Journal of Hydrology (2007) 335, 406– 418 available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/jhydrol The use of sensitivity analysis in on-line aquifer parameter estimation Yen-Chen Huang, Hund-Der Yeh * Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan Received 7 June 2006; received in revised form 20 October 2006; accepted 15 December 2006 KEYWORDS Groundwater; Pumping test; Sensitivity analysis; Parameter estimation model; Leaky aquifer; Unconfined aquifer Generally, a pumping test requires a lot of effort and expense to perform the test and the drawdown is measured and analyzed for determining the aquifer parameters. The estimated aquifer parameters obtained from graphical approaches may not be in good accuracy if the pumping time is too short to give a good visual fit to the type curve. Yet, the problems of long pumping time and required efforts can be significantly reduced if the drawdown data are measured and the parameters are simultaneously estimated on-line. However, the drawdown behavior of the leaky and unconfined aquifers in response to the pumping may have a time lag and the time to terminate the estimation may not be easily and quickly to decide when applying a parameter estimation model (PEM) on-line to analyze the parameters. This study uses the sensitivity analysis to explore the influence period of each aquifer parameter to the pumping drawdown and the influence period is used as a guide in terminating the estimation when applying the PEM for on-line parameter identification. In addition, the sensitivity analysis is also used to study the effects of different value of Sy and the distance between pumping well and observation well on the influence time of Sy during the pumping. ª 2006 Elsevier B.V. All rights reserved. Summary Introduction Groundwater hydrologists often conduct pumping tests to obtain aquifer parameters, such as hydraulic conductivity and storage coefficient, which are necessary information for quantitative groundwater studies. Theis (1935) obtained the solution for unsteady groundwater flow toward a pumping well in a confined aquifer by analogy to the problem of * Corresponding author. Fax: +886 3 5726050. E-mail address: [email protected] (H.-D. Yeh). heat conduction. Hantush and Jacob (1955) described non-steady radial flow to a well in a fully penetrating leaky aquifer under a constant pumping rate. In their model, the aquitard is overlain by an unconfined aquifer, and the main aquifer is underlain by an impermeable bed. Boulton (1954, 1963) developed the analytical solution by introducing the concept of delayed yield for unconfined formations. Neuman (1972, 1974) presented a solution that considers the effects of elastic storage and anisotropy of aquifers on drawdown behavior and recognized the existence of vertical flow components. Neuman’s model can fit observed pumping 0022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2006.12.007 The use of sensitivity analysis in on-line aquifer parameter estimation test data in many case studies and is very convenient to use in engineering practice. Moench (1997) presented a Laplace transform solution to a partially penetrating well of finite diameter in a slightly compressible water table aquifer. The solution, which accounts for the effects of well bore storage and skin, uses numerical inversion of Stehfest’s algorithm to obtain the dimensionless drawdown in time domain. In the past, the hydrogeological parameters were determined using graphical methods. Cooper and Jacob (1946) developed a method to approximate the Theis equation, together with a data analysis approach which does not require type-curve matching. Hantush (1964) developed a typecurve method for determining parameters of the leaky aquifer if the test period is long enough to reflect the influence of the leakage. Prickett (1965) described a systematic approach to determine the parameters, using a graphical procedure based on Boulton’s type curves. Neuman (1975) also gave a graphical type-curve solution procedure to determine the hydraulic parameters in unconfined aquifer. The aquifer parameters can also be obtained by parameter estimation model (PEM) which coupled an analytical solution or a numerical model in terms of aquifer drawdown along with a numerical approach such as nonlinear programming (e.g., Saleem, 1970), Marquardt algorithm (e.g., Chander et al., 1981), sensitivity matrix, (McElwee, 1980; Paschetto and McElwee, 1982), nonlinear least-squares and Newton’s method (e.g., Yeh, 1987; Yeh and Han, 1989), nonlinear regression (e.g., Lebbe, 1999), and extended Kalman filter (e.g., Leng and Yeh, 2003; Yeh and Huang, 2005). Some commercial softwares, like AQTESOLV (Duffield, 2002), also use nonlinear weighted least-squares approach to match the time–displacement data obtained from an aquifer test with type curves or straight lines for parameter estimation. Alternatively, heuristic optimization approaches such as Simulated Annealing (e.g., Yeh et al., in press) was proposed to couple with an analytical solution for determining the best-fit parameters. Recently, the sensitivity analysis is widely used in many fields. Cukier et al. (1973, 1975, 1978) as well as Schibly and Shuler (1973) developed a statistical approach for sensitivity analysis to nonlinear algebraic equations. Jiao and Rushton (1995) provided a sensitivity analysis of drawdown to parameters and drawdown’s influence on parameter estimation for pumping tests in large-diameter wells. They concluded that the well storage reduces the sensitivities of drawdown to transmissivity and storativity. Kabala and Milly (1990) used sensitivity analysis for analyzing the effect of parameter uncertainty and soil heterogeneity on the transport of moisture in unsaturated porous media. Kabala (2001) proposed logarithmic sensitivity to analyze the pumping test on a well with wellbore storage and skin. In addition, Kabala et al. (2002) also studied the logarithmic sensitivity, plausible relative errors, and deterministic parameter correlations in a simple semianalytic no-crossflow model of the transient flowmeter test (TFMT) that accounts for a thick skin around the wall. Vachaud and Chen (2002) analyzed a large-scale hydrologic model problem by sensitivity theory. Gooseff et al. (2005) performed sensitivity analysis of a conservative transient storage model and two different reactive solute transport models. The pumping test was commonly performed for a long period of time when applying a graphical approach to analyze the measurement data in the past. Otherwise, the esti- 407 mated result may not be in good accuracy if the pumping time is too short and the data points are too sparse to give a good visual fit to the type curve. In the leaky aquifer, the hydraulic head in the adjacent aquifer remains constant and that the two aquifers are in equilibrium at the beginning of the pumping. After pumping, the water is immediately withdrawn from the production aquifer and then the head difference between two aquifers induces a flow across the aquitard. Hence, the hydraulic parameters of the confining bed (aquitard) may not be accurately estimated if only first few drawdown data points are used. Physically, the drawdown in an unconfined aquifer can be divided into three segments (Charbeneau, 2000). In the early stage, water is instantaneously released from storage by the compaction of the aquifer and the expansion of the water. In the second stage, the vertical gradient near the water table causes drainage of the porous matrix. The vertical hydraulic conductivity Kz starts to contribute to the pumping and the rate of decline in the hydraulic head slows or stops after a period of time. Finally, when the flow is essentially horizontal and most of the pumping is supplied by the specific yield, Sy. Therefore, the analysis of Sy requires sufficient long drawdown data fallen at the third section. In some cases, the effect of well bore storage can not be neglected because the diameter of pumping well is large. The water is withdrawn from the well at the start of pumping, and consequently the groundwater flows into the well due to the head difference between the well and the formation. An on-line PEM for identifying aquifer parameter can facilitate the applicability of the pumping test. A practical question involved when using on-line PEM is: when is a suitable time to terminate the estimation? The results of parameter estimation may be inaccurate if the parameter estimation is terminated before the character of aquifer parameters starts to affect the drawdown. This study aims at providing a decision support using sensitivity analysis in terminating the estimation when applying the on-line PEM in determining the aquifer parameters. Three synthetic drawdown data sets, one for leaky aquifer (generated based on Hantush and Jacob’s model), and two for unconfined aquifer (generated based on Neuman’s model and Moench’s models). A PEM based on Simulated Annealing algorithm is applied to identify the parameters in both leaky and unconfined aquifers on-line using the synthetic and real field time-drawdown data sets. In addition, AQTESOLV is employed to identify the parameters of unconfined aquifer considering the effect of well bore storage using the synthetic data set. The influence period obtained from the sensitivity analyses is used as an indication to terminate the on-line estimation because the drawdown already reflects the effects from the aquifer parameters. Finally, two sensitivity analyses for different Sy values and different distance between pumping well and observation well are performed to study their affects on the influence period of the Sy. Drawdown of the pumping test in leaky and unconfined aquifers The Hantush and Jacob’s model describing the drawdown within a leaky aquifer in response to the pumping as a function of radial distance and time is (Hantush and Jacob, 1955) 408 r Q W u; ð1Þ 4pT B where s is drawdown, r is the distance between pumping well and observing well, u is dimensionless variable and it is defined as r2S/4Tt, K 0 is the vertical conductivity of leaky confining bed, b 0 is thickness of aquitard,pr/B = L is called ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ leakage coefficient and B is defined as T=ðK 0 =b0 Þ, Q is the pumping rate, and W(u,r/B) is the leaky well function. The leaky well function W(u, r/B) can be expressed as " # r Z 1 1 ðr=BÞ2 ¼ exp y W u; dy ð2Þ B y 4y u s¼ where y is a dummy variable. The solution for the equation describing the groundwater flow system in an unconfined aquifer developed by Neuman (1974) is " # Z 1 1 X q 1=2 sðr; z; tÞ ¼ 4yJ0 ðyb Þ u0 ðyÞ þ un ðyÞ dy ð3Þ 4pT 0 n¼1 where J0(x) is the zero order Bessel function of the first kind, b = Kzr2/Krb2 is a dimensionless parameter, y is a dummy variable, and f1 exp ts b y 2 r20 g cos hðr0 zD Þ u0 ðyÞ ¼ 2 ½y 2 þ ð1 þ rÞr20 y 2 r20 =r cos hðr0 Þ sin h½r0 ð1 d D Þ sin h½r0 ð1 lD Þ ð4Þ ðlD d D Þ sin hðr0 Þ 2 f1 exp ts b y þ r2n g cosðrn zD Þ un ðyÞ ¼ h 2 i y 2 ð1 þ rÞr2n y 2 þ r2n =r cosðrn Þ sin½rn ð1 d D Þ sin½rn ð1 lD Þ ðlD d D Þ sinðrn Þ r20 < y 2 Ksrw, and rD = r/rw. Notice that rw represents the well radius. The symbol en is the root of p ð11Þ en tanðen Þ ¼ ðrbw þ p=cÞ where r = S/Sy, c = a1bSy/Kz, and a1 is a fitting parameter for drainage from the unsaturated zone and has units of inverse time (1/T). A large value of a1 effectively eliminates this parameter from the solution. Sensitivity analysis of the aquifer parameters The sensitivity is defined as a rate of change in one factor with respect to a change in another factor. The parametric sensitivity may be expressed as (McCuen, 1985) Spi ¼ ð5Þ ð6Þ oO OðPi þ DPi ; P jjj6¼i Þ OðP1 ; P2 ; :::; P n Þ ¼ oPi DPi ð12Þ where O is the output function of the system (i.e., the aquifer drawdown) and Pi is the ith input parameter of the system. However, the values of the parametric sensitivity for various parameters are useless for making comparison if the unit and/or the order of magnitude of the parameters are different. Thus, the normalized sensitivity is used and defined as (Kabala, 2001) Si;t ¼ where ts = Tt/Sr2 represents the dimensionless time since pumping started, S equals Ss · b, zD = z/b is the dimensionless elevation of observation point, r = S/Sy is a dimensionless parameter, dD = d/b denotes the dimensionless vertical distance between the top of perforation in the pumping well and the initial position of 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 water table. The term of r0 and rn are respectively the roots of the following two equations: rr0 sin hðr0 Þ ðy 2 r20 Þ cos hðr0 Þ ¼ 0; Y.-C. Huang, H.-D. Yeh qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n ¼ n rD ¼ where W D ¼ pr2c =2pr2w Ss ðl dÞ, q e2n bw þ p , q qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 2 en b þ prD , bw ¼ rw K z =b K r , b = Kzr /Krb , Sw = Krds/ oO oO ¼ Pi oPi =P i oPi ð13Þ where Si,t is the normalized sensitivity of ith input parameter at time t. Note that O is a function of Pi and t. The partial derivative of this equation may be approximated by a forward differencing formula as oO OðPi þ DPi Þ OðP i Þ ¼ oPi DPi ð14Þ The increment in the denominator may be approximated by the parameter value times a factor of 103, i.e., DPi = 103Pi. Eq. (13) measures the influence that the fractional change in the parameter, or its relative error, exerts on the output. and rrn sinðrn Þ þ ðy 2 þ r2n Þ cosðrn Þ ¼ 0; ð2n 1Þðp=2Þ < rn < np ð7Þ Moench (1997) derived a Laplace transform solution for transient flow to a partially penetrating large-diameter well in an unconfined aquifer. The dimensionless drawdown is 2E D ðrD ; zD ; pÞ ¼ ð8Þ h pðlD d D Þ½1 þ pW D ðA þ Sw Þ with 1 X 2 K 0 ðqn Þfsin½en ð1 d D Þ sin½en ð1 lD Þg2 ð9Þ ðlD d D Þ n¼0 en qn K 1 ðqn Þ½en þ 0:5sinð2en Þ 1 X K 0 ðqn rD Þcosðen zD Þ½sinðen ð1 d D ÞÞ sinðen ð1 lD ÞÞ E ¼2 qn K 1 ðqn Þ½en þ 0:5sinð2en Þ n¼0 A¼ ð10Þ The objective function of the parameter estimation model bases on Simulated Annealing algorithm The aquifer parameters can be estimated for pumping test data based on Hantush and Jacob’s model (1955) for a leaky aquifer and Neuman’s model (1974) for an unconfined aquifer when minimizing the sum of square errors between the observed and predicted drawdowns. The objective function is defined as Minimize n X ðOhi Phi Þ2 ð15Þ i¼1 where n is the total time step and Ohi and P hi are respectively the observed and predicted drawdowns at time step i. Based on Eq. (15), Simulated Annealing method can deter- The use of sensitivity analysis in on-line aquifer parameter estimation mine the best-fit aquifer parameters to the observed drawdown data. Results and discussion Sensitivity analysis of aquifer parameters The synthetic time-drawdown data for a leaky aquifer listed in Table 1 was generated from Hantush and Jacob’s model (1955). The pumping rate Q is 3000 m3/day, the distance R between pumping well and observation well is 30 m, the transmissivity T is 1000 m2/day, storage coefficient S is 104, leakage coefficient L is 0.03. The observed pumping period ranges from 0.017 to 1000 min. The time-drawdown data and the normalized sensitivities are plotted in Figure 1. This figure indicates that the temporal distribution of each normalized sensitivity of the aquifer parameters reflects the temporal change of the drawdown in response to the relative change of each parameter. In other words, the non-zero periods in the normalized sensitivity curves imply that the aquifer parameters have influences on the drawdown at that time. In addition, this figure also indicates that all aquifer parameters have their own influence period to the drawdown. The influence period of parameter S increases from the start of pumping and decreases after 3 min. The drawdown is very sensitive to T except at the early period of the pumping and the normalized sensitivity is continuously increased through the end of the pumping. The parameter of leakage coefficient, L, appears to have influence on the drawdown from 1.5 min through the end of pumping. Such a phenomenon can be related to the physical behavior of the leaky aquifer. The normalized sensitivity of L keeps zero before 1.5 min, and it may ascribe to the fact that there is a time lag between the start of pumping and the response of the drawdown to the leakage effect. In contrast, the normalized sensitivities indi- Table 1 The synthetic drawdown data for the leaky aquifer No. Time (min) Drawdown (m) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.017 0.050 0.100 0.250 0.750 1.000 1.500 2.000 2.500 3.500 4.500 5.000 10.000 20.000 50.000 100.000 200.000 500.000 700.000 1000.000 0.013 0.099 0.203 0.380 0.621 0.687 0.781 0.847 0.899 0.977 1.035 1.059 1.215 1.365 1.539 1.640 1.702 1.728 1.730 1.730 Q = 3000 m3/day, R = 30 m. 409 cate that the parameters T and S have influence on the drawdown right at the beginning of pumping. In addition, the influence of S is larger than that of T at early pumping period. This result to some extend reflects the physical behavior of parameters T and S during the pumping. The time-drawdown data set 1 of an unconfined aquifer, generated by Neuman’s model (1974), for pumping starting from 1 to 176,360 s (49 h) in an unconfined aquifer are listed in Table 2. The thickness of the aquifer, b, is 10 m, pumping rate Q is 3000 m3/day, and the distance between the pumping well and observation well R is 10 m. The radial hydraulic conductivity Kr, vertical hydraulic conductivity Kz, storage coefficient S, and specific yield Sy are set to 1 · 103 m/s, 1 · 104 m/s, 1 · 104, and 1 · 101, respectively. The time-drawdown data and related normalized sensitivities are plotted in Figure 2. Similar to Figure 1, the temporal distribution of each normalized sensitivity reflects the temporal change of the drawdown in response to the relative change of each parameter, and all aquifer parameters affect the drawdown at different periods. The normalized sensitivity of parameter S starts from 1 to 10 s, Kz ranges from 1 to 1000 s, and Sy appears from 80 s to the end of pumping. The drawdown is most sensitive to the parameter Kr except at the early period of the pumping and the influence of Kr on the drawdown increases at the beginning and through the end of the pumping. The normalized sensitivity of S begins with highest value and drops quickly after the start of the pumping. The normalized sensitivity of Kz reaches its highest value in a range between 10 and 1000 s, implying that the slow decline of the water table is attributed to the contribution of the Kz at the moderate pumping time. The drawdown stops increasing when the normalized sensitivity of Kz approaches its maximum. The temporal distribution of Kr’s normalized sensitivity, displaying three segments during the pumping period, is similar to the drawdown curve. The second segment appears at 10 s and vanishes at 1000 s (16.67 min). Figure 2 shows that the drawdown increases in the third segment along with the decrease of Kz’s normalized sensitivity, clearly indicating rapid decrease of vertical drainage. The sensitivity curve demonstrates that the aquifer parameter Sy does not have any contribution in response to the pumping at the beginning of the test and starts to react at about 80 s (1.33 min). The time-drawdown data set 2 listed in Table 3 is generated by Moench’s model (1997). The pumping starts from 0.6 to 600,000 s (16.67 hours). The thickness of the aquifer, b, is 10 m, pumping rate Q is 1000 m3/day, and the distance between the pumping well and observation well R is 10 m. The parameters Kr, Kz, S, Sy, and well radius rw, are set to 1 · 103 m/s, 1 · 104 m/s, 1 · 104, 1 · 101, and 1 m respectively. The time-drawdown data and related normalized sensitivities are plotted in Figure 3. The upper panel of Figure 3 shows the same plot without the normalized sensitivity of parameter Kr because the magnitude of Kr’s normalized sensitivity is very large at the late time of pumping. Removing Kr’s normalized sensitivity is helpful to recognize the small change of other parameter’s normalized sensitivities at the early time of pumping. The normalized sensitivity of parameter rw varies from 2 to 2000 s, S changes from 0.6 to 1000 s, Kz ranges from 100 to 10,000 s, and Sy appears from 100 s toward the end of pump- 410 Y.-C. Huang, H.-D. Yeh Figure 1 Table 2 The time-drawdown data and the normalized sensitivities of the leaky aquifer parameters. The synthetic drawdown data set 1 for the unconfined aquifer No. Time (s) Drawdown (m) No. Time (s) Drawdown (m) No. Time (s) Drawdown (m) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 30 44 58 74 0.22 0.31 0.36 0.38 0.40 0.41 0.41 0.42 0.42 0.42 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 87 120 149 176 212 272 332 393 472 600 792 967 1143 1350 1723 2154 2632 3215 4385 0.44 0.44 0.44 0.44 0.45 0.45 0.46 0.46 0.47 0.48 0.49 0.50 0.52 0.53 0.55 0.58 0.61 0.64 0.70 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 6000 8000 9354 11,429 14,925 18,235 22,274 25,882 32,696 41,295 47,195 59,224 69,279 81,302 95,126 118,168 151,775 176,360 0.76 0.83 0.86 0.91 0.98 1.03 1.09 1.13 1.19 1.25 1.29 1.35 1.40 1.44 1.48 1.54 1.61 1.65 ing. The drawdown is very sensitive to the parameter Kr after pumping for 300 s. The influence of Kr on the drawdown starts at about 60 s and increases through the end of the pumping. The normalized sensitivity of parameter rw starts at the beginning of the pumping, reflecting the phenomenon that the well bore storage contributes to the drawdown imme- diately after pumping. The normalized sensitivity of S is relative small compared with those of other parameters. The normalized sensitivity of Kz reaches its highest value in the range between 600 and 2000 s. Similar to Figure 2, the drawdown slowly increases when the normalized sensitivity of Kz approaches its maximum, indicating that the slow decline of the water table is attributed to the contri- The use of sensitivity analysis in on-line aquifer parameter estimation Figure 2 Table 3 411 The time-drawdown data and the normalized sensitivities of the unconfined aquifer parameters (Neuman’s model). The synthetic drawdown data set 2 for the unconfined aquifer No. Time (s) Drawdown (m) No. Time (s) Drawdown (m) No. Time (s) Drawdown (m) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.6 1 2 3 4 5 6 7 8 9 10 12 14 16 18 21 0.0001 0.0003 0.0008 0.0014 0.0020 0.0028 0.0034 0.0040 0.0047 0.0055 0.0064 0.0075 0.0087 0.0101 0.0116 0.0134 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 24 27 31 36 41 47 54 63 72 82 95 125 189 249 497 655 0.0155 0.0178 0.0204 0.0234 0.0268 0.0306 0.0348 0.0396 0.0449 0.0507 0.0572 0.0722 0.0993 0.1199 0.1720 0.1892 33 34 35 36 37 38 39 40 41 42 43 44 45 46 1138 1722 1977 2992 5970 11,912 18,029 35,973 62,514 94,619 124,732 188,789 328,078 600,000 0.2130 0.2251 0.2290 0.2424 0.2741 0.3189 0.3507 0.4088 0.4577 0.4950 0.5200 0.5578 0.6084 0.6638 bution of the Kz at the moderate pumping time. Figure 3 also shows that the effect of well bore storage is larger than that of Kr at early pumping period. This phenomenon indicates that the water is withdrawn from the well first after pumping and the groundwater flows into the well due to the head difference between the well and the aquifer. Certainly, the parameter Sy still does not have any contribution in response to the pumping at the beginning of the test and starts to react at about 100 s (1.67 min). Compared Figure 2 with Figure 3, the normalized sensitivities of parameters Kr, Kz, S, and Sy have similar temporal distributions but different magnitudes. In Moench’s model, the effect of S is relative small, the influence periods of S and Kz are longer than that of Neuman’s model, and the effect of rw is larger than that of Kr at the beginning of pumping. 412 Y.-C. Huang, H.-D. Yeh Figure 3 The normalized sensitivities of the unconfined aquifer parameters (Moench’s model). Aquifer parameter identification using on-line PEM Table 4 lists the number of observations (drawdown data) used in the data analysis and the estimated parameters for a synthetic leaky aquifer case. The identification process starts with three observations (shown at the first column) since the number of unknown parameter is three. The target values of the parameters T, S and L are 1000 m2/day, 104, and 3 · 102, respectively. The parameter estimation indicates that the parameters T and S are correctly identified even at the beginning of the pumping. The results of estimated parameter L using three, four, five, and six observation data points have the same order of magnitude as the target value, and the relative errors of estimated L are 63%, 16%, 8.7%, and 2%, respectively. The parameters are stably identified using more than seven observation data, i.e., after 1.5 min. These results indicate that the aquifer parameters are determined when the corresponding normalized sensitivities start to respond to the pumping. Moreover, the temporal curve of estimated L exhibited in Figure 4 shows fluctuation at first few steps and approaches a constant value after about 1.5 min. These results imply that the on-line PEM can successfully identify the parameters of leaky aquifer when the estimated parameter L starts to be stabilized. Table 5 displays the field time-drawdown data and the estimated parameters for a leaky aquifer using different number of observations. The time-drawdown data measured from observation wells, as reported in Cooper, (1963) and Table 4 Number of observations used in the synthetic data analysis and the estimated parameters for a leaky aquifer Number of observations Time (min) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.10 0.25 0.75 1.00 1.50 2.00 2.50 3.50 4.50 5.00 10.00 20.00 50.00 100.00 200.00 500.00 700.00 1000.00 Estimated parameters T (m2/day) S · 104 L · 102 1000.53 1000.32 1000.52 999.93 1000.02 999.96 999.98 999.99 999.99 999.95 1000.06 1000.02 1000.01 1000.02 1000.02 1000.04 1000.06 1000.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 2.52 2.74 3.06 3.00 3.03 3.01 3.00 3.01 3.01 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 Target values: T = 1000 (m2/day), S = 104, and L = 3 · 102. cited by Lohman (1972, p. 31, Table 11), are selected for the data analysis. The distance between the pumping well and the observation well is 30.48 m. The pumping rate Q The use of sensitivity analysis in on-line aquifer parameter estimation Figure 4 case. culated based on the total number of observations (1239.28 m2/day for T and 9.80 · 105 for S), the relative errors of parameters T and S are both smaller than 5% when the number of observation is larger than 7. Similarly, the estimated values of parameter L remain almost the same when the number of the observation utilized by the on-line PEM is larger than 9. In this case, the on-line estimation can be terminated after 100 min. The on-line PEM saves tremendous 90% time and 3407 m3 groundwater resources if compared with total pumping time and pumped water volume required by conventional graphical approaches. Note that small fluctuation in the estimated parameters at the late period of pumping and a longer parameter estimation time than that of the synthetic case may be attributed to aquifer heterogeneity and/or measurement errors in the observed drawdowns. The identification results with different number of observation using on-line PEM for the synthetic unconfined aquifer data set 1 are listed in Table 6. The identification process starts with four observation data points because the number of unknown parameter is four. The target values of the parameters Kr, Kz, S, and Sy are 1 · 103 m/s, 1 · 104 m/s, 1 · 104, and 1 · 101, respectively. This table only lists the results when the number of observations is less than 20 because the estimated parameters are almost the same as the target values when the number of observation is larger than 20. Figure 2 shows that the normalized sensitivities of parameters Kr, Kz, and S have immediate response to the pumping and the parameter Sy has a time lag in response to the pumping. The identification results also reflect this phenomenon. The estimated Sy ranges from 4.44 · 102 to The estimated L versus time in the leaky aquifer is 5450.98 m2/day, the thickness of the aquifer is 30.48 m, and total pumping time is 1000 min (16.67 h). It is clear that the estimated values of parameters T and S do not fluctuate drastically when the number of observation using by on-line PEM is larger than 7, i.e. after 20 min. The estimated parameters T and S are 1203.80 m2/day and 1.04 · 104, respectively. Comparing with the estimated parameters calTable 5 413 The field time-drawdown data and the estimated parameters for a leaky aquifer using different number of observations Number of observations Time (min) Drawdown (m) 1 2 3 4 5 6 7 8 9 10 11 12 0.2 0.5 1 2 5 10 20 50 100 200 500 1000 0.536 0.838 1.094 1.298 1.609 1.798 1.972 2.109 2.167 2.195 2.198 2.198 Estimated results using on-line PEM Number of observations 4 5 6 7 8 9 10 11 12 Estimated values T (m2/day) S · 104 L · 102 1060.40 1182.30 1182.70 1203.80 1211.33 1222.18 1232.32 1236.93 1239.28 1.12 1.05 1.04 1.03 1.02 1.00 0.99 0.98 0.98 15.70 1.61 6.76 5.85 5.61 5.32 5.09 4.99 4.93 414 Table 6 Y.-C. Huang, H.-D. Yeh Number of observations used in the data analysis and the estimated parameters based on the synthetic data set 1 Number of observations 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time (s) 4 5 6 7 8 9 10 11 12 13 14 15 30 44 58 74 87 Estimated parameters Kr (m/s) · 103 Kz (m/s) · 104 S · 104 Sy · 101 0.997 1.000 1.000 0.997 1.000 0.999 1.000 0.998 1.000 1.000 1.000 0.998 1.000 0.998 1.000 1.000 1.000 1.006 0.999 0.999 1.010 0.998 1.000 1.000 1.000 0.995 0.997 0.998 0.998 1.000 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.612 0.616 1.190 0.444 1.570 0.933 0.972 0.712 2.010 1.140 1.220 0.816 1.040 0.987 1.010 0.993 1.000 Target values: Kr = 1 · 103 (m/s), Kz = 1 · 104 (m/s), S = 1 · 104, and Sy = 1 · 101. 2.01 · 101 and the largest relative errors are 101% when using 12 observation data. The identification results of Sy did not approach the target value until the number of observation is over 20, i.e., about 80 s. Therefore, the on-line PEM may not obtain accurate results of Sy if the time-drawdown data is too short to cover the response period of Sy. Similar to Figure 4, the curve of estimated Sy versus time displayed in Figure 5 shows dramatic fluctuation in the early period and converges to a constant value after about 80 s. Figure 2 demonstrates that the on-line PEM can successfully identify the aquifer parameters when Sy just starts to affect the drawdown. Therefore, the on-line estimation based on Figure 5 The estimated Sy versus time using the synthetic data set 1. Neuman’s model can be terminated once the identified parameters become stable. Similar to Table 6, the identification results for the synthetic data set 2 are listed in Table 7. The target values of the parameters Kr, Kz, S, Sy, and rw are 1 · 103 m/s, 1 · 104 m/s, 1 · 104, 1 · 101, and 1 m, respectively. The estimated parameters are all the same as the target values when the number of observation is larger than 30. The parameters Kr, Kz, S, and rw are accurately determined at first few seconds. The estimated Sy ranges from 1.00 · 102 to 2.91 · 101 and did not approach the target value until the pumping time is more than 125 s. The curve of estimated Sy versus time displayed in Figure 6 also shows dramatic fluctuation at the early time and converges to a fix value after about 125 s. Hence, the on-line estimation can be terminated even based on Moench’s model. Table 8 shows the estimated parameters when using different number of observations obtained from the field pumping test at an unconfined aquifer. The site of Cape Cod, Massachusetts (Moench et al., 2000) is selected for the study. The aquifer was composed of unconsolidated glacial outwash sediments that were deposited during the recession, 14,000–15,000 years before present, of the late Wisconsinan continental ice sheet. The depth of the pumping well was 24.4 m below the land surface. The top and bottom of the screen were located 4.0 and 18.3 m, respectively, below the initial water table, which was approximately 5.8 m below land surface. The aquifer saturated thickness was about 48.8 m. The well F507-080 was pumped at an average rate 1.21 m3/min for 72 hours. The data set of the observation well F505-032 is selected in this case. The distance between pumping well and observation well is 7.28 m. From Table 8, the estimated Kr ranges from 2.20 · 104 m/s to 1.97 · 103 m/s, the estimated Kz ranges from 1.0 · 106 m/s to 2.25 · 104 m/s, the estimated S ranges from 3.45 · 103 to 7.29 · 103, and the estimated Sy ranges from 0.016 to 0.3. This table demonstrates that The use of sensitivity analysis in on-line aquifer parameter estimation Table 7 Number of observations used in the data analysis and the estimated parameters based on the synthetic data set 2 Number of observations 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 415 Time (s) 7 8 9 10 12 14 16 18 21 24 27 31 36 41 47 54 63 72 82 95 125 189 249 Estimated parameters Kr (m/s) · 103 Kz (m/s) · 104 S · 104 Sy · 101 rw (m) 1.66 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.63 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.67 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.10 0.53 2.91 0.20 0.67 1.17 0.48 0.56 0.69 0.65 0.77 0.85 1.23 0.59 0.96 1.03 0.90 1.00 0.99 1.03 1.00 1.00 1.00 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Target values: Kr = 1 · 103 (m/s), Kz = 1 · 104 (m/s), S = 1 · 104, Sy = 1 · 101, and rw = 1 m. ate significantly at the early period of the pumping. Note that the estimated parameter Sy keeps the highest value (0.3) at early pumping period then dramatically decreases to a small value (0.016) after 20 min (18 observations). This result implies that Sy does not affect the estimation for other parameters before that time, i.e., the variation of parameter Sy does not significantly change the estimation result. Figure 7 displays the estimated Sy versus pumping time (different number of observations). In addition, the value of Sy versus logarithmic time is also shown in the upper panel of the figure. The estimated Sy keeps almost constant at first 20 min, then goes down rapidly and reaches a minimal at 100 min. After that the estimated Sy gradually increases and becomes flat after 1000 minutes (16.67 h), implying that the on-line estimation can be terminated at that time. In this case, the on-line PEM can save 77% pumping time if the test is terminated and 4041.4 m3 groundwater resources if compared with total pumping time and pumped water volume required by conventional graphical approaches. Figure 6 The estimated Sy versus time using the synthetic data set 2. The tests of other impacts to the influence period of the parameter Sy the ranges of estimated Kr and S are small as compared with those of the Kz and Sy. Such results may attribute to the fact that the parameters Kr and S have significant influence on the drawdown as the pumping starts and thus can be estimated using only few observations. In contrast, the influence periods of parameters Kz and Sy have some time lags after the start of pumping and the estimated results fluctu- The parameter Sy has the longest time lag in response to the drawdown than other parameters as indicated in Figures 1 and 2. The on-line PEM can correctly identify the aquifer parameters only when the parameters start to influence the drawdown. In the unconfined aquifer case, the Sy was assigned to 0.1 where the reasonable value is 0.01–0.3 (Batu, 1998). The normalized sensitivities reflect the sensi- 416 Table 8 Y.-C. Huang, H.-D. Yeh The estimated parameters for an unconfined aquifer (Cape Cod site) using different number of observations Number of observations 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Time (min) 0.15 0.22 0.32 0.47 0.68 1.00 1.47 2.15 3.17 4.75 6.75 10.10 14.90 21.90 31.90 46.90 67.90 99.90 151.00 221.00 325.00 492.00 675.00 1050.00 1470.00 2190.00 3100.00 4330.00 Figure 7 Estimated parameters Kr · 103 (m/s) Kz · 105 (m/s) S · 103 Sy · 101 0.65 0.73 0.91 0.88 0.96 0.51 0.32 0.22 0.24 0.30 0.26 0.25 0.44 1.01 1.39 1.74 1.92 1.96 1.97 1.92 1.82 1.70 1.60 1.54 1.50 1.47 1.46 1.45 0.10 1.05 0.98 1.18 1.19 1.89 2.15 2.23 2.41 2.78 2.60 2.20 2.34 1.51 0.74 0.48 0.39 0.37 0.36 0.38 0.42 0.46 0.49 0.51 0.52 0.53 0.54 0.54 7.18 7.29 7.45 7.17 7.16 5.61 4.28 3.32 3.52 4.31 3.83 3.64 5.64 8.14 7.00 6.21 5.65 5.52 5.49 5.63 5.98 6.39 6.71 6.94 7.10 7.19 7.26 7.29 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 2.91 3.00 3.00 2.95 3.00 1.49 0.56 0.27 0.18 0.16 0.16 0.18 0.24 0.33 0.43 0.52 0.59 0.64 0.68 0.69 The estimated Sy versus time in the field unconfined aquifer. The use of sensitivity analysis in on-line aquifer parameter estimation tivity of the drawdown in response to the relative change of each parameter at different time. Thus, it is interesting to examine the temporal distribution of normalized sensitivity for different value of Sy. Moreover, the distance between pumping well and observation well, R, is another problem deserved attention because the drawdown in response to the pumping becomes smaller when the distance from the pumping well goes farther. For investigating the effect of various value of Sy or R on the on-line parameter estimation, two tests are performed. The first test assigns three different values of Sy including two extreme values, i.e., 0.01 and 0.3, while the other parameters are kept the same as those given in previous unconfined aquifer case. The second test examines the effect of distance on the normalized sensitivity when the observation well is located at 10, 30, or 50 m from the pumping well. The normalized sensitivity of Sy versus time for the first test is demonstrated in Figure 8. The influence period starts slightly later when the Sy value gets larger. The Sy starts to influence the drawdown at 5 and 100 s when the value of Sy is 0.01 and 0.3, respectively, indicating that the time lag of the Sy may not be larger than 2 min in these two extreme cases. Figure 8 indicates that the largest normalized sensitivities are about the same in those cases because of the normalization of Sy. The results of the second test shown in Figure 9 indicate that a longer distance from the well has a slower response time. The shortest response time is about 10 seconds and the latest one is about 100 s. Comparing with the total pumping time of 176,360 s (2.04 days), the differences of the estimated parameters in these three cases may be negligible. In addition, the sensitivity analysis may be performed along with the on-line parameter estimation and provide a double check in terminating the pumping. Concluding remarks The sensitivity analysis is used to investigate the influence period of aquifer parameters in both leaky and unconfined 417 Figure 9 The normalized sensitivity of Sy for Sy = 0.1 and R = 10, 30, or 50 m. aquifers. The influences of parameters L and Sy on the drawdown are shown to have time lag in response to pumping in the leaky and unconfined aquifers, respectively. An on-line parameter estimation model is applied to estimate the parameters based on the data obtained from synthetic and field pumping tests for both leaky and unconfined aquifers. The results indicate that the on-line estimation can be terminated when the parameters are stabilized and their corresponding normalized sensitivities start to react the pumping. In the synthetic cases, the termination time of the on-line estimation is consistent with the influence period of the parameter which has longest time lag from the beginning of the pumping. This phenomenon indicates that the on-line estimation can be terminated if all identified parameters tend to be stabilized, i.e., the drawdown already reacts to the effect of aquifer parameters. In the field cases, the results indicate that the on-line parameter estimation model can save 90% pumping time in the leaky aquifer and 77% pumping time in the unconfined aquifer. Note that the small fluctuation in the estimated parameters at the late period of pumping and a longer on-line estimation time than that of the synthetic case occur. These results may be mainly caused by aquifer heterogeneity and/or measurement errors in the observed drawdown data. Finally, different values of the specific yield and distance between pumping well and observation well do not significantly affect the influence time of specific yield during the pumping. These results may provide a useful reference for on-line aquifer parameter estimation. Acknowledgements Figure 8 The normalized sensitivity of Sy for Sy = 0.01, 0.1, or 0.3 and R = 10 m. This study was partly supported by the Taiwan National Science Council under the grant NSC94-2211-E-009-015. The authors thank two anonymous reviewers for constructive comments and suggested revisions. 418 References Batu, V., 1998. Aquifer Hydraulics. John Wiley & Sons, New York. Boulton, N.S., 1954. Unsteady radial flow to a pumped well allowing for delayed yield from storage. Int. Ass. Sci. Hydrol., Publ 37, 472–477. Boulton, N.S., 1963. Analysis of data from non-equilibrium pumping tests allowing for delayed yield from storage. Proc. Inst. Civil Eng. 26, 469–482. Chander, S., Kapoor, P.N., Goyal, S.K., 1981. Analysis of pumping test data using Marquardt algorithm. Ground Water 19 (3), 275– 278. Charbeneau, R.J., 2000. Groundwater Hydraulics and Pollutant Transport. Prentice Hall, Upper Saddle River, NJ. Cooper Jr., H.H., Jacob, C.E., 1946. A generalized graphical method for evaluating formation constants and summarizing well field history. Transactions, American Geophysical Union 27 (IV), 526–534. Cooper, H.H., Jr., 1963. Type curves for nonsteady redial flow in an infinite leaky artesian aquifer, in Bentall, Ray, complier, Shortcuts and special problems in aquifer tests, U.S. Geol. Survey Water-supply Paper 1545–C. Cukier, R.I., Fortuin, C.M., Shuler, K.E., Petschek, A.G., Shaibly, J.H., 1973. Study of the Sensitivity of Coupled Reaction System to Uncertainties in Rate Coefficients: I. theory. J. Chem. Phys. 59 (8), 3873–3878. Cukier, R.I., Fortuin, C.M., Shuler, K.E., 1975. Study of the Sensitivity of Coupled Reaction System to Uncertainties in Rate Coefficients: III Analysis of the approximations. J. Chem. Phys. 63 (3), 1140–1149. Cukier, R.I., Levine, H.B., Shuler, K.E., 1978. Nonlinear sensitivity analysis of multiparameter mode systems. J. Comp. Phys. 26 (1), 1–42. Duffield, G.M., 2002. AQTESOLV for Windows. HydroSOLVE, Inc., Reston, VA. Gooseff, M.N., Bencala, K.E., Scott, D.T., Runkel, R.L., Mcknight, D.M., 2005. Sensitivity analysis of conservative and reactive stream transient storage models applied to field data from multiple-reach experiments. Adv. Water Resour. 28, 479–492. Hantush, M.S., Jacob, C.E., 1955. Non-steady radial flow in an infinite leaky aquifer. Trans. Amer. Geophys. Union 36, 95–100. Hantush, M.S., 1964. Hydraulics of Wells. Adv. Hydrosci. 1, 281– 442. Jiao, J.J., Rushton, K.R., 1995. Sensitivity of drawdown to parameters and its influence on parameter estimation for pumping tests in large-diameter wells. Ground Water 33 (5), 794–800. Kabala, Z.J., Milly, P.C.D., 1990. Sensitivity analysis of flow in unsaturated heterogeneous porous-media - theory, numericalmodel, and its verification. Water Resour. Res 26 (4), 593–610. Kabala, Z.J., 2001. Sensitivity analysis of a pumping test on a well with wellbore storage and skin. Adv. Water Res. 24, 483–504. Kabala, Z.J., El-Sayegh, H.K., Gavin, H.P., 2002. Sensitivity analysis of a no-crossflow model for the transient flowmeter test. Stochast. Environ. Res. Risk Assess. 16 (6), 399–424. Y.-C. Huang, H.-D. Yeh Lebbe, L.C., 1999. Hydraulic parameter identification. Springer, New York. Leng, C.H., Yeh, H.D., 2003. Aquifer parameter identification using the extended Kalman filter. Water Resour. Res. 39 (3), 1062. doi:10.1029/2001WR000840. Lohman, S.W., 1972. Ground-water hydraulics. U.S. Geological Survey professional paper 708. McCuen, R.H., 1985. Statistical Methods for Engineers. Prentice Hall, Englewood Cliffs, New Jersey. McElwee, C.D., 1980. Theis parameter evaluation from pumping tests by sensitivity analysis. Ground Water 18 (1), 56–60. Moench, A.F., 1997. Flow to a well of finite diameter in a homogenous anisotropic water table aquifer. Water Resour. Res. 33, 1397–1407. Moench, A.F., Garabedian, S.P., and LeBlanc, D.L., 2000. Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, Cape Cod, Massachusetts. USGS Open-File Report: 00-485. Neuman, S.P., 1972. Theory of flow in unconfined aquifers considering delayed response of the water table. Water Resour. Res. 8, 1031–1044. Neuman, S.P., 1974. Effects of partial penetration on flow in unconfined aquifers considering delayed aquifer response. Water Resour. Res. 10, 303–312. Neuman, S.P., 1975. Analysis of pumping test data from anisotropic unconfined aquifers considering delayed gravity response. Water Resour. Res. 11, 329–342. Paschetto, J., McElwee, C.D., 1982. Hand calculator program for evaluate Theis parameters from a pumping-test. Ground Water 20 (5), 551–555. Prickett, T.A., 1965. Type-curve solution to aquifer tests under water-table conditions. Ground Water 3, 5–14. Saleem, Z.A., 1970. A computer method for pumping-test analysis. Ground Water 8 (5), 21–24. Schibly, J.H., Shuler, K.E., 1973. Study of the sensitivity of coupled reaction system to uncertainties in rate coefficients: ii. applications. J. Chem. Phys. 59 (8), 3879–3888. Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Eos Trans. AGU 16, 519–524. Vachaud, G., Chen, T., 2002. Sensitivity of a large-scale hydrologic model to quality of input data obtained at different scales. J. Hydrol. 264, 101–112. Yeh, H.D., 1987. Theis’ solution by nonlinear least-squares and finite-difference Newton’s method. Ground Water 25, 710–715. Yeh, H.D., Han, H.Y., 1989. Numerical identification of parameters in leaky aquifers. Ground Water 27 (5), 655–663. Yeh, H.D., Huang, Y.C., 2005. Parameter estimation for leaky aquifers using the extended Kalman filter, and considering model and data measurement uncertainties. J. Hydrol. 302 (1– 4), 28–45. Yeh, H.D., Lin, Y.C., Huang, Y.C., in press. Parameter identification for leaky aquifers using global optimization methods. Hydrol. Process.

© Copyright 2017 ExploreDoc