A SEMIANALYTICAL SOLUTION FOR RESIDUAL DRAWDOWN AT A FINITE DIAMETER
WELL IN A CONFINED AQUIFER
1Hund-Der Yeh and Chih-Tse Wang2
ABSTRACT: After the end of pumping the water level in the observation well starts to recover and the reduced drawdown during the recovery period is named as the residual drawdown. Traditional approaches in analyzing the data of residual drawdown for estimating the aquifer hydraulic parameters are mostly based on the applica-tion of superposiapplica-tion principle and Theis equaapplica-tion. In addiapplica-tion, the effect of wellbore storage is commonly ignored in the evaluation even if the test well has a finite diameter. In this article, we develop a mathematical model for describing the residual drawdown with considering the wellbore storage effect and the existing draw-down distribution produced by the pumping part of the test. The Laplace-domain solution of the model is derived using the Laplace transform technique and the time-domain result is inverted based on the Stehfest algorithm. This new solution shows that the residual drawdown associated with the boundary and initial condi-tions are related to the well drawdown and the aquifer drawdown, respectively. The well residual drawdown will be overestimated by the Theis residual drawdown solution in the early recovery part if neglecting the wellbore storage. On the other hand, the Theis residual drawdown solution can be used to approximate the present resid-ual drawdown solution in the late recovery part of the test.
(KEY TERMS: aquifer characteristics; ground water hydrology; drawdown; wells; mathematical models; pump-ing tests.)
Yeh Hund-Der and Chih-Tse Wang, 2013. A Semianalytical Solution for Residual Drawdown at a Finite Diame-ter Well in a Confined Aquifer. Journal of the American WaDiame-ter Resources Association (JAWRA) 49(4): 966-972. DOI: 10.1111/jawr.12047
INTRODUCTION
In general, there are three steps involved in the estimation of hydraulic parameters of confined aqui-fers, such as the transmissivity and storativity. First, a continuous or instantaneous stress is applied to the test well. The response of the aquifer to the stress is then measured temporally or spatially at the same well and/or observation wells. Finally, the measured
response, i.e., the drawdown, is analyzed using the Theis equation (Theis, 1935) or Cooper-Jacob equa-tion (e.g., Batu, 1998).
The groundwater level will rise after the stoppage of pumping. The depth to the rising in water levels during the recovery period minus the depth to the static water level is known as the residual drawdown. The analysis of residual drawdown data can provide an independent check on hydraulic parameters deter-mined from the analysis of data observed in the 1
Paper No. JAWRA-11-0150-N of the Journal of the American Water Resources Association (JAWRA). Received November 30, 2011; accepted January 22, 2013.© 2013 American Water Resources Association. Discussions are open until six months from print publica-tion.
2
Respectively, Professor (Yeh), Institute of Environmental Engineering, National Chiao Tung University, 1001 University Road, Hsinchu, 30010, Taiwan; and Senior Engineer (Wang), Taiwan Branch, MWH Americas Inc., Taipei, 110, Taiwan (E-Mail/Yeh: [email protected]. tw).
Vol. 49, No. 4 AMERICAN WATER RESOURCES ASSOCIATION
pumping period (e.g., Todd and Mays, 2005). Gener-ally speaking, the residual data is more reliable than the drawdown data because there are no pumping effects involved. Traditional approaches in analyzing residual drawdown data are commonly developed based on the superposition principle and the Theis equation or Cooper-Jacob equation (e.g., Berg, 1975; Goode, 1997; Samani and Pasandi, 2003; Singh, 2003; Zheng et al., 2005). The use of Theis equation in the analysis of residual drawdown data however implies that the effects of well radius and wellbore storage are ignored. Willmann et al. (2007) suggested that the designed recovery period should not be shorter than twice the pumping duration if the Theis equa-tion is employed to analyze the recovery data. An equation describing the residual drawdown and tak-ing account of wellbore storage would be helpful in analyzing the early recovery test data.
Picking (1994) proposed a type-curve match method for analyzing recovery data based on tabular well function values computed from of Papadopulos and Cooper’s solution (1967). Shapiro et al. (1998) introduced a conceptual model for describing the early-time recovering water level following the termi-nation of pumping in a well subject to turbulent head losses. Their approximation to the recovering water level in the test well was obtained according to the principle of superposition and the solutions of Pap-adopulos and Cooper (1967) and Cooper et al. (1967), while the former solution is employed to deal with the response of the formation and the latter is used to handle the turbulent head loss. Moreover, the approximation of Shapiro et al. (1998, Equation 5) assumed that the pumping period is sufficiently large and the recovery period is very short when Papado-pulos and Cooper’s solution (1967) is used. No solu-tion for recovery in finite-diameter wells apart from those utilizing the principle of superposition has been presented before. A residual drawdown solution obtained from theoretical development rather than the superposition principle has its need in engineer-ing applications. Yeh and Wang (2009) introduced a residual drawdown model for the recovery period of the test after a constant head injection. Recently, Mills (2010) presented a method for data analysis based on Papadopulos and Cooper’s solution (1967) and Picking’s equation (1994) for water level recovery following pumping of confined aquifers.
The objective of this study is to develop a mathe-matical model for describing the residual drawdown taking into consideration the existing drawdown dis-tribution introduced by the previous pumping and the effects of well radius and wellbore storage. This model uses the well drawdown after the stoppage of pumping as the boundary condition along the well-bore and the drawdown distribution from prior
pump-ing as the initial condition. The Laplace domain solution for the residual drawdown to such an initial boundary value problem is obtained based on the Laplace transform technique and the time-domain result is evaluated using the Stehfest algorithm (Stehfest, 1970). This solution is applicable at any elapsed times in both the pumping and recovery peri-ods. In addition, this solution can also be employed to investigate the effect of wellbore storage on well residual drawdown or to determine the hydraulic parameters if coupled with an optimization algo-rithm. The option of using the Theis residual draw-down solution to approximate the present residual drawdown solution is also examined.
ANALYSIS METHODS
A radial groundwater flow equation describing the drawdown distribution in a homogeneous and isotropic confined aquifer of uniform thickness can be written as
@2s @r2þ 1 r @s @r¼ S T @s @t; rw r < 1 and t > 0 ð1Þ where s rð ; tÞ is the drawdown; t is the time; r is the radial distance from the centerline of the test well; rw is the radius of the well screen; and S and T are the storativity and the transmissivity of the aquifer, respectively.
Consider that a test with a constant pumping rate has been conducted at a finite-diameter well for a per-iod of time. The water level begins to recover when the pumping is terminated. Figure 1 shows the schematic diagram of drawdown distributions at the beginning of both the pumping period and recovery period of the test. The drawdown is denoted as s1ðr; t1Þ at pumping time t1and the residual drawdown is s2ðr; t2Þ at recov-ery time t2. The well drawdown in the pumping part and the well residual drawdown in the recovery part are expressed as H1ð Þ and Ht1 2ð Þ, respectively. Thet2 initial drawdown prior to pumping is zero everywhere (i.e., s1ðr; 0Þ ¼ 0) while the initial residual drawdown for the recovery part is equal to the final drawdown (i.e., s2ðr; 0Þ ¼ s1ðr; tpÞ) when the pumping is termi-nated at tp. A semi-analytical expression for residual drawdown is developed considering the effect of well-bore storage as following.
Drawdown Distribution in Pumping Period
Both the aquifer drawdown and well drawdown before pumping are assumed to be zero. The inner
boundary condition for maintaining a constant pump-ing rate Q at the test well with the effect of wellbore storage taken into consideration can be expressed as
2prwT@ s1 r; t1 ð Þ @ r r¼rw þpr2 c dH1ð Þt d t t¼t1 ¼ Q; t1> 0 ð2Þ where rc is the radius of the well casing. Zero draw-down at infinity is posed as the outer boundary condition during the pumping period. In addition, the well drawdown H1ð Þ is equal to the drawdown att1 the wellbore s1ðrw; t1Þ according to the continuity con-dition. The drawdown solution considering the wellbore storage given by Papadopulos and Cooper (1967) is s1ðq;s1Þ ¼ 2 Qa p2T Z1 0 1 exp s1 ax2 h i J0ð Þ xYqx½ 0ð Þ 2aYx 1ð Þx Y0ð Þ xJqx½ 0ð Þ 2aJx 1ð Þx xY0ð Þ 2aYx 1ð Þx ½ 2þ xJ 0ð Þ 2aJx 1ð Þx ½ 2 dx x2 ð3Þ where q ¼ r=rw is the dimensionless radial distance, s1¼ Tt1=r2c is the dimensionless pumping time, a ¼ r2
wS=r2c is the coefficient of wellbore storage, J0 and Y0 are the Bessel functions of the first and sec-ond kinds of order zero, respectively, J1 and Y1 are the Bessel functions of the first and second kinds of order one, respectively, and x is a dummy variable. The residual drawdown will, therefore, start with s1 q; sp
for 1 q < ∞ when pumping is ended at the dimensionless pumping timeτp (i.e.,τ1= τp). The pro-cedure of numerical evaluation for Equation (3) is similar to the one used in Yang et al. (2006), which includes a root search scheme, a numerical integra-tion method, and the Shanks method (Shanks, 1955).
In the numerical procedure, the Newton method is used to find the root of the integrand and the Gauss-ian quadrature is employed to perform the numerical integration within the interval between two consecu-tive roots. The integral is then transformed to an infi-nite series, which can be accelerated the convergence by the Shanks method when summing up the series.
If the effect of wellbore storage is negligible, the second term on the left-hand side (LHS) of boundary condition (i.e., Equation 2) should be removed. Then, the solution of Papadopulos and Cooper (1967), Equa-tion (3), reduces to (Carslaw and Jaeger, 1959, p. 338)
s1ðr; t1Þ ¼ Q p2r wT Z1 0 1 exp Tt1 S x 2 Y0ð ÞJr x 1ðrwxÞ J0ð ÞYr x 1ðrwxÞ J2 1ðrwxÞ þ Y21ðrwxÞ dx x2 ð4Þ
If the test well is treated as a line source, the well radius in the first term on the LHS of Equation (2) approaches zero (e.g., rw? 0) and the second term on the LHS of Equation (2) vanishes. Equation (3) will further reduce to the Theis equation.
Figure 2 shows the curves for the dimensionless well drawdown s1
q ¼ 1; s1
=Q=2pT vs. dimension-less pumping time τ1 plotted based on Equations (3 and 4) and the Theis equation for the coefficient of wellbore storagea ranging from 105to 101. Note that Equation (4) can also be evaluated using the same numerical procedure described above. It demonstrates that the difference in dimensionless well drawdown between Equation (4) and the Theis equation decreases rapidly asτ1increases and/ora decreases. In addition, the difference in dimensionless well draw-down between Equation (3) and the Theis equation also decreases with increasingτ1and/or decreasing a. Papadopulos and Cooper (1967) suggested that the Theis equation can approximate Equation (3) when τ1> 2.5 9 102. The difference in dimensionless draw-down between Equation (3) and the Theis equation is 2 9 102when τ1= 2.5 9 10
2
and less than 1 9 102 whenτ1 > 5 9 10
2
fora ranging from 105to 101. For real-world well-hydraulics problems, the radius of the well casing generally ranges from 0.05 m to 0.25 m and the hydraulic conductivity for fine sand is about 2 m/day (2.289 103cm/s) (Batu, 1998). Accordingly, τ1 equals 103 if t1 = 12 h, rc= 0.10 m (four inches), and the thickness of the confined aqui-fer is 10 m. In this case, the difaqui-ference in dimension-less well drawdown between Equation (3) and the Theis equation is less than 1 9 103m when Q is less than 12.5 m3/day. Under this circumstance, the Theis equation compares reasonably well with Equa-tion (3).
FIGURE 1. Schematic Diagram of Initial Conditions for an Aquifer Test in Pumping and Recovery Periods. The variables
with subscript 1 are in the pumping period while those with subscript 2 are in the recovery period.
Residual Drawdown Distribution in Recovery Period Recovery of water level follows the termination of the pumping. The following relationship is imposed according to the continuity requirement for the flow rate between the aquifer and test well
2prwT@ s2 r; t2 ð Þ @ r r¼rw ¼ pr2 c dH2ð Þt d t t¼t2 ; t2> 0 ð5Þ The initial conditions for residual drawdown away from the test well and the water level in the test well are, respectively, written as
s2ðr; 0Þ ¼ s1 r; tp ; rw r < 1 ð6Þ H2ð Þ ¼ s0 1 rw; tp ð7Þ Note that Equation (6) is a function of radial distance and pumping period. The inner and outer boundary con-ditions for the residual drawdown are, respectively,
s2ðrw; t2Þ ¼ H2ð Þ; tt2 2> 0 ð8Þ and
s2ð1; t2Þ ¼ 0; t2> 0 ð9Þ The detailed derivation for the solution describing the residual drawdown distribution derived using the Laplace transform technique is given in Appendix A and the result is
s2ðq; s2Þ ¼ L1fA1ðq; pÞ þ A2ðq; pÞg ð10Þ with A1ðq;pÞ ¼ s1 1;sp K0pffiffiffiffiffiffiapq h pð Þ a Zq 1 f pð ;xÞdx 2 4 3 5 I0 ffiffiffiffiffiffiap p q ð Þ A2ðq;pÞ ¼ 2pffiffiffiffiffiffiapI1papffiffiffiffiffiffi pI0pffiffiffiffiffiffiap h pð Þ a Z1 1 f pð ;xÞdx 2 4 3 5K0 ffiffiffiffiffiffiap p q ð Þ þ aZ 1 1 f pð ;xÞdx 2 4 3 5I0 ffiffiffiffiffiffiap p q ð Þ þ aZ q 1 g pð ;xÞdx 2 4 3 5K0 ffiffiffiffiffiffiap p q ð Þ f pð ; xÞ ¼ x s1x; spK0 ffiffiffiffiffiffiap p x ð Þ g pð ; xÞ ¼ x s1x; sp I0 ffiffiffiffiffiffiap p x ð Þ h pð Þ ¼ pK0 ffiffiffiffiffiffiap p ð Þ þ 2 ffiffiffiffiffiffipapK1 ffiffiffiffiffiffiap p ð Þ
where L1denotes the inverse Laplace transform oper-ator, p is the Laplace variable, s1 1; sp
and s1 x; sp
are the drawdowns in the test well and aquifer, respec-tively, atτp, I0and K0are the modified Bessel functions of the first and second kinds of order zero, respectively, and I1and K1are the modified Bessel functions of the first and second kinds of order one, respectively.
The first and second terms on the right-hand side (RHS) of Equation (10) are produced from the inner boundary condition and the initial condition, respec-tively. Equation (10) can be numerically inverted using the Stehfest algorithm (Stehfest, 1970; Chang and Yeh, 2009). Note that this solution can be employed to assess the effect of wellbore storage on well residual drawdown or to determine the hydraulic parameters if coupled with an optimization algorithm as presented, for example, in Chen and Yeh (2009) and Yeh and Chen (2007).
Traditional approaches in dealing with the recov-ery test generally assume a hypothetical recharge rate equaling the pumping rate at the termination of pumping (Todd and Mays, 2005). With the applica-tion of the Theis equaapplica-tion and the superposiapplica-tion prin-ciple, the residual drawdown distribution may be expressed as (Batu, 1998) s2ðq; s2Þ ¼ Q 4pT W aq2 4 spþ s2 ! W aq2 4s2 " # ð11Þ
where W is the Theis well function. Note that Equa-tion (11), referred to as the Theis residual drawdown solution hereinafter, is the same as the one presented in Picking (1994, Equation 2) but in a slightly different form. It is obvious that Equation (11) is much easier to evaluate than the present residual
FIGURE 2. The Curves of Dimensionless Well Drawdown vs. Dimensionless Pumping Time fora Ranging from 105to 101.
drawdown solution, Equation (10). One can approxi-mate Equation (10) by Equation (11) for all practical purposes if the effect of well radius is negligible. In the next section, we aim to examine the effect of well-bore storage on the well residual drawdown and com-pare the presented residual drawdown solution with the Theis residual drawdown solution.
ANALYSIS RESULTS
As discussed earlier, if the dimensionless pumping time τp is greater than 5 9 102, the drawdown, s1 q; sp
for 1 q < ∞, in Equation (10) can be approximated by the Theis equation. Assuming τp >5 9 102
and q = 1, the well residual drawdown inferred from Equation (10) can then be simplified as
s2 1; s2 ¼ Q 4pTL 1 ( W a 4sp K0 ffiffiffiffiffiffip ap hp þ 2a hp Z1 1 x W ax2 4sp K0 ffiffiffiffiffiffiap p xdx ) ð12Þ
The first and second terms on the RHS of Equa-tion (12) arise from the inner boundary condiEqua-tion and initial condition, respectively. In addition, Equa-tion (12) can be reduced to the well drawdown equa-tion given by Cooper et al. (1967, Equaequa-tion 7) if zero drawdown is assumed as the initial condition.
According to Equation (12), we define a normalized well residual drawdown as s2 1; s2 =bQ=2pT Wa=4sp c, where the denominator Q=2pTWa=4sp
is the double of well drawdown at τp and the numerator s2ð1; s2Þ in Equation (12) is evaluated using the Stehfest algorithm.
Figure 3 exhibits the behavior of normalized well residual drawdown as a function ofτ2fora ranging from 105to 101when the pumping is ended atτp= 103. This figure indicates that the present solution (i.e., Equa-tion 12) is close to the Theis residual drawdown soluEqua-tion (i.e., Equation 11) whenτ2> 50. In addition, this result also suggests that the effect of wellbore storage is negligi-ble because the normalized difference between these two residual drawdown solutions is less than 19 102after τ2> 50. Note that the curves shown on Figure 3 were computed from a Fortran code we developed based on Equation (12) and the Stehfest algorithm.
CONCLUSIONS
Traditionally, the analysis of residual drawdown from a recovery test involves applying the
superposi-tion principle and Theis equasuperposi-tion, which in fact ignores the effects of well radius and wellbore stor-age. In this study, we develop a semi-analytical model for describing the recovery (or residual draw-down) taking into consideration the effect of wellbore storage, where the existing drawdown distribution from the pumping part of the test is treated as the initial condition. The Laplace-domain solution of this model is obtained based on the Laplace transform technique and the time-domain result is inverted by the Stehfest algorithm. This solution can be applied to the problems with any elapsed times since the pumping stopped and recovery began. Based on the derived solution, the well residual drawdown is con-tributed from two parts; one is the inner boundary condition related to the well drawdown while the other is the initial condition related to the aquifer drawdown produced during the pumping part of the test. The presented residual drawdown solution reduces to the solution of Cooper et al. (1967) if a zero drawdown is used as the initial condition. In addition, this residual drawdown solution can be approximated by the Theis residual drawdown solu-tion in the case of large recovery time, when the effect of wellbore storage is negligible. When the dimensionless recovery time exceeds 50, the differ-ence in normalized well residual drawdown between the presented residual drawdown solution and Theis residual drawdown solution will be less than 1 9 102 for the dimensionless pumping time equal to 103.
FIGURE 3. The Curves of Normalized Well Residual Drawdowns vs. Dimensionless Recovery Time fora Ranging
APPENDIX A: DERIVATION OF EQUATION (10)
By taking the Laplace transform with respect to time, the subsidiary equations of Equation (1) and Equations (5-9) can be written as
d2s 2 d r2 þ 1 r ds2 d r ¼ S T ps2 s1 r; tp ðA1Þ ds2ðr; pÞ d r r¼rw ¼ r2c 2rwT p s2ðrw; pÞ s1 rw; tp ðA2Þ s2ð1; pÞ ¼ 0 ðA3Þ
where s2ðr; pÞ denotes the Laplace transform of s2ðr; tÞ:
The general solution to (A1) to (A3) can be expressed as (Kreyszig, 2006) s2ðr; pÞ ¼ c½ 1I0ð Þ þ cq r 2K0ð Þq r þ /½ 1I0ð Þ þ /q r 2K0ð Þq r ðA4Þ with q¼ ffiffiffiffiffiffiffiffi S Tp r c1¼ S T Z1 rw x s1 x; tp K0ð Þdxqx c2¼ r2 c 2rwTs1 rw; tp r2 c 2rwTp K0ðq rwÞ þ qK1ðq rwÞ þ q I1ðq rwÞ r2 c 2rwTp I0ðq rwÞ r2 c 2rwTp K0ðq rwÞ þ qK1ðq rwÞ S T Z1 rw x s1x; tp K0ð Þdxqx 2 4 3 5 /1¼ S T Zr rw x s1 x; tp K0ð Þ dxqx /2¼ S T Zr rw xs1 x; tp I0ð Þ dxqx
Equation (A4) is a Laplace-domain solution which can be reduced to that of Cooper et al. (1967,
Equa-tion 7) if s1 r; tp
¼ 0 for rw r < ∞. Being expressed in terms of dimensionless variables and taking the inverse Laplace transform, the time-domain solution for the residual drawdown is then given by Equa-tion (10).
ACKNOWLEDGMENTS
This study was partly supported the Taiwan National Science Council under the grants NSC 99-2221-E-009-062-MY3, NSC 101-3113-E-007-008, and NSC 101-2221-E-009-105-MY2. The authors would like to thank the associate editor and two anonymous reviewers for their valuable and constructive comments that help improve the clarity of our presentation. The Fortran codes devel-oped for generating the curves shown in Figures 2 and 3 are avail-able from the authors upon request.
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