The model of single-well push–pull (SWPP) test has been widely used to investigate reactive radial dispersion in remediation or parameter estimation of in situ aquifers. Previous analytical solutions only focused on a completely isolated aquifer for the SWPP test, excluding any influence of aquitards bounding the tested aquifer, and ignored the wellbore storage of the chaser and rest phases in the SWPP test. Such simplification might be questionable in field applications when test durations are relatively long because solute transport in or out of the bounding aquitards is inevitable due to molecular diffusion and cross-formational advective transport. Here, a new SWPP model is developed in an aquifer–aquitard system with wellbore storage, and the analytical solution in the Laplace domain is derived. Four phases of the test are included: the injection phase, the chaser phase, the rest phase and the extraction phase. As the permeability of the aquitard is much smaller than the permeability of the aquifer, the flow is assumed to be perpendicular to the aquitard; thus only vertical dispersive and advective transports are considered for the aquitard. The validity of this treatment is tested against results grounded in numerical simulations. The global sensitivity analysis indicates that the results of the SWPP test are largely sensitive (i.e., influenced by) to the parameters of porosity and radial dispersion of the aquifer, whereas the influence of the aquitard on results could not be ignored. In the injection phase, the larger radial dispersivity of the aquifer could result in the smaller values of breakthrough curves (BTCs), while there are greater BTC values in the chaser and rest phases. In the extraction phase, it could lead to the smaller peak values of BTCs. The new model of this study is a generalization of several previous studies, and it performs better than previous studies ignoring the aquitard effect and wellbore storage for interpreting data of the field SWPP test reported by Yang et al. (2014).

A single-well push–pull (SWPP) test could be applied for investigating aquifer properties related to reactive transport in the subsurface instead of the inter-well tracer test, due to its advantages of efficiency, low cost and easy implementation. The SWPP test is sometimes called the single-well injection–withdrawal test, single-well huff–puff test or single-well injection–backflow test (Jung and Pruess, 2012). A complete SWPP test includes the injection, the chaser, the rest and the extraction phase. The second and third phases are generally ignored in the analytical solutions but recommended in the field applications, since they could increase the reaction time for the injected chemicals with the porous media (Phanikumar and McGuire, 2010; Wang and Zhan, 2019).

Similar to other aquifer tests, the SWPP test is a forced-gradient groundwater tracer test, and analytical solutions are often preferred to determine the in situ aquifer properties, due to the computational efficiency. Currently, many analytical models are available for various scenarios of the SWPP tests (Gelhar and Collins, 1971; Huang et al., 2010; Chen et al., 2017; Schroth and Istok, 2005; Wang et al., 2018). However, these studies are based on a common underlying assumption, i.e., that the studied aquifer was isolated from adjacent aquitards. In other words, the aquitards bounding the aquifer are taken as two completely impermeable barriers for solute transport. To date, numerous studies have demonstrated that such an assumption might cause errors for groundwater flow (Zlotnik and Zhan, 2005; Hantush, 1967) and for reactive transport (Zhan et al., 2009; Chowdhury et al., 2017; Li et al., 2019). This is because even without any flow in the aquitards, molecular diffusion inevitably occurs when solute injected to the aquifer is close to the aquitard–aquifer interface. This is particularly true when a fully penetrating well is used for injection; thus a portion of injected solute is very close to the aquitard–aquifer interface and the SWPP test duration is relatively long, so the effect of molecular diffusion can materialize. Another important point to note is that the materials of the aquitard are usually clay and silt, which have strong absorbing capability for chemicals and great mass storage capacities (Chowdhury et al., 2017). To date, the influence of the aquitard on reactive transport in aquifers has attracted attention for several decades. As for radial dispersion, Chen (1985), Wang and Zhan (2013), and Zhou et al. (2017) presented analytical solutions for radial dispersion around an injection well in an aquifer–aquitard system. However, these models only focus on the first phase of the SWPP test (injection).

Another assumption included in many previous models of radial dispersion is that the wellbore storage is ignored for the solute transport. In the injection phase of the SWPP test, the wellbore storage refers to the mixing processes between the prepared tracer injected into the wellbore and the original (or native) water in the wellbore. As a result of the wellbore storage, the concentration inside the wellbore varies with time until reaching the same value as the injected concentration, as shown in Fig. 1a. When ignoring it, the concentration inside the wellbore is constant during the entire inject phase, which is certainly not true. Similarly, the wellbore storage in the chaser, rest and extraction phases refers to the concentration variation caused by mixing processes between the original solute in the wellbore and the tracer moving in or out the wellbore. The examples of ignoring wellbore storage include Gelhar and Collins (1971), Chen (1985, 1987), Moench (1989), Chen et al. (2007, 2012, 2017), Schroth et al. (2001), Tang and Babu (1979), Huang et al. (2010) and Zhou et al. (2017). Recently, Wang et al. (2018) developed a two-phase (injection and extraction) model for the SWPP test with specific considerations of the wellbore storage. In many field applications, the chaser and rest phases are generally involved and the mixing effect also happens in these two phases in the SWPP test, which will be investigated in this study.

The schematic diagram of the SWPP test.

Besides the abovementioned issues in previous studies, another issue is that the advection–dispersion equation (ADE) was used to govern the reactive transport of SWPP tests (Gelhar and Collins, 1971; Wang et al., 2018; Jung and Pruess, 2012). The validity of ADE has been challenged by numerous laboratory and field experimental studies before, when using a single representative value of advection, dispersion and reaction to characterize the whole system. In a hypothetical case, if great details of heterogeneity are known, one may employ a sufficiently fine mesh to discretize the porous media of concern and use ADE to capture anomalous transport characteristics fairly well (e.g., the early arrivals and/or heavy late-time tails of the breakthrough curves, BTCs). However, such a hypothetical case has rarely materialized in real applications, especially for field-scale problems. To remedy the situation (at least to some degree), the multi-rate mass transfer (MMT) model was proposed as an alternative to interpret the data of the SWPP test (Huang et al., 2010; Chen et al., 2017). In the MMT model, the porous media is divided into many overlapping continuums (Haggerty et al., 2000; Haggerty and Gorelick, 1995). A subset of MMT is the two overlapping continuums or the mobile–immobile model (MIM) in which the mass transfer between two domains (mobile and immobile) becomes a single parameter instead of a function. The MIM model can grasp most characteristics of MMT and is mathematically simpler than MMT. Besides the MMT model, the continuous time random walk (CTRW) model and the fractional advection–dispersion equation (FADE) model were also applied for anomalous reactive transport in SWPP tests (Hansen et al., 2017; Chen et al., 2017). Due to the complexity of the mathematic models of CTRW and FADE, it is very difficult or even impossible to derive analytical solutions for those two models, although both methods perform well in a numerical framework.

In this study, a new model of SWPP tests will be established by including both wellbore storage and the aquitard effect under the MIM framework. The reason for choosing MIM as the working framework is to capture the possible anomalous transport characteristics that cannot be described by ADE but at the same time to make the analytical treatment of the problem possible. Four stages of a SWPP test will be considered. The model of the wellbore storage will be developed using a mass balance principle in the chaser and rest phases. It does not seem difficult to solve this model of this study using numerical packages like MODFLOW-MT3DMS, TOUGH and TOUGHREACT, and FEFLOW. However, the numerical solutions may cause errors in treating the wellbore storage, since the volume of water in the wellbore was assumed to be constant (Wang et al., 2018), while in reality it changes with time and well discharge. Meanwhile, the numerical errors (like numerical dispersion and numerical oscillation) have to be considered in solving the ADE equation, especially for advection-dominated transport. In this study, an analytical solution will be derived to facilitate the data interpretation. Due to the format of analytical solutions, it is much easier to couple such solutions with a proper optimization algorithm (like a genetic algorithm). The analytical solution could serve as a benchmark to test the numerical solutions as well.

A single test well is assumed to fully penetrate an aquifer with uniform thickness. Both the aquifer and aquitards are homogeneous and extend laterally to infinity. Linear sorption and first-order degradation are included in the mathematic model of the SWPP test. Such assumptions might be oversimplified for cases in reality, while they are inevitable for the derivation of the analytical solution, especially for the aquifer homogeneity. For a heterogeneity aquifer, the solution presented here may be regarded as an ensemble-averaged approximation if the heterogeneity is spatially stationary. If the heterogeneity is spatially non-stationary, then one can apply non-stationary stochastic approach and/or Monte Carlo simulations to deal with the issue, which is out of the scope of this investigation.

The concept of homogeneity here deserves clarification. Despite the fact that the homogeneity assumption is commonly used in developing analytical and numerical models of subsurface flow and transport, one should be aware that a rigorous sense of homogeneity probably never exists in a real-world setting (unless the media are composed of idealized glass balls as in some laboratory experiments). Therefore, the homogeneity concept here should be envisaged as a media whose hydraulic parameters vary within relatively narrow ranges, or the so-called weak heterogeneity. The Borden site of Canada (Sudicky, 1988) is one example of weak aquifer heterogeneity. Wang et al. (2018) employed a stochastic modeling technique to test the assumption of homogeneity associated with the SWPP test and found that such an assumption could be used to approximate a heterogeneous aquifer when the variance of spatial hydraulic conductivity was small.

A cylindrical coordinate system is employed in this study, and the origin is
located at the well center, as shown in Fig. 1c. The

Considering advective effect, dispersive effect and first-order chemical
reaction in describing solute transport under the MIM framework, the
governing equations for the SWPP test are

where subscripts “u” and “l” refer to parameters in the upper and
lower aquitards, respectively; subscripts “m” and “im” refer to
parameters in the mobile and immobile domains, respectively;

The symbol of the advection term is positive in the extraction phase in
above equations, while it is negative before that. The dispersions are
assumed to be linearly changing with the flow velocity, and one has

Initial conditions are

Due to the concentration continuity at the aquifer–aquitard interface, one
has

where

The variation in the concentration with mixing effect in the injection phase
could be described by Wang et al. (2018):

As for the chaser phase, the models describing the concentration variation
in the wellbore could be obtained using mass balance principle:

In the extraction phase, the boundary condition is (Wang et al.,
2018)

The flow problem must be solved first before investigating the transport
problem of the SWPP test. The velocity involved in the advection and
dispersion terms of the governing Eqs. (1a) and (1b) is

Comparison of BTCs at the well screen computed by the solution of this study and Chen et al. (2017).

Comparison of the concentration distribution between the
analytical and numerical solutions along the

The water levels in the wellbore in Eqs. (12)–(14) could be calculated by
the models of Moench (1985):

In this study, the Laplace transform and Green's function methods will be
employed to derive the analytical solution of the new SWPP test models
described in Sect. 2. The dimensionless parameters are defined as follows:

As for the injection phase of the SWPP test, the solutions in the Laplace domain
are

In the chaser phase, the solutions of the SWPP test in the Laplace domain are

For the rest phase, the solutions of the SWPP test in the Laplace domain are

As for the extraction phase of the SWPP test, the solutions in the Laplace
domain are

Expressions of the coefficients in the solutions expressed in Eq. (25a)–(25f).

The vertical profiles (the

Expressions of the coefficients in the solutions expressed in Eq. (26a)–(26g).

Continued.

Expressions of the coefficients in the solutions expressed in Eqs. (28a)–(28g).

Continued.

Because the analytical solutions in the Laplace domain are too complex, it seems impossible to transform it into the real-time domain analytically. Alternatively, a numerical method will be introduced for the inverse Laplace transform. Currently, several methods are available, like the Stehfest model, Zakian model, Fourier series model, de Hoog model and Schapery model (Wang and Zhan, 2015). Here, the de Hoog method will be applied to conduct the inverse Laplace transform, since it performed well for radial-dispersion problems (Wang et al., 2018; Wang and Zhan, 2013).

The new SWPP test model is a generalization of several previous studies; for
instance, the new solution reduces to the solution of Gelhar and Collins
(1971) when

However, three assumptions still remain. First, the flow is in quasi-steady state; e.g., Eq. (15). Second, the groundwater flow is horizontal in the aquifer and is vertical in the aquitard. This treatment relies on the idea that the permeability of the aquitard is smaller than the permeability of the aquifer (Moench, 1985). Third, the model is simplified for the solute transport. For example, only vertical dispersion and advection effects are considered in the aquitard, and only radial dispersion and advection effects are considered in the aquifer. The validation of these assumptions will be discussed in the Sect. 4.2.

In this section, the newly derived analytical solutions will be tested from two aspects. Firstly, the new solution of this study could reduce to previous solutions under special cases, as the model established in this study is an extension of previous ones, and comparisons between them will be shown in Sect. 4.1. Secondly, although some assumptions included in previous models have been relaxed in the new model, some other processes of the reactive transport in the SWPP test have to be simplified in analytical solutions. Assumptions included in the new model have been discussed and their applicability is elaborated in Sect. 4.2.

To test the new solutions, the model of Chen et al. (2017) serves as a
benchmark; they ignored the aquitard effect and wellbore storage in the SWPP
test. Figure 2 shows the comparison of BTCs between them, and the parameters
used in such a comparison are

To test three assumptions outlined in Sect. 3.3, a numerical model will be established, where general three-dimensional transient flow and solute transport are considered in both aquifer and aquitards. A finite-element method with the help of COMSOL Multiphysics will be used to solve the three-dimensional model. The grid system is shown in Sect. S2.

In this study, four sets of aquitard hydraulic conductivities are employed, i.e.,

The initial drawdown and the initial concentration are 0 for aquifer and
aquitards. The hydraulic parameters are

As the first assumption in Sect. 3.3 has been elaborated on in Sect. S2.2,
the following discussion will only focus on the second and third
assumptions. Figure 3a, b and c represent the snapshots of
concentration distributions in the aquifer along the

Figure 4 shows the comparison of the analytical and numerical solutions for aquitards. Figure 4a1–c1 represent the snapshots of concentration distributions obtained from the analytical solution of this study at different times, and Fig. 4a2–c2 represent the snapshots of concentration distributions obtained from the numerical solution. One may find that the contour maps obtained from both solutions are almost the same in the aquifer, but very different in the aquitards. Therefore, the abovementioned third assumption in Sect. 3.3 is generally unacceptable in describing solute transport in the aquitard in the SWPP test but works well when the aquifer is of primary concern.

As mentioned in Sect.3.3, the new model is a generalization of many
previous models, and the conceptual model is closer to reality. However,
there are many parameters involved in this new model that have to be
determined first for applying this model. For instance, the involved
parameters for the aquitards include dispersivity (

Parameter estimation is an inverse problem, and it is generally conducted by an optimization model, such as a genetic algorithm and simulated annealing. Due to the ill-posedness of many inverse problems or insufficient observation data, the initial guess values of unknown parameters of interest are critical for finding the best values or real values of those parameters in the optimization model. Here, we recommend using values of parameters from the literature as the initial guesses for similar lithology. Table 4 lists some parameter values for sandy and clay aquifers in previous studies. When a result is not sensitive to a particular parameter of concern, the value from previous publications for similar lithology and/or situations could be taken as an estimated value of that parameter if there is no direct measurement of that particular parameter of concern. To prioritize the sensitivity of predictions with respect to the diverse parameters involved in the new model, a global sensitivity analysis is conducted in Sect. 5.2.

A partial list of parameters from the literature.

From the analytical solutions of Eqs. (26)–(28), one may find that BTCs
are affected by several parameters, like

A larger

Sensitivity analysis.

As shown in Sect. 4.2, the new analytical solution is a good approximation
for the numerical model in the aquifer when

Figure 6 shows the difference between the models with and without aquitards
for different flow velocities in the aquitard. The case of

Comparison of BTCs between the model with and without aquitards for different porosities.

Another important parameter is the radial dispersion in the aquifer. In this
section, three sets of the radial-dispersivity values will be used to
analyze its influence:

Figure 7 shows BTCs in the well face for different radial-dispersivity
values. Firstly, the difference is obvious among curves in all phases.
Secondly, a larger

BTCs in the wellbore for different

To test the performance of the new model, the field data reported in Chen et al. (2017) will be employed. Specifically, the experimental data conducted in the borehole TW3 will be analyzed. The reason for choosing
this dataset is that this borehole penetrated several layers, and it had
been interpreted by Chen et al. (2017) before (using a model that does not consider the aquitard effect and the wellbore storage). The physical
parameters of the SWPP test are

Figure 8a shows the fitness of the computed and observed BTCs. The
estimated parameters are

How accurate these parameters estimated by best fitting the observed data
are in representing the real aquifer will be discussed in the following.
The values of the retardation factor and reaction rate demonstrate that the
chemical reaction and sorption are weak for the tracer of KBr in the SWPP
test. It is not surprising since KBr is commonly treated as a
“conservative” tracer. The porosity of the real aquifer ranges from 0.01
to 0.1, according to the well log analysis (Yang et al., 2014), where the
estimated values are located. The estimated porosity represents the average
values of the aquifer and aquitards. The estimated dispersivity of the
aquifer is 0.7134 m by Chen et al. (2017), which is similar to ours. The
values of the water level in the test could be observed directly; however, these
data are not available, and they have to be estimated in this study. To
evaluate the uncertainty in the estimated parameters, the sensitivity of the
dispersivity to BTCs is analyzed, as shown in Fig. 8b. One may conclude
that the estimated values of this study seem to be representative of reality, since the error is smallest for

Fitness of observed BTC.

The single-well push–pull (SWPP) test could be applied to estimate the dispersivity, porosity and chemical reaction rates of in situ aquifers. However, previous studies mainly focused on an isolated aquifer, excluding all the possible effects of aquitards bounding the aquifer. In other words, the adjacent layers are assumed to be non-permeable, which is not exactly true in reality. In this study, a new analytical model is established and its associate solutions are derived to inspect the effect of overlying and underlying aquitards. Meanwhile, four stages are considered in the new model with wellbore storage: the injection phase, the chaser phase, the rest phase and the extraction phase. The anomalous behaviors of reactive transport in the test were described by a mobile–immobile framework.

To derive the analytical solution of the new model, some assumptions are
inevitable. For instance, only vertical advection and dispersion are
considered in the aquitard and only horizontal advection and dispersion are
considered in the aquifer, and the flow is quasi-steady state. Although
these assumptions have been widely used to describe the radial dispersion in
previous studies, the influences on reactive transport have not been
discussed in a rigorous sense before. In this study, numerical modeling
exercises are introduced to test the abovementioned assumptions of the
new model. Based on this study, several conclusions could be obtained.

A new model of the SWPP test is a generalization of many previous models by considering the aquitard effect, the wellbore storage and the mass transfer rate in both aquifer and aquitards. A sub-model of wellbore storage is developed.

The assumption of vertical advection and dispersion in the aquitard and horizontal advection and dispersion in the aquifer are tested by specially designed finite-element numerical models using COMSOL, and the result shows that this assumption is acceptable when the aquifer is of primary concern, provided that the ratios of the aquitard or aquifer permeability are less than 0.01, while such an assumption is generally unacceptable when the aquitards are of concern, regardless of the ratios of the aquitard or aquifer permeability.

The new model is more sensitive to

The performance of the new model is better than previous models that exclude the aquitard effect and the wellbore storage in terms of best fitting exercises with field data reported in Chen et al. (2017).

All data are available in the Supplement.

The supplement related to this article is available online at:

QW wrote the paper and developed the mathematic models; JW and WS derived the analytical solutions; HZ revised the paper.

The authors declare that they have no conflict of interest.

We thank the editor, associate editor and two anonymous reviewers for their constructive comments, which helped improve the quality of the paper.

The authors declare that they have no conflict of interest.

This research was partially supported by programs of the Natural Science Foundation of China (grant nos. 41772252, 41972250 and 41502229), Innovative Research Groups of the National Nature Science Foundation of China (grant no. 41521001), the Natural Science Foundation of Hubei Province, China (grant no. 2018CFA028), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (grant no. CUGGC07) and China Geological Survey (grant nos. DD20190263, 2019040022).

This paper was edited by Philippe Ackerer and reviewed by two anonymous referees.