Published online 17 July 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/hyp.6950
Obtaining the steady-state drawdown solutions
of constant-head and constant-flux tests
Chih-Tse Wang and Hund-Der Yeh*
Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan
Abstract:
The solutions of constant-head and constant-flux tests are commonly used to predict the temporal or spatial drawdown distribution or to determine aquifer parameters. Theis and Thiem equations, for instance, are well-known transient and steady-state drawdown solutions, respectively, of the constant-flux test. It is known that the Theis equation is not applicable to the case where the aquifer has a finite boundary or the pumping time tends to infinity. On the other hand, the Thiem equation does not apply to the case where the aquifer boundary is infinite. However, the issue of obtaining the Thiem equation from the transient drawdown solution has not previously been addressed. In this paper, the drawdown solutions for constant-head and constant-flux tests conducted in finite or infinite confined aquifers with or without consideration of the effect of the well radius are examined comprehensively. Mathematical verification and physical interpretation of the solutions to these two tests converging or not converging to the Thiem equation are presented. The result shows that there are some finite-domain solutions for these two tests that can converge to the Thiem equation when the time becomes infinitely large. In addition, the time criteria to give a good approximation to the finite-domain solution by the infinite-domain solution and the Thiem equation are investigated and presented. Copyright 2008 John Wiley & Sons, Ltd.
KEY WORDS groundwater; aquifer test; Theis equation; Thiem equation; wells Received 25 June 2007; Accepted 25 October 2007
INTRODUCTION
The Theis and Thiem equations are the well-known tran-sient and steady-state drawdown solutions, respectively, of a pumping test conducted in confined aquifers. Both solutions are very simple and easily used to predict the aquifer drawdown distribution in practical applications. The Theis equation is derived under the conditions of an infinite extended confined aquifer, neglecting the effect of the well radius. It is important to note that the Theis equation will give an infinite drawdown as the time approaches infinity. Chen (1984) proposed a modified Theis equation for drawdown distribution in a finite con-fined aquifer and gave a time criterion when applying the Theis equation.
The Thiem equation can be derived either from the continuity equation coupled with Darcy’s law (Todd and Mays, 2005) or the radial steady-state flow equation (Charbeneau, 2000). Many researchers have discussed the problem of steady-state flow and warn of erroneous results when applying the Thiem equation to problems of an infinite aquifer (Bear, 1979). Zaadnoordijk (1998) proposed an analytical algorithm based on the superposi-tion principle to simulate transient flow using the Theis equation between any two given steady-state groundwa-ter flows represented by the Thiem equation. His study
* Correspondence to: Hund-Der Yeh, Institute of Environmental Engi-neering, National Chiao Tung University, Hsinchu, Taiwan.
E-mail: [email protected]
implies that the steady-state condition is related to con-sideration of the effects of the well radius and the finite boundary. However, the issue of obtaining the Thiem equation from the transient drawdown solution has never previously been addressed.
The drawdown distribution may change in response to the constant-head or the constant-flux test. The former holds a specified drawdown in the test well while the latter keeps a constant pumping rate at the pumping well. Mathematically, these two tests can be formulated as different types of boundary value problem. With or without considering the effect of the well radius, many studies have been devoted to developing analytical solutions for these problems under various boundary conditions. The objectives of this current paper are (1) to examine the drawdown solutions obtained by considering or neglecting the effect of the well radius in a finite or infinite confined aquifer for both constant-head and constant-flux tests, (2) to give physical interpretation for the convergences of the transient drawdown solutions of these two tests to the Thiem equation after a long period of aquifer test, and (3) to present criteria for the time when the finite-domain solution can be approximated by the infinite-domain solution or the Thiem equation.
DRAWDOWN SOLUTIONS OF THE CONSTANT-HEAD TEST
This section considers a constant-head test conducted in a homogeneous and isotropic confined aquifer of constant
thickness. The one-dimensional radial flow equation describing the drawdown in this confined aquifer can be written as (Batu, 1998) ∂2s ∂r2 C 1 r ∂s ∂r D S T ∂s ∂t 1
where sr, t is the drawdown corresponding to the radial distance r from the test well and the time variable t, S is the storativity, and T is the transmissivity. The drawdown is initially assumed to be zero before the start of the aquifer test, i.e. sr, 0 D 0.
For the constant-head test, the drawdown in the test well is maintained constant and denoted sw. The rim
of the wellbore is selected as the inner boundary and the inner boundary condition for the drawdown can be expressed as srw, t D sw where rw is the well radius.
An outer boundary condition should be provided to solve the flow equation (1). Hereafter, we will present and discuss the solution for the outer boundary specified as a zero drawdown and located at either an infinite or a finite distance from the test well. In general, the zero-drawdown boundary plays a role in supplying water if the head in adjacent area is lower than the zero drawdown. The zero drawdown is often located at the boundary of a water body, such as a river, lake, or reservoir. In addition, the drawdown solution obtained by considering or neglecting the effect of well radius will also be presented.
Infinite domain with a finite well radius
It is assumed that the outer boundary is located at infinity and the outer boundary condition is expressed as s1, t D 0. The solution subject to the infinite outer boundary assumption is referred to as the infinite-domain solution. Using the method of Laplace transforms, the general Laplace-domain solution of (1) can be expressed in terms of bases I0 and K0, which are modified Bessel
functions of the first and second kinds of order zero, respectively. The function I0 tends to infinity under the
outer boundary condition and therefore must be excluded. Application of the inverse Laplace transforms under the inner boundary condition yields the drawdown distri-bution for the constant-head test as (Carslaw and Jaeger, 1959, p.335; Peng et al., 2002) sr, t D sw 1 2 1 0 exp T Sx 2t Y0rxJ0rwx J0rxY0rwx J20rwx C Y20rwx dx x 2
where J0 and Y0 are Bessel functions of the first and
second kinds of order zero, respectively, and x is a dummy variable.
The flow rate entering the test well can then be obtained by applying Darcy’s law to (2). Accordingly, the wellbore flux Qrw, t under the condition that the
aquifer has an infinite boundary and a finite well radius is (Peng et al., 2002) Qrw, t D 8Tsw 1 0 exp T Sx 2t 1 J20rwx C Y20rwx dx x 3
The exponential term on the right-hand side of (2) reduces to zero as time approaches infinity; the steady-state drawdown can thus be obtained as sw.
How-ever this result does not account for the outer bound-ary condition s1, t D 0. Such a result implies that the steady-state drawdown prevails within the entire domain and at the test well. Also, (3) shows that the wellbore flux decreases with increasing time and approaches zero for large times. This result suggests that under the infi-nite outer boundary condition there is not enough water to maintain a non-zero flow rate within the aquifer for a constant-head test. Thus (2) does not converge to the Thiem equation at large times.
Peng et al. (2002) proposed an improved algorithm for numerical evaluation of (2) and (3). Their results give dimensionless drawdown and dimensionless well-bore flux in tabular forms with better accuracy than Jaeger and Clarke (1942), Jacob and Lohman (1952), and Jaeger (1956). The solid line in Figure 1 is a plot of dimensionless wellbore flux (3), Qrw, t/2T sw,
ver-sus dimensionless time, Tt/r2
wS. It can be observed that
the wellbore flux decreases continuously and approaches zero rather than a constant flux as time becomes infinitely large. Similar results were also discussed in Yang and Yeh (2002, Figure 3) when considering the case of single-zone formation, and Zhan and Bian (2006, Figure 5) when decreasing the leakage across an aquitard as the time becomes large.
Figure 1. Dimensionless wellbore flux versus dimensionless time for the finite-domain solution (Equation (5)) with R/rwD10 and 102, the
infinite-domain solution (Equation (3)), and the Thiem equation. Note that both Equations (3) and (5) are solved for the constant-head test
Infinite Domain with Neglecting the Well Radius
When neglecting the effect of test well radius, that is, rw!0, the basis K0 of the general Laplace-domain
solution tends to infinity and must be excluded. There-fore, there is no solution for this case. The transient and steady-state solutions of the drawdown for the constant-head test with and without considering the effect of well radius under the infinite domain are listed in Table I.
Finite domain with a finite well radius
In this case, a well of finite radius is considered and a finite distance R from the well is selected to represent the outer boundary where the drawdown is kept at zero. Thus, the solution under a finite outer boundary condition, denoted as sR, t D 0, is referred to as a finite-domain solution. Carslaw and Jaeger (1959, p.332) gave the solution for this problem with Cauchy boundary conditions. By neglecting the flux components of the Cauchy boundary, a drawdown solution of the constant-head test for an aquifer with finite domain and well radius can be obtained as sr, t D sw lnR/r lnR/rw 1 nD1 exp T S˛ 2 nt J0rw˛nY0r˛n Y0rw˛nJ0r˛n [J20rw˛n J02R˛n]/J20R˛n 4 where ˛n are the roots of J0rw˛Y0R˛ Y0rw˛
J0R˛ D0. A detailed procedure for obtaining (4) is
given in Appendix A. The flow rate at wellbore can then be determined as Qrw, t D2T sw 1 lnR/rw C2 1 nD1 expT˛2nt/S [J20rw˛n J02R˛n]/J20R˛n 5 Obviously, the steady-state wellbore flux of (5) for time approaching infinity is Qrw D2Tsw/lnR/rw
which indeed is the Thiem equation. Figure 1 shows that the curve (5), i.e. the finite-domain solution, coincides with (3), i.e. the infinite-domain solution, at early stages of the constant-head test and asymptotically approaches the Thiem equation after a long period of time. In
other words, the infinite-domain solution can approxi-mate the finite-domain solution when time is less than the boundary-effect time criterion t1, implying that the
finite boundary has no effect on the wellbore flux. In addition, the finite-domain solution can be reduced to the Thiem equation when time is greater than steady-state time criterion t2, implying that the wellbore flux
can be considered at steady state. It is worth noting that both the Thiem equation and the infinite-domain solution have the advantage of computing the draw-down solution more easily than the finite-domain solu-tion.
Figure 1 indicates that both dimensionless time cri-teria Tt1/rw2 S and Tt2/rw2 S increase with the
dimen-sionless distance R/rw. The values of Tt1/rw2 S and
Tt2/rw2 S, for instance, are 4 and 4 ð 102 for R/rwD10
and 1 ð 103 and 3 ð 104 for R/rw D102, respectively,
if one defines that the absolute difference in dimension-less wellbore flux between (3) and (5) is dimension-less than 105.
Thus, the time criterion for t1 can be obtained
approxi-mately as R2S/10T based on a linear relationship between
Tt1/rw2 Sand R/rw for the constant-head test. Using the
same approach, t2 is 4R2S/T approximately. Table II
gives a list of time criteria for the finite-domain solu-tion.
Finite domain with neglecting the well radius
In this case, the outer boundary of the radial flow system is located at a finite distance from the test well and the inner boundary of the well radius is negligible. Note that the general Laplace-domain solution for drawdown in a constant-head test contains the bases of I0 and
Table II. The boundary-effect and steady-state time criteria for finite-domain solutions
Solution type Boundary-effect time criterion, t1 Steady-state time criterion, t2 Constant-head test 1 10 4 Constant-flux test 3 100 8 Constant-flux test when neglecting well radius 1 16 10 5 4 D R2S/T.
Table I. Transient and steady-state drawdown solutions of constant-head and constant flux tests
State Constant-head test Constant-flux test
Infinite domain Finite domain Infinite domain Finite domain
Considering well radius
Transient Equation (2) Equation (4) Equation (8) Equation (11)
Steady Ł Thiem equation Ł Thiem equation
Neglecting well radius
Transient Ł Ł Theis equation Equation (12)
Steady Ł Ł Ł Thiem equation
K0. The basis K0 has to be excluded from the general
solution as rw!0 and the remaining basis I0 cannot
satisfy the boundary conditions of fixed drawdown at wellbore and zero drawdown at the finite outer boundary simultaneously. Consequently, there is no solution for this case. Table I also lists the transient and steady-state drawdown solutions for the constant-head test when the outer boundary is finite, with and without considering the effect of well radius.
Interestingly, a heat problem presented in Carslaw and Jaeger (1959, p.327) is similar to the case mentioned above but admits a solution. Analogous to the groundwa-ter flow, their inner boundary condition has a finite draw-down instead of fixed drawdraw-down and the outer boundary condition at a finite distance R from the test well has a fixed drawdown rather than zero drawdown. The solu-tion can be written as (Carslaw and Jaeger, 1959, p.328, Equation (7)) sr, t D sR 2sR R 1 nD1 exp T S 2 nt J0rn nJ1Rn 6 where sR represents a fixed drawdown at finite distance
R, n are the roots of J0R D0 and J1 is the Bessel
function of the first kind of order one. This solution, (6), is extractable due to the fact that the inner boundary condition is finite and allowed to vary instead of being fixed in the constant-head test.
DRAWDOWN SOLUTIONS OF THE CONSTANT-FLUX TEST
This section considers a constant-flux test conducted at a confined aquifer. The initial condition for drawdown before the start of the pumping test is assumed to be zero. The drawdown solutions for (1) are presented for the cases that the outer boundary specified as zero drawdown is located at either an infinite or a finite domain and the inner boundary considers or does not consider the effect of well radius.
Infinite domain with finite well radius
The constant-flux test has a constant pumping rate Qp.
Thus, the inner boundary condition for the constant-flux test may be written as
2rwT ∂s ∂r rDrw D Qp 7
The drawdown solution of the constant-flux test subject to (7) and zero drawdown at infinity can be obtained by Laplace transforms as (Carslaw and Jaeger, 1959, p.338)
sr, t D Qp 2rwT 1 0 1 exp T Sx 2 t Y0rxJ1rwx J0rxY1rwx J21rwx C Y21rwx dx x2 8 where Y1 is the Bessel function of the second kind
of order one. Papadopulos and Cooper (1967) also
developed a drawdown solution to the constant-flux test by further considering the wellbore storage, and their inner boundary can be expressed as
2rwT ∂s ∂r rDrw r2wdH dt D Qp 9 where Ht is the well water level. Therefore, their drawdown solution can be reduced to (8) if one ignores the wellbore storage in (9). In addition, Cooper et al. (1967) also gave a well water level solution of the slug test by considering the wellbore storage.
Yeh et al. (2003) presented a closed-form solution with detailed numerical evaluations for a radial two-zone drawdown equation for groundwater under constant-flux pumping in a finite-radius well. The evaluation of (8) can be obtained if the transmissivities and storativities for the skin zone and formation zone are the same as in their article.
Figure 2 gives a graphical representation of dimension-less drawdown of (8) for dimensiondimension-less distance r/rwD
102 and dimensionless time Tt/r2
wS ranging from 102
to 109. The plot of drawdown distribution demonstrates
that drawdown increases with time and becomes infinity as time approaches infinity. This implies that the outer boundary cannot provide enough water to balance con-tinuous well pumping and the aquifer is overdrawn as time becomes infinitely large. This result indicates that the transient drawdown solution of the constant-flux test when considering the effects of well radius and the infi-nite domain does not reduce to the Thiem equation at very large times.
Infinite domain, neglecting well radius
The solution of the constant-flux test conducted in an infinite domain obtained by neglecting the effect of
Figure 2. Dimensionless drawdown versus dimensionless time for the finite-domain solution (Equation (11)) with R/rwD103 and 104 and
r/rwD102, the infinite-domain solution (Equation (8)), and the Thiem
equation. Note that both Equations (8) and (11) are solved for the constant-flux test
well radius is analysed in this case. By applying the Laplace transforms and the asymptotic form of modified Bessel function (Abramowitz and Stegun, 1979, p.375), one obtains the drawdown equation as
sr, t D Qp 4T 1 u expx x dx 10 where u D r2S/4T t, indicating that the value of u is inversely proportional to time.
Equation (10) is the well-known Theis equation and the integral in (10) is called the well function and expressed as Wu. This function tends to infinity when time approaches infinity and/or radial distance approaches zero; thus the drawdown of the Theis equation also becomes infinite. In addition, Cooper–Jacob’s equation, a special case of the Theis equation under the condition u <0Ð01, should also give infinite drawdown when u approaches zero. Note that the expression for the difference in the drawdowns at any two points within the aquifer is the same as the Thiem equation if the drawdown is expressed in terms of Cooper–Jacob’s equation. However, this expression is applicable only when the values of u for both points are less than 0Ð01. Accordingly, the locations of these two points should not be too far away from the pumping well and the expression for the drawdown difference based on Cooper–Jacob’s equation does not apply to the entire aquifer domain.
The transient and steady-state drawdown solutions of a constant-flux test conducted in an aquifer with an infinite outer boundary are compared in Table I. The results for the constant-head and constant-flux tests indicate that these two tests in an infinite domain do not have steady-state solutions; thus, (2), (8), and the Theis equation cannot reduce to the Thiem equation.
Finite domain with finite well radius
Similar to the development of (4), the drawdown solution for (1) subject to (7) and the condition of zero drawdown at a finite domain can be obtained as, according to Carslaw and Jaeger (1959, p.334),
sr, t D Qp 2T lnR r rw 1 nD1 exp T Sˇ 2 nt J1rwˇnY0rˇn Y1rwˇnJ0rˇn ˇn[J21rwˇn J20Rˇn]/J20Rˇn 11 where ˇn represent the roots of J1rwˇY0Rˇ
Y1rwˇJ0Rˇ D0. It is noteworthy that the
numer-ator of the second term in brackets of (11) equals 2/rwˇnwhen r D rw and the related drawdown
solu-tion was given in van Everdingen and Hurst (1949). The steady-state drawdown solution can easily be obtained by setting the time to infinity in (11), which is indeed the Thiem equation if the distance r is equal to the well radius.
Figure 2 shows the distribution of dimensionless draw-down of (11) versus dimensionless time for R/rw being
equal to 103 and 104 with r/rw D102. It also
demon-strates that the curve of (11), i.e. finite-domain solution, can be approximated by (8), i.e. infinite-domain solution, at early times and reduces to the Thiem equation after a long period of time. The former indicates the draw-down distribution that behaves as in an infinite aquifer before having the boundary effect. The latter implies that the drawdown distribution almost reaches its steady-state condition and the Thiem equation is applicable if the time is larger than the steady-state time criterion. Similar to the analysis used in the constant-head test, t1 is obtained as
3R2S/100T approximately for the constant-flux test when
considering the well radius if the absolute difference of dimensionless drawdown between (11) and (8) is less than 105. In addition, t
2 approximately equals 8R2S/T
if the absolute difference of dimensionless drawdown between (11) and Thiem equation is also less than 105.
Finite domain, neglecting well radius
In this case, we consider that the outer boundary is located at some finite distance and the well radius effect is negligible. Chen (1984) gave the drawdown solution as sr, t D Qp 4T[Wu WU C 2] 12 and D 1 nD1 J0bu/U1/2nc nJ1n Ð 1 0 exp U x 2n1 x 4U dx x 13
where U is R2S/4T t and n represents the roots of
Bessel function J0n D0.
Both the arguments u and U in (12) and (13) are inversely proportional to time. The functions Wu and WU can be expressed in terms of an infinite series and approximated as 0Ð5772 C ln u and 0Ð5772 C ln U, respectively, as u ! 0 and U ! 0 after a long period of time (Todd and Mays, 2005). The difference of Wu and WU reduces to 2 lnR/r where r should not be equal to zero. Moreover, the exponential term in the integrand of (13) approaches zero as U ! 0 and the value of is negligible. As a result, the draw-down of (12) approaches the Thiem equation when time approaches infinity. The results of transient and steady-state drawdown solutions for the constant-flux test with a finite outer boundary when considering or neglecting well radius effects are also shown in Table I.
Again, Equation (12), i.e. a finite-domain solution, can be approximated by the Theis equation (10) when U ½ 4 (Chen, 1984) and reduces to the Thiem equation when U 105 if the absolute difference of dimensionless
drawdown is less than 105. The time criteria t 1 and
t2 for the constant-flux test when neglecting the well
radius effect are obtained as R2S/16T and 105R2S/4T,
CONCLUDING REMARKS
This paper addresses the development of steady-state solutions from transient drawdown solutions of constant-head and constant-flux tests in a finite or infinite domain, with or without considering the well radius. The results show that the finite domain condition is sufficient for the development of a steady-state solution from a tran-sient drawdown solution. Such a condition ensures that the mass balance between extraction and supply is sat-isfied and the flow can reach its steady state condition within a finite domain. In addition, time criteria, which may be useful in practical applications, are provided for the approximation of the finite-domain solution by the infinite-domain solution or Thiem equation. It is found that the infinite-domain solutions can be used to deter-mine the drawdown distribution or the aquifer parameters if the time is smaller than the boundary-effect time cri-terion for an aquifer with a finite domain. Similarly, the Thiem equation is applicable whenever the time is larger than the steady-state time criterion.
ACKNOWLEDGEMENT
Research leading to this paper has been partially sup-ported by the grants from Taiwan National Science Coun-cil under contract number NSC 95-2221-E– 009-017. The authors sincerely thank two anonymous reviewers for constructive comments and suggested revisions.
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APPENDIX A: DERIVATION OF EQUATION (4) General solutions of the one-dimensional radial heat con-duction equation, which is analogous to the groundwater drawdown equation, subject to Cauchy boundary condi-tions at the edges of a hollow cylinder were given in Carslaw and Jaeger (1959, p.332). The Cauchy boundary conditions in terms of drawdown were expressed as
k1 ∂s ∂r k2s D k3, r D a A1 and k10∂s ∂r k 0 2s D k 0 3, r D b A2
where k1, k2, k3, k01, k20, k30 are constant; a and b are the
radial distance of the inner boundary and outer boundary in the considered region, respectively. Equations (A1) and (A2) represent the combination of the constant-head and constant-flux boundary conditions.
Using Laplace transforms, the drawdown solution based on (A1) and (A2) is
sr, t D A1 1 nD1 exp T S˛ 2 nt bk10˛nJ1b˛n k20J0b˛ncC0r, ˛n F˛n A2˛n A3 where A1D ak3bk01bk 0 2lnr/bc C bk 0 3[k1Cak2lnr/a] ak2k10 Cbk1k20Cabk2k20lnb/a A2Dk3bk10˛nJ1b˛n k20J0b˛nc k30[k1˛nJ1a˛n C k2J0a˛n] C0r, ˛n D J0r˛n[k1˛nY1a˛n C k2Y0a˛n] Y0r˛n[k1˛nJ1a˛n C k2J0a˛n] F˛n D k102˛2nCk 02 2[k1˛nJ1a˛n C k2J0a˛n]2 k12˛2nCk22[k01˛nJ1b˛n k02J0b˛n]2
and ˛n are the positive roots of
[k1˛J1a˛ C k2J0a˛]bk10˛Y1b˛ k20Y0b˛c
[k1˛Y1a˛ C k2Y0a˛][k10˛J1b˛ k02J0b˛] D 0
For the constant-head test with a finite well radius and outer boundary, the constants in (A1) and (A2) can be replaced by k1D0, k2D 1, k3 Dsw, k10 D0, k20 D1,
k0
3D0, a D rw, and b D R. By careful substitution, the
