Discussion of Papers/
Christopher Neuzil, Discussion Editor
‘‘Approximate Discharge for Constant Head Test with Recharging Boundary,’’ by Philippe Renard, May–June 2005 issue, v. 43, no. 3: 439–442.
Discussion by Hund-Der Yeh, Institute of Environmental Engineering, National Chiao Tung University, 300 Hsinchu, Taiwan, [email protected]; Shaw-Yang Yang, Depart-ment of Civil Engineering, Vanung University, 320 Chungli, Taiwan; and Yen-Ju Chen, Institute of Environmental Engi-neering, National Chiao Tung University, Hsinchu, Taiwan
Renard (2005) studied discharge in a constant head test with a recharging boundary in a radial confined aqui-fer. He proposed Laplace-domain solutions for the draw-down in an aquifer and the discharge for an aquifer with one discharging well and a recharge boundary repre-sented by one recharging well. In this comment, we wish to point out problems that exist with the unit step re-sponse function sDuand the drawdown sD, as given in the
equations 12 and 15 in Renard (2005). In addition, we de-rive a time-domain solution of the discharge for the same problem and suggest a numerical approach to evaluate the solution with accuracy to five decimal places.
The definitions of the symbols used herein are identi-cal to those given by Renard (2005). In the case of one discharging well and one recharging well separated by a distance 2lD, the observation well is at distance rD and
(2lD 2 rD) from the real well (discharging well) and
imaginary well (recharging well), respectively. By apply-ing the superposition principle, the unit step response function can be obtained as:
sDuðrD; pÞ ¼ K0ðrD ffiffiffip p Þ p ffiffiffippK1ð ffiffiffip p Þ 2 K0½ð2lD2 rDÞ ffiffiffip p p ffiffiffippK1ð ffiffiffip p Þ ð1Þ
where p is the Laplace variable and K0() and K1() are
the Bessel functions of the second kind of order zero and one, respectively. The first term on the right-hand side of Equation 1 represents the effect of discharge and the second term represents that of recharge. Equation 1 is valid only when the real, observation, and imaginary wells are along a straight line. Renard (2005) gave a dis-tance between the observation and the imaginary wells
as (2lD 2 1), which was incorrect. Therefore, equation
15 of Renard (2005) should read as:
sDðrD; pÞ ¼ K0ðrD ffiffiffip p Þ 2 K0½ð2lD2 rDÞ ffiffiffip p pfK0½ ffiffiffip p 2 K0½ð2lD21Þ ffiffiffip p g ð2Þ Renard (2005) presented the Laplace-domain solu-tion of the discharge from the well in his equasolu-tion 13 as:
qDðpÞ ¼ K1ð ffiffiffip p Þ ffiffiffi p p fK0½ ffiffiffip p 2 K0½ð2lD21Þ ffiffiffip p g ð3Þ
In addition, he also gave a simple approximate solu-tion of the discharge rate into a well using a weighted average of the two asymptotes plus a correction term as:
qDðtDÞ ¼ A lnð1 1 ffiffiffiffiffiffiffiptD p Þ1 B lnð2lD21Þ 1 C ð4Þ
In fact, the analytical solution in the time domain for Equation 3 can be derived using the Bromwich integral method (Peng et al. 2002; Yang and Yeh 2002), and the final result is:
qDðtDÞ ¼
2 p
ZN
0
e2tDu2J1ðuÞB2ðuÞ 2 Y1ðuÞB1ðuÞ B2
1ðuÞ 1 B22ðuÞ
du ð5Þ
where J0() and Y0() are, respectively, the Bessel
func-tions of the first and second kinds of order zero, and J1()
and Y1() are, respectively, the Bessel functions of the first
and second kinds of order one. In addition, B1(u) ¼
J0(u) 2 J0((2lD2 1)u) and B2(u)¼ Y0(u) 2 Y0((2lD2 1)u).
A numerical approach, including the singularity removal scheme, the Gaussian quadrature, and Shanks’ method (Peng et al. 2002; Yeh et al. 2003), can be used to eval-uate Equation 5 with accuracy to five decimal places for a very wide range of dimensionless time.
References
Peng, H.Y., H.D. Yeh, and S.Y. Yang. 2002. Improved numerical evaluation of the radial groundwater flow equation. Ad-vances in Water Resources 25, no. 6: 663–675.
Yang, S.Y., and H.D. Yeh. 2002. Solution for flow rates across the wellbore in a two-zone confined aquifer. Journal of Hydraulic Engineering ASCE 128, no. 2: 175–183. Yeh, H.D., S.Y. Yang, and 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. Advances in Water Resources 26, no. 7: 747–757.
Copyrightª 2007 The Author(s)
Journal compilationª 2007 National Ground Water Association. doi: 10.1111/j.1745-6584.2007.00386.x
