4.1 Simplified solution
When the well fully penetrates the entire thickness of the formation, i.e., ξ1 is zero and
ξ2 equals β, the drawdown and the well discharge can be obtained using Eqs. (11) and (37), respectively, with coefficients in Eqs. (31) – (34) as
0
* 1
) , ,
(ρ ξ ψ
p p
s = (38) and
p p H
Q( )= 0 (39) Equations (38) and (39) are identical to the solutions of drawdown and flow rate in Laplace domain given in Yang and Yeh [2006].
4.2 Numerical evaluations
Equation (11) contains single and double infinite series which consist of the summations of multiplication of integrals, trigonometric functions, and the modified Bessel functions of second kind. The integrals are in terms of trigonometric functions multiplying associated Legendre functions. This solution involves numerous complicated mathematical functions.
Therefore, numerical approaches including the Gaussian quadrature [Gerald and Wheatley, 1989], Shanks’ transform and Stehfest method are proposed to evaluate the solution. The Gaussian quadrature with 6 terms [Yang and Yeh, 2007] is first utilized to evaluate the
integrals in Eq. (11). Since the oscillation and slow convergence of the multiplication terms, the summations are difficult to evaluate accurately and efficiently. Therefore, the Shanks’
transform method [Shanks, 1955], a nonlinear iterative algorithm based on the sequence of partial sums, is used to compute the summations in Eq. (11). This method has been successfully devoted to efficiently computing the solutions arisen in the groundwater area [see, e.g., Peng et al. 2002; Yeh et al. 2003]. In addition, the Stehfest algorithm [Stehfest, 1970]
with eight weighting factors is further employed to inverse the Laplace domain solution into time domain solution. The proposed numerical approaches can accurately evaluate the drawdown solution to the mixed-type boundary value problem for a flowing partially penetrating well and the results are demonstrated in the following section.
4.3 Drawdown and well bore flux distribution
Figure 3 shows the dimensionless drawdown forβ =100, ξ1=50, ξ2 =100 and various ρ at τ =1, 100, 10 and 4 10 . As indicated in the figure, the dimensionless 6 drawdown is constant along the well screen and decreases with increasing dimensionless radial distance at τ =1 . In addition, the dimensionless drawdown increases with dimensionless time along the unscreened part of the well. Figure 4 shows the plots of the flux along the well screen forβ =100, ξ1 =50 and ξ2 =100 at τ =1, 100, 10 and 4 10 . 6 The dimensionless flux is non-uniformly distributed and large at the screen edge. The
vertical flow is induced by the presence of well partial penetration and the maximum flux at the screen edge occurs when the vertical flows enter the bottom of the well. The spatial dimensionless drawdown contours at τ =100, 10 and 3 10 are plotted in Figure 5. The 4 dimensionless drawdown increases with dimensionless time at a fixed radial distance and flow is horizontal when the dimensionless radial distance is large than 80 and the dimensionless time is 10 . Figures 6(a) and 6(b) show the spatial dimensionless drawdown 4 contours for various α2 with ξ1 =200 and ξ2 =250 at τ =105 and demonstrates the influence of anisotropy on the dimensionless drawdown. The flow is almost horizontal at lower part of the aquifer when the dimensionless radial distance is large than 400 for α2=1 and the vertical flow appears at lower part of the aquifer for α2=0.5. Figures 7(a) and 7(b) show the spatial dimensionless drawdown contours with same length of 50 but different locations of well screen. In Figure 7(a), the screen is symmetric with ξ1 =12.5 and
5 .
2 =37
ξ and in Figure 7(b) the screen extends from the top of the aquifer with ξ1 =25
and ξ2 =50 at τ =105. Since the screen is symmetric about the middle line of the aquifer, the drawdown contours are symmetric as demonstrated in Figure 7(a). Figure 8 illustrates the spatial dimensionless drawdown contours for ξ1=100 and ξ2 =150 at τ =107in an infinite aquifer. On the upper part of the aquifer, the direction of flow is downward when the radial distance is far from the test well and it is upward when the radial distance is close to the well screen. Since the aquifer is infinite, the drawdown at the upper screen ledge flows
upward and then flows toward the bottom of the aquifer and the drawdown at the lower screen ledge flows down toward the bottom of the aquifer.
4.4 Effect of penetration ratio
In order to explore the effect of partial penetration on the well discharge, Figure 9
illustrates the behavior of well discharge in response to four different penetration ratios β
λ
ω = / with λ =50. The well discharges responding to those four cases behave the
same at the small time; however, it decreases with increasing penetration ratio at large time.
If the penetration ratio is smaller than 0.01, the well discharge of this study agrees with that of constant head pumping test in Cassiani et al.[1999] in an aquifer of semi-infinite thickness.
In other words, if the aquifer thickness is greater than 100 times of the screen length, the aquifer can be considered as a semi-infinite aquifer. As the penetration ratio is equal to 1, the well discharge of this study is identical to that of Yang and Yeh [2006] for a fully penetrating well. In addition, well discharges of this study agree with those of Chang and Chen [2003] for ω =0.01 and ω =0.001 when λ =50. As indicated in Figure 9, there are no obvious differences in the well discharges in response to different penetration ratios until τ =104. For the cases of ω =0.1 and ω =0.01, the flow caused by the partial penetration did not reach to the bottom of the aquifer before τ =104. The aquifer thickness has influence on groundwater flow after τ is greater than 10 . The well discharge for 4
01 .
=0
ω is stabilized as τ increases to 10 and the well discharge for 6 ω =0.5 continues to decrease.