A novel analytical solution for a slug test conducted in a well with a finite-thickness skin

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A novel analytical solution for a slug test conducted

in a well with a ﬁnite-thickness skin

Hund-Der Yeh

a,*

, Shaw-Yang Yang

b

a_{Institute of Environmental Engineering, National Chiao Tung University, 75, Po-Ai Street, Hsinchu 30039, Taiwan} b_{Department of Civil Engineering, Vanung University, Chungli, Taiwan}

Received 17 August 2005; accepted 2 November 2005 Available online 5 January 2006

Abstract

An aquifer containing a skin zone is considered as a two-zone system. A mathematical model describing the head distribution is presented for a slug test performed in a two-zone confined aquifer system. A closed-form solution for the model is derived by Laplace transforms and Bromwich integral. This new solution is used to investigate the effects of skin type, skin thickness, and the contrast of skin transmissivity to formation transmissivity on the distributions of dimensionless hydraulic head. The results indi-cate that the effect of skin type is marked if the slug-test data is obtained from a radial two-zone aquifer system. The dimensionless well water level increases with the dimensionless positive skin thickness and decreases as the dimensionless negative skin thickness increases. In addition, the distribution of dimensionless well water level due to the slug test depends on the hydraulic properties of both the wellbore skin and formation zones.

Ó 2005 Published by Elsevier Ltd.

Keywords: Ground water; Slug test; Conﬁned aquifer; Skin eﬀect; Laplace transforms; Closed-form solution

1. Introduction

A slug test is one of the aquifer test methods com-monly used to investigate aquifer parameters. The test involves an instantaneous removal/injection of a small volume of water from/into a well[7]. An instantaneous head change is thus imposed within a well and the recov-ery/falloﬀ of water level is continuously measured using a pressure transducer that connects to a data logger. The aquifer parameters, e.g., transmissivity and storativity, can then be obtained if the slug-test data is analyzed.

Ferris and Knowles [12] originally introduced the analysis procedure from slug-test data. They derived an approximate solution for describing the water level change within the test well. The transmissivity is esti-mated based on a straight line, which represents residual

head versus inverse time. Employing an electrical analog model of the well-aquifer system, Bredehoeft et al. [4] demonstrated that Ferris and Knowles’ approximation is valid only for very late test time. Using the modified Thiem equation for the unconfined and steady state con-ditions, Bouwer and Rice [3]presented a procedure for determining the hydraulic conductivity or transmissivity for the unconfined aquifers. Using results from an elec-tric analog model, they obtained two empirical formulas related to the effective radius for the partially and fully penetrating wells. Later, Bouwer [2]provided informa-tion on using Bouwer and Rice’s method for testing the validity of falling level tests, the application of the method to the confined aquifers, the effect of well diam-eter, and the computer processing of field data. Cooper et al.[8] obtained a solution including the well storage analogous to a heat conduction problem provided by Carslaw and Jaeger [6]. Cooper et al.[8] applied their solution to a ground-water flow system and made a 0309-1708/$ - see front matter Ó 2005 Published by Elsevier Ltd.

doi:10.1016/j.advwatres.2005.11.002 *

Corresponding author. Fax: +886 3 5726050. E-mail address:[email protected](H.-D. Yeh).

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family of type curves. They used a matching approach for the slug-test data to estimate aquifer parameters. However, the aquifer parameters obtained by this tech-nique may be very rough because the shape of a type curve is rather insensitive to the value of aquifer storage

[19], especially, if the storage is very small. Kipp [17]

constructed a set of type curves that enables the well water level response data from the slug tests to be ana-lyzed if the inertial parameter is large. Pandit and Miner [23] provided an automatic fitting procedure to deter-mine the aquifer parameters when analyzing the slug-test data obtained from a confined aquifer. Marschall and Barczewski [20] presented an analysis of slug tests in the frequency domain for evaluating the solution of Cooper et al. [8]. Their solution is in terms of Kelvin functions [18], and the slug-test data is transformed using numerical Fourier transforms to determine aquifer parameters. Such an approach can avoid evaluating the integrand, which is an oscillatory function and difficult to evaluate.

Using the infinitesimally thin skin concept, Ramey and Agarwal [24] reported a solution to the drill-stem test (DST) problem with an inversion integral and related short- and long-time approximating forms. The skin effect describing the damage or improvement to the region surrounding a well is represented by a skin factor. Ramey et al. [25] presented semi- and double-log type curves, which combine the effects of the well storage and wellbore skin, to determine the formation permeability and skin effect by analyzing the slug-test data. Faust and Mercer[11]provided an infinite-aquifer solution for the response of slug tests to investigate the

effect of a finite-thickness skin. They assumed that the skin has a much lower permeability than that of the adjacent formation. Under this condition, the skin effect can lead to very low estimates of hydraulic conductivity if using the type-curve fitting method of Cooper et al.

[8]. Moench and Hsieh [21]commented on the

evalua-tion of slug tests in a finite-thickness skin by Faust and Mercer [11]. They showed that when the specific storage of the skin is negligibly small, the finite-thickness skin solution becomes equivalent to the infinitesimally thin skin solution. Under a finite-thickness skin condi-tion, the skin properties control the early time response, whereas the formation properties relate to the late time response. Further, Sageev[26]investigated the effects of the well storage and wellbore skin in a confined aquifer system. He obtained a similar result to that of Moench and Hsieh [21]. Karasaki et al. [16] developed various slug-tests models and related solutions for linear flow, radial flow with boundaries, two zone, and concentric composite aquifer systems. They provided type curves for each solution and noted that slug tests suffer the problem of non-uniqueness in matching the test data to type curves. Butler and Healey [5] investigated the estimate of hydraulic conductivity obtained through the pumping or slug test. They indicated that the hydraulic conductivity estimate from a pumping test is, on the average, larger than that from a series of slug test in the same formation.

An aquifer is considered as a radial two-zone (or composite) system if the formation properties near the well is apparently changed due to the well drilling or development. Well drilling causes the invasion of drilling Nomenclature

T Transmissivity

S Storativity

r Radial distance from the centerline of the well

rw Radius of the well

rc Radius of the standpipe

rs Radial distance from the well centerline to

the outer skin envelope t Time from the start of the test H Hydraulic head distribution

H0 Initial hydraulic head in the wellbore skin

and formation zones

H Hydraulic head in the Laplace domain

q2 pS/T

p Laplace variable

J0(Æ), Y0(Æ) Bessel functions of the ﬁrst and second

kinds of order zero

J1(Æ), Y1(Æ) Bessel functions of the ﬁrst and second

kinds of order one

I0(Æ), K0(Æ) Modiﬁed Bessel functions of the ﬁrst and

second kinds of order zero

I1(Æ), K1(Æ) Modiﬁed Bessel functions of the ﬁrst and

second kinds of order one j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT₁S₂=T₂S₁ f T2/T1 g S2/S1 a S2r2w=r2c b T2t=r2c q r/rw qc rc/rw qs rs/rw h H =H0 h H/H0 k2₁ fp/g k2₂ p Subscripts

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mud into the aquifer and may produce a positive well-bore skin that has lower permeability than that of the original formation. On the other hand, the extensive well development and/or substantial spalling and fracturing of the borehole wall may increase the permeability of the formation around a well. Under such circumstances, the disturbed formation is referred to as a negative well-bore skin. Moench and Hsieh[22]presented a Laplace-domain solution for the well response to a drill-stem test in the presence of skin. They found that the standard methods of analysis are adequate for open-well slug tests and may differ markedly in the pressure response for pressurized slug tests. Karasaki[15] used the time con-volution method of Duhamel’s theorem to evaluate the solution of Moench and Hsieh [22]. The systematized procedure and analysis method were proposed for a drill-stem test. Recently, Yang and Gates [31] structed a finite-element model for a slug test in a con-fined aquifer system considering the effect of a finite-thickness skin. They suggested that the effect of a wellbore skin on the estimate of hydraulic conductivity for low-permeability mediums could be minimized by the use of the late-time data.

A mathematical model describing the head distribu-tions in the skin and formation zones is presented for a slug test performed in a two-zone confined aquifer. The objective of this paper is to derive a new analytical solution in terms of hydraulic head for slug tests con-ducted in such a radial two-zone system. The solution is solved by applying Laplace transforms to the govern-ing equations and related boundary conditions of the model and the Bromwich integral [13] to the Laplace-domain solution. The solution in the time Laplace-domain is expressed in terms of an integral that covers a range from zero to infinity and has an integrand consisting of complicate products terms of Bessel functions. This newly derived solution is evaluated by numerical approaches and compared with that of Cooper et al.’s solution [8] for a uniform media and the results of numerical inversion from the Laplace-domain solution. The solution is employed to investigate the effects of skin type, skin thickness, and the contrast of skin trans-missivity to formation transtrans-missivity on the distribu-tions of dimensionless hydraulic head.

2. Mathematical derivations

2.1. Mathematical statement

Fig. 1displays the well and aquifer conﬁgurations for

a radial two-zone confined aquifer system. The assump-tions made for this aquifer system are: (1) the aquifer is homogeneous, isotropic, infinite-extent, and with a con-stant thickness, (2) the well is fully penetrating and with a finite radius, (3) the initial head is constant and

uni-form throughout the whole aquifer, and (4) the vertical ﬂow gradients are negligible. Under these assumptions, the governing equations for the head distributions in the skin and formation zones can respectively be written as o2H1 or2 þ 1 r oH1 or ¼ S1 T1 oH1 ot ; rw6r 6 rs ð1Þ and o2H2 or2 þ 1 r oH2 or ¼ S2 T2 oH2 ot ; rs6r <1 ð2Þ

where the subscripts 1 and 2, respectively, denote the skin and formation zones; H (or H(r, t)) is hydraulic head; r is radial distance from the well centerline; rwis

well radius; rsis radial distance from the well centerline

to the outer skin envelope; t is time from the start of test; S is storativity; and T is transmissivity.

The hydraulic heads are initially assumed to be zero in both the skin and formation zones, that is

H1ðr; 0Þ ¼ H2ðr; 0Þ ¼ 0; r > rw ð3Þ

The initial condition for hydraulic head in a well is

H1ðrw;0Þ ¼ H0 ð4Þ

where H0is the initial hydraulic head in aquifer. When

r = rw, the hydraulic head represents the well water level

if the well loss is negligible. The hydraulic head in the formation tends to zero as r approaches inﬁnity, that is

H2ð1; tÞ ¼ 0 ð5Þ

The conservation of mass at a well requires that pr2_c oH1 ot r¼rw ¼ 2prwT1 oH1 or r¼rw ð6Þ where rc is the standpipe radius. The hydraulic head is

continuous at the interface between the skin and forma-tion zones, i.e.,

Datum line T2 Impermeable layer Impermeable layer rw Formation zone Wellbore Skin zone r_c T1 Well screen rs H_i H0 Removal of water

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H1ðrs; tÞ ¼ H2ðrs; tÞ; t >0 ð7Þ

and the continuity of ﬂow rate between the skin and for-mation zones requires

T1 oH1ðrs; tÞ or ¼ T2 oH2ðrs; tÞ or ; t >0 ð8Þ 2.2. Closed-form solutions

The Laplace-domain solution for the hydraulic heads in both the skin and formation zones can be obtained by taking Laplace transforms of Eqs. (1)–(8). The results for H1 and H2 are respectively expressed as

H1¼ rwS2H0 T1q1 U1I0ðq1rÞ þ U2K0ðq1rÞ W1U1þ W2U2 ð9Þ and H2¼ rwS2H0 q₁rs K0ðq2rÞ W1U1þ W2U2 ð10Þ where q2 1¼ pS1=T1; q22¼ pS2=T2; g = S2/S1; a¼ S2r2w=r 2 c;

p is the Laplace variable[28]; I0(Æ) and K0(Æ) are

respec-tively the modiﬁed Bessel functions of the ﬁrst and sec-ond kinds of order zero; and I1(Æ) and K1(Æ) are

respectively the modiﬁed Bessel functions of the ﬁrst and second kinds of order one. Variables W1, W2, U1,

and U2are respectively deﬁned as

W1¼ gq1rwI0ðq1rwÞ þ 2aI1ðq1rwÞ ð11Þ W2¼ gq1rwK0ðq1rwÞ þ 2aK1ðq1rwÞ ð12Þ U1¼ T1q1 K1ðq1rsÞK0ðq2rsÞ þ ffiffiffiffiffiffiffiffiffiffi S2T2 S1T1 r K0ðq1rsÞK1ðq2rsÞ ð13Þ and U2¼ T1q1 I1ðq1rsÞK0ðq2rsÞ þ ffiffiffiffiffiffiffiffiffiffi S2T2 S1T1 r I0ðq1rsÞK1ðq2rsÞ ð14Þ When r = rw, the well water level in the Laplace domain,

Hw, obtained from Eq.(9) is

Hw¼ rwS2H0 T1q1 U1I0ðq1rwÞ þ U2K0ðq1rwÞ W1U1þ W2U2 ð15Þ The time-domain solutions of Eqs.(9) and (10)obtained using the Bromwich integral[13, p. 624]are respectively H1ðr; tÞ ¼ 2grwH0 p Z 1 0 eT 1S1u 2_t

A1ðr; uÞB1ðuÞ þ A2ðr; uÞB2ðuÞ B2₁ðuÞ þ B2 2ðuÞ du ð16Þ and H2ðr; tÞ ¼ 4grwH0 p2_r s Z 1 0 eT 1S1u2t

J0ðrjuÞB2ðuÞ Y0ðrjuÞB1ðuÞ B2 1ðuÞ þ B 2 2ðuÞ du u ð17Þ with and A1ðr; uÞ ¼ Y½ 1ðrsuÞY0ðrsjuÞJ0ðruÞ J1ðrsuÞY0ðrsjuÞY0ðruÞ

ffiffiffiffiffiffiffiffiffiffi S2T2

S1T1

r

Y0ðrsuÞY1ðrsjuÞJ0ðruÞ J0ðrsuÞY1ðrsjuÞY0ðruÞ

½ ð18Þ

A2ðr; uÞ ¼ J½ 1ðrsuÞJ0ðrsjuÞY0ðruÞ Y1ðrsuÞJ0ðrsjuÞJ0ðruÞ

S1T1

r

J0ðrsuÞJ1ðrsjuÞY0ðruÞ Y0ðrsuÞJ1ðrsjuÞJ0ðruÞ

½ ð19Þ

B1ðuÞ ¼ gðrwuÞ

J1ðrsuÞJ0ðrsjuÞY0ðrwuÞ Y1ðrsuÞJ0ðrsjuÞJ0ðrwuÞ

½ ffiffiffiffiffiffiffiS2T2 S1T1 q J0ðrsuÞJ1ðrsjuÞY0ðrwuÞ Y0ðrsuÞJ1ðrsjuÞJ0ðrwuÞ ½ ( ) 2a J1ðrsuÞJ0ðrsjuÞY1ðrwuÞ Y1ðrsuÞJ0ðrsjuÞJ1ðrwuÞ ½ ffiffiffiffiffiffiffiS2T2 S1T1 q

½

( )

ð20Þ

B2ðuÞ ¼ gðrwuÞ

J1ðrsuÞY0ðrsjuÞY0ðrwuÞ Y1ðrsuÞY0ðrsjuÞJ0ðrwuÞ

½

ffiffiffiffiffiffiffiS2T2

S1T1 q

½

( )

2a

½

ffiffiffiffiffiffiffiS2T2

S1T1 q

½

( )

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where u is a dummy variable and j¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT1S2=T2S1

p

. Note that J0(Æ) and Y0(Æ) are respectively the Bessel

func-tions of the ﬁrst and second kinds of order zero; and J1(Æ) and Y1(Æ) are respectively the Bessel functions of

the ﬁrst and second kinds of order one. Eqs.(16) and (17) are the closed-form solutions for hydraulic head distributions in the skin and formation zones, respec-tively. The detailed derivation of Eq. (16) is described

inAppendix A. Eq.(17)for hydraulic head distribution

in the formation can be obtained in a similar manner. From Eq.(16), the change of water level in the well, i.e., r = rw, is Hwðrw; tÞ ¼ 2grwH0 p Z 1 0 eT 1S1u2t

A1wðuÞB1ðuÞ þ A2wðuÞB2ðuÞ B2₁ðuÞ þ B2 2ðuÞ du ð22Þ with and 2.3. Dimensionless solutions

The dimensionless hydraulic head, h, is usually deﬁned as

h¼H0 H ðtÞ H0 Hi

ð25Þ where Hiis the well water level immediately after removal

or injection and H(t) is the well water level at time t. Dimensionless parameters are deﬁned as

b¼T2t r2 c ; f¼T2 T1 ; q¼ r rw ; qc¼ rc rw ; qs¼ rs rw ð26Þ where b is dimensionless time; f is a ratio of formation transmissivity to skin transmissivity; q is the dimension-less distance from the centerline of well; qcis the

dimen-sionless radius of standpipe; and qsis the dimensionless

radial distance from the centerline of well to the outer skin envelope. The positive skin is deﬁned for f > 1 and the negative skin for f < 1. Also, an aquifer is homogeneous and no skin exists when f = 1 and may be called a uniform aquifer.

The Laplace-domain solutions, Eqs. (9), (10), and (15), can respectively be expressed in dimensionless form as h1¼ f /1I0ðk1qÞ þ /2K0ðk1qÞ w1/1þ w2/2 ð27Þ h2¼ f q_s K0ðk2qÞ w₁/₁þ w2/2 ð28Þ and hw¼ f /1I0ðk1Þ þ /2K0ðk1Þ w1/1þ w2/2 ð29Þ where k2₁¼ fp=g; k22¼ p; and w₁¼ fpI0ðk1Þ þ 2ak1I1ðk1Þ ð30Þ w2¼ fpK0ðk1Þ þ 2ak1K1ðk1Þ ð31Þ /1¼ k1K1ðk1qsÞK0ðk2qsÞ þ fk2K0ðk1qsÞK1ðk2qsÞ ð32Þ and /2¼ k1I1ðk1qsÞK0ðk2qsÞ þ fk2I0ðk1qsÞK1ðk2qsÞ ð33Þ

Note that the Laplace-domain solutions of Eqs. (27)– (29)were also given in Moench and Hsieh[22].

Similarly, the time-domain solutions for head distri-butions in the skin and formation zones, Eqs. (16) and (17), can be respectively expressed in dimensionless form as h1ðq; bÞ ¼ 2g p Z 1 0 egbfasw 2 A1ðq; wÞB1ðwÞ þ A2ðq; wÞB2ðwÞ B2₁ðwÞ þ B2 2ðwÞ dw ð34Þ and h2ðq; bÞ ¼ 2g p Z 1 0 egbfasw2 2 pqs J0ðqjwÞB2ðwÞ Y0ðqjwÞB1ðwÞ B2₁ðwÞ þ B2 2ðwÞ dw w ð35Þ

A1wðrw; uÞ ¼ Y½ 1ðrsuÞY0ðrsjuÞJ0ðrwuÞ J1ðrsuÞY0ðrsjuÞY0ðrwuÞ

S1T1

r

Y0ðrsuÞY1ðrsjuÞJ0ðrwuÞ J0ðrsuÞY1ðrsjuÞY0ðrwuÞ

½ ð23Þ

A2wðrw; uÞ ¼ J½ 1ðrsuÞJ0ðrsjuÞY0ðrwuÞ Y1ðrsuÞJ0ðrsjuÞJ0ðrwuÞ

S1T1

r

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where w = rwu; and

and

In addition, the dimensionless water level at a well, Eq. (22), can also be expressed as

hwð1; bÞ ¼ 2g p Z 1 0 egbfaw 2 A1wðwÞB1ðwÞ þ A2wðwÞB2ðwÞ B2₁ðwÞ þ B2 2ðwÞ dw ð40Þ where

A1wð1;wÞ ¼ Y½ 1ðqswÞY0ðqsjwÞJ0ðwÞ J1ðqsuÞY0ðqsjwÞY0ðwÞ

pfgffiffiffiffiffi½Y0ðqswÞY1ðqsjwÞJ0ðwÞ J0ðqswÞY1ðqsjwÞY0ðwÞ

ð41Þ and A2wð1;wÞ ¼ J½ 1ðqswÞJ0ðqsjwÞY0ðwÞ Y1ðqswÞJ0ðqsjwÞJ0ðwÞ pfgffiffiffiffiffi½J0ðqswÞJ1ðqsjwÞY0ðwÞ Y0ðqswÞJ1ðqsjwÞJ0ðwÞ ð42Þ 3. Verification of solutions

The Laplace-domain solutions, Eqs.(9) and (10), and the time-domain solutions, Eqs.(34) and (35), for a two-zone aquifer system are compared with the existing solution for a uniform aquifer under the same well con-ﬁguration and geologic formation. The Laplace-domain solution for the hydraulic head in a uniform medium presented by Cooper et al.[8]is

H¼ H0rwSK0ðqrÞ Tq qr½ _wK0ðqrwÞ þ 2aK1ðqrwÞ

ð43Þ

It can be shown that both Eqs.(9) and (10)reduce to Eq. (43) if the hydraulic properties of the skin are equal to the hydraulic properties of the aquifer, i.e., f = g = 1. Similarly, Eqs.(34) and (35)reduce to

hðq;bÞ ¼2 p Z 1 0 ebaw2 J0ðqwÞ wY½ 0ðwÞ 2aY1ðwÞ Y0ðqwÞ wJ½ 0ðwÞ 2aJ1ðwÞ wJ0ðwÞ 2aJ1ðwÞ ½ 2þ wY½ 0ðwÞ 2aY1ðwÞ 2 dw ð44Þ

which indeed is the solution presented by Cooper et al. [8] for dimensionless hydraulic head distribution in a uniform medium.

The Laplace-domain solution of Eq. (29) and the time-domain solution of Eq. (40)for a well water level consist of products of Bessel functions. These functions are approximated by the formulas given in Abramowitz and Stegun [1]and Watson[29]and the function evalu-ations are accelerated using the Shanks method

[27,30,32]. The values of Bessel functions in Eqs. (29)

and (40) are computed at least to ten decimal places,

and thus have the same accuracy as those listed in Abra-mowitz and Stegun [1]. The work valuated the Bessel functions has been presented in Yang and Yeh [32]. The inverse Laplace transform of Eq. (29) is evaluated by the routine INLAP of IMSL [14] with accuracy to ﬁve decimal places. This routine was developed based on an algorithm originally proposed by Crump [9]and modiﬁed by de Hoog et al. [10]. This method approxi-mates the Laplace inversion by expressing the trans-formed function in a Fourier series.

Fig. 2illustrates the curves of the integrand in Eq.(40)

versus w for qc= 0.5, qs= 10, g = 1, b = 0.1, and

A1ðq; wÞ ¼ Y½ 1ðqswÞY0ðqsjwÞJ0ðqwÞ J1ðqsuÞY0ðqsjwÞY0ðqwÞ

pfgffiffiffiffiffi½Y0ðqswÞY1ðqsjwÞJ0ðqwÞ J0ðqswÞY1ðqsjwÞY0ðqwÞ ð36Þ

A2ðq; wÞ ¼ J½ 1ðqswÞJ0ðqsjwÞY0ðqwÞ Y1ðqswÞJ0ðqsjwÞJ0ðqwÞ pfgffiffiffiffiffi½J0ðqswÞJ1ðqsjwÞY0ðqwÞ Y0ðqswÞJ1ðqsjwÞJ0ðqwÞ ð37Þ B1ðwÞ ¼ gwð Þ J1ðqswÞJ0ðqsjwÞY0ðwÞ Y1ðqswÞJ0ðqsjwÞJ0ðwÞ ½ pfgffiffiffiffiffi½J0ðqswÞJ1ðqsjwÞY0ðwÞ Y0ðqswÞJ1ðqsjwÞJ0ðwÞ ( ) 2a ½J1ðqswÞJ0ðqsjuÞY1ðwÞ Y1ðqswÞJ0ðqsjuÞJ1ðwÞ pfgffiffiffiffiffi½J0ðqswÞJ1ðqsjwÞY1ðwÞ Y0ðqswÞJ1ðqsjwÞJ1ðwÞ ( ) ð38Þ

B2ðwÞ ¼ gwð Þ J½ 1ðqswÞY0ðqsjwÞY0ðwÞ Y1ðqswÞY0ðqsjwÞJ0ðwÞ

ffiffiffiffiffi fg p

J0ðqswÞY1ðqsjwÞY0ðwÞ Y0ðqswÞY1ðqsjwÞJ0ðwÞ

½

2a J½ 1ðqswÞY0ðqsjwÞY1ðwÞ Y1ðqswÞY0ðqsjwÞJ1ðwÞ

ffiffiffiffiffi fg p

J0ðqswÞY1ðqsjwÞY1ðwÞ Y0ðqswÞY1ðqsjwÞJ1ðwÞ

½

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a= 101 when f = 0.1, 1 or 10. The ﬁgure shows that these curves oscillate over some cycles and quickly die out if the skin presents (f 5 1). On the other hand, the curve only has a single peak for a uniform medium (f = 1). Thus, the closed-form solution for dimensionless water level at the well, Eq.(40), can be easily evaluated by using a numerical integration approach. The integration is carried out by the Gaussian quadrature for the ranges from zero to inﬁnite. The approach of numerical calcula-tions for Eq.(40)is similar to that of Yeh et al.[33].

Comparisons between the evaluated results of the closed-form solution of Eq. (40)and those obtained by a numerical inversion from Eq. (29) provide a cross check for the accuracy of both solutions. Under a uni-form medium condition (f = g = 1), Fig. 3 shows the

Symbol 10-2 10-1 100 101 102 103 Dimensionless time 0.00 0.20 0.40 0.60 0.80 1.00

Dimensionless well water level

Closed-form solution Numerical inversion solution Cooper et al.

= 10-1 = 10-5

α α

Fig. 3. Plots of dimensionless well water level versus dimensionless time (b) estimated by the closed-form solution, the numerical inversion from the Laplace-domain solution, and those given in Cooper et al.[8]

for qc= 0.5, qs= 10, and f = g = 1 when a = 101or 105.

Symbol 10-4 ₁₀-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 ₁₀4 Dimensionless time 0.00 0.20 0.40 0.60 0.80 1.00

10-1 10-2 10-3

10-4

10-5

Fig. 4. Plots of dimensionless well water level versus dimensionless time (b) estimated by the closed-form solution and the numerical inversion from the Laplace-domain solution for qc= 0.5, qs= 10, g= 1, and a = 101–105when f = 0.1. The line presents the closed-form solution and the circle presents the numerical inversion from the Laplace-domain solution. Symbol 10-4 ₁₀-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 ₁₀4 Dimensionless time 0.00 0.20 0.40 0.60 0.80 1.00

10-1 10-2 10-3 10-4 10-5 α

Fig. 5. Plots of dimensionless well water level versus dimensionless time (b) estimated by the closed-form solution and the numerical inversion from the Laplace-domain solution for qc= 0.5, qs= 10, g= 1, and a = 101–105 when f = 10. The line presents the closed-form solution and the circle presents the numerical inversion from the Laplace-domain solution. Symbol 0.0 0.4 0.8 1.2 1.6 2.0 w 0.0 4.0 8.0 12.0 16.0 Integrand in Eq. (40) 0.1 1 10

Fig. 2. A plot of the integrand in Eq. (40) versus w for qc= 0.5, qs= 10, g = 1, b = 0.1, and a = 0.1 when f = 0.1, 1 or 10.

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curves of dimensionless well water level versus dimen-sionless time (b) for qc= 0.5 and qs= 10 when

a= 101 or 105. The ﬁgure indicates that the results obtained by a numerical Laplace inversion agree extre-mely well with those of the closed-form solution and Cooper et al. [8]. Under a two-zone condition, the dimensionless well water level versus dimensionless time with qc= 0.5, qs= 10, g = 1, and a = 101–105 are

shown in Fig. 4 for f = 0.1 and in Fig. 5 for f = 10. The values of dimensionless well water level obtained by a numerical Laplace inversion are consistent with those of the closed-form solution to ﬁve decimal places. This indicates that the closed-form solution for a two-zone system yields correctly evaluated results when esti-mated by a numerical approach. Note that the modiﬁed Crump method fails to converge for the Laplace-domain

solution when the dimensionless time is very small (b < 0.001 for f = 0.1 and b < 0.01 for f = 10) as indi-cated inFigs. 4 and 5. In contrast, the numerical integra-tion approach applied to the time-domain soluintegra-tion works well for all range of dimensionless times.

4. Results and discussion

The curves of dimensionless well water level versus dimensionless time are developed to investigate the impacts of the skin type, skin thickness, and the contrast of skin transmissivity to formation transmissivity on dimensionless head distribution. For ease of compari-sons, g and qc are respectively chosen as one and 0.5.

In addition, all function evaluations for these solutions

1.E-4 1.E-3 1.E+1 1.E+0 1.E-1 1.E-2 = 0.1

Negative skin Formation

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Dimensionless distance 0.00 0.20 0.40 0.60 0.80 1.00

Dimensionless hydraulic head

1.E-4 1.E-3 1.E+1 1.E+0 1.E-1 1.E-2 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Dimensionless distance 0.00 0.20 0.40 0.60 0.80 1.00

No skin 1.E-4 1.E-3 1.E+1 1.E+0 1.E-1 1.E-2 = 10 Formation Positive skin 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Dimensionless distance 0.00 0.20 0.40 0.60 0.80 1.00

(a) (b)

(c)

β β

β

Fig. 6. Plots of dimensionless hydraulic head versus dimensionless time for qs= 5, a = 105, and b = 104to 10 when (a) f = 0.1, (b) f = 1, and (c) f= 10.

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are under double precision and accurate to ﬁve decimal places.

4.1. Eﬀect of skin type

Fig. 6displays the curves of dimensionless hydraulic

head versus dimensionless distance for qs= 5, a =

105, and b ranging from 104 to 10 when (a) f = 0.1, (b) f = 1, and (c) f = 10. For the case without a skin zone, the dimensionless hydraulic head gradually decreases when increasing radial distance as shown in Fig. 6b. If a ﬁnite-thickness skin the present, both Fig. 6a and c demonstrate that the relation of dimen-sionless hydraulic head versus dimendimen-sionless distance exhibits two curves with diﬀerent slope joined at the interface (qs= 5). A negative skin, which has a higher

transmissivity than the formation, has a curve with rel-ative mild slope in the skin zone and with steeper slope in the formation zone. In contrast, a positive skin has a very steep slope in the skin zone due to the lower trans-missivity and a relative ﬂat slope in the formation zone. The dimensionless hydraulic head at the well always decreases with increasing dimensionless time (b); on the other hand, the dimensionless hydraulic head in the formation zone increases at the beginning of the test, and then decreases at large-dimensionless time (say, b> 1 or 10). In addition, the dimensionless hydraulic head for an aquifer with a negative skin stabilizes more quickly than that with a positive skin.

4.2. Eﬀect of skin thickness

Fig. 7 displays two sets of curves to investigate the

inﬂuence of skin thickness (rs rw) on dimensionless well

water level for a = 105and qs= 5, 10, 50, and 100 when

f= 0.1 or 10. The ﬁgure indicates that the dimensionless skin thickness (i.e., qs 1) eﬀects the dimensionless well

water level at intermediate time, as b ranging from 0.01 to 10 for f = 0.1 and from 1 to 100 for f = 10. However, the dimensionless well water level diminishes to zero at large-dimensionless time. For a positive skin condition, the dimensionless well water level increases with dimen-sionless skin thickness. A larger dimendimen-sionless well water level reﬂects the eﬀect of a smaller hydraulic conductivity for a positive skin. In contrast, the dimensionless well water level decreases as dimensionless skin thickness increases if a negative skin is present.

4.3. Eﬀect of contrast of skin transmissivity to formation transmissivity

A plot of dimensionless well water level versus dimen-sionless time for qs= 10 and a = 105when f = 0.1, 0.5,

1, 5, and 10 is displayed inFig. 8. This figure shows the curves of dimensionless well water level for the system under the conditions with no skin (i.e., f = 1), negative skin (i.e., f = 0.1 or 0.5), and positive skin (i.e., f = 5 or 10). A smaller transmissivity of a positive skin (in contrast to aquifer transmissivity) produces a smaller flow rate from the well toward the formation and results in a higher dimensionless well water level. Therefore, a larger value of f has a higher dimensionless well water level. On the other hand, a larger transmissivity of a neg-ative skin (in contrast to aquifer transmissivity) yields a larger flow rate across the wellbore and results in a lower dimensionless well water level. Thus, a smaller value of f results in a lower dimensionless well water level. The

1.0E-4 1.0E-3 1.0E-2 1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4

Dimensionless time (β) 0.00 0.20 0.40 0.60 0.80 1.00

5 10 50 100 Symbol = 0.1 = 10 s ζ ζ

Fig. 7. A plot of dimensionless well water level versus dimensionless time for a = 105and qs= 5, 10, 50, and 100 when f = 0.1 or 10.

1.0E-2 1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 Dimensionless time (_β) 0.00 0.20 0.40 0.60 0.80 1.00

Symbol 0.1 ζ 0.5 1 5 10

Fig. 8. A plot of dimensionless well water level versus dimensionless time for qs= 10 and a = 105when f = 0.1, 0.5, 1, 5, and 10.

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figure demonstrates that the system with a positive skin has the highest dimensionless well water levels, the sys-tem without the skin give the second highest, and the one with a negative skin yield the lowest at the same dimensionless time. Fig. 8 also indicates that the differences of dimensionless well water level between the two-zone and uniform medium systems are negligi-bly small at small- and large-dimensionless times (say, b< 101 and b > 102). Contrarily, the differences of dimensionless well water level for the system under dif-ferent skin condition are quite large at intermediate-dimensionless time.

5. Conclusions

A new analytical solution (in the time domain) has been developed for slug tests in a well, which is installed in a radial confined aquifer system with a finite-thickness skin. This solution is derived from a radial two-zone ground-water flow equation using Laplace transforms and the Bromwich integral. In a uniform med-ium condition, the dimensionless well water levels com-puted from the closed-form solution agree very well with these of the Laplace-domain solution and Cooper et al. [8]. We have shown that numerical inversion fails to evaluate the Laplace-domain solution if the dimension-less time is very small. On the other hand, the present ana-lytical solution can be evaluated numerically for the entire time domain with accuracy to the fifth decimal place.

Under a radial two-zone condition, i.e., with a posi-tive or negaposi-tive skin, the dimensionless well water levels computed from the closed-form solution match with those of the Laplace-domain solution to at least five dec-imal places. This provides a double check to make sure that the closed-form and Laplace-domain solutions are correctly evaluated. The distributions of dimensionless hydraulic head in a uniform medium are significantly different from these in a two-zone aquifer with a positive or negative skin. The relation of dimensionless hydraulic head versus dimensionless distance exhibits two curves with different slope joined at the interface. The dimen-sionless hydraulic head of a negative skin more quickly stabilizes than that of a positive skin. Obviously, the magnitude of dimensionless hydraulic head strongly depends on the hydraulic properties of both the skin and formation zones.

The dimensionless skin thickness affects the dimen-sionless well water level at intermediate-dimendimen-sionless time. The dimensionless well water level increases with dimensionless positive skin thickness and decreases as dimensionless negative skin thickness increases. The dis-tributions of dimensionless well water level in a two-zone aquifer system significantly differ from those in a uni-form medium. The dimensionless well water levels are smaller for the system with a negative skin than those

with a positive skin at the same dimensionless time. The difference of dimensionless well water level between the two-zone and uniform medium systems decreases sig-nificantly with a because a smaller storage coefficient has less effect on dimensionless well water level.

Acknowledgements

This study was partly supported by the Taiwan Na-tional Science Council under the Grant NSC 92-2211-E-009-008. The authors appreciate the comments and suggested revisions of three anonymous reviewers that help improve the clarity of our presentation.

Appendix A. Derivation of Eq. (16)

The inverse Laplace transforms of Eq.(9)in the time domain can be obtained using the Bromwich integral

[13] as H1¼ 1 2pi Z nþi1 ni1 ept_H 1dp ðA:1Þ

where p = complex variable, i = imaginary unit, and n= large, real, and positive constant, so that all the poles lie to the left of line (n i1, n + i1).

A single branch point with no singularity (pole) at p = 0 exists in the integrand of Eq. (9). Thus, this inte-gral requires the Bromwich inteinte-gral for the Laplace inversion. The closed contour of integrand is shown in

y C A B D G H x F E R δ

Fig. 9. A plot of the closed contour integration of H for the Bromwich integral[13].

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Fig. 9with a cut of p plane along a negative real axis, where d is taken sufficiently small to exclude all poles from the circle about the origin. The closed contour con-sists of the part AB of the Bromwich line from minus infinity to infinity, semicircles BCD and GHA of radius R, lines DE and FG parallel to the real axis, and a circle EF of radius d about a origin. The integration along the small circle EF around an origin as d approaches zero is carried out using the Cauchy integral and the value of the integral is equal to zero. The integrals taken along BCD and GHA tend to zero as R approaches infinity. Consequently, Eq. (9) can be replaced by the sum of integrals along DE and FG. In other words, the integral can be written as H1¼ lim d!0 R!1 1 2pi Z DE ept_H 1dpþ Z FG ept_H 1dp ðA:2Þ For the first term on the right-hand-side (RHS) of

Eq. (A.2) along DE, we introduce the new variable

p = u2epiT1/S1and use the formulas [6, p. 490]

Kv ze 1 2pi ¼ 1 2pie 1 2vpi_½J vðzÞ iYvðzÞ ðA:3Þ and Iv ze 1 2pi ¼ e1 2vpi_J vðzÞ ðA:4Þ

where v = 0, 1, 2, . . .The ﬁrst term on the RHS of Eq.

(A.2)then leads to

H1DE¼ rwgH0 pi Z 1 0 eT 1S1u

2_t½A₂ðuÞ iA₁ðuÞ ½B1ðuÞ þ iB2ðuÞ

du ðA:5Þ

Likewise, introducing p = u2epiT1/S1, the integral

along FG gives minus the conjugate of Eq.(A.5)as H1FG¼ rwgH0 pi Z 1 0 eT 1S1u

2_t½A₁ðuÞ þ iA₂ðuÞ ½B1ðuÞ iB2ðuÞ

du ðA:6Þ

The closed-form solution of Eq. (16) can then be ob-tained by combining Eqs.(A.5) and (A.6).

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