An analytical solution for non-Darcian flow in a confined

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6 Z. Wen et al. / Advances in Water Resources xxx (2007) xxx–xxx
397 curves are not always obvious because of the non-linear
398 power law relationship between the discharge and the
399 hydraulic gradient. Furthermore, it is straightforward to
400 test the sensitivity of the solutions to two parameters n
401 and k in dimensional forms. Therefore, we prefer a dimen-
402 sional analysis for most parts of the following analysis. The
403 type curves are only used when the results are compared
404 with previous solutions of Darcian flow under the special
405 case of n =1.
406 Nevertheless, the type curves for non-Darcian flow
407 can be easily obtained based on the dimensional analysis
408 if the dimensionless terms can be adequately defined.
409 With the developed MATLAB program for the numeri-
410 cal Laplace inversion, we have obtained the drawdown
411 values against time which are presented in log–log scales,
412 as shown in Figs. 2–13. In the following discussion, we
413 only consider the case that n is larger than one, because
414 pre-linear flow is unlikely to occur near the pumping
415 wells.
416 3.1. Drawdowns without wellbore storage
417 3.1.1. Comparison of the non-linear type curves with Theis
418 curves
419 When n approaches one, flow is approaching Darcian.
420 The result of Eq. (18) will approach the classical Theis solu-
421 tion in the Laplace domain. We have compared our results
422 for n = 1 with the Theis type curves as shown in Fig. 2,
423 which has the same axes as those defined in Theis type
424 curves, i.e. u ¼ r2 S
and W ðuÞ ¼ 4pmk sðr; tÞ. It is clear to
4mkt
Q
425 see that our results agree perfectly with the Theis type
426 curves. This indicates that our MATLAB program based
Fig. 3. Comparison of the drawdowns obtained by the proposed
linearization and Laplace transform method and the Boltzmann transform
method with n = 2.0, Q =50m 3 /h, m =50m, k = 0.1(m/h) n , and
S = 0.001 for the distances r = 10 m and 100 m, respectively.
Fig. 4. Drawdowns versus r 2 /t with n = 1.5, Q =50m 3 /h, m =50m,
k = 0.1(m/h) n , and S = 0.001 for the distances r =10m,20m,50m,and
100 m, respectively.
UNCORRECTED PROOF
Fig. 2. Comparison of the type curves for non-Darcian flow and Theis
type curves.
on the numerical inversion is applicable. On the other
hand, it may suggest that the errors of the linearization
approximation are negligible at least when n is close to
one. In the following analysis, we consider the linearization
solutions as ‘‘quasi-exact’’ solutions when comparing to the
results of the Boltzmann method.
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Please cite this article in press as: Wen Z et al., An analytical solution for non-Darcian flow in a confined ..., Adv Water Resour
(2007), doi:10.1016/j.advwatres.2007.06.002