Comment on ‘‘Evaluation of the Hantush’s M(a, b) function using
binomial coefficients’’ by B. A. Mamedov and A. S. Ekenog˘lu
Shaw-Yang Yang1 and Hund-Der Yeh2
Received 13 August 2007; revised 25 September 2007; accepted 11 October 2007; published 7 December 2007.
Citation: Yang, S.-Y., and H.-D. Yeh (2007), Comment on ‘‘Evaluation of the Hantush’s M(a, b) function using binomial coefficients’’ by B. A. Mamedov and A. S. Ekenog˘lu, Water Resour. Res., 43, W12602, doi:10.1029/2007WR006431.
[1] Hantush [1961a] presented a solution of the
draw-down around a partially penetrating well with the M(a, b) function defined by the following definite integral [Trefry, 1998; Mamedov and Ekenog˘lu, 2006]:
Mða; bÞ ¼2 p Za 0 eb 1þyð 2Þ 1þ y2 dy ð1Þ
where y is a dummy variable and a and b are parameters related to the physical properties of unconfined aquifers. The values of the M(a, b) function was extensively tabulated by Hantush [1961b, 1964]. On the basis of binomial expansion theorem, Mamedov and Ekenog˘lu [2006] developed an interesting algorithm for evaluating the Hantush’s M function, which was expressed as
Mða; bÞ ¼ae b p Z1 0 ea2bx ffiffiffi x p 1þ a2x ð Þdx¼ eb p Nlim!1 XN i¼0 Fið1Þb iþ1=2ð Þg iþ 1=2; a2b for a2 1 1þe b p Nlim0!1 XN0 i¼0 Fið1Þbðiþ1=2Þg i 1=2; a2b for a2 1 8 > > > > > < > > > > > : with Fið1Þ ¼ 1 ð Þ 2ð Þ ið Þ i! ð3Þ and g i þ 1=2ð Þ; a2b¼ Za2b 0 ettð iþ1=2ð Þ1Þdt ð4Þ
where x = y2/a2. Equations (2) – (4) are evaluated by directly adding the infinite series; yet, the numerical evaluation is not straightforward and the accuracy of the results are not easy to evaluate because of the fact that the series involves the incomplete Gamma function and has a running sum from zero to infinity.
[2] In this comment, we provide a simple and efficient
numerical approach as an alternative to evaluate the Han-tush’s M function. The Gaussian quadrature is employed to perform the numerical integration of equation (1) piecewise along the y axis from (0,a) to (1, 1) where a change of variable has been used. An n-point Gaussian quadrature formula may be written as [Gerald and Wheatley, 1989]
Z 1 1 fðxÞdx ¼X n i¼1 WifðxiÞ ð5Þ
where Wiis a weighting factor andxiis an integration point.
Both the six- and ten-term formulas of the Gaussian quadrature are used to carry out the integration for the same area under the integrand with the step size Dy = a. The resulting values of the integration obtained from the six- and
ten-term formulas are defined as A6 and A10, respectively.
The absolute difference of these two results is defined by DA =jA10 A6j. If DA > CTOL, a half step size (a/2) will
be used and the same integration procedure will be repeated untilDA < CTOL which is a tolerance of accuracy. If DA < CTOL, a double step size (2Dy) is used for the next step. This procedure ensures that each numerical integration over a small step satisfies the required accuracy. Note that the last step size should be chosen such that the end of the step should be located right at a larger integrated range, i.e.,a. In short, the integrand in equation (1) is obtained simply by adding all the resulting integration values. This approach has been successfully applied in some groundwater related problems [see, e.g., Yang and Yeh, 2002, 2006, 2007; Yeh et al., 2003].
[3] For the case of M(2, 3/2), the resulting numerical
result for equation (2b) obtained by Mamedov and Ekenog˘lu [2006] is 0.0832546934534 and the required upper limit of summations, L, is 20 as shown in Table 1 (for their equation (24b)), while the present approach takes only two steps to obtain the result. For the case of M(3/5, 7/10), their approach requires L = 25 to converge the series of equation (2a) with the result of 0.1585691441918 shown in Table 2 (for their equation (24a)); however, the present approach also takes only two steps to obtain the result of
1
Department of Civil Engineering, Vanung University, Chungli, Taiwan.
2Institute of Environmental Engineering, National Chiao Tung
Uni-versity, Hsinchu, Taiwan.
Copyright 2007 by the American Geophysical Union. 0043-1397/07/2007WR006431
W12602
(2a)
(2b)
WATER RESOURCES RESEARCH, VOL. 43, W12602, doi:10.1029/2007WR006431, 2007
0.1585691441919 which has an accuracy to 13 decimal places. In addition, we also examine the case of M(13, 17) which is the extreme one given in Table 3 of Mamedov and Ekenog˘lu [2006]. Their algorithm requires L = 75 to obtain the result of 0.551120725 108 for equation (2b) (their equation (24b)), while the present approach needs just six steps to obtain the same result. The computation effort of the present approach in evaluating the Hantush’s M function is significantly less if compared with that of Mamedov and Ekenog˘lu [2006] approach using the transform function. Obviously, this present approach has the advantages of being easy, straightforward, and very efficient in evaluating the Hantush’s M function.
References
Gerald, C. F., and P. O. Wheatley (1989), Applied Numerical Analysis, 4th ed., Addison-Wesley, Boston, Mass.
Hantush, M. S. (1961a), Drawdown around a partially penetrating well, Proc. Am. Soc. Civil Eng., 87, 83 – 98.
Hantush, M. S. (1961b), Tables of the function M(m, b), Prof. Pap. 102, Res. Div., N. M. Inst. of Min. and Technol., Socorro, N. M.
Hantush, M. S. (1964), Hydraulics of wells, in Advances in Hydroscience, edited by V. T. Chow, pp. 281 – 432, Academic, New York.
Mamedov, B. A., and A. S. Ekenog˘lu (2006), Evaluation of the Hantush’s M(a, b) function using binomial coefficients, Water Resour. Res., 42, W09502, doi:10.1029/2006WR005005.
Trefry, M. G. (1998), Analytical series expressions for Hantush’s M and S function, Water Resour. Res., 34, 909 – 913.
Yang, S. Y., and H. D. Yeh (2002), Solution for flow rates across the wellbore in a two-layer confined aquifer, J. Hydraul. Eng., 128(2), 175 – 183.
Yang, S. Y., and H. D. Yeh (2006), A novel analytical solution for constant-head test in a patchy aquifer, Int. J. Numer. Anal. Methods Geomech., 30, 1213 – 1230, doi:10.1002/nag.523.
Yang, S.-Y., and H.-D. Yeh (2007), A closed-form solution for a confined flow into a tunnel during progressive drilling in a multi-layer ground-water flow system, Geophys. Res. Lett., 34, L07405, doi:10.1029/ 2007GL029285.
Yeh, H. D., S. Y. Yang, and H. Y. Peng (2003), A new closed-form solution for radial two-layer drawdown equation under constant-flux pumping in a finite-radius well, Adv. Water Resour., 26(5), 747 – 757.
S.-Y. Yang, Department of Civil Engineering, Vanung University, No. 1, Vannung Road, Chungli, Taoyuan, 320, Taiwan. (shaoyang@msa.mu. edu.tw)
H.-D. Yeh, Institute of Environmental Engineering, National Chiao Tung University, No. 75, Po-Ai Street, Hsinchu 300, Taiwan. (hdyeh@mail. nctu.edu.tw)
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