Analytical Solutions for Constant-Flux and Constant-Head
Tests at a Finite-Diameter Well in a Wedge-Shaped Aquifer
Yen-Ju Chen
1; Hund-Der Yeh
2; and Shaw-Yang Yang
3Abstract: Neglecting the effect of well radius may lead to a significant error in the predicted drawdown distribution near the pumping well area. New analytical solutions describing aquifer responses to a constant pumping or a constant head maintained at a finite-diameter well in a wedge-shaped aquifer are derived based on the image-well method and applicable to an arbitrarily located well in the system. The solutions are useful for quantifying groundwater exploitation from a wedge-shaped aquifer and for determining the hydrogeological parameters of a wedge-shaped aquifer in inverse problems.
DOI: 10.1061/共ASCE兲0733-9429共2009兲135:4共333兲
CE Database subject headings: Analytical techniques; Aquifers; Groundwater management; Wells; Parameters.
Introduction
The image-well method is useful in evaluating the impact of to-pographic boundaries existing around a well on the groundwater flow system. It removes aquifer boundaries and places image wells at judicious locations in order to reflect the effect of bound-aries. The drawdown in a test well can then be superposed as the sum of the individual drawdown due to the real well and the image wells共Ferris et al. 1962; Streltsova 1988; Kuo et al. 1994兲. Previous studies of the drawdown solutions are based on the su-perposition of the Theis solution共1935兲, which describes the tran-sient hydraulic response of an infinite confined aquifer to a line pumping.
Alternative drawdown solutions accounting for topographic boundaries can be based on integral transform methods共Chan et al. 1976, 1978; Yeh and Chang 2006兲. These can solve the prob-lem of a wedge-shaped aquifer for the constant-flux test 共CFT兲 with any wedge angle, but solutions are complicated and rather difficult to accurately evaluate 共Chan et al. 1976兲. Further, in those studies, the Dirac delta function was used to represent the pumping at the test well. Previous studies devoted to analytical solutions for the CFT in the vicinity of topographic boundaries all assume infinitesimal-diameter well, but ignoring the finite test-well diameter may lead to errors in the predicted drawdown near the test well.
Several studies have also examined the problem of constant-head test共CHT兲 in the vicinity of an aquifer boundary. Unlike the treatment in the CFT, one cannot directly apply the image-well
method to the CHT, because superposing the drawdown produced by each image well would violate the condition of constant draw-down at the test well. Based on an instantaneous source function and the convolution method, Murdoch and Franco 共1994兲 and Renard共2005兲 presented approximate expressions for various ge-ometries of flow problem, which are in accordance with the con-dition of constant well drawdown. Analytical solutions for the CHT in a wedge-shaped aquifer, to the writers’ knowledge, have never been addressed.
This technical note develops analytical solutions for the down distribution of CFT as well as well discharge rate and draw-down distribution of CHT in a wedge-shaped aquifer considering a finite well diameter. These new solutions describe the aquifer response to a finite-diameter well arbitrarily located in a wedge-shaped aquifer bounded by various combinations of recharging and impermeable boundaries. They are relatively easy to formu-late and evaluate once the test condition and the aquifer parameter values are known. The obtained drawdown solution for a CFT will be compared with two solutions in which the well diameter is neglected.
Mathematical Formulation
Mathematical Models of CFT and CHT in an Infinite Extent Aquifer
The assumptions related to the aquifer and well configurations are:共1兲 the confined aquifer is homogeneous, isotropic, and uni-form in thickness; and 共2兲 the test well with a finite diameter penetrates the entire thickness of the aquifer. Define the dimen-sionless variables as sD= s rw , rD= r rw , tD= Tt rw2S 共1兲
where sD= dimensionless drawdown; rD= dimensionless radial distance; tD= dimensionless time; s = drawdown; rw= radius of test well; r = radial distance from the centerline of test well; T = transmissivity of aquifer; S = storativity of aquifer; and t = time from the start of test. In radial coordinates, the governing equa-tion for the groundwater flow in a dimensionless form is 1
Graduate Student, Institute of Environmental Engineering, National Chiao Tung Univ., Hsinchu, Taiwan.
2
Professor, Institute of Environmental Engineering, National Chiao Tung Univ., Hsinchu, Taiwan 共corresponding author兲. E-mail: hdyeh@ mail.nctu.edu.tw
3
Professor, Dept. of Civil Engineering, Vanung Univ., Chungli, Taiwan.
Note. Discussion open until September 1, 2009. Separate discussions must be submitted for individual papers. The manuscript for this technical note was submitted for review and possible publication on October 30, 2007; approved on October 21, 2008. This technical note is part of the Journal of Hydraulic Engineering, Vol. 135, No. 4, April 1, 2009. ©ASCE, ISSN 0733-9429/2009/4-333–337/$25.00.
2s D rD 2 + 1 rD sD rD =sD tD 共2兲 For both the CFT and CHT, the initial dimensionless draw-down for the whole aquifer equals zero, that is
sD共rD,0兲 = 0 共3兲
The dimensionless drawdown at infinity is
sD共⬁,tD兲 = 0 共4兲
The boundary condition at the wellbore for the CFT is defined differently from the CHT. For the CFT, the boundary condition follows Darcy’s law and is described as
QDf= −
冏
sDf共rD,tD兲 rD
冏
rD=1, tD⬎ 0 共5兲
where the subscript f represents a CFT. Eq.共5兲 expresses the mass conservation at the rim of the wellbore. The dimensionless dis-charge QDfis defined as
QDf= Q 2rwT
共6兲 where Q = constant pumping rate. The Laplace-domain solution s
˜Dfto Eq.共2兲 subject to Eqs. 共3兲–共5兲 is 共van Everdingen and Hurst 1949; Carslaw and Jaeger 1959兲
s
˜Df=QDfK0共rD
冑
p兲 p冑
pK1共冑
p兲共7兲 where p⫽Laplace variable, and K0 and K1⫽modified Bessel functions of the second kind of order zero and one, respectively. For the CHT, the boundary condition at the wellbore is ex-pressed as
sDh共rD= 1,tD兲 = s0D, tD⬎ 0 共8兲 or expressed in the Laplace domain as
s
˜Dh共rD= 1, p兲 =s0D
p 共9兲
with the subscript h denoting the CHT and s0D⫽dimensionless initial drawdown defined as the initial drawdown normalized by rw. Eqs.共8兲 and 共9兲 state that the piezometric head at the wellbore is constant.
Based on Duhamel’s formula共Murdoch and Franco 1994兲, the drawdown produced by a CHT can be written as
sDh共rD,tD兲 =
冕
0 tD QDh共兲 sDfu共rD,tD−兲 tD d 共10兲 where QDh= dimensionless discharge at the test well during the CHT; sDfu= dimensionless unit step response function, which is defined as the drawdown under constant unit discharge共sDf/QDf兲. Taking the Laplace transform of Eq.共10兲 yieldss
˜Dh共rD, p兲 = pQ˜Dh共p兲s˜Dfu共rD, p兲 共11兲 Eq.共11兲 relates the solution of the CHT to that of the CFT. This relationship holds in any well model.
Configuration of an Image-Well System
The wedge angle of the wedge-shaped aquifer is assumed to be an aliquot part of 360°. It must be an aliquot part of 90° for aquifers
with boundaries that are either like or unlike; otherwise, it must be an aliquot part of 180° for aquifers with like boundaries共Ferris et al. 1962兲. This rule has its exception in the case where the wedge angle is an odd aliquot part of 360°, the test well is on the bisector of wedge angle, and the boundaries are both imperme-able共Ferris et al. 1962兲. The number of the image wells, n, re-quired in analyzing the flow toward the test well is given by the relation共Ferris et al. 1962兲
n =360°
− 1 共12兲
where refers to the wedge angle.
Fig. 1 shows the image-well system in a wedge-shaped aquifer where the image wells are named in a clockwise sequence 共Ii, i = 1 , 2 , . . . , n兲. The type of image well, whether it extracts water from, or injects water into an aquifer depends on the nature of the aquifer boundaries. Once the dimensionless distances lD1and lDn are known, the dimensionless distances lDmfrom the center of test well to the center of image wells can be obtained as
I1 I2 I3 I4 I5 I6 I7 β α PW Boundary Boundary l4 l5 l6 l7 l1 l3 l2 (a) I1 I2 I3 I4 I5 I6 I7 β α θ PW OW l1 r Boundary D1 Bou ndary D2 D3 D4 D5 D6 D7 (b)
Fig. 1. Configuration of an image-well system:共a兲 without observa-tion wells;共b兲 with an observation well. Note that PW and OW rep-resent pumping well and observation well, respectively.
lDm=
冦
冑
lDm−12 + lDn2 − 2lDm−1lDncos冋
− m 2册
, when m is even冑
lD12 + lDm−12 − 2lD1lDm−1cos冋
− m + 1 2 ␣ − m − 1 2 册
, when m is odd冧
共13兲 where␣=angle between the lower topographic boundary and the image line connecting the intersection point of the wedge-shaped aquifer to the test well, and is defined as −␣.The angle, shown in Fig. 1共b兲, is defined as the angle mea-sured from a line connecting the centers of the test well and the first image well共denoted by I1兲 to the line connecting the centers of the test well and observation well. The dimensionless distances,DDm, from the center of observation well to the center of image wells are
DDm=
冦
冑
rD2+ lDm2 − 2rDlDmcos冋
+m − 1 2 册
, when m is odd冑
rD 2 + lDm 2 − 2rDlDmcos冋
+ m − 2 2 ␣ + m 2册
, when m is even冧
共14兲Analytical Solutions for a CFT in a Wedge-Shaped Aquifer
For a CFT performed near aquifer boundaries, the use of the image-well method requires first to determine the location and operation types of the image wells, and then superpose the indi-vidual dimensionless drawdown s˜Df due to the real and each image well. Based on Eq.共7兲, the resultant s˜Dfcan be written as
s ˜Df共rD,,p兲 = QDf p
冑
pK1共冑
p兲冋
K0共rD冑
p兲 +兺
m=1 n 共− 1兲jK 0共DDm冑
p兲册
共15兲 where n represents the number of image wells. The variable j equals two for the case where boundaries are both impermeable and equals m for the case where boundaries are both recharging. In addition, j equals共m+1兲/2 when m is odd and m/2 when m is even for the case where the lower boundary is recharging and the upper boundary is impermeable; j equals共m+3兲/2 when m is odd and共m+4兲/2 when m is even for the case where the lower bound-ary is impermeable and the upper boundbound-ary is recharging. The lower boundary refers to the boundary next to the first image well and the upper boundary refers to that adjacent to the last image well.The time-domain solution, sDf, can be obtained by applying the Bromwich integral method共Yeh et al. 2003兲 to invert Eq. 共15兲 as
sDf共rD,,tD兲 =2QDf
冕
0 ⬁e−tDu2A1共rD,,u兲Y1共u兲 − A2共rD,,u兲J1共u兲
J12共u兲 + Y12共u兲 du u2 共16兲 where A1共rD,,u兲 = J0共rDu兲 +
兺
m=1 n 共− 1兲jJ 0共DDmu兲 共17兲 and A2共rD,,u兲 = Y0共rDu兲 +兺
m=1 n 共− 1兲jY 0共DDmu兲 共18兲 where J0and Y0= zero-order Bessel functions of the first and sec-ond kinds, respectively, and J1 and Y1= first-order Bessel func-tions of the first and second kinds, respectively.Analytical Solutions for a CHT in a Wedge-Shaped Aquifer
For a CHT performed adjacent to aquifer boundaries, it is re-quired to determine the location and operation types of the image wells before applying the convolution and the image-well method to solve for the discharge at the test well or the drawdown at the observation well. Mathematically, the well discharge is assigned at the center of test well, but the singularity of the unit step response function s˜Dfuat rD= 0 precludes the evaluation. Instead, s
˜Dfu is evaluated at the rim of the test well共rD= 1兲, and DDm is approximated as lDm− 1 for all the image wells. The following steps are therefore taken in the solutions.
• Step 1. Determine s˜Dfu共rD,,p兲 from Eq. 共15兲 corresponding to the same aquifer condition as the CHT.
• Step 2. Determine Q˜Dh共p兲 at the test well by substituting Eq. 共9兲 and s˜Dfu共rD,,p兲 with rD= 1 and DDm= lDm− 1 into Eq. 共11兲.
• Step 3. Determine s˜Dh共rD,,p兲 at the observation well by in-serting Q˜Dh共p兲 and its corresponding s˜Dfu共rD,,p兲 into Eq. 共11兲.
The dimensionless discharge at the test well obtained from Step 2 is expressed as Q˜Dh共p兲 = s0DK1共
冑
p兲冑
p兵K0共冑
p兲 + 兺m=1n 共− 1兲jK 0关共lDm− 1兲冑
p兴其 共19兲 which is inverted as QDh共tD兲 = 2s0D 冕
0 ⬁e−tDu2J1共u兲B2共u兲 − Y1共u兲B1共u兲
B12共u兲 + B22共u兲 du 共20兲 where B1共u兲 = J0共u兲 +
兺
m=1 n 共− 1兲jJ 0关共lDm− 1兲u兴 共21兲 and B2共u兲 = Y0共u兲 +兺
m=1 n 共− 1兲jY 0关共lDm− 1兲u兴 共22兲 Following Step 3, the dimensionless drawdown at the obser-vation well iss ˜Dh共rD,,p兲 =s0D p · K0共rD
冑
p兲 + 兺m=1 n 共− 1兲jK 0共DDm冑
p兲 K0共冑
p兲 + 兺m=1 n 共− 1兲jK 0关共lDm− 1兲冑
p兴 共23兲 Similarly, the dimensionless time-domain drawdown at the obser-vation well is obtained assDh共rD,,tD兲 = s0D+
2s0D
冕
0⬁
e−tDu2A1共rD,,u兲B2共u兲 − A2共rD,,u兲B1共u兲
B12共u兲 + B22共u兲
du u 共24兲
Comparison with Other Solutions
The present solution for a CFT is compared with two other solu-tions obtained in similar aquifer systems but assuming infinitesimal-diameter wells. The first solution is based on the superposition principle and the Theis solution共1935兲, whereas the second solution is given by Chan et al.共1978兲 for a wedge-shaped aquifer bounded by two recharging boundaries. Consider a CFT performed in the vicinity of two recharging boundaries intersect-ing at an angle of 45° as shown in Fig. 2. The aquifer data of T = 10−3m2/s and S=2.5⫻10−4are used in the simulation. Define r0as the length from the test well to the intersection point of the wedge-shaped aquifer. The test well is constructed at r0= 10 m and␣=30° with Q=10−2m3/s and r
w= 0.2 m. The dimensionless drawdowns, sDf, at rD= 1, 3, and 7 with=330° are calculated. The values of sDf, predicted by these three solutions plotted in Fig. 2, increase with tDand then reach to their steady-state values as tD approaches 103. The curve of sDf produced by the Theis-based solution is identical to that produced by Chan et al.’s solu-tion 共1978兲, but differ significantly from the present solution
when tD艋50 at rD= 1. The differences in sDfassuming an infini-tesimal diameter in the Theis-based and Chan et al.’s model are most important for a small rDand at early time.
Concluding Remarks
New solutions describing the drawdown distribution of CFT as well as discharge rate and drawdown distribution of CHT in a wedge-shaped confined aquifer are developed by considering the influence of well diameter. The present drawdown solution for a CFT is compared with the solutions obtained by neglecting the well diameter. It appears that ignoring the influence of the pres-ence of well diameter leads to a significant error in the predicted early time drawdown near the pumping well area.
Acknowledgments
This research was supported in part by a grant from Taiwan Na-tional Science Council under Contract No. NSC95-2221-E-009-017. The writers would like to thank the editor and four anonymous reviewers for their constructive comments and sug-gested revisions.
Notation
The following symbols are used in this technical note: DDi ⫽ dimensionless distance from observation well
to image well i; Ii ⫽ image well i;
J0, J1 ⫽ Bessel function of the first kind of order zero and order one, respectively;
K0, K1 ⫽ modified Bessel function of the second kind of order zero and order one, respectively; lDi ⫽ dimensionless distance from test well to
image well i; p ⫽ Laplace variable;
Q , QDf ⫽ constant discharge and dimensionless discharge produced by a CFT, respectively; QDh, Q˜Dh ⫽ dimensionless discharge produced by a CHT
in the time domain and Laplace domain, respectively;
r , rD ⫽ radial distance and dimensionless radial distance starting from the centerline of test well, respectively;
rw ⫽ effective radius of test well;
r0 ⫽ distance from the centerline of test well to the intersection point of the wedge-shaped aquifer; S ⫽ storativity of aquifer;
s , sD ⫽ drawdown and dimensionless drawdown, respectively;
sDf, s˜Df ⫽ dimensionless drawdown produced by a CFT in the time domain and Laplace domain, respectively;
sDfu, s˜Dfu ⫽ dimensionless unit step response drawdown in the time domain and Laplace domain, respectively;
sDh, s˜Dh ⫽ dimensionless drawdown produced by a CHT in the time domain and Laplace domain, respectively; 10-1 100 101 102 103 104 Dimensionless time, tD 0 5 10 15 20 25 Dimensionless draw down, sD rD= 1 rD= 3 rD= 7 Our solution Theis-based solution Chan et al.'s solution (1978)
15o 330o 30o OW PW r
Lower Recharging Boundary
Uppe rRec hargi ngB oundar y r0
Fig. 2. Comparison of present solution with the Theis-based solution, and Chan et al.’s solution 共1978兲. The above-left sketch shows the configuration of well and aquifer system. Note that PW and OW represent pumping well and observation well, respectively.
s0D ⫽ dimensionless initial drawdown;
T ⫽ transmissivity of aquifer;
t , tD ⫽ real time and dimensionless time from the start of test, respectively;
u ⫽ dummy variable;
Y0, Y1 ⫽ Bessel function of the second kind of order zero and order one, respectively;
␣ ⫽ angle between the lower topographic boundary and an image line connecting the intersected point of the wedge-shaped aquifer to test well;
 ⫽ −␣;
⫽ angle estimated from the line connecting test well and I1; and
⫽ wedge angle of wedged aquifer. References
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