Published online 31 January 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/hyp.6978
Semi-analytical solution for a slug test in partially
penetrating wells including the effect
of finite-thickness skin
Hund-Der Yeh,
1* Yen-Ju Chen
1and Shaw-Yang Yang
21Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan 2Department of Civil Engineering, Vanung University, Chungli, Taiwan
Abstract:
This paper presents a new semi-analytical solution for a slug test in a well partially penetrating a confined aquifer, accounting for the skin effect. This solution is developed based on the solution for a constant-flux pumping test and a formula given by Peres and co-workers in 1989. The solution agrees with that of Cooper and co-workers and the KGS model when the well is fully penetrating. The present solution can be applied to simulate the temporal and spatial head distributions in both the skin and formation zones. It can also be used to demonstrate the influences of skin type or skin thickness on the well water level and to estimate the hydraulic parameters of the skin and formation zones using a least-squares approach. The results of this study indicate that the determination of hydraulic conductivity using a conventional slug-test data analysis that neglects the presence of a skin zone will give an incorrect result if the aquifer has a skin zone. Copyright 2008 John Wiley & Sons, Ltd.
KEY WORDS groundwater; slug test; constant-flux pumping test; skin effect; partially penetrating well; Laplace transform; finite Fourier cosine transform
Received 14 April 2007; Accepted 27 November 2007
INTRODUCTION
A slug test is performed by suddenly removing/(adding) a small amount of water from/(into) the test well and measuring the change of water level at that well simul-taneously. The measured test data can then be analysed to determine the aquifer parameters, e.g. hydraulic con-ductivity and specific storage, representing the hydraulic characteristics around the vicinity of the wellbore. The slug test is commonly used in aquifer-site characterization because of the advantages of low cost, ease of imple-mentation and short test duration, absence of need for post-treatment of a large volume of contaminated water, and relatively minor disturbance to the groundwater flow system.
A skin is considered as a zone of distinct hydraulic conductivity adjacent to the wellbore face caused by a well drilling or completion. A positive skin (also called a low-permeability skin), which usually arises from dam-age caused by well drilling and incomplete well devel-opment, is a zone adjacent to the wellbore with lower hydraulic conductivity than the undisturbed formation. In contrast, a negative skin (also called a high-permeability skin) refers to a zone with higher hydraulic conductivity than the undisturbed formation owing to excessive well completion. Since the slug-test data reflects the hydraulic characteristics of the aquifer near the test well, ignoring
* Correspondence to: Hund-Der Yeh, Institute of Environmental Engi-neering, National Chiao Tung University, 75 Po-Ai Street, Hsinchu 30039, Taiwan. E-mail: hdyeh@mail.nctu.edu.tw
the existence of a wellbore skin can lead to significant errors in parameter determination from test-data analy-sis. In addition, the slug test well commonly partially penetrates the aquifer for most engineering practices. Because the water flux towards a partially penetrating well includes both horizontal and vertical components, the head response in a partially penetrating well differs from that in a fully penetrating well.
A number of mathematical models used to analyse slug test data have been published over the past four decades. For a fully penetrating well, Cooper et al. (1967) presented an analytical solution and proposed a corresponding type-curve method considering a finite-diameter well in a confined aquifer. However, the lack of sensitivity in the type-curve match limited the ability to correctly determine the storage coefficient. With regard to the wellbore-skin effect, Faust and Mercer (1984) investigated the effect of a finite-thickness skin on the response of slug tests using a simple analytical solution and numerical modelling. Following the suggestion of Faust and Mercer (1984), Moench and Hsieh (1985) presented a semi-analytical solution for the slug test and utilized a numerical inversion method to generate the type curves. They used type curves to illustrate the influence of a finite-thickness skin on the open well and pressurized slug tests. Yang and Gates (1997) analysed the effect of wellbore skin on slug test results by utilizing both finite-element modelling and field tests. Their results showed that the early- and late-time data, respectively, reflect the groundwater flow within the skin and undisturbed
formation zones. Yeh and Yang (2006) proposed an analytical solution and numerical results for a slug test performed in a confined aquifer with finite-thickness skin. Later, Yang and Yeh (2007) compared the numerical evaluation of the semi-analytical solution proposed by Moench and Hsieh (1985) with the finite-element model presented by Yang and Gates (1997), simulating a slug test performed in a two-zone confined aquifer.
For a partially penetrating well, Dougherty and Babu (1984) proposed a semi-analytical solution for a slug test in a confined aquifer system, taking into account the skin effect. In their mathematical model, a skin factor, without an exact skin thickness, was used to sum the effect of a disturbed zone around the wellbore. Hyder et al. (1994) extended the Dougherty and Babu (1984) solution and developed a mathematical model incorporating the effects of well partial penetration, aquifer anisotropy, and finite-thickness skin in a confined or unconfined aquifer system using a semi-analytical solution.
Peres et al. (1989) proposed a different approach for slug test data analysis. They developed a theoretical for-mula relating the slug test solution and the solution of constant-flux pumping test in a well fully penetrating a confined aquifer system by considering the effect of wellbore storage. The relationship describing the time derivative of the constant-flux solution was equivalent to the slug test solution. Moreover, using Duhamel’s the-orem, they pointed out that their formula was applicable to various well/aquifer constructions. Based on the for-mula given by Peres et al. (1989) and the solution of a constant-flux pumping test, this paper develops a new slug test solution to describe the head distribution in a confined aquifer system.
The literature regarding the constant-flux pumping test is briefly reviewed below. Novakowski (1989) proposed a semi-analytical solution for a constant-flux pumping test in a well fully penetrating a confined aquifer system, considering skin effects and wellbore storage. Further-more, he presented a series of type curves, obtained from a numerical inversion of the Laplace-domain solution, to illustrate the influences of skin, wellbore storage, and radial distances on the drawdown distributions. Yeh et al. (2003) developed an analytical solution for constant-flux pumping test conducted in a fully penetrating well in a two-zone confined aquifer system. Yang et al. (2006) developed an analytical solution for the constant-flux pumping test conducted in a partially penetrating well in a confined aquifer system. Note that the use of the for-mula given by Peres et al. (1989) in obtaining the slug test solution needs a constant-flux solution considering the effects of wellbore storage and the presence of a well partially penetrating a radial two-zone confined aquifer system.
The objective of this paper is to develop a new mathematical model for the slug test in a partially penetrating well, taking into consideration the effect of wellbore skin. The Laplace-domain solution of a constant-flux pumping test in a well partially penetrating a radial two-zone confined aquifer system taking into
account the effects of wellbore storage is derived first. A series of integral-transform techniques, i.e. Laplace transforms, finite Fourier cosine transforms, and inverse finite Fourier cosine transforms are applied to obtain the constant-flux solution. The slug test solution in the Laplace domain is then obtained by applying the formula given by Peres et al. (1989) to the solution of the constant-flux pumping test. A method of numerical Laplace inversion, the routine DINLAP of IMSL (1997) based on the modified Crump algorithm, is employed to generate the time-domain results. This routine has been applied successfully in other groundwater problems (Yang and Yeh, 2002, 2005; Yang et al., 2006). Finally, the slug test solution is used to investigate the effects of wellbore skin and the presence of a partially penetrating well on the water level at the test well.
MATHEMATICAL DEVELOPMENT
This section contains two parts: in the first part, a mathematical model is proposed for a constant-flux pumping test in a well partially penetrating the confined aquifer system, taking into account wellbore storage and the skin effect. In the second part, the solution of the constant-flux pumping test is converted into the slug test solution based on the formula given by Peres et al. (1989).
Mathematical model of constant-flux pumping test
Figure 1 illustrates the well and aquifer configura-tions. The assumptions for the constant-flux pumping test model are: (a) the wellbore skin is of finite thick-ness, homogeneous, and anisotropic throughout the entire thickness of the aquifer; (b) the undisturbed zone is con-fined, homogeneous, anisotropic, laterally infinite, and vertically finite with a constant thickness; (c) the test well is partially penetrated and has a finite radius; and (d) the initial hydraulic head is constant and uniform throughout the whole aquifer. The governing equations for the skin
Figure 1. Schematic representation of the partially penetrating well and radial two-zone aquifer system
zone and the formation zone are, respectively (Yang and Yeh, 2005) kr1 ∂2hc1 ∂r2 C 1 r ∂hc1 ∂r Ckz1 ∂2hc1 ∂z2 DSs1 ∂hc1 ∂t , rw r rs 1 and kr2 ∂2hc2 ∂r2 C 1 r ∂hc2 ∂r Ckz2 ∂2hc2 ∂z2 DSs2 ∂hc2 ∂t , rsr < 1 2
where hc is the hydraulic head during a constant-flux
pumping test, the subscripts 1 and 2 denote the skin zone and formation zone, respectively, r is the radial distance from the centreline of the pumping well, rw is
the well radius, rs is the outer radius of the skin zone,
z is the vertical direction, t is the time from the start of test, SS is the specific storage, and kr and kz are the
radial and vertical components of hydraulic conductivity, respectively.
Equations (1) and (2) are subject to the following boundary conditions. The initial hydraulic heads within the skin and formation zones are both equal to zero, i.e. hc1r, z,0 D hc2r, z,0 D 0, r > rw 3
At an infinite distance from the test well, the hydraulic head is also equal to zero.
hc21, z, t D0 4
For a confined aquifer system, impermeable boundaries exist at the bottom and top of the aquifer, thus
∂hc1r,0, t ∂z D ∂hc2r,0, t ∂z D0 5 and ∂hc1r, L, t ∂z D ∂hc2r, L, t ∂z D0 6
where L is the thickness of the aquifer. The conservation of mass at the interface between the skin and formation zones can be described by
kr1
∂hc1rs, z, t
∂r Dkr2
∂hc2rs, z, t
∂r 7
Furthermore, the conservation of mass at the test well is approximated as Q rc2∂hc1rw, z, t ∂t [Uz B1 Uz B2] D 2rwkr1B2B1 ∂hc1r, z, t ∂r rDrw 8 where Q is a constant discharge pumped from the test well, B1 and B2 denote the top and bottom z-coordinates
of the screen, respectively, U z Bi equals one when
z ½ Bi and zero when 0 z Bi for i D 1 or 2. Note
that Equation (8) assumes that the flow rate along the well screen is uniform. The second term on the left-hand side (LHS) of Equation (8) reflects the effect of wellbore storage and the term on the right-hand side (RHS) represents the total flux flows across the wellbore screen.
Applying the Laplace transform and finite Fourier cosine transform, Equations (1) and (2), respectively, can be reduced to ∂2hc1 ∂r2 C 1 r ∂hc1 ∂r D˛ 2h c1 9 and ∂2hc2 ∂r2 C 1 r ∂hc2 ∂r Dˇ 2h c2 10
where hcindicates the Fourier–Laplace domain solution
of hydraulic head, ˛ DSs1p C kz1wn2/kr1 and ˇ D
Ss2p C kz2wn2/kr2with wnDn/Lfor n D 1, 2, . . ..
The solutions of Equations (9) and (10) subject to the transformed boundary conditions of (4), (7), and (8) can be written, respectively, as hc1r, wn, p D QW1[CI0˛r C DK0˛r] rc2p2 2rwpkr1B2B1˛ 11 and hc2r, wn, p D QW1EK0ˇr rc2p2 2rwpkr1B2B1˛ 12 where W1 D 1 wn sin wnB2sin wnB1 13 C D K1˛rsK0ˇrs kr2 kr1 ˇ ˛ K0˛rsK1ˇrs 14 D D I1˛rsK0ˇrs C kr2 kr1 ˇ ˛ I0˛rsK1ˇrs 15 E D K0˛rsI1˛rs C K1˛rsI0˛rs 16 D CI0˛rw C DK0˛rw 17 and D CI1˛rw DK1˛rw 18
Furthermore, applying the finite Fourier cosine trans-forms to Equations (11) and (12), the hydraulic heads of the constant-flux pumping test within the skin and formation zones can then be described, respectively, as Qhc1r, z, p D 1 L QB2B1[C0I0˛0r C D0K0˛0r] rc2p202rwpkr1B2B1˛00 C 2 L 1 nD1 QW1[CI0˛r C DK0˛r] coswnz rc2p2 2rwpkr1B2B1˛ 19
and Qhc2r, z, p D 1 L QB2B1E0K0ˇ0r rc2p202rwpkr1B2B1˛00 C2 L 1 nD1 QW1 E K0ˇ rcoswnz rc2p2 2rwpkr1B2B1˛ 20 where the subscript 0 shown in the variables ˛0, ˇ0, C0,
D0, E0, 0, 0represents n equalling zero for the related
variables.
Solution of slug test
Taking wellbore storage and skin effect into consid-eration for a well fully penetrating a confined aquifer system, Peres et al. (1989) provide a formula relating the slug test and constant-flux solutions in the Laplace domain as
Qhr, z, p D rc2H0p
Q Qhcr, z, p 21
They showed that this relation is valid for any well/ aquifer construction when employing Duhamel’s the-orem. Based on Equation (21), the constant-flux solu-tions, i.e. Equations (19) and (20), reduce to the Laplace-domain solutions of a slug test for the skin and formation zones, respectively, as:
Qh1r, z, p D 1 L rc2H0B2B1[C0I0˛0r C D0K0˛0r] rc2p02rwkr1B2B1˛00 C 2 L 1 nD1 rc2H0W1[CI0˛r C DK0˛r] coswnz rc2p 2rwkr1B2B1˛ 22 and Qh2r, z, p D 1 L rc2H0B2B1E0K0ˇ0r rc2p02rwkr1B2B1˛00 C 2 L 1 nD1 rc2H0W1E K0ˇrcoswnz rc2p 2rwkr1B2B1˛ 23 In addition, the water level in the test well can be obtained through averaging the head in Equation (22) along the screened interval. That is
Qhwp D B2B1 L rc2H0[C0I0˛0rw C D0K0˛0rw] rc2p02rwkr1B2B1˛00 C 2 B2B1L 1 nD1 rc2H0W12[CI0˛rw C DK0˛rw] rc2p 2rwkr1B2B1˛ 24
Since Equations (22)–(24) are rather complicated, the inversions of these solutions to the time domain may not be tractable analytically. The DINLAP routine of IMSL (1997), which is a code of modified Crump algorithm,
with accuracy to five decimal places is employed to perform the numerical Laplace inversion.
RESULTS AND DISCUSSION
In this section, the newly derived solution (or model) is compared with the solutions of Cooper et al. (1967) and Hyder et al. (1994). The influence of a wellbore skin and the presence of a partially penetrating well on slug-test water level responses are investigated using the present solution. Assume that a slug test is operated in a confined and isotropic aquifer system. The aquifer thickness (L) is 20 m with hydraulic conductivity kr2 104 m s1
and specific storage Ss25 ð 105 m1. The radii of the
well (rw) and casing (rc) are 0Ð0915 m and 0Ð0508 m,
respectively. The initial water level in a test well, H0, is
1 m and the normalized head is defined as the well water level at any test time divided by the initial well water level.
Comparison with existing solutions
Cooper et al.’s solution (1967) was developed to describe the groundwater behaviour during a slug test performed in a finite-diameter well fully penetrating a confined aquifer system. Comparing the normalized head
response determined by the present solution with that of Cooper et al. (1967), Figure 2 shows good agreement of the normalized head with the result obtained for the case of a slug test is performed in a well fully penetrating a confined aquifer without the presence of skin zone.
Hyder et al.’s solution (1994), also called the KGS model, was developed for a partially penetrating well constructed in a confined or unconfined aquifer system and with or without the presence of skin zone. They used a different representation for the mass balance condition at the wellbore; that is
rc2dhwt dt [Uz B1 Uz B2] D 2rwkr1 B2B1 ∂hc1r, z, t ∂r rDrw 25
Notice that the RHS of Equation (25) is dependent on z, yet the LHS is not. The solution of Equation (1) with Equation (25) considered as the boundary condition for a slug test gives an approximate result for the well water level or hydraulic head in the skin zone. Figure 3 shows the simulated results given by the present solution and the KGS model (Hyder et al., 1994) for a slug test performed in a well with a wellbore skin. It is assumed that the skin zone has an outer radius of 0Ð305 m, hydraulic conductivity kr1103 m s1, 104 m s1 or
105 m s1, and specific storage Ss1 5 ð 105 m1.
The case kr1D104 m s1 implies that the wellbore
skin is absent. Figure 3a shows that the normalized head responses in a fully penetrating well simulated by the present solution coincide with those of the KGS model (Hyder et al., 1994) for positive skin, negative skin, and no skin cases. In contrast, the normalized head for a well partially penetrating a confined aquifer with screen length 6 m and 3 m are plotted in Figures 3b and 3c, respectively. The screens are located on the top of the confined aquifer. Figure 3c represents an extreme case, with penetration ratio only 0Ð15. In both figures, the normalized head simulated by the present solution is almost identical with those of the KGS model (Hyder
et al., 1994), except that the negative-skin case produced
by the present solution shown in Figure 3c gives slightly lower values at intermediate times compared with those of the KGS model (Hyder et al., 1994). These differences in normalized heads are attributed to the fact that the KGS model (Hyder et al., 1994) used the depth-averaging hydraulic head along the screened interval to represent the well water level and assumed the flux through the screen to be equal to the change of well water level with respect to time as the boundary condition of the screen part. In contrast, in the proposed mathematical model the hydraulic head was derived first and then the depth average of the hydraulic head was taken as the well water level. The differences in normalized heads simulated by the present solution and the KGS model (Hyder et al., 1994) may increase with the skin-zone hydraulic conductivity.
Effect of wellbore-skin type
Assume that a slug test is operated in a confined aquifer with a skin zone adjacent to the wellbore. The skin zone has an outer radius (rs) of 0Ð305 m and
specific storage (Ss1) of 5 ð 105 m1. Four different
radial hydraulic conductivities of the skin zone (kr1)
are considered: kr1 D103 m s1 for a negative skin
case, kr1D104 m s1 for a no skin case, and kr1D
105 m s1 or 106 m s1 for a positive skin case. The
partially penetrating screen is located at B1 D0 m and
B2D6Ð0 m. Figure 4a shows the influence of
wellbore-skin type on the slug test data for fully and partially penetrating wells. The normalized head difference shown in Figure 4b represents the differences in normalized head between the cases discussed in Figure 4a and the no skin case for a fully penetrating well (B1 D0 m,
B2D20Ð0 m and kr1 D104 m s1).
Figure 2. Plots of normalized head versus time determined by the present solution and Cooper et al.’s solution (1967) for a slug test performed in
a fully penetrating well
Under a fully or partially penetrating well condition, Figure 4a shows that the recovery rate of water level at the test well for the positive skin case is remarkably slower than that for the no skin case, while the recovery rate for a negative skin case is moderately swifter than that for the no skin case. The slow recovery of the well water level for the positive skin case is attributed to the low hydraulic conductivity of the skin zone. Accordingly, it is noteworthy that ignoring the presence of a positive skin will result in underestimation of hydraulic conduc-tivity determined from slug-test data analysis. On the other hand, the influence of a negative skin on the recov-ery rate of well water level is not so obvious; however, overestimation of hydraulic conductivity will occur if one disregards it. In short, the normalized head of a partially penetrating well is obviously slower than that of a fully penetrating well under the same aquifer condition. Thus, disregarding the presence of a partially penetrating well in the slug test data analysis will also cause underestimation of the hydraulic conductivity. Figure 4b shows that the largest normalized head difference is about 0Ð9, occurring at 20 s for the positive-skin case kr1D106 m s1 with
a partially penetrating well. Significant underestimation of hydraulic conductivity will occur if one ignores the presence of positive skin, and treats the partially pene-trating well as a fully penepene-trating well when analyzing slug test data.
Effect of wellbore skin thickness
The thickness of the skin zone may range from a few millimetres to several metres (Novakowski, 1989). Assume that a slug test performed in a confined aquifer system which has a skin zone outer radius (rs) of
0Ð150 m, 0Ð305 m or 0Ð610 m. Fully and partially pen-etrating wells are both considered in the investigation.
Figure 3. Plots of normalized head versus time determined by the present solution and the KGS model (Hyder et al., 1994) for (a) a fully penetrating well, (b) a partially penetrating well with B1D0 m and B2D6 m and aquifer thickness L D 20 m, and (c) a partially penetrating well with B1D0 m
and B2D3 m and aquifer thickness L D 20 m
The screen of the partially penetrating well is assumed to be perforated from B1D0 m to B2 D6Ð0 m. The
influence of skin thickness on well water level is pre-sented in Figure 5. Figure 5a illustrates the normalized head for a positive skin case with kr1D105 m s1,
while Figure 5b displays the negative skin case with kr1D103 m s1.
Figure 5a displays curves of normalized head versus time for both fully and partially penetrating well cases. The figure shows that the normalized head decreases with increasing skin thickness for the positive skin case, therefore, the normalized head takes more time to recover in a thicker positive skin zone. Figure 5b shows the recovery rate of well water level increasing with skin thickness for the negative skin case. On the other hand,
the recovery rate of well water level in a partially penetrating well is obviously slower than that in a fully penetrating well under the same aquifer conditions. In addition, the normalized head for the positive skin case shown in Figure 5a is stabilized more slowly than that for the negative skin case shown in Figure 5b, as mentioned in the previous section.
CONCLUSIONS
A new semi-analytical solution for a slug test in a partially penetrating well taking account of the wellbore skin effect has been developed via the formula of Peres
et al. (1989). The present solution for the simulated
Figure 4. Plots for a fully penetrating well (B2D20 m) or partially pen-etrating well (B2D6 m) with skin hydraulic conductivities Kr1D103, 104, 105 or 106 m s1 and formation hydraulic conductivity Kr2D104 m s1. (a) Normalized head versus time. (b) Normalized head difference versus time when compared with the results obtained for the case of no skin and a fully penetrating well (Kr1D104m s1,
B1D0 m and B2D20 m)
of Cooper et al. (1967) and the KGS model (Hyder et al., 1994) in the case of a well fully penetrating a confined aquifer system. For the case of a partially penetrating well, a difference between the present solution and the KGS model (Hyder et al., 1994) for the normalized head is due to different assumptions made for the boundary condition along the wellbore screen.
The new solution is used to examine the effect of wellbore skin on the well water level in a slug test. The recovery of water level during a slug test on an aquifer with a positive skin is slower than that of an aquifer without the skin; in contrast, the recovery of water level in an aquifer with a negative skin is more rapid than in an aquifer without the skin. The hydraulic conductivity of the formation determined from the recovery slug-test data analysis will be underestimated if one ignores the
Figure 5. Plots of normalized head versus time in a fully or partially penetrating well (B1D0 m and B2D6 m) with Kr2D104m s1and (a) a positive skin case with Kr1D105 m s1, (b) a negative skin case with Kr1D103m s1. The outer radius of the skin zone is 0Ð150, 0Ð305
or 0Ð610 m
presence of a positive skin; while it will be overestimated if one disregards the presence of a negative skin. In addition, when the skin thickness increases, the water level recovery rate decreases for the positive skin case; on the other hand, it increases for the negative skin case. Obviously, the presence of a partially penetrating well also has an effect on the water level recovery rate in the well.
ACKNOWLEDGEMENTS
This study is partially supported by the Taiwan National Science Council under the grant NSC 94-2211-E -009-012.
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NOTATION
B1 D distance from the top of aquifer to the top of
test-well screen
B2 D distance from the top of aquifer to the bottom of
test-well screen C D K1˛rsK0ˇrs kr2 kr1 ˇ ˛ K0˛rsK1ˇrs C0 DK1˛0rsK0ˇ0rs kr2 kr1 ˇ0 ˛0 K0˛0rsK1ˇ0rs D D I1˛rsK0ˇrs C kr2 kr1 ˇ ˛ I0˛rsK1ˇrs D0D I1˛0rsK0ˇ0rs C kr2 kr1 ˇ 0 ˛0 I0˛0rsK1ˇ0rs E D K0˛rsI1˛rs C K1˛rsI0˛rs E0 D K0˛0rsI1˛0rs C K1˛0rsI0˛0rs
H0Dinitial water level in a test well
hc1 Dhydraulic head of constant-flux pumping test within
a skin zone
hc2 Dhydraulic head of constant-flux pumping test within
a formation zone
Qh1D hydraulic head of slug test within a skin zone in
the Laplace domain
Qh2D hydraulic head of slug test within a formation zone
in the Laplace domain
Qhc1 Dhydraulic head of constant-flux pumping test within
a skin zone in the Laplace domain
Qhc2 Dhydraulic head of constant-flux pumping test within
a formation zone in the Laplace domain
QhwD well water level of slug test in the Laplace domain
hc1 Dhydraulic head of constant-flux pumping test within
a skin zone in the Fourier–Laplace domain hc2 Dhydraulic head of constant-flux pumping test within
a formation zone in the Fourier–Laplace domain I0 D Modified Bessel functions of the first kind of
order zero
I1 D Modified Bessel functions of the first kind of
order one
K0 DModified Bessel functions of the second kind of
order zero
K1 DModified Bessel functions of the second kind of
order one
kr1Dradial component of hydraulic conductivity in a
skin zone
kr2Dradial component of hydraulic conductivity in a
formation zone
kz1D vertical component of hydraulic conductivity in a
skin zone
kz2D vertical component of hydraulic conductivity in a
formation zone L D thickness of aquifer p D Laplace variable
Q D Constant-flow rate through the wellbore r D radial distance from the centreline of well rcD radius of casing
rsD outer radius of a skin zone
rwD effective radius of test well
Ss1 Dspecific storage within a skin zone
Ss2 Dspecific storage within a formation zone
t D time since the start of test U D unit step function
W1D 1w nsin wnB2sin wnB1 wnD nL , n D1, 2, . . . z D vertical direction ˛ D S s1 kr1 p C kz1 kr1 wn2, n D1, 2, . . . ˛0D S s1 kr1 p ˇ D S s2 kr2 p C kz2 kr2 wn2, n D1, 2, . . . ˇ0D S s2 kr2 p D CI0˛rw C DK0˛rw 0DC0I0˛0rw C D0K0˛0rw D CI1˛rw DK1˛rw 0DC0I1˛0rw D0K1˛0rw
