# Estimating Hydraulic Parameters of Aquifers Using Type Curve Analysis of Pumping Tests with Piecewise-Constant Rates

^{*}

## Abstract

**:**

_{1,D}), which depended on the hydraulic conductivity (K) and specific storage (S

_{s}) of the confined aquifers. On this basis, a new type curve method for estimating the aquifer K and S

_{s}was proposed by matching the observed drawdown data with a series of type curves dependent on t

_{1,D}. Furthermore, this method can handle recovery drawdown data. We applied this method to a field site in Wuxi City, Jiangsu Province, China, by analyzing the drawdown data from four pumping tests. The hydraulic parameters estimated using this method were in close agreement with those calibrated via PEST. The calibrated K values were further validated by comparing them with lithology-based results. In summary, the geometric means of K and S

_{s}were 6.62 m/d and 3.16 × 10

^{−5}m

^{−1}for the first confined aquifer and 0.92 m/d and 2.34 × 10

^{−4}m

^{−1}for the second confined aquifer.

## 1. Introduction

_{s}) of aquifers is essential for studying groundwater hydrology, including groundwater transport, depletion, rock seepage behaviors, and land subsidence [1,2,3,4,5,6,7]. There are three primary approaches for estimating these parameters: empirical methods [8,9], laboratory tests [10,11], and field tests [12,13,14,15]. Among the field test methods, pumping tests are widely used due to their ability to analyze a significant portion of the aquifer as a whole and obtain average parameter estimates [16].

_{s}) of an aquifer. While steady-state tests can only provide an estimate for K, it may also take a long time for the test to reach a steady state, leading to increased expenses. Theoretical methods for transient pumping tests have been developed since the classical work of Theis [17], including solutions for different aquifer types, pumping conditions, and boundary conditions [17,18,19,20]. For example, analytical solutions were proposed by Hantush [19] and Neuman [20] for leaky aquifers, and by Cooper and Jacob [18] for a well discharging into a confined aquifer at a constant rate. The majority of mathematical models for transient pumping tests assumed a constant pumping rate. Nevertheless, in field applications, it may not always be feasible to pump at a constant rate due to various head loss factors [21,22]. Thus, the observed drawdowns during the pumping period may not always be analyzed appropriately using traditional analytical models for constant-rate pumping tests.

_{s}) of the tested confined aquifer. Finally, this method was applied to analyze the drawdown data collected from multiple pumping tests conducted in a complex, multi-layered aquifer system in Wuxi City, Jiangsu Province, China. The estimated K and S

_{s}values were then calibrated using the analytical solution coupled with PEST.

## 2. Methodology

#### 2.1. Analytical Solution

_{1}and r

_{2}from the pumping well. The assumptions used in this study are similar to those of the Theis [17] model, with the exception that the pumping rate was assumed to vary with time. Several assumptions were made for this study: (1) the confined aquifer was homogeneous, isotropic, and uniformly thick, and had an infinite extent in the radial direction; (2) the initial head of the confined aquifer was uniformly distributed throughout the system; (3) the pumping well had an infinitesimal radius, and hence, the wellbore storage was negligible; (4) groundwater flow was primarily horizontal and followed Darcy’s law; and (5) the water was removed instantaneously as the head declines. The governing equation that describes transient flow toward the fully penetrating pumping well in the confined aquifer can be written as follows [17]:

_{w}(t)/(2πKb), where s(r, t) denotes the aquifer drawdown at time t and radial distance r [L]; Q

_{w}(t) denotes the pumping rate at time t [L

^{3}T

^{−1}]; S

_{s}denotes the specific storage [L

^{−1}]; K denotes the hydraulic conductivity [L

^{1}T

^{−1}]; r denotes the radial distance from the pumping well [L]; t denotes the time since the pumping started [T]; and b denotes the aquifer thickness [L].

_{w}(t) is represented by a piecewise-constant function, which was approximated by the average pumping rate Q

_{i}at each time step. Here, t

_{i}represents the inflection time at the junction between the ith and (i + 1)th step of the pumping history. It is worth noting that the initial inflection time was t

_{0}= 0, and the initial average pumping rate was Q

_{0}= 0. To simplify Equation (2), the following dimensionless variables for time and drawdown were introduced:

_{D}and s

_{D}are the dimensionless time and dimensionless drawdown, respectively. Then, the dimensionless form of Equation (2) could be expressed as

_{w,D}(t

_{D}) = Q

_{w}(t)/Q

_{1}denotes the dimensionless pumping rate (Q

_{1}≠ 0).

_{w}into n constant pumping rates (see Figure 2), Equation (4) can be rewritten as

_{0,D}= 0, φ

_{1}= 1, and β

_{0}= 0. Under the special condition of n = 1 (i.e., φ

_{2}= φ

_{3}= … = φ

_{n}= 0), Equation (5) becomes a new dimensionless form of the Theis [17] solution, which is numerically equivalent to the well function.

#### 2.2. Type Curve Method

_{i}and average pumping rate Q

_{i}for each step. As a result, one can calculate the dimensionless pumping rate increment φ

_{i}and inflection time ratio β

_{i}.

_{D}and s

_{D}relies on a set of t

_{1,D}values, given β

_{i}and φ

_{i}are fixed. To simplify the analysis, we considered a pumping test with three piecewise-constant pumping segments and examined the variations in s

_{D}versus t

_{D}for different t

_{1,D}values. The inflection times were set to t

_{1}= 2 days, t

_{2}= 4 days, and t

_{3}= 6 days. A fully penetrating well that pumped at rates of Q

_{1}= 4 m

^{3}/d from t

_{0}to t

_{1}, Q

_{2}= 2 m

^{3}/d from t

_{1}to t

_{2}, and Q

_{3}= 8 m

^{3}/d from t

_{2}to t

_{3}was considered. Thus, the involved dimensionless variables were calculated to be β

_{1}= 1, β

_{2}= 2, β

_{3}= 3, φ

_{1}= 1.0, φ

_{2}= −0.5, and φ

_{3}= 1.5. By substituting these values into Equation (5), we obtained a set of type curves for different first dimensionless inflection time (t

_{1,D}) values ranging from 20 to 500, as shown in Figure 3. The type curves of constant-rate pumping tests with their rates set at the first step rate (Q

_{1}), the second step rate (Q

_{2}), and the third step rate (Q

_{3}) are also depicted in this Figure 3.

_{1}). Once the pumping rate abruptly dropped to Q

_{2}for the second step, the influence of Q

_{2}became dominant and s

_{D}deviated from γ

_{1}, gradually approaching the lower asymptotic curve (γ

_{2}) corresponding to Q

_{w}= Q

_{2}. When the pumping rate suddenly changed from Q

_{2}to Q

_{3}for the third step, the type curves deviated from γ

_{2}and approached the upper asymptotic curve γ

_{3}in a manner that was mainly controlled by Q

_{w}= Q

_{3}. It was concluded that except for the first step, the type curves of subsequent steps will experience a sudden drawdown deviation at early and intermediate times, and gradually approach the curve of the constant-rate pumping case later on. The time corresponding to this drawdown deviation is related to the value of t

_{1,D}, and a larger t

_{1,D}will result in a later occurrence of the drawdown deviation.

_{i}and Q

_{i}are determined using piecewise-constant approximations of recorded variable pumping rates. Therefore, Equation (5) shows that s

_{D}is a function of t

_{D}, and is only influenced by different values of t

_{1,D}. Taking the logarithms of both sides of t

_{D}and s

_{D}in Equation (3) results in

_{D}versus t

_{D}. However, there is a shift of 4πKb/Q

_{1}along the vertical (s or s

_{D}) axis and 4K/(r

^{2}S

_{s}) along the horizontal (t or t

_{D}) axis. Equation (3) indicates that t

_{1,D}, which governs the type curve’s shape, is only linked to the confined aquifer’s hydraulic diffusivity (K/S

_{s}) when the radial distance r and the first step’s inflection time t

_{1}are given. Hence, the type curve method can be employed to estimate the hydraulic parameters in this study. The following outlines the new type curve method’s usage:

- (1)
- Plot the measured drawdown–time curve in a log–log graph.
- (2)
- Determine φ
_{i}and β_{i}, and prepare a series of type curves with different t_{1,D}values in a log–log graph of the same scale as the measured curve. - (3)
- Similar to Theis’s matching technique, match the measured drawdown–time curve with one of the type curves and choose the best matching curve.
- (4)
- Record the corresponding t
_{1,D}value, select a match point, and read the corresponding coordinates of s, t, s_{D}, and t_{D}.

_{s}, S

_{s}can be determined by substituting t

_{1,D}into the following expression:

_{s}obtained from different pumping tests or multiple observation wells of the same pumping test might differ. Variations could also be attributed to in situ pumping conditions (noise, temperature, etc.), resulting in uncertainties in the parameter estimation. Moreover, subjective errors in matching, such as the selection of matching points and reading inaccuracies, add uncertainty to parameter estimation. To mitigate the impact of such factors, most measured data should fall on the curved portion that characterizes type curves better than other parts. Furthermore, attention should be paid to the inflection point matching in each step of measured data.

_{s}from the value of t

_{1,D}corresponding to the type curve matched with the measured data is unreliable. This is because the matching between the measured data and the type curves depends on the shapes of the type curves. These shapes differ only minimally when t

_{1,D}varies by an order of magnitude. As a result, the accuracy of this method for determining S

_{s}is doubtful. When the measured data plot moves from one type curve to another, the determined value of K changes slightly compared with that of S

_{s}. To eliminate doubts regarding which type curve with a t

_{1,D}value to use for matching the measured data plot, we can estimate S

_{s}based on knowledge of the geologic conditions within an order of magnitude.

## 3. Field Application

#### 3.1. Background

_{S}) of a confined aquifer in Wuxi City, Jiangsu Province, China. The study area was located within the plain area adjacent to Tai Lake (Figure 4a), which is characterized by relatively flat and low-lying terrain with ground elevations ranging from −1.4 m to 2.9 m. The subsurface geology comprises a multi-layered aquifer system consisting of Quaternary unconsolidated sediments, with Paleocene sandstone underlying the lower bedrock, and Pliocene–Pleistocene Holocene clay, silt, and silty sand and Holocene clay overlaying it from bottom to top.

#### 3.2. Analysis of In Situ Pumping Test

_{1}= 1.67 days, t

_{2}= 3.29 days, and t

_{3}= 4.96 days, with time intervals corresponding to the three steps being 1.67, 1.63, and 1.67 days, respectively. Additionally, the average pumping rates of the three steps were Q

_{1}= 24.27 m

^{3}/d, Q

_{2}= 38.67 m

^{3}/d, and Q

_{3}= 54.47 m

^{3}/d.

_{1}= t

_{1}/t

_{1}= 1.00, β

_{2}= t

_{2}/t

_{1}= 1.97, and β

_{3}= t

_{3}/t

_{1}= 2.97. The dimensionless pumping rate increments for each step were φ

_{1}= 1.00, φ

_{2}= (Q

_{2}− Q

_{1})/Q

_{1}= 0.59, and φ

_{3}= (Q

_{3}− Q

_{2})/Q

_{1}= 0.65. We substituted these values into Equation (5) to derive a series of type curves that were functions of t

_{1,D}values, ranging from 100 to 2000, as depicted in Figure 7a. By matching the measured time–drawdown data to the type curves on a log–log graph, we found that the type curve with t

_{1,D}= 500 provided the best match, as shown in Figure 7a. We selected a matching point with the coordinates of t = 1.74 days, s = 6.53 m, t

_{D}= 516.72, and s

_{D}= 7.04 for the parameter estimation. By substituting these values into Equations (8) and (9), we obtained K = (7.04 × 24.27)/(4 × π × 2.5 × 6.53) = 0.83 m/d and S

_{s}= (7.04 × 24.27 × 1.74)/(π × 2.5 × 6.53 × 52 × 516.72) = 4.47 × 10

^{−4}m

^{−1}.

_{D}= 213.26, and s

_{D}= 6.02. By substituting these coordinate values into Equations (8) and (9), we obtained K = 0.84 m/d and S

_{s}= 1.26 × 10

^{−4}m

^{−1}. The type curve matching results for the other three pumping tests are presented in Figure 7c–h, following a similar process as that for pumping test 1. The values of the t

_{1,D}, coordinate values of the matching points, and estimated values of K and S

_{s}for the four pumping tests are listed in Table 1.

## 4. Discussion

_{p,i}and s

_{m,i}denote simulation drawdowns and measured drawdowns, respectively; and s

_{mc}denotes the average value of measured drawdowns. According to Nash and Sutcliffe [34], a higher CE value indicates that the simulated and measured data are more closely aligned. When the simulated and measured data match perfectly, the CE value is 1. Moreover, if CE exceeds the threshold of 0.5, it suggests that the deviation between the simulated and measured data is acceptable, which ameans that the estimated parameters can be considered reliable.

_{s}were reasonable. Notably, pumping tests 1 and 2 had higher CE values than the other two tests, indicating that the reliability of the K and S

_{s}estimates of tests 1 and 2 in the second confined aquifer was higher than that of tests 3 and 4 in the first confined aquifer. It is worth mentioning that although the CE value for pumping test 3 exceeded 0.90, there was a significant visible difference between the simulated and measured drawdowns (Figure 8c). The primary reason for this disparity could be attributed to the fact that the early period of pumping exhibited a continuous decrease in pumping rate over time, which did not perfectly align with the assumption of piecewise-constant pumping (Figure 6c).

_{s}estimates were also associated with narrow confidence intervals, even though those corresponding to pumping tests 3 and 4 were slightly wider. The differences between the K or S

_{s}estimates from various observation wells for the same pumping test may be attributed to the nonuniform thickness of the confined aquifer and spatial heterogeneity, which are commonly found in nature [31,32,33]. The influence of aquifer boundaries on parameter estimation may be ignored, as even the relatively small second confined aquifer had a lateral boundary range of 3000 m. Furthermore, for a specific pumping test, the difference in S

_{s}is typically greater than K, suggesting relatively higher heterogeneity in S

_{s}than K. Finally, it is noteworthy that the two aquifers of interest had noticeably different K and S

_{s}parameter estimates.

_{s}values obtained from the new type curve method were comparable to those provided by PEST, underscoring the reliability of this new method. To further validate the calibrated K estimates, a comparison was made with compiled experimental results reported in Ren and Santamarina [36], as shown in Figure 9.

_{s}were also calculated, yielding values of 6.62 m/d and 3.16 × 10

^{−5}m

^{−1}for the first confined aquifer and 0.92 m/d and 2.34 × 10

^{−4}m

^{−1}for the second confined aquifer.

## 5. Conclusions

_{s}). The proposed method was then applied to interpret field pumping tests carried out in an aquifer system located in Wuxi City, Jiangsu Province, China. The main conclusions drawn from this study are as follows:

- (1)
- The study introduced a new dimensionless transformation formula to simplify the analytical solution of variable-rate pumping tests, and a piecewise-constant function was further used to approximate the time-varying pumping rate records. Type curve analyses revealed that the time–drawdown curve of the first step was consistent with the Theis curve. However, the type curves of the subsequent steps deviated from the Theis curve and were associated with the first dimensionless inflection time (t
_{1,D}), which depended on the K and S_{s}of the confined aquifers. A large t_{1,D}resulted in a faster time for a sudden turn in the drawdown. - (2)
- A new type curve method was proposed to handle situations where the real pumping rate varies in a complicated pattern over time. One unique feature of this method is that the type curves depend on the pumping conditions rather than the observation conditions, making it applicable to drawdown data collected from various observation wells during a single pumping test. Furthermore, this new method could also be used to analyze recovery drawdown data by setting a zero pumping rate value for the corresponding shutdown period.
- (3)
- The hydraulic conductivity (K) and specific storage (S
_{s}) of the first and second confined aquifers at the field site were estimated using the pumping rate and drawdown records from four real pumping tests. The estimation results showed that the hydraulic parameters obtained from the newly proposed type curve method were close to the calibrated results reported by PEST, indicating the reliability and robustness of this new method. Moreover, the K estimates were further verified by comparing them with lithology-based results. The geometric means of K and S_{s}were 6.62 m/d and 3.16 × 10^{−5}m^{−1}for the first confined aquifer and 0.92 m/d and 2.34 × 10^{−4}m^{−1}for the second confined aquifer. - (4)
- The field pumping test results showed that the actual pumping rate may have an uncontrollable and short-duration decreasing trend at the early times of each step, resulting in uncertainty in the evaluation of aquifer hydraulic parameters. In addition, the heterogeneity of natural aquifers and the non-uniformity of their thickness also led to differences in the estimated hydraulic parameters of different observation wells in the same pumping test. Future studies will focus on characterizing the heterogeneity of aquifer systems from multiple pumping test data based on more realistic and refined pumping models.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Banks, D.; Odling, N.E.; Skarphagen, H.; Rohr-Torp, E. Permeability and stress in crystalline rocks. Terra Nova
**1996**, 8, 223–235. [Google Scholar] [CrossRef] - Liu, J.; Zhao, Y.; Tan, T.; Zhang, L.; Zhu, S.; Xu, F. Evolution and modeling of mine water inflow and hazard characteristics in southern coalfields of China: A case of Meitanba mine. Int. J. Min. Sci. Technol.
**2022**, 32, 513–524. [Google Scholar] [CrossRef] - Manoutsoglou, E.; Lazos, I.; Steiakakis, E.; Vafeidis, A. The Geomorphological and Geological Structure of the Samaria Gorge, Crete, Greece-Geological Models Comprehensive Review and the Link with the Geomorphological Evolution. Appl. Sci.
**2022**, 12, 10670. [Google Scholar] [CrossRef] - Stober, I.; Bucher, K. Origin of salinity of deep groundwater in crystalline rocks. Terra Nova
**1999**, 11, 181–185. [Google Scholar] [CrossRef] - Yuan, Z.; Zhao, J.; Li, S.; Jiang, Z.; Huang, F.A. Unified Solution for Surrounding Rock of Roadway Considering Seepage, Dilatancy, Strain-Softening and Intermediate Principal Stress. Sustainability
**2022**, 14, 8099. [Google Scholar] [CrossRef] - Zhao, Y.; Liu, Q.; Zhang, C.; Liao, J.; Lin, H.; Wang, Y. Coupled seepage-damage effect in fractured rock masses: Model development and a case study. Int. J. Rock Mech. Min.
**2021**, 144, 104822. [Google Scholar] [CrossRef] - Zhao, Y.; Luo, S.; Wang, Y.; Wang, W.; Zhang, L.; Wan, W. Numerical Analysis of Karst Water Inrush and a Criterion for Establishing the Width of Water-resistant Rock Pillars. Mine Water Environ.
**2017**, 36, 508–519. [Google Scholar] [CrossRef] - Chapuis, R.P.; Aubertin, M. On the use of the Kozeny–Carman equation to predict the hydraulic conductivity of soils. Can. Geotech. J.
**2003**, 40, 616–628. [Google Scholar] [CrossRef] - Ren, X.; Zhao, Y.; Deng, Q.; Li, D.; Wang, D. A relation of hydraulic conductivity-Void ratio for soils based on Kozeny-carman equation. Eng. Geol.
**2016**, 213, 89–97. [Google Scholar] [CrossRef] - Gallage, C.; Kodikara, J.; Uchimura, T. Laboratory measurement of hydraulic conductivity functions of two unsaturated sandy soils during drying and wetting processes. Soils Found.
**2013**, 53, 417–430. [Google Scholar] [CrossRef] - Masrouri, F.; Bicalho, K.V.; Kawai, K. Laboratory Hydraulic Testing in Unsaturated Soils. Geotech. Geol. Eng.
**2008**, 26, 691–704. [Google Scholar] [CrossRef] - Abdalla, F.; Mubarek, K. Assessment of well performance criteria and aquifer characteristics using step-drawdown tests and hydrogeochemical data, west of Qena area, Egypt. J. Afr. Earth. Sci.
**2018**, 138, 336–347. [Google Scholar] [CrossRef] - Hendrayanto; Kosugi, K.I.; Mizuyama, T. Field Determination of Unsaturated Hydraulic Conductivity of Forest Soils. J. For. Res.
**1998**, 3, 11–17. [Google Scholar] [CrossRef] - Neuman, S.P.; Witherspoon, P.A. Field determination of the hydraulic properties of leaky multiple aquifer systems. Water Resour. Res.
**1972**, 8, 1284–1298. [Google Scholar] [CrossRef] - Sethi, R. A dual-well step drawdown method for the estimation of linear and non-linear flow parameters and wellbore skin factor in confined aquifer systems. J. Hydrol.
**2011**, 400, 187–194. [Google Scholar] [CrossRef] - Butt, M.A.; Mcelwee, C.D. Aquifer-Parameter Evaluation from Variable-Rate Pumping Tests Using Convolution and Sensitivity Analysis. Groundwater
**1985**, 23, 212–219. [Google Scholar] [CrossRef] - Theis, C.V. 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. Am. Geophys. Union
**1935**, 16, 519–524. [Google Scholar] [CrossRef] - Cooper, H.H.; Jacob, C.E. A generalized graphical method for evaluating formation constants and summarizing well-field history. EOS Trans. Am. Geophys. Union
**1946**, 27, 526–534. [Google Scholar] [CrossRef] - Hantush, M.S. Nonsteady Flow to Flowing Wells in Leaky Aquifers. J. Geophys. Res.
**1959**, 64, 1043–1052. [Google Scholar] [CrossRef] - Neuman, S.P. Theory of Flow in Unconfined Aquifers Considering Delayed Response of the Water Table. Water Resour. Res.
**1972**, 8, 1031–1045. [Google Scholar] [CrossRef] - Rorabaugh, M. Graphical and theoretical analysis of step-drawdown test of artesian well. Proc. Am. Soc. Civ. Eng.
**1953**, 79, 1–23. [Google Scholar] - Sen, Z.; Altunkaynak, A. Variable discharge type curve solutions for confined aquifers. J. Am. Water Resour. Assoc.
**2004**, 40, 1189–1196. [Google Scholar] [CrossRef] - Butler, J.J.; McElwee, C.D. Variable-rate pumping tests for radially symmetric nonuniform aquifers. Water Resour. Res.
**1990**, 26, 291–306. [Google Scholar] [CrossRef] - Hantush, M.S. Drawdown around Wells of Variable Discharge. J. Geophys. Res.
**1964**, 69, 4221–4235. [Google Scholar] [CrossRef] - Singh, S.K. Well Loss Estimation: Variable Pumping Replacing Step Drawdown Test. J. Hydraul. Eng.
**2002**, 128, 343–348. [Google Scholar] [CrossRef] - Wen, Z.; Zhan, H.; Wang, Q.; Liang, X.; Ma, T.; Chen, C. Well hydraulics in pumping tests with exponentially decayed rates of abstraction in confined aquifers. J. Hydrol.
**2017**, 548, 40–45. [Google Scholar] [CrossRef] - Zhang, G. Type curve and numerical solutions for estimation of Transmissivity and Storage coefficient with variable discharge condition. J. Hydrol.
**2013**, 476, 345–351. [Google Scholar] [CrossRef] - Zhuang, C.; Zhou, Z.; Zhan, H.; Wang, J.; Li, Y. New graphical methods for estimating aquifer hydraulic parameters using pumping tests with exponentially decreasing rates. Hydrol. Process
**2019**, 33, 2314–2322. [Google Scholar] [CrossRef] - Luo, N.; Zhanfeng, Z.; Walter, A.I.; Steven, J.B. Comparative study of transient hydraulic tomography with varying parameterizations and zonations: Laboratory sandbox investigation. J. Hydrol.
**2017**, 554, 758–779. [Google Scholar] [CrossRef] - Doherty, J. PEST: Model Independent Parameter Estimation; Watermark Computing: Corinda, Australia, 2008. [Google Scholar]
- Copty, N.K.; Trinchero, P.; Sanchez-Vila, X.; Sarioglu, M.S.; Findikakis, A.N. Influence of heterogeneity on the interpretation of pumping test data in leaky aquifers. Water Resour. Res.
**2008**, 44, 2276–2283. [Google Scholar] [CrossRef] - Demir, M.T.; Copty, N.K.; Trinchero, P.; Sanchez-Vila, X. Bayesian Estimation of the Transmissivity Spatial Structure from Pumping Test Data. Adv. Water Resour.
**2017**, 104, 174–182. [Google Scholar] [CrossRef] - Sudicky, E.A.; Illman, W.A.; Goltz, I.K.; Adams, J.J.; Mclaren, R.G. Heterogeneity in hydraulic conductivity and its role on the macroscale transport of a solute plume: From measurements to a practical application of stochastic flow and transport theory. Water Resour. Res.
**2010**, 46, 489–496. [Google Scholar] [CrossRef] - Nash, J.; Sutcliffe, J. River flow forecasting through conceptual models: I. A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Chen, C.; Tao, Q.; Wen, Z.; Wörman, A.; Jakada, H. Step-drawdown test for identifying aquifer and well loss parameters in a partially penetrating well with irregular (non-linear increasing) pumping rates. J. Hydrol.
**2022**, 614, 128652. [Google Scholar] [CrossRef] - Ren, X.W.; Santamarina, J.C. The hydraulic conductivity of sediments: A pore size perspective. Eng. Geol.
**2018**, 233, 48–54. [Google Scholar] [CrossRef]

**Figure 1.**Conceptual model of a confined aquifer system with a pumping well and two observation wells. Q

_{w}represents the time-varying pumping rate; K and S

_{s}represent the hydraulic conductivity and specific storage of the confined aquifer, respectively; and r

_{1}and r

_{2}represent the radial distances between the two observation wells and the pumping well, respectively.

**Figure 2.**Approximation of the time-varying observed pumping rates (black curve) using a piecewise-constant function (red curve). Q

_{i}represents the average pumping rate of the ith step from t

_{i}

_{−1}to t

_{i}.

**Figure 3.**Type curves of dimensionless time t

_{D}versus dimensionless drawdown s

_{D}with a series of t

_{1,D}values. γ

_{1}denotes the Theis curve; γ

_{2}and γ

_{3}are the asymptotic curves for the type curves in the second and third steps, respectively.

**Figure 4.**The field site location for the pumping tests. (

**a**) The location of the study area; (

**b**) the locations of boreholes and pumping wells. A–A′ denotes the survey line.

**Figure 6.**Observed time-varying pumping rates (blue points) and corresponding piecewise-constant or step approximation (red line). (

**a**) First pumping test; (

**b**) Second pumping test; (

**c**) Third pumping test; (

**d**) Fourth pumping test. Q

_{i}denotes the average pumping rate of the ith step. Tests 1 and 2 were conducted in the 2nd confined aquifer, and tests 3 and 4 were conducted in the 1st confined aquifer.

**Figure 7.**Matching the scattered points of pumping time (t) versus measured drawdown (s) with the type curves (t

_{D}versus s

_{D}) depending on different t

_{1,}

_{D}values. (

**a**,

**b**) Observation wells 1 and 2 for the first pumping test; (

**c**,

**d**) Observation wells 1 and 2 for the second pumping test; (

**e**,

**f**) Observation wells 1 and 2 for the third pumping test; (

**g**,

**h**) Observation wells 1 and 2 for the fourth pumping test. Red circle and red solid dot denote the measured drawdown and matching point selected for the parameter estimation, respectively.

**Figure 8.**Simulated drawdowns (scattered points) and measured drawdowns (solid lines) of four pumping tests. (

**a**) First pumping test; (

**b**) Second pumping test; (

**c**) Third pumping test; (

**d**) Fourth pumping test. Each test result includes the measured drawdown data of two observation wells, where the blue points represent the drawdowns of observation well 1, and the green points represent the drawdowns of observation well 2.

**Figure 9.**Comparison of the estimated K in this study with compiled experimental results adapted with permission from Ren and Santamarina [36], Engineering Geology, published by Elsevier, 2018. CA represents a confined aquifer.

**Table 1.**Hydraulic conductivity K (m/d) and specific storage S

_{s}(×10

^{−4}m

^{−1}), as estimated using type curve analysis of pumping tests with piecewise-constant rates.

Test No. (Aquifer) | Obs. Well | t_{1,D} | Coordinate Values | K | S_{s} | |||
---|---|---|---|---|---|---|---|---|

t_{D} | s_{D} | t (d) | s (m) | |||||

1 (2nd CA *) | ow1-1 | 500 | 516.72 | 7.04 | 1.74 | 6.53 | 0.83 | 4.47 |

ow1-2 | 200 | 213.26 | 6.02 | 1.79 | 5.53 | 0.84 | 1.26 | |

2 (2nd CA) | ow2-1 | 1000 | 1020.38 | 8.73 | 2.13 | 3.66 | 1.07 | 3.58 |

ow2-2 | 200 | 217.55 | 7.05 | 2.27 | 3.55 | 0.89 | 1.66 | |

3 (1st CA) | ow3-1 | 20,000 | 481,359.64 | 30.96 | 1.06 | 0.54 | 4.98 | 0.02 |

ow3-2 | 1000 | 24,185.94 | 23.10 | 1.06 | 0.34 | 5.90 | 0.05 | |

4 (1st CA) | ow4-1 | 1000 | 1754.08 | 4.11 | 4.06 | 1.23 | 2.01 | 7.47 |

ow4-2 | 300 | 569.65 | 3.56 | 4.07 | 1.07 | 2.01 | 2.55 |

**Table 2.**PEST-estimated values of K and Ss and Nash–Sutcliffe coefficients (CE) for each pumping test.

Test No. (Aquifer) | Observation Well | K (m/d) | S_{s} (×10^{−4} m^{−1}) | CE |
---|---|---|---|---|

1 | ow1-1 | 0.84 [0.81,0.87] ^{a} | 4.28 [3.67,4.99] | 0.993 |

(2nd CA *) | ow1-2 | 0.87 [0.83,0.91] | 1.22 [1.03,1.44] | 0.988 |

2 | ow2-1 | 1.07 [0.99,1.14] | 3.75 [2.66,5.28] | 0.973 |

(2nd CA) | ow2-2 | 0.92 [0.85,0.98] | 1.53 [1.17,2.01] | 0.972 |

3 | ow3-1 | 4.96 [4.30,5.62] | 0.02 [0.004,0.08] | 0.933 |

(1st CA) | ow3-2 | 6.28 [5.73,6.83] | 0.04 [0.02,0.08] | 0.958 |

4 | ow4-1 | 1.98 [1.85,2.11] | 8.85 [6.63,11.81] | 0.819 |

(1st CA) | ow4-2 | 2.20 [2.03,2.37] | 1.78 [1.30,2.43] | 0.743 |

^{a}Values inside the bracket represent the 95% confidence intervals reported. * CA denotes confined aquifer

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Y.; Zhou, Z.; Zhuang, C.; Dou, Z.
Estimating Hydraulic Parameters of Aquifers Using Type Curve Analysis of Pumping Tests with Piecewise-Constant Rates. *Water* **2023**, *15*, 1661.
https://doi.org/10.3390/w15091661

**AMA Style**

Li Y, Zhou Z, Zhuang C, Dou Z.
Estimating Hydraulic Parameters of Aquifers Using Type Curve Analysis of Pumping Tests with Piecewise-Constant Rates. *Water*. 2023; 15(9):1661.
https://doi.org/10.3390/w15091661

**Chicago/Turabian Style**

Li, Yabing, Zhifang Zhou, Chao Zhuang, and Zhi Dou.
2023. "Estimating Hydraulic Parameters of Aquifers Using Type Curve Analysis of Pumping Tests with Piecewise-Constant Rates" *Water* 15, no. 9: 1661.
https://doi.org/10.3390/w15091661