Examining the large
‐time wellbore flux of constant head test
Hund
‐Der Yeh,
1Chih
‐Tse Wang,
2and Stefan G. Llewellyn Smith
3Received 24 October 2009; revised 9 March 2010; accepted 12 May 2010; published 14 September 2010.
[1]
The function 1/[ p ln ( p/l)] comes from the wellbore flux solution in the Laplace
domain for a constant head aquifer test when the Laplace variable p is small. The
resulting inverse Laplace transform v(lt), representing the large‐time wellbore flux,
grows exponentially with time, which does not agree with the physical behavior of the
wellbore flux. Based on this result, Chen and Stone (1993) asserted that the well
‐known
relationship of small p
–large t may fail to yield the correct large‐time asymptotic solution.
Yeh and Wang (2007) pointed out that the large
‐time wellbore flux is not n(lt) if the
inversion of 1/[ p ln ( p/l)] is subject to the constraint Re p > l. Chen (2009) subsequently
questioned the necessity of imposing this constraint on the Laplace transform in the
inversion of the large
‐time wellbore flux. Motived by Chen’s comment, we reexamine
the inversion of 1/[ p ln ( p/l)] and demonstrate that this contradictory issue originates
from a spurious pole introduced when applying the small p
–large t correspondence to the
Laplace domain solution. We explain why this occurs and why the actual wellbore flux at
large times is proportional to the function N(
lt), known as Ramanujan’s integral. The
function N(lt) does decay at large time, which agrees with the steady state wellbore flux of
the constant head test.
Citation: Yeh, H.‐D., C.‐T. Wang, and S. G. Llewellyn Smith (2010), Examining the large‐time wellbore flux of constant head test, Water Resour. Res., 46, W09602, doi:10.1029/2009WR008820.
1.
Introduction
[2] The Laplace transform is commonly used to transform
linear partial differential equations into ordinary differential equations. The time domain solution can then be obtained after inverting the Laplace domain solution obtained from the ordinary differential equation. To obtain the inverse Laplace transform, one may use tables [e.g., Abramowitz and Stegun, 1970; Sneddon, 1972; Oberhettinger and Badii, 1973; Hildebrand, 1976; O’Neil, 1991; Jeffrey, 2002] in con-junction with rules or methods like the shift theorem, partial fractions, and the convolution theorem. The inverse Laplace transform can also be obtained by Bromwich’s integral [e.g., Peng et al., 2002; Yang et al., 2006], occasionally in con-junction with the residue theorem and/or Jordan’s lemma [e.g., Spiegel, 1971; Duffy, 1998].
[3] Numerical inversion methods or approximate solutions
may be sought when the Laplace domain solution cannot be inverted analytically. Numerical inversion methods compute the time domain results at specified times [e.g., Davies, 2002; Yeh and Yang, 2006; Cohen, 2007]. In contrast, asymptotic approximations are useful for large or small times. For example, large‐time solutions can be obtained from invert-ing the Laplace domain solutions based on the relationship
of small Laplace variable p versus large time t [e.g., van Everdingen and Hurst, 1949; Yeh and Wang, 2007], some-times referred to as the small p–large t relation. However, the mathematical conditions to carry out such a relationship rely on knowledge of the functions beyond their transform alone; this is the realm of Abelian and Tauberian theorems [van der Pol and Bremmer, 1950].
[4] The constant flux test is usually performed to
char-acterize an aquifer in order to estimate the aquifer parameters. The Laplace domain wellbore flux of the constant head test in a confined aquifer can be expressed as [Chen and Stone, 1993, p. 208, equation (7); Carslaw and Jaeger, 1959, p. 335, equation (3)]
Q rðw; pÞ ¼ 2 rwswTq K1ðq rwÞ
pK0ðq rwÞ; ð1Þ where rw is the well radius of test well, Sw is the constant drawdown maintained at the test well, q =pffiffiffiffiffiffiffiffiffiffiffipS=T in which S and T are the storage coefficient and the transmissivity of the confined aquifer, respectively, and K0(·) and K1(·) are modified Bessel functions of the second kinds of order zero and order one, respectively. When p is small, with the limiting forms K0(x)∼ −[ln(x/2)+g] and K1(x)∼ 1/x, (1) can be reduced to [Chen and Stone, 1993; Yeh and Wang, 2007]
Q rð w; pÞ 4 swT 1
p ln p=ð Þ; ð2Þ
wherel = 4T/(e2grw2S) andg = 0.57722… is Euler’s constant. [5] Chen and Stone [1993] used an inverse Laplace
trans-form trans-formula listed by Oberhettinger and Badii [1973] to
1Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan.
2Taiwan Branch, MWH Americas Inc., Taipei, Taiwan. 3
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, University of California, San Diego, La Jolla, California, USA.
Copyright 2010 by the American Geophysical Union. 0043‐1397/10/2009WR008820
WATER RESOURCES RESEARCH, VOL. 46, W09602,doi:10.1029/2009WR008820, 2010
invert 1/[ p ln ( p/l)] on the right‐hand side of (2) and gave the result as Q rð w; tÞ 4 swT Z 1 0 t ð Þx G x þ 1ð Þdx ð3Þ
in which the integral on the right‐hand side of (3) is denoted as n(lt). They showed that n(lt) tends to infinity as time approaches infinity. Based on this result, they therefore concluded that the use of small p–large t relation fails to retrieve the correct large‐time wellbore flux.
[6] Yeh and Wang [2007] used the expansion given by
Ritchie and Sakakura [1956] to invert 1/[ p ln ( p/l)] at large time and gave the large‐time wellbore flux as
Q rðw; tÞ 4 swT ln tð Þ1 ln tð Þ ½ 2þ 2 2=6 ln tð Þ ½ 3 3 2=2 þ 3ð Þ ln tð Þ ½ 4 " # ; ð4Þ
wherex(3) = 1.2020569032 is the Riemann Zeta function. Equation (4) is in fact identical to the formula given in Carslaw and Jaeger [1959, p. 336, equation (10)] if the last two terms in the former are neglected. In other words, (4) gives more accurate large‐time results than Carslaw and Jaeger’s formula [1959, p. 336]. It must be emphasized that (4) tends to zero for large time. Yeh and Wang [2007] also argued that the constraint Re p >l should be imposed in deriving the Laplace inverse of 1/[ p ln ( p/l)] to obtain v(lt). [7] Recently, Chen [2009] questioned the necessity of the
constraint Re p >l. According to his argument, (3) is correct and hence the use of small p–large t relation fails to yield a correct result for the large‐time wellbore flux as concluded by Chen and Stone [1993]. Therefore, two paradoxical issues need to be clarified. One is whether (3) or (4) is the correct inverse result of (2), while the other is whether the physical character of (2) can represent that of (1). In this note, we will address these two issues.
2.
Inverse of 1/[
p ln ( p/l)]
2.1. Laplace Transform[8] The inverse Laplace transform of 1/[ p ln p] can
be found in Bateman [1954, p. 225, equation 4.26.(1)].
The derivation for 1/[ p ln ( p/l)] is presented in the fol-lowing. The inverse Laplace transform ofG(x + 1)/px + 1for Re p > 0 given in Oberhettinger and Badii [1973, p. 21, equation (3.3)] is
L1 G x þ 1ð Þ pxþ1
¼ tx; x > 1; ð5Þ whereG(·) is the Gamma function. The linearity property of the Laplace transform gives
L1 1 p= ð Þxþ1
" #
¼ G x þ 1ð Þð Þtx: ð6Þ Integrating both sides of (6) with respect to x from zero to infinity yields L1 1 Z1 0 1 p= ð Þxþ1dx 2 4 3 5 ¼Z 1 0 t ð Þx G x þ 1ð Þdx: ð7Þ
The right‐hand side of (7) is indeed the integral on the right‐ hand side of (3), denoted as v(lt). The integral on the left‐hand side of (7) becomes 1 Z1 0 1 p= ð Þxþ1dx ¼ 1 ln p=ð Þ x!1lim 1 p= ð Þxþ1 1 p= ð Þ " # : ð8Þ
The first term on the right‐hand side of (8) vanishes when Re p > l. The inverse Laplace transform of 1/[p ln (p/l)] thus givesn(lt). The region of validity of the Laplace trans-form of 1/[ p ln p] should be Re p > 1 rather than Re p > 0, which is the constraint mentioned in Bateman [1954, p. 225, equation 4.26.(1)]. By analytic continuation, the function 1/[ p ln ( p/l)] provides the Laplace transform of v(lt) in the whole complex plane minus the point Re p = l and the negative real axis (we take the branch cut for the logarithm along the negative real axis).
[9] The inverse Laplace transform can also be obtained
through the use of Bromwich’s integral. This approach leads to the result of Hardy [1940] and Llewellyn Smith [2000]. The Laplace inverse of 1/[ p ln ( p/l)] is expressed as
L1 1 p ln p=ð Þ ¼ 1 2i!!1lim Z cþi! ci! ezt z ln z=ð Þdz; ð9Þ
where c >l. The integration of (9) is along the line x = c parallel to the imaginary axis, i.e., the line (c− iw, c + iw) in the complex plane with real parameter w ranging from −∞ to∞. The integral in (9) can be evaluated by the Bromwich contour as shown in Figure 1 which includes a pole con-tribution from z =l and a branch point at z = 0. The former yields exp(lt). For the latter, one has z = xexp(−ip) on the bottom of the cut (line FG) and z = x exp(ip) on the top (line DE). Furthermore, a small circle of radius" about the branch point (circle EF) can be used, but the contribution from this circle vanishes. The branch cut integral then gives
1 2i Z1 0 ext ln ei1 x= ð Þþ 1 ln eð ix=Þ dx x ¼ Z1 0 etx ln x ð Þ2þ2 " # dx: ð10Þ
Figure 1. The contour used in Bromwich’s integral.
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The integral on the right‐hand side of (10) is denoted as N(lt) and is known as Ramanujan’s integral [Llewellyn Smith, 2000]. Putting this together thus gives the Laplace inverse of 1/[ p ln ( p/l)] as
L1 1 p ln p=ð Þ
¼ et N tð Þ; ð11Þ where N(lt) is defined for positive lt by the integral on the right‐hand side of (10).
[10] From the form of N(lt) in (11), it is clear that N(lt)
decays at large time. Hence, v(lt) grows exponentially for large time. In the Laplace variable, (11) can be expressed as
N pð Þ ¼ 1 p
1
p ln p=ð Þ: ð12Þ
2.2. Asymptotic Approximations and Abelian and Tauberian Theorems
[11] The behaviors of N(lt) for small and large t were
given by Llewellyn Smith [2000], which also included his-torical background. For small t, one can obtain
N tð Þ 1 ln 1=tð1 Þþ O 1 ln 1=tð Þ ½ 2
!
ð13Þ
which may be developed from using Watson’s lemma for N(lt) defined in (10). For large t, the result is
N tð Þ ln tð Þ1 þ O 1 ln tð Þ ½ 2
!
: ð14Þ
[12] The approach pioneered by van Everdingen and
Hurst [1949] uses the relation between the large‐time behavior of a function in time domain and the behavior of its Laplace transform for small p. The simplest relation is that pf ( p) ∼ f(t) (this relation also holds for large p–small t). The existence of such a relation is governed by Abelian and Tauberian theorems. The former go from the time variable to the Laplace variable and are simpler than the latter, which go in the opposite direction. In particular, the theorems are only meaningful when a limit exists. The simplest counter example is sin(t). Its Laplace transform is p/( p2+ 1), so that pL[sin(t)] vanishes for small p. However, sin (t) does not tend to zero for large time.
[13] Here we have not only p (p) ∼ 0, but also pN(p) ∼ 0
for small p. This is consistent with the limit of N(lt) for large t, but not for v(lt). The reason for this is that the Tauberian theorem necessary to transform from p to t does not hold for v(lt), in particular because the strip of con-vergence of the Laplace integral defining ( p) does not contain the imaginary axis [van der Pol and Bremmer, 1950, section VII.3. Theorem I ].
3.
Relation Between Large
‐Time Wellbore Flux
and 1/[
p ln (p/l)]
[14] We have seen thatn(lt) grows exponentially, which
is inconsistent with physical behavior of the wellbore flux.
To understand this apparent paradox, we return to the full expression for the wellbore flux (1). Carrying out the Bromwich integral gives
Q rðw; tÞ ¼ 8T sw Z1 0 exp T Srw2x 2t dx x J02ð Þ þ Yx 02ð Þx : ð15Þ
Clearly, the integral in (15) decays at large time.
[15] The fundamental problem is that in passing from (1)
to (2), which is indeed the small p expansion ofQ(rw, p), we have introduced a spurious pole at Re p = l in (2). The original form in (1) has no pole at Re p =l, and the original vertical Bromwich contour used to obtain (15) can be taken anywhere in the right half plane. However, the Bromwich contour used in obtaining (11) has to be to the right of Re p =l. This leads to the exponential term in (11).
[16] Hence, the constraint Re p >l imposed in the
devel-opment of the integral definition of(p) by Yeh and Wang [2007] is appropriate. This removes the pole contribution atl explicitly from (11). This essentially gives – N(lt), as shown in (12). Alternately, replacing the Bessel functions by their small x expansions simplifies the integral in (15), and the result is proportional to the integral definition of N(lt). In addition, the series obtained from the inverse Laplace transform of 1/[ p ln ( p/l)] given by Yeh and Wang [2007, equation (5)] should be understood as the series expansion of N(lt).
4.
Concluding Remarks
[17] The wellbore flux of the constant flux test is usually
used in the determination of hydraulic parameters of low‐ permeability aquifers. The wellbore flux of constant head test at large time can be obtained from the inverse of the Laplace domain wellbore flux for small value of the Laplace parameter p, provided one removes the spurious pole intro-duced at Re p =l The result is then proportional to N(lt), whose series expansion has been obtained previously [e.g., Ritchie and Sakakura, 1956; Carslaw and Jaeger, 1959; Bouwkamp, 1971; Llewellyn Smith, 2000; Yeh and Wang, 2007]. The solution N(lt) can then be used to estimate hydraulic parameters when coupled with an optimization approach to analyze the measurement of large‐time wellbore flux. The inversion of 1/[ p ln ( p/l)] without the constraint of Re p >l leads to the incorrect result n(lt) on the right‐ hand side of (7) (i.e., (1) in Chen’s [2009] comment). In fact, the physical large‐time wellbore flux should be pro-portional to N(lt) because the pole at Re p = l in(p) has been introduced artificially and must be excluded. This is a good example showing that the application of the small p–large t relation to obtain the large‐time solution from the Laplace domain solution should be carried out with care.
[18] Acknowledgments. This study was partly supported by the Taiwan National Science Council under the grants NSC 96‐2221‐E‐009‐ 087‐MY3 and NSC 99‐NU‐E‐009‐001. We are grateful to the Editor, Tissa Illangasekare, and two anonymous reviewers for their helpful comments and suggestions.
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