1 Introduction

A slug test is performed quickly at a relatively low cost in determining the hydraulic properties of aquifers. The tests involve measuring the recovery of well-water-level (WWL) data in a well after instantaneous injection/withdrawal of a small quantity of water into/from the well. Several mathematical models have been devoted to the analysis of a slug test, e.g., Hvorslev (1951), Cooper et al. (1967), Bouwer and Rice (1976), Springer and Gelhar (1991), Hyder et al. (1994), and Butler (1998).

Recently, the wellbore-skin effect has been considered in the analysis of a slug test. A positive skin is defined as a zone adjacent to the wellbore with a hydraulic conductivity smaller than that of the undisturbed formation (Yang and Yeh 2002). In contrast, a negative skin is referred to as a skin zone which has higher hydraulic conductivity than the undisturbed formation. Ramey et al. (1975) proposed an analytical solution for a slug test where the thickness of skin is infinitesimal. The assumption of infinitesimal skin thickness may introduce a large uncertainty because of the similarity in the shape of the type curves. Faust and Mercer (1984) investigated the effect of a finite-thickness skin on the response of slug test by a simple analytical solution and a numerical model. They pointed out that the effect of positive skin leads to an unreliable estimate in aquifer parameters. Following the concept of Faust and Mercer (1984), Moench and Hsieh (1985) presented a Laplace-domain solution with type curves and examined the influences of a finite-thickness skin on the open-well and pressurized slug tests. Yeh and Yang (2006) further extended their Laplace-domain solution to time domain based on the method of Bromwich integral.

Yeh and Chen (2007) combined Moench and Hsieh’s solution (1985) with simulated annealing (SA) (Lee et al. 2010; Rani and Moreira 2010) to determine three skin parameters (hydraulic conductivity k 1, specific storage S s1, and skin thickness d sk ) and two aquifer parameters (hydraulic conductivity k 2 and specific storage S s2) simultaneously from a slug test performed in a skin-affected confined aquifer system. The skin thickness d sk is equal to r s -r w , where r s and r w represent the outer radius of the wellbore-skin zone and the effective well radius, respectively. The WWL data were generated by Moench and Hsieh’s solution (1985) with a set of standard normally distributed noise added for both positive skin and negative skin scenarios. Some results in the positive skin scenarios showed that the parameters k 2 and d sk were inaccurately determined. They presented a figure of sensitivity analysis to demonstrate that the inaccuracy in these two parameter estimates was caused by insensitivity of aquifer parameters in response to the test and high correlation between the parameters k 1 and d sk . Moreover, the results also indicated that the problem of thin skin thickness causes the inaccuracy in parameter estimation.

Figure 1 shows the normalized sensitivities (Huang and Yeh 2007) of WWL change with respect to the parameter d sk over a 1,000-second interval. There are four curves representing the normalized sensitivities for different values of d sk with given values of the other four parameters (k 1, S s1, k 2, and S s2). The figure indicates that a smaller d sk has a higher normalized sensitivity and quicker response than those of larger d sk . Huang and Yeh (2007) pointed out that the parameters can be accurately estimated once the normalized sensitivity of parameter starts to respond to the change in drawdown. In other words, the aquifer parameters in the case of thin skin thickness should be estimated at least as well as those of thick skin-thickness cases. However, it seems that this finding is not applicable to the case of Yeh and Chen (2007) in which they used SA in searching for the optimal parameter values.

Fig. 1
figure 1

The normalized sensitivities to the parameter d sk ranged from 0.1085 to 4.9085 m over a time period of 1,000 s

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Yeh and Chen (2007, their Table 2) presented the synthetic WWL data generated by Moench and Hsieh’s solution (1985). The data, rounded-off to the third decimal, meet the measurement accuracy from engineering viewpoint. However, such inaccuracy in parameter estimation would lead to erroneous results because the algorithm of SA coupled with Moench and Hsieh’s solution (1985) is very sensitive to the measurement error in WWL data. In the case of Yeh and Chen (2007, Case 15a in Table 6b, the WWL data without adding noise), the estimated parameters k 1, k 2, S s1, S s2, and d sk based on SA are 2.31 × 10−5 m/s, 9.97 × 10−4 m/s, 1.71 × 10−5 1/m, 1.83 × 10−5 m/s, and 0.493 m, respectively. However, the target parameters are 1.00 × 10−5 m/s, 1.00 × 10−4 m/s, 1.00 × 10−4 1/m, 1.00 × 10−4 m/s, and 0.1085 m, respectively. The standard error of the estimate (SEE) for the predicted WWL is 3.16 × 10−4 m based on the estimated parameters and 3.31 × 10−4 m based on the target parameters.

The SEE is defined as \( \;{\left( {\sum\nolimits_{{j = 1}}^n {{{{{e_j}^2}} \left/ {v} \right.}} } \right)^{{ \frac{1}{2} }}} \), where e j represents the difference between the observed and the predicted WWL and v, the degree of freedom, is equal to the number of observed data points n minus the number of unknowns (Yeh 1987). Such a slightly larger SEE value form the target parameters is due to the round-off errors in WWL data. The relative errors (RE) of the estimated parameters k 1, k 2, S s1, S s2, and d sk are 131%, −0.3%, −82.9%, −81.7%, and 354%, respectively. Note that the RE is defined as the difference between the estimate and target values divided by the target value. This indicates that the SA gives a set of parameters which gives very good fit to the measured WWL data; yet, the estimated parameters are inaccurate in some positive-skin cases.

The problem of inaccuracy in parameter estimation could be improved to some extent by using a longer series of WWL data or analyzing WWL data of the test and observation wells simultaneously (Yeh and Chen 2007). However, the cost and labor spent in the slug test will inevitably increase if more measurements are needed; especially, when extra measurements are taken from observation wells. Moreover, the parameter estimation using SA took about 3.6 h when analyzing a set of 20 WWL data and about 7.6 h for analyzing a set of 47 WWL data when using a personal computer with 3.6 G Pentium IV CPU and 1 GB RAM.

Alternately, the method of extended Kalman filter (EKF) can give accurate parameter estimate because its algorithm accounts for the effects of system and measurement uncertainties in the system measurement model (Grewal and Andrews 1993), and the sequential data assimilation process is more efficient computationally. The Kalman filter was developed by R. E. Kalman in the late 1950s, and its most applications have been in control systems, tracking and navigation of all sorts of vehicles, as well as predictive design of estimation and control systems. After that, the EKF was proposed for dealing with nonlinear problems (Chou 2011). McLaughlin and Townley (1996) mentioned that the EKF was less likely to converge to an acceptable solution in the parameter estimation problems if the number of unknown parameters was large. They suggested that the EKF may only be able to deal with the case of relative small number of unknown parameters and large amount of the measurements. Drécourt (2003) concluded that the EKF may also be able to determine the parameter if the state and measurement equations were not highly nonlinear. Recently, the EKF was applied to the aquifer parameter and water table related estimations. Leng and Yeh (2003) used EKF and cubic spline to determine the aquifer parameters in both confined and unconfined aquifer systems. Yeh and Huang (2005) employed EKF to determine the aquifer parameters in leaky aquifer systems with and without considering the storage effect in the aquitard. From the analysis of field data, they demonstrated that the EKF can be applied to determine the aquifer parameters successfully. Goegebeur and Pauwels (2007) applied the EKF to a conceptual rainfall-runoff model with 10 parameters and demonstrated its robustness for parameter calibration, especially for problems with high observation errors, infrequent observations, and/or strongly erroneous initial parameters. Shamir et al. (2010) utilized ensemble EKF to link upstream watersheds and channels to main river channels and tributaries in a large regulated basin for flood forecasting. Nenna et al. (2011) applied the EKF approach to invert time-lapse electrical resistivity imaging data collected to observe changes in electrical conductivity under a recharge pond, which is a part of aquifer storage.

In the real-world problems, the skin zone is invisible, immeasurable, and usually very small; thus, the thickness of the skin zone is difficult to accurately determine, especially when the observed data contain measurement errors. Such a challenging task motivates the authors’ curiosity to use the EKF. The objective of this study is to investigate and resolve the problem of inaccuracy in the estimation of parameters k 2 and d sk in the positive skin scenarios. The procedure of the analyses to achieve the objective is given below:

  1. 1.

    Using the correlation analysis to quantify the strength of the relationship between the normalized sensitivity of WWL with respect to each of the aquifer parameters over a certain period of time.

  2. 2.

    Utilizing the sensitivity analysis to explore the problem of inaccuracy in the estimation of the parameter k 2 .

  3. 3.

    Developing an approach by coupling the EKF with Moench and Hsieh’s solution (1985) and analyzing those six positive-skin scenarios in Yeh and Chen (2007) where the parameter k 2 or d sk was not accurately determined.

2 Methodology

This section includes three parts: the first part briefly describes the theory of the EKF, the second part introduce the solution developed by Moench and Hsieh (1985). The third part presents the algorithm of combining EKF with Moench and Hsieh’s model (1985) to determinate the hydraulic parameters of the skin and formation zones.

2.1 Discrete Extended Kalman Filter

A nonlinear dynamic system may be expressed as (Grewal and Andrews 1993)

$$ {x_k} = f({x_{{k - 1}}},k - 1) + {w_k} $$
(1)

where x k is a state vector of the system at time step k, f(x k−1, k − 1) is a nonlinear function of the system, and w k is a state noise assumed to be normally distributed with a zero-mean white (uncorrelated) sequence with known covariance Q k .

The nonlinear implementation equation for the state vector is written as

$$ {\widehat{x}_k}( - ) = f({\widehat{x}_{{k - 1}}}( + ),k - 1) $$
(2)

where \( {\widehat{x}_k}( - ) \) denotes the a priori estimate at k step and \( {\widehat{x}_k}( + ) \) represents the a posteriori estimate at k-1 step.

Similarly, a measurement model of the system can be written as (Grewal and Andrews 1993)

$$ {z_k} = m\left( {{x_k},k} \right) + {\upsilon_k} $$
(3)

where m(x k , k) is a function for the measurement system and z k is a measurement vector at time step k. The measurement noise υ k is assumed to be a white noise with known constant covariance R k throughout the filtering process.

The nonlinear implementation equation for the measurement is

$$ {\widehat{z}_k} = m({\widehat{x}_k}( - ),k) $$
(4)

where \( {\widehat{z}_k} \) is a predicted measurement vector.

The recursive process of the EKF can be expressed as

$$ {P_k}( - ) = E\left[ {{e_k}( - )e_k^T( - )} \right] = E\left[ {\left( {{x_k} - {{\widehat{x}}_k}( - )} \right){{\left( {{x_k} - {{\widehat{x}}_k}( - )} \right)}^T}} \right] $$
(5)
$$ {P_k}( - ) = {\Phi_{{k - 1}}}{P_{{k - 1}}}( + )\Phi_{{k - 1}}^T + {Q_{{k - 1}}} $$
(6)
$$ {\overline K_k} = {P_k}( - )M_k^T{[{M_k}{P_k}( - )M_k^T + {R_k}]^{{ - 1}}} $$
(7)
$$ {P_k}( + ) = \left\{ {I - {{\overline K}_k}{M_k}} \right\}{P_k}( - ) $$
(8)
$$ {\widehat{x}_k}( + ) = {\widehat{x}_k}( - ) + {\overline K_k}({z_k} - {\widehat{z}_k}) $$
(9)

where P k ( − ) is a priori error covariance matrix, e k ( − ) is defined as \( {x_k} - {\widehat{x}_k}( - ) \), Φ k−1 is state transition matrix, \( {\overline K_k} \) is defined as Kalman gain, P k ( + ) is a posteriori covariance, M k is measurement matrix, and \( {\widehat{x}_k}( + ) \) is the updated estimate at step k.

The state transition matrix Φ k−1 and measurement matrix M k can be respectively expressed as

$$ {\Phi_{{k - 1}}} \approx {\left. {\frac{{\partial f(x,k - 1)}}{{\partial x}}} \right|_{{x = {{\widehat{x}}_{{k - 1}}}( - )}}} $$
(10)

and

$$ {M_k} \approx {\left. {\frac{{\partial m(x,k)}}{{\partial x}}} \right|_{{x = {{\widehat{x}}_k}( - )}}} $$
(11)

The initial estimates at time step k at some point are required and can be assigned based on the knowledge about the process. With initial estimates and Eqs. (2), (4)–(9), the recursive process of EKF is then established.

The EKF has two advantages over the SA: (1) the EKF can deal with system and measurement uncertainties in the algorithm and (2) the EKF is much more computationally efficient than the SA. The first advantage is that the EKF accounts for the uncertainties in both the system and measurement equations (i.e., w k in Eq. (1) and υ k in Eq. (3)). Hence, the impact of uncertainties can be reduced during the determination process if the w k and υ k are assigned properly. Contrarily, the SA determines the parameters when searching the optimal results for the objective function in terms of ordinary least squares. The second advantage is that the SA is a “batch” type algorithm that combines all available measurements (WWL in this study) in a single large measurement vector z (z = [z 1, z 2,…, z N ]T, where N is the total number of available measurements). The objective function is then evaluated by calculating entire measurement vector at each step. These evaluations involve huge computing burden, especially when the WWL data set is very long or the model is complicated. Oppositely, the EKF is much efficient since it updates the current estimate using only the newest measurement point at each parameter identification step.

2.2 Moench and Hsieh’s Solution (With Considering Skin Effect)

Moench and Hsieh (1985) developed a Laplace domain solution for the response to a drill-stem test in the presence of the skin with finite thickness. Their solution is adequate for the analyses of the open-well slug test in confined aquifers. The schematic diagram of the test is shown in Fig. 2 in which the hydraulic parameters of the skin zone (k 1, S s1, and d sk ) are considered. The assumptions leading to the solution are: (1) the aquifer is homogeneous, isotropic, infinite-extent, and of a constant thickness; (2) the well is fully penetrating and with a finite radius; (3) the initial head is constant and uniform throughout the whole aquifer; (4) the vertical flow gradients are negligible; (5) the skin zone is assumed homogeneous and isotropic. The dimensionless form of WWL solution in the Laplace domain can be written as

$$ \overline h = \frac{{\alpha \gamma \left[ {{\Delta_1}{K_0}\left( {q\beta } \right) - {\Delta_2}{I_0}\left( {q\beta } \right)} \right]}}{{{c_1}{\Delta_1} - {c_2}{\Delta_2}}} $$
(12)

with

$$ {\Delta_1} = \alpha \,{I_0}\left( {q\beta {r_{{ds}}}} \right)\,{K_1}\left( {q{r_{{ds}}}} \right) + \beta \,{I_1}\left( {q\beta {r_{{ds}}}} \right)\,{K_0}\left( {q{r_{{ds}}}} \right) $$
(13)
$$ {\Delta_2} = \alpha \,{K_0}\left( {q\beta {r_{{ds}}}} \right){K_1}\left( {q{r_{{ds}}}} \right) - \beta \,{K_1}\left( {q\beta {r_{{ds}}}} \right){K_0}\left( {q{r_{{ds}}}} \right) $$
(14)
$$ {c_1} = \alpha \gamma p{K_0}\left( {q\beta } \right) + \beta q{K_1}\left( {q\beta } \right) $$
(15)
$$ {c_2} = \alpha \gamma p{I_0}\left( {q\beta } \right) - \beta q{I_1}\left( {q\beta } \right) $$
(16)
$$ \alpha = {k_2}/{k_1} $$
(17)
$$ \beta = {\left( {\alpha {S_{{s1}}}/{S_{{s2}}}} \right)^{{1/2}}} $$
(18)
$$ \gamma = \frac{{\pi \,{r_c}^2}}{{2\pi r_w^2{S_{{s2}}}b}} $$
(19)
$$ {r_{{ds}}} = \frac{{{r_s}}}{{{r_w}}} $$
(20)

and

$$ q = {p^{{1/2}}} $$
(21)

where b is aquifer thickness and p is Laplace variable. Moreover, I 0(.) and I 1(.) express the modified Bessel functions of the first kind of order zero and one, respectively; and K 0(.) and K 1(.) are the modified Bessel functions of the second kind of order zero and one, respectively. The inverse Laplace transform of Eq. (12) is calculated by the routine INLAP of IMSL (2003) with the accuracy to five decimal places. This routine, developed based on an algorithm originally proposed by Crump (1976) and later modified by de Hoog et al. (1982), has been successfully applied in some groundwater problems (see, e.g., Chen et al. 1996). The time domain solution for the head H(t) is

$$ H(t){ } = {L^{{ - {1}}}}\left( {\overline h } \right) $$
(22)

where L −1 indicates the operator of inverse Laplace transform.

Fig. 2
figure 2

Schematic diagram of the well and aquifer configurations

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2.3 Application of Proposed Approach in Parameter Identification

The parameters in Eq. (2) at each time step can be expressed as following state vector

$$ {\widehat{x}_k}( - ) = {\left[ {\matrix{ {{k_1}} &{{S_{{s1}}}} &{{k_2}} &{{S_{{s2}}}} &{{d_{{sk}}}} \\ }<!end array> } \right]^T} $$
(23)

in the proposed approach. For applying to the nonlinear system, the EKF uses the first-order Taylor approximations of state transition and observation equations about the estimated state trajectory. The transition matrix is

$$ {\Phi_{{k - 1}}} \approx \left[ {\matrix{ {\frac{{\partial {k_1}}}{{\partial {k_1}}}} \hfill &{\frac{{\partial {k_1}}}{{\partial {S_{{s1}}}}}} \hfill &{\frac{{\partial {k_1}}}{{\partial {k_2}}}} \hfill &{\frac{{\partial {k_1}}}{{\partial {S_{{s2}}}}}} \hfill &{\frac{{\partial {k_1}}}{{\partial {d_{{sk}}}}}} \hfill \\ {\frac{{\partial {S_{{s1}}}}}{{\partial {k_1}}}} \hfill &{\frac{{\partial {S_{{s1}}}}}{{\partial {S_{{s1}}}}}} \hfill &{\frac{{\partial {S_{{s1}}}}}{{\partial {k_2}}}} \hfill &{\frac{{\partial {S_{{s1}}}}}{{\partial {S_{{s2}}}}}} \hfill &{\frac{{\partial {S_{{s1}}}}}{{\partial {d_{{sk}}}}}} \hfill \\ {\frac{{\partial {k_2}}}{{\partial {k_1}}}} \hfill &{\frac{{\partial {k_2}}}{{\partial {S_{{s1}}}}}} \hfill &{\frac{{\partial {k_2}}}{{\partial {k_2}}}} \hfill &{\frac{{\partial {k_2}}}{{\partial {S_{{s2}}}}}} \hfill &{\frac{{\partial {k_2}}}{{\partial {d_{{sk}}}}}} \hfill \\ {\frac{{\partial {S_{{s2}}}}}{{\partial {k_1}}}} \hfill &{\frac{{\partial {S_{{s2}}}}}{{\partial {S_{{s1}}}}}} \hfill &{\frac{{\partial {S_{{s2}}}}}{{\partial {k_2}}}} \hfill &{\frac{{\partial {S_{{s2}}}}}{{\partial {S_{{s2}}}}}} \hfill &{\frac{{\partial {S_{{s2}}}}}{{\partial {d_{{sk}}}}}} \hfill \\ {\frac{{\partial {d_{{sk}}}}}{{\partial {k_1}}}} \hfill &{\frac{{\partial {d_{{sk}}}}}{{\partial {S_{{s1}}}}}} \hfill &{\frac{{\partial {d_{{sk}}}}}{{\partial {k_2}}}} \hfill &{\frac{{\partial {d_{{sk}}}}}{{\partial {S_{{s2}}}}}} \hfill &{\frac{{\partial {d_{{sk}}}}}{{\partial {d_{{sk}}}}}} \hfill \\ }<!end array> } \right] $$
(24)

This matrix is time invariant, i.e., \( {\widehat{x}_k} = {\widehat{x}_{{k - 1}}} \). Based on Eq. (6), P k ( − ) can be estimated from the known transition matrix Φ k−1. To update the hydraulic parameters in Eq. (7), the Kalman gain \( {\overline K_k} \), estimated from the known M k and the prior covariance matrixP k ( − ), is first required. The measurement matrix M k is the partial derivatives of the estimated drawdown \( {\widehat{z}_k} \). The criterion chosen to terminate the recursive process is expressed as

$$ TO{L_i} = \left| {{P_i}\left( {k + 1} \right) - {P_i}(k)} \right| $$
(33)

where TOL i is the tolerance for parameter i and P i (k) is the value of parameter i at time step k. The process will be terminated when all TOL i meet the values assigned by the users. Figure 3 is a flow chart for the proposed approach in determining parameters when coupled the EKF with Moench and Hsieh’s solution (1985). After initializing the estimates of parameters, error covariance matrix P, and the measurement error covariance R, the recursive process starts to compute Eqs. (5)–(9). Note that a set of WWL data is repeatedly used from the beginning once the stepwise determination process of EKF has gone through all data. The estimates of parameters updated based on last measurement point in the data set will be treated as initial estimate for the next run of data assimilation.

Fig. 3
figure 3

Flowchart for the EKF for the proposed approach

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3 Results and Discussion

3.1 Correlation and Sensitivity Analyses in Positive Skin Scenarios

The normalized sensitivity of WWL changed with each of five parameters over 1,000 s in a positive skin scenario was demonstrated in Yeh and Chen (2007, Fig. 2b). In this case, the slug test is performed in a homogeneous and isotropic confined aquifer system. The test well fully penetrates the aquifer and the radius of effective well r w and well casing r c are 0.0915 m and 0.0508 m, respectively. The sudden drop of WWL is assumed 1 m while the aquifer thickness is 10 m. The WWL data are produced based on Moench and Hsieh’s solution (1985). The estimated values of parameters k 1, k 2, S s1, S s2, and d sk are 10−5 m/s, 10−4 m/s, 10−4 m−1, 10−4 m−1, and 0.3085 m, respectively. Their figure showed that the normalized sensitivities of parameters k 1, and d sk are symmetrical in shape on the horizontal axis but have different magnitudes, implying that they are highly correlated.

The correlation analysis is thus used to quantify the strength of the relationship between normalized sensitivities of WWL with respect to each aquifer parameter. The upper and lower parts of Table 1 respectively show the correlation matrices (correlation coefficients for each two variables) of five parameters’ normalized sensitivities over 15 and 1,000 s for the same scenario as given in Yeh and Chen (2007, Scenario 2). The bold numbers represent strong correlation between the normalized sensitivities of the parameters. The large value (positive or negative) of the correlation coefficient in these cases signifies that the normalized sensitivities to two parameters vary synchronously over 15 and 1,000 s. In other words, these two parameters begin and stop to influence the WWL almost simultaneously during the test. The upper part of Table 1 displays that the normalized sensitivities of all parameters are highly correlated with each other during first 15-second period. This may reflect that the parameters are rather difficult to accurately determine simultaneously using first 15-second data in Yeh and Chen (2007). The lower part of Table 1 shows that the normalized sensitivity to parameter S s1 is no longer highly correlated to those of the other parameters during 1,000 s, indicating that S s1 behaves differently if compared with those of the other parameters after 15 s.

Table 1 Two correlation matrices of normalized sensitivities to parameters over periods of 15 and 1,000 s
Full size table

Yeh and Chen (2007, Table 6a for Scenario 14) indicated that the skin parameters can be determined accurately if the parameters k 1 and k 2 have a distinct difference, i.e., k 2/k 1 is equal to 100. However, the parameter k 2 is still poorly estimated (the relative error RE is 502.06%). Figure 4a and b display the temporal variations of normalized sensitivities to parameters over 1,000 s for the parameter k 1 being equal to 10−5 and 10−6 m/s, respectively. The values of other parameters are kept the same and given in the figures. In Fig. 4b, the curves of normalized sensitivity to parameters k 2 and S s2 are almost invisible. Moreover, the shapes of normalized sensitivities of parameter k 1 and d sk shift to the right as shown in the figure. In other words, the changes of the WWL in response to the relative change of the parameter for k 1 and d sk are slow as compared with those shown in Fig. 4a. Therefore the inaccuracy in parameter k 2 estimation mentioned above may be caused by the problems of small value of k 1 and short WWL data. Obviously, the parameters k 2 and S s2 are difficult to accurately determine by analyzing only 15 s WWL data. Figure 5a and b show the normalized sensitivities for the negative skin cases where the values of k 1 are 10−4 and 10−3 m/s, respectively. The figures depict that the shapes of normalized sensitivity curves do not have significant change even the parameters k 1 and k 2 have distinct values. Hence, the predicted results in negative skin scenarios are more accurate than those of positive skin ones in Yeh and Chen (2007).

Fig. 4
figure 4

The normalized sensitivities to five parameters for a positive skin case with target parameter values listed in (a) Table 4 (b) Table 7

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Fig. 5
figure 5

The normalized sensitivities to five parameters for a negative skin case with k 2 = 10−5 m/s, S s1 = 10−4 m−1, S s2 = 10−4 m−1, d sk  = 0.3085 m, and (a) k 1 = 10−4 m/s (b) k 1 = 10−3 m/s

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3.2 Parameter Determination Using EKF

Six positive skin scenarios in Yeh and Chen (2007, Scenarios 17, 1, 2, 15, 10, and 14) with the RE of estimated k 2 or d sk larger than 100% are selected for testing the applicability of the EKF approach. The RE is an appropriate index for error analyses in these hypothetic cases because the target values of the parameters are known a priori. Tables 2, 3, 4, 5, 6 and 7 display the target values and predicted results in those scenarios analyzed by both SA and EKF. All WWL data sets (cases “a” to “e”) in these scenarios are the same as those of Yeh and Chen (2007). The cases “b” to “e” in the scenarios represent that four sets of standard normally distributed noise are added to the original WWL data.

Table 2 The target values and predicted results by analyzing the WWL data within 15 sec and d sk equals 1.3085 (Yeh and Chen 2007, Scenario 17)
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Table 3 The target values and predicted results by analyzing the WWL data within 15 s and d sk equals 0.9085 (Yeh and Chen 2007, Scenario 1)
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Table 4 The target values and predicted results by analyzing the WWL data within 15 s and d sk equals 0.3085 (Yeh and Chen 2007, Scenario 2)
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Table 5 The target values and predicted results by analyzing the WWL data within 15 s and d sk equals 0.1085 (Yeh and Chen 2007, Scenario 15)
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Table 6 The target values and predicted results for the same scenario as Table 2c while the analyzed WWL data lasts 180 s (Yeh and Chen 2007, Scenario 10)
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Table 7 The target values and predicted results for a positive skin aquifer with a distinct difference in the hydraulic conductivities of skin and formation zones (Yeh and Chen 2007, Scenario 14)
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Tables 2, 3, 4 and 5 display the predicted results of the five parameters for four positive-skin scenarios (Yeh and Chen 2007, Scenarios 17, 1, 2, and 15) when analyzing 15-second WWL data. The target values of d sk in the scenarios shown in Tables 2, 3, 4 and 5 are 1.3085, 0.9085, 0.3085, and 0.1085 m, respectively. The target values of parameters k 1, k 2, S s1, and S s2 are 1.00 × 10−5 m/s, 1.00 × 10−4 m/s, 1.00 × 10−4 m−1 and 1.00 × 10−4 m−1, respectively. In Tables 2, 3, 4 and 5, the initial estimates of parameters k 1, k 2, S s1, and S s2 for the EKF are 1.5 × 10−5 m/s, 1.5 × 10−4 m/s, 1.5 × 10−4 m−1, and 1.5 × 10−4 m−1. The initial estimates of d sk are 1.4085 m, 1.4085 m, 0.4085 m, and 0.4085 m for the cases of Tables 2, 3, 4 and 5, respectively. The TOL i for parameters k 1, k 2, S s1, S s2, and d sk are set as 10−9 (m/s), 10−8 (m/s), 10−8 (1/m), 10−8 (1/m), and 10−5 (m), respectively. In practice, reasonable initial estimates of parameters can generally be made based on the field geology and engineering experiences. Moreover, one can monitor the change of the parameters during the process on-line. Once the parameter values become negative or deviate far from reasonable ranges, the EKF process can be terminated and then restarted with a new set of initial estimates of parameters.

In Table 2, the results of SA show that the parameters k 1 and d sk are determined accurately but k 2 is determined inaccurately. The predicted results of EKF indicate that the parameters k 1, k 2, S s1, and d sk are all determined properly. However, there is slight inaccuracy in parameter S s2 estimate with a mean value of 1.38 × 10−4 m−1 and RE of 43.65%. Compared with the results of SA, the EKF provides more accurate estimates for all parameters, especially for the parameter k 2. The predicted values of parameter S s2 by the EKF are close to the initial estimate. Those results are attributed to the fact that the response of the WWL is generally less sensitivity to S s2 than to other parameters. Thus the predicted S s2 almost keeps the same value as the initial estimate during the EKF determination processes. In Table 3, the REs of predicted k 1, k 2, S s1 and d sk are 1.17%, 26.37%, −2.77% and 7.38%, respectively. Similar to Table 2, the mean value of 1.42 × 10−4 m−1 and RE of −42.46% for the parameter S s2 show a little inaccurate estimation. In Table 4, the mean values of predicted parameters k 1 and d sk using SA are significantly larger than the target values. Table 1 shows that the normalized sensitivity of parameters k 1 and d sk is significant negatively correlated. An increase in k 1 will reduce the values of WWL; oppositely, an increase in d sk will raise the values of WWL. Therefore, SA may provide a set either highly over-estimated or under-estimated value of the product of these two parameters which can give a very good prediction in WWL data. However, in this scenario, the results of EKF show that all parameters are accurately determined except for the parameter S s2. The REs of parameters k 1, k 2, S s1, S s2, and d sk are are −1.6%, 15.3%, 12.52%, 44.79%, and 1.53%, respectively. In Table 5, the SA results are not accurate for the parameters k 1 and d sk estimates. The REs of these two parameters are 157.00% and 601.75%, respectively. However, EKF gives more accurate estimates in k 1 and d sk . Tables 2, 3, 4 and 5 show that the EKF can accurately determine the parameters even in small d sk scenarios. In addition, the EKF can provide consistent estimates of parameters in five cases for each scenario. This behavior demonstrates that the EKF can reduce the uncertainties of measurements significantly.

Table 6 shows the predicted results by both SA and EKF for the same scenario listed in Table 4 but the measurement period of WWL data is extended to 180 s. The initial estimates and the TOL i of the parameters for the EKF are the same as those of Table 4. The results of SA show a little improvement for d sk , but the RE is still larger than 100%. The results of EKF are as good as those given in Table 4. The REs are −1.75%, 6.22%, 19.59%, 61.25%, and 1.53%, respectively. Table 7 shows the results for a positive skin scenario with largely different hydraulic conductivities between the skin zone and formation zone. The target values are the same as Table 4 except that k 1 = 10−6 m/s and the initial estimate of k 1 for the EKF is adjusted to 1.50 × 10−6 m/s. In this case, the results of k 1, S s1, and d sk are improved by using SA but the RE of k 2 is 502.06% which is inaccurately determined. In contrast, the results of EKF display that the parameters k 1, S s1, and d sk determined properly but the parameters k 2 and S s2 are slightly inaccurate. The REs of parameters k 1, k 2, S s1, S s2, and d sk are −0.13%, 50.88%, 1.46%, 50.12%, and 1.25%, respectively. The relatively inaccurate estimates of k 2 and S s2 are caused by the problem of small value of k 1 where the effect of aquifer properties can not reached to the WWL data within a short period.

The comparison of the predicted results from SA and EKF demonstrates that SA may not be able to handle the parameter estimation problem properly for the slug test with positive skin effect. In contrast, the EKF which accounts for the uncertainties of measurements in its algorithm gives accurate parameter estimates in these cases. Moreover, the results of EKF approach are not significantly affected by the parameter correlations. The slightly inaccurate estimates of parameters are due to the fact that the parameters are insensitive to the WWL or the relatively small value of k 1. Finally, the computing time using EKF in all scenarios are less than 30 s in a personal computer with 3.6 G Pentium IV CPU and 1 GB RAM while the SA needs 3.6 h and 7.6 h to obtain the optimal parameters when analyzing 20 and 47 WWL data, respectively. In these scenarios, the EKF saves at least 99.8% computing time when comparing with those of the SA.

4 Conclusions

This study aims at investigating the problem of inaccuracy in the skin thickness d sk and aquifer conductivity k 2 estimates for the positive skin scenarios given in Yeh and Chen (2007). The correlation analysis is used to quantify the strength of relationship between normalized sensitivities of WWL with respect to skin and aquifer parameters over 15- and 1,000-second periods. Moreover, the sensitivity analysis is employed to explore the problem of inaccuracy in the parameter k 2 determination. An approach of coupling the EKF with Moench and Hsieh solution (1985) is developed to determine five parameters, k 1, k 2, S s1, S s2, and d sk , in six positive skin scenarios where the k 2 and d sk were not accurately determined in Yeh and Chen (2007).

The results of correlation analysis demonstrate that the normalized sensitivity to parameter d sk correlates negatively with other parameters. Those high correction values indicate that the parameters k 1 and d sk may be difficult to accurately determine by SA shown in Yeh and Chen (2007). In addition, the results of sensitivity analysis show that the normalized sensitivities of WWL with respect to aquifer parameters (k 2 and S s2) decrease significantly if the skin conductivity k 1 is relatively small, say, 10−6 m/s, in the positive skin scenario. Consequently, the aquifer hydraulic properties can not respond to the test quickly and influence the WWL data. This may be the reason why the parameter k 2 was accurately determined in Yeh and Chen (2007, Scenario 14).

Tables 2, 3, 4, 5, 6 and 7 display the predicted results for six positive skin scenarios analyzed by both SA and EKF. The results indicate that the EKF provides much better estimates on parameters k 2 and d sk even the skin thickness is thin. Such results are attributed to the fact that the EKF accounts for the uncertainties of the measurement (round-off errors in WWL data set) in the algorithm. The predicted parameter S s2 using EKF is slightly inaccurate since it is very insensitive in response to the measurement error in WWL data. Moreover, the results listed in Table 7 using EKF show that the aquifer parameters k 2 and d sk are difficult to accurately determine when the skin conductivity k 1 is very small because the small value of k 1 retards the propagation of the change of WWL from the test well to the aquifer. The comparison of the computing time required by the EKF and the SA also indicates that the EKF is more efficient than the SA in parameter determination.