A semi-analytical solution of the drawdown distribution is developed for CHT performed in an unconfined aquifer with a partially penetrating well. The Laplace transforms and the method of separation of variables are employed to derive the transient drawdown in the Laplace domain for CHT. The Stehfest method is used to invert the solutions in time-domain and the Shanks method is applied to accelerate convergence in evaluating the infinite summations in the solution.
Large slopes of the drawdown distribution curves can be observed near the free surface boundary and the edge of the screen, which indicates that the vertical groundwater flows occur at these two areas. The dimensionless drawdown decreases with increasing S but y increases with the length of well screen. For different r , the drawdown decreases with the w increase of radial distance from pumping well and it might produce large error in drawdown if assuming the radius of pumping well is infinitesimal.
The present solution can be used for describing the transient drawdown distribution or investigating the effects of specific yield and conductivity ratio on the drawdown distribution in unconfined aquifers. In addition, the present solution can reduce to the solution for a fully penetrating well in either confined or unconfined aquifers under CHT.
APPENDIX A
Detailed derivations of Eqs. (14) and (15)
The dimensionless governing equations of Eqs. (1) and (3) can be expressed as The dimensionless initial conditions for regions 1 and 2 are
0 and the boundary conditions at the bottom and top of the aquifer for regions 1 and 2 in terms of dimensionless form can be written as
) 0
The dimensionless boundary conditions at ρ = 0 and infinity are respectively written as
l The dimensionless boundary condition along the screen is expressed as
0
0
The Laplace transform is defined as:
∫
∞ − dimensionless drawdown solutions can be obtained by taking Laplace transforms of governing equations Eqs. (A1) to (A2) using the initial condition (A3) and the results are* The transformed boundary conditions at the bottom and top of the aquifer for regions 1 and 2 can be written as
In the same fashion, the transformed boundary conditions at ρ= 0 and ∞ are After taking the Laplace transform, the boundary condition along the well screen is
0 and continuity conditions become
p l
Equations (A13) and (A14) can be, respectively, transformed as
1 Dividing throughout Equations (A23) and (A24) by F1G1 and F2G2, respectively, Equations (A23) and (A24) can then be separated into the following two systems of ordinary differential
1 0
The solutions of (A25) and (A27) subject to the boundary in (A15) are respectively ) respect to p . In addition, substituting (A29) into (A16) yields the following equation
0 )
sin(Ω1mζl = (A31) The eigenvalues Ω in Equation (A29) can then determined by solving Equation (A31) and 1m
the result is
Similarly, substituting (A30) into (A17) gives the following equation )
Equation (A33) can be numerically solved to obtain the eigenvalues Ω in Equation (A30). 2n
The general solutions of (A26) and (A28) are respectively
Then, substituting (A34) into (A18) and (A35) into (A19), respectively, yields
) The product of (A29) and (A38) gives the general solution of Equation (A23) as
)
) following equation obtained by substituting (A42) and (A43) into (A21) and (A20), respectively, as
Equations (A40) and (A41) are organized and expressed as
∑
∑
∞ The term on the left-hand side (LHS) of Eq. (A47) is a half-range Fourier cosine series of the function on the right-hand side (RHS) of Eq. (A47) for the region 0<ζ <1. The Carrying out the integration in (A49) and simplifying the result yields the coefficient A2n'(p) as expressed in Eq. (17).Similarly, substituting (A42) and (A43) into (A22), one can obtain
∑
APPENDIX B
Simplification of Eqs. (15) and (16) to the case of fully penetrating well in unconfined aquifers
Letting 0ζl = in Eqs. (16) and (17), the drawdown solution of Eq. (14) in region 1 is equal to zero and the Laplace-domain solution in Eq. (15) for dimensionless drawdown in region 2 pumping from a fully penetrating well in unconfined aquifers can be expressed as
)
Considering the skin effect in pumping system, Chen and Chang (2003) developed a Laplace-domain solution for describing the flow in an unconfined aquifer with pumping from a fully penetrating well under constant-head test. The solution is expressed as
⎭⎬ transform parameter and L−1{} is the Laplace inversion operator.
If skin effect is negligible (Sk =0), Eq. (B2) can be rearranged as
Since Eqs. (B1) and (B5) are both used for calculating the dimensionless drawdown in unconfined aquifers with fully penetrating wells; these two equations should be identical. The
identical to Eq. (B5).
Some definitions of variable in Eq. (B5) are different from that in Eq. (B1) and the relations are αw =β, Ω2n =εn and p+ω2n =χn. Eq. (B5) can be thus rearranged as
Substituting (B3) into Eq. (4), one can obtain
)
In addition, substituting (B6) into the term on the left-hand side (LHS) of Eq. (B5) results in
The RHS in Eq. (B8) can be further written as
⎥⎦
n n
n
n n
n 2 2
2
2 2
2 sin(2 ) 2
) sin(
2 )
) sin(
cos(
1
Ω + Ω
= Ω Ω + Ω Ω
(B12)
From Eq. (B7) to (B11), the following relation is established
n n
n
n
n 2 2
2
2 ) 2 sin(
) sin(
2 )
cos(
1
Ω + Ω
= Ω ε
λ (B13) Furthermore, base on Ω2n =εn and Eqs. (B6)-(B13), one can easily prove that Eq. (B1) is identical to Eq. (B5).
APPENDIX C
Simplification of Eq. (15) to the case of fully penetrating wells in confined aquifer
The Laplace-domain solution of Eq. (15) in region 2 for describing the flow due to pumping from a fully penetrating well in confined aquifers can be expressed as
)
Substituting Eq. (C2) into Eq. (C1) and using L'Hospital's rule, Eq. (C1) is simplified as
)
The non-dimensional form of Eq. (C3) is
) Eq. (C4) is further simplified as
)
Assuming the aquifer is confined, homogeneous and isotropic, the solution in Eq. (C5) is identical to the Laplace-domain solution in Hantush (1964) written as
) (
) ) (
, (
0 0
w
w
pK r
r s K
p r
s ⋅
⋅ ⋅
= λ
λ
(C6)where λ= (p⋅Ss)/K .
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Fig 1. Schematic representation of an unconfined aquifer with a partially penetrating well.
(a) (b)
(c) (d)
Fig 2. The dimensionless drawdown distributions at τ = (a) 1 , (b) 102 , (c) 104 , and (d) 106 .
Fig 3. The effect of specific yield on the dimensionless drawdown during CHT.
Fig 4. The dimensionless drawdown distributions at the well screen extended from ζ = to ζl β
ζ = in region 2.
(a)
(b)
Fig 5. Relationship for dimensionless drawdown versus dimensionless time with ζ =50, 75, and 100 at ρ =10 or 100 for S = (a) 0 and (b) 0.1 y
(a)
(b)
Fig 6. Spatial flow pattern in an unconfined aquifer with a partially penetrating well for κ =1,
=100
β , 25ζl = , S =10−4 at τ = 104 when S = (a) 0 and (b) 0.1. y
Fig 7. The effect of conductivity ratio (κ) of region 2 on the dimensionless drawdown during CHT.
Fig 8. Drawdown distribution for a well with three different well radii (r = 1, 0.1 and 0.01 m) w with σ =103, ζ =0.75, 5ζl =0. and κ =1.
Fig 9. Effect of α on drawdown in a 100 m thick aquifer when σ =103, 1κ = at ζ =0 and r = 100, 31.62, and 10 m for α = 100, 10-1 and 10-2, respectively.