Determination of vertical hydraulic conductivity of aquitards in a multilayered leaky system using water-level signals in adjacent aquifers

Highlights

• We apply harmonic analysis approach in a multilayered leaky system.

• We apply coherence analysis to identify leakage-induced water-level changes.

• K_v of aquitard is calculated using water-level fluctuations in the aquifers.

• Correlation between frequencies and phases support robust estimate of K_v.

Summary

This paper presents a methodology for determining the vertical hydraulic conductivity (K_v) of an aquitard, in a multilayered leaky system, based on the harmonic analysis of arbitrary water-level fluctuations in aquifers. As a result, K_v of the aquitard is expressed as a function of the phase-shift of water-level signals measured in the two adjacent aquifers. Based on this expression, we propose a robust method to calculate K_v by employing linear regression analysis of logarithm transformed frequencies and phases. The frequencies, where the K_v is calculated, are identified by coherence analysis. The proposed methods are validated by a synthetic case study and are then applied to the Westbourne and Birkhead aquitards, which form part of a five-layered leaky system in the Eromanga Basin, Australia.

Introduction

Determination of the vertical hydraulic conductivity (K_v) of aquitards is an important task for understanding hydraulic connection of an aquifer–aquitard systems (Eaton and Bradbury, 2003) and protecting groundwater from contamination (Hart et al., 2005, Remenda and van der Kamp, 1997). The K_v of an aquitard can be measured with laboratory tests (e.g. Arns et al., 2001, Timms and Hendry, 2008). However, these results may be several orders of magnitude different to the K_v required in the real-world study, because aquitards are generally heterogeneous and rock structures are disrupted during the sampling (Clauser, 1992, Schulze-Makuch et al., 1999). In contrast, in situ approaches are generally preferred as they can yield directly field-related values.

Commonly used in situ methods include pumping tests and slug tests (van der Kamp, 2001). During these tests, drawdowns are measured and plotted against elapsed time to produce an experimental curve. Hydraulic parameters of the aquifer and aquitard can be estimated by matching the experimental curve with a theoretical model. The theory supporting the analysis of K_v of the aquitard in a leaky aquifer system was developed by Hantush and Jacob (1955) and Hantush (1960). Neuman and Witherspoon, 1969a, Neuman and Witherspoon, 1969b improved the Hantush–Jacob solution by considering the storage ability of the aquitard and water-level responses in the unpumped aquifer. However, in a two-aquifer-one-aquitard leaky system, the drawdown in each aquifer depends on five dimensionless hydraulic parameters. In order to establish theoretical curves to cover the entire range of values necessary for the analysis of K_v, the ratio method is used (Neuman and Witherspoon, 1972, Wolff, 1970).

The ratio method, however, required drawdowns either increase or decrease regularly relating to the determined extraction/injection stresses. The current interest is to estimate the K_v of an aquitard based on arbitrary water-level fluctuations, which are caused by multiple underdetermined stresses. The deconvolution method was applied to such situation because water-level fluctuations induced by leakage via the aquitard follow the convolution relation (Neuman and Witherspoon, 1968):

$$s_2(t)=h(t)\otimes s_1(t)={\int }_0^t{s}_1(\tau )h(t-\tau )d\tau \text{,}$$

where s₁(t) and s₂(t) represent water-level fluctuations measured at different depths in one aquitard, and h(t) is a loss function expressed by means of Duhamel’s function (Neuman and Gardner, 1989, Neuman and Witherspoon, 1968).

The deconvolution approach proposed by Neuman and Gardner (1989) was carried out by minimizing differences between measured and theoretical drawdown. Those differences were a function of hydraulic diffusivity and background water-level fluctuations in the aquifer.

An alternative deconvolution method is based on the Fourier transform, and referred to as harmonic analysis method (Boldt-Leppin and Hendry, 2003). In this method, water-level fluctuations, measured at different depths in the aquitard, are decomposed into a sum of trigonometric components of different frequencies. These trigonometric components are defined as harmonic signals. The hydraulic diffusivity is expressed analytically either based on the amplitude or phase shift of harmonic signals. However, the harmonic analysis approach, by now, assumes that the thickness of the aquitard is half infinite, which limits its application.

In this study, we apply the harmonic analysis method in a multilayered leaky system where the thickness of the aquitard is finite and both the top and bottom of the aquitard is bounded by aquifers. The aim is to calculate K_v of the aquitard based on a pair of water-level signals measured in the two adjacent aquifers. The water-level fluctuations in the aquifers may be induced by many factors (e.g. pumping, recharge, leakage or earthquake). However, only leakage-induced water-level fluctuations can represent the properties of the aquitard and so can be used to infer the K_v of the aquitard. Therefore, it is desirable to find a method to identify the leakage-induced water-level fluctuations in the aquifers. Coherence analysis is proposed for this purpose.

Coherence was originally defined and used in signal processing, which analyses the cross correlation between two signals in the frequency domain (Carter, 1987). It was used in hydrogeology to understand the hydraulic connection in karstic aquifer systems (Larocque et al., 1998, Padilla and Pulido-Bosch, 1995). The coherence varies from 0 to 1.0 depending on the degree to which the convolution relationship in Eq. (1) is satisfied. In this study, its value is determined by the degree to which leakage-induced water-level signals are interrupted by other factors. A weak interruption corresponds to a large coherence value.

In this study, we first derive analytical expression for K_v by using the harmonic analysis method in the three-layered leaky system. Following this, the method to calculate phases and the definition of coherence are introduced briefly and a robust method to estimate K_v is proposed. As support, the methods are validated in a simulated case and are applied to the eastern Eromanga Basin, Queensland, Australia.