An area matching process to estimate the hydraulic parameters using transient constant-head test data

A constant-head test is a hydrogeological tool, in which groundwater is extracted by maintaining a constant head in a well, and the aquifer parameters (i.e. hydraulic conductivity, storativity) are estimated from the transient discharge measurements. Compared to the constant-head test, the constant-discharge test is a more frequently utilized technique for investigating the hydrogeological properties (Kruseman & de Ridder 1990). However, the use of traditional constant-discharge test may not be convenient, especially in a low-transmissivity formation, where it leads to the occurrence of a deep cone of drawdown in the vicinity of the pumping well which results in the dewatering of the pumping well in a very short period, therefore, adequate measurements cannot be gathered from the nearby observation wells (Freeze & Cherry 1979). On the other hand, this phenomenon is not encountered in the constant-head test, since a fixed water level is maintained at the well; hence this test is more suitable for aquifers of low-transmissivity. Wellbore storage effect is also needed to be considered as an additional unknown parameter in constant-discharge test, whereas this parameter is of no concern in constant-head tests (Mucha & Paulikova 1986). Since the head at the well is maintained at a constant level during the test, the head change in the well would not lead to any water release from wellbore storage (Bundschuh & Suárez 2010).

The first mathematical model describing the flow behavior under a constant-head test was provided by Jacob & Lohman (1952). The temporal variation of discharge is formulated by a cumbersome well function in this pioneering solution which involves a number of assumptions: (i) infinite-extent aquifer with horizontal flow; (ii) fully penetrating well; (iii) homogeneous and isotropic aquifer; and (iv) two-dimensional flow. Some analytical treatments for the problem at hand were proposed by Hantush (1959), Glover (1978) and Perrochet (2005). In a more recent work, Chang & Yeh (2009) developed an analytical solution for the constant-head test performed at a partially penetrating well.

Hydraulic parameters from the constant-head test data are traditionally estimated from a curve matching process by superimposing the theoretical type curve (⁠ vs ⁠) and the field data (⁠ vs ⁠) on a log-log scale (Jacob & Lohman 1952). Once the curves are overlapped, a matching point, selected anywhere on the superimposed curves, yields the corresponding ⁠, ⁠, and t. Using these estimated values, transmissivity (T) and storativity (S) are easily computed. However, an appropriate visual fit may not be attained in practice, as a result of measurement errors during data collection, the paucity in the collected test data, or the presence of heterogeneity. Alternatively, Lohman (1972) suggested a straight-line approximation which is similar to the method of Cooper & Jacob (1946) developed for a constant-discharge aquifer test. This approximation is valid for late-time data, however, there is no criterion describing the critical time when the late-time period begins, which may result in the misinterpretation of the test (Batu 1998).

Apart from these traditional techniques, there are a number of studies in the literature that focused on the simplification of the well function given by Jacob & Lohman (1952) with which the formation properties could be investigated. Glover (1978) proposed a fit equation to simplify the well function given by Jacob & Lohman (1952). Swamee et al. (2000) introduced a least-square minimization technique to estimate the aquifer parameters using the function proposed by Glover (1978). Ojha (2004) approximated the Jacob & Lohman (1952) well function by utilizing algebraic manipulations and presented an error minimization process to obtain aquifer parameters. Singh (2007) developed an alternative approximate well function form and formulated a Marquardt algorithm-based optimization method for the estimation of aquifer parameters. Singh (2009) also suggested an optimization scheme coupled with a Kernel method to discretize the time span into a number of uniform time steps for the purpose of assessing formation parameters.

Several techniques are readily available to be used for the estimation of aquifer parameters to analyze the classical constant-discharge data which would be collected from different porous medium formations. Derivative-based techniques developed for constant-discharge pump tests (i.e. Chow 1952; Sen 1986; Copty et al. 2011), for instance, could also be adapted to constant-head test to estimate the hydraulic properties. The well function given by Jacob & Lohman (1952) has, however, a sophisticated manner for performing derivative-based aquifer-parameter assessing methods to simplify the estimation procedure. Renard et al. (2009) and Avci et al. (2013) discussed the shortcomings of time derivation of the available field data for parameter estimation, since the time-derivative of the field data would result in highly erratic results due to some probable abnormalities in collected data. Recently, integration-based aquifer-parameter estimation methodologies were proposed by Avci et al. (2011) and Sahin (2012) to smooth the unreliable behavior appearing in derivative-based methods.

The purpose of this study is to formulate a simple parameter estimation method, as simple as the conventional curve matching procedure, to be used in lieu of other optimization based techniques, mentioned above. A new integration based parameter estimation methodology, called area matching process (AMP), was introduced to estimate aquifer parameters. This new technique (AMP) was established on an idea that an area within an integration window will be identical for both the theoretical type curve and the temporal flow rate curve obtained from the site measurements. The application procedure of AMP is straightforward, does not require any complicated calculations or curve matching process to acquire aquifer parameters. The AMP seems to have promising features to estimate aquifer parameters in a reliable, accurate and robust manner when compared to existing techniques as discussed in the following sections.

The performance of AMP was investigated through the numerical experiments conducted with different aquifer settings. The reliability and the performance of the proposed method were first examined with synthetically generated test data; to which, random noise was introduced in order to mimic the effects of possible measurement errors, fluctuations in the discharge rate, as well as the heterogeneity of the formation on the available test data. The contribution of the integration window on the estimation performance of AMP was also shown. In addition, the proposed method was implemented in numerical experiments, in which six different heterogeneous aquifer scenarios were simulated, in order to elaborate the variation of estimated transmissivity values. Finally, AMP was assessed with a real field test data set from the hydrogeology literature.

Eventually, regardless of which integration window is preferred, these hypothetical experiments verify that for homogeneous systems, the proposed AMP is capable of estimating aquifer parameters with a high accuracy even if the collected data suffer from substantial noise.

The previous studies on the conventional constant-discharge pump tests in the confined heterogeneous aquifer show that the interpreted transmissivity estimates eventually yield to some weighted average of the transmissivity values inside the cone of depression which expands with elapsed time from the beginning of pumping (i.e. Bibby 1979; Meier et al. 1998; Sanchez-Vila et al. 1999; Copty & Findikakis 2004; Wu et al. 2005; Sanchez-Vila et al. 2006; Copty et al. 2011; Avci & Sahin 2014). The same trend, the interpreted transmissivity values at the late time of constant discharge tests leading to the geometric mean of a normally distributed transmissivity field, was therefore verified by this test case example based on the available results.

The performance of the proposed AMP approach was also verified with a real field data set example of a constant-drawdown test obtained from Artesia Heights well near Grand Junction, Colorado in 1948 (Lohman 1965). The radius of the well of interest and the constant-head maintained at the well were reported to be 0.084 m and 28.14 m, respectively. The test results are tabulated in Table 3.

Figure 11 also shows the AMP would lead to almost the same transmissivity estimates (as a mean of 7.53 × 10⁻⁴m²/min), which seems independent from whichever data phase is employed to obtain aquifer parameters. In this regard, the results of AMP for the Lohman (1965) data are in line with those of synthetically generated test data studied in Test case 1. The AMP, therefore, could provide practitioners a great flexibility in the data analysis. A definite late-time criterion was not described by the Lohman (1972) straight-line method as discussed in Batu (1998). This potential drawback of the straight-line method can be overcome by performing AMP as delineated in Figure 11.

In this work, a new parameter estimation procedure, named as AMP, was formulated to analyze the transient constant-head data. The proposed AMP is as straightforward as the classical curve matching techniques; however, it avoids the need for superimposing the theoretical and the field curves. The idea behind this approach is founded on matching a unique area within an integration interval, which is indeed a common property for both the theoretical type curve and the field discharge curve. The reliability and the performance of the AMP were tested for various synthetically generated data sets augmented with random noise, as well as a real field example.

The following key conclusions can be drawn from these test cases. (i) The size of the integration window does not significantly influence the estimation performance of AMP. This feature implies that the available site data can be directly used to estimate aquifer parameters, without any need for interpolation or extrapolation that may lead to the misinterpretation of test results. (ii) For homogeneous aquifer settings, no matter which data phase (early- or late-time response) is utilized in the parameter estimation process; the AMP is able to estimate the hydrologic properties accurately, which in turn, provides a great flexibility to field practitioners while assessing aquifer parameters. (iii) For heterogeneous aquifer systems, the use of late time data was noticed to provide transmissivity values close to the geometric mean of a normally distributed transmissivity field. (iv) AMP is still applicable even with the use of two observation points, which allows this method to be utilized in circumstances when there is lack of sufficient data. (v) And finally, the parameter estimation capability of the AMP is as high as those of the present alternative methods.

In the light of these findings, the AMP can be evaluated as a viable approach for the interpretation of the transient constant-head test results. In order to simplify and automatize the curve-matching process, the established methodology based on area matching has the potential to be extended to other aquifer types such as leaky or unconfined aquifers.