A general analytical solution is developed for describing transient hydraulic head distribution induced from a fully-penetrating vertical well, horizontal well or RC well in an unconfined aquifer bounded by two parallel streams. The head solution is derived by means of double-integral transform, finite Fourier cosine transform, and Laplace transform. The boundary conditions at the interfaces where streambeds are connected to the aquifers are treated as third-type boundary conditions with different hydraulic parameters. The first-order free surface equation is used to describe the depletion of the water table. The aquifer is considered as a finite extent with no-flow boundary conditions in the y direction for the purpose of simplifying an infinite series in the SDR solution developed based on the head solution and Darcy’s law. The present solution is applied to predict the hydraulic head inside a horizontal well for the field case reported in Mohamed and Rushton [2006]. The solution is also applied to predict the hydraulic head near the caisson of the collector well for the field cases given in Schafer [2006]
and Jasperse [2009]. The predicted results seem to be reasonable when compared with these observed field data. Spatial variation in hydraulic head is investigated by using the present solution, and major conclusions drawn from that can be summarized as follows:
1. Depending on the ratio of Sy/Kv, the introduction of a fully-penetrating vertical well may result in significant vertical groundwater flow near water table during the early
the ratio is small. The transient head predicted by a model which neglects the vertical flow is significantly underestimated.
2. If the ratio of the streambed permeability over the aquifer one is less than 10-2, a deep and wide drawdown cone is eventually produced by a long period of pumping.
3. The RC well can produce small drawdown if the laterals are installed toward the stream.
4. Before the filtration, the largest drawdown occurs right at the center of a RC well.
Once the filtration starts to recharge the aquifer, the largest drawdown begins to move landward and away from the center of the well.
Some behaviors associated with temporal distributions of SDR are also examined, and the conclusions drawn from those observations are summarized as follows:
1. For an unconfined aquifer, the gravity drainage has a significant effect on temporal SDR. Neglecting the effect of the vertical flow described by the free surface equation tends to overestimate the temporal SDR. Such a conclusion is confirmed by the comparison of SDR predicted from the present solution with that taken from a field experiment executed by Hunt et al. [2001].
2. The ratio of K1'/K2' determines the distributions of filtration from the two parallel streams. When the boundary condition at RHS stream is regarded as a constant-head boundary condition, the RHS streambed has the same hydraulic conductivity as the aquifer (K2'=Kh). The steady-state SDR for the LHS stream depends only on the ratio of K /1' Kh. If K1'/Kh ≥10−2, the steady-state SDR is one. If K1'/Kh <10−7, the steady-state SDR is zero. If 10−7 <K1'/Kh ≤10−2, the steady-state SDR increases
from 0 to 1 with K /1' Kh.
3. A streambed with a lower permeability than an aquifer results in a much smaller SDR for a fixed time.
4. The curve of temporal SDR for an unconfined aquifer tends to be flat over the middle period of time due to gravity drainage from water table. However, this flat vanishes gradually with increasing Kv.
5. The collector well collects more SDR if the laterals are installed toward the stream.
6. The effect of the lateral number on SDR is insignificant if the laterals are symmetric to the center of a collector well.
REFERENCES
Asadi-Aghbolaghi, M., and H. Seyyedian (2010), An analytical solution for
groundwater flow to a vertical well in a triangle-shaped aquifer, J. Hydrol., 393(3-4), 341-348, doi:10.1016/j.jhydrol.2010.08.034.
Butler, J. J., B. A. Zlotnik, and M. S. Tsou (2001), Drawdown and stream depletion produced by pumping in the vicinity of a partially penetrating stream, Ground Water, 39(5), 651-659.
Butler, J. J., X. Zhan, and V. A. Zlotnik (2007), Pumping-induced drawdown and stream depletion in a leaky aquifer system, Ground Water, 45(2), 178-186,
doi:10.1111/j.1745-6584.2006.00272.x.
Fox, G. A., P. DuChateau, and D. S. Durnford (2002), Analytical model for aquifer response incorporating distributed stream leakage, Ground Water, 40(4), 378-384.
Freeze, R. A., and J. A. Cherry (1979), Groundwater, 604 pp., Table 2.2, Prentice-Hall, New Jersey.
Glover, R. E., and G. G. Balmer (1954), River depletion resulting from pumping a well near a river, Trans. Am. Geophys. Union, 35(3), 468-470.
Hantush, M. S. (1965), Wells near streams with semi-pervious beds, J. Geophys. Res., 70(12), 2829-2838.
Huang, C. S., Y. L. Chen, and H. D. Yeh (2011), A general analytical solution for flow to a single horizontal well by Fourier and Laplace transforms, Adv. Water Resour., 34(5), 640-648, doi:10.1016/j.advwatres.2011.02.015.
Huang, C. S., P. R. Tsou, and H. D. Yeh (2012), An analytical solution for a radial collector well near a stream with a low-permeability streambed, J. Hydrol., 446, 48-58, doi:10.1016/j.jhydrol.2012.04.028.
Hunt, B (1999), Unsteady stream depletion from ground water pumping, Ground Water, 37(1), 98-102.
Hunt, B., J. Weir, and B. Clausen (2001), A stream depletion field experiment, Ground Water, 39(2), 283-289.
Hunt, B. (2003), Unsteady stream depletion when pumping from semiconfined aquifer, J. Hydrol. Eng., 8(1), 12-19, doi:10.1061/(ASCE)1084-0699(2003)8:1(12).
Hunt, H. (2006), American experience in installing horizontal collector wells, Water Sci.
Hunt, B. (2008), Stream depletion for streams and aquifers with finite widths, J. Hydrol.
Eng., 13(2), 80-89, doi:10.1061/(ASCE)1084-0699(2008)13:2(80).
Hunt, B. (2009), Stream depletion in a two-layer leaky aquifer system, J. Hydrol. Eng., 14(9), 895-903, doi:10.1061/(ASCE)HE.1943-5584.0000063.
Intaraprasong, T., and H. Zhan (2009), A general framework of stream-aquifer interaction caused by variable stream stages, J. Hydrol., 373(1-2), 112-121, doi:10.1016/j.jhydrol.2009.04.016.
Jacob, C. E. (1950), Engineering Hydraulics, John Wiley & Sons, New York.
Jasperse, J. (2009), Planning, design and operations of collector 6, Sonoma County Water Agency, NATO Science for Peace and Security Series, 169-202,
doi:10.1007/978-94-007-0026-0_11.
Kim, G. B. (2010), Application of analytical solution for stream depletion due to groundwater pumping in Gapcheon watershed, South Korea, Hydrol. Process., 24(24), 3535-3546, doi:10.1002/hyp.7777.
Konikow, L. F., and E. Kendy (2005), Groundwater depletion: A global problem, Hydrogeol. J., 13(1), 317-320, doi:10.1007/s10040-004-0411-8.
Latinopoulos, P. (1985), Analytical solutions for periodic well recharge in rectangular aquifers with third-kind boundary conditions, J. Hydrol., 77, 293-306.
Mohamed, A., and K. Rushton (2006), Horizontal wells in shallow aquifers: Field experiment and numerical model, J. Hydrol., 329(1-2), 98-109, doi:
10.1016/j.jhydrol.2006.02.006.
Ravazzani, G., I. Giudici, C. Schmidt, and M. Mancini (2011), Evaluating the potential of quarry lakes for supplemental irrigation, J. Irrig. Drain. Eng.-ASCE, 137(8), 564-571, doi:10.1061/(ASCE)IR.1943-4774.0000321.
Schafer, D. C. (2006), Use of aquifer testing and groundwater modeling to evaluate aquifer/river hydraulics at Louisville Water Company, Louisville, Kentucky, USA, NATO Science Series IV Earth and Environmental Sciences, 60, 179-198,
doi:10.1007/978-1-4020-3938-6_8.
Sedghi, M. M., N. Samani, and B. Sleep (2009), Three-dimensional semi-analytical solution to groundwater flow in confined and unconfined wedge-shape aquifers, Adv.
Water Resour., 32(6), 925-935, doi:10.1016/j.advwatres.2009.03.004.
Shah, T., D. Molden, R. Sakthivadivel, and D. Seckler (2000), The global groundwater
situation: overview of opportunities and challenges, International Water Management Institute.
Sun, D., and H. Zhan (2006), Flow to a horizontal well in an aquitard-aquifer system, J.
Hydrol., 321(1-4), 364-376, doi:10.1016/j.jhydrol.2005.08.008.
Sun, D., and H. Zhan (2007), Pumping induced depletion from two streams, Adv. Water Resour., 30(4), 1016-1026, doi:10.1016/j.advwatres.2006.09.001.
Theis, C. V. (1941), The effect of a well on the flow of a nearby stream, Eos. Trans. Am.
Geophys. Union, 22, 734-738.
Todd, D. K., and L. W. Mays (2005), Groundwater Hydrology, 3rd ed. John Wiley &
Sons, New York.
Tsou, P. R., Z. Y. Feng, H. D. Yeh, and C. S. Huang (2010), Stream depletion rate with horizontal or slanted wells in confined aquifers near a stream, Hydrol. Earth Syst.
Sci., 14(8), 1477-1485, doi:10.5194/hessd-7-2347-2010.
Ward, N. D., and H. Lough (2011), Stream depletion from pumping a semiconfined aquifer in a two-layer leaky aquifer system, J. Hydrol. Eng., 16(11), 955-959, doi:10.1061/(ASCE)HE.1943-5584.0000382.
Wen, F., and X. Chen (2006), Evaluation of the impact of groundwater irrigation on streamflow in Nebraska, J. Hydrol., 327(3-4), 603-617,
doi:10.1016/j.jhydrol.2005.12.016.
Yeh, H. D., Y. C. Chang, and V. A. Zlotnik (2008), Stream depletion rate and volume of flow in wedge-shape aquifers, J. Hydrol., 349(3-4), 501-511,
doi:10.1016/j.jhydrol.2007.11.025.
Yeh, H. D., C. S. Huang, Y. C. Chang, and D. S. Jeng (2010), An analytical solution for tidal fluctuations in unconfined aquifers with a vertical beach, Water Resour. Res., 46, W10535, doi:10.1029/2009WR008746.
Zhan, H., and E. Park (2003), Horizontal well hydraulics in leaky aquifers, J. Hydrol., 281(1-2), 129-146, doi:10.1016/S0022-1694(03)00205-1.
Zlotnik, V. A., and D. M. Tartakovsky (2008), Stream depletion by groundwater pumping in leaky aquifers, J. Hydrol. Eng., 13(2), 43-50, doi:
10.1061/(ASCE)1084-0699(2008)13:2(43).
Zume, J., and A. Tarhule (2008), Simulating the impacts of groundwater pumping on stream-aquifer dynamics in semiarid northwestern Oklahoma, USA, Hydrogeol. J.,
APPENDIX A DEVELOPMENTOF EQUATIONS (10) AND (29)
Latinopoulos [1985] presented double-integral transform including various kernel
functions in a finite domain with any two of three boundaries such as first-type, second-type and third-type boundaries. Double-integral transform with the kernel function corresponding to a finite domain with two third-type boundaries, in our
notation, [Latinopoulos, 1985, Table I, p. 298] is
[
x c x]
dxwhere α is the variable of the transform and the roots of equation (20). Applying the i transform to a second-order differential ∂2h/ x∂ 2 with integration by parts and the boundary conditions, equations (5) and (6), results in
)
The formula for the inverse double-integral transform is
∑
∞Three different integral transforms applied to equations (3)-(9), respectively, leads to an ordinary differential equation (O.D.E.). Firstly, applying the double-integral transform defined by equation (A.1) to variable x in equation (3) with two boundary conditions, equations (5) and (6), results in a partial differential equation (P.D.E.) in terms of variables y, z and t. Secondly, applying finite Fourier cosine transform to y in the P.D.E. with two boundary conditions in equation (7) yields a P.D.E. in terms of z and
t. Lastly, applying Laplace transform to t in the P.D.E. and equations (8)-(9) with the
initial condition, equation (4), leads to a nonhomogeneous O.D.E. and two boundaries in a term of z as
where p is the variable in Laplace domain; ω, the variable in finite Fourier cosine domain, is defined as π j /Wy where j is an integer from 1, 2, 3…∞. Due to Dirac delta function, equation (4) is further separated to two nonhomogeneous O.D.E. as
D
The Dirac delta function introduces two required conditions at z=z0 as z0 Equation (A.11) is obtained based on head continuity requirement at z=z0. Equation (A.12) is derived by integrating equation (A.4) with respect to z from z=z0- to z=z0+ and reflects flux discontinuity at z=z0.
Solving equations (A.9) and (A.10) with two boundary conditions, equations (A.5) and (A.6), as well as two required conditions, equations (A.11) and (A.12), results in the solution in Laplace domain as
[ ]
Both equations (A.13) and (A.14) are a single-value function with respect to the variable p. This is because the function Ha( p) or Hb( p) gives the only result to a
where r' is an arbitrary positive value; θ is an arbitrary angle between 0 and 2π ; I represents imaginary unit. One can prove Ha(p+)=Ha(p−) or Hb(p+)=Hb(p−) for any value of R and θ if substituting equations (A.19) and (A.20) into equation (A.13) or equation (A.14).
Equations (A.13) and (A.14) have infinite simple poles at negative x axis in the complex plane. These poles are in fact the roots of the equation derived from letting the denominator of equation (A.13) to be zero. Note that equations (A.13) and (A.14) have the same denominator defined by equation (A.15). Obviously, two of these poles are
=0
p and
)
( 2 2
00=− α +ω
= i
s h
S p K
p (A.21)
which is found from λ(p)2 =0. The other poles are the roots of 0
)]
( sinh[
) ( )]
(
cosh[D p +K p D p =
S
p y λ vλ λ (A.22)
which is from equation (A.15). Equation (A.22) has only one root p lying between 0 p=0 and p= p00 at negative x axis. We let λ(p0)=β0 for an expression without a radical sign. Substituting p= p0 and λ(p0)=β0 into equation (A.22) results in equation (21). Note that β is the root of equation (21) and reflects the value of 0 p 0 through λ(p0)=β0. On the other hand, equation (A.22) has infinite roots p behind k
p00
p= at negative x axis. We let λ(pk)=Iβk for an expression without I and a radical sign. Substituting p= pk and λ(pk)=Iβk into equation (A.22) results in
equation (22). The value of p depends on k β through k λ(pk)=Iβk.
The inverse Laplace transform for a single-value function is the sum of its residue at each pole in the complex plane. The residue can be determined by the formula as
)
[
s( y k v)sin( k) v( s 2 y) kcos( k)]
{
sin[( ) ] cos[( ) ]}
The solution in x and y domains can be obtained by the inversion of finite Fourier cosine transform and double-integral transform. The inversion of finite Fourier cosine transform is derived by the formula as
Hb . The inverse double-integral transform is obtained by the formula as equation (A.3).
The solution for a fully-penetrating vertical well or RC well depends on the values of v and r. If v=1 and r=0, the solution reduces to equation (10) for a fully-penetrating vertical well. On the other hand, the solution reduces to equation (29) for a RC well by substituting v=0 and r=1 into the solution, then by integrating the result with respect to x0 and y0 along the component of each lateral in the x and y direction, respectively, then by taking the sum of the result for each lateral, eventually by dividing the result by the sum of each lateral length.
Table 1. Classification of original solutions involved in two-dimensional flow induced from a fully-penetrating vertical well
References Cited in the Text Aquifer Category Stream Treatment
Theis [1941]a confined aquifer first-type boundary condition
Glover and Balmer [1954]a confined aquifer first-type boundary condition
Hantush [1965]a confined aquifer third-type boundary condition
Hunt [1999]a confined aquifer source term of zero-width stream
Butler et al. [2001]b confined aquifer source term of finite-width stream
Fox et al. [2002]a confined aquifer source term of finite-width stream
Sun and Zhan [2007]a confined aquifer divided into three region for two low-permeability streambeds
two parallel streams treated as first-type boundary conditions
Zlotnik and Tartakovsky [2008]a leaky confined aquifer source term of zero-width stream Yeh et al. [2008]a wedge-shaped confined aquifer first-type boundary condition Intaraprasong and Zhan [2009]a confined aquifer divided into two region for
low-permeability streambed
first-type boundary condition with variable stage Asadi-Aghbolaghi and Seyyedian [2010]a triangle-shaped confined aquifer first-type boundary condition
The superscripts a and b represent the presentation of an analytical solution in time domain and a semi-analytical solution in Laplace domain, respectively.
Table 2. Classification of original solutions involved in quasi three-dimensional and three-dimensional flow
References Cited in the Text Aquifer Category Well Type Stream Treatment quasi three-dimensional groundwater flow
Hunt [2003]a semi-confined aquifer fully-penetrating vertical well source term of zero-width stream Butler et al. [2007]b leaky confined aquifer with
considering transient vertical flow in the lower aquitard
fully-penetrating vertical well source term of finite-width stream
Hunt [2008]b semi-confined aquifer extending finitely with two no-flow boundaries
fully-penetrating vertical well source term of finite-width stream
Hunt [2009]b two-layer aquifer system fully-penetrating vertical well in the upper aquifer
source term of zero-width stream in governing equation for the top aquifer Ward and Lough [2011]b two-layer aquifer system fully-penetrating vertical well in
the lower aquifer
source term of zero-width stream in governing equation for the top aquifer three-dimensional groundwater flow
Zhan and Park [2003]b leaky confined aquifer underlying water reservoir
horizontal well constant-head reservoir connecting the lower aquifer without low-permeability aquitard Sun and Zhan [2006]b leaky confined aquifer underlying
water reservoir
horizontal well constant-head reservoir connecting the lower aquifer by aquitard with effects of storage and low permeability
Sedghi et al. [2009]b wedge-shaped unconfined aquifer partially-penetrating vertical well
first-type boundary condition Tsou et al. [2010]a confined aquifer slanted well first-type boundary condition Huang et al. [2011]a unconfined aquifer horizontal well first-type boundary condition Huang et al. [2012]a unconfined aquifer RC well third-type boundary condition
The superscripts a and b represent the presentation of an analytical solution in time domain and a semi-analytical solution in Laplace domain,
Table 3. The default values and field data for aquifer parameters and well configurations Parameter Default Values Aquifer in Sarawak,
Malaysia
Aquifer near Ohio River in Louisville, Kentucky
Aquifer near Russian River in California
Aquifer near Doyleston in New (Kh, Kv) (0.1 m/day, 0.1Kh) (10, 0.06) m/day (119, 40) m/day (650, 217) m/day (3.78, 0.08) m/hour
Well Type none single horizontal
well
RC well with seven laterals RC well with ten laterals single vertical well Q 100 m3/day time-dependent pumping rate as described in the corresponding text 63 m3/s
(L1, L1,…LN) none 300 m (61, 61, 61, 73, 73, 73, 73)
Figure 1. Schematic diagram of an unconfined aquifer with a vertical well or a radial collector well; (a) and (c) top view; (b) and (d) cross section view
Figure 2. The patterns of the LHS and RHS functions from equation (20) for (a) K2'≠0 and (b) K2'=0 as well as from (c) equation (21) and (d) equation (22)
Figure 3. Contours of spatial head distributions induced from a fully-penetrating vertical well for various Sy/Kv when t=10 m/day
Figure 4. The spatial head distributions induced from a horizontal well for (a) 3D view and (b) top view
Figure 5. The predicted drawdown from the present solution and observed drawdown from Mohamed and Rushton [2006]
Figure 6. The contours of transient water table due to pumping in a radial collector well with three symmetrical laterals for various times
Figure 7. The contours of steady-state water table due to pumping in a radial collector well with four different configurations. (a) symmetry (b) non-symmetry (c) laterals toward stream (d) laterals landward
Figure 8. Temporal distribution curves of SDR for four different lateral configurations
Figure 9. Water levels predicted by the present solution and the observed field data from Schafer [2006]
Figure 10. Water levels predicted by the present solution and the observed field data from Jasperse [2009]
Figure 11. Type curve of steady-state SDR for the LHS stream versus K /1' Kh
Figure 12. Steady-state water table distributions at y=0 for various K /1' Kh
Figure 13. Temporal distribution curves of SDR for the LHS stream for various K /1' Kh
Figure 14. Temporal distribution curves of SDR due to pumping in a radial collector well with three symmetrical laterals for various K /v Kh
Figure 15. Temporal distribution curves of SDR for various lateral number and length
Figure 16. Comparison of temporal SDR predicted from the present solution, Theis’ solution [1941] and Hantush’s solution [1965] with field data taken from a field SDR experiment executed by Hunt et al. [2001]
VITA (作者簡歷)
姓 名 黃璟勝（Ching-Sheng Huang）
出生日期 民國 73 年 3 月 18 日
學 歷 93.09-97.01 取得國立交通大學土木工程學系學士學位
97.09-98.06 國立交通大學環境工程研究所碩士班 98.09 直升國立交通大學環境工程研究所博士班 98.09-102.01 國立交通大學環境工程研究所博士班 通訊電話 03-5712121＃55526
行動電話 0980-273694
地 址 973 花蓮縣吉安鄉和平路一段 142 巷 2 弄 10 號
電子郵件 [email protected]
榮譽事蹟 證照
1. 97 年土木工程高考三級及格 2. 98 年土木工程技師考試及格
3. TQC 工程製圖 Auto CAD 技能合格證書 獎狀
1. 第 19 屆水利研討會學生論文競賽博士組第二名獎狀 2. 97 學年度碩士班一年級第二學期書卷獎狀
獎學金
博士班期間榮獲 2012 年中技社科技研究獎學金 (15 萬元)。
學術表現
從 2010 到迄今，已接受的期刊論文共 7 篇，皆收錄在 SCI 水資源領域 Rank factor 10% 以內的期刊，如下 publication list 所示，其中 IF 和 RF 分別代表 impact factor 和 rank factor。
PUBLICATION LIST 期刊論文
1. Yeh, H.-D., C.-S. Huang, Y.-C. Chang, and D.-S. Jeng (2010), An analytical solution for tidal fluctuations in unconfined aquifers with a vertical beach, Water Resour. Res., 46, W10535, doi:10.1029/2009WR008746. (IF: 2.957; RF: 3.85%) 2. Chuang, M.-H., C.-S. Huang, G.-H. Li, and H.-D. Yeh (2010), Groundwater
fluctuations in heterogeneous coastal leaky aquifer systems, Hydrol. Earth Syst. Sci., 14, 1819-1826, doi:10.5194/hess-14-1819-2010. (IF: 3.148; RF: 2.56%)
3. Tsou, P.-R., Z-Y Feng, H.-D. Yeh, and C.-S. Huang (2010), Stream depletion rate with horizontal or slanted wells in confined aquifers near a stream, Hydrol. Earth Syst. Sci., 14, 1477-1485, doi:10.5194/hessd-7-2347-2010. (IF: 3.148; RF: 2.56%) 4. Kuo, C.-C., C.-S. Huang, and H.-D. Yeh (2011), Transient analysis for fluid
injection into a dome reservoir, Adv. Water Resour., 34, 1553-1562, doi:10.1016/j.advwatres.2011.08.006. (IF: 2.449; RF: 8.97%)
5. Huang, C.-S., Y.-L. Chen, and H.-D. Yeh (2011), A general analytical solution for flow to a single horizontal well by Fourier and Laplace transforms, Adv. Water Resour., 34, 640-648, doi:10.1016/j.advwatres.2011.02.015. (IF: 2.449; RF: 8.97%) 6. Huang, C.-S., P.-R. Tsou , and H.-D. Yeh (2012), An analytical solution for a radial
collector well near a stream with a low-permeability streambed, J. Hydrol., 446, 48-58, doi:10.1016/j.jhydrol.2012.04.028. (IF: 2.656; RF: 5.13%)
7. Huang, C.-S., H.-D. Yeh, and C.-H. Chang (2012), A general analytical solution for groundwater fluctuations due to dual tide in long but narrow islands, Water Resour.
Res., 48, W05508, doi:10.1029/2011WR011211. (IF: 2.957; RF: 3.85%)