Using the single-well push–pull (SWPP) test to determine the in situ biogeochemical reaction kinetics, a chase phase and a rest phase were recommended to increase the duration of reaction, besides the injection and extraction phases. In this study, we presented multi-species reactive models of the four-phase SWPP test considering the wellbore storages for both groundwater flow and solute transport and a finite aquifer hydraulic diffusivity, which were ignored in previous studies. The models of the wellbore storage for solute transport were proposed based on the mass balance, and the sensitivity analysis and uniqueness analysis were employed to investigate the assumptions used in previous studies on the parameter estimation. The results showed that ignoring it might produce great errors in the SWPP test. In the injection and chase phases, the influence of the wellbore storage increased with the decreasing aquifer hydraulic diffusivity. The peak values of the breakthrough curves (BTCs) increased with the increasing aquifer hydraulic diffusivity in the extraction phase, and the arrival time of the peak value became shorter with a greater aquifer hydraulic diffusivity. Meanwhile, the Robin condition performed well at the rest phase only when the chase concentration was zero and the solute in the injection phase was completely flushed out of the borehole into the aquifer. The Danckwerts condition was better than the Robin condition even when the chase concentration was not zero. The reaction parameters could be determined by directly best fitting the observed data when the nonlinear reactions were described by piece-wise linear functions, while such an approach might not work if one attempted to use nonlinear functions to describe such nonlinear reactions. The field application demonstrated that the new model of this study performed well in interpreting BTCs of a SWPP test.

A single-well push–pull (SWPP) test is a popular technique to characterize the in situ geological formations and to remedy the polluted aquifer by a series of biogeochemical reactions (Istok, 2012; Phanikumar and McGuire, 2010; Schroth and Istok, 2006). Therefore, the accuracy of the results is not only dependent on the experimental operation, but also on the conceptual model which is expected to properly represent the physical and biogeochemical processes. Unfortunately, most previous studies of the multi-species reactive transport models were based on some assumptions which may not be satisfied in actual applications, although those assumptions usually simplified the mathematical treatment of the problem (Istok, 2012; Wang et al., 2017).

As for the analytical solutions of the SWPP test, they have been widely used for applications, due to the high efficiency and great accuracy of the solutions, like the model of Gelhar and Collins (1971) for a fully penetrating well, the model of Schroth and Istok (2005) for a point source/sink well, and the model of Huang et al. (2010) for a partially penetrating well, assuming that the advection, the dispersion and the first-order reaction were involved in the transport processes. Haggerty et al. (1998) and Snodgrass and Kitanidis (1998) presented a simplified method based on a well-mixed reactor to estimate the first-order and zero-order reaction rate, without involving complex numerical modeling. Schroth and Istok (2006) provided two alternative models: one of them was a plug-flow model and the other was a variably mixed reactor model. Schroth et al. (2000) presented a simplified method for estimating retardation factors, based on the model of Gelhar and Collins (1971). Istok et al. (2001) extended the models of Haggerty et al. (1998) and Snodgrass and Kitanidis (1998) to estimate the Michaelis–Menten kinetic parameters which were used to describe the microbial respiration in the aquifer. Jung and Pruess (2012) presented a closed-form analytical solution for heat transport in a fractured aquifer involving a push-and-pull procedure. However, the above-mentioned analytical or semi-analytical solutions of the SWPP test were based on some over-simplified assumptions. For instance, the hydraulic diffusivity of the aquifer was assumed to be infinite, resulting in a time-independent flow velocity, where the hydraulic diffusivity is the ratio of the radial hydraulic conductivity over the specific storage. The wellbore storage effect on the flow field was assumed to be negligible as well. Therefore, how accurate parameter estimation could be needs to be tested. Recently, Wang et al. (2017) investigated the influences of a finite hydraulic diffusivity on the results and found that it might be significant, since both advective and dispersive transport were related to the flow velocity. One point to note is that the model of Wang et al. (2017) still contains an additional issue that has not been addressed: the wellbore storage influence on solute transport, which will be the focal point of this investigation.

The wellbore storage for solute transport refers to the variation of the
solute injected into the wellbore during the processes of the test. A
complete SWPP test contains four principle phases: injection of a prepared
solution (tracer) into a targeted aquifer; injection of a chaser; rest
period; extraction of the mixture solution. The second and third phases are
optional, but are recommended to extend the reaction time of the tracer in
the aquifer. In the injection phase, the concentration of the solute in the
wellbore is smaller than that of the original solution at the early stage,
since the original solute could be diluted by the original water in the
wellbore, due to the mixing effect. Therefore, excluding the wellbore storage
may overestimate the concentration in the wellbore at the early stage of the
injection phase before the pre-test water inside the wellbore is completely
flushed out of the borehole into the aquifer. In the chaser phase, the
concentration of the solute in the wellbore may be greater than the
concentration of the chaser, due to the mixing effect. The treatment of
excluding the wellbore storage could underestimate the concentration in the
wellbore at the early stage of the chase phase, due to the high concentration
of solutes in the wellbore at the end of the injection phase. When the chaser
phase is absent or the chaser concentration is not zero, the concentration
might not be zero in the early stage of the rest phase. As for the chaser
concentration, it is usually set to zero. However, under some circumstances,
investigators may use a non-zero concentration for the chase phase. For
example, Phanikumar and McGuire (2010) used 10 mg L

Actually, the above-mentioned assumptions used in the analytical and semi-analytical solutions can be relaxed in the numerical models, such as MODFLOW/MT3DMS (Harbaugh et al., 2000; Zheng and Wang, 1999), FEFLOW (Diersch, 2014), SUTRA (Voss, 1984), and STOMP (Nichols et al., 1997). Huang et al. (2010), Sun (2016), Haggerty et al. (1998), and Schroth and Istok (2006), respectively, employed such four software packages to carry out numerical simulations of SWPP tests, mainly involving advection, dispersion and first-order reaction. Unfortunately, the traditional three-dimensional models in the Cartesian coordinate system may create some errors in describing the wellbore storage of solute transport in the wellbore-confined aquifer, which is explained in the Supplement.

This study addresses multi-species reactive transport associated with SWPP tests with a better conceptual model that acknowledges the realistic circumstances that have been either overlooked or overly simplified in previous investigations. Firstly, we will employ a more realistic finite hydraulic diffusivity instead of an infinite hydraulic diffusivity to describe the flow field. Secondly, we will propose a better way to handle the boundary condition of transport at the wellbore by considering the wellbore storage effect for both groundwater and solute transport during the SWPP tests. Thirdly, the new model is tested using a field test dataset reported in McGuire et al. (2002). Fourthly, the sensitivity analysis and uniqueness analysis will be employed to investigate the assumptions used in previous studies on the parameter estimation.

A cylindrical coordinate system is adopted with the

The schematic diagram of the SWPP test at the beginning of the rest phase when the chase concentration is not 0.

The general form of the governing equation for a multi-species reactive SWPP
test is

As mentioned in the Introduction, several assumptions may be debatable in
previous studies and could be the source of errors for the actual
applications. Firstly, the transport model is composed of a set of
advection–dispersion equations (ADEs) built on the basis of flow velocity
which is assumed to be time-independent (Chen et al., 2017; Gelhar and
Collins, 1971; Huang et al., 2010; Phanikumar and McGuire, 2010):

The second assumption of the model is the boundary condition of the well
screen in the rest phase of the SWPP test, in which a Robin condition (or a
third-type condition) is employed to describe the aqueous solute transport
(Chen et al., 2017; Phanikumar and McGuire, 2010; Wang et al., 2017):

Thirdly, a constant solute concentration in the wellbore is applied in the
injection and chase phases without considering the solute diluted effect in
the wellbore (Chen et al., 2017; Gelhar and Collins, 1971; Istok, 2012;
Phanikumar and McGuire, 2010; Wang et al., 2017):

Fourthly, the solute transport caused by dispersion and advection was
assumed to be negligible in estimating the reaction rates. For instance, one
of the simplest models of such reactions may be the first-order reaction

Actually, the assumptions of Eqs. (2)–(9) are debatable for the actual applications, and may cause errors in modeling the solute transport in the SWPP test. The second and third assumptions relate to the wellbore storage of the solute transport in the SWPP test. In the following section, the new models will be proposed to investigate the potential errors when these assumptions are involved.

As for the first assumption in Sect. 2.1, Wang et al. (2017) demonstrated
that it might result in non-negligible errors in parameter estimation,
particularly for the estimation of dispersivity. A minor point to note is
that the model of Wang et al. (2017) mainly focused on conservative solute
transport, rather than reactive transport. Nevertheless, the pore velocity of
transient flow is calculated by Darcy's law:

As for the second assumption in the rest phase, as shown in Eq. (3), it
implies that the concentration of the solute is zero in the wellbore. This
assumption works when the chase concentration is zero and the prepared
solution is completely pushed out of the borehole into the aquifer at the end
of the chase phase. However, the chase concentration might be non-zero, as
demonstrated in Phanikumar and McGuire (2010) and McGuire et al. (2002).
Consequently, the concentration in the early stage of the rest phase, which
is close to the concentration at the end of the chase phase, is not zero.
This is because the water level in the wellbore is greater than the hydraulic
head in the surrounding aquifer due to the wellbore storage, resulting in a
positive flux from the wellbore into the aquifer. Correspondingly, when the
chase concentration is not zero or the prepared solution in the injection
phase is not completely pushed out of the wellbore, the concentration in the
wellbore may not be zero in the early stage of the rest phase. In this study,
we employed the Danckwerts condition for transport at the well screen in the
rest period (Danckwerts, 1953):

The third assumption mentioned in Sect. 2.1 seems not reasonable at the early
stage of the injection and chase phases, because the concentration of the
injected solute will be affected by the finite volume of water in the
wellbore. Take the chase phase as an example: it is impossible to immediately
reduce the solute concentration inside the wellbore from a certain level
during the tracer injection phase to zero when switching to the chase phase,
even when the solute concentration in the chase phase is zero. This is
because the wellbore with a finite radius contains a certain finite mass of
solute at the moment of switching from injection of a tracer to injection of
a chaser. Therefore, it will take some time to completely flush out the
residual tracer inside the wellbore after the start of the chase phase, and a
larger wellbore will take a longer time to flush out the residual tracer
inside the wellbore. This means that the concentration at the
wellbore–aquifer interface will not drop to zero immediately after the start
of the chase phase. Instead, it will take a finite period of time to
gradually approach zero during the chase phase. Similarly, the boundary
condition of the well screen in the injection phase might not be appropriate
in previous studies if the wellbore storage effect is of concern. Therefore,
the value of a solute concentration inside the wellbore should be smaller
than or equal to

Here, we will develop a new approach to take care of the concentration in
the wellbore in the injection and chase phases based on the mass balance
principle, i.e.,

In the chase phase, one has

Different from the model of Wang et al. (2017), the multi-species reactive transport models are used to describe the nonlinear biogeochemical reactive processes considering wellbore effects not only for groundwater flow, but also for solute concentrations. The new model of this study is an extension of Phanikumar and McGuire (2010) that ignored the wellbore storage for both groundwater flow and solute transport, and assumed that the aquifer hydraulic diffusivity was infinite. The Danckwerts condition rather than the Robin condition is applied at the well screen in the rest phase of this study. Therefore, the new model is more powerful in describing an arbitrary number of species and user-defined reaction rate expressions, including Monod/Michaelis–Menten kinetics.

In this study, we will use a finite-difference method to solve the model of the SWPP test, where the finite-difference scheme of the groundwater flow is the same as Wang et al. (2017), and the scheme of the transport governing equation (ADE) is similar to the model of Phanikumar and McGuire (2010). However, the flow velocity used in the advective term of ADE is computed by solving the model of groundwater flow rather than directly using Eq. (2), which was employed by Phanikumar and McGuire (2010).

To minimize numerical errors and to increase computational efficiency, we
employ a non-uniform grid system for simulations (Wang et al., 2014), which
is

Similarly, we logarithmically discretize the temporal domain:

As for the chase, one has

Comparison of BTCs between the solutions of Wang et al. (2017) and
of this study, where

By comparing the solution of this study with Wang et al. (2017), one may
conclude that the solution of this study appears to be accurate and reliable
since the mean square error between two solutions is smaller than 0.05 for
all cases in Fig. 2. In the wellbore (

It is also necessary to test the accuracy of the new models against the numerical software packages. Since the code of the original MODFLOW/MT3DMS package is open source and could be downloaded freely from the website of the United States Geological Survey, it is preferred by many modelers and is selected as the basis of comparison in this study. Unfortunately, such an open-source MODFLOW/MT3DMS package may create some errors in describing the solute transport in the wellbore-confined aquifer. The errors come from an assumption that the water volume in the wellbore is computed by a product of the wellbore cross section and the aquifer thickness, which is incorrect. The actual water volume in the wellbore should be computed by a product of the wellbore cross section and the water level in the wellbore (see the Supplement for a detailed explanation). Figure 3 shows the comparisons of BTCs between the open-source MODFLOW/MT3DMS package and the new model of this study. The water level of the wellbore is assumed to be equal to the aquifer thickness in the new models for the purpose of comparison, although it may not be true, and the other parameters used are the same as the ones in Fig. 2. Therefore, the agreement between the two models demonstrates the accuracy of the new model. Figure 3 shows that the concentration in the wellbore is not unit in the injection phase, and this is because the new model considers the wellbore storage for both groundwater flow and solute transport. It is worthwhile pointing out that an advanced version of MODFLOW/MT3DMS, namely MODFLOW-SURFACT, includes a fracture-well package (FWL4 and FWL5) to overcome the problems in the original open-source MODFLOW well package. The FWL4 and FWL5 packages calculate the water volume using simulated heads, not aquifer thicknesses (see the MODFLOW-SURFACT manual, Vol I, Sect. 3.2, Eq. 24 for details). FEFLOW also has a similar package, referred to as a discrete feature to simulate a pumping/extraction well, if one chooses to do so. Additionally, with a FEFLOW model, the model mesh can be highly discretized to accurately represent well dimensions using a subset of elements (in centers). The modeler can assign a porosity of the unit for those elements representing the wells, rather than assuming the same porosity of the surrounding materials. In the future, we will conduct a comprehensive comparative investigation of the method proposed in this study and those of MODFLOW-SURFACT and FEFLOW for understanding the effects of well mixing and wellbore storage for both flow and transport processes involving an aquifer–well system.

Comparison of BTCs between the solutions of MODFLOW/MT3DMS and of this study.

Revisiting the assumptions used in previous studies as mentioned in Sects. 2.1 and 2.2, one may find that the flow field and the wellbore storage are key factors for the SWPP test. This is not surprising, since the flow velocity is included not only in the advective term, but also in the dispersive term. The wellbore storage which is dependent on the volume of pre-test water in the wellbore may influence the concentration of the solute injected into the wellbore. As the influence of the hydraulic diffusivity solute transport in the SWPP test has been investigated in Wang et al. (2017), in this section, we mainly investigate the influence the wellbore storage on the reactive transport in the SWPP test in the transient flow field.

The variation of the transient flow field is mainly controlled by the
hydraulic diffusivity of the aquifer and the wellbore storage. In the
following discussion, we choose three representative types of porous media to
test the influence of the hydraulic diffusivity on the results of the SWPP
test, including fine sand, medium sand, and coarse sand. According to
Domenico and Schwartz (1990) and Batu (1998), one could obtain the values of
the hydraulic diffusivity for the above-mentioned three types of media:

The parameters related to the solute transport mainly come from the studies
of Phanikumar and McGuire (2010), who interpreted the field
experimental data of the SWPP test conducted by McGuire et al. (2002). Except for parameters specifically mentioned otherwise,
the default values used in the following section are

Figure 4a and b show the comparison of BTCs between the Robin and
Danckwerts conditions at the wellbore for different porous media, where

Figure 4a shows that the difference of BTCs between two boundary conditions
is significant at the early stage of the extraction phase when

Comparison of BTCs in the wellbore between the Robin and Danckwerts
conditions:

Figure 4b shows the comparison of BTCs for different boundary conditions in
the wellbore when

Figure 5 shows the comparison of BTCs in the wellbore for different boundary conditions and different porous media. The parameters used in this case are the same as the ones in Sect. 4.1. The initial head is 1 m. The boundary condition of the wellbore in the rest phase is described by the Danckwerts condition.

Two interesting observations can be seen from this figure. Firstly, the difference of BTCs between the two boundary conditions at the wellbore is obvious, and such a difference is larger for the medium sand than for the coarse sand, implying that it increases with the decreasing hydraulic diffusivity. Secondly, the values of BTCs obtained from Eqs. (16) to (17) are greater at the early stage of the extraction phase, while the peak values of BTC are smaller. In other words, the model of Eqs. (5)–(6) may underestimate the concentration in the early stage of the extraction phase while overestimating the peak values of BTCs.

These observations can be explained as follows. The model of Eqs. (5)–(6)
assumes that the volume of water in the wellbore is negligible, and the
concentration in the wellbore is close to 10.0 mg L

Physical and chemical parameters are important in predicting the contaminant
transport in the aquifer, and the values of these parameters are generally
estimated by best fitting the observed BTCs in the SWPP test using a
simplified model, ignoring a number of relevant factors such as the
influences of the flow field and the wellbore storage. The discussions in
Sect. 4 demonstrate that the negligence of such factors in reactive transport
might cause errors and invalidate the whole parameter estimation exercise.
Besides porosity, dispersivity, and reaction rates, the new model of this
study appears to be useful for estimating the values of hydraulic
conductivity and specific storage by best fitting the observed BTCs in the
SWPP test. For instance, the values could be determined by minimizing the sum
of absolute differences between the observed and calculated BTCs in the
wellbore:

The BTCs in the wellbore for the different boundary conditions at the wellbore in the injection and chase phases.

Although the number of observation points is usually much greater than the number of parameters needed to be estimated, one may still wonder whether Eq. (30) is practically reliable for estimation of multiple parameters simultaneously. To answer this question, two approaches are employed in the following: sensitivity analysis and uniqueness analysis. The sensitivity analysis is used to check whether the solution is sensitive to the parameters or not, while the uniqueness analysis is to check whether the multiple input parameter values could map to the same output results.

McCuen (1985) proposed a sensitivity model of a dependent variable, which was
normalized as (Kabala, 2001; Yang and Yeh, 2009)

From the mathematical models of the groundwater flow, one may find that both
hydraulic conductivity and specific storage could affect the flow field.
Since greater hydraulic conductivity or smaller specific storage could
shorten the time in approaching the steady state, we will employ the
hydraulic diffusivity for the sensitivity analysis, which is the ratio of the
two parameters. Figure 6 shows the sensitivity of the hydraulic diffusivity
on BTCs, and one may find that it is not sensitive to the hydraulic
diffusivity when the values of hydraulic diffusivity are sufficiently large.
This might be because the time in approaching the steady state is very short
when the hydraulic diffusivity values are sufficiently large (for instance,
greater than

Sensitivity analysis of the hydraulic diffusivity on BTCs in the extraction phase.

Besides the sensitivity analysis, the uniqueness analysis is also important for the parameter estimation, which is used to check whether there exist two or more sets of parameters for the same BTCs. Similar to the treatment in previous studies, we firstly use the transient model of this study to reproduce BTCs based on a set of given input parameters, and then estimate the values of parameters by best fitting such BTCs. If the values of the input parameters are different from the estimated parameter when the fitness is very good, one could conclude that the solution is not unique and the parameters estimated from Eq. (30) may not be reliable.

There are four physical parameters in the new model of this study, i.e., hydraulic conductivity, specific storage, dispersivity, and porosity, and one chemical parameter (reaction rate). Wang et al. (2017) investigated the uniqueness of solutions for the flow field, and the results showed that BTCs of the SWPP test were not unique for the flow-related parameters. For instance, BTCs with a steady-state flow field were almost the same as BTCs with a transient flow field, as shown in Figs. 10 and 11 of Wang et al. (2017). It implies that one may not inversely determine the hydraulic parameters of a flow field only by best fitting observed BTCs in the wellbore, and additional aquifer tests are required to supplement the SWPP test to determine the flow-related parameters. However, Wang et al. (2017) did not investigate the uniqueness of porosity and dispersion when the hydraulic parameters were given, which will be discussed in this study.

Sensitivity analysis of dispersivity and porosity on BTCs in the extraction phase.

Figure 8 shows comparison of BTCs for different dispersivities and porosities
but for the same hydraulic parameters, and one could see that the curves of

In summary, it seems impossible to determine all parameters (

Reaction parameters estimated by linear functions.

The models estimating the reaction rate are based on several assumptions in
previous studies, e.g., Eqs. (8)–(9) as demonstrated in Sect. 2.1. To test
the applicability of those equations, we will use the model of this study to
reproduce the data of

Comparison of BTCs for different dispersivities and porosities but for the same hydraulic parameters.

This simplified model of Eq. (9) has been widely used to estimate

Assuming that the extraction time since the rest phase ended could be divided
into

Fitness of

Computed

To test the influence of the hydraulic diffusivity on the accuracy of this
model in estimating

Therefore, one may conclude that

Computed

BTCs for the different porous media with a piece-wise linear
function to describe chemical reactions:

To test the model of this study, the field data of a SWPP test conducted in
a single well by McGuire et al. (2002) will be employed. In this
test, the prepared solution contains

Phanikumar and McGuire (2010) interpreted such data using a model containing
several assumptions mentioned in Sect. 2.1. The parameters used in their
model were

To demonstrate the importance of the wellbore storage of the solute
transport, which was ignored in Wang et al. (2017), the observed and computed
BTCs are compared based on the estimated parameters in Phanikumar and
McGuire (2010), as shown in Fig. 12a and b. The computed BTCs in Fig. 12a
and b are located at

The results showed that the fitness between the observed BTCs in the wellbore and computed BTCs by “PPTEST” was very good, as shown in Fig. 12a of this study or Fig. 5 of Phanikumar and McGuire (2010). However, by carefully checking the report of Phanikumar (2010), we found that the computed BTCs were at a radial distance of 0.15 m from the wellbore, rather than at the wellbore itself in Phanikumar (2010). They did not provide a convincing argument why to choose BTCs in the aquifer to represent BTCs in the wellbore, and thus the use of “0.15 m” in their analysis appears to be an artifact, rather than being physically based. Figure 12b shows the comparison of the computed and observed BTCs in the wellbore for different hydraulic diffusivities. Obviously, the new model ignoring the wellbore storage of the solute transport could not be used to interpret experimental data, since the computed BTCs are zero at the early stage of the extraction phase.

Spatial distribution of the flow velocity in the extraction phase.

Fitness of the field SWPP test data by the new model of this study.

From Fig. 12a and b, several interesting observations could be made. Firstly,
the difference of BTCs among different porous media is obvious. BTCs of the
coarse sand aquifer are close to the solution of “PPTEST”, as shown in
Fig. 12a. This is because the hydraulic diffusivity of the coarse sand
aquifer is the largest, which is close to the assumption used in “PPTEST”
that hydraulic diffusivity is infinity. Secondly, the wellbore concentration
is 10 mg L

We try to use the new model to interpret BTCs of the SWPP test, considering a
finite hydraulic diffusivity, a finite wellbore storage, and new boundary
conditions of the wellbore at the injection, chase and rest phases, assuming
the initial head of the flow field is 1 m. In a trial-and-error process of
best fitting the observed BTC data, we only estimate parameters of

A complete SWPP test includes injection, chase, rest and extraction phases,
where the second and third phases are not necessary but are recommended to
increase the duration of reaction. Due to the complex mechanics of
biogeochemical reactions, aquifer properties, and so on, previous
mathematical or numerical models contain some assumptions which may
oversimplify the actual physics; for instance, the hydraulic diffusivity of
the aquifer is infinite. The Robin or the third-type boundary condition was
often used in previous studies at the well screen in the injection, chase and
rest phases by ignoring the mixing effect of the volume of water in the
wellbore (namely, wellbore storage). In this study, we presented a
multi-species reactive SWPP model considering the wellbore storage for both
groundwater flow and solute transport, and a finite aquifer hydraulic
diffusivity. The models of wellbore storage for both solute transports are
derived based on the mass balance. The Danckwerts boundary condition instead
of the Robin condition is employed for solute transport across the well
screen in the rest phase. The robustness of the new model is tested by the
field data. Meanwhile, the sensitivity analysis and uniqueness analysis of
BTCs in wellbore are conducted. The following conclusions can be drawn from
this study.

The influence of wellbore storage for the solute transport increases with the decreasing hydraulic diffusivity in the injection and chase phases, and the model of Eqs. (16)–(17) underestimates the concentration in the early stage of the injection phase while overestimating the peak values of BTCs.

The values of

The Robin condition used to describe the wellbore flux in the rest phase works well only when the chase concentration is zero and the prepared solution in the injection phase is completely pushed out of the borehole into the aquifer, while the Danckwerts boundary condition performs better even when the chase concentration is not zero.

In the extraction phase, the peak values of BTCs increase with the decreasing hydraulic diffusivity, and the arrival time of the peak value becomes shorter when the hydraulic diffusivity is smaller.

It seems impossible to determine all parameters simultaneously by only
best fitting the observed BTCs in the wellbore of the SWPP test using Eq. (30). The hydraulic parameters needed to be estimated by supplementary
aquifer tests before determining the parameters related to the solute
transport. The value of

All data are available in the Supplement.

The supplement related to this article is available online at:

QRW and HBZ proposed the new models. QRW performed all computations and wrote the paper. HBZ revised the paper.

The authors declare that they have no conflict of interest.

This research was partially supported by the Program of the Natural Science Foundation of China (nos. 41502229 and 41772252), and Innovative Research Groups of the National Nature Science Foundation of China (no. 41521001). We sincerely thank editor Monica Riva and two anonymous reviewers for their constructive comments which helped us improve the quality of this paper.

This paper was edited by Monica Riva and reviewed by two anonymous referees.