Conclusions

The uses of the drawdown solution in predicting the drawdown distribution or determining the formation properties have been intensively studied in hydrology literature. Two well-known tests, the constant-head and constant-flux tests, are routinely used toward those goals. However, the derivations from the transient-state solution to the Thiem equation and the approximate solution at large period of time obtained from Laplace domain solution have not ever been thoroughly or correctly studied. Two main contributions from this dissertation can be summarized as follows.

First, the steady-state solutions developed from transient drawdown solutions for constant-head and constant-flux tests in a finite or infinite domain and with or without considering the effect of well radius have been presented. A new transient drawdown solution is derived for the constant-head test by considering the effect of well radius and maintaining a zero drawdown at a finite boundary. The results show that the finite domain condition is necessary for obtaining a steady-state solution from a transient solution for a groundwater flow problem. Such an imposed condition ensures that the mass balance is satisfied and the flow can reach steady state condition within a bounded domain. In addition, the time criteria are provided for the approximation of the finite-domain solution by the infinite-domain solution or Thiem

equation. The infinite-domain solutions can be used to determine the drawdown distribution or the aquifer parameters if the time is smaller than the boundary-effect time criterion. Similarly, the Thiem equation is valid if the time is greater than the steady-state time criterion.

Second, in regard to the large-time solution of wellbore flux in constant-head problem, this study found that the inverse Laplace transform formula in Oberhettinger and Badii [1973] adopted to invert Laplace domain solution by Chen and Stone [1993]

should be under the constraint p>

λ

. One will obtain an erroneous solution if applying that inverse Laplace transform formula without satisfying this necessary constraint. Therefore, the inconsistent results obtained by Chen and Stone [1993]

from the SPLT relationship and the Tauberian theorem that arose from a violation of the constraint from applying the inverse Laplace transform formula rather than using the SPLT relationship. In addition, a new large-time solution of the wellbore flux for the constant-head test is presented based on the SPLT relationship and the work of Ritchie and Sakakura [1956].

References

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Carslaw, H.S., and J.C. Jaeger (1959), Conduction of Heat in Solids, 2nd ed., Oxford University Press, London.

Charbeneau R.J. (2000), Groundwater hydraulics and pollutant transport, Prentice Hall, New York.

Chakrabarty, C., S.M.F. Ali, and W.S. Tortike (1993), Analytical solutions for radial pressure distribution including the effects of the quadratic-gradient term, Water Resour. Res., 29, 1171-1177.

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Chen, C.S. (1985), Analytical and approximate solutions to radial dispersion from an

injection well to a geological unit with simultaneous diffusion into adjacent strata, Water Resour. Res., 21, 1069-1076.

Chen, C.S. (1986), Solutions for radionuclide transport from an injection well into a single fracture in a porous formation, Water Resour. Res., 22, 508-518.

Chen, C.S., and W.D. Stone (1993), Asymptotic calculation of Laplace inverse in analytical solutions of groundwater problems, Water Resour. Res., 29, 207-210.

Crank, J. (1956), Mathematics of Diffusion, Oxford Univ. Press, New York.

Doetsch, G. (1961), Guide to the Applications of Laplace Transforms, Van Nostrand Reinhold, New York.

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Res., 26, 351-356.

Gerke, H.H., and M.T. van Genuchten (1993), Evaluation of first-older water transfer term for variably saturated dual-porosity flow modes, Water Resour. Res., 29, 1225-1238.

Hantush, M.S. (1960), Modification of the theory of leaky aquifers, J. Geophys. Res., 65, 3713-3725.

Jaeger, J.C. (1943), Heat flow in the region bounded internally by a circular cylinder, Proc. Roy. Soc. Edinburgh, 61, 223-229.

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penetrating well in a two-layer aquifer, Water Resour. Res., 19, 567-578.

Mathias, S.A., and R.W. Zimmernan (2003), Laplace transform inversion for late-time behavior of groundwater flow problems, Water Resour. Res., 39, 1283, doi:

10.1029/2003WR002246.

Neuman, S.P., and P.A. Witherspoon (1969), Theory of flow in a confined two-aquifer system, Water Resour. Res., 5, 803-817.

Oberhettinger, F., and L. Badii (1973), Tables of Laplace Transforms, Springer-Verlag, New York.

Peng, H. Y., H. D. Yeh, and S. Y. Yang (2002), Improved numerical evaluation for the radial groundwater flow equation, Adv. Water Resour., 25, 663-675.

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Res., 22, 1717-1727.

Ritchie, R.H., and A.Y. Sakakura (1956), Asymptotic expansions of solutions of the heat conduction equation in internally bounded cylindrical geometry, Appl. Phys., 27, 1453-1459.

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Yang, S.Y., and H.D. Yeh (2002), Solution for flow rates across the wellbore in a two-zone confined aquifer, J. Hydraulic Eng. ASCE, 128, 175-183.

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Yeh H.D., S.Y. Yang, H.Y. Peng (2003), A new closed-form solution for a radial two-layer drawdown equation for groundwater under constant-flux pumping in a finite-radius well, Adv. Water Resour., 26, 747-757.

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Appendixes

Appendix A: Examination of SPLT for dual-pore problem Appendix B: Derivation of Equation (7)

Appendix C: Derivation of Equation (9) Appendix D: Derivation of Equation (10)

Appendix A: Examination of SPLT for dual-pore problem

A fractured aquifer consisted of fractures, fissures, cracks, and macropores can be simulated as two separate but connected medium, one is associated with fracture medium and the other involves porous matrix. Such a fractured aquifer system is called dual-porosity system. Mathematically, a dual-porosity system involves two flow equations which are coupled by means of a transfer term of water flow [Dykhuizen, 1990]. Gerke and van Genuchten [1993] proposed a first-order model to account this transfer term which is assumed to be proportional to the difference in pressure head between the fracture medium and porous matrix. The specific value of water transferred to head difference is referred as water transfer coefficient.

For the dual-porosity media problem, Gerke and van Genuchten [1993] solved a large-time water transfer coefficient by comparing two Laplace domain solutions, one was the “first-order” flux equation between fracture and matrix and the other was Richards’s equation within the matrix block, while p became small. Moreover, Mathias and Zimmerman [2003] also obtained a large-time water transfer coefficient for the dual-porosity media problem based on the time domain approach. Their solution differs from the one obtained by Gerke and van Genuchten [1993] using the Laplace domain approach based on the SPLT relationship. When comparing both Laplace- and time domain solutions with the exact solution, they implicitly indicated

that the SPLT relationship did not hold. Therefore, the following context will go through the detailed mathematical derivations involved in the issues of Gerke and van Genuchten [1993] and resolve the argument on the validity of the SPLT relationship.

If the pressure head of the fracture pore is considered to be a constant in time, the water transfer coefficient can be obtained by comparing two rearranged flow equations in Laplace domain as [Gerke and van Genuchten, 1993]

( ) ( )

in the matrix block; h is the imposed head at the fracture boundary; and variables _f

( )

[

¹ wf Cmp Ka

]

^0.5

a −

ξ = and ζ = 1

(

−wf

)

Cmp αw while a is the characteristic half width of the matrix block, w is the fracture porosity, _f K is the hydraulic _a conductivity of the matrix block near the fracture/matrix interface, C is the specific _m water capacity at the matrix, and α_w is the water transfer coefficient.

Mathias and Zimmerman [2003] applied the Laurent-type expansion to Equations

formula of Doetsch [1961] when time is large. Based on the time domain approach, they obtained an exact large-time water transfer coefficient of

)

α which differs from the Laplace domain approach of 3K_a a2

w =

α obtained by Gerke and van Genuchten [1993] derived from Equations

(A1) and (A2) based on the SPLT relationship.

Once the water transfer coefficient is obtained, the normalized head difference for large value of time is [Mathis and Zimmerman, 2003]

( )

with the exact solution given in Crank [1956, p.48] which was shown in Mathis and Zimmerman [2003] as

It is clear that the normalized head difference approaches zero while dimensionless time Kat

[ ( )

2a ²(1−wf)Cm

]

on the RHS of Equation (A4) goes to a very large value.

After comparing with the exact solution given by Crank [1956], Mathias and

from the time domain approach and Laplace domain approach arises from the use of the SPLT relationship. However, this study finds that this relationship is indeed correct and the defect in Gerke and van Genuchten [1993] is mainly caused by ignoring the convergent requirements of series expansion for Equations (A1) and (A2).

Gerke and van Genuchten [1993] expanded tanh

( )

ξ and 1

(

1+ζ

)

in Equations (A1) and (A2) in terms of a series, respectively. Those two equations were then respectively expressed as

( ) ( ) ( ) ( )

compares Equations (A5) with (A6) and sets the second term on RHS of these two

and fourth terms are α_w = 15 2K_a a² ( ≅2.74K_a a²) and α_w =³ 31517K_a a² )

65 . 2

(≅ K_a a² , respectively. It seems that the estimated water transfer coefficient appears monotonously decreasing from 3 and asymptotically approaches to 2.47.

Notice that Gerke and van Genuchten [1993] truncated the third and remaining higher order terms of p in Equations (A5) with (A6) because of small p.

In fact, the series expansion for tanh

( )

ξ in Equation (A1) and 1

(

1+ζ

)

in Equation (A2) should be restricted to the convergent criteria ξ <π 2 and ζ <1

[Abramowitz and Stegun, 1970, p.15 and 85], respectively. The series expansion of

( )

ξ

tanh in Equation (A1) can be expressed as [Abramowitz and Stegun, 1970, p.85]

( )

ξ < . Subsequently, one uses the Fourier expansion of the Bernoulli number

[Abramowitz and Stegun, 1970, p.805] and obtains

( ) ( )

Therefore, the nth term of water transfer coefficient is obtained by letting Equation (A5) equal to Equation (A6) as

( ) ( )

where the limit of Riemann Zeta function,

∑

^∞=

− approach based on the SPLT relationship is exactly the same as that of Mathias and Zimmerman [2003] derived from the time domain approach. Therefore, the dispute in the discrepancy that was calculated using Laplace domain approach by making use of the SPLT relationship and those found by working in the time domain is clearly resolved.

Appendix B: Derivation of Equation (7)

By applying the Laplace transform, the integral function of Equation (7) can be expressed as

( ) ( ) ( )

The RHS of (B1) is a double integral and can be rearranged as

( ) ( ) ( )

The Gamma function is defined as [Abramowitz and Stegun, 1970, p.255]

(

⁺

)

⁼

∫

^{∞ −}

Γ

0

1 e u du

x ^{u x} (B3)

Replacing the second integral in Equation (B2) by Gamma function, the RHS of Equation (B2) after the integration gives

( )

reduces to

( ) ( ) (

λ

)

λ

p dx p

x L t

x

ln 1

0 1

=



 





+

∫

Γ

∞

(B5)

This derivation shows that Equation (7) is valid only under the condition thatp>λ.

Appendix C: Derivation of Equation (9)

The first four terms of Equation (8) can be rewritten using the notation of Ritchie and Sakakura [1956] as

( ) ( ) ( ) ( )

^_ properties of binomial coefficient, Gamma function, and Polygamma function applied to the following derivation are, respectively, [Abramowitz and Stegun, 1970]

( )

^s

( ) ( ) ( )

^{1 11 11 1}

Substituting Equations (C6) - (C9) into Equation (C1) gives

( ) ( ) ( )

Thus, the inverse Laplace transform of Equation (6) results in Equation (9) when truncating high-order terms of Equation (8).

Appendix D: Derivation of Equation (10)

General solutions of one-dimensional radial heat conduction equation in analogy to the groundwater drawdown equation subject to Cauchy boundary condition at the edges of hollow cylinder were given in Carslaw and Jaeger [1959]. The Cauchy boundary conditions in terms of drawdown were expressed as

a boundary and outer boundary in the considered region, respectively. Equations (D1) and (D2) represent the combination of the constant-head and constant-flux boundary conditions.

By using the Laplace transforms, the drawdown solutions based on Equations (D1) and (D2) is

[ ( ) ] [ ( ) ]

For the constant-head test with a finite well radius and outer boundary, the constants in Equations (D1) and (D2) can be replaced by k1=0, k2= -1, k3=sw, k1’=0, k2’=1, k3’=0, a=rw, and b=R. By carefully substitution, the drawdown solution, i.e., Equation (10) can be obtained from Equation (D3).

Table 1 Transient and steady-state drawdown solutions for constant-head and constant flux tests

Constant-head test Constant-flux test State

Infinite domain Finite domain Infinite domain Finite domain

Considering well radius

Transient Equation (3) Equation (10) Equation (13) Equation (15)

Steady s w Thiem equation Infinity Thiem equation

Neglecting well radius

Transient No solution No solution Theis equation Equation (16)

Steady No solution No solution Infinity Thiem equation

Table 2 The boundary-effect and steady-state time criteria for the finite-domain solutions

Note: τ =R²S T.

Solution type Boundary-effect time criterion, t₁

Steady-state time criterion, t₂

Constant-head test τ

10

1 4τ

Constant-flux test ₂τ 10

3 8τ

Constant-flux test when neglecting well radius

16τ

1 τ

4 10⁵

Outter boundary Inner

boundary h(r,t) or s(r,t)

r

s

h

rw

r s

h=hw

or s=sw

Initial condition h=ho or s=0

Q

Confined aquifer T, S

h

r

Figure1 Schematic diagram of drawdown distribution under constant-head or constant-flux test in a confined aquifer.

10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ Dimensionless time, Tt/(rw2S)

0.1 1.0

Dimensionless wellbore flux, Q/(2πTsw)

Graph 1 Eq. (5)

Eq. (11), R/rw=10 Eq. (11), R/rw=100 Thiem equation R/rw=1e5

Figure2 Dimensionless wellbore flux versus dimensionless time for the finite-domain solution, Equation (11) with R r_w = 10 and 10², the infinite-domain solution, Equation (5), and the Thiem equation. Note that both Equations (5) and (11) are solved for the constant-head test.

10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ Dimensionless time, Tt/rw2S

0 2 4 6

Dimensionless drawdown, s/(Q/2πT)

Graph 1 Eq. (13)

Eq. (15), R/rw=1E3 Eq. (15), R/rw=1E4 Thiem equation R/rw=1e6

Figure3 Dimensionless drawdown versus dimensionless time for the finite-domain solution, Equation (15), with R r_w = 10³ and 10⁴ and r r_w =10² , the infinite-domain solution, Equation (13), and the Thiem equation. Note that both Equations (13) and (15) are solved for the constant-flux test.

作者簡歷

姓名：王智澤 (Chih-Tse Wang) 性別：男

生日：1961 年 08 月 23 日 學歷：

國立交通大學環境工程研究所博士(2001-2007) 國立中興大學土木工程研究所碩士(1988-1990) 國立成功大學水利工程學系學士(1980-1984) 經歷：

普豐環保科技公司專案經理(2001-2003)

美商衛世敦國際公司台灣分公司資深工程師(1997-2001) 通訊地址：106 台北郵政 91-14 號信箱

聯絡電話：0960289264

電子郵件：[email protected]

著作：

期刊論文

Yeh, H.D., and C.T. Wang (2007), Large-time solutions for groundwater flow problems using the relationship of small p versus large t, Water Resources Research, 43(6), W06502, doi:10.1029/2006WR005472.

Wang, C.T., and H.D. Yeh (2007), Obtaining the steady-state drawdown solutions of constant-head and constant-flux tests, Hydrological Processes 21, doi:10.1002/hyp.6950.

研討會論文

Wang, C.T., and H.D. Yeh (2007), A Study for Residual Drawdown Solution with Considering the Wellbore Storage, American Geophysical Union (AGU) 2007 Fall Meeting, Poster H13B-1244, San Francisco, CA, December 10-14.

王智澤、葉弘德 (2006)，定水頭試驗井緣流量之近似解研析，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，論文摘要集 97 頁，論文集光碟版 320-330 頁。

葉弘德、王智澤 (2006)，雙孔隙介質系統中水移轉係數估算之研究，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，論文摘 要集96 頁，論文集光碟版 312-319 頁。

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案例研究，第十屆海峽兩岸環境保護研討會，台中市，論文摘要集15 頁。

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年度農業工程研討會，桃園市，論文摘要集244 頁，論文集光碟版 1515-1522

頁。

王智澤、葉弘德 (2004)，穩態定流量及定水頭之抽水洩降研析，第十四屆水

利工程研討會，新竹市，論文集F217-F221 頁。

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及水質保護研討會，台中市，論文集85-93 頁。

在文檔中 定水頭與定流量試驗洩降解之研究 (頁 36-61)