Investigating the Representative of Aquifer Transmissivity Determined by Passive Response Methods: A Comparison With Time‐Dependent Hydraulic Parameters Inferred From Different Stages of Pumping Tests

Aquifer pumping tests represent a standard method for estimating hydraulic characteristics, with practitioners often focusing on late period drawdown data because these are less affected by within‐ and near‐borehole effects (e.g., borehole‐storage and skin effects). Alternatively, groundwater responses to natural forcing (e.g., barometric pressure and earth tides) provide a passive method for estimating aquifer parameters at a low cost. However, to the best of our knowledge, no studies have compared parameters calculated from different periods within a pumping test with those from passive methods. Herein, we compare the aquifer transmissivity estimated using both active and passive methods in two wells located in the Beetaloo Region of Northern Australia. The active method estimates aquifer transmissivity during three periods (i.e., the early, middle, and late periods) of an aquifer pumping test, while the passive method employs groundwater responses to barometric‐pressure and earth‐tide fluctuations. We find that the range of best‐fit aquifer transmissivity is 1.18 × 10−5–1.79 × 10−5 m2/s and 1.73 × 10−5–2.14 × 10−5 m2/s for OW1 and OW2, respectively. The transmissivity estimated from the barometric pressure response method is the largest. The aquifer transmissivity using barometric pressure responses are consistent with early‐ and middle‐period estimates. This suggests that barometric pressure responses are more sensitive to within‐ and near‐borehole effects. The scales of the tidal response method are smaller than those of the pumping test method.

The water-level drawdown-time curves for the entire pumping period can be divided into early, intermediate, and stable periods (Grimestad, 1981).Early-and some intermediate-period data are usually ignored because estimated transmissivities vary considerably during these periods, and only stabilize in the later period.The Cooper and Jacob (1946) and Theis (1935) methods estimate the average values of aquifers based on late data or prolonged pumping test data (Oliver, 1993;Straface et al., 2007;Sun et al., 2013).While few studies estimate the transmissivity using early time data, these data provide the ability to evaluate the local versus regional heterogeneity and boundary effects (Wu et al., 2005), as well as the hydrogeological conditions in the vicinity of the pumping and observation wells.Therefore, investigating how parameters change dynamically during an aquifer pumping test provides additional hydrogeological information about the pumping wells, observation wells, and target aquifers.
Groundwater responses to earth and atmospheric tides (EAT) are increasingly used in aquifer characterization (Espejo et al., 2022;Hsieh et al., 1987;Hussein et al., 2013;Odling et al., 2015;Qi et al., 2023;Rasmussen & Crawford, 1997;Sun et al., 2020;Valois et al., 2022;H. Zhang et al., 2019;H. Zhang et al., 2021;Y. Zhang et al., 2021).Analytical solutions are now available to determine aquifer hydraulic parameters using these passive methods both in the time and frequency domain.Time-domain methods include regression deconvolution and harmonic least squares, which have been developed for unconfined, semi-confined, and confined aquifers; unsaturated zones; and soil moisture content (Rasmussen & Crawford, 1997;Rasmussen & Mote, 2007;Rau et al., 2020).Frequency domain methods using Fourier analysis have also been developed.Rojstaczer (1988) and Hussein et al. (2013) proposed barometric pressure response models that consider the responses in the aquifer, confining layer, and unsaturated zones.Different types of aquifers have different tidal response models.Hsieh et al. (1987) provided an analytical solution for the groundwater flow equation with harmonic disturbances in a confined aquifer.Wang (2000) considered the importance of the vertical diffusion of pore pressure and the responses to tides, proposing an analytical model by applying periodic loads on the half-space surface.Wang et al. (2018) proposed a tidal leaky response model that considers the impact of vertical leakage.
Active methods are conventionally used to estimate hydraulic parameters by inducing hydraulic stresses in boreholes that cause hydraulic perturbations in time and space (Alley et al., 2002;McMillan et al., 2019).However, active methods depend on expert execution and data interpretation, and are not suitable for continuous observations of the hydraulic parameters.However, passive methods are more convenient to implement than active methods, providing the ability to monitor aquifer hydraulic properties continuously while eliminating the need to directly stimulate the aquifer conditions (Allegre et al., 2016;McMillan et al., 2019;Sun et al., 2020;H. Zhang et al., 2019;S. Zhang et al., 2019).
The transmissivities derived by different methods represent different scales of aquifer characteristics.The scale for pumping tests is the distance between the pumping well and the monitoring well (Allegre et al., 2016;Elkhoury et al., 2006;Kitagawa & Kano, 2016;Xue et al., 2013).Sun et al. (2020) found that the transmissivity inferred from the Hsieh et al. (1987) model averages the aquifer properties at the scale of tens of meters; the Wang et al. (2018) model is consistent with Hsieh et al. (1987) model.Allegre et al. (2016) found that the scale effect of tidal response is smaller than that of the pumping test.H. Zhang et al. (2019) and S. Zhang et al. (2019) conclude that the scale effect of tidal response is slightly larger than that of the slug test.
Few studies have evaluated the ability of passive methods to characterize hydrogeologic systems.Several studies have compared aquifer parameters inferred from passive methods with those from slug tests (Qi et al., 2023;S. Zhang et al., 2019) and pumping tests (Allegre et al., 2016;Sun et al., 2020;Valois et al., 2022).Comparing the results of these passive methods with different pumping periods can provide more hydrogeological information for passive methods.In this study, we estimate aquifer transmissivity by analyzing groundwater data collected from wells in the Beetaloo Region, Northern Australia.Monitoring data includes time-drawdowns in two observation wells during the conventional aquifer-pumping test along with baseline data collected during non-pumping intervals.We compare the similarities and differences between active (AQTESOLV method) and passive (earth tide and barometric pressure responses) methods.And we investigate the scale effects on aquifer transmissivity by comparing the barometric pressure response model, and AQTESOLV program with dynamic parameters estimated using Newton's method during three periods of an aquifer pumping test.

Methodology
While conventional aquifer pumping tests are used to estimate parameters in the time domain, groundwater responses to natural forcing (e.g., earth tides and barometric pressure) can be estimated both in the time and frequency domains.Both earth and atmospheric tides (EAT) are periodic, in that they oscillate at specific frequencies; the amplitude ratios and phase lags between natural forcing and observed groundwater responses can be used to estimate aquifer parameters.

Earth Tide Responses
Earth tides commonly affect the groundwater levels observed in wells as the deformation of the solid earth is transferred to the entrained fluids (Melchior, 1966).Changing fluid pressures within geologic media equilibrate with fluid pressures in boreholes, resulting in changes in the groundwater levels.The M 2 tidal constituent is usually used for tidal analyses as it is larger and more reliable than others (Bower, 1983), and is relatively independent of atmospheric tides.
The tidal amplitude ratio between fluid pressures within geologic media relative to theoretical tidal amplitudes is related to the poroelastic properties of the geologic media.Further, equilibration between the aquifer and borehole introduces a time-constant, or phase lag, which reflects the permeability of the hydrogeologic unit.Thus, we can estimate aquifer hydraulic properties using the amplitude ratio and phase lag of the tidal response (Allegre et al., 2016;Turnadge et al., 2019).Wang et al. (2018) extend the tidal model developed by Hsieh et al. (1987) for confined aquifers to include vertical leakage through an adjacent confining layer using the aquitard conductance, σ′ = K′/b′.
The governing equation for groundwater flow in the leaky aquifer is where h is the hydraulic head above a common reference; r is the radial distance from the well, T and S are the aquifer transmissivity and storage coefficient, respectively; ε is the tidal oscillation volumetric strain of the aquifer; B is the Skempton's coefficient of the aquifer; K u is the undrained bulk modulus of the aquifer; K′ and b′ are the vertical hydraulic conductivity and the thickness of the aquitard, respectively; ρ is the density of groundwater; and g is the acceleration due to gravity.It is worth noting that BK u is usually unknown and has large uncertainties (Wang et al., 2018).
Periodic groundwater fluctuations within the well are found using where h w,o is the complex amplitude, given by The amplitude ratio (A) and phase lag (η) of the tidal response are where ω is the frequency, r c is the radius of the well casing, r w is the radius of the well, and K 0 and K 1 are the zeroth and first order of the second modified Bessel function, respectively.

Barometric Pressure Responses
The barometric force exerted on the ground surface varies with time.This changing surface load is rapidly transmitted through the solid earth, affecting fluid pressures as a function of the elasticity of the mineral skeleton and entrained fluids (Jacob, 1940).In situ fluid pressures (along with their corresponding hydraulic heads) respond rapidly to these induced forces.Hydraulic heads in open wells respond instantaneously to barometric pressure changes, creating an imbalance in hydraulic heads between boreholes and the aquifer.On the other hand, water levels in wells that is isolated from the atmosphere using a seal (a well cap) are not immediately affected by barometric pressure changes.
Further, imbalances in the hydraulic head between sealed boreholes and the aquifer leads to an inverse response to that observed in an open well (Rasmussen & Crawford, 1997).In both cases, the resulting imbalance in hydraulic heads requires time to equilibrate, which provides the opportunity to estimate aquifer hydraulic properties (Rasmussen & Crawford, 1997;Rojstaczer, 1988;Rojstaczer & Agnew, 1989;Sun et al., 2019Sun et al., , 2020;;H. Zhang et al., 2021;Y. Zhang et al., 2021).Hussein et al. (2013) determined the groundwater response to barometric pressure changes based on the Rojstaczer (1988) model, which considers vertical airflow through the unsaturated zone, attenuation by the capillary fringe T cf , vertical flow through confining layers, and horizontal flow between the aquifer and wellbore.The model establishes three dimensionless parameters using the well radius r w , thickness of the unsaturated zone L unsat , vertical pneumatic diffusivity of the unsaturated zone D unsat , aquifer horizontal transmissivity T, vertical pneumatic diffusivity of the confining layer D con , confining layer thickness b′, and angular frequency ω: If the theoretical barometric response model that includes the three dimensionless parameters R, Q, and W can be fitted with the observed barometric response function, hydraulic parameters in the three dimensionless parameters can be estimated (see Text S1 in Supporting Information S1 for more details).
The amplitude ratio and phase lag provide information about the barometric efficiency (BE) and the response phase (θ) of the three dimensionless parameters: where 14) 10.1029/2022WR033952 5 of 20 where p 0 is the pore pressure of the aquifer (the influence of the well can be ignored); S 0 is the drawdown in the well; A and x 0 denote the amplitude of the atmospheric load and the water level fluctuation in the well, respectively; ρ is the density of water; g is the acceleration due to gravity; S and S′ are the storage coefficients of the aquifer and the confining layer, respectively; K 0 is the modified Bessel function of the second kind of order zero; i is an imaginary number; and T cf is an attenuation factor.

Aquifer Pumping Test Responses
Traditional pumping tests are considered to yield the average hydraulic properties over a wide range of geologic media by fitting pumping test data to analytical solutions (Straface et al., 2007).The estimation of aquifer parameters from pumping tests is a dynamic process, where the T and S estimated from pumping tests vary with increasing pumping duration and eventually converge to a stable value.

Hantush-Jacob Solution
The Hantush-Jacob analytic solution (Hantush & Jacob, 1955) is commonly used to estimate the aquifer transmissivity T and aquifer storage coefficient S, as well as the vertical hydraulic conductivity of aquitard K′ in leaky confined aquifers.The solution assumes no storage in incompressible leaky aquitards.In a homogeneous, isotropic, and leaky confined aquifer of infinite extent, the Hantush-Jacob equation is as follows: where s is the groundwater drawdown at any point and time, Q is the pumping rate, and W (u, r/B) is the Hantush-Jacob well function for leaky confined aquifers, given by where u = r 2 /4Dt is the dimensionless well argument, r is the distance to the pumping well, D = T/S is the aquifer hydraulic diffusivity, and t is the elapsed time.The hydraulic properties are found by fitting this analytic solution to observed time-drawdown data.

Dynamic Process of the Pumping Test
The Hantush and Jacob (1955) solution assumes that the aquifer is homogeneous, isotropic, and has an infinite lateral extent.However, aquifers are heterogeneous and bounded in nature, so it is generally assumed that the hydraulic properties estimated by the Hantush-Jacob solution are the spatial averages over a cone of depression (Meier et al., 1998;Sanchez-Vila et al., 1999).
Estimating the hydraulic properties of aquifers from pumping tests is a dynamic process, and the effective transmissivity estimate for each pumping test varies with the observed location (Huang et al., 2011).The transmissivity and storage coefficient values of the aquifer estimated by the pumping test method also vary with pumping time.
Each drawdown measurement from a pumping test is associated with a particular area of influence, depending on the location of the pumping well and the observation well, the duration of pumping, and the flow characteristics of the formation (Oliver, 1993).Yeh (1998) postulated that the value of the effective transmissivity varies with time based on the concept of diffusion (Gupta, 1959).
The cross-correlation analysis method provides insight into the meaning of the estimated T and S from traditional aquifer test analyses (Sun et al., 2013;Wu et al., 2005;Zhu & Yeh, 2005).Wu et al. (2005) used cross-correlation analyses to conclude that (a) effective transmissivity values obtained using early pumping test data are greater than the geometric mean for the entire aquifer, and then approach and the geometric mean over a larger period of time.(b) The effective storage coefficient values also vary with time, approaching the arithmetic mean of the aquifer over a long period of time (Wu et al., 2005).
The drawdown-time curve can be divided into the early, middle, and late periods (Grimestad, 1981).The early period is the initial period of the pumping test.The transmissivity is limited to the area between the observation well and pumping wells, and mainly reflects the value of the storage coefficient in the restricted area between the pumping and observation wells (Wu et al., 2005).In the middle period, the effect of S on the head weakens and the flow is moving toward a steady state.In the late period, the head is strongly affected by the transmissivity and has little relation with the storage coefficient.The estimated transmissivity is close to certain average values of the aquifer only when the pumping lasts long enough.In contrast, the estimated S is dominated by the local average S between the pumping and observation wells (Liu et al., 2002;Zhu & Yeh, 2005).

Application
The objective of this study is to evaluate two passive methods (groundwater responses to barometric pressure and tidal fluctuations) and one active method (aquifer pumping test) for estimating the aquifer transmissivity.

Study Site Description
This study utilizes groundwater data from one pumping and two observation wells (i.e., PW, OW1, and OW2, respectively) located in the Beetaloo Region of Northern Australia (Table S1 in Supporting Information S1; Figure 1a).The area is located far from the coast, at a distance of over 300 km.The lithology and aquifer thickness at each well are given in Table 1.Note that the borehole diameters and litho-stratigraphy are the same in all wells.PW, OW1, and OW2 are located in the semi-confined aquifer.The unsaturated zone in the confining layer consists of red-brown topsoil, the saturated zone in the confining layer is mainly composed of claystone and siltstone, and the aquifer is mainly composed of basalt.PW was drilled through the entire aquifer.The PW is 263 m deep, of which 0-68 m is a 0.26 m diameter steel casing.Observation well OW1 is 253 m deep, of which 0-68 m is a steel casing with a 0.26 m diameter, while observation well OW2 is 274.7 m deep, of which 0-70 m is a steel casing with a 0.26 m diameter (Figure 1c).
Groundwater levels are monitored in the two observation wells; OW1, located south of the pumping well at a distance of 89.8 m, and OW2, located at a distance of 979.1 m north of the pumping well (Figure 1b).

Data Collection
The water level was recorded using a Level TROLL 400 absolute (unvented) pressure transducers with a measurement accuracy of ±0.1% F.S. and a resolution of ±0.005% F.S. The water level measurement system consists of a pressure sensor and a data collector with a sampling rate of 1 hr.The theoretical earth tide strain time series for the geographic locations of sites OW1 and OW2 are calculated using ETGTAB (Venedikov et al., 2003).
Both observation wells are used to estimate aquifer properties during the passive (baseline) and active (pumping) periods.The passive period utilizes data from 26 November 2020 to 6 May 2021.During this period, groundwater levels are influenced by both earth tides and barometric pressure (Figure 2), which provides data for estimating the aquifer hydraulic parameters with natural disturbances.The active period extends from 07:30 on 19 November 2020 to 18:50 on 22 November 2020, when PW is pumped at a rate of Q = 16 m 3 /day.The measured pumping rates and resulting drawdowns are shown in Figure 3.Note that the two wells exhibit different behaviors, with OW1 having a larger response than OW2.

Influence of the Earth Tide
We calculated the volumetric strain for two observation wells, OW1 and OW2, using ETGTAB.The sampling intervals were both 1 hr.Ocean tides have been observed to affect groundwater levels tens of kilometers from the shoreline (Agnew, 1997;Beaumont & Berger, 1975), our site lies over 300 km from the nearest shoreline, but we still analyzed the ocean tide effects.The results show that ocean tide has less effect on the earth tide response in our study (see Text S2 in Supporting Information S1 for more details).
When estimating aquifer hydraulic parameters, Wang et al. (2018)'s tidal leaky response model assigns a fixed value to one of the three parameters and then solves for the remaining parameters.Many studies often assign the  aquifer storage coefficient first; then, the amplification factor and phase lag equations are iterated to fit appropriate transmissivity and leakage coefficients (Sun et al., 2020;H. Zhang et al., 2019;S. Zhang et al., 2019).In addition, the barometric pressure response method also requires a given aquifer storage coefficient for estimating aquifer parameters; to ensure uniformity among methods, the tidal response method also estimates the S first.Therefore, we assign S = 4.60 × 10 −5 and 1.09 × 10 −6 for OW1 and OW2, respectively, based on the results of the pumping test program-AQTESOLV estimation.Groundwater fluctuations are then inverted to determine the aquifer transmissivity and aquitard conductance.
Harmonic analysis is performed using the M 2 tide (1.9327 cpd) component.The chosen window size is 30 days, which is the minimum time required to separate M 2 tidal frequencies (Allegre et al., 2016;Sun et al., 2020;Xue et al., 2016).We use the Baytap08 program (Tamura et al., 1991) for the harmonic analysis of earth tide to obtain the phase lag for earth tide and well water levels.OW1 exhibits an M 2 phase lag of η = 28.46° with RMSE = 1.60°, and an amplitude ratio of A = 0.31 (mm/10 −9 ) with RMSE = 0.001 (mm/10 −9 ).Further, OW2 exhibits an M 2 phase lag of η = 28.45° with RMSE = 0.75°, and an amplitude ratio of A = 0.35 (mm/10 −9 ) with RMSE = 0.001 (mm/10 −9 ).Note that the phase is positive at both wells, indicating that the groundwater level responses precede tidal forcing.This phenomenon is attributed to the high rates of leakage (Wang et al., 2018) or negative skin effects (Valois et al., 2022).

Influence of the Barometric Pressure
The groundwater level and barometric pressure are monitored using an absolute (unvented) pressure sensor, which compensates for the barometric pressure using measurements from a barometer.The sampling rate of the groundwater level and barometric pressure is 1 hr.
The regression deconvolution method (Rasmussen & Crawford, 1997;Rasmussen & Mote, 2007) is used to mitigate the effect of earth tides on groundwater levels in the OW1 and OW2 wells from 26 November 2020 to 6 May 2021 (Figure S2, see Text S3 in Supporting Information S1 for more details).
These preprocessed groundwater and barometric pressure data are analyzed and fitted to the barometric pressure response model proposed by Hussein et al. (2013).The barometric pressure response model has seven parameters (BE, D unsat , D con , T, T cf , S, and S′).S′ is not sensitive to the fitting process and changing this value does not affect the other parameters (Hussein et al., 2013;Sun & Xiang, 2020, Figure S3 in Supporting Information S1); we roughly use S′ = 1 × 10 −4 for this calculation.The barometric pressure response model is also insensitive to the aquifer storage coefficient (Hussein et al., 2013;Sun & Xiang, 2020; Figure S4 in Supporting Information S1), and we use aquifer storage coefficient estimated by the AQTESOLV method, S = 4.60 × 10 −5 and 1.09 × 10 −6 in the calculations for OW1 and OW2, respectively.The other five parameters (BE, D unsat , D con , T, and T cf ) in the barometric pressure response model were obtained by fitting (see Text S1 in Supporting Information S1 for more details).
The result of the fitting is shown in Figure 5 and Table 2.At the best fit of the barometric pressure response function, the barometric efficiencies for wells OW1 and OW2 are BE = 0.66 and 0.70, respectively.The aquifer transmissivity estimates using the barometric pressure response method are T = 1.79 × 10 −5 and 2.14 × 10 −5 m 2 /s, and the vertical hydraulic conductivities within the confining layer are K′ = 7.75 × 10 −9 and 8.54 × 10 −9 m/s.The vertical pneumatic diffusivities of the confining layer are D con = 3.72 × 10 −3 and 4.27 × 10 −3 m 2 /s and the vertical pneumatic diffusivities of the unsaturated zone are D unsat = 2.90 × 10 −3 and 5.54 × 10 −3 m 2 /s.The attenuation factors are T cf = 0.94 and 0.97.
There are three hydraulic parameters to be estimated in the Hantush-Jacob solution (1955): aquifer transmissivity T, storage coefficient S, and the vertical hydraulic conductivities within the confining layer K′.The confining layer is relatively thin in comparison to the aquifer.Further, the flow in the aquifer is horizontal and the flow in the confining layer is vertical.Therefore, we mainly focus on the dynamic processes of aquifer parameters.The pumping test drawdown curves of OW1 and OW are insensitive to δ (δ = r/2B) (Figure S5 in Supporting Information S1), so we use δ = 0.046 and 0.155 estimated by AQTESOLV method as the initial values for the analysis of the dynamic process of the pumping tests.
We then dynamically estimate the aquifer transmissivity and hydraulic diffusivity throughout the pumping test using Newton's method (Kreyszig, 2000) (see Text S4 in Supporting Information S1 for more details).This approach uses observed drawdowns at two different times, s 1 = s(t 1 ) and s 2 = s(t 2 ), that are separated by a time increment, ∆t = t 2t 1 .This yields two equations suitable for sequentially estimating the two unknowns (T, D) throughout the pumping test, with the storage coefficient being found using their ratio, S = T/D.The dynamically estimated transmissivity, diffusivity, and storage coefficient of OW1 and OW2 throughout the aquifer test are shown in Figure 6.To observe the change trend more clearly, the time is normalized by using the ratio of pumping time t to the square of the distance r between the observation and the pumping wells as the standardized time.
With the progression of the pumping process, the drawdown and the cone of depression continue to expand (Boulton, 1954).The OW1 well is closer to the pumping well than the OW2 well, so the OW1 well responds to the pumping test before the OW2 well.As the pumping test continues, the transmissivity T and hydraulic diffusivity D gradually decrease, and the storage coefficient S increases gradually (Sun et al., 2013).
The hydraulic diffusivity can be obtained by calculating the transmissivity and storage coefficient (D = T/S).Therefore, the estimated transmissivity and storage coefficients are then analyzed as horizontal and vertical coordinates to ascertain their dynamic changes during the pumping test.The early, middle, and late periods are clearly distinguishable, as shown in Figure 7.This staging relationship provides us with the possibility to evaluate the periods of the aquifer parameter estimation methods.This can help understand the scales represented by different methods and provides more information for the analytical solution models that originally simplify the hydrogeological conditions.Due to their different distances from the pumping well, the two observation wells differ in their period observability in the pumping test.
Observation well OW1, near the pumping well is less obvious than observa- tion well OW2, away from the pumping well, in terms of observability in the periods.During the pumping tests, the distance between the observation wells and the pumping wells should be within a reasonable range.When the observation well is too far from the pumping well, the response of the water level to the pumping test is not obvious; when it is too close, the aquifer characteristics cannot be fully reflected (Neuman, 1972).Therefore, to better study the dynamic process of the pumping tests in the target aquifer in the future, the observation well, with a suitable distance from the pumping well, should be selected for research, and a suitable value for this distance warrants further investigation.

Discussion
The aquifer hydraulic parameters estimated by two passive methods as well as the active method, are shown in Table 3.This section compares these results to identify the similarities and differences between the methods to establish the appropriate scales of influence that might affect the observed differences, and to evaluate the parameter uncertainties associated with each method.

Comparison Between the Methods
In order to compare and analyze the reasons for similarities and differences in the estimation of hydraulic parameters by different methods.And to determine the influence of the simplifications of the analytical solution model in the passive methods on the estimated transmissivity, the pumping test method is compared with the tidal and barometric pressure responses analytical solution models to evaluate the applicability of each method.In

Note.
And statistical summary of estimated parameter fits and their expected scale of influence.The pumping test period is from 07:30 on 19 November 2020 to 18:50 on 22 November 2020.The period for the barometric pressure and earth tide responses methods is from 26 November 2020 to 6 May 2021.The bold-italic values represent the storage coefficient estimated by the AQTESOLV method.Both the earth tide response method and the barometric pressure response method use the storage coefficient estimated by the AQTESOLV method.

Table 3 Hydraulic Parameters (T = Transmissivity (m 2 /s), K′ = Aquitard (Vertical) Hydraulic Conductivity (m/s), S = Storage Coefficient (dimensionless)) of OW1 and OW2
Obtained Using Three Methods addition, we compare the tidal and barometric pressure responses with the staged pumping tests to explore the range that they represent.

Comparison Between the Original Models
Many assumptions and simplifications are used in the establishment of the original models to obtain the analytic solutions, which may lead to differences in the hydraulic parameters estimated by different methods (Qi et al., 2023;Sun et al., 2020).
Solutions for estimating aquifer hydraulic parameters from pumping tests are available for a variety of boundary conditions and geometries (Bredehoeft, 1967;Cooper et al., 1967;Fischer, 1992).In contrast, the analytical solution for the tidal response is suitable for a homogeneous, isotropic, infinitely porous medium through which a finite-radius well penetrates completely (Hsieh et al., 1987;McMillan et al., 2019;Merrit, 2004).The analytical solution model of the barometric pressure response assumes that the aquifer compressibility, porosity, and Poisson's ratio are uniform in both the vertical and horizontal directions.Moreover, the three-dimensional problem of flow between the aquifer and well is decomposed into two types of vertical flows and a single type of horizontal flow (Hussein et al., 2013;Rojstaczer, 1988).
The earth tide response method and the barometric pressure response method allow the use of a compressible matrix, whereas the Hantush and Jacob (1955) model, assumes that the matrix is incompressible (Bastias Espejo et al., 2022;Hantush & Jacob, 1955;Rojstaczer, 1988;Valois et al., 2022;Wang et al., 2018).
The confined pore pressure generated by earth tide strain is a mechanical phenomenon caused by the elastic deformation of the porous matrix (Wang & Manga, 2021).The barometric pressure response model investigates three one-dimensional flow problems, unlike the earth tide response which is a direct action on the matrix.Barometric loading is a top-down phenomenon, and pressure variations must diffuse from the ground surface to the aquifer.In contrast, tidal strain variations are instantaneous at all depths (Sohn & Matter, 2023).Barometric response models and tidal response models are fundamentally different.
The barometric response model is the similar to the solution given by Hantush and Jacob (1955) for aquifer response to pumpage under conditions of leakance; the difference is that the wells discharge at a periodic rate rather than at a constant rate (Rojstaczer, 1988).

Comparison of Hydraulic Parameters
The RMSE values and r b values for the best fitting of different methods are shown in Table 3 (see Text S5 in Supporting Information S1 for more details).For wells OW1 and OW2, the RMSE values for the AQTESOLV and barometric pressure responses methods are small, so the differences between the observed and theoretical values do not cause an order of magnitude difference in the estimated aquifer transmissivity T. Uncertainty in BK u increases the errors in estimating aquifer transmissivity using the earth tide response method.
By analyzing and comparing the best-fit transmissivity calculated by the three methods, the range of aquifer transmissivity is 1.18 × 10 −5 -1.79 × 10 −5 m 2 /s and 1.73 × 10 −5 -2.14 × 10 −5 m 2 /s for OW1 and OW2, respectively.It can be found that the differences in the transmissivity calculated by the three methods are not very significant, and all are of the same order of magnitude.Among the three methods for estimating aquifer parameters, the transmissivity estimated from the barometric pressure response method is the largest; the aquifer transmissivities for OW1 and OW2 are T = 1.79 × 10 −5 and 2.14 × 10 −5 m 2 /s, respectively.The aquifer transmissivity estimated from OW1 and OW2 differ slightly, and the results suggest that the hydrogeological conditions of the aquifers in the two wells are heterogeneous.There are certain differences in the estimated aquifer transmissivity using AQTESOLV, earth tide, and barometric pressure methods; it could be that the different methods act at different scales (Allegre et al., 2016;Qi et al., 2023;Sun et al., 2020;Valois et al., 2022), and could also be due to differences in the fundamental theories and assumptions (Bastias Espejo et al., 2022;Sun et al., 2020;Valois et al., 2022).And the uncertainty of BK u in the tidal response method leads to larger uncertainty (Valois et al., 2022) in the estimated parameters.
The barometric pressure response method considers the storage coefficient of the confining layer, while both the Hantush and Jacob (1955) solution and the earth tide response method assume no storage in incompressible leaky aquitards.However, the barometric pressure response function is not sensitive to the storage coefficient of the confining layer, as shown in Figure S3 of the Supporting Information S1.Therefore, whether the storage coefficient of the confining layer is considered in the models of these three methods has little effect on our analysis.
Both passive methods require a given aquifer storage coefficient when estimating the aquifer parameters.The storage coefficients of the aquifers in OW1 and OW2 are around 10 −5 and 10 −6 , and both response functions are insensitive to the storage coefficients of the aquifers in this range (Figures S4 and S6 in Supporting Information S1).
The K′ estimated by the AQTESOLV method and the barometric pressure response method are more similar.However, the K′ estimated by the earth tide response method is two to three orders of magnitude larger than the AQTESOLV and barometric pressure response methods for well OW1, and the K′ estimated by the earth tide response method is one to two orders of magnitude larger than the AQTESOLV and barometric pressure response methods for well OW2.Wang et al.'s (2018) tidal leaky response model requires undrained conditions.In general, when the hydraulic conductivity of the confining layer exceeds 5 × 10 −5 m/s, this will lead to drained conditions and may result in errors in the use of analytical solutions (Espejo et al., 2022).The hydraulic conductivity of the confining layers in both OW1 and OW2 estimated using the tidal response are less than 5 × 10 −5 m/s, which satisfies the undrained condition.

Representativeness of the Aquifer Transmissivity Determined by Different Methods
The estimation of the hydraulic parameters of aquifers by pumping tests is a dynamic change process, and the drawdown-time curve can be divided into three periods (the early, middle and late periods) (Grimestad, 1981), with different periods measured at different scales.Passive methods generalize the actual hydrogeological processes to mathematical analytical solution models, which may lead to the omission of some hydrogeological information.Therefore, we compare the parameters from the AQTESOLV method, barometric pressure and earth tide responses with the dynamic variation of parameters during the different periods of the pumping test, and identify which periods these passive methods are located in.
The aquifer transmissivities of OW1 and OW2 obtained by different methods are shown in the staging curves (Figure 7).For these wells, the estimated aquifer transmissivity points from the AQTESOLVE method are located in the late period of the dynamic estimation of aquifer parameters in the pumping test.It is confirmed that the conventional pumping tests estimation is based on the late data of pumping test or long-term pumping to estimate the average parameter values of the aquifer (Straface et al., 2007).
The aquifer transmissivities estimated by the barometric pressure response are located in the early and middle periods of the dynamic estimation of aquifer parameters during the pumping test.Therefore, the aquifer parameters estimated by the barometric pressure response of OW1 and OW2 reflect the geological conditions near the observation well.
In the OW1 and OW2 wells, the different period of the dynamic estimation of aquifer parameters are distributed over a small range.Due to the differences in the fundamental theory and assumptions and the uncertainty of BK u in the earth tide response method, it leads us to not be able to determine the representativeness of the estimated aquifer transmissivity in the OW1 and OW2 wells by using the dynamic estimation of aquifer parameters in the pumping test.The earth tide response method for estimating the representativeness of aquifer transmissivity deserves further research.

Scale Effect
The transmissivity of aquifers has a scale effect (Vesselinov et al., 2001), with the transmissivity increasing approximately linearly with the observed scale for a certain range and remaining constant thereafter (Sánchez-Vila et al., 1996).The transmissivities obtained by different methods represent different scale ranges of aquifer characteristics (Rovey & Cherkauer, 1995).
In this study, we use a one-dimensional metric to quantify the scale effects of the earth tide and pumping test methods.No method is currently available for characterizing the scale effects of the barometric pressure response method, to the best of our knowledge.
The tidal leaky response model of Wang et al. (2018) shares the same scale with Hsieh's model (Sun et al., 2020).The drawdown in the aquifer in Hsieh's model could be expressed as (Hsieh et al., 1987), where where Ker (α) and Kei (α) are the Kelvin functions of order zero, and Ker 1 and Kei 1 are the Kelvin functions of order one.Q 0 exp (iωt) is the discharge from the aquifer to the well.r is the distance from the well center, r w is the radius of the well, and r c is the radius of the well casing.T is the average transmissivity of the aquifer.S is the aquifer storage coefficient.ω is the angular frequency.x o is the complex amplitude of the water level fluctuations.
The value of α w calculated by Equation 25 is usually relatively small (<0.1), in which case both Ker 1 (α w ) and Kei 1 (α w ) can be approximated by −1/(2 1/2 α w ).This leads to ϕ ≈ 1 and ψ ≈ 0 (Hsieh et al., 1987), and Equation 21becomes, We define the zone of influence using an attenuation of 5% relative to its maximum (Xue et al., 2013), which reflects the measurement scale for the parameters.Thus, the most sensitive area is within 42.7 m of OW1 and within 114.2 m of OW2 (Figure S7 in Supporting Information S1).
The approximate scale of the pumping test data is the distance between the pumping well and the monitoring well (Allegre et al., 2016;Kitagawa & Kano, 2016).OW1 is 89.8 m away from pumping well PW and OW2 is 979.1 m away from pumping well PW.We consider the radius of influence for the pumping tests of these two wells to be 89.8 and 979.1 m, respectively.Thus, the scales of the tidal response method are smaller than those of the pumping test method (Table 3).

Uncertainties Analysis
There are several sources of uncertainties that originate from the raw data, the analytical process and the model assumptions (Guo et al., 2021;Sun et al., 2020;Valois et al., 2022).
First, the uncertainty from the raw data.Uncertainties due to pressure sensor resolution (0.1%F.S.) and sampling frequency (1 hr) and regression deconvolution are low, and improving data quality can improve these uncertainties (Valois et al., 2022).
Based on the available borehole information, we only know that the PW well is complete penetrating well and do not know the lithology beneath the OW1 and OW2 (Figure 1c).In this study, we consider the aquifers observed in the boreholes as our target aquifers and consider wells OW1 and OW2 as complete penetrating wells.Whether OW1 and OW2 are actually complete penetrating wells introduces some uncertainty.In the AQTESOLV method, if the observation wells are partially penetrating and the anisotropy ratio is set to 0.5 (Valois et al., 2022), the aquifer transmissivity at best fit is: T = 1.46 × 10 −5 and 1.62 × 10 −5 m 2 /s for wells OW1 and OW2 (see Text S6 in Supporting Information S1 for more details).The difference compared to complete penetrating wells is 4.79% and 6.36%, respectively, which has little effect on the T estimates.The analysis of natural disturbance processes is chosen for the non-pumping period.And the natural disturbances act directly on the aquifer, resulting in less water level fluctuations around the observation wells (Valois et al., 2022).Therefore, we consider that the flow direction and velocity of the earth tide and the barometric pressure are less disturbed in partially penetrating wells.
The second part is analytical process.In the earth tide method, the amplitude ratio and phase lag of the estimated tidal components, and the uncertainty in the value of BK u introduce errors.Xue et al. (2013) examined the inversion error inherent in the tidal response method.The calculated phase shifts have a subsample resolution because of the long-time windows.When the window length is 30 days, the RMSE of the M 2 phase shift of wells OW1 and OW2 are 1.60° and 0.75°.As shown in Figure 4, these errors do not cause an order of magnitude difference in the aquifer transmissivity.We use the general range of BK u values (5-50 GPa) to estimate hydraulic parameters.
The uncertainty in the BK u values has an effect on the estimation of T and K′, resulting in an order of magnitude difference in the estimated hydraulic parameters.This uncertainty can be reduced if supported by relevant experimental data.
Uncertainty is introduced in the barometric pressure response method when the aquifer storage coefficient is given before inverting the hydraulic parameters with the barometric pressure response function.However, as shown in Figure S4 of the Supporting Information S1, the barometric pressure response function is insensitive to the aquifer storage coefficient.
The third part is model uncertainty.The formal errors in the AQTESOLV, tidal response, and barometric pressure response methods are introduced by the model assumptions.Both the earth tide response model and the Hantush and Jacob (1955) model do not consider the unsaturated zone.The barometric pressure response considers the unsaturated zones with L unsat = 20 m.The vertical hydraulic conductivity of the unsaturated zone and confining layers at the best fit of OW1 and OW2 are K unsat = 1.45 × 10 −8 and 2.77 × 10 −8 m/s, K′ = 7.75 × 10 −9 and 8.54 × 10 −9 m/s, respectively.The values of the vertical hydraulic conductivity in the unsaturated zone and confining layer of OW1 and OW2 do not differ significantly.The barometric pressure response model takes into account the storage coefficient of the confining layers, which is not considered by the other two methods.The estimated parameters average the hydraulic properties at the scale that the models can measure, and whether it is consistent with the real value depends on the homogeneity of the aquifer (Sun et al., 2020).
However, the inherent uncertainty due to model selection is difficult to estimate, and model uncertainties can be assessed by comparing the results of different models (Valois et al., 2022).

Conclusions
The responses of groundwater to the earth tides and barometric pressure can be used to estimate the aquifer transmissivity.However, this approach is based on a simplified analytical solution of the groundwater flow equation, which has various assumptions and less inclusion of geological system characteristics.To assess the scale of investigation of different methods, we compare the aquifer transmissivity estimated by the passive methods with the dynamic process of pumping tests.
The aquifer transmissivities determined from the different methods fall within the range of estimates obtained during different periods of an aquifer pumping test conducted in the Beetaloo Region of Northern Australia.For OW1 and OW2, the transmissivity estimated by the barometric pressure response method is the largest.
The aquifer parameters inferred from the AQTESOLV method are located in the late period along the dynamic parameter curve, confirming that the conventional interpretation that relies on late-period data provides the best aquifer transmissivity value.The aquifer transmissivities estimated using the barometric pressure method, however, are located in the early and middle periods along the dynamic parameter curve, reflecting conditions within and near the observation well.Thus, the barometric pressure method represents a smaller region of the hydrogeological properties than the pumping-test methods.The representative estimation of aquifer transmissivity by the tidal response method deserves further study.These conclusions are based on existing studies, and whether they are applicable to all aquifers requires further research.

Figure 1 .
Figure 1.(a) Location of study area (black triangle) in the Beetaloo Region of Northern Australia.(b) Location of observation wells OW1 and OW2 relative to pumping well PW.(c) Schematic diagram of the structure and aquifers of wells PW, OW1, and OW2.

Figure 4 .
Figure 4.The relationship of transmissivity, T, and specific leakage, σ (σ′ = K′/b′), on the amplification factor (a) and phase lag (b) in degrees when using M 2 in Wang et al. (2018)'s tidal leaky response model, with S = 1 × 10 −5 , r c = r w = 0.13 m and BK u = 9.8 GPa.Hydraulic parameter estimates using the tidal response method for wells OW1 (c) and OW2 (d).

Figure 5 .T
Figure 5. Barometric pressure response function fitting using amplitude gain for wells OW1 (a) and OW2 (b), and phases for wells OW1 (c) and OW2 (d).Solid blue line is the theoretical Gain-ω or θ-ω curve fitted based on theoretical models.Red lines are error bars.

Figure 6 .
Figure 6.Estimated transmissivity, diffusivity, and storage coefficients for wells OW1 and OW2 using parameters dynamically estimated during an aquifer pumping test.

Figure 7 .
Figure 7. Dynamic variation of aquifer parameters (transmissivity and storage coefficient) in three periods (i.e., early, middle, late) of a pumping test for wells (a) OW1 and (b) OW2.Comparison of aquifer transmissivity obtained from the barometric pressure and AQTESOLV method with estimates from three periods (i.e., early, middle, late) of a pumping test for Wells (c) OW1 and (d) OW2.