Oscillatory pumping tests (OPTs) provide an alternative to constant-head and
constant-rate pumping tests for determining aquifer hydraulic parameters when
OPT data are analyzed based on an associated analytical model coupled with an
optimization approach. There are a large number of analytical models presented
for the analysis of the OPT. The combined effects of delayed gravity
drainage (DGD) and the initial condition regarding the hydraulic head are
commonly neglected in the existing models. This study aims to develop a new
model for describing the hydraulic head fluctuation induced by the OPT in an
unconfined aquifer. The model contains a groundwater flow equation with the
initial condition of a static water table, Neumann boundary condition specified
at the rim of a partially screened well, and a free surface equation
describing water table motion with the DGD effect. The solution is derived
using the Laplace, finite-integral, and Weber transforms. Sensitivity
analysis is carried out for exploring head response to the change in each
hydraulic parameter. Results suggest that the DGD reduces to instantaneous
gravity drainage in predicting transient head fluctuation when the dimensionless
parameter

Numerous attempts have been made by researchers to the study of the oscillatory pumping test (OPT) that is an alternative to constant-rate and constant-head pumping tests for determining aquifer hydraulic parameters (e.g., Le Vine et al., 2016; Christensen et al., 2017; Watlet et al., 2018). The concept of the OPT was first proposed by Kuo (1972) in petroleum literature. The process of the OPT contains extraction stages and injection stages. The pumping rate, in other words, varies periodically as a sinusoidal function of time. Compared with traditional constant-rate pumping, the OPT in contaminated aquifers has the following advantages: (1) it is low cost because it does not dispose contaminated water from the well, (2) it has a reduced risk of treating contaminated fluid, (3) it has smaller contaminant movement, and (4) it has a stable signal that is easily distinguished from background disturbance such as the tide effect and varying river stage (e.g., Spane and Mackley, 2011). However, the disadvantages of the OPT include the need of an advanced apparatus producing periodic rate. Oscillatory hydraulic tomography adopts several oscillatory pumping wells with different frequencies (e.g., Yeh and Liu, 2000; Cardiff et al., 2013; Zhou et al., 2016; Muthuwatta et al., 2017). Aquifer heterogeneity can be mapped by analyzing multiple data collected from observation wells. Cardiff and Barrash (2011) reviewed articles associated with hydraulic tomography and classified them according to nine categories in a table.

Various groups of researchers have worked with analytical and numerical models for the OPT; each group has its own model and investigation. For example, Black and Kipp (1981) assumed the response of confined flow to the OPT to be simple harmonic motion (SHM) in the absence of the initial condition. Cardiff and Barrash (2014) built an optimization formulation strategy using the Black and Kipp analytical solution. Dagan and Rabinovich (2014) also assumed hydraulic head fluctuation to be SHM for the OPT at a partially screened well in unconfined aquifers. Cardiff et al. (2013) characterized aquifer heterogeneity using the finite element-based COMSOL software that adopts SHM hydraulic head variation for the OPT. In contrast, Rasmussen et al. (2003) found that hydraulic head response tends toward SHM at a late period of pumping time when considering the initial condition prior to the OPT. Bakhos et al. (2014) used the Rasmussen et al. (2003) analytical solution to quantify the time after which hydraulic head fluctuation can be regarded as SHM since the OPT began. As mentioned above, most of the models for the OPT assume hydraulic head fluctuation to be SHM without the initial condition, and all of them treat the pumping well as a line source with infinitesimal radius.

Field applications of the OPT for determining aquifer parameters have been conducted in recent years. Rasmussen et al. (2003) estimated aquifer hydraulic parameters based on 1 or 2 h period of the OPT at the Savannah River site. Maineult et al. (2008) observed spontaneous potential temporal variation in aquifer diffusivity at a study site in Bochum, Germany. Fokker et al. (2012, 2013) presented spatial distributions of aquifer transmission and the storage coefficient derived from curve fitting based on a numerical model and field data from experiments at the southern city limits of Bochum, Germany. Rabinovich et al. (2015) estimated aquifer parameters of equivalent hydraulic conductivity, specific storage, and specific yield at the Boise Hydrogeophysical Research Site by curve fitting based on observation data and the Dagan and Rabinovich (2014) analytical solution. They conclude that the equivalent hydraulic parameters can represent the actual aquifer heterogeneity of the study site.

Although a large number of studies have been made in developing analytical models for the OPT, little is known about the combined effects of delayed gravity drainage (DGD), a finite-radius pumping well, and the initial condition prior to the OPT. An analytical solution to such a question will not only have important physical implications but will also shed light on OPT model development. This study builds an improved model describing hydraulic head fluctuation induced by the OPT in an unconfined aquifer. The model is composed of a typical flow equation with the initial condition of a static water table, an inner boundary condition specified at the rim of the partially screened well for incorporating the finite-radius effect, and a free surface equation describing the motion of the water table with the DGD effect. The analytical solution of the model is derived by the methods of the Laplace transform, finite-integral transform, and Weber transform. Based on the present solution, sensitivity analysis is performed to explore the hydraulic head in response to the change in each hydraulic parameter. The effects of DGD and instantaneous gravity drainage (IGD) on the head fluctuations are compared. The quantitative criterion for treating the well radius as infinitesimal is discussed. The effect of the initial condition on the phase of head fluctuation is investigated. In addition, curve fitting of the present solution to head fluctuation data recorded at the Savannah River site is presented.

Consider an OPT in an unconfined aquifer, illustrated in Fig. 1. The aquifer
is of unbound lateral extent with a finite thickness

Schematic diagram for oscillatory pumping test at a partially screened well of finite radius in an unconfined aquifer.

The flow equation describing spatiotemporal head distribution in aquifers
can be written as

The Laplace transform and finite-integral transform are applied to solve
Eqs. (8)–(13) (Latinopoulos, 1985; Liang et al., 2017, 2018). The
transient solution can then be expressed as

The eigenvalues

When

A pseudo-steady-state (PSS) solution

The Weber transform, defined in Eq. (B1) of the supporting material, may be
considered to be a Hankel transform with a more general kernel function. It can
be applied to diffusion-type problems with a radial-symmetric region from a
finite distance to infinity. For groundwater flow problems, it can be used
to develop the analytical solution for the flow equation with a Neumann
boundary condition specified at the rim of a finite-radius well (e.g., Lin
and Yeh, 2017; Povstenko, 2015). Taking the transform and the formula of

Applying the finite Fourier cosine transform to the model, Eqs. (18)–(22)
with

Table 1 classifies the present solution (i.e., Solution 1) and its special cases (i.e., Solutions 2 to 6) according to transient or PSS flow, unconfined or confined aquifer, and IGD or DGD. Each of the solutions (Solutions 1 to 6) reduce to a special case for a fully screened well. Existing analytical solutions can be regarded as special cases of the present solution, as discussed in Sect. 3.4 (e.g., Black and Kipp, 1981; Rasmussen et al., 2003; Dagan and Rabinovich, 2014).

The present solution and its special cases.

Solution 1 consists of Eqs. (14a)–(14k) with the roots of Eq. (15)
and

Sensitivity analysis evaluates hydraulic head variation in response to the
change in each of

The following sections demonstrate the response of the hydraulic head to
oscillatory pumping using the present solution. The default values in
calculation are

Influence of delayed gravity drainage on the dimensionless amplitude

Relative error (RE) of the dimensionless amplitudes

Previous analytical models for the OPT consider either confined flow (e.g.,
Rasmussen et al., 2003) or unconfined flow with IGD effect (e.g., Dagan and
Rabinovich, 2014). Little attention has been paid to the consideration of
the DGD effect. This section addresses the difference among these three
models. Figure 2 shows the curve of the dimensionless
amplitude

The normalized sensitivity coefficient

Existing analytical models for the OPT mostly treated the pumping well as a line
source with an infinitesimal radius (e.g., Rasmussen et al., 2003; Dagan and
Rabinovich, 2014). The finite difference scheme for the model also treats
the well as a nodal point by neglecting the radius. This will lead to
significant error when a well has a radius ranging from 0.5 to 2 m (Yeh
and Chang, 2013). This section discusses the relative error in predicted
amplitude, defined as

Head fluctuations at

The temporal distributions of the normalized sensitivity coefficient

Head fluctuations at

Figure 5 demonstrates head fluctuations predicted by DGD Solution 1 and IGD
Solution 2, expressed as

Comparison of field observation data with head fluctuations predicted by the present solution. Solutions 3 and 6 represent transient and PSS confined flows, respectively. PSS Solutions 4 and 5 stand for DGD and IGD conditions, respectively.

Figure 6 displays head fluctuations predicted by transient Solution 3
expressed as

Rasmussen et al. (2003) conducted field OPTs in a three-layered aquifer
system containing one surficial aquifer, the Barnwell–McBean aquifer in
the middle, and the deepest Gordon aquifer at the Savannah River site. Two clay
layers dividing these three aquifers may be regarded as impervious strata.
For the OPT at the surficial aquifer, the formation has an average thickness of 6.25 m near the test site. The fully screened pumping well has a 7.6 cm
outer radius. The pumping rate can be approximated as

The aquifer hydraulic parameters are determined based on Solutions 3 to 6
coupled with the Levenberg–Marquardt algorithm provided in the Mathematica
function FindFit (Wolfram, 1991). Note that a robust Gauss–Newton algorithm
(Qin et al., 2018a, b) or an automatic optimization method “SCE-UA”
(Duan et al., 1992; Wang et al., 2017) provides an alternative to the parameter estimation.
Solutions 4 and 5 are used to predict depth-averaged head, expressed as

Hydraulic parameters estimated by the present solution and the Rasmussen et al. (2003) solution for OPT data from the Savannah River site.

A variety of analytical models for the OPT have been proposed so far, but little
attention is paid to the joint effects of DGD, the initial condition, and the finite
radius of a pumping well. This study develops a new model for describing
hydraulic head fluctuation due to the OPT in unconfined aquifers. A static
hydraulic head prior to the OPT is regarded as an initial condition. A Neumann
boundary condition is specified at the rim of a finite-radius pumping well.
A free surface equation accounting for the DGD effect is considered to be the
top boundary condition. The solution of the model is derived from the Laplace
transform, finite-integral transform, and Weber transform. The sensitivity
analysis of the head response to the change in each of hydraulic parameters
is performed. The observation data obtained from the OPT at the Savannah
River site are analyzed by the present solution when coupling the
Levenberg–Marquardt algorithm to estimate aquifer hydraulic parameters. Our
findings are summarized as follows:

When

Assuming a large-diameter well as a line source with an infinitesimal radius
underestimates the amplitude of head fluctuation in the domain of

The sensitivity analysis suggests that the changes in four parameters,

Analytical solutions for the OPT are generally expressed as the sum of the
exponential and the harmonic functions of time (i.e.,

The data sets of these solutions in Figs. 2–7 are available upon request. The OPT data in Fig. 7 were provided by Todd C. Rasmussen.

The supplement related to this article is available online at:

CSH conceived the presented idea, developed the present solution and code for the model, and performed the results in the figures. YHT developed the code for the simulations of those existing solutions. HDY and TY supervised the findings of this work. All authors discussed the results and contributed to the final paper.

The authors declare that they have no conflict of interest.

Research leading up to this paper has been partially supported by the grants from the Fundamental Research Funds for the Central Universities (2018B00114), the National Natural Science Foundation of China (51809080, 51879068, and 41561134016), National Key Research and Development Program (2018YFC0407900), and the Taiwan Ministry of Science and Technology under the contract number MOST 107-2221-E-009-019-MY3. The authors are grateful to Todd C. Rasmussen for kindly providing the OPT data obtained from the Savannah River site. Edited by: Graham Fogg Reviewed by: Todd Rasmussen and two anonymous referees