at www.sciencedirect.com

Advances in Water Resources 31 (2008) 383398
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Wellbore damage or improvement also influences theresults of interwell hydraulic tests. The term skin effect isused to describe the damage or improvement to the regionsurrounding the well that is caused by drilling practices.Wellbore skin effects can be examined mathematically intwo ways: (1) by assuming the skin to be infinitesimallythin, and (2) by assuming it to be of finite thickness [8].The effect of both wellbore storage and the skin regionon the results of pumping tests has long been investigatedin the petroleum industry [10,24,28] and in groundwater

studies [5,13,1922,25,27]. These solutions account forwellbore storage and an infinitesimally thin skin in bothpumping and observation wells. The concept of an infini-tesimally thin skin is in certain circumstances realistic aswell as being a mathematically convenient means of linkingthe hydraulic head in the formation to the hydraulic headin the borehole. In such cases, the skin is assumed to haveno storativity. This assumption permits an estimate of theskin factor if the storage coefficient of the formation isknown. The effects of the infinitesimal skin tend to be indis-tinguishable from the effects of a decrease in the storagecoefficient in the analytical solution [14].

Clearly, a more versatile treatment of wellbore skineffects is achieved using composite analytical models inwhich the skin region, like the formation region, has a finitethickness, permeability and storativity. This type of com-posite model was used by numerous authors[2,4,6,11,12,14,17,18,2933] for the analysis of confinedaquifers.

Perina and Lee [23] have presented a general well func-tion for groundwater flow toward an extraction well par-tially penetrating a confined, leaky or unconfined aquifer.They have taken a non-uniform radial flux along the screenand a finite-thickness skin into account but they did notanalyze wellbore storage and finite-thickness skin effects

on the flow to these kinds of aquifers. Moreover, they

assumed that drainage from the zone above the water tableoccurs instantaneously in response to a decline in the eleva-tion of the water table [16]. Moench [15] presented aLaplace transform solution for the problem of flow to apartially penetrating well of finite diameter in a slightlycompressible phreatic aquifer. This solution allows theevaluation of both pumped well and observation piezome-ter data. It accounts for effects of wellbore storage and athin skin and also takes into account the non-instantaneousand instantaneous release of water from the unsaturated

zone. Because the skin region is usually of finite thickness,development of a model is necessary for the problem of aphreatic aquifer in which the medium is represented bytwo regions of radial flow each with individual hydrody-namic properties and thickness.

The following mathematical development is for the flowto a partially penetrating well in a phreatic aquifer wherethe influence of wellbore storage and a skin region of finitethickness are present. This model is similar to the solutionby Moench [15], with the exception of the finite thicknessskin. The solution is obtained in Laplace space andinverted numerically using the Stehfest method [26]. Thissolution can be used to investigate the effects of skin type,skin thickness, wellbore storage, partial penetration, non-instantaneous and instantaneous release of water fromthe unsaturated zone and distance to the observation pointon the drawdown distribution in a radial two-region phre-atic aquifer flow system. The major purpose of this paper isto demonstrate how this solution can be used to developinsight into pumping-induced flow in non-uniform aqui-fers. In other words, the emphasis is on the developmentof insight into how aquifer and well characteristics affectpumping-induced drawdown, so that field practitioners willhave a better idea of what to expect when working in nat-ural systems. A comparison to the case of infinitesimally

thin skin [15] is made and the influence of the finite thick-

Nomenclature

r radial distance from the axis of pumping well, Lrw radius of the well, Lrc inside radius of the pumped well over the range

of changing water levels during pumpage, Lrs radius of the skin, Lz vertical distance above the bottom, Lb initial saturated thickness of aquifer, Ld vertical distance from initial water table to top

of well screen, Ll vertical distance from initial water table to bot-

tom of well screen, Ls1 drawdown in the skin region, Ls2 drawdown in the formation region, Lsw drawdown in the pumping well, Lt time since start of pumping, TQ pumping rate, L3 T1

Kr1, Kr2 hydraulic conductivity in the horizontal direc-tion, L T1

Kz1, Kz2 hydraulic conductivity in the vertical direction,

L T

1

Sy1, Sy2 specific yieldSs1, Ss2 specific storage, L

1

Cs1 Kr1Ss1 ;Cs2 Kr2Ss2 Hydraulic diffusivity, L2 T1

a1, a2 empirical constant describing the drainage fromthe unsaturated zone, T1

P Laplace transform variableI0(), K0(), I1(), K1() Modified Bessel functionsc Eulers constant (0.5772)

Subscripts

1,2 skin zone, formation

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ness skin and its parameters on the results of pumping testswill also be investigated.

2. Analytical solution

The assumptions regarding the well and aquifer configu-

rations for a radial two-region phreatic aquifer system asshown in Fig. 1 are [7,15,16]: (1) the aquifer is homogeneousand anisotropic, of infinite lateral extent, and with uniformthickness horizontally. (2) Vertical flow across the lowerboundary of the aquifer is negligible. (3) The well dischargesat a constant rate from a specified zone below an initiallyhorizontal water table. (4) The head within the well doesnot vary spatially. (5) The radial flux from the aquifer tothe well does not vary along the length of the screened sec-tion [9]. (6) The vertical flux from the aquifer through thebase of the well is negligible. (7) The porous medium andfluid are slightly compressible and have constant physicalproperties. (8) The initial hydraulic head is spatially uni-form. (9) The finite radius well is partially penetrating (10)In addition to the instantaneous elastic response of aquifercompression and water expansion, gravity drainage of thepores is considered. (11) The drawdown is considered smallcompared with the saturated thickness so that boundaryconditions appropriate for an initially horizontal water tablecan be imposed. (12) It is also assumed that the vertical fluxof water into the aquifer from the unsaturatedzone occurs ina manner that varies exponentially with time in response to astep decline in hydraulic head at the water table.

Under these assumptions the governing equations forthe skin and formation regions are given in dimensional

format as, respectively,

o2s1

or2 1

r

os1

or KD1 o

2s1

oz2 1

Cs1

os1

ot; rw 6 r6 rs; 1

and

o2s2

or2 1

r

os2

or KD2 o

2s2

oz2 1

Cs2

os2

ot; rP rs; 2

where the subscripts 1 and 2, respectively, denote the skinand formation regions; s is drawdown [L]; r is radial dis-tance from the well centerline [L]; rw is well radius [L]; rsis radial distance from the well centerline to the outer skinboundary [L]; t is time since the start of pumping [T];KD1 Kz1Kr1, KD2

Kz2Kr2

, Cs1 Kr1Ss1 and Cs2 Kr2Ss2

where Kr andKz are the hydraulic conductivity in horizontal and verticalplane, respectively [L T1] and Ss is specific storage [L

1].The phreatic surface is initially horizontal, with zero

drawdown in both the skin and the formation,

s1r;z; 0 s2r;z; 0 0; 3

with the same condition for the drawdown in the well:swrw;z; 0 0: 4The head at infinity is fixed, so the boundary condition forthe formation zone is:

s21;z; t 0: 5The boundary condition representing the extension of thecasing is [15]:

os1

or

rrw

0; z< b l; z> b d; 6

where z, b, l and d are shown in Fig. 1. The continuity

equation at the well face is approximated as [7,15]:

Fig. 1. Schematic diagram of the well and the aquifer.

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2prwl dKr1 os1or

rrw

Cdswdt

Q; b l 6 z6 b d;

7where l d is the length of the screen, C is the wellborestorage coefficient [L2], which accounts for the wellbore

storage capacity, equal top

r

2

c for open wells (rc is the radiusof the standpipe in the interval where water levels arechanging; rc may be different to the radius of the well). Inthis case, C is the cross-sectional area of the free surfacein the well. The fluid compressibility is neglected in thewellbore accumulation terms because volume changes due

to liquid level dominate. The radial flow from the aquiferto the well at the sand face is assumed to be independentof vertical coordinate z and varies only with time [7,15].Q is the constant discharge rate [L3 T1], sw is the averagedrawdown in the wellbore. The no-flow boundary condi-

tions at the bottom impervious boundary are:

os1

oz

z0

os2oz

z0

0: 8

The conditions at the water table, which approximate therate of drainage per unit area from the unsaturated zonesof the two regions, are [15]:

Kz1os1

oz

zb

a1Sy1Zt

0

os1

ot0ea1tt

0 dt0; rw 6 r6 rs and

9

Kz2

os2

oz

zb

a2Sy2Z

t

0

os2

ot0ea2tt

0 dt0; rP rs: 10

The rate of exponential decline in the skin and formationregions is controlled by empirical relaxation coefficientsa1 and a2, respectively [1,15,22]. Sy1 and Sy2 are specificyields pertaining to the two regions.

In order to ensure continuity of flow between the skinregion and the undisturbed formation, auxiliary conditionsat the skin-formation boundary rs must also be met. Con-tinuity between the well and the skin region is given by:

swrw;z; t

s1

rw;z; t

:

11

The drawdown is continuous at the interface between theskin zone and the undisturbed formation:

s1rs;z; t s2rs;z; t; 12while there is also conservation of flux at this interface:

Kr1

os1

rs;z; t

or K

r2

os2

rs;z; t

or:

13

A solution to (1) and (2) with respect to the conditions (3)(13) can be found in the Laplace transform domain. Detailsof the solution are given in the Appendix. The results forthe dimensionless drawdowns in the Laplace domain fora partially penetrating well configuration are expressed as

where Pis the Laplace variable, sD1, sD2 and swD are the La-place-domain solutions for the skin region, formation re-gion and the pumping well, respectively (all dimensionlessparameters in above and proceeding equations are definedin Table 1) and

sD1 X1n0

f1K0qn1rDw1 I0qn1rDw2CDPKrRI0qn1w1 CDPK0qn1w22cosen1d2 d1en1dD lD qn1f2K1qn1w1 qn1I1qn1w2

; 14

sD2 X1

n0

q1K0

qn2rD

f1

K0

qn1rsD

I1

qn1rsD

I0

qn1rsD

K1

qn1rsD

CDPKrRI0qn1w1 CDPK0qn1w22cosen1d2 d1en1dD lD qn1f2K1qn1w1 qn1I1qn1w2 15and

swD X1n0

f1K0qn1rwDw1 I0qn1rwDw2CDPKrRI0qn1w1 CDPK0qn1w22cosen1d2 d1en1dD lD qn1f2K1qn1w1 qn1I1qn1w2

; 16

Table 1Dimensionless variables

rD r/rwrwD rw/brsD rs/rwzD

zb

dDdb

lDlb

sD14pbKr2s1

Q

sD24pbKr2s2

Q

swD4pbKr2sw

Q

tDtCs2r2w

KD1Kz1

Kr1

KD2Kz2Kr2

CsRCs2Cs1

KrR,KzRKr2Kr1

;Kz2Kz1

CDC

2pr2wldSs2

c1a1bSy1Kz1

c2a2bSy2Kz2

bw1 KD1r2wD

bw2 KD2r2wD

b1 bw1r2D

b2 bw2r2D

r1Ss1bSy1

r2Ss2bS

y2

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f1 4KrR cosen1zDfcosen1sinen1dD sinen1lD sinen1cosen1lD cosen1dDg=PdD lD; 17

f2 en1 sinen1 cosen1; 18w1 qn1K0qn2rsDI1qn1rsD KrRqn2K1qn2rsDI0qn1rsD;

19

w2 qn1K0qn2rsDK1qn1rsD KrRqn2K1qn2rsDK0qn1rsD;

20d1 sin2en1dD sin2en1lD=2 and 21

d2 sinen1 cos2en1dD cos2en1lD2

!

cosen1 sin2en1lD sin2en1dD2

!: 22

For a fully penetrating well the foregoing equations reduceto:

where

n1 qn1K1qn1 CDPKrRK0qn1; 26n2 qn1I1qn1 CDPKrRI0qn1; 27qn1 bw1e2n1 CsRP1=2 and 28qn2 bw2e2n2 P1=2 29

en1 and en2 are the roots of, respectively:

en1 tanen1 PCsRr1bw1 Pc1

and 30

en2 tanen2 Pr2bw2 Pc2

; 31

where K0, K1, I0 and I1 are modified Bessel functions.

3. Early time drawdown approximation

Drawdown in phreatic aquifers commonly shows athree-stage profile, i.e., an early, an intermediate and a latestage. The short time approximation of drawdown is brieflysummarized below.

The dimensionless time is inversely proportional to theLaplace transform parameter P and small t corresponds

to large values of P. For this en1 and en2 approach (n 1/2)p, qn1 and qn2 approach (CsRP)

1/2 and (P)1/2, respectively,and sin(en1) can be approximated by (1)n1. Then sD2given by (24) reduces to

sD1 X1n0

4KrR sinen1 cosen1zDK0qn1rDw1 I0qn1rDw2Pf2n2w2 n1w1

; 23

sD2 X1n0

4KrRqn1 sinen1 cosen1zDK0qn2rDK0qn1rsDI1qn1rsD I0qn1rsDK1qn1rsDPf2n2w2 n1w1

24

and

swD X1n0

4KrR sinen1 cosen1zDK0qn1rwDw1 I0qn1rwDw2Pf2n2w2 n1w1

; 25

sD2 4KrrR ffiffiffiffiffiffiffiffiffiffiCsRP

pK0 ffiffiffiP

prD K0 ffiffiffiffiffiffiffiffiffiffiCsRP

prsD I1 ffiffiffiffiffiffiffiffiffiffiCsRP

prsD

I0 ffiffiffiffiffiffiffiffiffiffiCsRPp

rsD K1 ffiffiffiffiffiffiffiffiffiffiCsRPp

rsD Pn2w2 n1w1 X

1

n0 1

n1 cos

n

0:5

pzD

n 0:5p ;32

where

n1 ffiffiffiffiffiffiffiffiffiffiCsRP

pK1

ffiffiffiffiffiffiffiffiffiffiCsRP

p CDPKrRK0


p ; 33

n2 ffiffiffiffiffiffiffiffiffiffiCsRP

pI1


p CDPKrRI0


p ; 34

w1 ffiffiffiffiffiffiffiffiffiffiCsRP

pK0

ffiffiffiP

prsD

I1


prsD

ffiffiffiP

pKrRK1

ffiffiffiP

prsD

I0


prsD

and 35

w2 ffiffiffiffiffiffiffiffiffiffiCsRP

pK0

ffiffiffiP

prsD

K1


prsD

ffiffiffiP

pKrRK0


prsD

K1

ffiffiffiP

prsD

: 36

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The summation term in (32) can be calculated by using thefollowing relation [3]:

X1n0

1n1 cosn 0:5pzDn 0:5p

12

0 6 zD 6 1

0 zD 1

&: 37

For a large argument z, I0(z)

!I1(z)

!ez/(2pz)1/2, and

K0(z) ! K1(z) ! (2pz)1/2ez with this approximation, sD2can be represented by the following formula

sD2 4

ffiffiffiffiffiffiffiCsR

pexp ffiffiffiPp rD rsD ffiffiffiffiffiffiffiCsRp rsD ffiffiffiffiffiffiffiCsRp

CDP2 ffiffiffiffiffirDp KrR ffiffiffiffiffiffiffiCsRp

for rsD > 1 38

sD2 2exp rD 1

ffiffiffiP

p CDP

2ffiffiffiffiffiffiffiffiffiffiffiffiffi2 rD

p for rsD 1 39

By performing Laplace inversion the following closed formequations are obtained:

sD2 4 ffiffiffiffiffiffiffiCsRp erfc 1

2ffitprD

rsD

ffiffiffiffiffiffiffiCsRp

rsD ffiffiffiffiffiffiffiCsRp h i

CDffiffiffiffiffirD

pKrR

ffiffiffiffiffiffiffiCsR

p for rsD > 1 40

sD2 2erfc rD1

2ffit

p

CDffiffiffiffiffiffiffiffiffiffiffiffiffi

2 rDp for rsD 1 41

4. Late time drawdown approximation

Large t corresponds to small values of P. In this limit,

e01 and e02 approach ffiffiffiffiffiffiffiffiffiffiffiffiffiPCsRr1bw1q and ffiffiffiffiffiffiffiffiffiPr2bw2q , q01 and q02approach P

CsRr1 CsRP

1=2

and Pr2

P

1=2, respectively,

and sin(e01) can be approximated by e01. For n > 0, en1and en2 approach np and

ffiffiffiffiffiffiffiffiffiP

r2bw2

qrespectively, qn1 and qn2

approach en1ffiffiffiffiffiffiffibw1

pand en2

ffiffiffiffiffiffiffibw2

p, respectively, and sin(en1)

and cos(en1) can be approximated by1nP

en1CsRr1bw1and (1)n,

respectively. Then sD2 given by (24) reduces to

where

n01 q01K1q01 CDPKrRK0q01; 43n02 q01I1q01 CDPKrRI0q01; 44w01 q01K0q02rsDI1q01rsD q02KrRK1q02rsDI0q01rsD;

45w02 q01K0q02rsDK1q01rsD q02KrRK0q01rsDK1q02rsD;

46

nn1 npffiffiffiffiffiffiffibw1

pK1np


p CDPKrRK0np


p; 47

nn2 npffiffiffiffiffiffiffibw1

pI1np


p CDPKrRI0np


pand 48

wn1 npffiffiffiffiffiffiffibw1

pK0np


prsDI1np


prsD

npffiffiffiffiffiffiffibw2

pKrRK1np


prsDI0np


prsD: 49

As P is small, the summation term in (38) is negligiblysmall relative to the first term and the late time drawdownsolution can appropriately be described with first term.

For a small argument z, I0(z) ! 1, I1(z) ! z/2,K0(z) ! ln(z), K1(z) ! 1/z and, after the establishmentof late time radial flow field, sD2 can be approximated bythe following formula

sD2 2 ln

ffiffiffiffiffiffiffiffiffiffiffiffiPr2

Pq

rD

P

: 50

By performing Laplace inversion the following closed formequation is obtained:

sD2 lnt c 2 lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 r2

r2rD

s" #51

or

sD2 ln 1:78tr21 r2r2D

!52

where c = Eulers constant (0.5772). The early time solu-tion (Eq. (40)) and the late time solution (Eq. (52)) arecompared graphically with the overall response of the phre-atic aquifer in the presence of wellbore storage and finitethickness skin (Eq. (24)) in Fig. 3. As can be observed,

for tD=r2D P 3 10

4

Eq. (47) approximates Eq. (24) andthe corresponding curves coincide with each other.

5. Results and discussion

To illustrate the effect of the finite thickness skin on theaquifer response, some typical type curves (dimensionlessdrawdown versus dimensionless time) are plotted using

realistic aquifer parameters following Moench [15] (rc =rw = 0.1 m, b = l= 25 m, d= 0 (full penetration), Kr2 =Kz2 = 10

4 m s1, Sy1 = Sy2 = 0.25, Ss1 = Ss2 = 2 106

m1, c ! 1). The influence of skin type, skin thickness,and the contrast of skin hydraulic conductivity relative tothe formation hydraulic conductivity, partial penetration,and observation point on expected field results are investi-

gated by generating the type curves in a manner similar to

sD2 2KrRP

q01K0q02rDK0q01rsDI1q01rsD I0q01rsDK1q01rsDn02w02

n01w01&

2Pffiffiffiffiffiffiffibw1

pCsRr1bw1

X1n1

1n cosnpzDK0qn2rDK0qn1rsDI1qn1rsD I0qn1rsDK1qn1rsDnpnn2wn2 nn1wn1

); 42

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Novakowski [17]. These type curves may be employed toevaluate the effects of the individual or combination ofparameters identical to the field situation.

For the purpose of validating the solution, theoreticalresponses produced for the case of an infinitesimally thinskin, due to Moench [15], are compared with the solution

developed here for the finite thickness skin case (Eq.(24)). The comparison is made between these two solutionsfor various values ofCD from 10

2 to 106, at rD = 100, whilethe skin region is thin (rsD = 1.1). Differences are not dis-cernable in the plotted curves (Fig. 2). This comparisonsuggests that the solution is correct and that the approxi-

mation for an infinitesimally thin skin can be used to ana-lyze pumping test results where the skin thickness is verysmall relative to the radial distance to the observation well.Fig. 3 demonstrates the early time (Eq. (32)) and the latetime (Eq. (47)) approximations of Eq. (24). Note that cor-responding curves coincide with each other.

We next investigate pumping tests taking into accountthe influence of a finite thickness skin and wellbore storageeffects. A set of type curves were produced with fixeddimensionless parameter values of KrR = KzR = 100,rD = 100, and rsD = 10 for five successively larger valuesof CD in the range 1010

5. This arrangement is equivalent

10-1

100

101

102

103

104

105

106

107

108

10-3

10-2

10-1

100

101

102

tD/r

D2

sD2

Finite Thickness Skin,CD

=102


=103


=104


=105


=106

Infinitesimally Thin Skin, CD

=102


=103


=104


=105


=106

Fig. 2. Comparison of the solution accounting for wellbore storage and infinitesimally thin skin to the solution developed here for the finite thickness skin.In this case rsD = 1.1, KrR and KzR = 100, rD = 100, with CD ranging from 10

2 to 106.

10-1

100

101

102

103

104

105

106

107

108

109

10-3

10-2

10-1

10

0

101

102

tD/r

D

2

sD2

CD

=103

CD=10

5

Early Time, CD

=103

Early Time, CD

=105

Late Time, CD

=103

Late Time, CD

=105

Fig. 3. Comparison of the derived general solution with the early and late time approximations.

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to a field setting in which an observation well is 10 m awayfrom a pumping well of 0.1 m radius whose skin region is 1-m thick and has a hydraulic conductivity two orders ofmagnitude less than that of the formation region. The gen-erated curves are compared with the type curves for aninfinitesimally thin skin with rsD = 1.1 or rs = 0.11 m

(Fig. 4). As can be seen, the skin effects are apparent forthe early time segment of the type curves. The effects ofincreasing skin thickness act to delay the early time draw-down at the observation well. Accurate early time data

are needed to characterize the finite thickness skin region.Further, contrary to the confined aquifer [17], the infinites-imally thin skin solution does not approximate the finitethickness solution even in the presence of large wellborestorage.

To examine the phreatic aquifer response to pumping

from wells with negative or positive skin, type curvesshown in Fig. 5 were generated for KrR = KzR ranging from104 to 104 at fixed values of CD = 10

4, rD = 100, andrsD = 10. KrR = KzR values of 10

4 to 1 specify positive skin

10-1

100

101

102

103

104

105

106

107

108

109

10-4

10-3

10-2

10-1

100

101

102

tD/r

D

2

sD2

Finite Thickness Skin, CD

=10


=102


=103


=104


=105

Infiniesimally Thin Skin , CD

=10


=102


=103


=104


=105

Fig. 4. Comparison of the solution accounting for wellbore storage and infinitesimally thin skin to the solution developed for the finite thickness skin (KrR,KzR = 100 for CD ranging from 10 to 10

5, rD = 100, rsD = 10).

10-2

10-1

100

101

102

103

104

105

106

107

10-4

10-3

10-2

10-1

100

101

tD/r

D

2

sD2

KrR

=10-3

KrR=10

-2

KrR

=10-1

KrR

=100

KrR

=101

KrR

=102

KrR

=103

KrR

=104

Fig. 5. Type curves developed for the finite thickness skin with various conductivity contrasts between skin and formation (CD = 104, rD = 100 and

rsD = 10, KrR and KzR range from 10

3 to 104).


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or low permeable skin region relative to the formationregion, while values less than 1 define the negative skinor high permeable skin region relative to the formationregion. In Fig. 5 two sets of curves are distinguished. Oneset representing the positive skin and another depictingthe negative skin. The effects of positive skin are reflectedin the early time segment of the type curves. When an aqui-fer has a positive skin and is subjected to pumping, thereplenishment from the formation is slower so that for suc-cessively larger values of KrR = KzR from 1 to 10

4 thecurves are shifted progressively one order of magnitudeto the right along the tD=r

2D axis. At the intermediate and

late time segment of curves, the positive skin effects disap-

pear as all curves coincide and follow the same trend along

a common asymptote. In the second set of curves the effectsof negative skin are hardly reflected on the early time seg-ment of curves and the position of type curves is ratherinsensitive to the negative skin. Similar behavior wasobserved for the case of a confined aquifer [17]. But in con-trast to the positive skin, the intermediate and late time seg-ments of the curves, although they follow similar trends,are sharply separated and are highly sensitive to the nega-tive skin. The higher the permeability contrast the higherthe separation will be. This behavior is because of the lar-ger permeability and faster replenishment from the forma-tion and also greater effect of delayed drainage.

To explore the phreatic aquifer response to pumping

from a well of negative skin of various thicknesses, the

10-2

10-1

100

101

102

103

104

105

106

107

108

10-3

10-2

10-1

100

101

102

tD/r

D

2

sD2

rsD

=1.1, CD

=0

rsD

=1.1, CD

=104

rsD

=10, CD

=0

rsD

=10, CD

=104

rsD

=30, CD=0

rsD

=30, CD

=104

Fig. 6a. Type curves developed for the finite negative skin of different thicknesses ( CD = 0 and 104, rD = 100 and, KrR, KzR = 103, rsD = 1.1, 10, 30).

10-1

100

101

102

103

104

105

106

107

108

10-1

100

101

102

tD/r

D

2

sD2

rsD

=1.1, CD

=0

rsD

=1.1, CD

=104

rsD

=10, CD

=0

rsD

=10, CD

=104

rsD

=30, CD

=0

rsD=30, C

D=10

4

Fig. 6b. Type curves developed for the finite positive skin of different thicknesses (CD = 0 and 104, rD equals 100 and, KrR, KzR 10

3, rsD of 1.1, 10, 30).


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dimensionless timedrawdown curves illustrated in Fig. 6awere produced for KrR = KzR = 10

3, CD = 0 and 104,

rD = 100 and rsD = 1.1, 10 and 30. While the increase inthe wellbore storage shifts the curves to the right less thanan order of magnitude as in Fig. 4 and similarly to the con-fined aquifer [17], the early time segment of the type curvesis quite insensitive to the thickness of the negative skin at afixed value ofCD. However, the results for the intermediateand late time segments show differences. At larger CD thenegative skin advances the rate of drawdown at a ratherdifferent slope than the case for the smaller CD and the dif-ference between the timedrawdown curves is more evi-

dent. The influence of gradual variation in rsD causes a

sharp variation in the intermediate responses. The thickerthe negative skin the lower the intermediate and late timedrawdown will be. Seemingly, the thicker the negative skinthe longer the delayed drainage will persist. This pointemphasizes the importance to well development operationsin the improvement of permeability contrast in the skinregion to enhance the well efficiency. The increase in slopeof curves at tD=r

2D > 10

7 is due to the low permeabilityaquifer region relative to high permeable skin, the formertending to behave as a barrier flow boundary.

Fig. 6b is plotted with the same parameter values asFig. 6a except that the skin is positive with KrR =

KzR = 103. As can be observed, only the early time segment

10-1

100

101

102

103

104

105

106

107

108

10-2

10-1

100

101

102

tD/rD

2

sD2

CD

=0, rD

=100

CD

=104, rD

=100

CD

=0, rD

=300

CD

=104, r

D=300

CD

=0, rD

=500

CD

=104, r

D=500

CD

=0, rD

=1000

CD

=104, r

D=1000

CD

=0, rD

=2000

CD

=104, r

D=2000

Fig. 7a. Type curves for the positive skin at different observation points (rD = 100, 300, 500, 1000 and 2000, rsD = 10, CD = 0 and 104, KrR, KzR = 103).

10-2

10-1

100

101

102

103

104

105

106

107

10-4

10-3

10-2

10-1

100

101

tD/r

D

2

sD2

CD

=0, rD

=100

CD

=104, r

D=100

CD

=0, rD

=300

CD=10

4

, rD=300

CD

=0, rD

=500

CD

=104, r

D=500

CD

=0, rD

=1000

CD

=104, r

D=1000

CD

=0, rD

=2000

CD

=104, r

D=2000

Fig. 7b. Type curves for the negative skin at different observation points (rD = 100, 300, 500, 1000 and 2000, rsD = 10, CD = 0 and 104, KrR, KzR = 10

3).


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of the drawdown curve is affected by the thickness of posi-tive skin and its role is to move the curves to the right. Thedisplacement is over two orders of magnitude for thedimensionless skin thickness of 1.130. Note also thatagain the wellbore storage moves the curves further tothe right with flatter slopes and tends to mask the interme-diate drawdown or the effect of delayed yield, particularlyfor the thicker skin. Moreover, for larger wellbore storagethe curves are less sensitive to the variations of skinthickness.

To test the influence of positive and negative finite thick-

nesses skin on the response of the aquifer at various obser-

vation points away from the pumped well, e.g., rD from 100to 2000, type curves for CD values of 0 and 10

5 andrsD = 10 are presented in Fig. 7. As for the confined aquifer[17] the influence of the positive finite thickness skin can beidentified in a fairly wide range of radial distances from thepumping well. All type curves move towards a commonasymptote at large tD=r

2D (Fig. 7a). However, the early

and intermediate drawdowns reduce at larger rD so thatthe effect of the finite skin is diminished. In other wordsbecause of extended cone of depression, at large rD thedelayed drainage takes place in an extended area of the for-

mation region and the rate of drawdown decreases. At

10-1

100

101

102

103

104

10-1

100

101

102

tD

sD2

lD

=0.2

lD

=0.4

lD

=0.6

lD

=0.8

lD

=1

Theis

Fig. 8a. The effect of partial penetration for a well of infinitesimal diameter without skin effect based on Neuman [16] (rD = 25, zD = 0, r1 = r2 = 102,

b1 = b2 = 103, dD = 0, KrR, KzR = 1, KD1 = KD2 = 0.1).

10-1

100

101

102

103

104

105

106

10-2

10-1

100

101

102

tD

s

D2

lD

=0.2, CD

=0

lD

=0.2, CD

=104

lD

=0.4, CD

=0

lD

=0.4, CD

=104

lD

=0.6, CD

=0

lD

=0.6, CD

=104

lD

=0.8, CD

=0

lD

=0.8, CD

=104

lD

=1, CD

=0

lD

=1, CD

=104

Theis

Fig. 8b. The effect of partial penetration for positive skin (rD = 25, rsD = 10, zD = 0, r1 = r2 = 102, b1 = b2 = 10

3, dD = 0, CD = 0 and 104,

KD1 = KD2 = 0.1, KrR, KzR = 103).


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rD = 2000 the early and intermediate drawdown becomesso small that the corresponding type curve reaches itsasymptotic behavior.

In the case of negative skin (Fig. 7b), the permeable skin(in contrast to aquifer permeability) results display the typ-ical behavior of a phreatic aquifer. The less permeable for-mation region acts the same as a barrier to flow andincreases the rate of dimensionless drawdown (fromtD=r

2D 102 to tD=r2D 105). The influence of the formation

region gradually gets more distinct at farther observationpoints, i.e., from rD = 100 to 2000. Also note that while

in Fig. 7a the effect of wellbore storage is well distin-

guished, in Fig. 7b it is almost masked by the negative skinas demonstrated already in Fig. 5.

Fig. 4 of Neuman [16] is regenerated by our solution(Fig. 8a). This figure illustrates the effects of partial pene-tration for a phreatic aquifer pumped by a well havingno wellbore storage and skin. It is observed that as the wellpenetrates nearer to the point of observation, an earlierresponse is observed and the drawdown is greater. Neuman[16] argues that as the depth of penetration (penetrationratio) decreases the importance of gravity effects increases.This increase is of course expected because gravity associ-

ated with the vertical component of the flow velocity vector

10-2

10-1

100

101

102

103

104

105

106

10-3

10-2

10-1

100

101

102

tD

sD2

lD

=0.2, CD

=0

lD

=0.2, CD

=104

lD

=0.4, CD

=0

lD

=0.4, CD

=104

lD

=0.6, CD

=0

lD

=0.6, CD

=104

lD

=0.8, CD

=0

lD

=0.8, CD

=104

lD

=1, CD

=0

lD

=1, CD

=104

Theis

Fig. 8c. The effect of partial penetration for negative skin (rD = 25, rsD = 10, zD = 0, r1 = r2 = 102, b1 = b2 = 10

3, dD = 0, CD = 0 and 104,

KD1 = KD2 = 0.1, KrR, KzR = 103).

10-1

100

101

102

103

104

105

106

107

108

10-2

10-1

100

101

102

tD/r

D

2

sD2

Delay Index=0.01, CD=0

Delay Index=0.01, CD

=104


=0


=104


=0


=104

Fig. 9a. The effect of delayed drainage for positive skin (rD = 100, rsD = 10, CD = 0 and 104, KrR, KzR = 10

3).


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becomes more important when the depth of penetration issmall.

To study the effects of partial penetration on theresponse of the phreatic aquifer pumped by a well of finitethickness skin, type curves shown in Figs. 8b and 8c areplotted for different values of penetration ratios (ld). Forthe purpose of comparison Figs. 8b and 8c are generatedfor some parameter values as Fig. 4 in [16], these arer1 = r2 = 10

2, b1 = b2 = 103, dD = 0, KD1 = KD2 = 0.1

and the observation point is located at rD = 25 andzD = 0. In these generations wellbore storage and posi-tive/negative skin effect are taken into account. In the caseof both positive and negative skins, also a decrease in thepenetration ratio causes lower drawdown in the aquifer.In comparison to Fig. 8a, Figs. 8b and 8c demonstratethe same results and indicate that with the presence ofeither positive or negative skin, the penetration ratio hasthe same effects on the response of phreatic aquifer whenCD = 0, i.e., its decrease delays and decreases the overalldrawdown. Fig. 8b further demonstrates that wellborestorage CD = 10

4 delays and decreases the early time draw-down (as expected). However, in contrast to the abovearguments, the drawdown decreases as the penetrationratio increases from 0.2 to 0.6 for the case of a positiveskin. In other words the gravity flow loses its role indecreasing the drawdown. For the penetration ratio greaterthan 0.6 the drawdown increases because the depth of pen-etration is close to the observation point. Note that theabove range of penetration ratio values for the negativeskin is 0.20.8 (Fig. 8c).

For the type curves presented so far, it was assumed thatthe release of water from the unsaturated zone is instanta-neous (delay index a ! 1). Figs. 9a and 9b are plotted tocompare the response of the phreatic aquifer pumped by a

well of finite thickness skin for various values ofa ranging

from 102 to 106 for the negative and positive skin,respectively. Higher values ofa both for negative and posi-tive skin reduce the early and intermediate drawdowns andthe persistence of the intermediate drawdown is increased.The effect for the negative skin is more pronounced. Againthe wellbore storage delays and reduces the drawdown asexpected.

6. Conclusions

A mathematical model is developed for describing theground water flow in phreatic aquifers with two regions ofradial flow in response to a constant rate pumping test.The model incorporates the effects of partial penetration,well storage, anisotropy, finite radius well skin and non-instantaneous release of water from the unsaturated zone.The solution is first derived using Laplace transforms.Time-domain solutions of dimensionless drawdowntime-time curves were then obtained from the numerical inversionof the Laplace transforms. This solution can be used either topredict the drawdown or to investigate the effects of the well-bore storage, skin type, skin thickness, and the permeabilitycontrast between skin and formation region on drawdowndistribution at the pumping well as well as any observationpoint in the skin and formation regions. The interpretationof the type curves leads to the following conclusions:

The infinitesimally thin skin solution developed byMoench [15] can approximate the developed finite thick-ness skin solution where the skin region is thin. Whenthe skin region is thick the infinitesimally thin skin solu-tion can never approximate the developed finite skinsolution even for large wellbore storage.

The positive skin effects are reflected in the early time

results and disappear in the intermediate and late time

10-2

10-1

100

101

102

103

104

105

106

107

108

10-3

10-2

10-1

100

101

102

tD/r

D

2

sD2


=0


=104


=0


=104


=0


=104

Fig. 9b. The effect of delayed drainage for negative skin (rD = 100, rsD = 10, CD = 0 and 104, KrR, KzR of 10

3).


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aquifer responses. The negative skin effects are howeverdisguised at the early time and manifested at the inter-mediate and late time aquifer responses. In general thehigher the permeability contrast between skin and for-mation the higher the deviation from the uniformsolution.

The thick negative skin lowers the overall drawdown inthe aquifer and causes the delayed drainage to persistlonger. This highlights the importance of well develop-ment operations. The timedrawdown curve of an aqui-fer pumped by a well of thick negative skin shows thetypical behavior of a phreatic aquifer affected by a bar-rier flow boundary. The thicker the skin the sharper theeffect of the boundary. In general the negative skin tendsto disguise the wellbore storage effect. The thick positiveskin delays the early time drawdown and shortens theintermediate time response.

The observation point distance from the pumping wellwith finite skin has a pronounced effect on the time

drawdown curves. For the negative skin the formationregion acts as a barrier flow boundary. At distant obser-vation points the effects of a positive skin are too smallto be reflected on the early and intermediate aquiferresponses and the type curve takes its asymptotic form.

Partial penetration decreases the drawdown in the caseof both positive and negative skin; however this effectmay be reversed in the presence of wellbore storagefor some particular range of penetration ratio.

An increase in delayed drainage for negative and posi-tive skin reduces the early and intermediate drawdownsand causes an increased persistence of intermediate

drawdown. This effect is more pronounced for the neg-ative skin.

Acknowledgement

Financial support provided by the Ministry of Science,Research and Technology of Iran, Shiraz University andEPFL, Switzerland for the first author are acknowledged.

Appendix. Analytical solution

The governing equations in dimensionless format are

o2

sD1or2D

1rD

osD1orD

bw1 o2

sD1oz2D

CsR osD1otD

; 1 6 rD 6 rsD;

A1o

2sD2

or2D 1

rD

osD2

orD bw2

o2sD2

oz2D osD2

otD; rD P rsD: A2

For the definition of the dimensionless parameters refer toTable 1. The associated dimensionless inner and outerboundary conditions are given by

sD1rD;zD; 0 sD2rD;zD; 0 0; A3swDrwD;zD; 0 0; A4

sD21;zD; tD 0; A5

osD1

orD

rD1

0; zD < 1 lD;zD > 1 dD; A6

osD1

orD

rD1

KrR CD dswDdtD

2lD dD

;

1 lD 6 zD 6 1 dD; A7osD1

ozD

zD0

osD2ozD

zD0

0; A8

osD1

ozD

zD1

c1ZtD

0

osD1

ot0DeCsRc1r1bw1tDt

0D

dt0D;

1 6 rD 6 rsD; A9osD2

ozD

zD1

c2ZtD

0

osD2

ot0Dec2r2bw2tDt

0D

dt0D;

rD P rsD; A10swD1;zD; tD sD11;zD; tD; A11sD

1r

sD;zD; tD

sD

2r

sD;zD; tD

and

A12

osD1rsD;zD; tD

orD KrR osD2rsD;zD; tD

orD: A13

The solution for the dimensionless drawdowns in both theskin and formation zones can be obtained by taking La-place transforms of Eqs. (A1)(A3).

o2sD1

or2D 1rD

osD1

orD bw1

o2sD1

oz2D CsRPsD1; 1 6 rD 6 rsD;

A14o

2sD2

or2D 1rD

osD2

orD bw2

o2sD2

oz2D PsD2; rD P rsD; A15

sD21;zD;P 0; A16osD1

orD

rD1

0; for zD < 1 lD;zD > 1 dD; A17dsD1drD

rD1

KrR CDPswD 2PlD dD

; 1 lD 6 zD 6 1 dD;

A18osD1

ozD

zD0

osD2ozD

zD0

0; A19osD1

ozD

zD1

PsD1CsRr1bw1 Pc1

; 1 6 rD 6 rsD; A20

osD2

ozDzD1

PsD2

r2bw2 Pc2 ; rD P rsD; A21

swD1;zD;P sD11;zD;P; A22sD1rsD;zD;P sD2rsD;zD;P and A23osD1rsD;zD;P

orD KrR osD2rsD;zD;P

orD: A24

Solutions to (A14) and (A15) that satisfy (A19)(A21) are

sD1 X1n0

fn1rD;P cosen1zD and A25

sD2

X1

n0

fn2

rD;P

cos

en2zD

;

A26

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where en1 and en2 are the roots of

en1 tanen1 PCsRr1bw1 Pc1

and en2 tanen2

Pr2bw2 Pc2

: A27

Substitution of (A25) and (A26) into (A14) and (A15)yields

X1n0

&o

2fn1rD;Por2D

1rD

ofn1rD;PorD

:

bw1e2n1 CsRPfn1rD;P'

cosen1zD 0 and A28X1n0

o

2fn2rD;Por2D

1rD

ofn2rD;PorD

:

bw2e2n2 Pfn2rD;P!cosen2zD 0: A29

Hence fn1 and fn2 must satisfy

o2fn1rD;P

or2D 1

rD

ofn1rD;PorD

bw1e2n1 CsRPfn1rD;P 0 and A30o

2fn2rD;Por2D

1rD

ofn2rD;PorD

bw2e2n2 Pfn2rD;P 0: A31The general solution of (A30) is

fn1rD;P gnPK0qn1rD wnPI0qn1rD A32and because of (A16), the general solution of (A30) is

fn2rD;P unPK0qn2rD: A33Substitution of (A32) and (A33) to (A25) and (A26) yields

sD1 X1n0

gnPK0qn1rD

wnPI0qn1rD cosen1zD and A34

sD2 X1n0

unPK0qn2rD cosen2zD; A35

where qn1 bw1e2n1 CsRP1=2, qn2 bw2e2n2 P1=2 and K0and I0 are the zero-order modified Bessel functions of thesecond and first kind, respectively, and gn, wn and un arecoefficients to be determined.

If the boundary condition (A23) is employed

gnPK0qn1rsD wnPI0qn1rsD cosen1zD unPK0qn2rsD cosen2zD: A36

Boundary condition (A24) gives

wnPqn1I1qn1rsD gnPqn1K1qn1rsD cosen1zD KrRunPqn2K1qn2rsD cosen2zD: A37

Substituting (A34) into (A18) and letting the substitution

ofswD based on (A22) yields

X1n0

wnPqn1I1qn1 gnPqn1K1qn1 cosen1zD

CDPKrRX1n0

wnPI0qn1 gnPK0qn1

cos

en1zD

2KrR

PlD dD : A38

Applying (A17) one obtains

X1n0

wnPqn1I1qn1 gnPqn1K1qn1 cosen1zD 0:

A39Multiplying (A38) by cos(em1zD) and integrating over theindicated interval, one obtains

X1

n

0

wnPqn1I1qn1gnPqn1K1qn1Z1dD

1

lD

cosen1zDcosem1zDdzD CDPKrRX1n0

wnPI0qn1

gnPK0qn1Z1dD

1lDcosen1zDcosem1zDdzD 2KrR

PlD dD :

A40Multiplying (A39) by cos(em1zD) and integrating over thebelow and above the screen, one obtainsX1n0

wnPqn1I1qn1 gnPqn1K1qn1

Z1lD0

cosen1zD cosem1zD 0 and A41X1n0

wnPqn1I1qn1 gnPqn1K1qn1

Z0

1dDcosen1zD cosem1zD 0: A42

Adding (A41) and (A42) to the left-hand side of (A40) andconsidering that all terms in the sum on the left-hand sideof the result are zero except those for which n = m, the setcos(en1zD) is orthogonal over the interval 0, 1 and so oneobtains

wnPqn1I1qn1 gnPqn1K1qn1Z

1

0

cos2en1zDdzD CDPKrRwnPI0qn1 gnPK0qn1

Z1dD

1lDcos2en1zDdzD 2KrR

PlD dD

Z1dD

1lDcosen1zDdzD: A43

The Eqs. (A36), (A37) and (A42) can be solved to deter-mine the constants gn, wn and un which are then substitutedinto (A34) and (A35) to obtain the solutions for sD1 and.

sD2, respectively.

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References

[1] Boulton NS. Analysis of data from non-equilibrium pumping testsallowing for delayed yield from storage. Proc Inst Civil Eng1963;26:46982.

[2] Butler JJ. Pumping tests in nonuniform aquifers-the radially sym-metric case. J Hydrol 1988;101:1530.

[3] Chen CS, Chang CC. Theoretical well hydraulics theory and dataanalysis for the constant head test in an unconfined aquifer with skineffect. Water Resour Res 2003;39(6):1121. doi:10.1029/2002WR001516.

[4] Chen CS, Chang CC. Theoretical evaluation of non-uniform skineffect on aquifer response under constant rate pumping. J Hydrol2006;317:190201.

[5] Chu WC, Garcia-Rivera J, Raghaven R. Analysis of interference testdata influenced by wellbore storage and skin at the flowing well. J PetTechnol 1980;32:2639.

[6] Cimen M. Type curves for unsteady flow to a large-diameter well inPatchy aquifer. J Hydraul Eng 2005;10(3):2004.

[7] Dougherty DE, Babu DK. Flow to a partially penetrating well in adouble-porosity reservoir. Water Resour Res 1984;20(8):111622.

[8] Earlougher Jr RC. Advances in well test analysis. Monogr. Ser, vol.

5. Dallas: Society of Petroleum Engineers; 1977.[9] Hantush MS. Hydraulics of wells. Advances in hydroscience, vol.

1. California, San Diego: Academic; 1964. p. 281432.[10] Hawkins Jr MF. A note on the skin effect. Trans Am Inst Min Metal

Pet Eng 1956;207:35667.[11] Karasaki K. Well test analysis in fractured media. PhD thesis,

University of California, Berkeley; 1986.[12] Markle JM, Rowe RK, Novakowski KS. A model for the constant-

head pumping test conducted in vertically fractured media. Int JNumer Anal Meth Geomech 1995;19:45773.

[13] Moench AF. Transient flow to a large-diameter well in an aquiferwith storative semiconfining layers. Water Resour Res 1985;21(8):112131.

[14] Moench AF, Hsieh PA. Analysis of slug test data in a well with finitethickness skin. In: Proceedings of the 17th international congress, Int

Assoc Hydrogeol, Tucson, Ariz; 1985.[15] Moench AF. Flow to a well of finite diameter in a homogeneous,

anisotropic phreatic aquifer. Water Resour Res 1997;33(6):1397407.[16] Neuman SP. Effects of partial penetration on flow in unconfined

aquifers considering delayed aquifer response. Water Resour Res1974;10(2):30312.

[17] Novakowski KS. A composite analytical model for analysis ofpumping tests affected by well bore storage and finite thickness skin.Water Resour Res 1989;25(9):193746.

[18] Novakowski KS. Interpretation of the transient flow rate obtainedfrom constant-head tests conducted in situ in clays. Can Geotech J1993;30:6006.

[19] Ogbe DO, Brigham WE. A model for interference testing withwellbore storage and skin effects at both wells. In: Proceedings of the59th annual technical conference, Soc Pet Eng, Houston, TX; 1984.

[20] Papadopoulos IS, Cooper Jr HH. Drawdown in a well of largediameter. Water Resour Res 1967;3(1):2414.

[21] Park E, Zhan H. Hydraulics of a finite-diameter horizontal well withwellbore storage and skin effect. Adv Water Resour 2002;25(4):389400.

[22] Park E, Zhan H. Hydraulics of horizontal wells in fractured shallowaquifer systems. J Hydrol 2003;281:14758.

[23] Perina T, Lee TC. General well function for pumping from aconfined, leaky, or unconfined aquifer. J Hydrol 2006;317:23960.

[24] Ramey Jr HJ, Agarwal RG. Annulus unloading rates and wellborestorage. Soc Pet Eng J 1972;24:45362.

[25] Sandal HM, Horne RN, Ramey HJ, Williamson JW. Interferencetesting with wellbore storage and skin effects at the produced well. In:Proceedings of the 53rd annual technical conference, Soc Pet Eng,Houston, TX; 1978.

[26] Stehfest H. Numerical inversion of Laplace transforms. CommunACM 1970;13(1):479.

[27] Tongpenyai Y, Raghaven R. The effect of wellbore storage and skinon interference test data. J Pet Technol 1981;33:15160.

[28] Van Everdingen AF, Hurst W. The application of the Laplacetransformation to flow problems in reservoirs. Trans Am Inst MinMetall Pet Eng 1949;186:30524.

[29] Wang JSY, Narasimhan TN, Tsang CF, Witherspoon PA, Transientflow in tight fractures. Report LBL-7027, Lawrence Berkeley Lab,Berkeley, CA; 1978. p. 10316.

[30] Yang SY, Yeh HD. Laplace-domain solutions for radial two-zoneflow equations under the conditions of constant-head and partiallywell. J Hydraul Eng 2005;131(3):20915.

[31] Yang SY, Yeh HD. Solution for flow across the wellbore in a two-zone confined aquifer. J Hydraul Eng 2002;128(2):17583.

[32] Yeh HD, Yang SY, Peng HY. A new closed-form solution for a radialtwo-layer drawdown equation for groundwater under constant-fluxpumping in a finite-radius well. Adv Water Resour 2003;26:74757.

[33] Yeh HD, Yang SY. A novel analytical solution for a slug testconducted in a well with a finite-thickness skin. Adv Water Resour2006;29(10):147989.

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