This paper developed new semi-analytical solutions for the aquifer system in response to the constant head test at a partially penetrating well in a confined aquifer of infinite radial extent and finite vertical extent. The Laplace and finite cosine Fourier transforms is first used to reduce the original partial differential equation with mixed-type boundary and initial conditions for a partially penetrating well in an aquifer of finite thickness to the dual or triple series equations.
The present solutions for a fully penetrating well in an aquifer of finite thickness are identical to the solutions of the drawdown and well discharge given in Chen and Stone [1993].
It is found that the solution of Cassiani et al. [1999] for well response to a constant head pumping test in a semi-infinite aquifer approximates the solution for the case where the aquifer thickness of a finite aquifer is 100 times greater than the length of well screen. In addition, the flux is non-uniformly distributed along the screen and with a local peak at the edge, due to the vertical flow induced by well partial penetration.
The new semi-analytical solutions provide accurate description of the response of the aquifer system to a constant head pumping test performed at a partially penetrating well in a confined aquifer of infinite radial extent and finite vertical extent. Those solutions are particularly attractive for practical applications since they can be used to evaluate the
sensitivities of the input parameters in a mathematical model. In addition, the solution can be used to calculate the flow rates during the constant-head test and plots specific drawdown (drawdown divided by the flow rate) versus time to identify the hydraulic parameters if coupling with an optimization approach in the analysis of aquifer data, and to verify a numerical solution.
APPENDIX A
The Laplace and finite cosine Fourier transforms are first used to solve the mixed-type boundary value problem. The definition of Laplace transform is [Sneddon, 1972]：
[
→]
=∫
∞ − transform of Eqs. (6) and (8) – (10) and using the initial condition in Eq. (7), the problem reads:In order to eliminate the ξ coordinate, the finite cosine Fourier transform is used as follows [Sneddon, 1972]:
[
→]
=∫
Substituting Eq. (A6) into Eqs. (A2), (A3) and (A5) results in the Bessel differential equation
as with the boundary condition
0
The general solution of Eq. (A7) with the boundary condition Eq. (A8) is [Carslaw and Jaeger, 1959, p. 193] where A(n,p) can be found from using the mixed-type boundary condition Eq. (A4). The inverse of the finite cosine Fourier transform is [Sneddon, 1972, p.425]
)
Thus, the solution in ξ domain obtained by inserting Eq. (A9) into Eq. (A10) is )
with its derivative with respect toρ given by
)
Substituting Eq. (A11) into Eq. (A4a) and Eq. (A12) into Eq. (A4b) results in a system of the dual series equations (DSE)
K p
We define that The DSE of (A13) can be arranged as [Sneddon, 1966, p.161]:
nx p
Our goal is to determine the coefficients B(0, p) and B(n, p) appearing in Eq. (A15). The pair of dual series equations (DSE) can be solved by following the procedure given in Sneddon
[1966]. Assume that when 0≤ x≤μ1
The coefficient B0 and Bn in Eq. (A15) are respectively given by the equations [Sneddon, 1966, p. 161, Eqs. (5.4.56) and (5.4.57)] Integrating (A15b), one can obtain
∫
Substituting Eqs. (A17) and (A18) into (A19), one can find that h1(y)satisfies the following
equation: [Sneddon, 1966, p. 161, Eq. (5.4.58)]
[ ]
The summation term on the left-hand side of Eq. (A20) can be expressed as [Sneddon, 1966, p.
59, Eq. (2.6.31)] where )Heav( X is the Heaviside unit step function which is of different value for different range of X such as
Substituting (A21) into (A20), it yields
⎪⎭
Using the property of Heaviside unit step function in Eq. (A22), an equivalent integral equation of (A23) can be obtained
∫
− ⎪⎩⎪⎨⎧∫
− −∑ ∫
⎪⎭⎪⎬⎫By integrating Eq. (A25) and substituting it into Eqs. (A17) and (A18), the coefficients B0
and Bn can be expressed as Eqs. (12) and (13), respectively.
For computational convenience, the coefficients can be written in the matrix form as
⎥⎥
APPENDIX B
Similar to the procedure in Appendix A, the problem with the boundary in Eqs. (27) and (28) results in a set of triple series equations as
nx p
We split Eq. (B1) into the following equations
0
Equations (B2) and (B3) can be regarded as dual series relations by means of which the coefficients C0, D0, Cn and Dn can be determined.
The pair of dual series equations (DSE), i.e., Eq. (B2), can be solved by the procedure given in Appendix A and the coefficients can be written in the matrix form as
⎥⎥
p (B3) is rewritten as
0
with the elements
)
⎥⎦
⎢ ⎤
⎣
⎡ Ω ⋅ − −Ω ⋅ −
= (−12)−
∫
0 ( ) ( 1, ) 1( 2) 2( 1, 2) 21 0 1 1
2 μ μ
μ dy f i
dy y i y df H
p YY
i
i (B19)
⎥⎦
⎢ ⎤
⎣
⎡ Ω − ⋅ − −Ω − ⋅ −
−
−
= (−1)(− (1)1)− − −
∫
0 ( , 1) ( 1, ) 2( 2, 1) 2( 1, 2) 22 1 1 1
1 2
μ μ
λ μ dy j f i
dy y i j df
y j H
YY j j
i j
ij (B20)
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Figure 1 Schematic representation of a partially penetrating well with the screen extends from the top of the aquifer in a confined aquifer
Figure 2 Schematic representation of a partially penetrating well with arbitrary location of well screen in a confined aquifer
Figure 3a The drawdown distribution at dimensionless time for various ρ
Figure 3b The drawdown distribution at dimensionless timeτ =100 for various ρ
Figure 3c The drawdown distribution at dimensionless timeτ =10 for various 4 ρ
Figure 3d The drawdown distribution at dimensionless timeτ =106 for various ρ
Figure 4 The distribution of flux along the well screen at different dimensionless time
Figure 5(a) The spatial drawdown contours at dimensionless time τ =100
Figure 5(b) The spatial drawdown contours at dimensionless time τ =10 3
Figure 5(c) The spatial drawdown contours at dimensionless time τ =104
Figure 6(a) The spatial drawdown contours at dimensionless time τ =105 for α2 =1.0
Figure 6(b) The spatial drawdown contours at dimensionless time τ =105 for α2 =0.5
Figure 7(a) The spatial drawdown contours at dimensionless time τ =106 for ξ1 =12.5 and 5
.
2 =37
ξ with β =50
Figure 7(b) The spatial drawdown contours at dimensionless time τ =106 for ξ1 =25and
2 =50
ξ with β =50
Figure 8 The spatial drawdown contours as at dimensionless time τ =107 for ξ1 =100 and
2 =150 ξ
Figure 9 The influence of the penetration ratio on the flux
VITA (個人簡歷)
姓 名 張雅琪
性 別 女
生 日 民國69 年 03 月 14 日
1998-2002 學士，交通大學土木工程學系 2002-2003 交通大學環境工程研究所碩士班 2003-2009 交通大學環境工程研究所博士班 學經歷
2007-2008 Visiting Scholar, Environmental Science and Engineering, University of North Carolina, USA
行動電話 0988314404
通訊電話 03-5712121#55527
通訊地址 300 新竹市交通大學環境工程研究所
E-mail [email protected]
PUBLICATION LIST
(A) Journal Papers
1. Yeh, H. D., and Y. C. Chang, 2006, New Analytical Solutions for Groundwater Flow in Wedge-shaped Aquifers with Various Topographic Boundary Conditions, Vol. 29, No. 3, 471-480, Advances in Water Resources. (SCI )
2. Y. C. Chang and H. D. Yeh, 2007, Optimum Allocation for Soil Contamination
Investigations in Hsinchu, Taiwan by Double Sampling, Soil Science Society of America Journal, 71(5), 1585-1592, doi: 10.1061/sssaj1006.0130. (SCI)
3. Y. C. Chang, H. D. Yeh and Y. C. Hung, 2008, Determination of the parameter pattern and values for a one-dimensional multi-zone unconfined aquifer, Hydrogeology Journal, 16, 205-214. doi :10.1007/s10040-007-0228-3 (SCI)
4. Y. C. Chang and H. D. Yeh, 2007, Analytical solution for groundwater flow in an anisotropic sloping aquifer with arbitrarily located multiwells, Journal of Hydrology, 347, 143-152, 10.1016/j.jhdrol.2007.09.012. (SCI)
5. H. D. Yeh, Y. C. Chang and V. A. Zlotnik, 2008, Stream Depletion in Wedge-Shaped Aquifers, Journal of Hydrology, 349, 501-511. (SCI)
6. H. D. Yeh, S. B. Wen, Y. C. Chang and C. S. Lu, 2008, A new approximate solution for chlorine decay in pipes, Water Research, 42, 2787-2795. (SCI)
7. Chang, Y. C., H. D. Yeh, 2009, New solutions to the constant-head test performed at a partially penetrating well. Journal of Hydrology, doi:10.1016/j.jhydrol.2009.02.016 (SCI; IF:2.161)
8. Chang, Y. C., H. D. Yeh, 2009, Solutions for mixed boundary value problem in a constant-head test aquifer. Water Resources Research (In review)
9. Chang, Y. C., H. D. Yeh, K. F. Liang and M. C. T. Kuo, 2009, Scale dependency of fractional flow dimension in a fractured formation, Journal of Hydrology (In
10. Chang, Y. C., H. D. Yeh, Solutions for radial two-zone flow in unconfined aquifer under constant-head test. (In preparation)
11. Chang, Y. C., H. D. Yeh, G. Y. Chen, The flow rate across the wellbore in a two-zone unconfined aquifer. (In preparation)
(B) Conference papers
1. 張雅琪、葉弘德，92 年 4 月，離群值的檢定―以新竹市土壤污染數據為對象，第四 屆環境管理研討會, 嘉義，台灣，論文集光碟版 2002.
2. 張雅琪、葉弘德，92 年 11 月，使用雙重採樣法決定最佳採樣樣本數─以新竹市土壤 污染數據為對象，環工學會第十五屆年會及第一屆土壤與地下水技術研討會，國立中 興大學，台中，論文摘要集頁，論文集光碟版。
3 張雅琪、葉弘德，楔形含水層在定水頭邊界條件下之解析解，第 14 屆水利工程研討會, 新竹，台灣， 2004.
4. 張雅琪、葉弘德，93 年 10 月，水平多層自由含水層之地下水流研析，九十三年度農 業工程研討會，中國農業工程學會，桃園，論文摘要集 245 頁，論文集光碟版 1523-1529 頁。
5. Yeh, H. D., Y. C. Chang, and Y. C. Huang, 2005, Identifying horizontal multi-zone unconfined aquifer parameters using simulated annealing, AOGS 2nd annual meeting, Singapore, 58-HS-A0504.
6. 張雅琪、葉弘德，94 年月，應用多變量統計分析近海工業區之地下水污染，第九屆
土壤與地下水污染整治研討會，台北，論文集203～214 頁。
7. 張雅琪、葉弘德，94 年 9 月，楔形含水層在不同邊界條件下之地下水流研析，九十 四年電子計算機於土木水利工程應用研討會，中國土木水利工程學會，國立成功大 學，台南，論文集(III)491-496 頁。
8. 張雅琪、葉弘德，94 年 11 月，應用多變量統計分析地下水污染源:以台灣南部某污 染場址為案例，九十四年度農業工程研討會，中國農業工程學會，桃園，論文摘要集 111 頁，論文集光碟版。
9. 張雅琪、葉弘德，94 年 11 月，利用統計方法區分地下水污染降解模式，第三屆土壤
與地下水研討會，環境工程學會，桃園，論文摘要集487 頁。
10. Chang, Y. C., and H. D. Yeh, 2006, Analytical Solution for Anisotropic Sloping Aquifers with Arbitrarily Located Multiwells and Transient Recharge, AGU Western Pacific Meeting, Beijing, China, H41D-0084, WP96.
11. 張雅琪、葉弘德、梁康阜、郭明錦、范愷軍，95 年 10 月，破碎帶含水層流場的流 動幾何形狀及水力特性之檢定，九十五年度農業工程研討會，中國農業工程學會，國
立成功大學，台南市，論文集98-99 頁，論文集光碟版 331.PDF
12. 張雅琪、葉弘德，96 年 8 月，計算楔形含水層的河川消耗速率及體積，第十六屆水 利工程研討會，中國土木水利工程學會，國立聯合大學理工學院，苗栗，論文摘要集 100 頁，論文集檔案 PDF：542-548。