2.1 Revisit of the previous model

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

where C_i is the aqueous-phase concentration of the ith reactive solute, S_i is the solid-phase concentration of the ith reactive solute, t is the time in the SWPP test, ρ_b is the bulk density, θ is the porosity, H is the Heaviside step function, λ_j and n_j are the constant and orders, N is the number of the segment, $t_j^{\ast }$ and $t_{j+\mathrm{1}}^{\ast }$ are the times at two ends of segment j, and F_j is Monod/Michaelis–Menten kinetics. For the purpose of simplicity, we only present the reactive processes of the chemicals as described by Eq. (1), while the expressions of the transport (e.g., dispersion, diffusion, and advection) could be seen in Phanikumar and McGuire (2010), who used it to describe biogeochemical reactive transport of an arbitrary number of species including Monod/Michaelis–Menten kinetics, and the sorption models could be isotherm (Freundlich, Langmuir and linear sorption), one-site kinetic and two-site kinetic. As for the studies of Gelhar and Collins (1971), Schroth and Istok (2005), Huang et al. (2010), Haggerty et al. (1998), Snodgrass and Kitanidis (1998), Schroth and Istok (2006), Schroth et al. (2000), Istok et al. (2001), Jung and Pruess (2012), Wang et al. (2017), and so on, the governing equation is a special case of Eq. (1).

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):

where r_w is the well radius; r is the radius distance from the center of the well; B is the aquifer thickness; Q is the flow rate of the well; $v_{\mathrm{r}}=u_{\mathrm{r}}/\mathit{\theta }$ is the average radial pore velocity and u_r is the radial Darcian velocities. Equation (2) implies that the hydraulic diffusivity of the aquifer is infinite; thus, the flow velocity is independent of time. Meanwhile, the wellbore storage is negligible or the well radius r_w is assumed to be infinitesimal in formulating Eq. (2).

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):

where t_inj, t_cha, t_res, and t_ext represent the durations of the injection, chase, rest, and extraction phases, respectively; C is the resident concentration of the aqueous phase to represent C_i in Eq. (1); D_r is the dispersion coefficient, which is

in which α_r is the radial dispersivity; D₀ is the effective diffusion coefficient in the aquifer.

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):

where $C_{\mathrm{0}}^{\mathrm{inj}}$ and $C_{\mathrm{0}}^{\mathrm{cha}}$ represent the solute concentrations injected into the wellbore during the injection and chase phases, respectively. A detailed discussion about the above-mentioned assumptions can be seen in Phanikumar and McGuire (2010) or 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

where λ is the first-order reaction rate constant. Besides the first-order reaction, Eq. (7) could be used to describe the first-order biodegradation and radioactive decay. Haggerty et al. (1998) presented a simplified method to estimate λ for the SWPP test:

where t^∗ is the time since the end of injection; C_rec(t^∗) is the reactant concentration; C_tra(t^∗) is the concentration of a conservative tracer. To obtain the value of λ, the reactant and the conservative tracer should be fully mixed and injected into the aquifer simultaneously to conduct the SWPP test. After measuring the data of C_rec(t^∗) and C_tra(t^∗) in the extraction phase, one could fit the data of $\ln \left(\frac{C_{\mathrm{rec}}\left(t^{\ast }\right)}{C_{\mathrm{tra}}\left(t^{\ast }\right)}\right)\sim t^{\ast }$ using a linear function, and the slope of t^∗ is the estimation of λ. Snodgrass and Kitanidis (1998) derived a similar model for estimating λ:

Comparing Eq. (8) with Eq. (9), one could find that the difference is the first terms on the right-hand sides of equations, while λ is the slope for both Eqs. (8) and (9). Although the accuracy of both models has been tested by a number of investigators, previous studies on reactive transport were based on an assumption that the aquifer hydraulic diffusivity was infinite (e.g., Eq. 1 of Reinhard et al., 1997, and Eq. 2 of Haggerty et al., 1998).

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.

2.2 A revised model with a finite hydraulic diffusivity

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:

where K_r is the radial hydraulic conductivity; s is drawdown which could be obtained by solving the following mass balance equation with the proper initial and boundary conditions:

where S_s is the specific storage of the aquifer; s_w is the drawdown inside the wellbore.

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):

Actually, Eq. (15) acknowledges the continuity of concentration and continuity of mass flux simultaneously across the well screen, namely ${\left.C\right|}_{r\to r_{\mathrm{w}}^{-}}={\left.C\right|}_{r\to r_{\mathrm{w}}^{+}}$ and ${\left.\left(v_{\mathrm{r}}C\right)\right|}_{r\to r_{\mathrm{w}}^{-}}={\left.\left(v_{\mathrm{r}}C-D_{\mathrm{r}}\frac{\partial C}{\partial r}\right)\right|}_{r\to r_{\mathrm{w}}^{+}}$, where the − and + signs in the subscript of r_w represent approaching of the well screen from inside the well and outside the well, respectively.

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 $C_{\mathrm{0}}^{\mathrm{inj}}$ in the injection phase and greater than or equal to $C_{\mathrm{0}}^{\mathrm{cha}}$ in the chase phase.

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.,

where Δm represents the mass entering into the well during time interval Δt; $C_{\mathrm{w}}^t$ and $C_{\mathrm{w}}^{t+\mathrm{\Delta }t}$ represent the solute concentrations in the wellbore at times t and t+Δt, respectively; V^t represents the volume of water in the wellbore at time t. The initial values of $C_{\mathrm{w}}^t$ and V^t at the injection phase are

where H_w represents the water depth of the wellbore.

In the chase phase, one has

where the − and + signs in the subscripts of Eqs. (20)–(21) hereinafter represent approaching of the limit from the left- and right-hand sides of t_inj, respectively.

2.3 Capability of the new SWPP model of this study

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.

4.1 The rest phase

Figure 4a and b show the comparison of BTCs between the Robin and Danckwerts conditions at the wellbore for different porous media, where $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{10}$ mg L⁻¹ in Fig. 4a and $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{0.0}$ mg L⁻¹ in Fig. 4b. For the purpose of comparison, the boundary conditions at the wellbore in the injection and chase phases are still described by Eqs. (5)–(6).

Figure 4a shows that the difference of BTCs between two boundary conditions is significant at the early stage of the extraction phase when $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{10}$ mg L⁻¹, and BTCs of the Danckwerts condition are above BTCs of the Robin condition. With time going, such a difference becomes negligible. As for the curves of the Robin condition, the solute concentration in the wellbore is 0 in the chase phase; correspondingly, the concentration starts from 0 at the early stage of the extraction phase. Actually, the solute concentration in the wellbore may be non-zero in the rest phase due to the wellbore storage and finite hydraulic diffusivity when $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{10}$ mg L⁻¹. Another interesting observation is that the properties of the porous media could also influence the difference of BTCs between two boundary conditions. Obviously, a smaller hydraulic diffusivity would result in a larger difference between them; e.g., such a difference is greater for the fine sand aquifer.

Figure 4Comparison of BTCs in the wellbore between the Robin and Danckwerts conditions: (a) $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{10.0}$ mg L⁻¹; (b) $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{0}$.

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Figure 4b shows the comparison of BTCs for different boundary conditions in the wellbore when $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{0.0}$, and one could find that the difference of BTCs between the Robin and Danckwerts conditions is negligible, which implies that the Robin condition performs well when $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{0.0}$, while this is not for the case when $C_{\mathrm{0}}^{\mathrm{cha}}\ne \mathrm{0.0}$.

4.2 The injection and chase phases

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⁻¹ in the rest phase, due to $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{10.0}$ mg L⁻¹. As for the model of Eqs. (16)–(17), the volume of water in the wellbore is non-negligible and could dilute the concentration in the injection phase; i.e., the solute concentration in the wellbore could not immediately rise to $C_{\mathrm{0}}^{\mathrm{inj}}$ at the early stage of the injection phase, thus resulting in smaller peak values of BTCs. Similarly, the concentration in the wellbore could not immediately reduce to $C_{\mathrm{0}}^{\mathrm{cha}}$ at the early stage of the chase phase, which makes the concentration larger at the early stage of the extraction phase based on the model of Eqs. (16)–(17).

5.1 Sensitivity analysis

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

where SC_i,j is the sensitivity coefficient of the jth parameter I_j at the ith time; C_i is the concentration at the ith time. In this study, the differentiation of Eq. (31) will be approximated by a finite-difference scheme:

where ΔI_j is a small increment.

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 4.17×10² m² h⁻¹), and the influence of the transient flow could be ignored. Therefore, the steady-state assumption could be used to approximate the flow field in the SWPP test when the hydraulic diffusivity is greater than 4.17×10² m² h⁻¹. Otherwise, the steady-state assumption is not recommended. Figure 7 shows that the BTCs in the wellbore are sensitive to both dispersivity and porosity.

Figure 6Sensitivity analysis of the hydraulic diffusivity on BTCs in the extraction phase.

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5.2 Uniqueness analysis of physical parameters

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.

Figure 7Sensitivity analysis of dispersivity and porosity on BTCs in the extraction phase.

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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 α_r=0.01 m and θ=0.33 are almost the same as the curves of α_r=0.006 m and θ=0.9. Therefore, Eq. (30) may not be used to determine α_r and θ simultaneously. Fortunately, the porosity could be measured in the laboratory from core samples or determined by the SWPP test with drift flow (Hall et al., 1991; Paradis et al., 2018). When the values of K_r, S_s, and θ are given, the dispersivity could be determined uniquely by Eq. (30).

In summary, it seems impossible to determine all parameters (K_r, S_sα_r, and θ) simultaneously by only best fitting the observed BTCs in the wellbore of the SWPP test using Eq. (30). Therefore, before determining the parameters related to the solute transport (α_r and θ), the hydraulic parameters (K_r and S_s) needed to be estimated by supplementary aquifer tests, or by best fitting the pressure data measured during the SWPP test, e.g., the pumping phase. The value of α_r could be determined by Eq. (30) when the porosity is given.

Table 1Reaction parameters estimated by linear functions.

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5.3 Chemical parameter estimation

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 $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$ based on a set of given parameters, and then using Eqs. (8)–(9) (which is based on an infinite hydraulic diffusivity presumption) to estimate λ (denoted as $\stackrel{\mathrm{̃}}{\mathit{\lambda }}$) by best fitting $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$. Two species involved in this case are Cl⁻¹ and ${\mathrm{SO}}_{\mathrm{4}}^{-\mathrm{2}}$, in which C_tra and C_rec represent the concentrations of Cl⁻¹ and ${\mathrm{SO}}_{\mathrm{4}}^{-\mathrm{2}}$, respectively. Figure 9 shows the fitness of the simulated $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$ in the wellbore using a linear function, with the detailed information shown in Table 1. Two sets of λ are employed in the discussions for the reactant, e.g., λ=0.1 and 0.2 h⁻¹. One may conclude that the simplified models of Eqs. (8)–(9) with an infinite hydraulic diffusivity perform well in the estimation of λ for reactive transport under the finite hydraulic diffusivity condition.

Figure 8Comparison of BTCs for different dispersivities and porosities but for the same hydraulic parameters.

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This simplified model of Eq. (9) has been widely used to estimate λ, due to the advantages that λ could be determined directly by best fitting the observed $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$, without knowledge of the aquifer properties, such as porosity, dispersivity, and hydraulic diffusivity. However, this model is proposed based on the first-order reaction assumption, which is a linear function as shown in Eq. (7). Whether this model works for nonlinear reactions or not is still unknown and will be investigated in the following section.

Assuming that the extraction time since the rest phase ended could be divided into N−1 segments, Phanikumar and McGuire (2010) employed the Heaviside unit step function to describe a type of nonlinear biogeochemical reaction:

where λ_j is the reaction constant in the temporal segment j, and the Heaviside step function H(⋅) is

Equation (33) is a series of piece-wise linear (n_j=1) or nonlinear (n_j≠1) functions, which are an extension of Eq. (7).

Figure 9Fitness of $\ln \left[C_{\mathrm{rec}}/C_{\mathrm{tra}}\right]\sim t^{\ast }$ produced by the numerical solution of this study with the first-order reaction in the different porous media.

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Figure 10Computed $\ln \left[C_{\mathrm{rec}}/C_{\mathrm{tra}}\right]\sim t^{\ast }$ by the model of this study using a piece-wise linear function to describe the nonlinear chemical reactions.

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To test the influence of the hydraulic diffusivity on the accuracy of this model in estimating λ_j for the nonlinear reactions, the model of this study is used to reproduce the data of $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$ with a set of specific λ_j, n_j and $t_j^{\ast }$ for three types of porous media. Figures 10 and 11 represent the computed $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$ based on the model of the chemical reactions described by the piece-wise linear and nonlinear functions, respectively. The values of λ_j and $t_j^{\ast }$ of Fig. 10 are obtained by best fitting the observation data using a piece-wise linear function (e.g., n_j=1) proposed by Phanikumar and McGuire (2010). The circle represents the experiment data observed by McGuire et al. (2002). The parameters related to the chemical reactions in Fig. 11 are from Phanikumar and McGuire (2010) by best fitting the observation data using a nonlinear function: λ_j=0.25, n_j=0.25, $N=j=\mathrm{1}$. Comparing Figs. 10 and 11, one may find that the influence of the hydraulic diffusivity on the computed $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$ is negligible for the chemical reaction described by the piece-wise linear function, which is similar to the first-order reaction as shown in Fig. 9. However, the influence of the hydraulic diffusivity on the relationship of $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$ cannot be ignored if one attempts to use nonlinear functions to describe such a chemical reaction. The difference between the curves of different porous media is obvious in Fig. 11. The agreement between the observed and computed data is satisfactory for the medium and coarse sands, but not for the fine sand in Fig. 11. This is because the hydraulic diffusivity values of the medium and coarse sands are larger than that of the fine sand, and thus are close to the assumption of an infinite hydraulic diffusivity used in Phanikumar and McGuire (2010).

Therefore, one may conclude that λ_j, n_j and $t_j^{\ast }$ could be determined by directly best fitting the observed $\ln \left(C_{\mathrm{rec}}/C_{\mathrm{tra}}\right)\sim t^{\ast }$ when the nonlinear reactions are described by the piece-wise linear functions, in a similar way to estimating the linear reaction rate by Eq. (7). However, such an approach may not work if one attempts to use nonlinear functions to describe such reactions.

Figure 11Computed $\ln \left[C_{\mathrm{rec}}/C_{\mathrm{tra}}\right]\sim t^{\ast }$ by the model of this study using a nonlinear function to describe the nonlinear chemical reactions.

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Figure 12BTCs for the different porous media with a piece-wise linear function to describe chemical reactions: (a) Cl¹⁻ and ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ in the aquifer at $r=r_{\mathrm{w}}+\mathrm{0.15}$ m, (b) Cl¹⁻ in the wellbore.

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6.1 Revisit of the previous model

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 B=0.1 m, r_w=0.0125 m, α_r=0.001 m, θ=0.33 m, D₀=0 m² h⁻¹, t_inj=0.6 h, t_cha=0.067 h, t_res=0.0333 h, t_ext=3.6 h, Q_inj=0.0333 m³ h⁻¹, Q_cha=0.0255 m³ h⁻¹, and $Q_{\mathrm{ext}}=-\mathrm{0.011}$ m³ h⁻¹. The concentrations of NaCl were $C_{\mathrm{0}}^{\mathrm{inj}}=\mathrm{100}$ mg L⁻¹ in the injection phase and $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{10}$ mg L⁻¹ in the chase phase. As for the reactant of Na₂SO₄, the concentrations were $C_{\mathrm{0}}^{\mathrm{inj}}=\mathrm{20}$ and $C_{\mathrm{0}}^{\mathrm{cha}}=\mathrm{2}$ mg L⁻¹.

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 $r=r_{\mathrm{w}}+\mathrm{0.15}$ m and r=r_w, respectively. The legend of “PPTEST” represents the solution of Phanikumar and McGuire (2010), and the others are produced by the new model, ignoring the wellbore storage effect on the solute transport.

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.

Figure 13Spatial distribution of the flow velocity in the extraction phase.

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Figure 14Fitness of the field SWPP test data by the new model of this study.

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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⁻¹ at the early stage of the extraction phase for Cl⁻. This is mainly due to the chosen boundary condition at the well screen, which has been discussed in detail in Sect. 4.1. Thirdly, the peak values of BTCs increase with the decreasing hydraulic diffusivity, and the arrival times of peak values increase with the decreasing hydraulic diffusivity. Such an observation is also found in Fig. 4a and b. Fourthly, the configuration of BTCs in the aquifer (at $r=r_{\mathrm{w}}+\mathrm{0.15}$ m) computed by the model of this study shows that the concentration firstly decreases with time and then increases with time, as shown in Fig. 12a. This observation could be explained by the corresponding flow field, as shown in Fig. 13. Looking at the flow velocity in the aquifer at $r=r_{\mathrm{w}}+\mathrm{0.15}$ m, one may find that the flow direction is still outward from the wellbore in the early stage of the extraction phase, due to the finite hydraulic diffusivity. The outward flow will persist for a finite period of time, depending on the value of the hydraulic diffusivity, and then reverse its direction to flow towards the wellbore for the rest of the extraction phase. This feature is very different from the results with an infinite hydraulic diffusivity assumption, in which the flow direction is always towards the wellbore for the entire extraction phase.

6.2 Fitness of this study

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 K_r, S_s and α_r, while the other parameters are the same as those used to produce Fig. 5 of Phanikumar and McGuire (2010). Figure 14 demonstrates the fitness of the observed BTC data in the wellbore when K_r=1.0 m h⁻¹, $S_{\mathrm{s}}=\mathrm{1.0}\times {\mathrm{10}}^{-\mathrm{5}}$ m⁻¹ and α_r=0.015 m. Since the hydraulic diffusivity of this case is greater than the hydraulic diffusivity of medium sand (4.17×10² m² h⁻¹), the influence of the flow field could be negligible. In this study, we mainly estimated the value of dispersivity, where the porosity is fixed and comes from the reference of Phanikumar and McGuire (2010). Therefore, the dispersivity is uniquely determined.