# Determination of Aquitard Storage from Pumping Tests in Leaky Aquifers

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theory

^{2}T

^{−1}] is the aquifer transmissivity, ${S}_{s}$ [L

^{−1}] is the specific storage coefficient of the aquitard, $K$ [L T

^{−1}] is the hydraulic conductivity of the aquitard, $r$ [L] is the radial distance measured from the center of the well, $z$ [L] is the elevation, and $t$ [T] is the time since pumping started. The initial condition is

^{3}T

^{−1}] is the pumping rate, while infinitely far away from the well, the drawdown remains zero.

^{−1}], as

#### 2.2. Test Case

^{3}/d and averaged 380 m

^{3}/d. Groundwater drawdowns were observed at six locations, specifically in three fully penetrating observation wells at 14 m, 45 m and 85 m distance from the pumping well and in three piezometers with screens of 2 m length located in the middle of the aquifer at 13 m, 86 m and 262 m distance from the pumping well. Water levels in the observation wells and piezometers were measured manually with portable water level sensors, initially at one-minute intervals for observations near the pumping well and later gradually expanded to larger intervals.

## 3. Results

^{−8}in double precision.

^{2}are plotted on log-log paper, a Theis type-curve can be fitted as an upper bound as shown in Figure 3. The model parameters are obtained as explained in the literature [14,15], resulting in T ≈ 75 m

^{2}/d and S ≈ 3 × 10

^{−4}. In the late phase of the pumping test, the aquifer behaves as if it is semi-confined and the flow becomes stationary. The drawdown can thus be described by the steady-state Hantush well flow equation [8], which depends on a leakage factor $B=\sqrt{T/\mathrm{C}}$ related to the radius of influence of the pumping well. Considering the local scale of the pumping test, we can assume that B = 300 m and the leakage coefficient can be estimated as C = T/B

^{2}≈ 8. × 10

^{−4}d

^{−1}. Lastly, the starting value for the aquitard storage coefficient is assumed to be the same as for the aquifer, so S’ ≈ 3 × 10

^{−4}.

^{−5}m

^{−1}for the aquifer and 1.1 × 10

^{−4}m

^{−1}for the aquitard, which is also within expectations for these type of porous materials, e.g., [15]. The specific storage of the aquitard is about four times larger than that of the aquifer. Yeh and Huang [23] analyzed different data sets from pumping tests in leaky aquifers, and suggested that aquitard storage can be ignored only if the ratio of the aquitard storage to the aquifer storage is less than 10

^{−3}.

## 4. Discussion

^{−4}= 7.84 × 10

^{−4}, but this is larger than the value obtained by fitting the three-parameter model. The reason may be due to the fact that all observations are used in the fitting procedure, rather than just the observations at large distances. Nevertheless, these results indicate that estimates of aquifer storage can be inaccurate when storage in the aquitard is ignored. Rushton [24] studied the effect of aquitard storage using a numerical model and also concluded that unreliable parameter values may be deduced in a pumping test analysis if aquitard storage is ignored.

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

- Alley, W.M.; Healy, R.W.; Labaugh, J.W.; Reilly, T.E. Flow and storage in groundwater systems. Science
**2002**, 296, 1985–1990. [Google Scholar] [CrossRef] [PubMed] - Van der Kamp, G. Methods for determining the in situ hydraulic conductivity of shallow aquitards—An overview. Hydrogeol. J.
**2001**, 9, 5–16. [Google Scholar] [CrossRef] - Burbey, T.J. Use of time-subsidence data during pumping to characterize specific storage and hydraulic conductivity of semi-confining units. J. Hydrol.
**2003**, 281, 3–22. [Google Scholar] [CrossRef] - Zhuang, C.; Zhou, Z.; Zhan, H.; Wang, G. A new type curve method for estimating aquitard hydraulic parameters in a multi-layered aquifer system. J. Hydrol.
**2015**, 527, 212–220. [Google Scholar] [CrossRef] - Hart, D.J.; Bradbury, K.R.; Feinstein, D.T. The vertical hydraulic conductivity of an aquitard at two spatial scales. Groundwater
**2006**, 44, 201–211. [Google Scholar] [CrossRef] [PubMed] - Konikow, L.F.; Neuzil, C.E. 2007. A method to estimate groundwater depletion from confining layers. Water Resour. Res.
**2007**, 43, W07417. [Google Scholar] [CrossRef] - Smith, L.A.; van der Kamp, G.; Hendry, M.J. A new technique for obtaining high-resolution pore pressure records in thick claystone aquitards and its use to determine in situ compressibility. Water Resour. Res.
**2013**, 49, 732–743. [Google Scholar] [CrossRef] - Hantush, M.S.; Jacob, C.E. Non-steady radial flow in an infinite leaky aquifer. Trans. Am. Geophys. Union
**1955**, 36, 95–100. [Google Scholar] [CrossRef] - Hantush, M.S. Modification of the theory of leaky aquifers. J. Geophys. Res.
**1960**, 65, 3713–3725. [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] - De Smedt, F. Constant-rate pumping test in a leaky aquifer with water release from storage in the aquitard. Groundwater
**2020**, 58, 487–491. [Google Scholar] [CrossRef] [PubMed] - Moench, A.F. Transient flow to a large-diameter well in an aquifer with storative semiconfining layers. Water Resour. Res.
**1985**, 21, 1121–1131. [Google Scholar] [CrossRef] - Feng, Q.; Zhan, H. On the aquitard–aquifer interface flow and the drawdown sensitivity with a partially penetrating pumping well in an anisotropic leaky confined aquifer. J. Hydrol.
**2015**, 521, 74–83. [Google Scholar] [CrossRef] - Kruseman, G.P.; de Ridder, N.A. Analysis and Evaluation of Pumping Test Data, 2nd ed.; Publication 47; International Institute for Land Reclamation and Improvement: Wageningen, The Netherlands, 1994. [Google Scholar]
- Batu, V. Aquifer Hydraulics: A Comprehensive Guide to Hydrogeologic Data Analysis; John Wiley & Sons Inc.: New York, NJ, USA, 1998. [Google Scholar]
- Cheng, A.H.-D. Multilayered Aquifer Systems-Fundamentals and Applications; Marcel Dekker: New York, NY, USA; Basel, Switzerland, 2000. [Google Scholar]
- Walton, W.C. Aquifer Test Modelling; CRC Press: Boca Raton, FL, USA, 2007. [Google Scholar]
- Boonstra, H.; Soppe, R. Well hydraulics and aquifer tests. In Groundwater Engineering, 2nd ed.; Delleur, J.W., Ed.; CRC Press: Boca Raton, FL, USA, 2007; pp. 10-1–10-35. [Google Scholar]
- Sneddon, I.N. The Use of Integral Transforms; Tata McGraw-Hill Publ. Co. Ltd.: New Delhi, India, 1974; 539p. [Google Scholar]
- Davies, B.; Martin, B. Numerical inversion of the Laplace transform: A survey and comparison of methods. J. Comput. Phys.
**1979**, 33, 1–32. [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 groundwater storage. Trans. Am. Geophys. Union
**1935**, 16, 519–524. [Google Scholar] [CrossRef] - IMSL Numerical Libraries. Available online: https://www.imsl.com/ (accessed on 9 March 2023).
- Yeh, H.D.; Huan, Y.C. Parameter estimation for leaky aquifers using the extended Kalman filter, and considering model and data measurement uncertainties. J. Hydrol.
**2005**, 302, 28–45. [Google Scholar] [CrossRef] - Rushton, K.R. Impact of aquitard storage on leaky aquifer pumping test analysis. Q. J. Eng. Geol. Hydrogeol.
**2005**, 38, 325–336. [Google Scholar] [CrossRef] - Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Burnham, K.P.; Anderson, D.R. Multimodel inference: Understanding AIC and BIC in model selection. Sociol. Method. Res.
**2004**, 33, 261–304. [Google Scholar] [CrossRef]

**Figure 1.**Schematic cross-section of a pumping well in a leaky aquifer, where r is the radial distance from the center of the well, z is the elevation from the base of the aquitard and Q is the pumping rate of the well.

**Figure 2.**Schematic map of the pumping test site showing the location of the pumping well, observations wells, piezometers and watercourses.

**Figure 3.**Plot of the observed drawdown s against t/r

^{2}and the Theis well flow equation [21] fitted as an upper bound; green symbols correspond to observation wells, blue symbols to piezometers, and the red solid line to the Theis well flow equation.

**Figure 4.**Plot of fitted (solid and dotted lines) and observed (symbols) drawdown s against time t; the solid lines correspond to the four-parameter model for well flow in a leaky aquifer including storage in the aquitard, given by Equation (14), the dotted lines to the three-parameter model ignoring storage in the aquitard, given by Equation (21); the green symbols correspond to observation wells and the blue symbols to piezometers.

**Table 1.**Calibrated aquifer and aquitard parameters (mean estimates and 95% confidence intervals) with storage in the aquitard, Equation (14), and without storage in the aquitard, Equation (21); also given are model fitting criteria: degree of freedom (DF), residual sum of squares (RSS), residual standard error (RSE), Akaike information criterion (AIC), and Bayesian information criterion (BIC).

Parameter | Units | Equation (14) | Equation (21) |
---|---|---|---|

T | m^{2}/d | 71.6 ± 0.9 | 75.1 ± 1.8 |

S | 10^{−4} | 2.73 ± 0.16 | 4.75 ± 0.29 |

C | 10^{−3} d^{−1} | 1.96 ± 0.11 | 2.07 ± 0.22 |

S’ | 10^{−4} | 15.4 ± 1.6 | 0 |

DF | - | 171 | 172 |

RSS | m^{2} | 0.203 | 0.862 |

RSE | m | 0.034 | 0.071 |

AIC | - | −676 | −425 |

BIC | - | −660 | −412 |

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 author. 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**

De Smedt, F.
Determination of Aquitard Storage from Pumping Tests in Leaky Aquifers. *Water* **2023**, *15*, 3735.
https://doi.org/10.3390/w15213735

**AMA Style**

De Smedt F.
Determination of Aquitard Storage from Pumping Tests in Leaky Aquifers. *Water*. 2023; 15(21):3735.
https://doi.org/10.3390/w15213735

**Chicago/Turabian Style**

De Smedt, Florimond.
2023. "Determination of Aquitard Storage from Pumping Tests in Leaky Aquifers" *Water* 15, no. 21: 3735.
https://doi.org/10.3390/w15213735