Analysis of pumping test data for determining uncon
fined-aquifer
parameters: Composite analysis or not?
Hund-Der Yeh&Yen-Chen Huang
Abstract Recently, composite analysis (CA), which si-multaneously analyzes all drawdown data from multiple observation wells, has been applied to determine the hydraulic parameters of an unconfined aquifer. Moench (1994) claimed that the value of specific yield (Sy)
determined from non-composite analysis (nonCA) is sometimes unrealistically low as compared with that obtained by water-balance calculation, and results from CA are better representative of aquifer properties than those from nonCA. To examine the validity of this assertion, the drawdown data from a pumping test conducted at Cape Cod, Massachusetts, USA, were analyzed using both nonCA and CA methods. The results show that the mean estimates of hydraulic conductivity and Sydetermined from CA are close to those determined
from nonCA. In some cases the analysis based on CA also results in low estimates of Sy as compared with those
determined based on nonCA. A hypothetical case study is presented, which examines the effect of measurement errors on the estimated parameters. The results indicate that the CA method also gives poorer estimates of Sythan
the nonCA method if the pumping test data contain measurement errors. Moench AF (1994) Specific yield as determined by type-curve analysis of aquifer-test data. Ground Water, 32(6):949–957.
Keywords Groundwater hydraulics . Unconfined aquifer . Hydraulic testing . Composite analysis . USA
Introduction
Groundwater hydrologists often conduct pumping tests and data analyses to obtain aquifer parameters which are necessary information for quantitative groundwater stud-ies. For an unconfined aquifer system, the analysis of pumping test data is slightly complicated because the transient drawdown curve exhibits three segments in response to the pumping. During thefirst segment, which occurs at early pumping time, water is released from storage instantaneously. During the second segment, the vertical gradient near the water table induces drainage of the porous matrix and, consequently, the decline rate of the hydraulic head slows down and may even stop after a period of time. Finally, when the flow is essentially horizontal, most of the pumping is supplied by the specific yield, Sy, in the third segment. Boulton (1954, 1963)
developed an analytical solution for the uncon fined-aquifer flow equation by introducing the concept of delayed yield. Prickett (1965) described a systematic approach for the determination of hydraulic parameters using a graphical procedure based on Boulton’s method. Cooley and Case (1973) indicated that Boulton’s equation described aflow system with a rigid phreatic aquitard on top of the main aquifer where the effect of the unsaturated zone above the phreatic surface was neglected. Neuman (1972, 1974) developed a solution that considers the effects of elastic storage and anisotropy of aquifers on the drawdown behavior. Neuman (1975) also gave a graphical type-curve match procedure to determine the hydraulic parameters of unconfined aquifers. Moench (1995) com-bined the Boulton and Neuman models forflow toward a well in an unconfined aquifer. He used a non-physical parameterα1to represent the delayed decline of the water
table during pumping. Grimestad (2002) reanalyzed transient drawdown data from two pumping tests con-ducted in unconfined aquifers, one at Cape Cod, Massa-chusetts, USA (Moench 1994) and the other at Borden, Texas, USA (Nwankwor et al. 1984). Grimestad (2002) concluded that a portion of the water pumped from the aquifers was derived from other sources. Zhan and Zlotnik (2002) discussed how a solution forflow to a horizontal or slanted well in an unconfined aquifer can be obtained. Hunt (2006) used a meaningful aquifer parameter instead of an empirical constant in the equation of Zhan and Zlotnik (2002) to describe flow to a well when a number
Received: 20 December 2007 / Accepted: 21 November 2008 Published online: 14 January 2009
# Springer-Verlag 2008
H.-D. Yeh ())
:
Y.-C. HuangInstitute of Environmental Engineering, National Chiao Tung University,
No. 75, Po-Ai Street, Hsinchu, 300, Taiwan e-mail: hdyeh@mail.nctu.edu.tw
Tel.: +886-3-5731910 Fax: +886-3-5726050 Y. Huang
of overlying aquitards exist between the pumped aquifer and free surface.
In the past, the pumping drawdown data obtained from a single observation well were commonly used for the aquifer parameter determination. Based on Boulton’s solution for large-time data, Mania and Sucche (1978) employed the least-squares approach to determine the unconfined-aquifer parameters. Followed the concept of Ferris et al. (1962), Moench (1994) employed the composite analysis (CA) and graphical type-curve method (called composite plot) to determine the specific yield. He pointed out that the Sy determined from analyzing the
drawdown data from a single observation well were sometimes unrealistically low as compared with those determined by water-balance calculations from field data or controlled laboratory experiments performed on sam-ples of aquifer material. He interpreted that the unrealis-tically low values of Sy were generally caused by (1)
improper procedures, (2) bad data, or (3) aquifer hetero-geneity. Moreover, he showed that the effect of partial penetration should be included in the analysis of the drawdown data and the composite plot has to be used with a single match point for all measured drawdown data. Finally, Moench concluded that the determination of Sy
using CA is consistent with that obtained byfield water-balance calculation in a relatively homogeneous, uncon-fined aquifer. Based on the drawdown data from multiple observation wells, Heidari and Moench (1997) determined the best-fit parameters using the nonlinear least squares approach instead of the composite plot.
Meier et al. (1998) used Cooper and Jacob’s solution (Cooper and Jacob1946) to determine effective transmis-sivity values based on the analysis of synthetic drawdown data in a confined aquifer with heterogeneous transmis-sivity and homogeneous storativity. The results indicated that the transmissivity values determined from the analysis of simulated drawdowns from individual observation wells are all very close to the effective transmissivity value. Wu et al. (2005) presented two approaches (distance drawdown and spatial moment analyses) in determining effective transmissivity and storage coef fi-cient in a synthetically heterogeneous confined aquifer. The results indicated that the estimate of transmissivity needed long pumping time to converge its geometric mean. Wu et al. (2005) concluded that the analyzed results using the drawdown data from a single observation well may be difficult to interpret because of the heterogeneity of the aquifers. Illman and Neuman (2001, 2003) and Illman and Tartakovsky (2005) analyzed a series of cross-hole air injection tests conducted in unsaturated fractured tuffs. The type-curve, steady-state, and asymptotic analy-ses were used to determine the equivalent permeability and porosity of fractured tuffs. The results showed that the geometric mean of the permeability obtained by analyzing cross-hole measurements is larger by a factor 50 than that obtained by single-hole pneumatic injection tests.
Chen and Ayers (1998) determined four parameters of the unconfined aquifer based on both Neuman’s (1974) and Moench’s (1995) analytical solutions. They first
analyzed the drawdown data from individual wells (also called nonCA in this study), and the data sets from two or more wells were then combined into a large data set and analyzed simultaneously. In the Chen and Ayers (1998) study, observation wells along a line or randomly chosen were analyzed by CA. The results indicated that the parameterα1in Moench’s solution (1995) for representing the delay yield might be not important and difficult to determine properly in the analysis of their pumping test data. Moreover, they found that the value of Sydetermined
from CA might be lower than the normal range for sand and gravel of the test site. Kollet and Zlotnik (2005) presented the analysis of transient drawdown data from a pumping test. The results showed that highly uncertain and physically unrealistic estimates of Sy and vertical
hydraulic conductivity (Kz) might be due to the
heteroge-neity of the aquifer and the return flow of the test. They suggested that both the nonCA and CA analyses were necessary for analyzing the pumping test data in examining the consistency and reliability of parameter estimates. Tartakovsky and Neuman (2007) developed an analytical solution for the delayed response process characterizing flow to a partially penetrating well in an unconfined aquifer. The solution generalized those of Neuman (1972,1974) by accounting for unsaturatedflow above the water table. The field data from Cape Cod (Moench et al. 2001) were analyzed using both CA and nonCA. The results indicate that the estimates of Syand storage are respectively smaller
and larger than those of Moench et al. (2001).
Simulated annealing (SA) is a stochastic technique for solving optimization problems. SA wasfirst proposed by Kirkpatrick et al. (1983) as a method for solving combina-torial optimization problems. Subsequently, utilization of SA in optimization problems has been widely applied in hydrological engineering. For example, this method has been employed to design the strategies of groundwater remediation (Dougherty and Marryott1991; Marryott et al.
1993), determine the hydraulic parameters of aquifers (Yeh et al. 2007a; Yeh and Chen 2007), and identify the groundwater contamination source (Yeh et al.2007b).
The objective of this study is to examine whether aquifer parameter determination using CA with different spatial distribution and types of observation wells gives a better or higher estimate of Sy than nonCA in analyzing
pumping test data obtained from unconfined aquifers. The computer method based on simulated annealing and Neuman’s solution (Neuman 1974) (also called SANS) (Huang et al. 2008) is used to determine the uncon fined-aquifer parameters of thefield case and hypothetical case studies using both CA and nonCA approaches. In thefield case study, the drawdown data sets from each of 20 observation wells arefirst analyzed using nonCA and then these wells are classified into seven groups to examine whether the spatial distribution and the type of observa-tion wells affect the results of CA. In the hypothetical case study, a pumping test which has one pumping well and two observation wells is conducted. The synthetic tran-sient drawdown data are generated based on Neuman’s solution (Neuman1974). In each observation well, 20 sets
of noise generated from normal distribution with zero mean and variance of 10−4arefirst added to the drawdown data and then the nonCA and CA are employed to determine the hydraulic parameters. The purpose of this case study is to investigate the effect of measurement errors on the parameter estimation. This article provides extensive case studies which may be helpful in choosing the right method, CA or nonCA, for analyzingfield pumping test data.
Methodology and case studies
Methodology of non-composite and composite
analyses
The aquifer parameters can be determined by the least-squares approach when minimizing the sum of square residuals between the observed and predicted drawdowns. That is Minimize X n i¼1 Ohi Phi ð Þ2 ð1Þ
where Ohi and Phi are respectively the observed and
predicted drawdowns at different time steps and n is the total number of observed drawdown data. Heidari and Moench (1997) suggested using observed drawdown data obtained from multiple wells to simultaneously determine the best-fit aquifer parameters. The objective function they used for a CA is defined as
MinimizeX nw j¼1 Xn i¼1 Ohi;j Phi;j 2 ð2Þ where nw is the number of observation wells and n is the number of observed drawdown data at each well.
The computer method based on simulated annealing
and Neuman
’s solution (SANS)
Simulated annealing (SA) is known as an optimization algorithm for simulating a material crystallized in the process of annealing. The arrangement of the material molecules is initially disordered at high temperature. The system is gradually cooled; meanwhile, the arrangement becomes more ordered and the system approaches a thermodynamic equilibrium. Based on this concept, the solution, which may not be the best one, is accepted to avoid the solution becoming trapped in a local optimum during the optimization procedure. The details of the SA can be found in Huang et al. (2008).
The analytical solution (Neuman 1974), considering the effects of delayed yield and well partial penetration describing the groundwaterflow system in an unconfined aquifer, is s r; z; tð Þ ¼ q 4T Z 1 0 4yJ0 y1=2 u0ð Þ þy X1 n¼1 unð Þy " # dy ð3Þ
where q is pumping rate, J0 (x) is the zero order Bessel
function of thefirst kind, ¼ Kzr2Krb2 is a dimension-less parameter, Kris the horizontal hydraulic conductivity, r
is the distance between pumping well and observation well, b is the thickness of the aquifer, y is a dummy variable. The functions u0(y) and un(y) are respectively defined as
u0ð Þ ¼y 1exp t½ s yð2r02Þ f gcosh rð0zDÞ y2þ 1þð Þr2 0 y2r 2 0 ð Þ2 cosh rð Þ0 sinh r½0ð1dDÞsinh r½0ð1lDÞ lDdD ð Þ sinh rð Þ0 ð4Þ unð Þ ¼y 1exp t½ s yð2þrn2Þ f gcos rðnzDÞ y2 1þð Þr2 n yð2þr2nÞ 2 cos rð Þn sin r½nð1dDÞsin r½nð1lDÞ lDdD ð Þ sin rð Þn ð5Þ
where ts¼ TtSr2 denotes the dimensionless time since pumping started, T is transmissivity which equals Kr× b,
S is the storage coefficient, t represents the pumping time, dD=d/b represents the dimensionless vertical distance
between the top of perforation in the pumping well and the initial position of the water table, and lD=l/b is the
dimensionless vertical distance between the bottom of perforation in the pumping well and the initial position of the water table. The variables of r0and rnare respectively
the roots of the following two equations r0sinh rð Þ y0 2 r02 cosh rð Þ ¼ 0; r0 02< y 2 ð6Þ and rnsin rð Þ þ yn 2þ rn2 cos rð Þ ¼ 0; 2n 1n ð Þ =2ð Þ < rn< n ð7Þ The SANS approach was employed to determine the hydraulic parameters of the unconfined aquifer when using Eqs. (1) and (3) for nonCA and Eqs. (2) and (3) for the CA.
Description of the field case study
The site of Cape Cod shown in Fig.1was selected for the study. The aquifer system consists of unconsolidated glacial outwash sediments deposited during the recession, 14,000–15,000 years BP, of the late Wisconsinan conti-nental ice sheet (Moench et al. 2001). The depth of the pumping well is 24.4 m below the ground surface. The top and bottom of the screen are located 4.0 and 18.3 m, respectively, below the initial water table, which is∼5.8 m below ground surface. The aquifer-saturated thickness is ∼48.8 m. The total number of the observation wells is 20, which includes three sets of well clusters. The clusters F504 and F505 both have three observation wells each where the piezometers are separately located at shallow, mid-depth, and deep-depth. Details with regard to the
exact radial and vertical positions, the lengths of the well screens, and the characteristics of the observation wells are given in Table1(Moench et al.2001). The terms PW, SP, MP, DP, LS, SDT, and LDT represent the pumping well, shallow-depth piezometer, mid-depth piezometer, deep-depth piezometer, long-screened well, short radial distance between pumping well and observation well, and long radial distance between pumping well and observation well, respectively. The farthest observation well (F376-037) is located 69.37 m from the pumping well. The structure and distribution of the piezometers in the observation wells were listed in a table of Moench et al. (2001). Well F507-080 was pumped (1.21 m3/min) for 72 h. In the case study, 20 observation wells are classified into the following seven groups based on the number of wells used, the length of well screen, and the distance from the pumping well to the observation well:
1. Two wells which are randomly chosen from 20 observation wells
2. A well cluster which consists of two or three observation wells
3. Three or four wells with different radial distances from the pumping well
4. Three or four wells with long screens
5. Three or four wells with different depth of piezometers 6. Two or three well clusters
7. Twenty wells (global analysis)
The type of observation for the hydraulic head of the aquifer includes the piezometer and the observation well with different length of screen. Accordingly, there are 50, 3, 2, 2, 5, and 4 cases for groups 1–6, respectively, analyzed by the present method of CA.
The hypothetical case study
A synthetic pumping data set generated based on Neu-man’s solution (Neuman 1974) was used to explore the effect of measurement errors on the estimated parameters. The pumping test was conducted with a constant pumping rate of 1,000 m3/day in an unconfined aquifer of 10 m thickness. Two observation wells were installed, observa-tion well one (OW1) was located at 10 m away from the pumping well, while the second (OW2) was located 30 m. The depth of the pumping well was 10 m and the screen length was 5 m where the top of the screen was 5 m below the initial water table. The depth of the two observation wells was 6 m and the screen length was 1 m where the top of the screen was the same as that of the pumping well. The pumping period was 1,000 min and the total number of drawdown data was 48 in each observation well. The“true” parameter values are: Kr=10−3m/s, Kz=
10−4m/s, S=10−4, and Sy=10−1. Forty sets of noise were
first generated by the routine RNNOF of IMSL (1997), which produces normally distributed random numbers
with zero mean and unit variance. They were then divided by 100 to represent the measurement errors with the magnitude on the order of centimeter. In each observation well, twenty sets of noise were added into the drawdown
data and the SANS was employed to determine the hydraulic parameters based on nonCA. Then the draw-down data from two observation wells were analyzed based on CA.
Table 1 Locations of the observation wells
Well No. Radial
distancea(m) Depthb(m) Number of observed data Screen length (m) Type of wellc F507-080 0.10 4.02 31 14.33 PW F505-032 7.28 3.26 32 0.61 SP, SDT F505-059 5.94 9.33 32 2.74 MP, SDT F505-080 6.58 17.80 33 0.61 DP, SDT F504-032 14.20 2.93 31 0.61 SP, SDT F504-060 15.18 9.14 33 2.74 MP F504-080 16.18 17.53 32 0.61 DP F377-037 25.94 4.05 30 0.61 SP F383-032 28.35 3.69 18 0.61 SP F383-061 28.32 12.16 24 0.61 MP F383-082 28.90 18.84 18 0.61 DP F383-129 29.47 32.92 17 0.61 DP F384-033 41.85 4.82 23 0.61 SP F381-056 48.71 6.10 20 0.61 SP, LDT F347-031 68.79 4.51 18 0.61 SP, LDT F434-060 11.77 0.61 15 11.89 LS F450-061 20.21 0.52 16 11.89 LS F476-061 19.99 0.67 16 11.89 LS F478-061 30.88 0.67 16 11.89 LS F385-032 68.46 3.05 17 0.61 SP, LDT F376-037 69.37 4.02 20 0.61 SP, LDT a
Distance from the center of pumping well b
Depth below the initial water table to the top of the screen c
PW pumping well; SP shallow-depth piezometers; MP mid-depth piezometers; DP deep-depth piezometers; LS long-screened well; SDT short radial distance between pumping well and observation well; LDT long radial distance between pumping well and observation well
Table 2 The parameters determined from the SANS using nonCA (20 cases)
Well No. Kr(m/min) ×10−2 Kz(m/min) ×10−2 S ×10−3 Sy×10−1 SEE ×10−3(m)
F505-032 8.58 0.98 8.38 0.65 7.06 F505-059 7.66 3.25 5.23 1.76 8.55 F505-080 7.42 3.40 5.39 2.01 7.02 F504-032 8.99 1.51 8.42 0.91 5.02 F504-060 8.31 2.99 4.80 1.29 3.80 F504-080 8.12 3.27 5.29 1.60 3.54 F377-037 9.11 2.17 8.66 1.44 2.65 F383-032 9.46 1.08 34.70 1.90 3.64 F383-061 7.96 2.76 7.34 1.89 3.34 F383-082 7.99 2.16 9.45 1.67 2.05 F383-129 8.05 1.88 5.11 1.11 1.82 F384-033 8.59 1.72 7.14 1.96 2.82 F381-056 9.77 1.67 3.87 1.47 3.36 F347-031 9.61 1.05 26.80 1.73 1.53 F434-060 7.90 2.26 61.80 1.60 4.02 F450-061 8.90 1.74 41.50 1.52 3.41 F476-061 8.85 1.74 57.30 1.36 4.36 F478-061 9.43 1.60 51.10 1.51 2.21 F385-032 11.79 0.94 2.29 1.73 2.34 F376-037 13.10 0.96 5.90 1.32 7.06 Mean 8.98×10−2 1.96×10−2 1.80×10−2 1.52×10−1 SD 1.38×10−2 8.10×10−3 1.98×10−2 3.49×10−2 CV 0.15 0.41 1.10 0.23 Mean (log10) −1.05 −1.75 −1.97 −0.83 SD (log10) 0.06 0.19 0.44 0.12
Results and discussion
Field case study
Non-composite analysis
Table 2 lists four estimated parameters for the pumping test data obtained from those 20 observation wells based on nonCA. Note that the standard error of the estimate (SEE) listed in the last column of Table 2 is defined as
ð
Pnj¼1e 2 j
=
vÞ
1=2
, where ej represents the difference between
the observed drawdown and the drawdown predicted by Neuman’s solution (Neuman 1974) with the estimated parameters, and v, the degree of freedom, is equal to the number of observed data points minus the number of unknowns (Yeh 1987). The estimated Kr ranges from
7.42×10−2to 1.31×10−1 m/min. The Kz is about half or
one order of magnitude less than Kr. Note that the
estimated S at observation wells with long screen length (F434-060, F450-061, F476-061, and F478-061) is about one order larger than that of other observation wells. The lowest Sy(6.50×10−2) is obtained from the well F505-032
and the highest one (2.01×10−1) is from the well F505-080 as shown in Table2. In addition, the estimated Syis
larger than 0.1 for all wells except the wells F505-032 and F504-032. The SEE values range from 1.53×10−3to 8.55× 10−3m, indicating that the nonCA can accurately determine the hydraulic parameters. The last five rows of Table 2
display the mean (i.e., arithmetic mean), standard deviation (SD), coefficient of variation (CV), mean of logarithmic parameter values (Mean(log10)), and the SD of logarithmic parameter values, SD (log10), of each parameter. The CV, defined as the ratio of the SD to the mean, is a measure of dispersion of a probability distribution. The SDs of estimated parameters demonstrate that the estimate of parameter S has high variation at the test site. Figure2displays the estimated parameters Kr, Kz, S, and Sy versus the radial distance
between pumping well and observation well. Those results indicate that estimated parameters are quite uncorrelated with the distance between pumping well and observation well. Figure3 shows the observed drawdown and the predicted drawdown curve drawn based on the parameters of Kr=
7.42×10−2 m/min, Kz=3.40×10−2 m/min, S=5.39×10−3,
and Sy=2.01×10−1obtained from analyzing the data of well
F505-080 using the SANS of nonCA. This figure demon-strates that the predicted drawdown curvefits the observed data quite well and thus the aquifer parameters are accurately determined.
Composite analysis
The drawdown data of the seven groups were employed to investigate whether the spatial distribution and the type of
observation wells affect the results of CA. Table 3shows 50 cases which are divided into three sets for group 1 (two observation wells which are randomly chosen from 20 observation wells) and their drawdown data were simul-taneously analyzed based on CA. Sets 1–3 represent situations where the Sy determined by the CA is smaller
than, falls between, and larger than Sy of nonCA,
respectively. The total number of cases analyzed in each of these three sets is respectively 2, 38, and 10 and the estimated parameters are displayed in Tables4,5, and 6, for sets 1–3, respectively. Note that the statistics of nonCA results are also listed at the bottom of Tables4–6for the comparison of nonCA and CA results. The means of estimated Krdetermined from CA listed in Tables4–6are
9.31×10−2, 8.30×10−2, and 8.10×10−2m/min, respective-ly. Those values are very close to that determined from nonCA, i.e., 8.98×10−2 m/min. This may be due to the fact that the aquifer is quite homogeneous in the horizontal direction (Masterson et al. 1997). Note that the SD is not given in Table 4 since there are only two case results. In these three sets, the estimated Sy ranges
from 7.40×10−2to 2.27×10−1and the mean of each set is 1.36×10−1, 1.46×10−1, and 1.60×10−1, respectively. The Sydetermined from nonCA shown in Table2ranges from
6.50×10−2 to 2.01×10−1 with a mean of 1.52×10−1. Obviously, the mean of Sy obtained from nonCA is not
significantly different from those of three sets determined from CA. Similarly, the means and SDs of logarithmic parameter values obtained from the results of CA listed in Tables 4–6 are also close to those determined from the results of nonCA. These results demonstrate that the average values of parameters Kr and Sy analyzed by the
SANS of CA and nonCA are close. Notice that the results
Table 3 The estimatedSyin well group 1 determined from CA in comparison to those from nonCA Composite wells
Set 1a, number of cases: 2
F377-037, F381-056 F476-061, F376-037
Set 2b, number of cases: 38
F505-032, F505-080 F505-032, F383-129 F434-060, F476-061 F381-056, F347-031 F505-032, F384-033 F450-061, F476-061 F383-032, F383-061 F505-032, F377-037 F476-061, F478-061 F383-032, F384-033 F505-032, F450-061 F504-060, F434-060 F383-032, F505-080 F504-032, F504-060 F504-060, F450-061 F383-061, F383-082 F504-060, F383-032 F505-059, F434-060 F384-033, F381-056 F504-060, F504-080 F377-037, F505-080 F385-032, F376-037 F504-060, F505-080 F381-056, F505-080 F504-032, F383-129 F505-032, F434-060 F505-080, F504-080 F504-032, F504-080 F505-032, F504-060 F505-059, F450-061 F505-032, F383-032 F505-032, F505-059 F505-059, F476-061 F505-032, F383-061 F505-059, F505-080 F505-059, F504-060 F505-032, F383-082 F434-060, F450-061
Set 3c, number of cases: 10
F383-129, F504-060 F505-059, F376-037 F504-060, F476-061
F385-032, F505-080 F434-060, F478-061 F504-060, F478-061
F505-032, F376-037 F450-061, F478-061 F505-059, F478-061
F505-032, F504-032 a
Syis smaller than that of two nonCA cases b
Syfalls between that of two nonCA cases c
Syis larger than that of two nonCA cases
Fig. 3 The drawdown curve predicted by Neuman’s solution (Neuman 1974) with parameters determined from the nonCA for well F505-080
Table 4 The estimated parameters of set 1 in Table3(two cases) Kr(m/min) ×10−2 Kz(m/min) ×10−2 S ×10−3 Sy×10−1 F377-037, F381-056 9.35 1.96 8.67 1.42 F476-061, F376-037 9.27 1.53 57.90 1.29 Mean 9.31×10−2 1.75×10−2 3.33×10−2 1.36×10−1 CV 6.08×10−3 1.74×10−1 1.05 6.78×10−2 Mean (log10) −1.03 −1.76 −1.65 −0.87
Statistics of nonCA results
Mean 8.98×10−2 1.96×10−2 1.80×10−2 1.52×10−1
SD 1.38×10−2 8.10×10−3 1.98×10−2 3.49×10−2
CV 0.15 0.41 1.10 0.23
Mean (log10) −1.05 −1.75 −1.97 −0.83
SD (log10) 0.06 0.19 0.44 0.12
SD standard deviation; CV coefficient of variation
Table 5 The estimated parameters of set 2 in Table3(38 cases)
Well No. Kr(m/min) ×10−2 Kz(m/min) ×10−2 S × 10−3 Sy× 10−1
F505-032, F505-080 8.46 2.67 5.04 0.74 F381-056, F347-031 9.65 1.43 8.12 1.64 F383-032, F383-061 8.38 2.47 8.71 1.87 F383-032, F384-033 9.18 14.30 12.30 1.94 F383-032, F505-080 7.55 3.21 5.35 2.10 F383-061, F383-082 8.04 2.65 7.91 1.70 F384-033, F381-056 9.20 1.63 6.20 1.71 F385-032, F376-037 12.50 9.06 6.17 1.50 F504-032, F383-129 8.77 1.98 6.54 0.89 F504-032, F504-080 8.75 3.04 5.24 0.90 F505-032, F383-032 7.50 1.97 7.25 1.85 F505-032, F383-061 7.68 3.01 5.68 1.30 F505-032, F383-082 7.94 1.85 7.20 1.20 F505-032, F383-129 8.65 1.24 7.71 0.64 F505-032, F384-033 7.61 1.77 7.44 1.55 F505-032, F377-037 7.70 2.64 5.60 1.34 F505-032, F450-061 7.83 3.29 5.57 1.20 F504-032, F504-060 8.61 3.00 4.80 0.94 F504-060, F383-032 8.14 2.75 5.20 1.77 F504-060, F504-080 8.38 2.99 5.18 1.36 F504-060, F505-080 7.85 3.02 5.13 1.59 F505-032, F434-060 7.89 3.53 5.52 1.06 F505-032, F504-060 8.08 3.35 4.94 0.95 F505-032, F505-059 8.25 3.12 5.05 0.82 F505-059, F505-080 7.59 3.28 5.26 1.86 F434-060, F450-061 8.23 2.01 55.30 1.58 F434-060, F476-061 8.26 2.01 59.40 1.50 F450-061, F476-061 8.86 1.77 47.70 1.44 F476-061, F478-061 8.98 1.63 59.40 1.47 F504-060, F434-060 7.98 3.11 4.80 1.50 F504-060, F450-061 8.34 2.86 4.96 1.47 F505-059, F434-060 7.68 3.25 5.18 1.71 F377-037, F505-080 7.81 3.01 5.20 1.72 F381-056, F505-080 7.83 2.98 5.20 1.63 F505-080, F504-080 7.49 3.35 5.35 2.10 F505-059, F450-061 7.82 3.07 5.29 1.75 F505-059, F476-061 7.83 3.06 5.35 1.71 Mean 8.30×10−2 3.12×10−2 1.12×10−2 1.46×10−1 SD 8.78×10−3 2.23×10−2 1.55×10−2 3.77×10−2 CV 0.11 0.71 1.38 0.26 Mean (log10) −1.08 −1.56 −2.13 −0.85 SD (log10) 0.04 0.19 0.31 0.13
Statistics of nonCA results
Mean 8.98×10−2 1.96×10−2 1.80×10−2 1.52×10−1
SD 1.38×10−2 8.10×10−3 1.98×10−2 3.49×10−2
CV 0.15 0.41 1.10 0.23
Mean (log10) −1.05 −1.75 −1.97 −0.83
SD (log10) 0.06 0.19 0.44 0.12
of sensitivity analysis performed by Huang and Yeh (2007) indicated that the drawdown in an unconfined aquifer was more sensitive to parameters Kr and Sy than
the other two parameters in response to pumping. Accordingly, the drawdown curve predicted based on the parameters obtained by the CA and nonCA will be similar because the parameters Krand Sydominate the drawdown
behavior in response to the pumping.
Table 7 lists the results of groups 2–4 and Table 8
shows the results of groups 5 and 6 when applying CA. Also, the statistics of the nonCA results are given at the bottom of these two tables. The estimated Sy in group 2
from CA always falls between the lowest and highest ones of those of the related nonCA cases. For example, the estimated Syof cluster F505 is 9.30×10−2and the nonCA
results of wells F505-032, F505-059, and F505-080 are 6.50×10−2, 1.76×10−1and 2.01×10−1, respectively. Thus, the lowest value is 6.50×10−2 and the highest one is 2.01×10−1. In group 3, the mean is 7.50×10−2m/min for Krand 1.83×10−1 for Sy, respectively. The mean of Kris
slightly less than that of the nonCA cases. Oppositely, the mean of Syis larger than that from the nonCA. In group 4,
the estimates of S for these two cases are about one order larger than those of other groups. Notably, these results are similar to those in Table 2. The statistics of estimated Kr
and Syin this table are close to those obtained by nonCA.
Table8displays the results of well groups 5 and 6. For group 5, the results are obtained by analyzing the drawdown data from three or four piezometers installed at deep, middle, or shallow depths. The mean values of parameters Kr and Sy are 7.83×10−2 m/min and 1.56×
10−1, respectively. The mean of Kris slightly less than that
of the nonCA cases; on the other hand, the mean of Kzis
larger than that from the nonCA. The small SDs of each parameter imply that the estimated parameters will be
close despite using the data from the observation wells with different depths of piezometers. In this table, the means of estimated parameters are close to that deter-mined from the nonCA. Table 9 shows four statistics of estimated parameters determined from nonCA and CA based on the data of groups 1–6. The means of estimated Kr and Sy are close and those of Kz and S show little
variation. Table10displays the results of analyzing all the data simultaneously based on three different analytical solutions (i.e., Moench 1997; Tartakovsky and Neuman
2007; and Neuman, 1974). The result shown on the first row are reported by Moench et al. (2001), the second ones are adopted from Tartakovsky and Neuman (2007; Table 3), and the last ones are obtained from group 7 by the SANS. Moench et al. (2001) gives the largest value of Sy (0.26) and smallest value of S (2.21×10−3) while
Tartakovsky and Neuman (2007) yields the largest value of Kz(4.88×10−2m/min) and smallest value of Kr(6.12×
10−2m/min) and the SANS produces the largest value of Kr (7.77×10−2 m/min) and smallest value of Sy (0.17).
Overall, the differences among those three sets of estimated parameters are minor. The statistics of nonCA results are listed at the bottom of this table. The means of estimated Krand Syby the nonCA are slightly larger and
smaller than those obtained by the SANS, respectively. This is reasonable because a greater Kraccompanied by a
smaller Sycan produce a similar drawdown curve to that
produced by a smaller Krwith a greater Sy. Note that both
works of Moench (1997) and Tartakovsky and Neuman (2007) contain some additional parameters in their solutions which are not included in Neuman’s solution (Neuman1974) of SANS and thus are not listed.
Figure4displays the comparison between the observed drawdown data in wells F505-080 and F505-032 and the predicted drawdown based on the estimated parameters
Table 6 The estimated parameters of set 3 in Table3(10 cases)
Kr(m/min) ×10−2 Kz(m/min) ×10−2 S × 10−3 Sy× 10−1 F383-129, F504-060 8.34 2.99 4.84 1.27 F385-032, F505-080 7.39 3.38 5.43 2.27 F505-032, F376-037 7.78 1.45 8.48 1.40 F505-032, F504-032 8.36 1.70 7.27 0.93 F505-059, F376-037 7.71 3.13 5.33 1.87 F434-060, F478-061 8.15 1.97 62.30 1.67 F450-061, F478-061 9.02 1.65 47.70 1.56 F504-060, F476-061 8.34 2.91 4.95 1.42 F504-060, F478-061 8.19 2.82 5.06 1.66 F505-059, F478-061 7.68 3.13 5.30 1.97 Mean 8.10×10−2 2.51×10−2 1.57×10−2 1.60×10−1 SD 4.68×10−3 7.32×10−3 2.10×10−2 3.79×10−2 CV 0.06 0.29 1.34 0.24 Mean (log10) −1.09 −1.62 −2.05 −0.81 SD (log10) 0.02 0.14 0.42 0.11
Statistics of nonCA results
Mean 8.98×10−2 1.96×10−2 1.80×10−2 1.52×10−1
SD 1.38×10−2 8.10×10−3 1.98×10−2 3.49×10−2
CV 0.15 0.41 1.10 0.23
Mean (log10) −1.05 −1.75 −1.97 −0.83
SD (log10) 0.06 0.19 0.44 0.12
Table 8 The estimated parameters determined from well groups 5 and 6 based on CA
Wells for CA Estimated parameters
Kr(m/min) ×10−2 Kz(m/min) ×10−2 S × 10−3 Sy× 10−1 Group 5 (three or four observation wells with different depth of piezometers)
F505-080, F504-080, F383-082 7.71 3.20 5.30 1.78
Three deep piezometers
F505-080, F504-080, F383-082, F383-129 7.76 3.17 5.26 1.71
Four deep piezometers
F505-059, F504-060, F383-061 7.86 3.03 5.19 1.68
Three mid piezometers
F505-032, F504-032, F377-037 8.05 2.14 6.45 1.23
Three shallow piezometers
F505-032, F504-032, F377-037, F383-032 7.78 2.83 5.34 1.41
Four shallow piezometers
Mean 7.83×10−2 2.87×10−2 5.51×10−3 1.56×10−1
SD 1.33×10−3 4.36×10−3 5.29×10−4 2.33×10−2
Mean (log10) −1.11 −1.55 −2.26 −0.81
SD (log10) 0.01 0.07 0.04 0.07
Group 6 (two or three well clusters)
Clusters F504, F505 8.06 3.15 5.07 1.13 Clusters F505, F383 7.72 3.16 5.39 1.55 Clusters F504, F383 8.26 3.01 5.16 1.36 Clusters F504, F505, F383 7.85 3.17 5.15 1.43 Mean 7.97×10−2 3.12×10−2 5.19×10−3 1.37×10−1 SD 2.37×10−3 7.54×10−4 1.38×10−4 1.77×10−2 Mean (log10) −1.10 −1.51 −2.28 −0.87 SD (log10) 0.01 0.01 0.01 0.06
Statistics of nonCA results
Mean 8.98×10−2 1.96×10−2 1.80×10−2 1.52×10−1
SD 1.38×10−2 8.10×10−3 1.98×10−2 3.49×10−2
Mean (log10) −1.05 −1.75 −1.97 −0.83
SD (log10) 0.06 0.19 0.44 0.12
SD standard deviation
Table 7 The estimated parameters determined from well groups 2, 3, and 4 based on CA
Wells for CA Estimated parameters
Kr(m/min) ×10−2 Kz(m/min) ×10−2 S × 10−3 Sy× 10−1 Group 2 (an observation well cluster which consists of two or three observation wells)
Cluster F505 8.17 3.13 5.06 0.93 Cluster F504 8.66 3.01 5.10 0.99 Cluster F383 8.07 3.20 5.40 1.65 Mean 8.30×10−2 3.11×10−2 5.19×10−3 1.19×10−1 SD 3.16×10−3 9.61×10−4 1.86×10−5 4.00×10−2 Mean (log10) −1.08 −1.51 −2.29 −0.94 SD (log10) 0.02 0.01 0.02 0.14
Group 3 (three or four observation wells with different radial distances from the pumping well)
F505-032, F384-033, F376-037 7.51 1.74 7.43 1.76
F505-032, F383-032, F381-056, F376-037 7.49 1.96 6.49 1.89
Mean 7.50×10−2 1.85×10−2 6.96×10−3 1.83×10−1
Mean (log10) −1.12 −1.73 −2.16 −0.74
Group 4 (three or four observation wells with long screens)
F434-060, F476-061, F478-061 8.35 1.87 61.40 1.58
F434-060, F450-061, F476-061, F478-061 8.46 1.84 56.90 1.57
Mean 8.41×10−2 1.86×10−2 5.92×10−2 1.58×10−1
Mean (log10) −1.08 −1.73 −1.23 −0.80
Statistics of nonCA results
Mean 8.98×10−2 1.96×10−2 1.80×10−2 1.52×10−1
SD 1.38×10−2 8.10×10−3 1.98×10−2 3.49×10−2
Mean (log10) −1.05 −1.75 −1.97 −0.83
SD (log10) 0.06 0.19 0.44 0.12
determined from both nonCA (dashed line) and CA (solid line). Judging from the curve fitting (SEE values) to the observed data, the analysis of data based on nonCA gives more precisely parameter estimation (SEE=7.02×10−3m for well F505-080 and 7.06×10−3m for well F505-032) than that based on CA (SEE=9.51×10−3m).
In fact, the mathematical model describing the response of an aquifer to well pumping generally assumes that the aquifer material is homogeneous. Mathematically, the analysis of data based on CA suggests that the geology of the observation well near the pumping well has greater weight than that far away from the pumping well in a multiple observation well system. In other words, the geological properties of the observation wells located near the pumping well will have greater influence on the parameter estimation than geological properties far away from the pumping well. If the aquifer is fairly homoge-neous, the estimated parameters based on the CA and nonCA should be about the same. However, in a heterogeneous aquifer, the application of CA to the determination of parameters is very likely to give biased
estimate of the hydraulic parameters. In addition, the principle of least squares requires that the measurement errors are mutually independent (Neter et al. 1996). For sampling points installed very close together or located at the same place but at different depths, there is a tendency for neighboring observations to be correlated if the data are collected sequentially. Under those circumstances, the assumption that the measurement errors are mutually independent is very likely violated. In other words, autocorrelation may exit in the drawdown data of multiple observation wells. Thus, the application of CA may give a biased estimation for the hydraulic parameters from a statistical viewpoint (Berthouex and Brown2002).
Hypothetical case study
Tables 11–13 display the results of the hypothetical case study when analyzing the drawdown data from OW1, OW2, and both of these two observation wells, respec-tively. From these tables, the means of estimated param-eters are very close to the target values and the differences
Table 10 The global analysis obtained by three different approaches
Kr(m/min) × 10−2 Kz(m/min) × 10−2 S × 10−3 Sy
Moench et al. (2001) 7.02 4.27 2.21 0.26
Tartakovsky and Neuman (2007) 6.12 4.88 5.10 0.18
SANS 7.77 3.14 5.28 0.17
Statistics of nonCA results
Mean 8.98×10−2 1.96×10−2 1.80×10−2 1.52×10−1
SD 1.38×10−2 8.10×10−3 1.98×10−2 3.49×10−2
Mean (log10) −1.05 −1.75 −1.97 −0.83
SD (log10) 0.06 0.19 0.44 0.12
SD standard deviation
Table 9 The statistics of estimated parameters based on nonCA and CA for data of groups 1–6
Parameter Kr(m/s) × 10−2 Kz(m/s) × 10−2 S × 10−2 Sy× 10−1 Kr(m/s) Kz(m/s) S Sy Statisitics Mean SD NonCA 8.98 1.96 1.80 1.52 1.38 8.10 1.98 3.49 Group 1 Set 1 9.31 1.75 3.33 1.36 NA NA NA NA Set 2 8.30 3.12 1.12 1.46 8.78×10−3 2.23×10−2 1.55×10−2 3.77×10−2 Set 3 8.10 2.51 1.57 1.60 4.68×10−3 7.32×10−3 2.10×10−2 3.79×10−2 Group 2 8.30 3.11 0.52 1.19 3.16×10−3 9.61×10−4 1.86×10−5 4.00×10−2 Group 3 7.50 1.85 0.70 1.83 NA NA NA NA Group 4 8.41 1.86 5.92 1.58 NA NA NA NA Group 5 7.83 2.87 0.55 1.56 1.33×10−3 4.36×10−3 5.29×10−4 2.33×10−2 Group 6 7.97 3.12 0.52 1.37 2.37×10−3 7.54×10−4 1.38×10−4 1.77×10−2
Mean (log10) SD (log10)
NonCA −1.05 −1.75 −1.97 −0.83 0.06 0.19 0.44 0.12 Group 1 Set 1 −1.03 −1.76 −1.65 −0.87 2.64×10−3 0.08 0.58 0.03 Set 2 −1.08 −1.56 −2.13 −0.85 NA NA NA NA Set 3 −1.09 −1.62 −2.05 −0.81 0.02 0.14 0.42 0.11 Group 2 −1.08 −1.51 −2.29 −0.94 0.02 0.01 0.02 0.14 Group 3 −1.12 −1.73 −2.16 −0.74 NA NA NA NA Group 4 −1.08 −1.73 −1.23 −0.80 NA NA NA NA Group 5 −1.11 −1.55 −2.26 −0.81 0.01 0.07 0.04 0.07 Group 6 −1.10 −1.51 −2.28 −0.87 0.01 0.01 0.01 0.06
in estimated parameters are also minor when compared with the target ones. This result may be due to the fact that there is plenty of observed drawdown data so that the effect of measurement errors, which are normally distrib-uted with zero mean, is minor. The SEE value is about 0.01 which is equal to the SD of the measurement errors. In Table 13, the categories A, B, and C represent the situation whereby the Sy determined from the CA is
smaller than, falls between, and is greater than Sy of
nonCA, respectively. Most of the sets (17 sets) belong to the category B and this result indicates that the CA generally gives an average estimate of Sy. However, the
total number of categories A and C is 3 and this result indicates that there is 15% chance that the CA gives larger or smaller estimates of Sythan those of the nonCA cases
in this hypothetical case study. Obviously, the parameter determination obtained from the CA still gives poorer estimate of Sy than that determined from nonCA if the
drawdown data contain measurement errors.
Conclusions
This study uses a computer method named SANS devel-oped by Huang et al. (2008) to analyze the drawdown data of real and hypothetical cases for determining the hydraulic parameters of unconfined aquifers. In the field cases, the drawdown data of the pumping tests are taken from the famous experimental site of Cape Cod, Massa-chusetts for the comparison of parameter estimation based on nonCA and CA approaches. On the other hand, the hypothetical case is designed to explore the effect of measurement errors on the estimated parameters.
The drawdown data obtained from 20 observation wells at the Cape Cod site were analyzed separately based on the present method of nonCA. In addition, the drawdown data from those observation wells were classified into seven groups and analyzed by the present method of CA. Those seven groups are: (1) two wells randomly chosen from 20 observation wells; (2) a well cluster consisting of two or three observation wells; (3) three or four wells with different radial distances from the
Table 11 The results of nonCA using the data from OW1
Noise set Kr(m/s) × 10−3 Kz(m/s) × 10−4 S × 10−4 Sy× 10−1 SEE (m) × 10−2
1 1.014 0.985 0.991 0.958 1.067 2 1.001 1.033 0.947 1.011 1.115 3 0.991 1.024 1.006 1.035 1.071 4 0.968 1.052 1.014 1.118 1.025 5 1.049 0.976 0.917 0.819 0.967 6 1.007 1.013 0.984 0.918 0.925 7 0.975 1.015 1.062 1.092 0.936 8 1.026 0.950 0.972 0.853 1.018 9 1.005 0.986 1.029 0.944 1.120 11 0.979 1.010 0.980 1.163 0.979 12 0.996 1.013 0.928 1.022 0.856 13 1.097 0.934 1.007 0.658 0.985 14 0.999 1.002 1.005 1.001 1.003 15 1.044 0.935 1.059 0.824 0.979 16 0.985 0.985 0.900 1.097 1.078 17 1.021 0.958 0.997 0.922 0.869 18 0.979 1.040 1.077 1.126 1.023 19 0.999 0.997 0.944 1.020 0.902 20 0.980 1.061 1.012 1.167 0.887 Mean 1.008×10−3 9.956×10−4 9.903×10−5 9.792×10−2 SD 3.173×10−5 3.792×10−6 4.751×10−6 1.344×10−2 SD standard deviation
Fig. 4 The drawdown curves predicted by Neuman’s solution (Neuman1974) with parameters determined from nonCA and CA for the wells F505-032 and F505-080
pumping well; (4) three or four wells with long screen; (5) three or four wells with different depth of piezometers; (6) two or three well clusters; and (7) all wells (global analysis). The results were used to examine whether the spatial distribution and the type of observation well affect the determination of parameters when applying the present method of CA. For well group 1, 50 cases were analyzed and the Sydetermined by the CA can be divided into three
categories; i.e., the Sy is smaller than, in between, and
larger than those of the counterpart, nonCA. The means of Kr and Sy in these three categories are very close to that
determined from the nonCA. This result indicates that both CA and nonCA give similar results since the drawdown behavior is much more sensitive to these two parameters than the other two parameters in response to the pumping. For well groups 2–6, the estimated Sy always falls between the lowest and highest values of Sy
from nonCA when analyzing the drawdown data from
Table 13 The results of CA using the data from OW1 and OW2
Noise set Kr(m/s) ×10−3 Kz(m/s) ×10−4 S × 10−4 Sy× 10−1 SEE (m) × 10−2 Category
1 1.003 0.977 1.002 1.043 0.936 B 2 1.017 1.007 0.960 0.958 1.043 B 3 0.996 1.026 0.991 1.014 0.993 B 4 0.996 1.020 0.993 0.993 1.099 B 5 1.015 1.015 0.946 0.944 0.957 B 6 0.985 1.026 0.988 1.028 0.951 B 7 1.003 0.964 1.065 1.005 0.974 B 8 0.985 0.973 1.018 1.040 1.029 B 9 0.986 0.990 1.005 1.055 1.047 B 11 1.008 0.985 0.982 0.999 1.056 A 12 0.994 0.993 0.940 1.029 0.959 B 13 1.026 0.988 1.078 0.894 1.092 B 14 1.006 0.994 0.972 0.986 1.030 B 15 1.010 0.956 1.093 0.959 0.958 B 16 1.000 0.969 0.909 1.019 1.084 B 17 1.009 0.972 1.005 0.970 0.978 C 18 1.009 1.005 1.059 0.983 1.071 B 19 1.000 1.003 0.965 1.001 0.910 B 20 1.023 1.006 1.013 0.948 0.953 A Mean 1.004×10−3 9.916×10−5 1.000×10−4 9.930×10−2 SD 1.176×10−5 2.150×10−6 4.735×10−6 3.961×10−3
Category A is the estimated Syusing CA is smaller than those of two nonCA cases; category B is the estimated Syusing CA falls between those of two nonCA cases; category C is the estimated Syusing CA is larger than those of two nonCA cases; SD standard deviation Table 12 The results of nonCA using the data from OW2
Noise set Kr(m/s) × 10−3 Kz(m/s) × 10−4 S × 10−4 Sy× 10−1 SEE (m) × 10−2
1 0.989 0.933 1.294 1.071 0.883 2 1.045 0.987 1.109 0.907 1.005 3 1.023 1.065 0.928 0.950 0.934 4 1.027 1.022 0.948 0.927 1.198 5 1.001 1.035 1.038 0.979 0.964 6 0.963 0.993 0.927 1.109 0.982 7 1.063 0.925 1.160 0.903 0.912 8 0.959 0.912 1.259 1.147 0.966 9 0.968 0.944 0.733 1.139 0.907 11 0.992 0.978 1.110 1.005 1.143 12 0.904 0.895 0.992 1.275 0.959 13 0.950 0.933 1.287 1.093 1.065 14 1.047 1.010 0.759 0.911 1.013 15 0.973 0.916 1.188 1.067 0.959 16 0.988 0.951 1.026 1.036 1.122 17 1.021 0.992 1.040 0.944 1.127 18 1.015 0.991 1.039 0.961 1.160 19 0.996 1.020 1.107 0.995 0.959 20 0.997 0.954 1.117 0.985 0.980 Mean 9.961×10−4 9.696×10−5 1.070×10−4 1.022×10−1 SD 3.740×10−5 4.647×10−6 1.649×10−5 9.703×10−3 SD standard deviation
three and four observation wells under different data selection strategies. For well group 7, the estimated parameters have the largest value of Kr and the smallest
value of Sy when compared with those analyzed by
different analytical solutions in the previous studies. In the hypothetical case, two sets of pumping draw-down data are generated. One set is from observation well 1 (OW1) and the other is from observation well 2 (OW2). Forty sets of normally distributed noise were generated to represent the measurement errors with the magnitude on the order of centimeter and two 20 noise sets were added to the drawdown data of OW1 and OW2, respectively. The SANS was first used to determine the hydraulic parameters based on nonCA using the drawdown data from OW1 or OW2. Then the drawdown data from two observation wells were analyzed based on CA. In this case study, the estimates of Syare larger or smaller than those
determined from nonCA in 3 out of total 20 cases. Obviously, the parameter determination using CA also gives poorer estimates of Sy than that determined using
nonCA when the data contain measurement errors. Based on the results of field and hypothetical case studies, the parameter determination based on the CA does not always give better result than those obtained based on the nonCA. The biased estimates of Sy are probably
attributed to one or more of problems, including improper procedures, bad data, and aquifer heterogeneity (Moench
1994) and have nothing to do with the use of CA or nonCA.
Acknowledgements Research leading to this paper has been partially supported by grants from Taiwan National Science Council under the contract number NSC96–2221–E–009–087-MY3. The authors would also like to thank the associate editor and three anonymous reviewers for their valuable and constructive comments.
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